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\begin{document} \title[\resizebox{3in}{!}{Long title}]{Analytic and geometric properties of scattering from periodically modulated quantum-optical systems} \author{Rahul Trivedi}\email{[email protected]} \affiliation{E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA} \author{Alex White} \affiliation{E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA} \author{Shanhui Fan} \affiliation{E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA} \author{Jelena Vu\v{c}kovi\'c} \affiliation{E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA} \date{\today} \begin{abstract} We study the scattering of photons from periodically modulated quantum-optical systems. For excitation-number conserving quantum optical systems, we connect the analytic structure of the frequency-domain $N$-photon scattering matrix of the system to the Floquet decomposition of its effective Hamiltonian. Furthermore, it is shown that the first order contribution to the transmission or equal-time $N-$photon correlation spectrum with respect to the modulation frequency is completely geometric in nature {\it i.e.}~it only depends on the Hamiltonian trajectory and not on the precise nature of the modulation being applied. \end{abstract} \maketitle \section{Introduction} Quantum information processing and communication systems rely strongly on the generation and manipulation of non-classical states of light \cite{lodahl2015interfacing,ding2016demand,michler,senellart2017high,zhang2018strongly, kok2007linear,o2009photonic,roy2017colloquium,reiserer2015cavity,duan2010colloquium,sangouard2011quantum,nemoto2014photonic}. Implementing quantum systems for such applications often involves interfacing a localized quantum system (e.g.~a few-level system such as a quantum dot or color center) with bosonic baths (such as optical fibers or waveguides). Significant control over the states of light emitted by the localized system into the bosonic bath can be gained by engineering the coupling between the two \cite{englund2005controlling,daveau2017efficient, pelton2002efficient}, and by controlling the excitation of the localized system \cite{he2013demand, fischer2017signatures, hanschke2018quantum}. Recently, the ability to modulate the localized system on frequency-scales comparable to or exceeding the decay rate of the localized system into the bosonic bath has been demonstrated in various quantum-optical platforms such as quantum dots \cite{metcalfe2010resolved} and color centers \cite{miao2019electrically}. \edit{This has opened up the possibility of engineering the spectral content of the photons scattered by the localized system into the bosonic bath by engineering the modulation applied on the localized system. Such spectral engineering could enable quantum networks of localized systems that are robust to variations in their physical characteristics, unlock quantum information protocols relying on high-dimensional entangled photon states \cite{pichler2016photonic} and realize non-reciprocal photon transport \cite{yuan2015achieving}. From a theoretical standpoint, it has opened up the question of how to calculate and understand the scattering properties of the modulated localized system.} The scattering properties of time-independent (unmodulated) localized systems can be completely described by its scattering matrix. Significant progress has been made in developing single and two-photon scattering matrices for specific localized systems (e.g.~two-level systems, Jaynes Cumming systems) by adapting a variety of different techniques from quantum field-theory \cite{shen2007strongly, shi2013two, shi2009lehmann, shi2015multiphoton}. The problem of systematically calculating scattering and emission from a general time-independent Markovian localized system was addressed in refs.~\cite{caneva2015quantum, xu2015input}, and it was shown that the computation of scattering matrices only required diagonalization of an effective non-hermitian Hamiltonian that is completely restricted to the Hilbert space of the localized system. The introduced formalism can be used to derive explicit relationships between the few-photon scattering properties of the localized system and the spectrum of its effective Hamiltonian \cite{xu2013analytic, trivedi2019photon} and this has been employed to understand a number of experimentally relevant quantum systems \cite{trivedi2019photon, xu2018generate}. While most of the efforts in calculating and understanding scattering matrices were restricted to time-independent localized systems, a procedure for calculating the propagator from pulsed localized systems (i.e.~systems whose Hamiltonian has time-dependence only within a finite time-window) was recently developed \cite{trivedi2018few, fischer2017scattering}. It was shown that it is possible to define a scattering matrix for a time-dependent system provided it is asymptotically time-independent, and a recipe for its computation was provided \cite{trivedi2018few}. This procedure was applied to understand scattering of a single-photon from a two-level system driven by a pulsed laser, and the scattering matrix was shown to have significantly different structure from that of a time-independent two-level system \cite{trivedi2018few}. In this paper, we consider the problem of calculating the scattering matrix for a periodically modulated localized system. We focus exclusively on localized systems which are excitation number conserving even in the presence of periodic modulation, and relate the scattering matrices to the Floquet decomposition of the non-Hermitian effective Hamiltonian of the localized system. Special attention is paid to the difference in the analytic properties of the resulting scattering matrix from the scattering matrix of time-independent systems. Finally, we consider the slow modulation regime and study the properties of the equal-time $N-$photon correlation function. It is shown that this correlation function, to the zeroth order in the modulation frequency, is equal to the time-average of the instantaneous correlation function obtained by assuming the system to be time independent and that the first order correction is completely geometric in nature. This paper is organized into three major sections --- section \ref{sec:model} introduces the mathematical model of the system under consideration along with a review of the frequency-domain scattering matrix. Section \ref{sec:smat} presents the construction and general properties of the $N-$photon scattering matrix for a periodically modulated quantum system. As an example, single and two-photon scattering from a cavity with Kerr-nonlinearity is studied. Finally, in section \ref{sec:slow_mod_smat}, we study the properties of the equal-time $N-$photon correlation function in the slow modulation regime. \section{Model and prelimnaries}\label{sec:model} This section is intended to introduce the model for the system under consideration and also provide a review of scattering theory for open quantum systems. The analysis in this section closely follows that of refs.~\cite{xu2015input, trivedi2018few}. We consider a general class of time-dependent systems which have a periodic localized system interacting with two bosonic baths schematically depicted in Fig.~\ref{fig:schematic}a. The Hilbert space of the two bosonic baths is described by frequency-dependent annihilation operators $a_\omega$ and $b_\omega$. These operators satisfy the bosonic commutation relations: $[a_\omega, a_\nu] = 0, [b_\omega, b_\nu] = 0, [a_\omega, a_\nu^\dagger] = \delta(\omega - \nu), [b_\omega, b_\nu^\dagger] = \delta(\omega - \nu)$ and $[a_\omega, b_\nu] = [a_\omega, b_\nu^\dagger] = 0$. The dynamics of this system are governed by the following time-dependent Hamiltonian: \begin{align}\label{eq:basic_hamiltonian} H(t) = H_s(t) + \sum_{s \in \{a, b\}} \int_{-\infty}^\infty \omega s_\omega^\dagger s_\omega d\omega + \sum_{s\in \{a, b\}} \int \textrm{i}\big(L s_\omega^\dagger - s_\omega L^\dagger \big)\frac{d\omega}{\sqrt{2\pi}} \end{align} Here $L$ is the operator through which the localized system couples to the bosonic baths. Throughout this paper, we will consider the bath described by $a_\omega$ as the input bath and that described by $b_\omega$ as the output bath. For simplicity, we assume that the two baths couple equally to the localized system. \begin{figure} \caption{\textbf{Schematic:} \textbf{a.} Schematic of the a modulated localized system coupling to the bosonic baths. The Hilbert space of the localized system is denoted by $\mathcal{H}_s$ and the Hilbert space of the bosonic baths are described by the frequency-dependent annihilation operators $a_\omega$ and $b_\omega$. The Hamiltonian of the localized system is denoted by $H_s(t)$, and $L$ is the system operator through which the localized system couples bosonic baths. b) Schematic of the level-structure of an excitation-number conserving localized system. The Hilbert space of the system can be divided into $\mathcal{H}_s^0, \mathcal{H}_s^1, \mathcal{H}_s^2\dots$, which correspond to space of states with excitation numbers 0, 1, 2 $\dots$. The operator $L$ maps $\mathcal{H}_s^n$ to $\mathcal{H}_s^{n-1}$ for $n\geq 1$ and annihilate the states in $\mathcal{H}_s^0$ while the operators $L^\dagger$ maps $\mathcal{H}_s^n$ to $\mathcal{H}_s^{n + 1}$.} \label{fig:schematic} \end{figure} We will also assume the localized system and its coupling to the bosonic baths to be excitation number conserving --- this requires that (a) the Hilbert space $\mathcal{H}_s$ of the localized system can be expressed as a direct sum of subspaces: $\mathcal{H}_s = \mathcal{H}_s^0 \oplus \mathcal{H}_s^1 \oplus \mathcal{H}_s^2\dots $ such that each subspace $\mathcal{H}_s^n$ is invariant under evolution with respect to the system Hamiltonian $H_s(t)$ and (b) $L$ maps the subspace $\mathcal{H}_s^n$ to $\mathcal{H}_s^{n-1}$ for $n \geq 1$, $L^\dagger$ maps the subspace $\mathcal{H}_s^{n}$ to $\mathcal{H}_s^{n+1}$ for $n \geq 0$ and $L$ annihilates $\mathcal{H}_s^0$ {\it i.e.}~$\mathcal{H}_s^0$ is within the null-space of $L$. Throughout this paper, we will refer to $\mathcal{H}_s^n$ as the $n^\text{th}$ excitation subspace and associate with it an excitation number $n$. The operator $L$ then decreases the excitation number of the localized system's state by 1 and $L^\dagger$ increases it by 1. Furthermore, evolving a state in $\mathcal{H}_s^0$ with respect to $H(t)$ is identical to evolving it with respect to $H_s(t)$ without interacting with the bosonic bath --- $\mathcal{H}_s^0$ is therefore the space of the ground states of the localized system. In this paper, we will restrict ourselves to systems with a single ground state $\ket{g}$ {\it i.e.}~$\mathcal{H}_s^0 = \{\ket{g}\}$. As is shown in appendix \ref{app:ph_no_cons}, for an excitation-number conserving system with a single ground state, $N$ photons incident on the localized system can only scatter into $N$ outgoing photons, and consequently the scattering properties of the system can be described by the $N$-photon scattering matrix: \begin{align}\label{eq:smat_elem} S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) = \bra{\text{vac}; g}\bigg( \prod_{i=1}^N b_{\omega_i} \bigg)\hat{\textrm{S}} \bigg(\prod_{i=1}^N a_{\nu_i}^\dagger \bigg)\ket{\text{vac}; g} \end{align} where $\hat{\textrm{S}}$ is the scattering-matrix defined via \cite{taylor2006scattering}: \begin{align}\label{eq:smat_def} \hat{\textrm{S}} = \lim_{\substack{t_+ \to \infty \\ t_- \to -\infty}} U_0(t_0, t_+) U(t_+, t_-) U_0(t_-, t_0) \end{align} Here $U(\cdot, \cdot)$ is the propagator corresponding to the Hamiltonian $H(t)$ and $U_0(\cdot, \cdot)$ is the propagator corresponding to the Hamiltonian $H_0(t)$ corresponding to the uncoupled localized system and bosonic baths: \begin{align}\label{eq:ref_hamil} H_0(t) = H_s(t) + \sum_{s\in \{a, b\}} \int_{-\infty}^\infty \omega s_\omega^\dagger s_\omega d\omega \end{align} Additionally, $t_0$ is a time-reference that is used for defining the input and output asymptotes corresponding to the states incident and scattered from the localized system \cite{taylor2006scattering}. While this time reference does not affect the scattering matrix of a time-independent system, it encodes the `time of arrival' of the incident photon wave-packet and thus is relevant for time-dependent system. Using the input-output formalism \cite{gardiner1985input, xu2015input}, it can easily be shown that the scattering matrix element in Eq.~\ref{eq:smat_elem} is related to the Heisenberg-picture system operator $L(t) = U(t_0, t) L\ U(t, t_0)$ via (refer to appendix \ref{app:smat_to_gfunc} for derivation): \begin{align}\label{eq:gfunc_to_smat} S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) =(-1)^N e^{-\textrm{i}\phi(t_0)} \int_{-\infty}^\infty \dots \int_{-\infty}^\infty G(t_1, t_2 \dots t_N; s_1, s_2 \dots s_N) \prod_{i=1}^N e^{\textrm{i}(\omega_i t_i - \nu_i s_i)}\frac{dt_i ds_i}{2\pi} \end{align} where $\phi(t_0) = \sum_{i=1}^N(\omega_i - \nu_i)t_0 $ and we have introduced the time-domain system Green's function: \begin{align}\label{eq:gfunc} G(t_1, t_2 \dots t_N; s_1, s_2 \dots s_N) = \bra{\text{vac}; g} \mathcal{T} \bigg[\prod_{i=1}^N L(t_i) \prod_{i=1}^N L^\dagger(s_i) \bigg] \ket{\text{vac}; g} \end{align} where $\mathcal{T}[\cdot]$ indicates time-ordering in its arguments. An application of the quantum regression theorem can allow us to evaluate these Green's functions entirely within the Hilbert space of the localized system by replacing the Heisenberg operators $L(t)$ and $L^\dagger(t)$ with respect to the Hamiltonian $H(t)$ with Heisenberg operators $\tilde{L}(t)$ and $\tilde{L^\dagger}(t)$ with respect to the non-Hermitian effective Hamiltonian $H_\text{eff}(t) = H_s(t) - \textrm{i}L^\dagger L $ {\it i.e.} \begin{subequations}\label{eq:gfunc_eff_hamil} \begin{align} G(t_1, t_2 \dots t_N; s_1, s_2 \dots s_N) = \bra{g} \mathcal{T}\bigg[\prod_{i=1}^N \tilde{L}(t_i) \prod_{i=1}^N \tilde{L^\dagger}(s_i) \bigg] \ket{g}, \end{align} where \begin{align} \tilde{O}(t) = U_\text{eff}(0, t) O U_\text{eff}(t, 0)\ \text{for} \ O \in \{L, L^\dagger\}, \end{align} \end{subequations} with $U_\text{eff}(t, s)$ being the propagator corresponding to $H_\text{eff}(t)$. It can be noted that both $H_\text{eff}(t)$ and $U_\text{eff}(t, s)$ do not effect the excitation number of the state that they act on, and can therefore be described by their restrictions $H_\text{eff}^n(t)$ and $U_\text{eff}^n(t, s)$ respectively within the $n^\text{th}$ excitation subspace $\mathcal{H}_s^n$. Eqs.~\ref{eq:gfunc_to_smat} and \ref{eq:gfunc_eff_hamil} are used in the following sections to study the computation and properties of the frequency-domain scattering matrix. \section{Scattering matrices}\label{sec:smat} In this section, we explore the systematic construction of the frequency-domain scattering matrices (Eq.~\ref{eq:gfunc_to_smat}) of the modulated quantum system. Special attention is paid to the similarities and differences that arise in these scattering matrices relative to the time-independent case. Explicit results are provided for single and two-photon scattering matrices. \subsection{Construction and analytic properties} The discrete time-translation symmetry of the periodically modulated quantum system imposes a fundamental constraint on the structure of the $N-$photon frequency-domain scattering matrix. In particular, the form of the scattering matrix should conserve the total photon frequency modulo $\Omega$ --- this implies that the scattering matrix $S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$ can be written as a sum of terms proportional to $\delta(\sum_{i=1}^N \omega_i - \sum_{i=1}^N \nu_i - k\Omega)$: \begin{align}\label{eq:smat_general_decomp} S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) = \sum_{k=-\infty}^{\infty} e^{-\textrm{i}k\Omega t_0} S_k(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)\delta\bigg(\sum_{i=1}^N \omega_i - \sum_{i=1}^N \nu_i - k\Omega\bigg). \end{align} Note that here we have explicitly shown the dependence on the time-reference $t_0$ used for defining the scattering matrix (Eq.~\ref{eq:smat_def}) that enters Eq.~\ref{eq:gfunc_to_smat} as a phase factor depending on the difference between the total input and output frequencies under consideration --- the discrete time-translation symmetry of the Hamiltonian ensures that the scattering matrix is periodic in $t_0$ with period $2\pi / \Omega$. This general form of the scattering matrix can be contrasted with the scattering matrix for time-independent systems, which would be proportional to $\delta(\sum_{i=1}^N \omega_i - \sum_{i=1}^N \nu_i)$ since it conserves the total photon frequency and consequently be independent of the time-reference $t_0$. The functions $S_k(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$ in Eq.~\ref{eq:smat_general_decomp} can, in general, be further decomposed into a sum of a non-singular function, denoted by $S_k^\text{C}(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$, and terms with delta-function singularities. The \emph{connected part} of the $N-$photon scattering matrix, $S^\text{C}(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$, can then be defined as: \begin{align} S^\text{C}(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) = \sum_{k=-\infty}^\infty e^{-\textrm{i}k\Omega t_0} S_k^\text{C}(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)\delta\bigg(\sum_{i=1}^N \omega_i - \sum_{i=1}^N \nu_i - k\Omega\bigg). \end{align} From a physical standpoint, $S^\text{C}(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$ accounts for all the nonlinear interactions between the $N$ incident photons that are induced by the localized quantum system --- while it conserves the total photon frequency modulo $\Omega$, it can in general lead to a change in the individual photon frequencies. Furthermore, an application of the cluster decomposition principle allows us to construct the full $N-$photon scattering matrix from its connected part and the connected part of fewer photon scattering matrices \cite{xu2013analytic, xu2015input}: \begin{align}\label{eq:smat_cluster_decomp} S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) = \sum_{\mathcal{B}}\sum_P \prod_{k=1}^{|\mathcal{B}|} S^\text{C}(\omega_{\mathcal{B}_kP(1)}, \omega_{\mathcal{B}_kP(2)} \dots; \nu_{\mathcal{B}P_k(1)}, \nu_{\mathcal{B}P_k(2)} \dots ), \end{align} where $\mathcal{B}$ is an ordered partition of $\{1, 2, 3\dots N\}$ into smaller subsets, $P$ is a permutation of $\{1, 2, 3\dots \}$ and $\mathcal{B}{P}$ is the partition $\mathcal{B}$ applied on $\{P(1), P(2) \dots P(N)\}$. For time-independent localized system, it can be shown that the frequency domain scattering matrix is completely determined by the spectral decomposition of the effective Hamiltonian of the localized system. In particular, the position and linewidth of resonances in the $N-$photon scattering matrix are determined by the complex eigenvalues of the effective Hamiltonian, and the amplitude of the scattering matrix at these resonances is determined by its eigenvectors. For periodically modulated localized systems, a similar relationship can be established between the scattering matrices and the Floquet decomposition of the effective Hamiltonian. Since the effective Hamiltonian is non-Hermitian, its Floquet decomposition within the $n-$excitation subspace requires the solution of the following eigenvalue equations \cite{longhi2017floquet}: \begin{subequations}\label{eq:floquet_problem} \begin{align} &H_\text{eff}^n(t)\ket{\phi_k^n(t)} + \textrm{i}\frac{d}{dt} \ket{\phi_k^n( t)} = \lambda_k^n \ket{\phi_k^n(t)} \\ &\big(H_\text{eff}^n(t)\big)^\dagger\ket{\chi_k^n(t)} - \textrm{i}\frac{d}{dt} \ket{\chi_k^n(t)} = \big(\lambda_k^n\big)^* \ket{\chi_k^n(t)} \end{align} \end{subequations} where $\lambda_k^n$ is the $k^\text{th}$ Floquet eigenvalue of $H_\text{eff}^n(t)$ and $(\ket{\phi_k^n(t)}, \ket{\chi_k^n(t)})$ are the $k^\text{th}$ birothogonal Floquet eigenvectors of $H_\text{eff}^n$. We note that both $\ket{\phi_k^n(t)}$ and $\ket{\chi_k^n(t)}$ are periodic with periodicity of the system Hamiltonian: $\ket{\phi_k^n(t + T)} = \ket{\phi_k^n(t)}$ and $\ket{\chi_k^n(t + T)} = \ket{\chi_k^n(t)}$. They also satisfy $\bra{\chi_k^n(t)}\phi_l^n(t)\rangle = \delta_{k, l}$ for all $t \in (0, T]$. The Floquet eigenvalue $\lambda_k^n$ will be, in general, a complex number and can be expressed in terms of its real and imaginary parts: $\lambda_k^n = \varepsilon_k^n - \textrm{i}\kappa_k^n / 2$. We note that $\varepsilon_k^n$ can only be uniquely specified to modulo $\Omega$. Provided such biorthogonal states exist, the propogator $U_\text{eff}^n(t, s)$ in the $n^\text{th}$ excitation subspace can be expressed as: \begin{align}\label{eq:floquet_decomp} U_\text{eff}^n(t, s) = \sum_k \ket{\phi_k^n(t)} \bra{\chi_k^n(s)} \exp(-\textrm{i}\lambda_k^n(t - s)). \end{align} This decomposition of the propagator along with Eq.~\ref{eq:gfunc_eff_hamil} can be used to relate the frequency domain scattering matrices to the Floquet decomposition of the effective Hamiltonian. Due to the periodic time-dependence of the Floquet states, the frequency-domain scattering matrices has resonances at $\varepsilon_k^n + p\Omega$ for $p \in \mathbb{Z}$ with linewidths $\kappa_k^n$. Furthermore, the amplitudes of these resonances are determined by the Fourier components of the periodic Floquet eigenstates $\ket{\phi_k^n(t)}$ and $\ket{\chi_k^n(t)}$. This is made more explicit for single and two-photon scattering matrices in the following subsection. \subsection{Single and two-photon scattering matrices} Of particular interest are the single and two-photon scattering matrices, since they can often be easily probed experimentally with transmission and two-photon correlation experiments. Following the procedure outlined above for the single-photon scattering matrix, we obtain (details in appendix \ref{app:smat}): \begin{subequations}\label{eq:conn_part_one_ph} \begin{align} S(\omega; \nu) = \sum_{k\in \mathbb{Z}} e^{-\textrm{i}k\Omega t_0} S_k(\nu) \delta(\omega - \nu - k\Omega), \end{align} with \begin{align} S_k(\nu) = \sum_{m\in \mathbb{Z}}\text{L}^{1\to 0}_{k+m} \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^1 + m\Omega - \nu) + \upkappa^1 / 2} \bigg) \text{L}^{0\to 1}_m. \end{align} \end{subequations} Here $\upvarepsilon^1$ and $\upkappa^1$ are vectors of $\varepsilon_n^1$ and $\kappa_n^1$ respectively and $\text{D}(\cdot)$ constructs a diagonal matrix from a vector that is passed as its argument. $\text{L}^{1\to0}_k$ is a row vector and $\text{L}^{0\to 1}_k$ is a column vector, and their elements are given by: \begin{align}\label{eq:L_fc_def_single_ex} \big[\text{L}^{1\to 0}_k\big]_n = \int_0^T \bra{g} L \ket{\phi_n^1(t)} e^{\textrm{i}k\Omega t} \frac{dt}{T} \ \text{and} \ \big[\text{L}^{0\to 1}_k\big]_n = \int_0^T \bra{\chi_n^1(t)} L^\dagger \ket{g}e^{-\textrm{i}k\Omega t}\frac{dt}{T} \end{align} Clearly, the form of the single-photon scattering matrix implies that a photon at frequency $\nu$ is in general scattered into photons at frequencies differing from $\nu$ by an integer multiple of $\Omega$. Furthermore, the amplitude of transmission at these sidebands would in general depend on the Fourier series components of the Floquet states $\ket{\phi_n^1(t)}$ and $\ket{\chi_n^1(t)}$. A similar procedure can be followed for the computation of the two-photon scattering matrix. As is shown in appendix \ref{app:smat}, the connected part of the two-photon scattering matrix, $S^\text{C}(\omega_1, \omega_2; \nu_1, \nu_2)$ can be expressed as a sum of two components: $S^{\text{C}, 1}(\omega_1, \omega_2; \nu_1, \nu_2)$ which is completely determined by the Floquet-decomposition of $H_\text{eff}^1(t)$ and $S^{\text{C}, 2}(\omega_1, \omega_2; \nu_1, \nu_2)$ which depends on the Floquet-decomposition of $H_\text{eff}^2(t)$: \begin{subequations}\label{eq:conn_part_two_ph} \begin{alignat}{2} S^{\text{C}, 1}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{1}{2\pi \textrm{i}}\sum_{P, Q}\sum_{p, m, n \in \mathbb{Z}} \bigg[&&\text{L}^{1\to 0}_p \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^1 - \omega_{P(1)} + p\Omega) + \upkappa^1/2} \bigg)\text{L}^{0\to 1}_{n + p - k} \mathcal{P}\frac{1}{\omega_{P(2)} - \nu_{Q(2)}-n\Omega} \nonumber \\& &&\text{L}^{1\to 0}_{m + n} \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^1 - \nu_{Q(2)} + m\Omega) + \upkappa^1 / 2} \bigg)\text{L}^{0\to 1}_{m}\bigg],\\ S^{\text{C}, 2}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{1}{2\pi}\sum_{P, Q}\sum_{p, m, n \in \mathbb{Z}} \bigg[&&\text{L}^{1\to 0}_p \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^1 - \omega_{P(1)} + p\Omega) + \upkappa^1/2} \bigg)\text{L}^{2\to 1}_{n - p + k} \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^2 - \nu_1 - \nu_2 + n\Omega) + \upkappa^2/2} \bigg) \nonumber \\& &&\text{L}^{1\to 2}_{n - m} \text{D}\bigg(\frac{1}{\textrm{i}(\upvarepsilon^1 - \nu_{Q(2)} + m\Omega) + \upkappa^1 / 2} \bigg)\text{L}^{0\to 1}_{m}\bigg], \end{alignat} \end{subequations} where $P, Q$ are permutations of the two-element set $\{1, 2\}$, $\mathcal{P}$ indicates the principal part, $\upvarepsilon^2$ and $\upkappa^2$ are vectors of $\varepsilon_n^2$ and $\kappa_n^2$ and $\textrm{L}^{2\to 1}_n, \textrm{L}^{1\to2}_n$ are matrices whose elements are given by: \begin{align}\label{eq:L_fc_def_two_ex} \big[\text{L}^{2\to 1}_k]_{m, n} = \int_0^T \bra{\chi_m^1(t)} L \ket{\phi_n^2(t)} e^{\textrm{i}k\Omega t} \frac{dt}{T} \ \text{and} \ \big[\text{L}^{1\to 2}_k]_{m, n} = \int_0^T \bra{\chi_m^2(t)}L^\dagger \ket{\phi_n^1(t)}e^{-\textrm{i}k\Omega t}\frac{dt}{T} \end{align} The full two-photon scattering matrix can be constructed from the connected parts in Eqs.~\ref{eq:conn_part_one_ph} and \ref{eq:conn_part_two_ph} by an application of the cluster decomposition principle (Eq.~\ref{eq:smat_cluster_decomp}). We note that while it appears that $S^{\text{C}, 1}(\omega_1, \omega_2; \nu_1, \nu_2)$ has singularites corresponding to principle parts --- as is shown in Appendix \ref{app:smat}, a proper evaluation of the summation removes these singularities. As an illustrative example of this procedure, we consider the computation of the single- and two-photon scattering matrices for a cavity with Kerr-nonlinearity and a periodically modulated resonance frequency. The Hamiltonian of the localized system under consideration here is given by: \begin{align} H_s(t) = \Delta(t)a^\dagger a + \chi(a^\dagger)^2 a^2, \end{align} with a coupling operator $L = \sqrt{\kappa / 2} \ a$. Here, $\Delta(t)$ is the periodic modulation applied on the cavity mode, $\chi$ is the photon-photon repulsion in the cavity due to the Kerr nonlinearity and $\kappa$ is the decay rate of the cavity. We assume that the mean of $\Delta(t)$ over one period is 0. Since the $N-$excitation subspace for this system has dimensionality $1$, there is only one solution to the Floquet problem in Eq.~\ref{eq:floquet_problem}: \begin{align} \ket{\phi^N_1(t)} = \ket{\chi^N_1(t)} = e^{-\textrm{i}N \varphi(t)} \frac{(a^\dagger)^N}{\sqrt{N!}}\ket{g} \ \text{and} \ \varepsilon^N_1 = - \frac{\textrm{i}N\kappa}{2} + \chi N (N - 1), \end{align} where $\varphi(t) = \int_0^t \Delta(t')dt'$. With this choice of Floquet states, $\text{L}^{1\to 0}_k, \text{L}^{0\to 1}_k, \text{L}^{2 \to 1}_k$ and $\text{L}^{1 \to 2}_k$ in Eqs.~\ref{eq:L_fc_def_single_ex} and \ref{eq:L_fc_def_two_ex} reduce to scalars given by: \begin{align} \text{L}^{1\to 0}_k = \sqrt{\kappa}\alpha_k, \ \text{L}^{2\to 1}_k = \sqrt{2\kappa}\alpha_k, \ \text{L}^{1\to 2}_k = \sqrt{2\kappa}\alpha_k^*, \text{ and } \text{L}^{0\to 1}_k = \sqrt{\kappa}\alpha_k^* \end{align} where $\alpha_k$ are the Fourier-series components of $e^{-\textrm{i}\varphi(t)}$: \begin{align} e^{-\textrm{i}\varphi(t)} = \sum_{k\in \mathbb{Z}} \alpha_k e^{-\textrm{i}k\Omega t}. \end{align} Therefore, the single-photon scattering matrix (Eq.~\ref{eq:conn_part_one_ph}) evaluates to \begin{align} S_k(\nu) = \sum_{m\in\mathbb{Z}} \frac{\alpha_{k + m} \alpha_m^*}{\textrm{i}( m\Omega - \nu) + \kappa / 2}. \end{align} Similarly, using Eqs.~\ref{eq:conn_part_two_ph}, the two-photon scattering matrices evaluate to: \begin{subequations} \begin{alignat}{2} S^{\text{C}, 1}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{\kappa^2}{2\pi \textrm{i}}\sum_{P, Q}\sum_{p, m, n \in \mathbb{Z}} \bigg[ \frac{\alpha_p \alpha^*_{n + p -k}\alpha_{m + n}\alpha^*_m}{\big(\textrm{i}(p\Omega - \omega_{P(1)}) + \kappa / 2\big)\big(\textrm{i}(m\Omega - \nu_{Q(2)}) + \kappa/2 \big)} \mathcal{P}\frac{1}{\omega_{P(2)} - \nu_{Q(2)}-n\Omega} \bigg],\\ S^{\text{C}, 2}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{\kappa^2}{\pi}\sum_{P, Q}\sum_{p, m, n \in \mathbb{Z}} \bigg[\frac{\alpha_p \alpha_{m-p+k}\alpha^*_{n-m}\alpha^*_m}{\big(\textrm{i}(p\Omega- \omega_{P(1)}) + \kappa/2\big)\big(\textrm{i}(n\Omega + 2\chi - \nu_1 - \nu_2) + \kappa\big)\big(\textrm{i}(m\Omega- \nu_{Q(2)} ) + \kappa/2\big)}\bigg], \end{alignat} Numerical studies of the single and two-photon transport through a modulated Kerr cavity with $\Delta(t) = \Delta_0 \sin \Omega t$ are shown Figure \ref{fig:chi3_cavity_scat}. Figure~\ref{fig:chi3_cavity_scat}\textbf{a} shows the total single-photon transmission $T(\nu) = \sum_{k\in \mathbb{Z}} |S_k(\nu)|^2$ through the cavity for slow modulation ($\Omega \ll \kappa$) and fast modulation ($\Omega \gg \kappa$) of its resonant frequency. In the fast modulation regime, the transmission spectra shows resonances at integer multiples of $\Omega$ with the transmission being smaller than the resonant transmission for unmodulated cavity. In the slow modulation regime, moderate transmissions are achieved if the input photon is within the resonant frequencies achieved by the periodic modulation ($[-\Delta_0, \Delta_0]$). The amplitude $|S_k(\nu)|$ of a photon at frequency $\nu$ scattering into a photon at frequency $\nu + k\Omega$ within the slow and fast modulation regime is shown in Figs.~\ref{fig:chi3_cavity_scat}\textbf{b}. We point out that the single-photon transmissions obtained here are unaffected by the non-linearity $\chi$ in the cavity mode --- they are identical to the classical transmission that would be obtained through a linear cavity with the same modulation \cite{minkov2017exact}. \begin{figure} \caption{\textbf{Single photon scattering from a modulated Kerr cavity.} \textbf{a.} The total single-photon transmission as a function of the input frequency $\nu$ for different modulation frequencies. \textbf{b.} Amplitude of scattering a single-photon at frequency $\nu$ into an output photon at frequency $\nu + k\Omega$ as a function of $\nu$ and $k$. $\Delta_0 = 3\kappa$ has been assumed in all simulations.} \label{fig:chi3_cavity_scat} \end{figure} The connected part of the two-photon scattering matrix corresponding to the $k^\text{th}$ side-band under excitation by two photons at frequencies $\nu_1 = \nu_2 = 0$, $S_k^\text{C}(\omega_1, \omega_2; \nu_1 = 0, \nu_2 = 0)$, is shown in Fig.~\ref{fig:chi3_cavity_scat_two_ph}. Since the output frequencies $\omega_1$ and $\omega_2$ of the two photons emitted into this sideband are constrained to satisfy $\omega_1 + \omega_2 = k\Omega$, they can be completely parametrized by their frequency different $\delta = \omega_1 - \omega_2$. As can be seen from Fig.~\ref{fig:chi3_cavity_scat_two_ph} --- the amplitude of the connected part increased on increasing the nonlinearity $\chi$. This is intuitively expected since the connected part captures the photon-photon interactions induced by the localized system. Furthermore, we note that there is an asymmetry in the amplitudes of the connected part corresponding to $k = 1$ and $k = -1$ --- this can be attributed to the fact that the nonlinearity $\chi$ results in an \emph{increase} in the cavity resonant frequency with the number of photons in the cavity and thus has larger contribution to one side-band as opposed to the other. Indeed, in the two-level system limit ($\chi\to \infty$), it can be seen from Fig.~\ref{fig:chi3_cavity_scat_two_ph} that both the sidebands have identical connected part amplitudes. \begin{figure} \caption{\textbf{Two photon scattering from cavity with Kerr non-linearity:} The connected part of the two-photon scattering matrix for the $k = -1, 0, 1$ sidebands with the two input photons being at $\nu_1 = \nu_2 = 0$ as a function of the frequency difference $\delta$ between the output photons. Note that the two output photon scattered into the $k^\text{th}$ sideband are constrained have a mean frequency of $k{\Omega} / 2$. Parameter values of $\Omega= 2.5\kappa$ and $\Delta_0 = 3\kappa$ have been assumed in all simulations.} \label{fig:chi3_cavity_scat_two_ph} \end{figure} \end{subequations} \section{Geometric properties in slow modulation regime} \label{sec:slow_mod_smat} In a number of physical systems, the modulation period is significantly smaller than the timescale of evolution of the localized system \cite{xiao2005berry, xiao2006berry}. Such systems are considered to be in the slow modulation regime and have been a subject of significant theoretical interest. In particular, for closed quantum systems, observables such as the Berry phase \cite{xiao2010berry} can be defined which only depend on the geometry of the Hamiltonian being modulated and are independent of the modulation being applied to the Hamiltonian. In this section, we study scattering from a slowly modulated quantum system. In particular, it is shown that the equal time $N-$photon correlation function, to the zeroth order in modulation frequency, is equal to the time-average of the instantaneous correlation function obtained by assuming the system to be time independent. Furthermore, we show that the first-order correction to the $N-$photon correlation function is purely geometric in nature {\it i.e.}~it is independent of the precise form of the modulation applied on the Hamiltonian. We consider a localized system with Hamiltonian dependent on a set of parameters $\text{p} = \{p_1, p_2 \dots p_M\}$: $H_s(\text{p})$. These parameters are varied along a closed loop $\mathcal{C}$ within the space of allowed parameters periodically to yield a Hamiltonian $H_s(t) = H_s(\text{p}(t))$. The equal-time $N-$photon correlation $G_N(\nu)$ at frequency $\nu$ is defined in terms of the $N-$photon scattering matrix via: \begin{align} G_N(\nu) = \frac{1}{N! T}\int_0^T \bigg|\int_{-\infty}^\infty \dots \int_{-\infty}^\infty S(\omega_1, \omega_2 \dots \omega_N; \nu, \nu \dots \nu)\prod_{i=1}^N e^{-\textrm{i}\omega_i t} d\omega_i \bigg|^2 dt, \end{align} or equivalently in terms of the $N-$excitation Green's function via: \begin{align}\label{eq:nph_corr_gfunc} G_N(\nu) = \frac{1}{N! T}\int_0^T \bigg| \int_{-\infty}^\infty \dots \int_{-\infty}^\infty G(t, t \dots t; s_1, s_2 \dots s_N) \prod_{i=1}^N e^{-\textrm{i}\nu s_i}ds_i \bigg|^2 dt. \end{align} For $N = 1$, from Eq.~\ref{eq:conn_part_one_ph} this correlation function is identical to the total transmission $\sum_{k=-\infty}^\infty |S_k(\nu)|^2$ through the localized system. For $N\geq 2$, this correlation function can be measured with $N-$photon coincidence counts on the emission from the localized system. We now consider the calculation of a perturbative expansion for $G_N(\nu)$ with respect to $\Omega$. As is shown in appendix \ref{app:geometry}, it follows from the definition of the $N-$excitation Green's function that \begin{subequations}\label{eq:pert_exp} \begin{align} \frac{1}{N!}\int_{-\infty}^\infty \dots \int_{-\infty}^\infty G(t, t \dots t; s_1, s_2 \dots s_N) \prod_{i=1}^N e^{-\textrm{i}\nu s_i}ds_i = e^{-\textrm{i}N\nu t}\bigg[\mathcal{G}_N^{(0)}(\text{p} (t); \nu) + \mathcal{G}_N^{(1)}(\text{p}(t); \nu)\cdot\frac{d\text{p}(t)}{dt} + O(\Omega^2)\bigg], \end{align} where $\mathcal{G}_N^{(0)}(\text{p}; \nu)$ is zeroth order in the modulation frequency $\Omega$ and is given by: \begin{align} \mathcal{G}_N^{(0)}(\text{p}; \nu) &= (-\textrm{i})^N\bra{g} L^N \bigg[\prod_{n=N}^1 (H_\text{eff}^n(\text{p}) - n\nu)^{-1}L^\dagger\bigg] \ket{g} \end{align} and $\mathcal{G}_N^{(1)}(\text{p}; \nu)$, also zeroth order in $\Omega$, is given by: \begin{align} \mathcal{G}_N^{(1)}(\text{p}; \nu) &= (-\textrm{i})^{N-1}\sum_{k=1}^N \bra{g}L^N\bigg[ \prod_{n=N}^{k+1}(H_\text{eff}^n(\text{p}) - n\nu)^{-1}L^\dagger\bigg](H_\text{eff}(\text{p}) - k\nu)^{-1}\nabla_\text{p}\bigg[\prod_{n=k}^1 (H_\text{eff}^n(\text{p}) - n\nu)^{-1} L^\dagger\bigg] \ket{g}. \end{align} \end{subequations} Here $H_\text{eff}^n(\text{p}) $ is the $n-$excitation effective Hamiltonian as a function of the parameters $\text{p}$. The equal-time $N-$photon correlation function can now be expanded into a perturbative series in $\Omega$: $G_N(\nu) = G_N^{(0)}(\nu) + \Omega G_N^{(1)}(\nu) + O(\Omega^2)$ where both $G_N^{(0)}(\nu)$ and $G_N^{(1)}(\nu)$ are zeroth order in $\Omega$. It follows from Eqs.~\ref{eq:nph_corr_gfunc} and \ref{eq:pert_exp} that the zeroth order contribution $G_N^{(0)}(\nu)$ is given by: \begin{align} G_N^{(0)}(\nu) = \int_0^T\big|\mathcal{G}_N^{(0)}(\text{p}(t); \nu)\big|^2\frac{dt}{T}. \end{align} It can be noted that $|\mathcal{G}_N^{(0)}(\text{p}; \nu)|^2$ is the equal-time $N-$photon correlation function that would be measured from the emission of a time-independent localized system with Hamiltonian $H_s(\text{p})$ and consequently to zeroth order $G_N(\nu)$ is simply a time-average of the instantaneous correlation function $|\mathcal{G}_N^{(0)}(\text{p}; \nu)|^2$. Furthermore, $G_N^{(0)}(\nu)$ is dynamical in nature {\it i.e.}~it is dependent on the precise modulation of the parameters $\text{p}$. The first order contribution, $G_N^{(1)}(\nu)$, is given by: \begin{align} G_N^{(1)}(\nu) = \frac{1}{\pi} \text{Re}\bigg[\int_0^T \big[\mathcal{G}_N^{(0)}(\text{p}(t); \nu)\big]^* \mathcal{G}_N^{(1)}(\text{p}(t); \nu)\cdot \frac{d\text{p}(t)}{dt} dt \bigg] = \frac{1}{\pi} \text{Re}\bigg[\oint_\mathcal{C} \big(\mathcal{G}_N^{(0)}(\text{p}; \nu)\big)^* \mathcal{G}_N^{(1)}(\text{p}; \nu) \cdot d\text{p}\bigg]. \end{align} It can immediately be seen that the first order correction $G_N^{(1)}(\nu)$ is completely geometric in nature {\it i.e.}~it only depends on the loop $\mathcal{C}$ in the parameter space that the parameters $\text{p}$ trace during modulation. \begin{figure} \caption{\textbf{Scattering from a slowly modulated Jaynes Cumming system:} \textbf{a.} Modulation trajectories considered in our calculations --- the dashed line indicates the loop in the complex plane along which the cavity-TLS coupling constant $g$ is varied in one modulation period. The thickness of the colored shaded region around any point on the loop indicates how fast $g$ is changing at that point. \textbf{b.} Zeroth order and first order contribution to the total single-photon transmission $T(\nu) = G_1(\nu)$ through the Jaynes Cumming system as a function of the input frequency $\nu$. \textbf{c.} Zeroth order and first order contribution to the equal-time two photon correlation $G_2(\nu)$ in the output of the Jaynes Cumming system as a function of the input frequency $\nu$. } \label{fig:slow_mod} \end{figure} As an illustrative example, we consider scattering from a Jaynes Cumming system formed by coupling a cavity with resonant frequency $\omega_c$ to a two-level system at frequency $\omega_e$: \begin{align} H_s(g) = \omega_e \sigma^\dagger \sigma + \omega_c a^\dagger a + (g a\sigma^\dagger + g^* a^\dagger \sigma), \end{align} where we modulate the complex cavity-TLS coupling strength $g$ periodically as a function of time to obtain a time-dependent Hamiltonian. We assume that this system coupled to the bosonic bath with through the cavity mode {\it i.e.}~$L = \sqrt{\kappa / 2}\ a$ where $\kappa$ is the decay rate of the cavity. We consider three different modulations of $g$ as depicted in Fig.~\ref{fig:slow_mod}\text{a} which traverse the same loop in the complex plane per period. The shaded regions in Fig.~\ref{fig:slow_mod}a indicate the rate of change of $g$ with time at different points on the loop. Figure \ref{fig:slow_mod}b shows the zeroth and first order contributions to the transmission spectrum $T(\nu) = G_1(\nu)$ for the three different choices of $g(t)$. We clearly see that the zeroth order contribution $T^{(0)}(\nu)$ is dependent on the time-dependence of the modulation applied on $g$ whereas the first order contribution $T^{(1)}(\nu)$ is identical for the three different modulation schemes {\it i.e.}~it is completely geometric in nature. A similar behavior can be seen for the zeroth and first order contributions to the equal-time two-photon correlation. \section{Conclusion} In this paper, we studied scattering of photons from periodically modulated quantum systems. A procedure for constructing $N-$photon scattering matrices and relating them to the Floquet decomposition of the effective Hamiltonian of the quantum system was outlined. Furthermore, we studied the properties of the equal time $N-$photon correlation function in the slow modulation regime and show that the first order correction with respect to the modulation frequency is completely geometric in nature. The formalism and results presented in this paper are of fundamental interest in the study of time-dependent open systems as well as for simulating quantum systems relevant for building quantum information processing systems. \section{Photon number conservation by the scattering matrix}\label{app:ph_no_cons} Let $\Pi_s^n$ be the projector onto the $n^\text{th}$ excitation subspace $\mathcal{H}_s^n$. The excitation number operator $\mu_s$ can be constructed from $\Pi_s^n$ via: \begin{align} \mu_s = \sum_{n=0}^\infty n\Pi_s^n. \end{align} By construction, $\mu_s = \mu_s^\dagger$ and $\mu_s \ket{\phi} = n\ket{\phi}$ for $\ket{\phi} \in \mathcal{H}_s^n$. Additionally \begin{align}\label{eq:comm_l_mu} [L, \mu_s] = L. \end{align} To see this, suppose $\ket{\phi} \in \mathcal{H}_s^n$ for any $n \geq 1$ then $L\ket{\phi} \in \mathcal{H}_s^{n-1}$. Therefore, \begin{align} L \mu_s \ket{\phi} = n L\ket{\phi} \text{ and } \mu_s L \ket{\phi} = (n - 1)L \ket{\phi} \implies [L, \mu_s]\ket{\phi} = L\ket{\phi}. \end{align} Furthermore, for $\ket{\phi} \in \mathcal{H}_s^0$, since $\mu_s \ket{\phi} = 0$ and $L\ket{\phi} = 0$ it follows that $[L, \mu_s] \ket{\phi} = 0 = L\ket{\phi}$. This shows that the operators $L$ and $\mu_s$ satisfy Eq.~\ref{eq:comm_l_mu} Finally, consider the excitation number operator $\mu$ for the full system constructed by adding $\mu_s$ with the photon number operator for the two baths, \begin{align}\label{eq:ex_num_op} \mu = \mu_s + \sum_{l \in \{a, b\}} \int_{-\infty}^\infty l_\omega^\dagger l_\omega d\omega. \end{align} From the commutator $[L, \mu_s] = L$ it follows that $[H(t), \mu] = 0$ {\it i.e.}~the observable corresponding to $\mu$ is a conserved quantity. Since a state with $N$ photons in the bosonic baths and the system in $\ket{g}$ is an eigenstate of $\mu$ with eigenvalue $N$, this conservation law immediately implies that it can only scatter into a state with $N$ photons in the bosonic baths. \section{Relating the scattering matrix elements to the Green's function}\label{app:smat_to_gfunc} In this appendix, we derive the relationship between the Green's function $G(t_1, t_2 \dots t_N; s_1, s_2 \dots s_N)$ and the scattering matrix element $S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$ (Eq.~\ref{eq:gfunc_to_smat}). Using the fact that $H_s(t)\ket{g} = 0$ and $L\ket{g} = 0$, it follows from Eq.~\ref{eq:basic_hamiltonian} that $H(t)\ket{g; \text{vac}} = 0$. Noting that the propagator $U_0(t_f, t_i)$ with respect to the Hamiltonian $H_0(t)$ (Eq.~\ref{eq:ref_hamil}) satisfies $U_0(t_i, t_f) l_\omega U_0(t_f, t_i) = l_\omega e^{-\textrm{i}\omega (t_f - t_i)} \ \forall \ l \in\{a, b\}$, the scattering matrix element $S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)$ can be expressed as: \begin{align}\label{eq:smat_in_terms_hop} S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N) = e^{-\textrm{i}\phi(t_0)} \lim_{\substack{t_+\to \infty \\ t_- \to -\infty}}e^{\textrm{i}\sum_{i=1}^N (\omega_i t_+ - \nu_i t_-)} \bra{g;\text{vac}} \mathcal{T}\bigg[\bigg(\prod_{i=1}^N b_{\omega_i}(t_+)\bigg) \bigg(\prod_{i=1}^N a_{\nu_i}^\dagger(t_-)\bigg)\bigg]\ket{g;\text{vac}}, \end{align} where $a_\nu(t_-) = U(t_0, t_-) a_\nu U(t_-, t_0)$ and $b_\omega(t_+) = U(t_0, t_+) b_\omega U(t_+, t_0)$ with $U(\cdot, \cdot)$ being the propagator with respect to the Hamiltonian $H(t)$ (Eq.~\ref{eq:basic_hamiltonian}) and $\mathcal{T}[\cdot]$ indicates a time-ordering with respect to its arguments. Note that since $t_+ \geq t_-$, this time-ordering is effectively an identity operation in Eq.~\ref{eq:smat_in_terms_hop}. Next, we use the Heisenberg equations of motion for $a_\nu(t)$ and $b_\omega(t)$ --- from Eq.~\ref{eq:basic_hamiltonian}, it follows that: \begin{align} \frac{d}{dt} \begin{pmatrix} a_\nu(t) \\ b_\omega(t) \end{pmatrix} = -\textrm{i}\begin{pmatrix} \nu a_\nu(t) \\ \omega b_\omega(t) \end{pmatrix} + \frac{1}{\sqrt{2\pi}} \begin{pmatrix} L(t) \\ L(t) \end{pmatrix} \end{align} These equations of motion can easily be integrated from $t_-$ to $t_+$ to yield the following: \begin{subequations} \begin{align} &a_\nu^\dagger(t_-) = a_\nu^\dagger(t_+)e^{-\textrm{i}\nu(t_+ - t_-)} - \int_{t_-}^{t_+} L(s) e^{-\textrm{i}\nu (s - t_-)}\frac{ds}{\sqrt{2\pi}}\label{eq:a_init_final}\\ &b_\omega(t_+) = b_\omega(t_-)e^{-\textrm{i}\omega(t_+ - t_-)} + \int_{t_-}^{t_+} L(t) e^{-\textrm{i}\omega( t_+ - t)}\frac{dt}{\sqrt{2\pi}}\label{eq:b_init_final} \end{align} \end{subequations} Substituting Eq.~\ref{eq:b_init_final} into Eq.~\ref{eq:smat_in_terms_hop} and noting that any term with $b_{\omega_i}(t_-)$ goes to 0 since the time-ordering operator places it to the right of $L(t)\ \forall \ t \in (t_-, t_+)$, $b_{\omega_i}(t_-)$ commutes with $a_{\nu_i}(t_-)$ and annihilates $\ket{g; \text{vac}}$, we obtain: \begin{align}\label{eq:smat_partial} &S(\omega_1, \omega_2 \dots \omega_N; \nu_1, \nu_2 \dots \nu_N)\nonumber\\ &= e^{-\textrm{i}\phi(t_0)} \lim_{\substack{t_+\to\infty \\ t_- \to -\infty}} \int_{t_-}^{t_+}\dots \int_{t_-}^{t_+} \bra{g; \text{vac}} \mathcal{T}\bigg[\bigg(\prod_{i=1}^N L(t_i)\bigg)\bigg(\prod_{i=1}^N a_{\nu_i}^\dagger(t_-)\bigg) \bigg]\ket{g;\text{vac}}\prod_{i=1}^N e^{\textrm{i}\omega_i t_i} \frac{dt_i}{\sqrt{2\pi}} \end{align} Similarly, substituting Eq.~\ref{eq:a_init_final} into Eq.~\ref{eq:smat_partial} and noting that any term with $a_{\nu_i}(t_+)$ goes to 0 since the time-ordering operator places it to the left of $L(t), L(t) \ \forall \ t \in(t_-, t_+)$ and that $a^\dagger_{\nu_i}(t_+)$ annihilates $\bra{g; \text{vac}}$, we obtain the result in Eq.~\ref{eq:gfunc_to_smat}. \section{Scattering matrix calculation}\label{app:smat} \subsection{Single-photon scattering matrix} The starting point for the calculation of the single-photon scattering matrix is the evaluation of the single-photon Green's function which is given by: \begin{align}\label{eq:single_ex_gfunc} G(t; s) = \bra{ g} \mathcal{T}[\tilde{L}(t) \tilde{L^\dagger}(s)]\ket{g} = \bra{g} L {U}_\text{eff}^1(t, s)L^\dagger\ket{g}\Theta(t\geq s). \end{align} Using the Floquet decomposition of $U_\text{eff}^1(t, s)$ (Eq.~\ref{eq:floquet_decomp}), this can be expressed as: \begin{align} G(t; s) = \big(\text{L}^{1\to 0}(t) \text{D}\big[e^{-\textrm{i}\lambda^1(t - s)}\big] \text{L}^{0\to 1}(s)\big)\Theta(t\geq s), \end{align} where $\text{L}^{1\to0}(t)$ is a row-vector and $L^{0 \to 1}(s)$ is a column-vector, and their elements are given by: \begin{align}\label{eq:td_ann_mat_elm} \big[\text{L}^{1\to 0}(t)\big]_n = \bra{g} L \ket{\phi_n(t)} \ \text{ and } \ \big[\text{L}^{0 \to 1}(s)\big] = \bra{\chi_n^1(s)} L^\dagger \ket{g} \end{align} We note that $\text{L}^{1\to 0}_k$ and $\text{L}^{0\to 1}_k$ defined in Eq.~\ref{eq:L_fc_def_single_ex} of the main text are simply the Fourier series coefficients of $\text{L}^{1 \to 0}(t)$ and $\text{L}^{0\to 1}(s)$ respectively: \begin{align}\label{eq:fseries} \text{L}^{1\to 0}(t) = \sum_{k \in \mathbb{Z}} \text{L}^{1\to 0}_k e^{-\textrm{i}k\Omega t} \ \text{and} \ \text{L}^{0\to 1}(s) = \sum_{k\in \mathbb{Z}} \text{L}^{0\to 1}_k e^{\textrm{i}k\Omega s}. \end{align} From Eqs.~\ref{eq:gfunc_to_smat}, \ref{eq:single_ex_gfunc} and \ref{eq:fseries}, it follows that the single-photon scattering matrix $S(\omega; \nu)$ is given by Eq.~\ref{eq:conn_part_one_ph} in the main text. \subsection{Two-photon scattering matrix} \label{app:two_ph_smat} The two-excitation Green's function $G(t_1, t_2; s_1, s_2)$, given by Eq.~\ref{eq:gfunc_eff_hamil} with $N = 2$, is symmetric under the swap operations $t_1\leftrightarrow t_2$ and $s_1 \leftrightarrow s_2$. Defining $\mathcal{G}(t_1, t_2; s_1, s_2) = G(t_1, t_2; s_1, s_2) \Theta(t_1\geq t_2 \ \text{and} \ s_1\geq s_2)$, it follows that: \begin{align}\label{eq:ordered_gfunc} G(t_1, t_2; s_1, s_2) = \sum_{P, Q} \mathcal{G}(t_{P(1)}, t_{P(2)}; s_{Q(1)}, s_{Q(2)}), \end{align} where $P, Q$ are permutations of the two-element set $\{1, 2\}$. It thus follows from Eq.~\ref{eq:gfunc_to_smat} that: \begin{align}\label{eq:full_smat_partial_smat} S(\omega_1, \omega_2; \nu_1, \nu_2) = \sum_{P, Q}e^{-\textrm{i}\phi(t_0)} \mathcal{S}(\omega_{P(1)}, \omega_{P(2)}; \nu_{Q(1)}, \nu_{Q(2)}), \end{align} where \begin{align}\label{eq:def_new_S} \mathcal{S}(\omega_1, \omega_2; \nu_1, \nu_2) = \int_{-\infty}^\infty \dots \int_{-\infty}^\infty \mathcal{G}(t_1, t_2; s_1, s_2) \prod_{i=1}^2 e^{\textrm{i}(\omega_i t_i - \nu_i s_i)}\frac{dt_i ds_i}{2\pi}. \end{align} From Eq.~\ref{eq:gfunc_eff_hamil}, it follows that: \begin{align} \mathcal{G}(t_1, t_2; s_1, s_2) = \mathcal{G}^1(t_1, t_2; s_1, s_2) + \mathcal{G}^2(t_1, t_2; s_1, s_2), \end{align} where \begin{subequations} \begin{align} \mathcal{G}^1(t_1, t_2; s_1, s_2) &= \bra{g} L(t_1) L^\dagger(s_1) L(t_2) L^\dagger(s_2) \ket{g} \Theta(t_1\geq s_1 \geq t_2 \geq s_2) \nonumber \\ & =\bra{g}L U_\text{eff}^1(t_1, s_1) L^\dagger \ket{g}\bra{g}L U_\text{eff}^1(t_2, s_2) L^\dagger \ket{g}\Theta(t_1\geq s_1 \geq t_2 \geq s_2) \\ \mathcal{G}^2(t_1, t_2; s_1, s_2) &= \bra{g}L(t_1) L(t_2) L^\dagger(s_1) L^\dagger(s_2)\ket{g} \Theta(t_1\geq t_2 \geq s_1 \geq s_2) \\ &=\bra{g}L U_\text{eff}^1(t_1, t_2) L^\dagger U_\text{eff}^2(t_2, s_1) L U_\text{eff}^1(s_1, s_2)L^\dagger\ket{g}\Theta(t_1\geq t_2 \geq s_1 \geq s_2). \end{align} \end{subequations} Using the Floquet decomposition of $U_\text{eff}^{1,2}(t, s)$ (Eq.~\ref{eq:floquet_decomp}), it follows that: \begin{subequations}\label{eq:g2_explicit} \begin{align} \mathcal{G}^1&(t_1, t_2; s_1, s_2) =\nonumber\\ &\big(\text{L}^{1 \to 0}(t_1) \text{D}\big[e^{-\textrm{i}\lambda^1(t_1 - s_1)}\big] \text{L}^{0 \to 1}(s_1)\big)\big(\text{L}^{1 \to 0}(t_2) \text{D}\big[e^{-\textrm{i}\lambda^1(t_2 - s_2)}\big] \text{L}^{0 \to 1}(s_2)\big) \Theta(t_1\geq s_1 \geq t_2 \geq s_2), \\ \mathcal{G}^2&(t_1, t_2; s_1, s_2) = \nonumber \\ &\big(\text{L}^{1\to 0}(t_1) \text{D}\big[e^{-\textrm{i}\lambda^1(t_1 - t_2)}\big] \text{L}^{2\to 1}(t_2) \text{D}\big[e^{-\textrm{i}\lambda^2(t_2 - s_1)}\big] \text{L}^{1\to 2}(s_1) \text{D}\big[e^{-\textrm{i}\lambda^1(s_1 - s_2)}\big] \text{L}^{1\to 0}(s_2) \big) \Theta(t_1 \geq t_2 \geq s_1 \geq s_2), \end{align} where $\text{L}^{1\to 0}(t), \text{L}^{0\to 1}(s)$ are defined in Eq.~\ref{eq:td_ann_mat_elm} and $L^{2\to 1}(t), L^{1\to 2}(s)$ are matrices with elements \begin{align} \big[\text{L}^{2\to 1}(t)\big]_{m, n} = \bra{\chi^1_m(t)} L \ket{\phi^2_n(t)} \ \text{and} \ \big[\text{L}^{1\to 2}(s)\big]_{m, n} = \bra{\chi_m^2(s)}L^\dagger \ket{\phi_n^1(s)}. \end{align} \end{subequations} It can be noted that $\text{L}^{2\to1}_k$ and $\text{L}^{1\to 2}_k$ defined in Eq.~\ref{eq:L_fc_def_two_ex} of the main text are simply the Fourier series coefficients of $\text{L}^{2\to 1}(t)$ and $\text{L}^{1\to 2}(s)$: \begin{align} \text{L}^{2\to 1}(t) = \sum_{k \in \mathbb{Z}} L_k^{2\to 1} e^{-\textrm{i}k\Omega t} \ \text{and} \ \text{L}^{1\to 2}(s) = \sum_{k\in \mathbb{Z}} L_k^{1\to 2} e^{\textrm{i}k\Omega s}. \end{align} Using Eq.~\ref{eq:g2_explicit} to evaluate the integral in Eq.~\ref{eq:def_new_S}, we obtain: \begin{subequations}\label{eq:partial_smat} \begin{align} \mathcal{S}(\omega_1, \omega_2; \nu_1, \nu_2) = \sum_{\substack{k\in \mathbb{Z}, \\ j \in\{1,2\}}} \mathcal{S}^j_k(\omega_1, \omega_2; \nu_1, \nu_2) \delta(\omega_1 + \omega_2 - \nu_1 - \nu_2 - k\Omega), \end{align} where \begin{alignat}{2} \mathcal{S}^{1}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{1}{2\pi \textrm{i}}\sum_{p, m, n \in \mathbb{Z}} \bigg[&&\text{L}^{1\to 0}_p \text{D}\bigg(\frac{1}{\textrm{i}(\uplambda^1 - \omega_1 + p\Omega)} \bigg)\text{L}^{0\to 1}_{n + p - k} \bigg(\frac{1}{\textrm{i}(\omega_2 - \nu_2-n\Omega) - \textrm{i} 0^+}\bigg) \nonumber \\& &&\text{L}^{1\to 0}_{m + n} \text{D}\bigg(\frac{1}{\textrm{i}(\lambda^1 - \nu_2 + m\Omega) } \bigg)\text{L}^{0\to 1}_{m}\bigg],\\ \mathcal{S}^{2}_k(\omega_1, \omega_2; \nu_1, \nu_2) &= \frac{1}{2\pi}\sum_{p, m, n \in \mathbb{Z}} \bigg[&&\text{L}^{1\to 0}_p \text{D}\bigg(\frac{1}{\textrm{i}(\lambda^1 - \omega_1 + p\Omega)} \bigg)\text{L}^{2\to 1}_{n - p + k} \text{D}\bigg(\frac{1}{\textrm{i}(\lambda^2 - \nu_1 - \nu_2 + n\Omega) } \bigg) \nonumber \\& &&\text{L}^{1\to 2}_{n - m} \text{D}\bigg(\frac{1}{\textrm{i}(\lambda^1 - \nu_2 + m\Omega)} \bigg)\text{L}^{0\to 1}_{m}\bigg]. \end{alignat} \end{subequations} The full two-photon scattering matrix can be constructed from Eqs.~\ref{eq:full_smat_partial_smat} and \ref{eq:partial_smat}. To explicitly extract the connected part of the two-photon scattering matrix, we note that $1 / (x - \textrm{i}0^+) = \mathcal{P}(1/x) + \pi \delta(x)$ --- the connected part of the scattering matrix can thus be obtained by replacing $1 / (x - \textrm{i}0^+)$ by $\mathcal{P}(1/x)$ in the resulting expressions for the scattering matrix. This yields the results in Eq.~\ref{eq:conn_part_two_ph} in the main text. Finally, we show that $S^{\text{C},1}_k(\omega_1, \omega_2; \nu_1, \nu_2)$ defined in Eq.~\ref{eq:conn_part_two_ph} is not singular despite containing the principle parts. We begin by rewriting it as \begin{alignat}{2} S^{\text{C}, 1}_k(\omega_1, \omega_2;& \nu_1, \nu_2) = \sum_{P, Q} \sum_{n\in \mathbb{Z}} S_{k - n}(\omega_{P(1)} - (k - n)\Omega) S_n(\nu_{Q(2)}) \mathcal{P}\frac{1}{\omega_{P(2)} - \nu_{Q(2)} - n\Omega}, \nonumber \\ =&\sum_{n\in \mathbb{Z}}\sum_{i=1}^2 \bigg[ S_{k - n}(\omega_i - (k - n)\Omega) S_n(\bar{\nu}_i) \mathcal{P}\frac{1}{\bar{\omega}_i - \bar{\nu}_i - n\Omega} + S_{k- n }(\bar{\omega}_i - (k - n)\Omega) S_n({\nu}_i) \mathcal{P}\frac{1}{{\omega}_i - {\nu}_i - n\Omega}\bigg] \nonumber \\ =&\sum_{n\in \mathbb{Z}} \sum_{i=1}^2 \bigg[ S_{n}(\omega_i - n\Omega) S_{k-n}(\bar{\nu}_i) \mathcal{P}\frac{1}{\bar{\omega}_i - \bar{\nu}_i - (k - n)\Omega} + S_{k - n}(\bar{\omega}_i - (k - n)\Omega) S_{n}({\nu}_i) \mathcal{P}\frac{1}{{\omega}_i - {\nu}_i - n\Omega}\bigg] \end{alignat} where $\bar{\lambda}_{1, 2} = \lambda_{2, 1}$ for $\lambda \in \{\omega, \nu\}$ and $S_k(\nu)$ is the scattering amplitude of a single-photon at frequency $\nu$ into frequency $\nu + k\Omega$ defined in Eq.~\ref{eq:conn_part_one_ph}. We note that since the arguments of $S^{\text{C},1}_k$ are constrained to satisfy $\omega_1 + \omega_2 - \nu_1 - \nu_2 = k\Omega$, it follows that: \begin{align} S_k^{\text{C}, 1}(\omega_1, \omega_2; \nu_1, \nu_2) = \sum_{n\in \mathbb{Z}} \sum_{i=1}^2 \bigg[S_{n}(\nu_i - (\nu_i - \omega_i + n\Omega))S_{k-n}(\bar{\nu}_i)- S_{k-n}(\bar{\nu}_i + (\nu_i - \omega_i + n\Omega))S_n(\nu_i)\bigg]\mathcal{P}\frac{1}{\nu_i - \omega_i + n\Omega} \end{align} Noting that \begin{align} S_n(\nu - \delta) S_{m}(\bar{\nu}) - S_n(\nu) S_{m}(\bar{\nu} + \delta) = M_{n, m}(\nu, \bar{\nu}, \delta) \delta \end{align} where $M_{n, m}(\nu, \bar{\nu}, \delta)$, defined below, is a smooth and finite function of its arguments: \begin{align} M_{n, m}(\nu, \bar{\nu}, \delta) = \sum_{i, j}\sum_{p, q \in \mathbb{Z}}\frac{\big[\text{L}_{p}^{0\to 1} \big]_i \big[\text{L}_{p + n}^{1\to 0}\big]_i\big[\text{L}_q^{0\to1}\big]_j\big[\text{L}^{1\to0}_{q+m}\big]_j (\lambda_j^1 + \lambda_i^1 + (p + q)\Omega - \bar{\nu} - \nu)}{(\lambda_i^1 + p\Omega - (\nu - \delta))(\lambda_j^1 + q\Omega - \bar{\nu})(\lambda_i^1 + p\Omega - \textrm{i}\nu)(\lambda_j^1 + q\Omega - (\bar{\nu} + \delta))}, \end{align} it follows that \begin{align} S_k^{\text{C}, 1}(\omega_1, \omega_2; \nu_1, \nu_2) = \sum_{n\in \mathbb{Z}}\sum_{i= 1}^2 M_{n, k-n}(\nu_i, \bar{\nu}_i; \nu_i - \omega_i + n\Omega). \end{align} This shows that $S_k^{\text{C}, 1}(\omega_1, \omega_2; \nu_1, \nu_2)$ is indeed a well defined and finite function of the input and output frequencies subject to the constraint $\omega_1 + \omega_2 - \nu_1 - \nu_2 = k\Omega$. \section{Equal time $N$-photon correlation function in slow modulation regime}\label{app:geometry} In this appendix, we provide a derivation of Eq.~\ref{eq:pert_exp} of main text. Noting that the $N-$excitation Green's function $G(t_1, t_2 \dots t_N; s_1, s_2\dots s_N)$ is symmetric under permutation of the times $s_1, s_2 \dots s_N$, it follows that: \begin{align}\label{eq:g_ordered} \frac{1}{N!} \int_{-\infty}^\infty \dots \int_{-\infty}^\infty G(t, t \dots t; s_1, s_2 \dots s_N) \prod_{i=1}^N e^{-\textrm{i}\nu s_i} ds_i = \int_{-\infty}^{\infty} \dots \int_{-\infty}^{s_3}\int_{-\infty}^{s_2} G(t, t \dots t; s_1, s_2 \dots s_N) \prod_{i=1}^N e^{-\textrm{i}\nu s_i} ds_i. \end{align} Furthermore, from Eq.~\ref{eq:gfunc_eff_hamil}, we obtain that if $s_1 \leq s_2 \dots \leq s_N$ then: \begin{align} G(t, t \dots t; s_1, s_2 \dots s_N) = \langle g | L^N \bigg[\prod_{n = {N}}^1 U_\text{eff}^n(s_{n + 1}, s_n) L^\dagger\bigg]_{s_{N+1} = t} \ket{g} \end{align} To proceed further, we consider the evaluation of \begin{align} \int_{-\infty}^t U_\text{eff}^n(t, s) O(s) e^{-\textrm{i} \nu s} ds, \end{align} where $O(s)$ is a time-dependent operator which is assumed to be slowly varying. Since $U_\text{eff}^n(t, s)$ is the propagator corresponding to the Hamiltonian $H_\text{eff}^n(t)$, it follows that: \begin{align} \int_{-\infty}^t U_\text{eff}^n(t, s) O(s) e^{-\textrm{i} \nu s} ds = -\textrm{i}\int_{-\infty}^t \bigg[\frac{\partial}{\partial s}\big(U_\text{eff}^n(t, s) e^{-\textrm{i} \nu s}\big)\bigg] (H_\text{eff}^n(s) - \nu)^{-1}O(s) ds. \end{align} Applying integration by parts, we obtain: \begin{align}\label{eq:one_by_parts} \int_{-\infty}^t U_\text{eff}^n(t, s) O(s) e^{-\textrm{i} \nu s} ds= -\textrm{i}(H_\text{eff}^n(t) - \nu)^{-1}O(t)e^{-\textrm{i}\nu t} + \textrm{i}\int_{-\infty}^t U_\text{eff}^n(t, s) \frac{\partial}{\partial s}\big[ (H_\text{eff}^n(s) - \nu)^{-1}O(s)\big] e^{-\textrm{i}\nu s}ds. \end{align} Repeating a similar calculation for the integral on the right of Eq.~\ref{eq:one_by_parts} and neglecting terms that are second order in the derivatives of $H_\text{eff}^n(t)$ and $O(t)$, we obtain: \begin{align} \int_{-\infty}^t U_\text{eff}^n(t, s) O(s) e^{-\textrm{i} \nu s} ds \approx -\textrm{i}(H_\text{eff}^n(t) - \nu)^{-1}O(t) e^{-\textrm{i}\nu t} + (H_\text{eff}^n(t) - \nu)^{-1} \frac{\partial}{\partial t} \big[ (H_\text{eff}^n(s) - \nu)^{-1}O(s)\big] e^{-\textrm{i}\nu t} \end{align} Repeated application of this to the integral in Eq.~\ref{eq:g_ordered} together with neglecting any terms that second order or higher in the derivatives of effective Hamiltonian, we obtain the result in Eq.~\ref{eq:pert_exp}. {} \end{document}

\begin{document} \title{A Bayesian analysis of classical shadows} \author{Joseph M. Lukens} \email{[email protected]} \affiliation{Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA} \author{Kody J. H. Law} \affiliation{School of Mathematics, University of Manchester, Manchester, M13 9PL, UK} \author{Ryan S. Bennink} \affiliation{Quantum Computational Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA} \date{\today} \begin{abstract} The method of classical shadows heralds unprecedented opportunities for quantum estimation with limited measurements [H.-Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. \textbf{16}, 1050 (2020)]. Yet its relationship to established quantum tomographic approaches, particularly those based on likelihood models, remains unclear. In this article, we investigate classical shadows through the lens of Bayesian mean estimation (BME). In direct tests on numerical data, BME is found to attain significantly lower error on average, but classical shadows prove remarkably more accurate in specific situations---such as high-fidelity ground truth states---which are improbable in a fully uniform Hilbert space. We then introduce an observable-oriented pseudo-likelihood that successfully emulates the dimension-independence and state-specific optimality of classical shadows, but within a Bayesian framework that ensures only physical states. Our research reveals how classical shadows effect important departures from conventional thinking in quantum state estimation, as well as the utility of Bayesian methods for uncovering and formalizing statistical assumptions. \end{abstract} \maketitle \section*{Introduction} \label{sec:intro} Measurement and characterization of quantum systems comprise a long-standing problem in quantum information science~\cite{James2001}. However, the exponential scaling of Hilbert space dimension with the number of qubits makes full characterization extremely challenging, inspiring a plethora of approaches designed to estimate properties of quantum states with as few measurements as possible, such as compressed sensing~\cite{Gross2010, Flammia2012}, adaptive tomography~\cite{Huszar2012, Kravtsov2013, Granade2017}, matrix product state formulations~\cite{Cramer2010}, and neural networks~\cite{Torlai2018, Carrasquilla2019, Lohani2020}. Very recently, a groundbreaking approach known as classical shadows was proposed and analyzed~\cite{Huang2020}. Building on and simplifying ideas from ``shadow tomography''~\cite{Aaronson2018}, the classical shadow was shown to provide accurate predictions of observables with a fixed number of measurements, including simulated examples for quantum systems in excess of 100 qubits~\cite{Huang2020}. Astonishingly simple, the classical shadow is formed by collecting the results of random measurements on a repeatedly prepared input state, and inverting them through an appropriate virtual quantum channel. However, several features of the classical shadow remain enigmatic, including its highly nonphysical nature, optimality with respect to alternative cost functions, and relationship to more conventional likelihood-based tomographic techniques. One such method, Bayesian mean estimation (BME)~\cite{Blume2010}, provides a conceptually straightforward path to estimate a quantum state given measured data, making use of prior knowledge and providing meaningful error bars for any experimental conditions. BME appears particularly well suited for contextualizing classical shadows, since it returns a principled estimate under any number of measurements (even zero), and is optimal in terms of minimizing average squared error~\cite{Robert1999}. In this work, we directly compare the estimates of classical shadows and BME for identical simulated datasets. For particular observables with relatively improbable values from the perspective of BME, shadow is found to reach the ground truth with significantly fewer measurements. However, after properly reformulating the problem under test for consistency with the Bayesian prior, the situation reverses, with BME returning estimates possessing lower error on average. In the latter portion of our investigation, we seek to construct a BME model emulating the key features of the classical shadow, but with positive semidefinite states as support. While complicated by the shadow's nonphysical nature, we ultimately propose an observable-oriented pseudo-likelihood that rates quantum states by their observable values with respect to those of shadow. Our pseudo-likelihood successfully mimics the dimension-independence of shadow, with the advantage of delivering entirely physical estimates for any number of measurements. \section*{Results} \label{sec:res} \subsection*{Problem Formulation} \textit{Classical Shadows.} For our analysis, we invoke the setup of the original classical shadow proposal~\cite{Huang2020}. Consider a $D$-dimensional Hilbert space occupied by a ground truth quantum state $\rho_g$ that can be repeatedly prepared. On each preparation $m$, $\rho_g$ is subjected to a randomly chosen $D\times D$ unitary $U_m$ and one measurement is performed in the computational basis, leaving result $\ket{b_m}$. Defining $\ket{\psi_m} = U_m^\dagger\ket{b_m}$, the classical snapshot associated with measurement $m$ follows as $\mathcal{M}^{-1}(\ket{\psi_m}\bra{\psi_m})$, where $\mathcal{M}(\cdot)$ is the quantum channel defined by averaging over all possible unitaries and outcomes. We assume the $U_m$ are drawn from the set of $D\times D$ Haar-random unitaries, in which case $\mathcal{M}^{-1}(\ket{\psi_m}\bra{\psi_m}) = (D+1)\ket{\psi_m}\bra{\psi_m} - I_D$, with $I_D$ the $D\times D$ identity matrix~\cite{Huang2020}. (This channel holds for the more restricted class of random Cliffords as well~\cite{Webb2016, Zhu2017}.) Averaging over $M$ measurements yields the shadow estimator \begin{equation} \label{eq:shad} \rho_s = \frac{D+1}{M} \sum_{m=1}^M \ket{\psi_m}\bra{\psi_m} - I_D. \end{equation} (In what follows, the phrases ``classical shadow,'' ``shadow estimator,'' and simply ``shadow'' refer interchangeably to this estimator as well as the procedure more generally.) In this form, the simplicity of $\rho_s$ is evident: it is merely a scaled and recentered average of all observed outcomes. Interestingly, though, $\rho_s$ is in general not positive semidefinite; for $M<D$, $\rho_s$ possesses at least $D-M$ eigenvalues equal to $-1$. Accordingly, in the targeted regime for classical shadows of $M\ll D$, $\rho_s$ is highly nonphysical. Understanding the role the shadow estimator's negativity on estimation forms a central goal of the present study. Finally, defining $\lambda$ as the expectation of the observable $\Lambda$ ($\lambda = \Tr\rho\Lambda$), the shadow estimate thereof follows as \begin{equation} \label{eq:s} \lambda^{(s)} = \Tr\rho_s\Lambda, \end{equation} to be compared to the ground truth $\lambda^{(g)} = \Tr\rho_g\Lambda$. As an aside, we note that Ref.~\cite{Huang2020} employed an additional statistical technique, ``median of means,'' to reduce the impact of outliers by partitioning the $M$ outcomes into $K$ subsets and taking the median as the estimate $\lambda^{(s)}$. In the interests of simplicity and ease of comparison, we focus on $K=1$ in Eq.~(\ref{eq:shad}). We expect the benefits of selecting $K>1$ will prove similar in both the shadow and Bayesian cases~\cite{Orenstein2019}, but work on this is beyond the scope of the present investigation. \textit{Bayesian Mean Estimation.} In the Bayesian paradigm, the same set of measurement outcomes ${\bm{\mathcal{D}}} = \{\ket{\psi_1}, \ket{\psi_2},...,\ket{\psi_M} \}$ is related to a possible density matrix $\rho(\mathbf{x})$ via a likelihood consisting of the product of probabilities set by Born's rule: \begin{equation} \label{eq:LL} L_{\bm{\mathcal{D}}}(\mathbf{x}) = \prod_{m=1}^M \braket{\psi_m|\rho(\mathbf{x})|\psi_m}, \end{equation} that is, $L_{\bm{\mathcal{D}}}(\mathbf{x})\propto \Pr({\bm{\mathcal{D}}}|\rho)$---the probability of receiving the dataset ${\bm{\mathcal{D}}}$ given quantum state $\rho$. Some prior distribution $\pi_0(\mathbf{x})$ is also assumed, defined for parameters $\mathbf{x}$ such that $\rho(\mathbf{x})$ is always physical: trace-one, Hermitian, and positive semidefinite. Then the posterior describing the distribution of $\mathbf{x}$ given the observed data ${\bm{\mathcal{D}}}$ ensues from Bayes' rule: \begin{equation} \label{eq:post} \pi(\mathbf{x}) = \frac{1}{\mathcal{Z}} L_{\bm{\mathcal{D}}}(\mathbf{x}) \pi_0 (\mathbf{x}). \end{equation} Note that the randomness of the chosen unitaries $U_m$ does not enter the Bayesian model; only the outcomes $\ket{\psi_m}$ play a role. The selection of unitary $U_m$ is independent of the (unknown) density matrix, i.e., $\Pr(U_m=U|\rho) = \Pr(U_m=U)$; thus any probabilities would cancel out through the normalization factor $\mathcal{Z}$ in Eq.~(\ref{eq:post}). Intuitively, in the Bayesian view the experimenter knows the unitaries exactly post-experiment, regardless of how they were chosen, so imposing uncertainty on them in the estimation process proves superfluous. Consequently, while the uncertainty of BME depends strongly on the variety of measurements chosen, the theory does not, a conspicuous departure from shadow where the distribution of $U_m$ enters directly through the inverted quantum channel $\mathcal{M}^{-1}(\cdot)$. Formally, the posterior distribution in Eq.~(\ref{eq:post}) completes the Bayesian model. From this, one can estimate any function of $\rho(\mathbf{x})$. For the most direct comparison with the classical shadow, here we focus on BME specifically, which for some observable $\Lambda$ is the point estimate defined as \begin{equation} \label{eq:B} \begin{split} \lambda^{(B)} & = \braket{\Tr \rho\Lambda}_\rho \\ & = \int d\mathbf{x}\, \pi(\mathbf{x}) \, \Tr \rho(\mathbf{x})\Lambda \\ & = \Tr\left\{ \left[ \int d\mathbf{x}\, \pi(\mathbf{x}) \rho(\mathbf{x}) \right] \Lambda \right\} \\ & = \Tr \rho_B\Lambda, \end{split} \end{equation} where the last two lines follow, respectively, from the linearity of the trace operation and defining the Bayesian mean $\rho_B = \int d\mathbf{x}\, \pi(\mathbf{x}) \rho(\mathbf{x})$. This convenient simplification, in which the Bayesian mean of a quantity is simply its value at $\rho_B$, holds for linear functions of $\rho$, which includes all quantum observables and which we focus on in this article. Moreover, $\lambda^{(B)}$ is the function of ${\bm{\mathcal{D}}}$ which minimizes the mean-squared error (MSE) averaged over all possible states and outcomes. That is, \begin{equation} \label{eq:opt} \lambda^{(B)} = \argmin_{\lambda({\bm{\mathcal{D}}})} \int d{\bm{\mathcal{D}}} \int d\mathbf{x}\, \pi(\mathbf{x},{\bm{\mathcal{D}}}) \left[ \lambda({\bm{\mathcal{D}}}) - \Tr \rho(\mathbf{x})\Lambda \right]^2, \end{equation} with $\pi(\mathbf{x},{\bm{\mathcal{D}}})$ the joint distribution over data and parameters~\cite{Robert1999}. This optimality is nonasymptotic, holding for any number or collection of unitaries $\{U_1, U_2,..., U_M\}$. Considering the widely different expressions for $\rho_s$ [Eq.~(\ref{eq:shad})] and $\rho_B$ [Eq.~(\ref{eq:B})], we found it remarkable just how well $\rho_s$ performed in Ref.~\cite{Huang2020} in light of BME's optimality in Eq.~(\ref{eq:opt}); it was this feature which initially inspired us to develop a thorough comparison between shadow and BME. \textit{Simulated Experiments} In general, comparing the performance of estimators derived from classical (frequentist) statistics---like $\rho_s$---with those from Bayesian methods proves tricky business, since they view uncertainty in functionally different ways. Therefore we adopt a pragmatic view which aligns with the interests of experimentalists: perform experiments, compute the associated shadow and BME estimators, and calculate their error with respect to actual values. While the final step is not always possible in practice, it is in numerical simulation, where the ground truth $\rho_g$ is known exactly. Doing so enables us to illuminate the advantages and disadvantages of both approaches on equal footing. We employ the approach described in the ``Methods'' section for obtaining simulated datasets ${\bm{\mathcal{D}}}$. \begin{figure*} \caption{Comparison of shadow and BME estimates of $\lambda_n$ for (a) $D=32$ and (b) $D=256$. Results from fifty trials for each dimension are plotted, assuming a fixed ground truth state $\ket{0}$.} \label{fig1} \end{figure*} \subsection*{Comparing Classical Shadows and BME} \label{sec:compare} \textit{Picture 1: Fixed Ground Truth.} As our first benchmark, we compare the performance of $\rho_s$ and $\rho_B$ in estimating three rank-1 observables, of which fidelities and entanglement witnesses form an important and experimentally relevant subset. Specifically, we consider $\Lambda_n = \ket{\phi_n}\bra{\phi_n}$ ($n=0,1,2$) where \begin{equation} \label{eq:states} \begin{split} \ket{\phi_0} & = \ket{0}\\ \ket{\phi_1} & = \frac{1}{\sqrt{2}}\ket{0} + \frac{1}{\sqrt{2(D-1)}}\sum_{j=1}^{D-1} \ket{j}\\ \ket{\phi_2} & = \ket{1}. \end{split} \end{equation} These possess ground truth values equally spaced within the physically allowed range for trace-one, rank-one observables: $\lambda_0^{(g)}=1$, $\lambda_1^{(g)}=\frac{1}{2}$, and $\lambda_2^{(g)}=0$. The shadow estimator is readily obtained from Eq.~(\ref{eq:shad}), so we compute $\rho_s$ for all $M\in\{1,2,...,1000\}$, where $M$ defines the set containing the first $M$ measurements: ${\bm{\mathcal{D}}}=\{\ket{\psi_m};\; m=1,2,...,M\}$. On the other hand, $\rho_B$ requires evaluation of the high-dimensional integral $\int d\mathbf{x}\,\pi(\mathbf{x})\rho(\mathbf{x})$. To that end, we summon Markov chain Monte Carlo (MCMC) methods, several of which have been explored in the context of quantum state estimation, including Metropolis--Hastings~\cite{Blume2010, Mai2017}, Hamiltonian Monte Carlo~\cite{Seah2015}, sequential Monte Carlo (SMC)~\cite{Granade2016}, and slice sampling~\cite{Williams2017, Lu2019a}. We select the preconditioned Crank--Nicolson algorithm~\cite{Cotter2013} applied in Ref.~\cite{Lukens2020}, which to our knowledge is the most efficient BME approach currently available for density matrix recovery. Finally, because of our assumed pure state ground truth, we take as prior all pure states $\rho=\ket{\psi}\bra{\psi}$ uniformly distributed on the complex $D$-dimensional unit hypersphere. Numerically, the parameters $\mathbf{x}$ reduce to a $D$-dimensional complex column vector, so we have $\pi_0(\mathbf{x})\propto \exp\left(-\frac{1}{2}\mathbf{x}^\dagger\mathbf{x}\right)$, $\rho(\mathbf{x}) = \frac{\mathbf{x}\mathbf{x}^\dagger}{|\mathbf{x}|^2}$, and $d\mathbf{x} = \prod_{l=1}^{D} d(\R x_l) d(\I x_l)$ with $x_l$ denoting a single component of $\mathbf{x}$. The use of pure states is not central to the BME formalism whatsoever, but does permit us to simulate in higher dimensions than otherwise possible. With pure states only, our parameterization entails $2D$ real numbers, compared to $2D^2+D$ for mixed states. As an example, for $D=256$, the pure state prior, and likelihood of Eq.~(\ref{eq:LL}), each MCMC chain takes about ten minutes to converge on our desktop computer, which for the 400 settings involved in Fig.~\ref{fig1}(b) amounts to $\sim$2.5 days. Based on previous studies~\cite{Lukens2020} the mixed state version would therefore have been completely unfeasible at this dimension with our computational resources, likely taking weeks (or more) to complete~\footnote{Incorporating some of the methods suggested in Ref.~\cite{Lukens2020} in further research, such as embedding within SMC samplers and parallelization, should permit the extension to significantly larger $D$ and mixed states}. With pure states, then, we can focus more directly on dimensional scaling and the statistics from many trials. For each trial, we perform BME for eight collections of measurements $M\in\{1,50,100,200,400,600,800,1000\}$. We keep $R=2^{10}$ samples from each chain of length $RT$, where we select the thinning factor $T$ empirically to obtain convergence. Figure~\ref{fig1} plots the estimates for all 50 trials obtained by both shadow and BME for $D=32$ [Fig.~\ref{fig1}(a)] and $D=256$ [Fig.~\ref{fig1}(b)]. A thinning value of $T=2^9$ ($T=2^{12}$) is used for $D=32$ ($D=256$). Each column corresponds to a particular expectation value $\lambda_n$; the bottom row shows the MSE with respect to the ground truth, averaged over all trials defined as $\braket{|\lambda_n^{(\cdot)} - \lambda_n^{(g)}|^2}_\mathrm{trials}$ with $\cdot=s$ for the shadow and $\cdot=B$ for BME. The classical shadows show wide variation for low $M$, including highly nonphysical estimates ($\lambda_n^{(s)}>1$ or $<0$), but they converge to ground truth values rapidly, with nearly identical rates for all observables and dimensions. This is confirmed quantitatively in the MSE curves that attain values of $\sim$10$^{-3}$ by $M=1000$ for all cases. The behavior proves vastly different for BME. While physical estimates are always returned, the number of measurements needed to reach the ground truth varies strongly both with observable $\lambda_n$ and with dimension $D$. Intriguingly, shadow shows significantly lower MSE for $\lambda_0$ and $\lambda_1$, widening as $D$ increases. On first glance, this presents a paradox: Eq.~(\ref{eq:opt}) implies that $\lambda_n^{(B)}$ should possess the lowest possible MSE for any $n$ and $M$, and yet $\lambda_n^{(s)}$ convincingly surpasses it these cases. Yet this dilemma can be resolved by studying the prior $\pi_0(\mathbf{x})$. When the Bayesian model assigns equal \emph{a priori} weights to all possible states---a sensible choice for an uninformative prior---this by implication makes observable values such as $\lambda_0^{(g)}=1$ highly unlikely, since only one state in the domain attains this. On the other hand, expectations for any rank-1 projector $\Lambda$ on the order of $\lambda\sim \frac{1}{D}$ are to be expected initially since $\int d\mathbf{x}\,\pi_0(\mathbf{x}) \Tr \rho(\mathbf{x})\Lambda = \frac{1}{D}$. This manifests itself in Fig.~\ref{fig1} in BME's much lower MSE for $\lambda_2$, whose ground truth value $\lambda_2^{(g)}=0$ is much more probable. Thus, by running 50 repeated trials with the \emph{same ground truth} $\rho_g=\ket{0}\bra{0}$, the situation over which we average does not accurately reflect the uninformative prior; the conditions for BME optimality are not met. \textit{Picture 2: Random Ground Truth.} To accurately reflect uninformative prior knowledge, we therefore must prepare \emph{random} ground truth states in our simulations. To do so, we leverage the equivalence between (i) randomly prepared input states with a fixed observable---the situation of interest---and (ii) random selection of an observable for a fixed input. Consider the expectation of observable $\Lambda$, where the quantum state is rotated by some random unitary $U$: \begin{equation} \label{eq:equiv} \Tr \left[(U\rho U^\dagger)\Lambda\right]= \Tr \left[\rho(U^\dagger\Lambda U)\right]. \end{equation} Thus one can emulate the effect of a randomized state by randomly rotating the observable and evaluating it on a fixed state. Practically speaking, we are free to employ the same simulated datasets and estimators $\rho_s$ and $\rho_B$ above, but select at random a different projector $\Lambda=\ket{\phi}\bra{\phi}$ for each trial. This is equivalent to performing all trials with a random ground truth but a fixed observable. We call this randomized evaluation ``Picture 2'' to distinguish it from the fixed ground truth case above (Picture 1). \begin{figure*} \caption{Estimating rank-1 observable $\Lambda$ for randomly chosen ground truth states (Picture 2). (a) $D=32$ case. (b) $D=256$ case. The first four columns show $\lambda$ values for each trial; the last column plots MSE with respect to ground truth over all trials.} \label{fig2} \end{figure*} Results appear in Fig.~\ref{fig2} for (a) $D=32$ and (b) $D=256$. The first column plots the ground truth value $\lambda^{(g)}$ for each trial, the next three columns plot the shadow and BME estimates for increasing numbers of measurements, and the final column presents the MSE with respect to the ground truth. Now BME returns much more accurate estimates than shadow on average, and the paradox regarding Bayesian optimality is solved: the Bayesian mean gives the lowest MSE as long as the prior accurately reflects the true uncertainty of the system under test. Accordingly, this BME study clarifies an underlying assumption in selecting observables in Picture 1: being able to ``guess'' an observable with such high overlap to the ground truth suggests that one is not really operating under the neutrality implied by a uniform prior; an informative prior would more accurately reflect the situation. This observation brings to light an interesting question of motivation in a given quantum experiment. In the sense of ensuring that any estimate is adequately justified by the data, the idea of ``baking in'' a prior favoring some subset of quantum states is undesirable. And yet, in many situations the researcher \emph{does} have strong beliefs---or at least hopefulness---about the state being prepared, and wants to verify this by computing an observable, such as fidelity, where it is desired that $\lambda^{(g)} \sim 1$. In this case, one wishes to validate such high values quickly with few measurements, but likely does not care so much about how well the procedure can estimate the ground truth when it is \emph{low} (e.g., when $\lambda^{(g)}\sim \frac{1}{D}$), since this situation suggests a poorly prepared state anyway. Accordingly, the felt cost is stronger when error is higher for situations with $\lambda^{(g)} \gg \frac{1}{D}$ than when $\lambda^{(g)} \sim \frac{1}{D}$, which is not captured by the standard MSE as expressed in Eq.~(\ref{eq:opt}). And as shown in our tests here, it is precisely these improbable situations wherein shadow excels over BME. Thus our simulations reveal one surprising reason classical shadows are so powerful: they perform well within those subspaces of the entire Hilbert space which are of interest to a high-fidelity system. \begin{figure*} \caption{Bayesian inference results utilizing the pseudo-likelihood in Eq.~(\ref{eq:PL1}) for (a) $D=32$ and (b) $D=256$. The overlap with shadow, $\Tr\rho_B\rho_s$ is plotted in (c) for $D=32$ and (d) for $D=256$.} \label{fig3} \end{figure*} \subsection*{Emulating Classical Shadows with BME} \label{sec:emulate} \textit{Pseudo-Likelihood Formulation.} The dimension-independence and rapid convergence of classical shadows for cases of interest indicate the value of a Bayesian version with similar features, both to gain further insight into shadow itself and to improve thereon by ensuring only physically acceptable states. A simple approach for custom Bayesian models, gaining traction in ``probably approximately correct'' (PAC) learning~\cite{Guedj2019}, proposes use of a pseudo-likelihood that rates a prospective state's suitability through a cost function, instead of a full likelihood based on a physical model. In quantum state tomography in particular, quadratic costs of the form $\lVert \rho-\tilde{\rho} \rVert_F^2$ have been explored~\cite{Mai2017, Lukens2020}, where $\tilde{\rho}$ signifies some point estimator and $\lVert A \rVert_F=\sqrt{\Tr A^\dagger A}$ the Frobenius norm. Therefore, to obtain a physical state with properties similar to $\rho_s$, we first suggest the pseudo-likelihood \begin{equation} \label{eq:PL1} L_{\bm{\mathcal{D}}}(\mathbf{x}) = \exp\left(- \frac{K}{2} \lVert \rho(\mathbf{x})-\rho_s \rVert_F^2\right). \end{equation} The constant $K$ establishes the relative weight of prior and likelihood. Previously, we suggested $K=M$ for reasonable uncertainty quantification~\cite{Lukens2020}; here we consider $K=MD$ to impart dimension-independence. (Incidentally, we have found no significant modifications to the results below when testing with $K\gg MD$.) Figure~\ref{fig3}(a) and (b) show the BME results obtained for $D=32$ and $D=256$, respectively, where we again return to Picture 1 with fixed ground truth for all trials. For the tests here, thinning of $T=2^8$ ($T=2^{10}$) is used for the $D=32$ ($D=256$) MCMC chains. Compared to the shadow results of Fig.~\ref{fig1}, the BME predictions still do not reach ground truth values for $\lambda_0$ and $\lambda_1$ efficiently. This proves intriguing, since $\lVert \rho-\rho_s \rVert_F^2$ with $\rho=\ket{\psi}\bra{\psi}$ is minimized precisely by states for which $\braket{\psi|\rho_s|\psi}$ is large. So if $\lambda_0^{(s)}=\braket{g|\rho_s|g}\sim 1$ (cf. Fig.~\ref{fig1}), it is odd that predictions using a BME value maximizing $\braket{\psi|\rho_s|\psi}$ looks so different for $D=256$. The origin of this discrepancy, however, lies in $\rho_s$'s nonphysicality. \begin{figure*} \caption{Bayesian estimation using the pseudo-likelihood of Eq.~(\ref{eq:PL2}) with $N=3$. (a) Results for $D=32$. (b) Results for $D=256$. The MSE values for shadow from Fig.~\ref{fig1} are reproduced for comparison.} \label{fig4} \end{figure*} Plotting the average overlap between shadow and Bayesian samples ($\Tr\rho_B\rho_s$) in Fig.~\ref{fig3}(c) and (d), we find that $\rho_B$ overlaps with $\rho_s$ \emph{more strongly than the ground truth} $\rho_s=\ket{g}\bra{g}$. Because $\rho_s$ is not positive semidefinite, $\Tr\rho_B\rho_s > 1$ for all cases examined. Thus the BME procedure succeeds in finding states with strong overlap to the shadow, but the closest physical state to $\rho_s$ is not the ground truth, even though $\braket{g|\rho_s|g}\sim 1$. Intuitively, this nonphysicality helps explain why observables with highly improbable values from the Bayesian view are estimated so much more efficiently with shadow. For a parameterization over physical states and rank-1 observable $\Lambda$, only a single state in the Hilbert space attains $\lambda=1$, and since this represents the maximum value possible for any valid quantum state, it can only be approached from below. On the other hand, a continuum of shadow estimators $\rho_s$ permit $\lambda=1$, for $\rho_s$ is constrained only by Hermiticity and unit-trace---not positive semidefiniteness. Therefore the estimate $\lambda^{(s)}$ can err on either the high or low side (cf. Fig.~\ref{fig1}), pulling the shadow more rapidly to the ground truth in these extreme cases. This discloses the second central finding of our investigation: the nonphysicality of $\rho_s$ is not a deficiency, but rather critical to obtaining dimension independence. Thus the key features of the shadow are not necessarily translated onto physical projections like the BME model here~\footnote{As an additional check, we performed the algorithm of Ref.~\cite{Smolin2012} to determine the closest physical density matrix to $\rho_s$, finding very similar results as Fig.~\ref{fig3}. This indicates that our projection conclusions are not an artefact of the pure state prior, but hold for general mixed states as well.} or, for that matter, alternative projected-least-squares approaches~\cite{Smolin2012, Guta2020}. While strange from the conventional wisdom of maximum likelihood and Bayesian mean estimation, nonphysical states are actually beneficial for classical shadows. \textit{Observable-Oriented Pseudo-Likelihood.} Deriving a positive semidefinite model emulating classical shadows remains an intriguing question, however, to eliminate unphysical estimates while retaining the favorable scaling features. With projecting directly onto $\rho_s$ proving unfruitful to this end, we note that, indeed, $\rho_s$ was never intended to serve as an accurate substitute for the true $\rho_g$; instead it facilitates estimates of observables~\cite{Huang2020}. Accordingly, we propose the ``observable-oriented pseudo-likelihood'' \begin{equation} \label{eq:PL2} L_{\bm{\mathcal{D}}}(\mathbf{x}) = \exp\left(-\frac{K}{2} \sum_{n=0}^{N-1} \left|\Tr\rho(\mathbf{x})\Lambda_n - \lambda_n^{(s)}\right|^2 \right), \end{equation} where we insert the estimates $\lambda_n^{(s)}$ of $N$ observables from $\rho_s$. This formalism ensures only physical values are returned [through the prior $\pi_0(\mathbf{x})$], and rates the fitness of proposed states through their overlap with respect to shadow's predictions of observables only. For dimension-independence, we again set $K=MD$ and perform BME for all simulated datasets and $N=3$ above, thinning to $T=2^{10}$ ($T=2^{13}$) for $D=32$ ($D=256$). The results follow in Fig.~\ref{fig4}. Now BME shows very similar behavior to shadow: the MSE with respect to the ground truth matches shadow results from Fig.~\ref{fig1} closely, though BME still outperforms for $\lambda_2$. Yet unlike shadow, BME here always gives physically permissible estimates ($\lambda_n^{(B)}\in[0,1]$). This pseudo-likelihood therefore attains the goal of a BME model commensurate with classical shadows. Yet it is important to emphasize that this approach depends heavily on the quality of the classical shadow. It refines estimates from the shadow with its positive semidefinite requirement, but it does not do markedly better at estimating the ground truth state---at least for arbitrary observables. As an example, we repeat the inference procedure for an observable-oriented pseudo-likelihood based solely on $\Lambda_1$, i.e., \begin{equation} \label{eq:PL3} L_{\bm{\mathcal{D}}}(\mathbf{x}) = \exp\left(-\frac{K}{2} \left|\Tr\rho(\mathbf{x})\Lambda_1 - \lambda_1^{(s)}\right|^2 \right), \end{equation} which has ground truth value $\lambda_2^{(g)}=\frac{1}{2}$. Results for the $D=32$ case appear in Fig.~\ref{fig5}, where we plot the Bayesian estimates for all three observables even though the psuedo-likelihood is based on $\lambda_1$ only. The estimate $\lambda_1^{(B)}$ closely matches shadow as designed, and $\lambda_2^{(B)}$ agrees with the ground truth well, due to the fact its value is highly probable for a uniform prior. But $\lambda_0^{(B)}\rightarrow \sim\frac{1}{4}$, far from $\lambda_0^{(g)}=1$. When using the pseudo-likelihood above, all quantum states with identical overlap to $\Lambda_1$ are equally probable, of which the ground truth $\rho_g$ represents just one possibility. The estimate of $\lambda_0$ given only $\lambda_1$ information reflects the inherent uncertainty within this specification. So to summarize, our observable-oriented pseudo-likelihood builds physicality into shadow, yet it can only (in general) accurately predict the $N$ observables injected into it: to infer quantities beyond these $N$ can prove unreliable. \begin{figure} \caption{Bayesian inference results employing the psuedo-likelihood in Eq.~(\ref{eq:PL3}), for $D=32$. The shadow MSE values from Fig.~\ref{fig1} are reprinted for clarity.} \label{fig5} \end{figure} \section*{Discussion} \label{sec:disc} Our numerical investigations here have elucidated two fascinating features of classical shadows: \begin{enumerate} \item Classical shadows perform extremely well at predicting ``unlikely'' observables, i.e., those which obtain high values only on a restricted subset of states within the complete Hilbert space. \item The nonphysicality of classical shadows is critical to their dimension-independence and accuracy under few measurements. \end{enumerate} These findings do not contradict the optimality of Bayesian methods expressed in Eq.~(\ref{eq:opt}): BME with a full likelihood minimizes MSE for any number and collection of measurements, provided the prior distribution accurately reflects the true knowledge involved. The predictive power of $\rho_s$, then, derives from the fact that the situations in which it is much more accurate that BME are often of particular interest in practice, such as verification of a high-fidelity or highly entangled quantum state. Desiring to extend these features in the Bayesian context, we proposed an observable-oriented pseudo-likelihood that attains shadow's dimension-independence and state-specialized accuracy, with the advantage of guaranteed physicality. Nonetheless, in all these explorations there remains one prominent sense in which classical shadows unquestionably eclipse BME: computational efficiency. The shadow estimator $\rho_s$ is formed directly from measurements for any dimension $D$; yet computing $\rho_B$ requires tedious MCMC methods, with the number of parameters increasing linearly (quadratically) with $D$ for a pure (mixed) state prior. Here we considered up to $D=256$, a far cry from the $D=2^{120}$ example in Ref.~\cite{Huang2020}, where there is no hope for BME with a parameterization such as ours. Moving forward, it would therefore seem profitable to explore simplified Bayesian models that maintain a fixed parameter dimensionality even as the Hilbert space grows exponentially. For example, if one could specify a prior and likelihood on an observable $\lambda$ only, to the effect of $\pi(\lambda)\propto L_{\bm{\mathcal{D}}}(\lambda)\pi_0(\lambda)$, the inference procedure would not be limited directly by exponentially large Hilbert spaces. In this way, Bayesian methods could be extendable to the types of quantum systems sought for practically useful quantum computation. Overall, our analyses have revealed the value of BME as a tool for shedding light on estimation procedures which formally have no connection to the Bayesian paradigm. The numerical simulations here reveal the complementary strengths of classical shadow and Bayesian tomographic approaches in the efficient estimation of quantum properties. And so we expect valuable opportunities for both methods as quantum information processing resources continue to mature in size and complexity. \section*{Methods} \subsection*{Data Simulation Approach} The method of classical shadows introduced in Ref.~\cite{Huang2020} involves application of a Haar-random (or effectively Haar-random) unitary $U$ followed by measurement in the computational basis. We exploit the fact that our target state is pure to substantially reduce the complexity of simulating this procedure. In particular, our simulation method requires the generation of only size-$D$ random vectors rather than $D\times D$ random unitaries. Without loss of generality we work in a rotated basis such that the first basis state coincides with the ground truth: $\rho_g = \ket{0}\bra{0}$. Then the probability of observing outcome $j$ depends only on $|\braket{j|U|0}|^2 = |U_{j0}|^2 = |(U^\dagger)_{0j}|^2$. That is, the distribution of outcomes depends only on the first row of $U^\dagger$. Now, when $U$ is Haar-random, each individual row and column of $U^\dagger$ is a uniformly distributed length-1 vector $u$. Furthermore, given any component $u_j$, the remaining components are a uniformly distributed vector of length $\sqrt{1-|u_j|^2}$. A uniformly random vector $u$, corresponding to the first row of $U^\dagger$, may be obtained by generating $D$ complex normal random values and normalizing them to yield a unit length vector. An outcome $n\in \{0,1,\ldots,D-1\}$ is then chosen with probability $|u_n|^2$. This selects the $n$th column of $U^\dagger$. Since this column (whichever it is) is uniformly distributed, its remaining elements are uniformly distributed with length $\sqrt{1-|u_n|^2}$. The explicit procedure is as follows: \begin{enumerate} \item Posit a measurement unitary $U_m^\dagger = [\tilde{\varphi}_0 \cdots \tilde{\varphi}_{D-1}]$, where each $\tilde{\varphi}_n$ is a column vector corresponding to one of the $D$ possible output states. \item Generate $D$ complex normal samples $w_n \stackrel{\textrm{i.i.d.}}{\sim}\mathcal{N}(0,1) + i\mathcal{N}(0,1)$ and normalize \begin{equation} \label{eq:row} u_n = \frac{w_n}{\sqrt{\sum \limits_{n^\prime=0}^{D-1} |w_{n^\prime}|^2}}. \end{equation} These define projections of the unitary's basis states on the ground truth: $u_n = \braket{0|\tilde{\varphi}_n}$, or in other words, the elements in the first row of $U_m^\dagger$. \item Select an integer $n\in\{0,1,...,D-1\}$ at random with probability $|u_n|^2$. This implies that the state $\tilde{\varphi}_n$ is detected. \item Generate $D-1$ complex normal samples $v_j \stackrel{\textrm{i.i.d.}}{\sim}\mathcal{N}(0,1) + i\mathcal{N}(0,1)$ ($j=1,2,...,D-1$). These set the remaining coefficients of the detected state $\tilde{\varphi}_n$. \item Finally, take \begin{equation} \label{eq:col} \ket{\psi_m} = u_n\ket{0} + \sqrt{ \frac{1-|u_n|^2} {\sum\limits_{j^\prime=1}^{D-1} |v_{j^\prime}|^2}} \sum_{j=1}^{D-1} v_j \ket{j} \end{equation} as the measured state. \end{enumerate} Utilizing this method, we performed 50 independent trials with 1000 measurements each, for Hilbert space dimensions $D=32$ and $D=256$, giving a total of 100 datasets which are used in all subsequent tests above. The two values of $D$ were selected specifically to clarify how classical shadows and BME differ in their scaling with dimension. \section*{Acknowledgments} This work was funded by the U.S. Department of Energy, Office of Advanced Scientific Computing Research, through the Quantum Algorithm Teams and Early Career Research Programs. This work was performed in part at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under contract no. DE-AC05-00OR22725. \begin{thebibliography}{29} \makeatletter \providecommand \@ifxundefined [1]{ \@ifx{#1\undefined} } \providecommand \@ifnum [1]{ \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \@ifx [1]{ \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \natexlab [1]{#1} \providecommand \enquote [1]{``#1''} \providecommand \bibnamefont [1]{#1} \providecommand \bibfnamefont [1]{#1} \providecommand \citenamefont [1]{#1} \providecommand \href@noop [0]{\@secondoftwo} \providecommand \href [0]{\begingroup \@sanitize@url \@href} \providecommand \@href[1]{\@@startlink{#1}\@@href} \providecommand \@@href[1]{\endgroup#1\@@endlink} \providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax} \providecommand \@@startlink[1]{} \providecommand \@@endlink[0]{} \providecommand \url [0]{\begingroup\@sanitize@url \@url } \providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }} \providecommand \urlprefix [0]{URL } \providecommand \Eprint [0]{\href } \providecommand \doibase [0]{http://dx.doi.org/} \providecommand \selectlanguage [0]{\@gobble} \providecommand \bibinfo [0]{\@secondoftwo} \providecommand \bibfield [0]{\@secondoftwo} \providecommand \translation [1]{[#1]} \providecommand \BibitemOpen [0]{} \providecommand \bibitemStop [0]{} \providecommand \bibitemNoStop [0]{.\EOS\space} \providecommand \EOS [0]{\spacefactor3000\relax} \providecommand \BibitemShut [1]{\csname bibitem#1\endcsname} \let\auto@bib@innerbib\@empty \bibitem [{\citenamefont {James}\ \emph {et~al.}(2001)\citenamefont {James}, \citenamefont {Kwiat}, \citenamefont {Munro},\ and\ \citenamefont {White}}]{James2001} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~F.~V.}\ \bibnamefont {James}}, \bibinfo {author} {\bibfnamefont {P.~G.}\ \bibnamefont {Kwiat}}, \bibinfo {author} {\bibfnamefont {W.~J.}\ \bibnamefont {Munro}}, \ and\ \bibinfo {author} {\bibfnamefont {A.~G.}\ \bibnamefont {White}},\ }\href {\doibase 10.1103/PhysRevA.64.052312} {\bibfield {journal} {\bibinfo {journal} {Phys. 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Sci.}\ }\textbf {\bibinfo {volume} {28}},\ \bibinfo {pages} {424} (\bibinfo {year} {2013})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Lukens}\ \emph {et~al.}(2020)\citenamefont {Lukens}, \citenamefont {Law}, \citenamefont {Jasra},\ and\ \citenamefont {Lougovski}}]{Lukens2020} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont {Lukens}}, \bibinfo {author} {\bibfnamefont {K.~J.~H.}\ \bibnamefont {Law}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Jasra}}, \ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Lougovski}},\ }\href {\doibase 10.1088/1367-2630/ab8efa} {\bibfield {journal} {\bibinfo {journal} {New J. Phys.}\ }\textbf {\bibinfo {volume} {22}},\ \bibinfo {pages} {063038} (\bibinfo {year} {2020})}\BibitemShut {NoStop} \bibitem [{Note1()}]{Note1} \BibitemOpen \bibinfo {note} {Incorporating some of the methods suggested in Ref.~\cite {Lukens2020} in further research, such as embedding within SMC samplers and parallelization, should permit the extension to significantly larger $D$ and mixed states}\BibitemShut {NoStop} \bibitem [{\citenamefont {Guedj}(2019)}]{Guedj2019} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Guedj}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv:1901.05353}\ } (\bibinfo {year} {2019})}\BibitemShut {NoStop} \bibitem [{Note2()}]{Note2} \BibitemOpen \bibinfo {note} {As an additional check, we performed the algorithm of Ref.~\cite {Smolin2012} to determine the closest physical density matrix to $\rho _s$, finding very similar results as Fig.~\ref {fig3}. This indicates that our projection conclusions are not an artefact of the pure state prior, but hold for general mixed states as well.}\BibitemShut {Stop} \bibitem [{\citenamefont {Smolin}\ \emph {et~al.}(2012)\citenamefont {Smolin}, \citenamefont {Gambetta},\ and\ \citenamefont {Smith}}]{Smolin2012} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont {Smolin}}, \bibinfo {author} {\bibfnamefont {J.~M.}\ \bibnamefont {Gambetta}}, \ and\ \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Smith}},\ }\href {\doibase 10.1103/PhysRevLett.108.070502} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. 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\begin{document} \title{\bf\Large A Product Version of the Hilton-Milner Theorem} \date{} \author{Peter Frankl$^1$, Jian Wang$^2$\\[10pt] $^{1}$R\'{e}nyi Institute, Budapest, Hungary\\[6pt] $^{2}$Department of Mathematics\\ Taiyuan University of Technology\\ Taiyuan 030024, P. R. China\\[6pt] E-mail: $^[email protected], $^[email protected] } \maketitle \begin{abstract} Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in\mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem. \end{abstract} \section{Introduction} For a positive integer $n$ let $[n]$ denote the standard $n$-set $\{1,2,\ldots,n\}$. For $1\leq i\leq j\leq n$ set also $[i,j]=\{i,i+1,\ldots,j\}$. For an integer $k$ let $\binom{[n]}{k}$ denote the collection of all subsets of $[n]$. Subsets of $\binom{[n]}{k}$ are called {\it $k$-graphs} or {\it $k$-uniform families}. A $k$-graph $\mathcal{F}$ is called {\it $t$-intersecting} if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal{F}$. Analogously, two $k$-graphs $\mathcal{F}$ and $\mathcal{G}$ are called {\it cross $t$-intersecting} if $|F\cap G|\geq t$ for all $F\in \mathcal{F}$ and $G\in \mathcal{G}$. In case of $t=1$ we omit the 1 and use the term {\it cross-intersecting}. One of the central results of extremal set theory is the Erd\H{o}s-Ko-Rado Theorem. {\noindent\bf Erd\H{o}s-Ko-Rado Theorem (\cite{EKR}).} Let $n>k>t>0$ and suppose that $\mathcal{F}\subset \binom{[n]}{k}$ is $t$-intersecting then for $n\geq n_0(k,t)$, \begin{align}\label{ineq-ekr} |\mathcal{F}| \leq \binom{n-t}{k-t}. \end{align} We should mention that the exact value of $n_0(k,t)$ is $(k-t+1)(t+1)$. For $t=1$ it was determined already in \cite{EKR}, for $t\geq 15$ it is due to \cite{F78}. Finally Wilson \cite{W3} closed the gap $2\leq t\leq 14$ with a proof valid for all $t$. Let us note that the {\it full $t$-star}, $\left\{F\in \binom{[n]}{k}\colon [t]\subset F\right\}$ shows that \eqref{ineq-ekr} is best possible. In general, for a set $T\subset[n]$ let $\mathcal{S}_T=\left\{S\in \binom{[n]}{k}\colon T\subset S\right\}$ denote the {\it star of $T$}. For the case $t=1$ the classical Hilton-Milner Theorem is a strong stability result. Recall that a family $\mathcal{F}$ is called {\it non-trivial} if $\cap \{F\colon F\in \mathcal{F}\}=\emptyset$. {\noindent\bf Hilton-Milner Theorem (\cite{HM67}).} If $n> 2k$ and $\mathcal{F}\subset \binom{[n]}{k}$ is non-trivial intersecting, then \begin{align}\label{ineq-nontrival} |\mathcal{F}| \leq \binom{n-1}{k-1}- \binom{n-k-1}{k-1} +1 =:h(n,k). \end{align} By now there are many different proofs known for this important result, cf. \cite{Alon, Borg,FFuredi,FT,F2017,GH,Mors} etc. Let \[ \mathcal{H}\mathcal{M}(n,k) =\left\{F\in \binom{[n]}{k}\colon 1\in F,\ F\cap [2,k+1]\neq \emptyset\right\}\cup \{[2,k+1]\}. \] Clearly, $\mathcal{H}\mathcal{M}(n,k)$ is a non-trivial intersecting family and it shows that \eqref{ineq-nontrival} is best possible. Let us state our main result. \begin{thm}\label{main} Suppose that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial cross-intersecting families, $n\geq 4k$, $k\geq 8$. Then \begin{align}\label{ineq-hfhg2} |\mathcal{F}||\mathcal{G}|\leq h(n,k)^2=\left(\binom{n-1}{k-1}- \binom{n-k-1}{k-1} +1\right)^2. \end{align} \end{thm} We can prove the same statement for $n\geq 5k$, $k \geq 7$ or $n\geq 8k$, $k\geq 6$ as well using the same proof. Besides $\mathcal{F}=\mathcal{G}=\mathcal{H}\mathcal{M}(n,k)$, for any $A,B \in {[2,n]\choose k}$ with $A\cap B\neq \emptyset$ the construction \begin{align*} \mathcal{F} = \left\{F\in \binom{[n]}{k}\colon 1\in F,\ F\cap A \neq \emptyset\right\}\cup\{B\},\\[5pt] \mathcal{G} = \left\{G\in \binom{[n]}{k}\colon 1\in G,\ G\cap B \neq \emptyset\right\}\cup \{A\} \end{align*} also attains equality in \eqref{ineq-hfhg2}. In the next section we will state several theorems involving the product of the sizes of cross-intersecting families. Some are important and powerful. However, to the best of our knowledge \eqref{ineq-hfhg2} is the first such result that implies the Hilton-Milner Theorem (just set $\mathcal{F}=\mathcal{G}$). In our proofs, we need two simple inequalities involving binomial coefficients. \begin{prop} Let $n,k,i$ be positive integers. Then \begin{align} &\binom{n-i}{k} \geq \left(\frac{n-k-(i-1)}{n-(i-1)}\right)^i \binom{n}{k}, \label{ineq-key}\\[5pt] &\binom{n-i}{k-i} \binom{n}{k}\leq \binom{n-i+1}{k-i+1} \binom{n-1}{k-1}, \mbox{\rm \ for } n\geq k\geq i\geq 2.\label{ineq-key2} \end{align} \end{prop} \begin{proof} Since \[ \frac{\binom{n-i}{k}}{\binom{n}{k}} = \frac{(n-k)(n-k-1)\ldots(n-k-(i-1))}{n(n-1)\ldots(n-(i-1))}\geq \left(\frac{n-k-(i-1)}{n-(i-1)}\right)^i, \] we have \eqref{ineq-key} holds. For \eqref{ineq-key2}, simply note that \[ \frac{\binom{n-i}{k-i} \binom{n}{k}}{\binom{n-i+1}{k-i+1} \binom{n-1}{k-1}} =\frac{n(k-i+1)}{k(n-i+1)}=\frac{kn-(i-1)n}{kn-(i-1)k}\leq 1, \] and the inequality follows. \end{proof} Let us recall the following common notations: $$\mathcal{F}(i)=\{F\setminus\{i\}\colon i\in F\in \mathcal{F}\}, \qquad \mathcal{F}(\bar{i})= \{F\in\mathcal{F}: i\notin F\}.$$ Note that $|\mathcal{F}|=|\mathcal{F}(i)|+|\mathcal{F}(\bar{i})|$. For $P\subset Q\subset [n]$, let \[ \mathcal{F}(P,Q) = \left\{F\setminus Q\colon F\in\mathcal{F},\ F\cap Q=P \right\}. \] We also use $\mathcal{F}(\bar{Q})$ to denote $\mathcal{F}(\emptyset, Q)$. For $\mathcal{F}(\{i\},Q)$ we simply write $\mathcal{F}(i,Q)$. \section{Earlier product theorems and some tools} In this section, we review some earlier theorems concerning the product of cross-intersecting families. We also recall Hilton's Lemma and give some corollaries that will be used later. \begin{thm}[Pyber \cite{Pyber86}] Suppose that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are cross-intersecting, $n\geq 2k$. Then \begin{align}\label{ineq-pyber} |\mathcal{F}||\mathcal{G}| \leq \binom{n-1}{k-1}^2. \end{align} \end{thm} \begin{thm}[Matsumoto-Tokushige \cite{MT}] Let $k,\ell$ be positive integers, $n\geq 2k\geq 2\ell$. Suppose that $\mathcal{F}\subset \binom{[n]}{k}$ and $\mathcal{G}\subset \binom{[n]}{\ell}$ are cross-intersecting. Then \begin{align}\label{ineq-mt} |\mathcal{F}||\mathcal{G}| \leq \binom{n-1}{k-1}\binom{n-1}{\ell-1}. \end{align} \end{thm} Note that for the case $n\geq 3k\geq 3\ell$, \eqref{ineq-mt} was already proved by Pyber \cite{Pyber86}. For a short proof of \eqref{ineq-pyber} cf. \cite{FK2017}. In that paper some more precise product results are proven. \begin{example} Let $2\leq s\leq k+1$ and define two families \[ \mathcal{A}_s =\left\{A\in \binom{[n]}{k}\colon 1\in A, A\cap [2,s]=\emptyset\right\},\ \mathcal{B}_s=\mathcal{S}_{\{1\}} \cup \left\{B\in \binom{[n]}{k}\colon [2,s]\subset B\right\}. \] \end{example} It is easy to check that $\mathcal{A}_s$ and $\mathcal{B}_s$ are cross-intersecting. \begin{thm}[\cite{FK2017}] Let $3\leq s\leq k+1$, $n\geq 2k$. Suppose that $\mathcal{A},\mathcal{B} \subset \binom{[n]}{k}$ are cross-intersecting and \[ |\mathcal{B}|\geq \binom{n-1}{k-1}+\binom{n-s}{k-s+1}. \] Then \begin{align}\label{ineq-FK17} |\mathcal{A}||\mathcal{B}| \leq \left(\binom{n-1}{k-1}-\binom{n-s}{k-1}\right)\left(\binom{n-1}{k-1}+\binom{n-s}{k-s-1}\right)=|\mathcal{A}_s||\mathcal{B}_s|. \end{align} \end{thm} An important tool for proving the above results is the Kruskal-Katona Theorem (\cite{Kruskal,Katona}, cf. \cite{F84} or \cite{Keevash} for short proofs of it). Daykin \cite{daykin} was the first to show that the Kruskal-Katona Theorem implies the $t=1$ case of the Erd\H{o}s-Ko-Rado Theorem. Hilton \cite{Hilton} gave a very useful reformulation of the Kruskal-Katona Theorem. To state it let us recall the definition of the lexicographic order on $\binom{[n]}{k}$. For two distinct sets $F,G\in \binom{[n]}{k}$ we say that $F$ {\it precedes} $G$ if \[ \min\{i\colon i\in F\setminus G\}<\min\{i\colon i\in G\setminus F\}. \] E.g., $(1,7)$ precedes $(2,3)$. For a positive integer $b$ let $\mathcal{L}(n,b,m)$ denote the first $m$ members of $\binom{[n]}{b}$. {\noindent\bf Hilton's Lemma (\cite{Hilton}).} Let $n,a,b$ be positive integers, $n>a+b$. Suppose that $\mathcal{A}\subset \binom{[n]}{a}$ and $\mathcal{B}\subset \binom{[n]}{b}$ are cross-intersecting. Then $\mathcal{L}(n,a,|\mathcal{A}|)$ and $\mathcal{L}(n,b,|\mathcal{B}|)$ are cross-intersecting as well. For a family $\mathcal{A} \subset\binom{[n]}{a}$ and an integer $b$ define the family of transversals (of size $b$) $\mathcal{T}^{(b)}(\mathcal{A})$ by \[ \mathcal{T}^{(b)}(\mathcal{A}) =\left\{B\in \binom{[n]}{b}\colon B\cap A\neq \emptyset \mbox{ for all } A\in \mathcal{A}\right\}. \] Note that $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting iff $\mathcal{B} \subset \mathcal{T}^{(b)}(\mathcal{A})$. Let us use this notation to prove three corollaries of Hilton's Lemma. \begin{cor} Let $t\geq 1$. \begin{align}\label{hiltonLem-1} \mbox{If } |\mathcal{A}| \geq \binom{n-1}{a-1}+\ldots+\binom{n-t}{a-1} \mbox{ then } |\mathcal{B}|\leq \binom{n-t}{k-t}. \end{align} \end{cor} \begin{proof} Note that \begin{align*} \mathcal{L}(n,a,|\mathcal{A}|)\supset \mathcal{L}\left(n,a,\binom{n-1}{a-1}+\ldots+\binom{n-t}{a-1}\right)&=\left\{A\in \binom{[n]}{a}\colon A\cap [t]\neq \emptyset\right\}\\[5pt] &=:\mathcal{E}(n,a,t). \end{align*} Since $\mathcal{T}^{(b)}(\mathcal{E}(n,a,t))$ is the $t$-star $\{B\in\binom{[n]}{b}\colon [t]\subset B\}$, the statement follows. \end{proof} \begin{cor} \begin{align}\label{hiltonLem-2} \mbox{If } |\mathcal{A}| > \binom{n-1}{a-1}-\binom{n-b-1}{a-1} \mbox{ then } |\mathcal{B}|\leq \binom{n-1}{b-1}. \end{align} \end{cor} \begin{proof} Suppose indirectly $|\mathcal{B}|\geq \binom{n-1}{b-1}+1$. Consider \[ \mathcal{L}\left(n,b,\binom{n-1}{b-1}+1\right)=\left\{B\in \binom{[n]}{b}\colon 1\in B\right\}\cup \{[2,b+1]\}. \] Noting that \[ \mathcal{T}^{(a)}\left(\mathcal{L}\left(n,b,\binom{n-1}{b-1}+1\right)\right) = \left\{A\in \binom{[n]}{a}\colon 1\in A, A\cap [2,b+1]\neq \emptyset\right\} \] has size $\binom{n-1}{a-1}-\binom{n-b-1}{a-1}$ the desired contradiction follows. \end{proof} \begin{cor} \begin{align}\label{hiltonLem-3.1} \mbox{If } |\mathcal{A}| > \binom{n-1}{a-1}+\binom{n-2}{a-1}+\binom{n-4}{a-2} \mbox{ then } |\mathcal{B}|\leq \binom{n-3}{b-3}+\binom{n-4}{k-3}. \end{align} \end{cor} \begin{proof} Note that \begin{align*} \mathcal{L}(n,a,|\mathcal{A}|)\supset &\mathcal{L}\left(n,a,\binom{n-1}{a-1}+\binom{n-2}{a-1}+\binom{n-4}{a-2}\right)\\[5pt] =&\left\{A\in \binom{[n]}{a}\colon 1\in A \mbox{ or }2\in A\mbox{ or }\{3,4\}\subset A \right\}=:\mathcal{D}(n,a). \end{align*} Then clearly \[ \mathcal{T}^{(b)}(\mathcal{D}(n,a)) =\left\{B\in\binom{[n]}{b}\colon \{1,2,3\}\subset B \mbox{ or } \{1,2,4\}\subset B\right\} \] and the statement follows. \end{proof} For later use let us prove one more consequence of Hilton's Lemma. We state it in the special case that we need in Section 3. \begin{lem} Suppose that $\mathcal{F}, \mathcal{G}\subset \binom{[n]}{k}$ are cross-intersecting, $n>2k$. Let \[ \binom{n-4}{k-4}<|\mathcal{F}|\leq \binom{n-3}{k-3}, \] and \[ \binom{n-1}{k-1}+ \binom{n-2}{k-1}+\binom{n-3}{k-1}<|\mathcal{G}|\leq \binom{n-1}{k-1}+ \binom{n-2}{k-1}+\binom{n-3}{k-1}+\binom{n-4}{k-1}. \] Define \[ f=|\mathcal{F}|-\binom{n-4}{k-4}\mbox{ and } g= |\mathcal{G}| -\binom{n-1}{k-1}- \binom{n-2}{k-1}-\binom{n-3}{k-1}. \] Then \begin{align}\label{hiltonLem-3} \frac{g}{\binom{n-4}{k-1}}+\frac{f}{\binom{n-4}{k-3}}\leq 1. \end{align} \end{lem} \begin{proof} Note that all \[ F\in \mathcal{L}(n,k,|\mathcal{F}|)\setminus \mathcal{L}\left(n,k,\binom{n-4}{k-4}\right) \] satisfy $F\cap [4]=[3]$. Let $\mathcal{F}_0$ be the family of size $f$ formed by the corresponding sets $F\setminus [4]\in \binom{[5,n]}{k-3}$. Also, for all \[ G\in \mathcal{L}(n,k,|\mathcal{G}|)\setminus \mathcal{L}\left(n,k,\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}\right), \] $G\cap [4]=\{4\}$ follows from $g\leq \binom{n-4}{k-1}$ (a consequence of \eqref{hiltonLem-1}). Let $\mathcal{G}_0\subset \binom{[5,n]}{k-1}$ be formed by the corresponding sets $G\setminus [4]$. Obviously, $\mathcal{F}_0$ and $\mathcal{G}_0$ are cross-intersecting. The inequality \eqref{hiltonLem-3} is essentially due to Sperner \cite{Sperner} but let us repeat the simple argument. Define a bipartite graph with partite sets $\binom{[5,n]}{k-3}$ and $\binom{[5,n]}{k-1}$ by putting an edge between $F\in\binom{[5,n]}{k-3}$ and $G\in \binom{[5,n]}{k-1}$ iff $F\cap G=\emptyset$. This bipartite graph is bi-regular implying that the neighborhood $\mathcal{N}(\mathcal{F}_0)$ satisfies \[ |\mathcal{N}(\mathcal{F}_0)|/\binom{n-4}{k-1}\geq |\mathcal{F}_0|/\binom{n-4}{k-3}. \] Since $\mathcal{F}_0\cup \mathcal{G}_0$ is an independent set, $\mathcal{N}(\mathcal{F}_0)\cap \mathcal{G}_0 =\emptyset$ implying \[ \frac{|\mathcal{G}_0|}{\binom{n-4}{k-1}}+\frac{|\mathcal{F}_0|}{\binom{n-4}{k-3}}\leq 1. \] \end{proof} \section{Some size restrictions for the general case and the proof for shifted-resistant families} Throughout the proof we assume that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial, cross-intersecting and \begin{align}\label{indirectAssum} |\mathcal{F}||\mathcal{G}| \geq h(n,k)^2. \end{align} \begin{prop} For $n\geq 2k-1$, $k\geq 3$ \begin{align}\label{ineq-f1} \left(\binom{n-3}{k-3}+\binom{n-4}{k-3}\right)& \left(\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}\right)\nonumber\\[5pt] &\qquad<\left(\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}\right)^2. \end{align} \end{prop} \begin{proof} Note that \[ \binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1} \leq \frac{n-1}{k-1}\left(\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}\right). \] Since $n\geq 2k-1$ implies $\frac{n-1}{k-1}\leq \frac{n-3}{k-2}<\frac{n-2}{k-2}$, we infer \begin{align*} \frac{n-1}{k-1}\left(\binom{n-3}{k-3}+\binom{n-4}{k-3}\right)&\leq \frac{n-2}{k-2}\binom{n-3}{k-3}+\frac{n-3}{k-2}\binom{n-4}{k-3}\leq \binom{n-2}{k-2}+ \binom{n-3}{k-2}. \end{align*} Thus \eqref{ineq-f1} follows. \end{proof} \begin{prop} In proving Theorem \ref{main} we may assume that \begin{align} &\min \left\{|\mathcal{F}|,|\mathcal{G}|\right\}> \binom{n-3}{k-3}+\binom{n-4}{k-3},\label{ineq-1}\\[5pt] &\max \left\{|\mathcal{F}|,|\mathcal{G}|\right\}\leq \binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-4}{k-2}.\label{ineq-2} \end{align} \end{prop} \begin{proof} By symmetry assume that $|\mathcal{F}|\leq |\mathcal{G}|$. If $|\mathcal{F}|< \binom{n-4}{k-4}$, then applying \eqref{ineq-key2} twice we obtain \[ |\mathcal{F}||\mathcal{G}|<\binom{n-4}{k-4}\binom{n}{k} <\binom{n-2}{k-2}^2 <h(n,k)^2. \] If $|\mathcal{G}|> \binom{n}{k}-\binom{n-4}{k}$ then by \eqref{hiltonLem-2} we have $|\mathcal{F}|< \binom{n-4}{k-4}$. Thus we may assume that $|\mathcal{F}|\geq \binom{n-4}{k-4}$ and $|\mathcal{G}|\leq \binom{n}{k}-\binom{n-4}{k}$. Let \[ |\mathcal{F}|=\binom{n-4}{k-4}+\alpha\binom{n-4}{k-3},\ |\mathcal{G}| =\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}+\beta\binom{n-4}{k-1}. \] Note that \eqref{hiltonLem-3} implies $\alpha+\beta\leq 1$ and thereby \[ |\mathcal{F}| \leq \binom{n-3}{k-3}-\beta \binom{n-4}{k-3}. \] Since $\frac{n-k}{n-3}>\frac{n-k-2}{n-3}$ implies \[ \frac{\binom{n-4}{k-3}}{\binom{n-3}{k-3}} >\frac{\binom{n-4}{k-1}}{\binom{n-3}{k-1}}>\frac{\binom{n-4}{k-1}}{\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}}, \] it follows that \begin{align} |\mathcal{F}||\mathcal{G}| &\leq\left(\binom{n-3}{k-3}-\beta \binom{n-4}{k-3}\right)\left(\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}+\beta\binom{n-4}{k-1}\right)\nonumber\\[5pt] &\leq \binom{n-3}{k-3}\left(\binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}\right).\label{ineq-hfhgab} \end{align} By \eqref{ineq-key2} we know \[ \binom{n-3}{k-3}\binom{n-1}{k-1}<\binom{n-2}{k-2}^2. \] Moreover, \begin{align*} \binom{n-3}{k-3}\binom{n-2}{k-1} &\leq \binom{n-2}{k-2}\binom{n-3}{k-2} \frac{k-2}{n-2}\frac{n-2}{k-1} < \binom{n-2}{k-2}\binom{n-3}{k-2},\\[5pt] \binom{n-3}{k-3}\binom{n-3}{k-1} &\leq \binom{n-2}{k-2}\binom{n-4}{k-2} \frac{k-2}{n-2}\frac{n-3}{k-1} < \binom{n-2}{k-2}\binom{n-4}{k-2}. \end{align*} Thus from \eqref{ineq-hfhgab} and $k\geq 3$ we obtain \begin{align*} |\mathcal{F}||\mathcal{G}| \leq \binom{n-2}{k-2}\left(\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}\right)<h(n,k)^2, \end{align*} contradicting \eqref{indirectAssum}. Thus we may assume that \[ |\mathcal{F}|\geq \binom{n-3}{k-3} \mbox{ and } |\mathcal{G}|\leq \binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-3}{k-1}. \] If $|\mathcal{F}|\leq \binom{n-3}{k-3}+\binom{k-4}{k-3}$, then by \eqref{ineq-f1} \[ |\mathcal{F}||\mathcal{G}|\leq \left(\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}\right)^2< h(n,k)^2, \] contradicting \eqref{indirectAssum} again. Thus we may further assume $|\mathcal{F}| > \binom{n-3}{k-3}+\binom{n-4}{k-3}$. Then by \eqref{hiltonLem-3.1} it implies that \[ |\mathcal{G}|\leq \binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-4}{k-2}. \] \end{proof} We need the following computational bound to estimate the size of $\mathcal{F}$ and $\mathcal{G}$ below. \begin{lem} For $n\geq 4k$ and $k\geq 8$, \begin{align} h(n,k)&>\frac{32}{9}\binom{n-2}{k-2}>3\binom{n-2}{k-2},\label{ineq-hmn-2k-22}\\[5pt] h(n,k)&> \frac{9}{4}\binom{n-2}{k-2}+\frac{9}{4}\binom{n-4}{k-2}.\label{ineq-hmn-2k-2} \end{align} \end{lem} \begin{proof} Since $n\geq (j-1)k/2$ implies $\frac{n-k-j+3}{n-j+1}\geq \frac{n-k}{n}$, by \eqref{ineq-key} and $n\geq 4k$ we infer \[ \frac{\binom{n-j}{k-2}}{\binom{n-2}{k-2}}\geq \left(\frac{n-k-j+3}{n-j+1}\right)^{j-2} \geq \left(\frac{n-k}{n}\right)^{j-2}\geq \left(\frac{3}{4}\right)^{j-2},\ 2\leq j\leq 9. \] It follows that \begin{align*} \sum_{j=2}^9 \binom{n-j}{k-2}&\geq \binom{n-2}{k-2}\sum_{j=2}^9 \left(\frac{3}{4}\right)^{j-2}=\frac{1-\left(\frac{3}{4}\right)^8}{1-\frac{3}{4}} \binom{n-2}{k-2}> \frac{32}{9}\binom{n-2}{k-2}. \end{align*} Thus by $k\geq 8$ we have \[ h(n,k)>\binom{n-1}{k-1}-\binom{n-k-1}{k-1}\geq \sum_{j=2}^9 \binom{n-j}{k-2}> \frac{32}{9}\binom{n-2}{k-2}. \] By \eqref{ineq-key} and $n\geq 4k$ we also have \begin{align}\label{ineq-3} \sum_{j=2}^4 \binom{n-j}{k-2}&\geq \binom{n-2}{k-2} \left(1+\frac{3}{4}+\left(\frac{3}{4}\right)^2\right)>\frac{9}{4}\binom{n-2}{k-2}. \end{align} Moreover, \[ \frac{\binom{n-j}{k-2}}{\binom{n-4}{k-2}}\geq \left(\frac{n-k-j+3}{n-j+1}\right)^{j-4} \geq \left(\frac{n-k}{n}\right)^{j-4}\geq \left(\frac{3}{4}\right)^{j-4},\ 5\leq j\leq 9. \] We infer \begin{align}\label{ineq-4} \sum_{j=5}^9 \binom{n-j}{k-2}&\geq \binom{n-4}{k-2}\left(\frac{3}{4}+\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3 +\left(\frac{3}{4}\right)^4+\left(\frac{3}{4}\right)^5\right)> \frac{9}{4}\binom{n-4}{k-2}. \end{align} Adding \eqref{ineq-3} and \eqref{ineq-4} we obtain \eqref{ineq-hmn-2k-2}. \end{proof} To prove the theorem we apply shifting, a powerful method that can be traced back to \cite{EKR}. For $\mathcal{F}\subset{[n]\choose k}$ and $1\leq i< j\leq n$, define the shift $$S_{ij}(\mathcal{F})=\left\{S_{ij}(F)\colon F\in\mathcal{F}\right\},$$ where $$S_{ij}(F)=\left\{ \begin{array}{ll} (F\setminus\{j\})\cup\{i\}, & j\in F, i\notin F \text{ and } (F\setminus\{j\})\cup\{i\}\notin \mathcal{F}; \\[5pt] F, & \hbox{otherwise.} \end{array} \right. $$ It is well known (cf. \cite{F87}) that shifting preserves the cross-intersecting property. Let us define the {\it shifting partial order} $\prec$. For two $k$-sets $A$ and $B$ where $A=\{a_1,\ldots,a_k\}$, $a_1<\ldots<a_k$ and $B=\{b_1,\ldots,b_k\}$, $b_1<\ldots<b_k$ we say that $A$ precedes $B$ and denote it by $A\prec B$ if $a_i\leq b_i$ for all $1\leq i\leq k$. A family $\mathcal{F}\subset \binom{[n]}{k}$ is called {\it shifted} (or {\it initial}) if $A\prec B$ and $B\in \mathcal{F}$ always imply $A\in \mathcal{F}$. By repeated shifting one can transform an arbitrary $k$-graph into a shifted $k$-graph with the same number of edges. The only problem with shifting is that it might destroy the non-triviality of $\mathcal{F}$ or $\mathcal{G}$ or both. \begin{fact}\label{fact-3.1} If $S_{ij}(\mathcal{F})\subset \mathcal{S}_{\{i\}}$, then \begin{itemize} \item[(i)] $\mathcal{F}(\overline{ij})=\emptyset$ and \item[(ii)] $\mathcal{F}(i)\cap \mathcal{F}(j)=\emptyset$. (Note that this is equivalent with $\mathcal{F}(\overline{i}j)\cap \mathcal{F}(i\overline{j})=\emptyset$). \end{itemize} \end{fact} With all this preparation we are ready to state and prove the main result of this section. \begin{prop}\label{lem-2.4} Let $n\geq 4k$, $k\geq 8$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. Suppose that there exist disjoint pairs $(a,b), (c,d)$ such that $S_{ab}(\mathcal{F})\subset \mathcal{S}_{a}$ and $S_{cd}(\mathcal{F})\subset \mathcal{S}_{c}$. Then \begin{align}\label{ineq-main} |\mathcal{F}||\mathcal{G}|< h(n,k)^2. \end{align} \end{prop} \begin{proof} For notational convenience assume that $(a,b)=(1,2)$ and $(c,d)=(3,4)$. Arguing indirectly we assume \eqref{indirectAssum}. For $i=2,3,4$, define \[ \mathcal{F}_i = \{F\setminus [4]\colon F\in \mathcal{F},\ |F\cap [4]|=i\}. \] Since $S_{ij}(\mathcal{F})\subset \mathcal{S}_{\{i\}}$ for $(i,j)=(1,2)$ and $(3,4)$ implies $(1,2),(3,4)\in \mathcal{T}^{(2)}(\mathcal{F})$, we have \begin{align}\label{ineq-fcap5n} |F\cap [5,n]|\leq k-2 \mbox{ for all }F \in \mathcal{F}. \end{align} \begin{claim}\label{claim-1} For every $E\in \mathcal{F}_2$, there are at most two choices for $S\in \binom{[4]}{2}$ with $E\cup S\in \mathcal{F}$. For every $T\in \mathcal{F}_3$, there are at most two choices for $R\in \binom{[4]}{3}$ with $T\cup R\in \mathcal{F}$. \end{claim} \begin{proof} Suppose that for some $E\in \mathcal{F}_2$ there are three choices for $S\in \binom{[4]}{2}$ with $E\cup S\in \mathcal{F}$. Note that Fact \ref{fact-3.1} (i) implies $S\neq (1,2)$ and $S\neq (3,4)$. Choose $S_1,S_2$ such that $S_1\cap S_2\neq \emptyset$. Without loss of generality, assume that $S_1=(1,3)$ and $S_2=(1,4)$, then $E\cup \{1\} \in \mathcal{F}(3)\cap \mathcal{F}(4)$, contradicting Fact \ref{fact-3.1} (ii). Similarly, suppose that for some $T\in \mathcal{F}_3$ there are three choices for $R\in \binom{[4]}{3}$ with $T\cup R\in \mathcal{F}$. Then we may choose $R_1,R_2$ such that $R_1\cap R_2= (1,2)$ or $(3,4)$. Without loss of generality, assume that $R_1=(1,2,3)$ and $R_2=(1,2,4)$, then $T\cup \{1,2\} \in \mathcal{F}(3)\cap \mathcal{F}(4)$, contradicting Fact \ref{fact-3.1} (ii). \end{proof} By Claim \ref{claim-1} we have \begin{align}\label{ineq-hfhp} |\mathcal{F}| \leq 2|\mathcal{F}_2|+2|\mathcal{F}_3|+|\mathcal{F}_4|. \end{align} It follows that \begin{align}\label{ineq-hfupbound} |\mathcal{F}| &\leq 2\binom{n-4}{k-2}+2\binom{n-4}{k-3}+\binom{n-4}{k-4}=\binom{n-2}{k-2}+\binom{n-4}{k-2}. \end{align} By \eqref{ineq-hmn-2k-2}, for $n\geq 4k$, $k\geq 8$ we have $|\mathcal{F}| < \frac{4}{9}h(n,k)$. Then the indirect assumption \eqref{indirectAssum} implies \begin{align}\label{ineq-2.5hnk} |\mathcal{G}| >\frac{9}{4}h(n,k). \end{align} Let \[ \mathcal{P}=\left\{P\in \binom{[4]}{2}\colon P\cap (1,2)\neq \emptyset, P\cap (3,4)\neq \emptyset\right\}. \] \begin{claim}\label{claim-new3.0} \begin{align}\label{ineq-new3.0} \sum_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|> 4\binom{n-5}{k-3}. \end{align} \end{claim} \begin{proof} Suppose for contradiction that $\sum\limits_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|\leq 4\binom{n-5}{k-3}$. Note that Fact \ref{fact-3.1} (i) implies $|\mathcal{F}(\{1,2\},[4])|=|\mathcal{F}(\{3,4\},[4])|=0$. Then by Claim \ref{claim-1} we have \begin{align*} |\mathcal{F}|&\leq \sum_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|+2|\mathcal{F}_3|+|\mathcal{F}_4|\\[5pt] &\leq 4\binom{n-5}{k-3}+2\binom{n-4}{k-3}+ \binom{n-4}{k-4}\\[5pt] &=4\binom{n-5}{k-3}+\binom{n-4}{k-3}+\binom{n-3}{k-3}\\[5pt] &< 3\binom{n-3}{k-3}+3\binom{n-5}{k-3}. \end{align*} By \eqref{ineq-hmn-2k-2} it follows that \[ |\mathcal{F}| \leq 3\left(\frac{k-2}{n-2}\binom{n-2}{k-2}+\frac{k-2}{n-4}\binom{n-4}{k-2}\right)< \frac{4(k-1)}{3(n-1)} h(n,k). \] By \eqref{indirectAssum} and \eqref{ineq-hmn-2k-22}, \begin{align}\label{ineq-sumtotal} |\mathcal{G}| >\frac{3(n-1)}{4(k-1)} h(n,k)>\frac{3(n-1)}{4(k-1)} 3\binom{n-2}{k-2}>2\binom{n-1}{k-1}, \end{align} contradicting \eqref{ineq-2}. \end{proof} \begin{claim} \begin{align}\label{ineq-hg4barub} |\mathcal{G}(\overline{[4]})|\leq \binom{n-7}{k-3}+\binom{n-8}{k-3}. \end{align} \end{claim} \begin{proof} By Claim \ref{claim-1} and \eqref{ineq-new3.0}, we infer \[ 2|\mathcal{F}_2| \geq \sum_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|> 4\binom{n-5}{k-3}. \] It follows that $|\mathcal{F}_2|>2\binom{n-5}{k-3}> \binom{n-5}{k-3}+\binom{n-6}{k-3}+\binom{n-8}{k-4}$. Since $\mathcal{F}_2$, $\mathcal{G}(\overline{[4]})$ are cross-intersecting, by \eqref{hiltonLem-3.1} we obtain \eqref{ineq-hg4barub}. \end{proof} \begin{claim} For $i\in [4]$, \begin{align}\label{ineq-newhgi} |\mathcal{G}(i,[4])|\leq \binom{n-4}{k-1} - \binom{n-2-k}{k-1}. \end{align} \end{claim} \begin{proof} Since $\mathcal{F}$ is non-trivial, there exists $F\in \mathcal{F}$ such that $i\notin F$. Let $E=F\cap [5,n]$. By \eqref{ineq-fcap5n}, $|E|\leq k-2$. Now the cross-intersection implies $E\cap E'\neq \emptyset$ for each $E' \in \mathcal{G}(i,[4])$. Thus the claim follows. \end{proof} \begin{claim}\label{claim-new3} At most five of $\mathcal{G}(P,[4])$, $P\in \binom{[4]}{2}$ have size greater than $\binom{n-5}{k-3}-\binom{n-3-k}{k-3}$. \end{claim} \begin{proof} Suppose for contradiction that $|\mathcal{G}(P,[4])|>\binom{n-5}{k-3}-\binom{n-3-k}{k-3}$ for all $P\in \binom{[4]}{2}$. Then for each $P\in \binom{[4]}{2}$ let $P'=[4]\setminus P$. Apply \eqref{hiltonLem-2} with $a=b=k-2$ to the cross-intersecting families $\mathcal{G}(P,[4])$ and $\mathcal{F}(P',[4])$, we infer $|\mathcal{F}(P',[4])|\leq \binom{n-5}{k-3}$. Then \[ \sum_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|\leq 4\binom{n-5}{k-3}, \] contradicting \eqref{ineq-new3.0}. \end{proof} By Claim \ref{claim-new3}, we infer that \begin{align}\label{ineq-claim2p2} \sum_{P\subset[4], |P|\geq 2} |\mathcal{G}(P,[4])|&\leq 5\binom{n-4}{k-2}+\binom{n-5}{k-3}-\binom{n-3-k}{k-3}+4\binom{n-4}{k-3}+\binom{n-4}{k-4}\nonumber\\[5pt] &<2 \binom{n-4}{k-2}+3\left(\binom{n-4}{k-2}+\binom{n-4}{k-3}\right)+\left(\binom{n-4}{k-3} +\binom{n-4}{k-4}\right)\nonumber\\[5pt] &\qquad+\binom{n-5}{k-3}\nonumber\\[5pt] &=2 \binom{n-4}{k-2}+3\binom{n-3}{k-2}+\binom{n-3}{k-3}+\binom{n-5}{k-3}\nonumber\\[5pt] &=2 \binom{n-4}{k-2}+2\binom{n-3}{k-2}+\binom{n-2}{k-2}+\binom{n-5}{k-3}\nonumber\\[5pt] &<2\binom{n-2}{k-2}+2\binom{n-3}{k-2} +\binom{n-4}{k-2}. \end{align} \begin{claim}\label{claim-2} Exactly two of $\mathcal{G}(1,[4])$, $\mathcal{G}(2,[4])$, $\mathcal{G}(3,[4])$, $\mathcal{G}(4,[4])$ have size greater than $\binom{n-5}{k-2}-\binom{n-3-k}{k-2}$. \end{claim} \begin{proof} Suppose that at least three of $\mathcal{G}(1,[4])$, $\mathcal{G}(2,[4])$, $\mathcal{G}(3,[4])$, $\mathcal{G}(4,[4])$ have size greater than $\binom{n-5}{k-3}-\binom{n-3-k}{k-3}$. Then for each $P\in \mathcal{P}$ there exists $i\in [4]$ such that $i\notin P$ and $|\mathcal{G}(i,[4])|> \binom{n-5}{k-2}-\binom{n-3-k}{k-2}$. Apply \eqref{hiltonLem-2} to $\mathcal{G}(i,[4])$ and $\mathcal{F}(P,[4])$ with $a=k-1$ and $b=k-2$, we infer $|\mathcal{F}(P,[4])|\leq \binom{n-5}{k-3}$ for all $P\in \mathcal{P}$, contradicting \eqref{ineq-new3.0}. Thus, at most two of $\mathcal{G}(1,[4])$, $\mathcal{G}(2,[4])$, $\mathcal{G}(3,[4])$, $\mathcal{G}(4,[4])$ have size greater than $\binom{n-5}{k-3}-\binom{n-3-k}{k-3}$. Suppose that at most one of them has size greater than $\binom{n-5}{k-2}-\binom{n-3-k}{k-2}$. Then by \eqref{ineq-newhgi} we have \begin{align*} \sum_{ 1\leq i\leq 4} |\mathcal{G}(i,[4])|\leq \binom{n-4}{k-1} - \binom{n-2-k}{k-1}+3\left(\binom{n-5}{k-2}-\binom{n-3-k}{k-2}\right). \end{align*} Using \[ \binom{n-5}{k-2}-\binom{n-3-k}{k-2}\leq\binom{n-4}{k-2}-\binom{n-2-k}{k-2}, \] we get \begin{align}\label{ineq-claim2p1} \sum_{ 1\leq i\leq 4} |\mathcal{G}(i,[4])|&\leq \binom{n-4}{k-1} - \binom{n-2-k}{k-1}+\binom{n-4}{k-2} - \binom{n-2-k}{k-2}\nonumber\\[5pt] &\qquad+2\left(\binom{n-5}{k-2}-\binom{n-3-k}{k-2}\right)\nonumber\\[5pt] &< \binom{n-3}{k-1} - \binom{n-1-k}{k-1}+2\binom{n-5}{k-2}\nonumber\\[5pt] &< h(n,k)- \binom{n-2}{k-2}- \binom{n-3}{k-2} +2\binom{n-5}{k-2}. \end{align} Adding \eqref{ineq-hg4barub}, \eqref{ineq-claim2p2} and \eqref{ineq-claim2p1}, \begin{align*} |\mathcal{G}|=&\sum_{\emptyset\neq P\subset[4]} |\mathcal{G}(P,[4])|+|\mathcal{G}(\overline{[4]})|\\[5pt] <&h(n,k)+\binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-4}{k-2}+2\binom{n-5}{k-2} +\binom{n-7}{k-3}+\binom{n-8}{k-3}. \end{align*} Note that $k\geq 7$ implies $h(n,k)\geq \sum\limits_{2\leq i\leq 8} \binom{n-i}{k-2}$. It follows that \begin{align*} |\mathcal{G}|<&2h(n,k)+\binom{n-5}{k-2}-\binom{n-6}{k-2}-\binom{n-7}{k-2}-\binom{n-8}{k-2} +\binom{n-7}{k-3}+\binom{n-8}{k-3}\\[5pt] =&2h(n,k)+\binom{n-6}{k-3}+\binom{n-7}{k-3}+\binom{n-8}{k-3} -\binom{n-7}{k-2}-\binom{n-8}{k-2}\\[5pt] \leq &2h(n,k)+2\binom{n-6}{k-3}-2\binom{n-8}{k-2}. \end{align*} Since $n\geq 4k$ and $k\geq 3$ imply \begin{align*} \frac{\binom{n-8}{k-2}}{\binom{n-6}{k-3}} &=\frac{(n-k-3)(n-k-4)(n-k-5)}{(n-6)(n-7)(k-2)}\\[5pt] &\geq \frac{(3k-3)(3k-4)(3k-5)}{(4k-6)(4k-7)(k-2)}>\left(\frac{3}{4}\right)^2\times3>1. \end{align*} we infer $|\mathcal{G}|<2h(n,k)$, contradicting \eqref{ineq-2.5hnk}. \end{proof} Now consider a pair $(i,P)$, $P\in \binom{[4]}{2}$, $i\in [4]\setminus P$ and $|\mathcal{G}(i,[4])|> \binom{n-5}{k-2}-\binom{n-3-k}{k-2}$. Apply \eqref{hiltonLem-2} to $\mathcal{G}(i,[4])$ and $\mathcal{F}(P,[4])$ with $a=k-1$, $b=k-2$, we infer $|\mathcal{F}(P,[4])|\leq \binom{n-5}{k-3}$. By Claim \ref{claim-2}, exactly two of $\mathcal{G}(1,[4])$, $\mathcal{G}(2,[4])$, $\mathcal{G}(3,[4])$, $\mathcal{G}(4,[4])$ have size greater than $\binom{n-5}{k-2}-\binom{n-3-k}{k-2}$. Hence, at most one $P\in \{(1,3),(1,4),(2,3),(2,4)\}$ satisfies $|\mathcal{F}(P,[4])|> \binom{n-5}{k-3}$. It follows that \[ \sum_{P\in \mathcal{P}} |\mathcal{F}(P,[4])| \leq \binom{n-4}{k-2}+3\binom{n-5}{k-3}. \] Then by Claim \ref{claim-1} and the identity $\binom{n-2}{k-2}=\binom{n-4}{k-2}+2\binom{n-4}{k-3}+\binom{n-4}{k-4}$, \begin{align*} |\mathcal{F}| &\leq \sum_{P\in \mathcal{P}}|\mathcal{F}(P,[4])|+2|\mathcal{F}_3|+|\mathcal{F}_4|\\[5pt] &\leq 3\binom{n-5}{k-3}+\binom{n-4}{k-2}+2\binom{n-4}{k-3}+ \binom{n-4}{k-4} \\[5pt] &=\binom{n-2}{k-2}+ 3\binom{n-5}{k-3}. \end{align*} Since for $n\geq 4k$, \begin{align*} |\mathcal{F}|\leq \binom{n-2}{k-2}+ 3\binom{n-5}{k-3}=\binom{n-2}{k-2}+\frac{3(k-2)}{n-4}\binom{n-4}{k-2}<\binom{n-2}{k-2}+\frac{3}{4}\binom{n-4}{k-2}, \end{align*} by \eqref{ineq-hmn-2k-22} and \eqref{ineq-hmn-2k-2} we infer \begin{align}\label{ineq-hf1} |\mathcal{F}| &< \frac{1}{4}\binom{n-2}{k-2}+ \frac{3}{4} \left(\binom{n-2}{k-2} + \binom{n-4}{k-2}\right)\nonumber\\[5pt] &<\frac{1}{4}\times \frac{9}{32}h(n,k)+\frac{3}{4}\times \frac{4}{9}h(n,k)\nonumber\\[5pt] & = \frac{155}{384}h(n,k)<\frac{32}{73}h(n,k). \end{align} By Claim \ref{claim-2} and \eqref{ineq-newhgi}, \begin{align}\label{ineq-claim3p1} \sum_{1\leq i\leq 4}|\mathcal{G}(i,[4])| &\leq 2\left(\binom{n-4}{k-1} - \binom{n-2-k}{k-2}\right)+2\left(\binom{n-5}{k-2}-\binom{n-3-k}{k-2}\right)\nonumber\\[5pt] &\leq 2\left(\binom{n-3}{k-1} - \binom{n-1-k}{k-2}\right)\nonumber\\[5pt] &= 2h(n,k)-2\binom{n-2}{k-2}-2\binom{n-3}{k-2}. \end{align} Adding \eqref{ineq-hg4barub}, \eqref{ineq-claim2p2} and \eqref{ineq-claim3p1}, \begin{align}\label{ineq-hg1} |\mathcal{G}|&=|\mathcal{G}(\overline{[4]})|+\sum_{\emptyset\neq P\subset[4]} |\mathcal{G}(P,[4])|\nonumber\\[5pt] & \leq 2h(n,k)+\binom{n-4}{k-2}+\binom{n-7}{k-3}+\binom{n-8}{k-3}\nonumber\\[5pt] &\leq 2h(n,k)+\binom{n-2}{k-2}\overset{\eqref{ineq-hmn-2k-22}}< \frac{73}{32}h(n,k). \end{align} But now \eqref{ineq-hf1} and \eqref{ineq-hg1} imply $|\mathcal{F}||\mathcal{G}|<h(n,k)^2$, contradicting \eqref{indirectAssum}. This concludes the proof of the proposition. \end{proof} \section{The shifted case and the proof of the main theorem} In this section, we determine the maximum product of sizes of non-trivial shifted cross-intersecting families. First, we determine the maximum sum of sizes of non-trivial shifted cross-intersecting families $\mathcal{F},\mathcal{G}$ with $(2,5,7,\ldots,2k-1,2k+1)\notin \mathcal{F}\cup \mathcal{G}$ by modifying an injective map introduced in \cite{F2017}. \begin{prop} Let $n\geq k>0$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be cross-intersecting. Suppose that both $\mathcal{F}$ and $\mathcal{G}$ are non-trivial shifted families. Moreover $(2,5,7,\ldots,2k-1,2k+1)\notin \mathcal{F}\cup \mathcal{G}$. Then \begin{align}\label{ineq-hfhg2hm} |\mathcal{F}|+|\mathcal{G}|\leq 2h(n,k). \end{align} \end{prop} \begin{proof} We claim that for every $H\in \mathcal{F}\cup \mathcal{G}$ there exists $\ell$ such that $|H\cap [2\ell]|\geq \ell$. Moreover, if $1\notin H$ then $\ell \geq 2$. Arguing indirectly, assume that no such $\ell$ exists for $H=\{a_1,a_2,\ldots,a_k\}$ where $a_1<a_2<\ldots<a_k$. For $\ell=1$ this implies $a_1\geq 2$. For $\ell\geq 2$ we infer $a_\ell> 2\ell$ for all $\ell=2,\ldots,k$. By shiftedness, we see that $(2,5,7,\ldots,2k+1)\in \mathcal{F}\cup \mathcal{G}$, a contradiction. Let $\ell(H)$ be the maximal $\ell$ such that $|H\cap [2\ell]|\geq \ell$. Recall the notation $A\triangle B = (A\setminus B)\cup (B\setminus A)$, the symmetric difference. Define the function $\phi\colon \phi(H) = H\triangle [2\ell(H)]$. \begin{claim}\label{claim-4} For $G,G'\in \mathcal{G}$, $\phi(G)\neq \phi(G')$. \end{claim} \begin{proof} If $\ell(G)=\ell(G'):=\ell$ then $\phi(G)\triangle [2\ell]=G\neq G'=\phi(G') \triangle [2\ell]$ implies $\phi(G)\neq \phi(G')$. On the other hand if $\ell(G)>\ell(G')$ then \[ |\phi(G) \cap [2\ell(G)]|=\ell >|\phi(G') \cap [2\ell(G)]| \] by the maximality of $\ell(G')$. \end{proof} \begin{claim}\label{claim-5} For $G\in \mathcal{G}(\bar{1})$, $\phi(G)\setminus \{1\}\notin \mathcal{F}(1)$. \end{claim} \begin{proof} Set $\ell=\ell(G)$. Let $G=(x_1,x_2,\ldots,x_k)$ with $x_1<x_2<\ldots<x_k$. The maximal choice of $\ell$ implies \[ [2\ell]\cap G =\{x_1,\ldots,x_\ell\}, \ x_{\ell+1}>2\ell+2,\ldots,x_k>2k. \] By shiftedness $(x_1,\ldots,x_\ell)\cup (2\ell+2,2\ell+4,\ldots,2k)\in \mathcal{G}$. Note that $G\in \mathcal{G}(\bar{1})$ implies $x_1\geq 2$. If $\phi(G)\setminus \{1\}=([2,2\ell]\setminus (x_1,\ldots,x_\ell))\cup (2\ell+2,2\ell+4,\ldots,2k)\in \mathcal{F}(1)$ then by shiftedness $([2\ell]\triangle (x_1,\ldots,x_\ell))\cup (2\ell+1,2\ell+3,\ldots,2k-1)\in \mathcal{F}$ and it contradicts the cross-intersecting property. \end{proof} \begin{claim}\label{claim-6} If $H\neq [2,k+1]$, $H\in \mathcal{F}\cup \mathcal{G}$, $1\notin H$ then $\phi(H)\cap [2,k+1]\neq \emptyset$. \end{claim} \begin{proof} There are two cases. Let $\ell=\ell(H)$. If $2\ell\geq k+1$ then $[2,k+1]\subset [2\ell]$. Consequently, we can choose $x\in [2,k+1]\setminus H$ since $H\neq [2,k+1]$. Thus $x\in [2\ell]\setminus H\subset H\triangle [2\ell]$, i.e., $x\in [2,k+1]\cap \phi(H)$. As we noted before, $1\notin H$ and $(2,5,7,\ldots,2k-1,2k+1)\notin \mathcal{F}\cup \mathcal{G}$ imply that $\ell\geq 2$. If $k+1>2\ell$ then $\ell\geq 2$ implies the existence of $x\in [2,2\ell]\setminus H$ and hence $x\in H\triangle [2\ell]$. It follows that $x\in [2,k+1]\cap \phi(H)$. \end{proof} Note that the non-triviality and the shiftedness imply $[2,k+1]\in \mathcal{F}\cap \mathcal{G}$. From Claims \ref{claim-4}, \ref{claim-5} and \ref{claim-6}, we infer \begin{align} |\mathcal{F}(1)|+|\mathcal{G}(\bar{1})| &=|\mathcal{F}(1)|+|\phi\left(\mathcal{G}(\bar{1})\setminus \{[2,k+1]\}\right)|+1\nonumber\\[5pt] &\leq \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1.\label{ineq-hf1hgbar} \end{align} Switching the roles of $\mathcal{F}$ and $\mathcal{G}$, we obtain \begin{align} &|\mathcal{F}(\bar{1})|+|\mathcal{G}(1)| \leq \binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1.\label{ineq-hfbarhg1} \end{align} Adding \eqref{ineq-hf1hgbar} and \eqref{ineq-hfbarhg1}, we get \eqref{ineq-hfhg2hm}. \end{proof} Recall the following inequality from \cite{F78}, for a proof using linear algebra cf. \cite{F2022}. \begin{lem}[\cite{F78}]\label{lem-walk} Let $\mathcal{F}\subset \binom{[n]}{k}$ be a shifted family with $0\leq t<k$. If $[t]\cup \{t+2,t+4,\ldots,2k-t\}\notin \mathcal{F}$, then \begin{align}\label{ineq-walk} |\mathcal{F}| \leq \binom{n}{k-t-1}. \end{align} \end{lem} \begin{prop}\label{lem-2.6} Let $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial and cross-intersecting, $n\geq 4k$, $k\geq 8$. If both $\mathcal{F}$ and $\mathcal{G}$ are shifted, then \begin{align}\label{ineq-shiftedhfhg2} |\mathcal{F}||\mathcal{G}|\leq h(n,k)^2. \end{align} \end{prop} \begin{proof} We may assume that $(1,3,5,\ldots,2k-1)\notin \mathcal{F}\cap \mathcal{G}$. Indeed, if $(1,3,5,\ldots,2k-1)\in \mathcal{F}\cap \mathcal{G}$, then by cross-intersection we infer $(2,4,\ldots,2k)\notin \mathcal{F}\cup\mathcal{G}$. By shiftedness, it follows that $(2,5,7,\ldots,2k-1,2k+1)\notin \mathcal{F}\cup \mathcal{G}$. By \eqref{ineq-hfhg2hm}, \[ |\mathcal{F}||\mathcal{G}|\leq \left(\frac{|\mathcal{F}|+|\mathcal{G}|}{2}\right)^2\leq h(n,k)^2 \] and we are done. By symmetry assume that $(1,3,5,\ldots,2k-1)\notin \mathcal{F}$. Then by \eqref{ineq-walk} we have \begin{align}\label{ineq-f781} |\mathcal{F}| \leq \binom{n}{k-2}. \end{align} By \eqref{ineq-1} and $n\geq 4k$, we obtain that \[ |\mathcal{F}|>\binom{n-3}{k-3}+\binom{n-4}{k-3}=\frac{n-3}{k-3}\binom{n-4}{k-4}+\frac{n-k}{k-3} \binom{n-4}{k-4} >7\binom{n-4}{k-4}. \] By \eqref{ineq-key} and $n\geq 4k$, \[ |\mathcal{F}|> 7\binom{n-4}{k-4}>\left(\frac{4}{3}\right)^4\binom{n-4}{k-4}> \left(\frac{n-3}{n-k+1}\right)^4\binom{n-4}{k-4}> \binom{n}{k-4}. \] Then Lemma \ref{lem-walk} implies that $(1,2,3,5,7,\ldots,2k-3)\in \mathcal{F}$. We distinguish two cases. {\noindent \bf Case 1.} $(1,2,4,6,\ldots,2k-2)\notin \mathcal{F}$. In view of \eqref{ineq-walk} the assumption $(1,2,4,6,\ldots,2k-2)\notin \mathcal{F}$ implies for $n\geq 4k$ \begin{align}\label{ineq-f782} |\mathcal{F}| \leq \binom{n}{k-3}\overset{\eqref{ineq-key}}{<}\left(\frac{n-2}{n-k+1}\right)^3\binom{n-3}{k-3} <\left(\frac{4}{3}\right)^3\binom{n-3}{k-3}<3\binom{n-3}{k-3}. \end{align} By \eqref{ineq-2}, \begin{align}\label{ineq-f783} |\mathcal{G}| \leq \binom{n-1}{k-1}+\binom{n-2}{k-1}+\binom{n-4}{k-2}< 2\binom{n-1}{k-1}. \end{align} From \eqref{ineq-f782} and \eqref{ineq-f783}, we obtain \begin{align}\label{ineq-hfhg3} |\mathcal{F}||\mathcal{G}|<6\binom{n-1}{k-1}\binom{n-3}{k-3}\overset{\eqref{ineq-key2}}{\leq} 6\binom{n-2}{k-2}^2\overset{\eqref{ineq-hmn-2k-22}}<h(n,k)^2. \end{align} {\noindent \bf Case 2.} $(1,2,4,6,\ldots,2k-2)\in \mathcal{F}$. By cross-intersection, $(3,5,\ldots,2k+1)\notin \mathcal{G}$. Using that $\mathcal{F}$ is non-trivial, $[2,k+1]\in \mathcal{F}$ follows. By cross-intersection again $G\cap [2,k+1]\neq \emptyset$ for all $G\in \mathcal{G}$. Consequently \[ |\mathcal{G}(1)|\leq \binom{n-1}{k-1}-\binom{n-k-1}{k-1}<h(n,k). \] Consider $\mathcal{G}(\bar{1},2)\subset \binom{[3,n]}{k-1}$. Since $\mathcal{F}(\bar{2})\neq \emptyset$ and $\mathcal{F}$ is shifted, $\mathcal{F}(1,\bar{2})\neq \emptyset$. Thus, \[ |\mathcal{G}(\bar{1},2)|\leq \binom{n-2}{k-1} - \binom{n-k-1}{k-1}. \] For $G\in \mathcal{G}(\bar{1},\bar{2})$ define $\tilde{G}=\{x-2\colon x\in G\}$ and set \[ \tilde{\mathcal{G}}=\{\tilde{G}\colon G\in \mathcal{G}(\bar{1},\bar{2})\}. \] Note that $(3,5,\ldots,2k+1)\notin \mathcal{G}$ implies $(1,3,\ldots,2k-1)\notin \tilde{\mathcal{G}}$. Applying \eqref{ineq-walk} with $t=1$ yields \begin{align}\label{ineq-g1bar2bar} |\mathcal{G}(\bar{1},\bar{2})|\leq \binom{n-2}{k-2}. \end{align}Thus, \begin{align}\label{ineq-hg} |\mathcal{G}| \leq |\mathcal{G}(1)|+|\mathcal{G}(\bar{1},2)|+|\mathcal{G}(\bar{1},\bar{2})|<2h(n,k). \end{align} By \eqref{ineq-f781} and \eqref{ineq-key}, we obtain for $n\geq 4k$ \begin{align}\label{ineqhfnk-2} |\mathcal{F}|\leq \binom{n}{k-2}\leq \left(\frac{n-1}{n-k+1}\right)^2\binom{n-2}{k-2}< \left(\frac{4}{3}\right)^2\binom{n-2}{k-2}=\frac{16}{9}\binom{n-2}{k-2}. \end{align} Combining \eqref{ineqhfnk-2} and \eqref{ineq-hg}, we obtain \begin{align*} |\mathcal{F}||\mathcal{G}|<\frac{32}{9}\binom{n-2}{k-2}h(n,k) \overset{\eqref{ineq-hmn-2k-22}}{<} h(n,k)^2. \end{align*} \end{proof} In order to prove Theorem \ref{main} let us introduce the notion of shift-resistant pair. For a pair of families $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$, define the quantity \[ w(\mathcal{F},\mathcal{G}) =\sum_{F\in \mathcal{F}}\sum_{i\in F} i + \sum_{G\in \mathcal{G}}\sum_{j\in G} j. \] Let us fix $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ so that $\mathcal{F},\mathcal{G}$ are non-trivial cross-intersecting, $|\mathcal{F}||\mathcal{G}|$ is maximal, moreover among such pairs $w(\mathcal{F},\mathcal{G})$ is minimal. If in such pair $\mathcal{F}$ and $\mathcal{G}$ are not both shifted then we say that $(\mathcal{F},\mathcal{G})$ is shift-resistant. \begin{prop}\label{prop-4.4} Suppose that $(\mathcal{F},\mathcal{G})$ is a shift-resistant cross-intersecting pair. Then for $\mathcal{H}=\mathcal{F}$ or $\mathcal{H}=\mathcal{G}$ there exist disjoint pairs $(a,b)$, $(c,d)$ such that $S_{ab}(\mathcal{H})\subset \mathcal{S}_a$ and $S_{cd}(\mathcal{H})\subset \mathcal{S}_c$. \end{prop} \begin{proof} Since $\mathcal{F}$ and $\mathcal{G}$ are not both shifted, without loss of generality suppose $S_{ab}(\mathcal{F})\subset \mathcal{S}_a$. Note that this implies $F\cap (a,b)\neq \emptyset$ for all $F\in \mathcal{F}$. Hence \[ \mathcal{S}_{\{a,b\}}=\left\{S\in \binom{[n]}{k}\colon (a,b)\subset S\right\}\subset \mathcal{G} \] follows from the maximality of $|\mathcal{F}||\mathcal{G}|$. Consider an arbitrary pair $(c,d) \subset [n]\setminus (a,b)$. First note that $S_{cd}(\mathcal{G})\subset \mathcal{S}_c$ cannot hold. Indeed, $\mathcal{S}_{\{a,b\}}\subset \mathcal{G}$ implies $\mathcal{G}(\bar{c},\bar{d})\neq \emptyset$ and thereby $S_{cd}(\mathcal{G})\not\subset \mathcal{S}_c$. If $S_{cd}(\mathcal{F})\subset \mathcal{S}_c$ then we are done. Thus we may assume that $S_{cd}(\mathcal{G})\not\subset \mathcal{S}_c$. Now the minimality of $w(\mathcal{F},\mathcal{G})$ implies $S_{cd}(\mathcal{F})=\mathcal{F}$ and $S_{cd}(\mathcal{G})=\mathcal{G}$ for all $(c,d)\subset [n]\setminus (a,b)$. Let $D=(d_1,d_2,\ldots,d_{k-1})$ be the $k-1$ smallest elements of $[n]\setminus (a,b)$. Then $\mathcal{F}(\bar{a},\bar{b})=\emptyset$ and $\mathcal{F}(\bar{a})\neq \emptyset$ imply $\mathcal{F}(\bar{a},b) \neq \emptyset$, moreover $S_{cd}(\mathcal{F})=\mathcal{F}$ for all $(c,d) \subset [n]\setminus (a,b)$ implies $D\in \mathcal{F}(\bar{a},b)$. Similarly, $D\in \mathcal{F}(a,\bar{b})$. Thus $\mathcal{F}(\bar{a},b)\cap \mathcal{F}(a,\bar{b})\neq \emptyset$, contradicting $S_{ab}(\mathcal{F})\subset \mathcal{S}_a$. \end{proof} \begin{proof}[Proof of Theorem \ref{main}] Let $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be a pair of non-trivial cross-intersecting families with $|\mathcal{F}||\mathcal{G}|$ maximal. Moreover, we assume that $w(\mathcal{F},\mathcal{G})$ is minimal among all such pairs. Then the minimality of $w(\mathcal{F},\mathcal{G})$ implies that either both of $\mathcal{F}$, $\mathcal{G}$ are shifted or $(\mathcal{F},\mathcal{G})$ forms a shift-resistant pair.If $\mathcal{F}$ and $\mathcal{G}$ are both shifted then by Proposition \ref{lem-2.6} we conclude that $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. If $(\mathcal{F},\mathcal{G})$ forms a shift-resistant pair, then by Proposition \ref{prop-4.4} there exist disjoint pairs $(a,b)$, $(c,d)$ such that either $\mathcal{F}$ or $\mathcal{G}$ (call it $\mathcal{H}$) satisfies $S_{ab}(\mathcal{H})\subset \mathcal{S}_a$ and $S_{cd}(\mathcal{H})\subset \mathcal{S}_c$. By symmetry assume that $S_{ab}(\mathcal{F})\subset \mathcal{S}_a$ and $S_{cd}(\mathcal{F})\subset \mathcal{S}_c$. Then by Proposition \ref{lem-2.4} we conclude that $|\mathcal{F}||\mathcal{G}|< h(n,k)^2$. \end{proof} \section{ Further improvements and concluding remarks} The most intriguing problem is whether \eqref{ineq-hfhg2} of Theorem \ref{main} holds for the full range, that is, for all $n,k$ satisfying $n\geq 2k\geq 4$. Note that for $n=2k$, $|\mathcal{F}|+|\mathcal{G}|\leq {2k \choose k}=2h(n,k)$ is obvious. Consequently, $|\mathcal{F}| |\mathcal{G}| \leq h(n,k)^2$. By a different argument, we can also prove Theorem \ref{main} for $n\geq 3k$ and $k\geq 15$ as well. Let us first prove a statement of some independent interest. \begin{prop}\label{prop-5.1} Suppose that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are cross-intersecting, $n\geq 2k$ and $\min\{|\mathcal{F}|,|\mathcal{G}|\}\geq \binom{n-3}{k-3}+\binom{n-4}{k-3}$. Then \begin{align} |\mathcal{F}|+|\mathcal{G}|\leq 2\binom{n-1}{k-1}. \end{align} \end{prop} \begin{proof} By Hilton's Lemma, without loss of generality, assume that $\mathcal{F}=\mathcal{L}(n,k,|\mathcal{F}|),\mathcal{G}=\mathcal{L}(n,k,|\mathcal{G}|)$, $|\mathcal{F}|\leq |\mathcal{G}|$ and $|\mathcal{G}|>\binom{n-1}{k-1}$. Thus $|\mathcal{G}(1)|=\binom{n-1}{k-1}$ and $|\mathcal{F}(1)|=|\mathcal{F}|$. If $|\mathcal{G}(\bar{1})|\leq\binom{n-2}{k-1}$ then \[ |\mathcal{F}|+|\mathcal{G}|= |\mathcal{F}(1,2)|+|\mathcal{F}(1,\bar{2})|+|\mathcal{G}(1)|+|\mathcal{G}(\bar{1},2)|. \] Since $\mathcal{F}(1,\bar{2}),\mathcal{G}(\bar{1},2)\subset\binom{[3,n]}{k-1}$ are cross-intersecting, we have \[ |\mathcal{F}(1,\bar{2})|+|\mathcal{G}(\bar{1},2)| \leq \binom{n-2}{k-1}. \] It follows that \begin{align*} |\mathcal{F}|+|\mathcal{G}|&= |\mathcal{F}(1,2)|+|\mathcal{G}(1)|+(|\mathcal{F}(1,\bar{2})|+|\mathcal{G}(\bar{1},2)|)\\[5pt] &\leq \binom{n-2}{k-2}+\binom{n-1}{k-1}+\binom{n-2}{k-1}\\[5pt] &=2\binom{n-1}{k-1}. \end{align*} Thus we may assume that $|\mathcal{G}|>\binom{n-1}{k-1}+\binom{n-2}{k-1}$. Now \[ |\mathcal{G}|=\binom{n-1}{k-1}+\binom{n-2}{k-1}+|\mathcal{G}(\bar{1},\bar{2})|\mbox{ and }|\mathcal{F}|=|\mathcal{F}(1,2)|. \] By the assumption on $|\mathcal{F}|$, $\mathcal{F}(1,2)\subset\binom{[n-2]}{k-2}$ satisfies $|\mathcal{F}(1,2)|\geq \binom{n-3}{k-3}+\binom{n-4}{k-3}$. Consequently $\{3,4\}\subset G$ for all $G\in \mathcal{G}(\bar{1},\bar{2})$. We infer $|\mathcal{G}(\bar{1},\bar{2},3,4)|=|\mathcal{G}(\bar{1},\bar{2})|$ and \[ |\mathcal{F}(1,2)|= \binom{n-3}{k-3}+\binom{n-4}{k-3}+|\mathcal{F}(1,2,\bar{3},\bar{4})|. \] As $\mathcal{F}(1,2,\bar{3},\bar{4})$ and $\mathcal{G}(\bar{1},\bar{2},3,4)$ are cross-intersecting, we have \[ |\mathcal{F}(1,2,\bar{3},\bar{4})|+|\mathcal{G}(\bar{1},\bar{2},3,4)|\leq \binom{n-4}{k-2}. \] Consequently, \begin{align*} |\mathcal{F}|+|\mathcal{G}|&= \binom{n-3}{k-3}+\binom{n-4}{k-3}+|\mathcal{F}(1,2,\bar{3},\bar{4})|+\binom{n-1}{k-1} +\binom{n-2}{k-1}+|\mathcal{G}(\bar{1},\bar{2},3,4)|\\[5pt] &\leq \binom{n-1}{k-1}+ \binom{n-2}{k-1}+ \binom{n-3}{k-3}+\binom{n-4}{k-3}+\binom{n-4}{k-2}\\[5pt] &=2\binom{n-1}{k-1}. \end{align*} \end{proof} \begin{lem}\label{lem-5.2} Let $n\geq 2k$ and let $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. Let $|\mathcal{F}|=\alpha h(n,k)$ and suppose that $\alpha <1$. Set $f(\alpha) =\frac{1+\alpha^2}{(1-\alpha)^2}$. Then $|\mathcal{F}||\mathcal{G}|> h(n,k)^2$ implies \begin{align}\label{ineq-keyfalpha} f(\alpha) \geq \prod_{1\leq i\leq k-1} \frac{n-i}{n-k-i}. \end{align} \end{lem} \begin{proof} By \eqref{ineq-1} and Proposition \ref{prop-5.1}, we infer $|\mathcal{F}|+|\mathcal{G}|\leq 2\binom{n-1}{k-1}$. Note that $|\mathcal{F}||\mathcal{G}|> h(n,k)^2$ implies $|\mathcal{G}|\geq h(n,k)/\alpha$. Therefore, \begin{align*} 2\binom{n-1}{k-1} \geq |\mathcal{F}|+|\mathcal{G}| &\geq \left(\alpha+\frac{1}{\alpha}\right)h(n,k)\geq \left(\alpha+\frac{1}{\alpha}\right)\left(\binom{n-1}{k-1}-\binom{n-k-1}{k-1}\right). \end{align*} By rearranging, \[ \left(\alpha+\frac{1}{\alpha}-2\right)\binom{n-1}{k-1}\leq \left(\alpha+\frac{1}{\alpha}\right)\binom{n-k-1}{k-1}. \] Multiplying both sides by $\alpha$ we get \[ (1-\alpha)^2\binom{n-1}{k-1}\leq (1+\alpha^2)\binom{n-k-1}{k-1}. \] Thus, \begin{align*} f(\alpha)=\frac{1+\alpha^2}{(1-\alpha)^2} \geq \frac{\binom{n-1}{k-1}}{\binom{n-k-1}{k-1}} =\frac{(n-1)(n-2)\ldots(n-k+1)}{(n-k-1)(n-k-2)\ldots(n-2k+1)}. \end{align*} \end{proof} In case of shift-resistant families and $k\geq 9$ we succeeded to extend the proof to the whole range. \begin{prop}\label{prop-5.3.0} Let $n\geq 2k+1$, $k\geq 6$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. Suppose that $(\mathcal{F},\mathcal{G})$ forms a shift-resistant pair. Set $f(\alpha) =\frac{1+\alpha^2}{(1-\alpha)^2}$. If \[ \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i} > f(0.64)\approx 10.8765, \] then \begin{align*} |\mathcal{F}||\mathcal{G}|\leq h(n,k)^2. \end{align*} \end{prop} \begin{proof} By Proposition \ref{prop-4.4} and \eqref{ineq-hfupbound}, we have \begin{align} |\mathcal{F}| \leq \binom{n-2}{k-2}+\binom{n-4}{k-2}. \end{align} For $n=2k+1$, \begin{align*} \frac{\binom{n-1}{k}+\binom{n-4}{k}}{\binom{n}{k}} &=\frac{k+1}{2k+1}\left(\frac{2k(2k-1)2(k-1)+k(k-1)(k-2)}{2k(2k-1)2(k-1)}\right)\\[5pt] &=\frac{k+1}{2k+1}\frac{3(3k-2)}{4(2k-1)} =\frac{3}{4} \frac{3k^2+k-2}{4k^2-1} >\frac{9}{16}. \end{align*} Since $\binom{x-d}{k}/\binom{x}{k}$ is an increasing function of $x$, for all $n\geq 2k+1$, \begin{align}\label{ineq-key3} \binom{n}{k}+\binom{n-1}{k}+\binom{n-4}{k}>\frac{25}{16}\binom{n}{k}. \end{align} By \eqref{ineq-key3} we infer for $(n-2)\geq 2(k-2)+1$, \[ \binom{n-2}{k-2}+\binom{n-3}{k-2}+\binom{n-6}{k-2}>\frac{25}{16}\binom{n-2}{k-2}. \] For $(n-4)\geq 2(k-2)+1$, \[ \binom{n-4}{k-2}+\binom{n-5}{k-2}+\binom{n-7}{k-2}>\frac{25}{16}\binom{n-4}{k-2}. \] Therefore, \[ h(n,k) \geq \sum_{2\leq i\leq 7} \binom{n-i}{k-2}\geq \frac{25}{16}\left(\binom{n-2}{k-2}+\binom{n-4}{k-2}\right)\geq \frac{25}{16} |\mathcal{F}|, \] implying that $\alpha =\frac{|\mathcal{F}|}{h(n,k)} \leq \frac{16}{25}=0.64$. Since $f(\alpha)$ is increasing in $[0,1]$, we obtain \[ f(\alpha)\leq f\left(0.64\right)< \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i}. \] Then Lemma \ref{lem-5.2} implies $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{proof} \begin{prop} For $n\geq 2k$, \begin{align}\label{ineq-key4} \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i} >\left(\frac{n-\frac{k}{2}}{n-\frac{3k}{2}}\right)^{k-1}. \end{align} \end{prop} \begin{proof} Note that for $m>d>i>0$, \begin{align}\label{ineq-key5} \frac{m-d-i}{m-i}\cdot \frac{m-d+i}{m+i} <\left(\frac{m-d}{m}\right)^2. \end{align} Equivalently, \begin{align*} &\frac{(m-d)^2-i^2}{(m-d)^2}<\frac{m^2-i^2}{m^2}, \mbox{ that is} \\[5pt] &\left(\frac{i}{m}\right)^2<\left(\frac{i}{m-d}\right)^2, \end{align*} which is true for $m>d>0$. Applying \eqref{ineq-key5} repeatedly with $m=n-\frac{k}{2}$ and $d=k$, we obtain \[ \frac{(n-k-1)(n-k-2)\ldots (n-2k+1)}{(n-1)(n-2)\ldots (n-k+1)} <\left(\frac{n-\frac{3k}{2}}{n-\frac{k}{2}}\right)^{k-1} \] and \eqref{ineq-key4} follows. \end{proof} \begin{cor}\label{prop-5.2} Let $n\geq 2k+1$, $k\geq 9$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. If $(\mathcal{F},\mathcal{G})$ forms a shift-resistant pair, then $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{cor} \begin{proof} By Theorem \ref{main} we may assume that $2k\leq n\leq 4k$. By \eqref{ineq-key4} and $k\geq 9$, it follows that \[ \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i} >\left(\frac{n-\frac{k}{2}}{n-\frac{3k}{2}}\right)^{k-1}\geq \left(\frac{4k-\frac{k}{2}}{4k-\frac{3k}{2}}\right)^8=\left(\frac{7}{5}\right)^8\approx 14.7579>f(0.64). \] By Proposition \ref{prop-5.3.0}, we conclude that $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{proof} \begin{cor}\label{prop-5.2.2} Let $2k+1 \leq n\leq 3.13k$, $k\geq 6$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. If $(\mathcal{F}, \mathcal{G})$ forms a shift-resistant pair, then $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{cor} \begin{proof} By \eqref{ineq-key4} and $k\geq 6$, it follows that \[ \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i} >\left(\frac{n-\frac{k}{2}}{n-\frac{3k}{2}}\right)^{k-1}\geq \left(\frac{3.13k-\frac{k}{2}}{3.13k-\frac{3k}{2}}\right)^5=\left(\frac{2.63}{1.63}\right)^5\approx 10.9356>f(0.64). \] By Proposition \ref{prop-5.3.0}, we conclude that $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{proof} Similarly, we can prove the same statement for $2k+1 \leq n\leq 3.54k$, $k=7$ or $2k+1 \leq n\leq 3.96k$, $k=8$. \begin{prop}\label{prop-5.4} Let $n\geq 3k$, $k\geq 14$ and $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ be non-trivial cross-intersecting. If both $\mathcal{F}$ and $\mathcal{G}$ are shifted, then \begin{align*} |\mathcal{F}||\mathcal{G}|\leq h(n,k)^2. \end{align*} \end{prop} \begin{proof} By Theorem \ref{main} we may assume that $3k\leq n\leq 4k$. Suppose that $|\mathcal{F}|\leq |\mathcal{G}|$. By \eqref{ineq-f781} we have $|\mathcal{F}|\leq \binom{n}{k-2}$. Set $\alpha=\frac{|\mathcal{F}|}{h(n,k)}$. Then \[ \frac{1}{\alpha}=\frac{h(n,k)}{|\mathcal{F}|} \geq \frac{\binom{n-1}{k-1}-\binom{n-k-1}{k-1}}{\binom{n}{k-2}}. \] Note that $3k\leq n\leq 4k$ and $k\geq 14$ imply \[ \frac{\binom{n-1}{k-1}}{\binom{n}{k-2}} = \frac{(n-k+2)(n-k+1)}{(k-1)n} \geq \frac{4}{3} \] and \begin{align*} \frac{\binom{n-k-1}{k-1}}{\binom{n}{k-2}} &=\frac{(n-k-1)(n-k-2)\ldots(n-2k+1)}{(k-1)n\ldots(n-k+3)}\\[5pt] &\leq \frac{n-k-1}{k-1} \left(\frac{n-k-2}{n}\right)^{k-2}\\[5pt] &\leq \left(3+\frac{2}{k-1}\right)\left(\frac{3}{4}\right)^{k-2}\\[5pt] &< 3.16\times\left(\frac{3}{4}\right)^{12}<\frac{1}{9}. \end{align*} It follows that $\frac{1}{\alpha} \geq \frac{4}{3}-\frac{1}{9}=\frac{11}{9}$, implying $\alpha \leq \frac{9}{11}$. Thus, \[ f(\alpha) \leq f\left(\frac{9}{11}\right)=50.5. \] By \eqref{ineq-key4} and $k\geq 14$, it follows that \[ \prod\limits_{1\leq i\leq k-1} \frac{n-i}{n-k-i} >\left(\frac{n-\frac{k}{2}}{n-\frac{3k}{2}}\right)^{k-1}\geq \left(\frac{4k-\frac{k}{2}}{4k-\frac{3k}{2}}\right)^{13}=\left(\frac{7}{5}\right)^{13}\approx 79.3917>f\left(\frac{9}{11}\right). \] By Lemma \ref{lem-5.2} the proposition follows. \end{proof} Applying Corollary \ref{prop-5.2}, Proposition \ref{prop-5.4} and repeating the proof of Theorem \ref{main}, we obtain the following result. \begin{thm}\label{main3} Suppose that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial cross-intersecting families, $n\geq 3k$, $k\geq 14$. Then $|\mathcal{F}||\mathcal{G}|\leq h(n,k)^2$. \end{thm} The next proposition rules out many potential constructions that may prevent Theorem \ref{main} holding for the full range. \begin{prop}\label{prop-5.3} Let $n\geq 2k\geq 4$. Suppose that $\mathcal{F},\mathcal{G}\subset\binom{[n]}{k}$, non-trivial, cross-intersecting. If $\mathcal{T}^{(2)}(\mathcal{F})\cap\mathcal{T}^{(2)}(\mathcal{G})\neq \emptyset$ then \begin{align}\label{ineq-prop-0} |\mathcal{F}|+|\mathcal{G}|\leq 2h(n,k). \end{align} \end{prop} \begin{proof} Without loss of generality, assume $(1,2)\cap H\neq \emptyset$ for all $H\in \mathcal{F}\cup \mathcal{G}$. Obviously, \begin{align}\label{ineq-prop-3} |\mathcal{F}([2])|+|\mathcal{G}([2])|\leq 2\binom{n-2}{k-2}. \end{align} Note that $\mathcal{F}(1,\bar{2})$, $\mathcal{G}(1,\bar{2})$, $\mathcal{F}(\bar{1},2)$ and $\mathcal{G}(\bar{1},2)$ are non-empty by non-triviality. $\mathcal{F}(1,\bar{2})$ and $\mathcal{G}(\bar{1},2)$ are cross-intersecting $(k-1)$-graphs on $[3,n]$. Consequently, by a classical result of Hilton and Milner \cite{HM67} concerning non-empty cross-intersecting families we obtain \begin{align}\label{ineq-prop-1} |\mathcal{F}(1,\bar{2})|+|\mathcal{G}(\bar{1},2)|\leq\binom{n-2}{k-1}-\binom{n-k-1}{k-1}+1. \end{align} The same is true for $\mathcal{F}(\bar{1},2)$ and $\mathcal{G}(1,\bar{2})$: \begin{align}\label{ineq-prop-2} |\mathcal{F}(\bar{1},2)|+|\mathcal{G}(1,\bar{2})|\leq \binom{n-2}{k-1}-\binom{n-k-1}{k-1}+1. \end{align} Summing up \eqref{ineq-prop-3}, \eqref{ineq-prop-1} and \eqref{ineq-prop-2} yields \[ |\mathcal{F}|+|\mathcal{G}|\leq 2\left(\binom{n-2}{k-2}+\binom{n-2}{k-1}-\binom{n-k-1}{k-1}+1\right)=2h(n,k). \] \end{proof} Let us mention that the case $k=2$, that is, if $\mathcal{F}$ and $\mathcal{G}$ are non-trivial cross-intersecting graphs then $|\mathcal{F}||\mathcal{G}|\leq 9$ is easy to prove. If $\mathcal{F}$ and $\mathcal{G}$ share an edge then $|\mathcal{F}|+|\mathcal{G}|\leq 6$ follows from Proposition \ref{prop-5.1} and thereby $|\mathcal{F}||\mathcal{G}|\leq 9$. Arguing indirectly assume $|\mathcal{F}|\geq 4$ and let $(x,y)$ be an edge of $\mathcal{G}$. Non-triviality of $\mathcal{G}$ implies that both $x$ and $y$ have degree two in $\mathcal{F}$. Let $u$ and $v$ be the neighbors of $x$ in $\mathcal{F}$. Then the only candidate for an edge in $\mathcal{G}$ that does not contain $x$ is $(u,v)$ whence $(u,v)\in\mathcal{G}$. By cross-intersection the two neighbors of $y$ in $\mathcal{F}$ must be $u$ and $v$ as well. Now cross-intersection implies $\mathcal{G}=\{(x,y),(u,v)\}$ whence $|\mathcal{F}||\mathcal{G}|=8<9$. {\bf\noindent Remark.} There are many cases of equality in \eqref{ineq-prop-0}. Namely, let $A,B\in \binom{[3,n]}{k-1}$. Set \[ \mathcal{F}(1,\bar{2}) = \{ A\},\ \mathcal{G}(\bar{1},2)=\left\{G\in \binom{[3,n]}{k-1}\colon G\cap A\neq\emptyset\right\}. \] There are two possibilities for $B$. \begin{itemize} \item[(a)] $\mathcal{F}(\bar{1},2) = \{B\},\ \mathcal{G}(1,\bar{2})=\left\{G\in \binom{[n]}{k-1}\colon G\cap B\neq\emptyset\right\}$, \item[(b)] $\mathcal{F}(\bar{1},2) = \left\{F\in \binom{[n]}{k-1}\colon F\cap B\neq\emptyset\right\},\ \mathcal{G}(1,\bar{2})=\{B\}$. \end{itemize} Add $\mathcal{F}(1,2)=\mathcal{G}(1,2)=\binom{[3,n]}{k-2}$. Clearly $|\mathcal{F}||\mathcal{G}|=h(n,k)^2$ holds only in case (b). It is natural to consider the analogous product version of the more general Hilton-Milner-Frankl Theorem. To state this result let us define two types of families \begin{align*} &\mathcal{H}(n,k,t) =\left\{H\in \binom{[n]}{k}\colon [t]\subset H, H\cap [t+1,k+1]\neq \emptyset\right\}\cup \left\{[k+1]\setminus \{j\}\colon 1\leq j\leq t\right\},\\[5pt] &\mathcal{A}(n,k,t) =\left\{A\in \binom{[n]}{k}\colon |A\cap [t+2]|\geq t+1\right\}. \end{align*} \begin{thm} Suppose that $\mathcal{F}\subset \binom{[n]}{k}$ is non-trivial and $t$-intersecting, $n>(k-t+1)(t+1)$. Then \begin{align}\label{thm-5.1} |\mathcal{F}| \leq \max\left\{|\mathcal{H}(n,k,t)|,|\mathcal{A}(n,k,t)|\right\}. \end{align} \end{thm} For $t=1$, $|\mathcal{H}(n,k,t)|\geq |\mathcal{A}(n,k,t)|$ and \eqref{thm-5.1} reduces to the Hilton-Milner Theorem. For $t=2$ and $n>n_0(k,t)$ it was proved in \cite{F782}. For $t\geq 20$ and all $n>(k-t+1)(t+1)$ it follows from \cite{FFuredi2}, however it was first proved in full generality by \cite{AK}. Let us conclude this paper by announcing the corresponding product version. \begin{thm}[\cite{FW2022-2}] Suppose that $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial and cross $t$-intersecting, $n\geq4(t+2)^2k^2$, $k\geq 5$. Then \begin{align}\label{thm-5.2} |\mathcal{F}||\mathcal{G}| \leq \max\left\{|\mathcal{H}(n,k,t)|^2,|\mathcal{A}(n,k,t)|^2\right\}. \end{align} \end{thm} \end{document}

\begin{document} \title{Tilted Cone and Cylinder, Cone and Tilted Sphere} \author{Mehmet Kirdar} \maketitle \begin{abstract} In this note, we will consider two classical volume problems related to elliptic integrals. The first problem has a neat formula by means of elliptic integrals. We remade it with details. In the second problem, we found a messy formula. On the other hand, it seems to be useful to find a good approximation for the volume. \textit{Key Words. Cone, cylinder, sphere, elliptic, integral.} \textit{Mathematics Subject Classification. [2020] 51M25, 33E05.} \end{abstract} \section{Introduction} In this note, I discuss two classical volume problems. The first problem which, I saw in [2], dates back to 1932, has a neat solution formula by means of elliptic integrals. I reproduced the formula for the case $k<1$ with some details for elliptic integrals. There is a key identity which also appeared in the second problem but is not available in [2]. I believe that Rhodes did this computations somewhere else. His purpose in this article was Landen transformations but I believe that they are sometimes complications. The solution for the second problem I found is not very neat. I used Maclaurin's series expansions of elliptic integrals of the first kind and the second kind and wrote the solution as an infinite series of trigonometric integrals. It seems to be useful to find a good approximation for the volume. I do not know whether this formula was known before. I have not seen. I must also mention the beautiful book of Harris Hancock, [1], which helped me to understand the tricky identities about elliptic integrals. WolframAlpha helped me a lot during my research. Its abilities are amazing. \section{Tilted cone and cylinder} Consider the cylinder\textbf{\ }$x^{2}+y^{2}=1$ and the cone $z=\cot \alpha \sqrt{(x-k)^{2}+y^{2}}$, $0\leq k\leq 1.$ We want to find the volume of the bounded region inside the cylinder, under the cone and above $z=0$. Here $ \alpha $ is the fixed angle of the cone, the angle between the cone and its axis, $0\leq \alpha \leq \frac{\pi }{2}.$ Let the origin be $O=(0,0,0),$ the vertex of the cone be $T=(k,0,0)$ and let $P=(\cos \theta ,\sin \theta ,0)$, $0\leq \theta \leq 2\pi ,$ be a point one the unit circle of the $xy$-plane. Let the angle between $TP$ and positive side of the $x$-axis be\ $\phi $, $0\leq \phi \leq 2\pi $. See [2] for some figures about this problem. If $TP=R$ then by law of cosines, $R=\sqrt{ 1-k^{2}\sin ^{2}\phi }-k\cos \phi $. The perpendicular from $P$ to $xy$ -plane cuts the cone with height $R\cot \alpha $. Therefore, the parameterization of the region in tilted cylindrical coordinates is $0\leq r\leq R,0\leq \phi \leq 2\pi $ and $0\leq z\leq r\cot \alpha $. And in tilted coordinates volume differential is $dV=rdrd\phi dz$. With two successive integrations, the volume integral can be reduced to $V=\dfrac{ 2\cot \alpha }{3}\dint\limits_{0}^{\pi }R^{3}d\phi $. Now, by putting $R^{3}$ and observing that \[ \dint\limits_{0}^{\pi }(-k^{3}\cos ^{3}\phi -3k\cos \phi +3k^{3}\allowbreak \cos \phi \sin ^{2}\phi )d\phi =\allowbreak 0 \] we have \[ V=\dfrac{4\cot \alpha }{3}\dint\limits_{0}^{\dfrac{\pi }{2} }(3k^{2}+1-4k^{2}\sin ^{2}\phi )\sqrt{1-k^{2}\sin ^{2}\phi }d\phi . \] By the definition of the elliptic integral of the second kind $E(k)$, $V$ becomes \[ V=\dfrac{4(3k^{2}+1)\cot \alpha }{3}E(k)-\dfrac{16k^{2}\cot \alpha }{3} \dint\limits_{0}^{\dfrac{\pi }{2}}\sin ^{2}\phi \sqrt{1-k^{2}\sin ^{2}\phi } d\phi . \] Next, $E_{2}(k)=$ $\dint\limits_{0}^{\dfrac{\pi }{2}}\sin ^{2}\phi \sqrt{ 1-k^{2}\sin ^{2}\phi }d\phi $ must be computed in terms of elliptic integrals. Let $\Delta =\sqrt{1-k^{2}\sin ^{2}\phi }$, $S=\sin \phi $ and $ C=\cos \phi $. The tricky identity is \[ S^{2}\Delta =\frac{2k^{2}-1}{3k^{2}}\Delta +\frac{1-k^{2}}{3k^{2}}\dfrac{1}{ \Delta }+\left[ -\frac{1}{3}(1-2S^{2})\Delta +\frac{k^{2}}{3}S^{2}C^{2}\frac{ 1}{\Delta }\right] . \] Integrating from $0$ to $\dfrac{\pi }{2}$, since \[ \dint\limits_{0}^{\pi /2}\left[ -\frac{1}{3}(1-2S^{2})\Delta +\frac{k^{2}}{3} S^{2}C^{2}\frac{1}{\Delta }\right] d\phi =\left[ -\frac{1}{3}SC\Delta \right] _{0}^{\pi /2}=0, \] we find \[ E_{2}(k)=\frac{2k^{2}-1}{3k^{2}}E(k)+\frac{1-k^{2}}{3k^{2}}K(k)\text{ } (\star ) \] and \[ V=\frac{4}{9}\cot \alpha \left[ (k^{2}+7)E(k)+4(k^{2}-1)K(k)\right] \] where $K(k)=\dint\limits_{0}^{\dfrac{\pi }{2}}\left( 1-k^{2}\sin ^{2}\phi \right) ^{-\frac{1}{2}}d\phi .$ This formula is obtained in [2]. There, formula for $k>1$ case is also obtained and then they are combined with a Landen transformation interpretation. \section{Cone and tilted sphere} Let us find the volume of the bounded region between the tilted sphere $ (x+k)^{2}+y^{2}+z^{2}=1,$ $0\leq k\leq 1$ and the cone $z=\cot \alpha \sqrt{ x^{2}+y^{2}}.$The set-up of the volume integral is easier than that of the first problem. So, we can skip figures. The sphere in spherical coordinates is $\rho ^{2}+2k\rho \cos \theta \sin \phi +k^{2}-1=0$ and the cone is $\phi =\alpha $. Thus, the volume of the region \[ 0\leq \rho \leq -k\cos \theta \sin \phi +\sqrt{1-k^{2}+k^{2}\cos ^{2}\theta \sin ^{2}\phi },\text{ }0\leq \theta \leq 2\pi ,\text{ }0\leq \phi \leq \alpha \] is found as \[ V=\dint\limits_{0}^{2\pi }\dint\limits_{0}^{\alpha }\left\{ \begin{array}{c} (k^{3}\cos \theta \sin ^{2}\phi -k\cos \theta \sin ^{2}\phi -\frac{4}{3} k^{3}\allowbreak \cos ^{3}\theta \sin ^{4}\phi )+ \\ \left( \frac{4}{3}k^{2}\cos ^{2}\theta \sin ^{3}\phi +\frac{1-k^{2}}{3}\sin \phi \right) \sqrt{1-k^{2}+k^{2}\cos ^{2}\theta \sin ^{2}\phi } \end{array} \right\} d\phi d\theta . \] Since \[ \dint\limits_{0}^{2\pi }(k^{3}\cos \theta \sin ^{2}\phi -k\cos \theta \sin ^{2}\phi -\frac{4}{3}k^{3}\allowbreak \cos ^{3}\theta \sin ^{4}\phi )d\theta =\allowbreak 0 \] and due to symmetry, we find \[ V=\dint\limits_{0}^{\alpha }\dint\limits_{0}^{\dfrac{\pi }{2}}\left( \frac{16 }{3}k^{2}\cos ^{2}\theta \sin ^{3}\phi +\frac{4}{3}(1-k^{2})\sin \phi \right) \sqrt{1-k^{2}+k^{2}\cos ^{2}\theta \sin ^{2}\phi }d\theta d\phi . \] Let us define \[ K=\dfrac{k\sin \phi }{\sqrt{1-k^{2}\cos ^{2}\phi }} \] and thus, \begin{eqnarray*} V &=&\dint\limits_{0}^{\alpha }\left( \frac{16}{3}k^{2}\sin ^{3}\phi +\frac{4 }{3}(1-k^{2})\sin \phi \right) \sqrt{1-k^{2}\cos ^{2}\phi }E(K)d\phi \\ &&-\dint\limits_{0}^{\alpha }\text{ }\frac{16}{3}k^{2}\sin ^{3}\phi \sqrt{ 1-k^{2}\cos ^{2}\phi }\left( \dint\limits_{0}^{\dfrac{\pi }{2}}\sin ^{2}\theta \sqrt{1-K^{2}\sin ^{2}\theta }d\theta \right) d\phi . \end{eqnarray*} By using the star identity, $(\star )$ of the first problem, it can be written as \[ V=\frac{4}{9}\dint\limits_{0}^{\alpha }(8k^{2}\sin ^{3}\phi +7(1-k^{2})\sin \phi )\sqrt{1-k^{2}\cos ^{2}\phi }E(K)d\phi -\frac{16}{9}\dint\limits_{0}^{ \alpha }\sin \phi \sqrt{1-k^{2}\cos ^{2}\phi }K(K)d\phi . \] We can now insert infinite series of $E(K)$ and $K(K)$ and do term by term integration to obtain a formula which involves trigonometric integrals. Let us recall that $E(K)=\dfrac{\pi }{2}\dsum\limits_{n=0}\dfrac{c_{n}}{1-2n} K^{2n}$ and $K(K)=\dfrac{\pi }{2}\dsum\limits_{n=0}c_{n}K^{2n}$ where $ c_{n}=\left( \dfrac{(2n)!}{2^{2n}(n!)^{2}}\right) ^{2}.$ Putting these and $ K $ in the last equation we obtain \[ V=\frac{2\pi }{9}\dsum\limits_{n=0}^{\infty }\frac{c_{n}k^{2n}}{1-2n} \dint\limits_{0}^{\alpha }\frac{8k^{2}\sin ^{2n+3}\phi +(3-7k^{2}+8n)\sin ^{2n+1}\phi }{(1-k^{2}\cos ^{2}\phi )^{n-\frac{1}{2}}}d\phi . \] The zeroth term of the series gives the following approximation of the volume for small $k$: \[ \frac{2\pi }{9}\left( (1+2k^{2})\sqrt{1-k^{2}}-\cos \alpha (1+4k^{2}-2k^{2}\cos \alpha )\sqrt{1-k^{2}\cos ^{2}\alpha }+(2-3k^{2})\frac{ \arcsin k-\arcsin (k\cos \alpha )}{k}\right) . \] This gives the exact value $\dfrac{2\pi }{3}(1-\cos \alpha )$ in the limit $ k\rightarrow 0.$ \end{document}

\begin{document} \pagestyle{plain} \title{A Fourier-Chebyshev Spectral Method for \\ Cavitation Computation in Nonlinear Elasticity \thanks{The research was supported by the NSFC projects 11171008 and 11571022.}} \author{Liang Wei, \hspace{1mm} Zhiping Li\thanks{Corresponding author, email: [email protected]} \\ {\small LMAM \& School of Mathematical Sciences, Peking University, Beijing 100871, China}} \date{} \maketitle \begin{abstract} A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions. \end{abstract} \noindent \textbf{Key words}: Fourier-Chebyshev spectral method, cavitation, nonlinear elasticity, interpolation error analysis, energy error estimate, convergence. \section{Introduction} In 1958, Gent and Lindley \cite{GentLindley1958} established the well known defective model for the cavitation in nonlinear elasticity characterizing the phenomenon as material instability associated to the dramatic growth of pre-existing micro voids under large hydrostatic tensions, which very well matched the experimental observation of sudden void formation in vulcanized rubber. Using the defective model, Gent et.al. \cite{GentPark1984}, Lazzeri et.al. \cite{Bucknall1993,Bucknall1994}, and many other researchers studied the cavitation phenomenon in elastomers containing rigid spherical inclusions as well as in the standard model problems. In 1982, Ball \cite{Ball1982} established the famous perfect model, in which cavitations form in an originally intact body as an absolute energy minimizing bifurcation solution, and produced the same cavitation criterion. The profound relationship of the two models are studied by Sivaloganathan et.al. \cite{Sivaloganathan2002,Sivaloganathan2006} and Henao \cite{Henao2009}. Since the perfect model is known to be seriously challenged by the Lavrentiev phenomenon \cite{Lavrentiv}, the defective model is chosen by most researchers in numerical studies of the cavitation phenomenon, using mainly a variety of the finite element methods (see Xu and Henao \cite{XuHenao}, Lian and Li \cite{LianLi2011dual,LianLi2011force}, Su and Li \cite{SuLi} among many others). A spectral collocation method \cite{NegronMarrero}, which approximates the cavitation solution with truncated Fourier series in the circumferential direction and finite differences in the radial direction, is also found some success. In a typical 2-dimensional defective model with a prescribed displacement boundary condition, one considers to minimize the stored energy of the form \begin{align} \label{note:E} E( \mathbf{u} ) = \int_{\Omega_\varepsilon} W( \nabla \mathbf{u} (\mathbf{x}) ) \mathrm{d} \mathbf{x} , \end{align} in the set of admissible deformations \begin{align} \label{note:A_epsl} \mathcal{A}_\varepsilon = \left\{ \mathbf{u} \in W^{1,p}( \Omega_\varepsilon ) : \mathbf{u}~\text{is one-to-one}~ a.e.,~ \mathbf{u} \left|_{\partial \Omega} \right.=\mathbf{u}_0,~ \det \nabla \mathbf{u} > 0 ~a.e. \right\} , \end{align} where $\Omega_\varepsilon = \Omega \setminus \overline{\mathbb{B}_\varepsilon (\mathbf{x}_0 )} \subset \mathbb{R}^2$ is a domain occupied by the compressible hyperelastic material in its reference configuration, with $\Omega$ being a regular simply-connected domain and $\mathbb{B}_\varepsilon ( \mathbf{x}_0 ) =\{\mathbf{x} \in \mathbb{R}^2: |\mathbf{x}|< \varepsilon \}$ being a pre-existing circular defect of radius $\varepsilon \ll 1$ centered at $\mathbf{x}_0$, and where $W:\mathbb{M}_{+}^{2\times 2} \rightarrow \mathbb{R}^{+}$ is the stored energy density function of the hyperelastic material, and $\mathbb{M}_{+}^{2\times 2}$ denotes the set of $2\times 2$ matrices with positive determinant. The Euler-Lagrange equation of the above minimization problem is the following displacement/traction boundary value problem: \begin{equation} \label{EL:original} \begin{cases} \mathrm{div} \displaystyle \frac{\partial W (\nabla \mathbf{u})}{\partial \nabla \mathbf{u}} = \mathbf{0}, & \text{in}~ \Omega_\varepsilon ; \vspace*{3mm} \\ \displaystyle \frac{\partial W (\nabla \mathbf{u})}{\partial \nabla \mathbf{u}} \cdot \mathbf{n} = \mathbf{0}, &\text{on}~ \partial \mathbb{B}_\varepsilon( \mathbf{x}_0 );\vspace*{1mm} \\ \mathbf{u} (\mathbf{x}) = \mathbf{u}_0 (\mathbf{x}), & \text{on}~\partial \Omega, \end{cases} \end{equation} where $\mathbf{n}$ is the unit exterior normal with respect to $\Omega_\varepsilon$. In the present paper, without loss of generality \cite{Ball1982,PietroStefania2013}, we consider the stored energy density function $W(\cdot)$ of the form \begin{equation} \label{note:W} W( \nabla \mathbf{u} ) = \kappa | \nabla \mathbf{u} |^p + h( \det \nabla \mathbf{u} ), \qquad \nabla\mathbf{u}\in \mathbb{M}_{+}^{2\times 2},~ 1<p<2 , \end{equation} where $\kappa$ is a positive material constant, $|\cdot |$ denotes the Frobenius norm of a matrix and $h \in \mathcal{C}^3( (0,+\infty) )$ is a strictly convex function satisfying \begin{equation} \label{note:h} h(t)\rightarrow +\infty ~\text{as}~ t \rightarrow 0^+, ~\text{and}~ \frac{h(t)}{t} \rightarrow +\infty ~\text{as}~ t \rightarrow +\infty . \end{equation} Since the cavitation solution is generally considered to have high regularity except in a neighborhood of the defects, where the material experiences large expansion dominant deformations, we restrict ourselves to a simplified reference configuration $\Omega_{(\varepsilon,\gamma)} = \mathbb{B}_\gamma( \mathbf{0}) \setminus \overline{\mathbb{B}_\varepsilon( \mathbf{0}) }$ $(0<\varepsilon \ll \gamma \leq 1)$, and denote \begin{equation}\begin{split} \label{note:A_epsl_gamma} \mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 ) = \{ \mathbf{u} \in W^{1,p}( \Omega_{(\varepsilon,\gamma)} ) :~ \mathbf{u}~\text{is one-to-one}~ a.e.,~& \mathbf{u} \left|_{\partial \mathbb{B}_\gamma( \mathbf{0})} \right. = \mathbf{u}_0, \\ & \det \nabla \mathbf{u} > 0 ~a.e. \} . \end{split}\end{equation} Taking the advantages of the smoothness of the cavitation solutions in the defective model when $\mathbf{u}_0$ is sufficiently smooth and the high efficiency and accuracy of spectral methods in approximating smooth solutions of partial differential equations (see Li and Guo \cite{LiGuo}, Shen \cite{Shen,ShenTang} etc.), we develop a Fourier-Chebyshev spectral method to solve the Euler-Lagrange equation \eqref{EL:original}, which approximates the cavitation solution with truncated Fourier series in the circumferential direction and truncated Chebyshev series in the radial direction. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is derived, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing highly accurate numerical cavitation solutions. We would like to point out here, even though the reference domain is restricted to a circular ring $\Omega_{(\varepsilon,\gamma)}$, to further exploring its highly efficient feature in a neighborhood of a cavity surface, our method can be coupled with a domain decomposition method, especially in combining with some finite element methods to extend the application to more general situations with multiple pre-existing tiny voids. The structure of the rest of the paper is as follows. In \S 2, we rewrite the Euler-Lagrange equation of the cavitation problem in a proper computing coordinates. In \S 3, the Fourier-Chebyshev spectral method is applied, the corresponding discrete equilibrium equation is derived, and an algorithm to solve the nonlinear equation is presented. \S 4 is devoted to the analysis of the interpolation error of the cavitation solution, the elastic energy error bound and the convergence of the discrete cavitation solution. In \S 5, numerical experiments and results are presented to show the efficiency and accuracy of our method. \section{The Euler-Lagrange Equation} In the Cartesian coordinate system, an admissible deformation $\mathbf{u} \in \mathcal{A}_{(\varepsilon,\gamma)}(\mathbf{u}_0)$ is written as $\mathbf{u}( \mathbf{x} ) = \left[ u_1(x_1,x_2), u_2(x_1,x_2) \right]^T $. Denote \begin{align} \label{note:DFg} D(\mathbf{u}) := \det \nabla \mathbf{u}, \quad F(\mathbf{u}) := \frac{1}{2} |\nabla \mathbf{u}|^2, \quad g(t) := \kappa \left( \sqrt{2t} \right)^p, \end{align} and to futher simplify the notation, $D(\mathbf{u})$ and $F(\mathbf{u})$ will be denoted below as $D$, $F$ wherever no ambiguity is caused. For the elastic energy density function $W(\cdot)$ given by \eqref{note:W} and the elastic energy $E(\cdot)$ given by \eqref{note:E}, we have \begin{align} \label{note:E_u} E( \mathbf{u} ) = \int_{\Omega_{(\varepsilon,\gamma)}} \left[ g\left( F(\mathbf{u}) \right) + h\left( D(\mathbf{u}) \right) \right] \mathrm{d} \mathbf{x} . \end{align} For the convenience of the implementation of the Fourier-Chebyshev spectral method, we introduce a $(\rho,\phi)$-coordinate system defined on the computational domain $\Omega' :=(-1,1) \times (0,2\pi)$, by coupling the Cartesian to polar coordinates transformation \begin{equation} \begin{cases}\label{coordinate:polar} x_1 = r \cos \theta, \\ x_2 = r \sin \theta, \end{cases} \text{and} \quad \begin{cases} u_1 = R(r,\theta) \cos \Theta(r,\theta), \\ u_2 = R(r,\theta) \sin \Theta(r,\theta), \end{cases} \end{equation} defined on the domain $(\varepsilon,\gamma) \times (0,2\pi)$, with a transformation defined by \begin{equation} \begin{cases}\label{coordinate:rho_phi} r = \frac{\gamma+\varepsilon}{2}+\frac{\gamma-\varepsilon}{2} \rho,\\ \theta = \phi, \end{cases} \text{and} \quad \begin{cases} R(r,\theta) = P(\rho,\phi), \\ \Theta(r,\theta) = Q(\rho,\phi)+\phi, \end{cases} \end{equation} defined on the computational domain $\Omega' =(-1,1) \times (0,2\pi)$. In $(\rho,\phi)$-coordinates, $D(\mathbf{u})= \det \nabla \mathbf{u}$, $F(\mathbf{u})= |\nabla \mathbf{u}|^2 /2$ defined in \eqref{note:DFg} can be rewritten as functions of $P(\rho,\phi)$, $Q(\rho,\phi)$: \begin{subequations} \label{note:DF} \begin{align} \label{note:D} D(P,Q) &= \frac{\rho_r}{r} P \left[ P_\rho (Q_\phi+1) - P_\phi Q_\rho \right] ,\\ F(P,Q) &= \frac{\rho_r^2}{2} ( P_\rho^2 + P^2 Q_\rho^2 ) + \frac{1}{2 r^2} \left[ P_\phi^2 + P^2 (Q_\phi+1)^2 \right] , \end{align} \end{subequations} where $\rho_r = 2/(\gamma-\varepsilon)$; the elastic energy $E(\mathbf{u})$ in \eqref{note:E_u} can be expressed as \begin{align} \label{energy} E( P,Q ) = \int_{\Omega'} \left[ g\left( F(P,Q) \right) + h\left( D(P,Q) \right) \right] \frac{r}{\rho_r} \mathrm{d} \rho \mathrm{d} \phi, \end{align} and the set of admissible deformation $\mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$ (see \eqref{note:A_epsl_gamma}) is reformulated as \begin{equation}\begin{split} \mathcal{A}_{\Omega'}( \mathbf{u}_0 ) = \{ (P,Q):~ \exists~ \mathbf{u} \in \mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 ), \, & \text{{\it s.t.} } \mathbf{u}~\text{is mapped to}~(P,Q) \\ & \text{by the transformations \eqref{coordinate:polar} and \eqref{coordinate:rho_phi}} \}. \end{split}\end{equation} Thus, in $(\rho,\phi)$-coordinates, the cavitation solution $(P,Q)\in \mathcal{A}_{\Omega'}( \mathbf{u}_0 )$ is characterized as the minimizer of $E( P,Q )$ in $\mathcal{A}_{\Omega'}( \mathbf{u}_0 )$, {\it i.e.} \begin{align} \label{problem} (P,Q)=\arg\min_{ (P,Q)\in \mathcal{A}_{\Omega'}( \mathbf{u}_0 ) } E(P,Q), \end{align} or alternatively, as the solution to the Euler-Lagrange equation of \eqref{problem}: \begin{align} \label{EL:body} \left\{\begin{aligned} \int_{\Omega'} f_1(P,Q;\bar{P}) \mathrm{d} \rho \mathrm{d} \phi=0,\\ \int_{\Omega'} f_2(P,Q;\bar{Q}) \mathrm{d} \rho \mathrm{d} \phi=0, \end{aligned}\right. \quad \forall~(\bar{P},\bar{Q}) \in \mathcal{A}_{\Omega'}( \mathbf{0} ), \end{align} where, by the definition and direct calculations, we have \begin{align*} f_1 &:= \bar{P} \left[ g'(F)\left( r\rho_r P Q_\rho^2 + \frac{1}{r\rho_r} P (Q_\phi+1)^2 \right) + h'(D) \left( P_\rho (Q_\phi+1) - P_\phi Q_\rho \right) \right] \\ &\qquad + \bar{P}_\rho \left[ g'(F) r\rho_r P_\rho + h'(D) P (Q_\phi+1) \right] + \bar{P}_\phi \left[ g'(F) \frac{1}{r\rho_r} P_\phi - h'(D) P Q_\rho \right] ,\\ f_2 &:= \bar{Q}_\rho P \left[ g'(F) r\rho_r P Q_\rho - h'(D) P_\phi \right] + \bar{Q}_\phi P \left[ g'(F) \frac{1}{r\rho_r} P (Q_\phi+1) + h'(D) P_\rho \right]. \end{align*} \section{The Fourier-Chebyshev Spectral Method} To discretize the Euler-Lagrange equation \eqref{EL:body} defined on $\Omega' =(-1,1) \times (0,2\pi)$ in $(\rho,\phi)$-coordinates, we first approximate the unknowns $(P(\rho,\phi), Q(\rho,\phi))$ by the finite Fourier-Chebyshev polynomials: \begin{subequations} \label{note:PQ_NM} \begin{align} P^{NM}(\rho,\phi) &:= \sum_{j=0}^M \left( \sum_{k=0}^{N/2} \alpha_{k,j} \cos k\phi + \sum_{k=1}^{N/2-1} \beta_{k,j} \sin k\phi \right) T_j(\rho), \vspace*{2mm} \\ \label{note:Q_NM} Q^{NM}(\rho,\phi) &:= \sum_{j=0}^M \left( \sum_{k=0}^{N/2} \xi_{k,j} \cos k\phi + \sum_{k=1}^{N/2-1} \eta_{k,j} \sin k\phi \right) T_j(\rho), \end{align} \end{subequations} where $T_j$ is the Chebyshev polynomial of the first kind of degree $j$, defined as \[ T_j(x) = \cos(j \arccos x), \] with $T_0(x) = 1, T_1(x) = x$ and satisfying the recurrence relation \cite{ShenTang} \[ T_{j+1}(x) = 2x T_j(x) - T_{j-1}(x), \quad j\geq 1. \] \begin{rem} We use the trigonometric polynomials to approximate $Q=\Theta-\theta$ instead of $\Theta$ (see \eqref{coordinate:rho_phi} and \eqref{note:Q_NM}) so that the Gibbs phenomenon can be avoided (see e.g. \cite{Gibbs}), since the periodic extension of $Q=\Theta-\theta$ from $[0, 2\pi)$ to $\mathbb{R}^1$ is smooth, while that of $\Theta$ is a sawtooth function with jump discontinuities at $2k\pi$, $k=0,1,\dotsc$. \end{rem} The discretized problem of solving the Euler-Lagrange equation \eqref{EL:body} is then read as: find $(P^{NM},Q^{NM}) \in \mathcal{B}^{NM}$ such that \begin{align} \label{EL:final} \left\{\begin{aligned} \int_{\Omega'} f_1(P^{NM},Q^{NM};\bar{P}) \mathrm{d} \rho \mathrm{d} \phi=0,\\ \int_{\Omega'} f_2(P^{NM},Q^{NM};\bar{Q}) \mathrm{d} \rho \mathrm{d} \phi=0, \end{aligned}\right. \quad \forall~(\bar{P},\bar{Q}) \in \mathcal{B}^{NM}_0, \end{align} where $\mathcal{B}^{NM}$ and $\mathcal{B}^{NM}_0$ are the discrete trial and test function spaces defined as \begin{subequations} \label{note:space} \begin{align} \mathcal{B}^{NM} &:= \left\{ (P^{NM},Q^{NM}):\text{the Fourier-Chebyshev polynomials \eqref{note:PQ_NM} satisfying} \right. \nonumber\\ &\qquad \left. P^{NM}(1,\phi_n) = P_0(1,\phi_n),~ Q^{NM}(1,\phi_n) = Q_0(1,\phi_n),~ 0\leq n\leq N-1 \right\}, \label{note:B_NM} \\ \mathcal{B}^{NM}_0 &:= \left\{ (P^{NM},Q^{NM}):\text{the Fourier-Chebyshev polynomials \eqref{note:PQ_NM} satisfying} \right. \nonumber\\ &\qquad\qquad\qquad\qquad\;\; \left. P^{NM}(1,\phi_n) = 0,~ Q^{NM}(1,\phi_n) = 0,~ 0\leq n\leq N-1 \right\}, \end{align} \end{subequations} where, in \eqref{note:space}, $\phi_n = 2\pi n/N,~ 0\leq n\leq N-1$, and the Dirichlet boundary condition $(P_0(1,\phi_n), Q_0(1,\phi_n))$ is defined by $\mathbf{u}_0$ via the coordinates transformations \eqref{coordinate:polar} and \eqref{coordinate:rho_phi}. To solve the equation \eqref{EL:final} numerically, we need to replace the integrals in \eqref{EL:final} by proper numerical quadratures. Let $\left\{ \rho_{m'}, \omega^C_{m'} \right\}_{m'=0}^{M'}$ and $\left\{ \phi_{n'}, \omega^F_{n'} \right\}_{n'=0}^{N'-1}$ be the sets of Gauss-Chebyshev and Fourier quadrature nodes and weights respectively, {\it i.e.} \cite{ShenTang} \begin{align} \begin{array}{lll} \rho_{m'} = \cos \DF{(2m'+1) \pi}{2M'+2}, & \omega^C_{m'} = \DF{\pi}{M'+1}, & 0\leq m'\leq M', \\ \phi_{n'} = \DF{2\pi n'}{N'}, & \omega^F_{n'} = \DF{2\pi}{N'}, & 0\leq n'\leq N'-1, \end{array} \end{align} then we are led to the following discretized Euler-Lagrange equation: find $(P^{NM},Q^{NM}) \in \mathcal{B}^{NM}$ such that for all $(\bar{P},\bar{Q}) \in \mathcal{B}^{NM}_0$ \begin{align} \label{EL:final_D} \left\{\begin{aligned} \sum_{n'=0}^{N'-1} \sum_{m'=0}^{M'} f_1(P^{NM}(\rho_{m'}, \phi_{n'}),Q^{NM}(\rho_{m'}, \phi_{n'});\bar{P}(\rho_{m'}, \phi_{n'})) \sqrt{1-\rho^2_{m'}} \omega^C_{m'} \omega^F_{n'}=0,\\ \sum_{n'=0}^{N'-1} \sum_{m'=0}^{M'} f_2(P^{NM}(\rho_{m'}, \phi_{n'}),Q^{NM}(\rho_{m'}, \phi_{n'});\bar{Q}(\rho_{m'}, \phi_{n'})) \sqrt{1-\rho^2_{m'}} \omega^C_{m'} \omega^F_{n'}=0. \end{aligned}\right. \end{align} Let $\{a_k,b_k\}$ and $\{c_k,d_k\}$ be the discrete Fourier coefficients of $P_0(1,\phi)$ and $Q_0(1,\phi)$ respectively, then the boundary condition in \eqref{note:B_NM} can be expressed as \begin{align} \label{note:alpha_k0} \begin{array}{cc} \begin{aligned} \alpha_{k,0} &= -\sum_{j=1}^M \alpha_{k,j} + a_k,\\ \beta_{k,0} &= -\sum_{j=1}^M \beta_{k,j} + b_k, \end{aligned} & \begin{aligned} \xi_{k,0} &= -\sum_{j=1}^M \xi_{k,j} + c_k, \quad 0 \leq k \leq N/2 ,\\ \eta_{k,0} &= -\sum_{j=1}^M \eta_{k,j} + d_k, \quad 1 \leq k \leq N/2-1. \end{aligned} \end{array} \end{align} Noticing also that the following $N \times M$ functions \begin{equation} \label{note:basis} \left\{ \cos k\phi \cdot (T_{j}(\rho)-1) \right\}_{1\leq j\leq M}^{0\leq k\leq N/2} ,\quad \left\{ \sin k\phi \cdot (T_{j}(\rho)-1) \right\}_{1\leq j\leq M}^{1\leq k\leq N/2-1}, \end{equation} form a set of bases for $\mathcal{B}^{NM}_0$, we conclude that the discrete Euler-Lagrange equation \eqref{EL:final_D} consists of $2NM$ nonlinear algebraic equations, which, for the simplicity of the notations, will be denoted as $\mathbf{f}( \mathbf{y} ) = \mathbf{0}$, with $2NM$ unknowns $\mathbf{y}=$ $\left\{ \alpha_{k,j}, \xi_{k,j} \right\}_{1 \leq j \leq M}^{0 \leq k \leq N/2} \cup$ $\left\{ \beta_{k,j}, \eta_{k,j} \right\}_{1 \leq j \leq M}^{1 \leq k \leq N/2-1}$. Denote $E(\mathbf{y})$ as the discrete elastic energy defined by replacing the integral in $E(P^{NM},Q^{NM})$ (see \eqref{energy}) with the numerical quadrature, then $\mathbf{f}( \mathbf{y} )$ may be viewed as the gradient of the discrete elastic energy $E(\mathbf{y})$. In our numerical experiments, the discrete equilibrium equations $\mathbf{f}( \mathbf{y} ) = \mathbf{0}$, {\it i.e.} \eqref{EL:final_D}, are solved by an algorithm combined a gradient type method with a damped quasi-Newton method \cite{quasiNewton}. More specifically, we use a gradient type method, which calculates a descent direction of the energy and conducts a incomplete line search in each iteration, to provide an appropriate initial cavity deformation for a damped quasi-Newton method with Broyden's correction, which will then produce a reasonably accurate numerical cavity solution. The algorithm is summarized as follows, where the determinant of the deformation $(P^{NM},Q^{NM})$ corresponding to $\mathbf{y}$ is denoted as $D(\mathbf{y}) := D(P^{NM},Q^{NM})$ (see \eqref{note:D}). \vspace*{2mm} {\bf Algorithm:} \begin{description} \item[Step 1] Given $\mathbf{y}^G_0$, set $TOL = 10^{-1}$, compute $\mathbf{f}(\mathbf{y}^G_0)$ and $E(\mathbf{y}^G_0)$. \item[Step 2] If $TOL < 10^{-10}$, then output $\mathbf{y}^G_0$ and stop; else, set $t^G_{-1} = 1$ and $j:=0$. \item[Step 3] For $j\geq 0$, if $|\mathbf{f}(\mathbf{y}^G_j)| < TOL$, then go to Step 6; else, set $t^G_j = 4\cdot t^G_{j-1}$. \item[Step 4] Set $\mathbf{y}^G_{j+1} = \mathbf{y}^G_{j} - t^G_j \cdot \mathbf{f}(\mathbf{y}^G_j)$, compute $\mathbf{f}(\mathbf{y}^G_{j+1})$, $E(\mathbf{y}^G_{j+1})$ and $D(\mathbf{y}^G_{j+1})$. \item[Step 5] If $t^G_j < 10^{-16}$, then output $\mathbf{y}^G_j$ and stop; else if $E(\mathbf{y}^G_{j+1}) < E(\mathbf{y}^G_{j})$ and $D( \mathbf{y}^G_{j+1} ) >0$, then set $j:=j+1$ and go to Step 3; else, set $t^G_j := t^G_j /2$ and go to Step 4. \item[Step 6] Set $\mathbf{y}^N_0 := \mathbf{y}^G_j$, compute $\mathbf{f}(\mathbf{y}^N_0)$ and $\mathbf{B}_0 = [ \nabla \mathbf{f}( \mathbf{y}^N_0 ) ]^{-1}$, set $t^N_{-1} = 1$ and $k:=0$. \item[Step 7] For $k\geq 0$, if $|\mathbf{f}(\mathbf{y}^N_k)| < 10^{-10}$, then output $\mathbf{y}^N_k$ as the solution and stop; else, set $t^N_k = 4\cdot t^N_{k-1}$. \item[Step 8] Set $\mathbf{y}^N_{k+1} = \mathbf{y}^N_{k} - t^N_k \ cdot \mathbf{B}_k \cdot \mathbf{f}(\mathbf{y}^N_k)$, compute $\mathbf{f}(\mathbf{y}^N_{k+1})$ and $D(\mathbf{y}^N_{k+1})$. \item[Step 9] If $t^N_k < 10^{-16}$, then go to Step 2 with $\mathbf{y}^G_0 := \mathbf{y}^N_k$ and $TOL := TOL/10$; else if $|\mathbf{f}(\mathbf{y}^N_{k+1})| < |\mathbf{f}(\mathbf{y}^N_{k})|$ and $D(\mathbf{y}^N_{k+1}) >0$, then go to Step 10; else, set $t^N_k := t^N_k /2$ and go to Step 8. \item[Step 10] Compute $\mathbf{s}_k = \mathbf{y}^N_{k+1} - \mathbf{y}^N_k$, $\mathbf{z}_k = \mathbf{f}(\mathbf{y}^N_{k+1}) - \mathbf{f}(\mathbf{y}^N_k)$, and \begin{align} \mathbf{B}_{k+1} = \mathbf{B}_k + \frac{ \left( \mathbf{s}_k - \mathbf{B}_k \mathbf{z}_k \right) \mathbf{s}_k^T \mathbf{B}_k }{ \mathbf{s}_k^T \mathbf{B}_k \mathbf{z}_k}. \end{align} Set $k:=k+1$ and go to Step 7. \end{description} \section{Error Analysis and the Convergence Theorem} In this section, we analyze the interpolation error of the discrete Fourier-Chebyshev spectral method for the cavitation solutions, which will enable us to derive the elastic energy error estimate for the discrete cavitation solution, and prove the convergence of the method. Before analyzing the interpolation error of a cavitation solution, we first introduce some notations. Let $\mathcal{B} := \{ (P,Q) \in C^1(\overline{\Omega'}): P(1,\phi)=P_0(1,\phi),~ Q(1,\phi)=Q_0(1,\phi)\}$, let $\mathcal{B}_{+} := \{ (P,Q) \in \mathcal{B}: D(P,Q)>0\}$ (see \eqref{note:D}), and denote $\mathcal{B}^{NM}_+=\mathcal{B}^{NM}\cap \mathcal{B}_+$ (see \eqref{note:B_NM}). Let $\omega(\rho) := (1-\rho^2)^{-1/2}$, $\Lambda := (0,2\pi)$, $I := (-1,1)$, and recall that $(\rho,\phi) \in \Omega' = I \times \Lambda$. For given integers $\sigma \geq 0$ and $\mu \geq 0$, denote $H^\sigma_\omega(I) = \{ \psi: \norm{\psi}_{ H^\sigma_\omega(I) } < \infty \}$ the weighted Hilbert space with the norm defined as \begin{align*} \norm{\psi}_{H^\sigma_\omega(I)} = \left( \sum^\sigma_{j=0} \int_I \left| \frac{d^j \psi}{d \rho^j} \right|^2 \omega \mathrm{d} \rho \right)^{1/2}, \end{align*} and denote $H^\mu( \Lambda; H^\sigma_\omega(I) )= \{ v: \norm{v}_{ H^\mu( \Lambda; H^\sigma_\omega(I) ) } < \infty \}$ the Hilbert space equipped with the norm defined as \begin{align*} \norm{v}_{ H^\mu( \Lambda; H^\sigma_\omega(I) ) } = \left( \sum^\mu_{k=0} \int_\Lambda \norm{ \frac{\partial^k v}{\partial \phi^k} }_{H^\sigma_\omega(I)}^2 \mathrm{d} \phi \right)^{1/2}. \end{align*} \begin{defn} \label{defn:INM} Define the interpolation operator $I^{NM}: \mathcal{B} \rightarrow \mathcal{B}^{NM}$ as $$ [I^{NM} (P, Q)](\rho_m,\phi_n)=(P,Q)(\rho_m,\phi_n), \;\; \forall\; 0\leq n \leq N-1,~0\leq m\leq M, $$ where $\rho_m = \cos( m\pi / M )$, $\phi_n=2n \pi /N$. \end{defn} The interpolation operator $I^{NM}$ is shown to have the following error estimates (see Lemma~5 in \cite{LiGuo}). \begin{lem} \cite{LiGuo} \label{lem:Liguo} If $v \in H^{\beta}(\Lambda;H^\sigma_\omega(I)) \cap H^{\mu}(\Lambda;H^\alpha_\omega(I)) \cap H^{\mu'}(\Lambda;H^{\sigma'}_\omega(I)),~ 0\leq \alpha \leq \sigma,\sigma',~ 0\leq \beta \leq \mu,\mu',~ \sigma,\sigma' >\frac{1}{2} ~\text{and}~ \mu,\mu' >1$, then, there exists a constant $c>0$ independent of $P$, $Q$, $M$ and $N$, such that \begin{equation*}\begin{split} \norm{I^{NM}v-v}_{H^{\beta}(\Lambda;H^\alpha_\omega(I))} &\leq cM^{2\alpha-\sigma} \norm{v}_{H^{\beta}(\Lambda;H^\sigma_\omega(I))} + cN^{\beta-\mu} \norm{v}_{H^{\mu}(\Lambda;H^\alpha_\omega(I))} \\ &\qquad \qquad +cq(\beta) M^{2\alpha-\sigma'} N^{\beta -\mu'} \norm{v}_{H^{\mu'}(\Lambda;H^{\sigma'}_\omega(I))}, \end{split}\end{equation*} where $q(\beta)=0$ for $\beta>1$ and $q(\beta)=1$ for $\beta \leq 1$. \end{lem} \begin{thm} \label{thm:PQ} Let $(P,Q)\in \mathcal{B} \cap H^s( \Lambda; H^l_\omega(I) )$, $l>2$, $s>1$. Then there exists a constant $c>0$ independent of $P$, $Q$, $M$ and $N$, such that \begin{align*} \norm{I^{NM}P - P}_{H^{\beta}(\Lambda;H^\alpha_\omega(I))} &\leq c \|P\|_* \left( M^{2\alpha-l} + N^{\beta-s} \right),\\ \norm{I^{NM}Q - Q}_{H^{\beta}(\Lambda;H^\alpha_\omega(I))} &\leq c \|Q\|_* \left( M^{2\alpha-l} + N^{\beta-s} \right), \end{align*} where $\alpha,\beta=0,1$ and $\|\cdot\|_*$ is a norm defined by: \begin{align*} \|v\|_* &:= \max\left\{ \norm{v}_{H^\beta( \Lambda,H^l_\omega(I) ) }, \norm{v}_{H^s ( \Lambda,H^\alpha_\omega(I) ) }, \norm{v}_{H^s ( \Lambda,H^l_\omega(I) ) } \right\}. \end{align*} Furthermore, if $(P,Q)\in \mathcal{B} \cap H^{l}(\Lambda;H^{l}_\omega(I))$ with $l> 6$, then \begin{equation}\begin{split}\label{det_error} &\| D(I^{NM}P,I^{NM}Q) -D(P,Q) \|_{C(\overline{\Omega'})} \\ &\qquad \qquad \le c (\|P\|_{H^{l}(\Lambda;H^{l}_\omega(I))} +\|Q\|_{H^{l}(\Lambda;H^{l}_\omega(I))}) (M^{6-l} + N^{3-l}). \end{split}\end{equation} \end{thm} \begin{proof} The first half of the theorem is a direct consequence of Lemma \ref{lem:Liguo} by setting $\sigma=\sigma'=l$, $\mu=\mu'=s$, and taking $\alpha=0$ or $1$ and $\beta=0$ or $1$ respectively. By \eqref{note:D}, the error estimate \eqref{det_error} follows from Lemma~\ref{lem:Liguo} with $\sigma=\sigma'=\mu=\mu'=l$, $\alpha=\beta=3$, and the fact that $H^3_\omega(\Omega') \hookrightarrow H^3(\Omega') \hookrightarrow C^1(\overline{\Omega'})$. \end{proof} In what follows below, we always assume that, for a cavitation solution $(P,Q)$, the following hypotheses hold: \begin{description} \item[(H1)] $(P,Q)\in \mathcal{B}_+ \cap H^{l}(\Lambda;H^{l}_\omega(I))$ with $l> 6$ and $(P,Q)$ is the energy minimizer in $\mathcal{B}_+$. \item[(H2)] there exists constants $c_F>1$ and $c_D>1$ such that $F$ and $D$ (see \eqref{note:DF}) satisfies \begin{align*} c_F^{-1} \leq 2r^2 F \leq c_F, \quad c_D^{-1} \leq D \leq c_D, \quad \text{on}~ \overline{\Omega'}. \end{align*} \end{description} \begin{rem} Notice that, by \eqref{note:DF} \begin{subequations} \label{note:rDrF} \begin{align} rD & = \rho_r P \left[ P_\rho (Q_\phi+1) - P_\phi Q_\rho \right] ,\\ 2r^2F & = r^2 \rho_r^2 ( P_\rho^2 + P^2 Q_\rho^2 ) + P_\phi^2 + P^2 (Q_\phi+1)^2, \end{align} \end{subequations} and for a cavitation solution $P\ge P_0>0$, and in the radially symmetric case $Q_{\phi}=0$. the hypothesis $c_F^{-1} \leq 2r^2 F$ is not too harsh a requirement on a general solution. While the other bounds are the direct consequences of $(P,Q) \in C^1(\overline{\Omega'})$. \end{rem} To estimate the error on the elastic energy of the interpolation function of a cavitation solution, we will making use of an auxiliary grid in radial direction on which the elastic energy of the cavitation solution is radially quasi-equi-distributed in the sense given in Lemma~\ref{lem:grid}. The properties of such grids are given by the following two lemmas. \begin{lem} \label{lem:energy} Let $(P,Q)\in \mathcal{B}_+$. Let $D$ and $F$ be defined by \eqref{note:DF}. Then, there exists a constant $c\geq 1$ such that, for all grid $\varepsilon = r_0 < r_1 < \cdots < r_K = \gamma$, the elastic energy \begin{align*} E_{(a,b)} := \int_{\Omega_{(a,b)}} \left[ g(F) + h(D)\right] r \mathrm{d} r \mathrm{d} \theta = \int_a^b \int_0^{2\pi} \left[ \kappa r^{1-p} \cdot \left( 2r^2 F \right)^{p/2} + r h \left( D \right) \right] \mathrm{d} r \mathrm{d} \theta \end{align*} satisfies \begin{align} \label{energy_bound} c^{-1} r_i^{1-p} \tau_i \leq E_{(r_{i-1},r_i)} \leq c r_{i-1}^{1-p} \tau_i, \quad 1\leq i \leq K. \end{align} \end{lem} \begin{proof} It follows from the convexity of $h(\cdot)$ and the hypothesis (H1) that $$ 0 < h(D) \leq \max \left\{ h(c_D^{-1}), h(c_D) \right\} \triangleq c_h, \quad \text{on}~ \overline{\Omega'}. $$ Since $r_{i-1} < \gamma \leq 1$ and $1< p <2$, the hypothesis (H2) implies \begin{align*} E_{(r_{i-1},r_i)} &\leq 2\pi \kappa c_F^{p/2} \int_{r_{i-1}}^{r_i} r^{1-p} \mathrm{d} r + 2\pi c_h \int_{r_{i-1}}^{r_i} r \mathrm{d} r \\ &\leq 2\pi \kappa c_F^{p/2} \cdot r_{i-1}^{1-p} \tau_i + 2\pi c_h \cdot r_{i-1} \tau_i \\ &\leq 2\pi \left( \kappa c_F^{p/2} + c_h \right) \cdot r_{i-1}^{1-p} \tau_i, \\ E_{(r_{i-1},r_i)} &\geq 2\pi \kappa c_F^{-p/2} \int_{r_{i-1}}^{r_i} r^{1-p} \mathrm{d} r \geq 2\pi \kappa c_F^{-p/2} \cdot r_i^{1-p} \tau_i. \end{align*} Hence, the conclusion \eqref{energy_bound} follows by taking $$ c = \max \left\{ 1, 2\pi \left( \kappa c_F^{p/2} + c_h \right), \left( 2\pi \kappa c_F^{-p/2} \right)^{-1} \right\}. $$ \end{proof} \begin{lem} \label{lem:grid} Let $K \gg \varepsilon^{-1}$ be a sufficiently large integer. Let $\varepsilon = r_0 < r_1 < \cdots < r_K = \gamma$ be a given grid satisfying \begin{align}\label{energy_equal} 2r_{K-1}>r_{K} \;\;\;\; \text{and} \;\;\;\; r_{i-1}^{1-p} \tau_i = r_{i}^{1-p} \tau_{i+1}, \quad 1\leq i \leq K-1, \end{align} where $\tau_i := r_i - r_{i-1}$, $1\leq i\leq K$. Then, we have \begin{equation} \label{ineq:rtau} \sum_{i=1}^{K} \frac{1}{\tau_i} \left( \frac{r_i}{r_{i-1}} \right)^{2p-2} < 2^{2p} \gamma^{-1} K^2, \end{equation} and the energy is radially quasi-equi-distributed on the grid, {\it i.e.} \begin{equation} \label{ineq:KE} 2^{1-p} c^{-2}<\frac{ K \cdot E_{(r_{i-1},r_i)} }{ E_{(\varepsilon,\gamma)} } < 2^{p-1} c^2, \quad \forall i=1,\dotsc,K, \end{equation} where $c$ is the same constant in Lemma~\ref{lem:energy}. \end{lem} \begin{proof} By \eqref{energy_equal}, we have \begin{align} \label{eq:upsilon} \frac{\tau_{i+1}}{\tau_i} = \left( \frac{r_i}{r_{i-1}} \right)^{p-1} = \left( 1+ \frac{\tau_i}{r_{i-1}} \right)^{p-1} = (1+ \Upsilon_i)^{p-1} , \quad 1\leq i \leq K-1, \end{align} where denote $\Upsilon_i := \frac{\tau_i}{r_{i-1}} >0$, $1\leq i\leq K-1$. Notice that, by definition, \begin{equation}\label{e:Upsilon>} \Upsilon_{i+1} = \frac{\tau_{i+1}}{r_{i}} = \frac{\tau_i (1+ \Upsilon_i)^{p-1}}{r_{i-1}+ \tau_i} = \frac{ \frac{\tau_i}{r_{i-1}} (1+ \Upsilon_i)^{p-1}}{1+ \frac{\tau_i}{r_{i-1}}} = \Upsilon_i (1+ \Upsilon_i)^{p-2} < \Upsilon_i, \end{equation} {\it i.e.} $\Upsilon_i$ is a strictly deceasing function of $i$. On the other hand, $\frac{\tau_{i+1}}{\tau_i}=(1+ \Upsilon_i)^{p-1} >1$ implies that $\tau_i$ is a strictly increasing function of $i$, and as a consequence, we have $K \Upsilon_1 \varepsilon = K \tau_1 < \sum_{i=1}^K \tau_i = \gamma - \varepsilon < K \tau_K$, which yields, since $K \gg \varepsilon^{-1}$, \begin{equation}\label{e:Upsilon_1} \Upsilon_1 < \frac{\gamma-\varepsilon}{K\varepsilon} \ll 1, \;\; \text{and} \;\; \tau_K > \frac{\gamma-\varepsilon}{K}. \end{equation} By \eqref{eq:upsilon} and \eqref{e:Upsilon>}, we also have, for all $1\leq i\leq K-1$, \begin{align*} \frac{1}{\tau_i} = \frac{1}{\tau_{i+1}} (1+ \Upsilon_i)^{p-1} < \frac{1}{\tau_{i+1}} (1+ \Upsilon_{1})^{p-1} < \cdots < \frac{1}{\tau_{K}} (1+ \Upsilon_{1})^{(p-1)(K-i)}. \end{align*} Now, express the left-hand side of \eqref{ineq:rtau} as \begin{equation} \label{eq:taur} \sum_{i=1}^{K} \frac{1}{\tau_i} \left( \frac{r_i}{r_{i-1}} \right)^{2p-2} = \sum_{i=1}^{K-1} \frac{1}{\tau_i} (1+ \Upsilon_i)^{2p-2} + \frac{1}{\tau_K} \left( \frac{r_K}{r_{K-1}} \right)^{2p-2}. \end{equation} For the first term on the right hand side of \eqref{eq:taur}, by \eqref{eq:upsilon}, \eqref{e:Upsilon>} and \eqref{e:Upsilon_1}, we have \begin{align*} \sum_{i=1}^{K-1} \frac{1}{\tau_i} (1+ \Upsilon_i)^{2p-2} &< (1+ \Upsilon_1)^{2p-2} \sum_{i=1}^{K-1} \frac{1}{\tau_i} < (1+ \Upsilon_1)^{2p-2} \frac{1}{\tau_K} \sum_{i=1}^{K-1} (1+ \Upsilon_{1})^{(p-1)(K-i)} \\ &= (1+ \Upsilon_1)^{2p-2} \frac{1}{\tau_K} \left[ \frac{ 1- (1+ \Upsilon_1)^{(p-1)K} }{ 1- (1+ \Upsilon_1)^{p-1} } -1\right] < \frac{2^{2p-2} (2K-1)}{\tau_K} . \end{align*} Since $r_K/r_{K-1}<2$ (see \eqref{energy_equal}), by \eqref{e:Upsilon_1} and \eqref{eq:taur}, this yields \eqref{ineq:rtau}. Next, it follows from \eqref{energy_bound}, \eqref{energy_equal} and $\Upsilon_i \ll 1$, $\forall~i$ that \begin{align*} K \cdot E_{(r_{j-1},r_j)} & \leq c K \cdot r_{j-1}^{1-p} \tau_j = c \sum_{i=1}^K r_{i-1}^{1-p} \tau_i = c \sum_{i=1}^K r_{i}^{1-p} \tau_i \cdot \left( \frac{r_{i-1}}{r_i} \right)^{1-p} \\ &= c \sum_{i=1}^K r_{i}^{1-p} \tau_i \cdot (1+ \Upsilon_i)^{p-1} < 2^{p-1} c \sum_{i=1}^K r_{i}^{1-p} \tau_i \\ &\leq 2^{p-1} c^2 \sum_{i=1}^K E_{(r_{i-1},r_i)} = 2^{p-1} c^2 E_{(\varepsilon,\gamma)}, \quad \forall j=1,\dotsc,K. \end{align*} This proves the second inequality of \eqref{ineq:KE}. Notice that, by \eqref{e:Upsilon>} and \eqref{e:Upsilon_1}, we have $$ \frac{r_{i-1}^{1-p}}{r_{i}^{1-p}}=(1+\Upsilon_i)^{p-1}<2^{p-1}, \quad \forall i=1,\dotsc,K. $$ Thus, by \eqref{energy_bound} and \eqref{energy_equal}, \begin{equation*} \label{e:E_subequi} E_{(r_{i-1},r_i)} < 2^{p-1}c^2 E_{(r_{j-1},r_j)}, \quad \forall i, j=1,\dotsc,K. \end{equation*} Denote $E_{max} = \max_{1\le i \le K} E_{(r_{i-1},r_i)}$, then $2^{p-1}c^2 K E_{(r_{i-1},r_i)} > K E_{max} \ge E_{(\varepsilon,\gamma)}$. This proves the first inequality of \eqref{ineq:KE}. \end{proof} \begin{rem} For $K$ sufficiently large, it is not difficult to show that there exists an auxiliary grid $\varepsilon = r_0 < r_1 < \cdots < r_K = \gamma$ such that \eqref{energy_equal} holds. \end{rem} The theorem below gives the relative and absolute errors of the elastic energy $E(P,Q)=E_{(\varepsilon,\gamma)}$ when a cavitation solution $(P,Q)$ is replaced by its interpolation functions $(I^{NM}P, I^{NM}Q)$. \begin{thm} \label{thm:E_tilde} Let $(P,Q)$ be a cavitation solution satisfying the hypotheses (H1) and (H2). Then, for $M$, $N$ sufficiently large, there exists a constant $C$ such that \begin{align*} \frac{ \left| E(I^{NM}P, I^{NM}Q) - E(P,Q) \right| }{E(P,Q)} & \leq C \left( M^{2-l} + N^{1-l} \right) ,\\ \left| E(I^{NM}P, I^{NM}Q) - E(P,Q) \right| & \leq C \left( M^{2-l} + N^{1-l} \right) . \end{align*} \end{thm} \begin{proof} To simplify the notation, we denote (see \eqref{note:DF}) $$ \widetilde{P}:= I^{NM}P,~ \widetilde{Q}:= I^{NM}Q, \quad \widetilde{D}:= D(\widetilde{P},\widetilde{Q}),~ \widetilde{F}:= F(\widetilde{P},\widetilde{Q}). $$ By \eqref{note:DFg} and \eqref{energy}, the energy error can be bounded as follows \begin{align}\label{e:energy_error_bounds} \left| E(\widetilde{P},\widetilde{Q}) - E(P,Q) \right| &\leq \int_0^{2\pi} \int_{\varepsilon}^\gamma r \left[ \left| g(\widetilde{F}) - g(F) \right| + \left| h(\widetilde{D}) - h(D) \right| \right] \mathrm{d} r \mathrm{d} \theta \nonumber \\ &\leq \int_0^{2\pi} \int_{\varepsilon}^\gamma r^{1-p} \kappa \left| \left( 2r^2 \widetilde{F} \right)^{p/2} - \left( 2r^2 F \right)^{p/2} \right| \mathrm{d} r \mathrm{d} \theta \nonumber \\ &\qquad + \int_0^{2\pi} \int_{\varepsilon}^\gamma r \left| h(\widetilde{D}) - h(D) \right| \mathrm{d} r \mathrm{d} \theta \triangleq I + II. \end{align} By the hypotheses (H1), (H2), and as a consequence of Theorem~\ref{thm:PQ}, we have $(P,Q)$, $(\widetilde{P},\widetilde{Q})$ and their first order derivatives are all bounded, and \begin{subequations} \label{error_INM} \begin{align} \| \widetilde{P} - P \|_{\omega,\Omega'} &\leq c \left( M^{-l} + N^{-l} \right) ,& \| \widetilde{Q} - Q \|_{\omega,\Omega'} &\leq c \left( M^{-l} + N^{-l} \right) ,\\ \| \widetilde{P}_\rho - P_\rho \|_{\omega,\Omega'} &\leq c \left( M^{2-l} + N^{-l} \right) ,& \| \widetilde{Q}_\rho - Q_\rho \|_{\omega,\Omega'} &\leq c \left( M^{2-l} + N^{-l} \right) ,\\ \| \widetilde{P}_\phi - P_\phi \|_{\omega,\Omega'} &\leq c \left( M^{-l} + N^{1-l} \right) ,& \| \widetilde{Q}_\phi - Q_\phi \|_{\omega,\Omega'} &\leq c \left( M^{-l} + N^{1-l} \right) , \end{align} \end{subequations} where $\| \cdot\|_{\omega,\Omega'}$ is the weighted $L^2$-norm on $\Omega'$. Thus, by \eqref{note:rDrF} and recalling $\rho_r = 2/(\gamma-\varepsilon)$, we have \begin{align*} \left| r\widetilde{D}-rD \right| &\leq \rho_r \left( \left| \widetilde{P} \widetilde{P}_\rho (\widetilde{Q}_\phi+1) - P P_\rho (Q_\phi+1) \right| + \left| \widetilde{P} \widetilde{P}_\phi \widetilde{Q}_\rho - P P_\phi Q_\rho \right| \right) \\ &\leq c \left( \left| \widetilde{P} - P \right| + \left| \widetilde{P}_\rho - P_\rho \right| + \left| \widetilde{P}_\phi - P_\phi \right| + \left| \widetilde{Q}_\rho - Q_\rho \right| + \left| \widetilde{Q}_\phi - Q_\phi \right| \right) ,\\ \left| 2r^2\widetilde{F} - 2r^2F \right| &\leq r^2 \rho_r^2 \left| \widetilde{P}_\rho^2 + \widetilde{P}^2 \widetilde{Q}_\rho^2 - ( P_\rho^2 + P^2 Q_\rho^2 ) \right| + \left| \widetilde{P}_\phi^2 - P_\phi^2 \right| \\ &\qquad + \left| \widetilde{P}^2 (\widetilde{Q}_\phi+1)^2 - P^2 (Q_\phi+1)^2 \right| \\ &\leq c \left( \left| \widetilde{P} - P \right| + \left| \widetilde{P}_\rho - P_\rho \right| + \left| \widetilde{P}_\phi - P_\phi \right| + \left| \widetilde{Q}_\rho - Q_\rho \right| + \left| \widetilde{Q}_\phi - Q_\phi \right| \right) , \end{align*} and as a consequence, it follows from \eqref{error_INM} that \begin{subequations} \begin{align} \| r\widetilde{D}-rD \|_{\omega,\Omega'} &\leq c \left(M^{2-l}+N^{1-l} \right), \label{e:rD_error} \\ \label{e:2r2F_error} \| 2r^2\widetilde{F} - 2r^2F \|_{\omega,\Omega'} &\leq c \left(M^{2-l}+N^{1-l} \right). \end{align} \end{subequations} By hypothesis (H1), (H2) and Theorem~\ref{thm:PQ}, both $D>0$ and $\widetilde{D}>0$ are bounded away from $0$ and $+\infty$, hence, by \eqref{e:rD_error}, we have \begin{align}\label{e:II} II &= \int_0^{2\pi} \int_{\varepsilon}^\gamma r |h'(\vartheta_1)| \left| \widetilde{D} - D \right| \mathrm{d} r \mathrm{d} \theta \leq c \int_0^{2\pi} \int_{-1}^1 \left| r \widetilde{D} - r D \right| \mathrm{d} \rho \mathrm{d} \phi \nonumber \\ & \leq c \| r \widetilde{D}-r D \|_{\omega,\Omega'} \leq c \left(M^{2-l}+N^{1-l} \right), \end{align} where $\vartheta_1$ is between $\widetilde{D}$ and $D$, and thus $h'(\vartheta_1)$ is bounded. On the other hand, let $\varepsilon = r_0 < r_1 < \cdots < r_K = \gamma$ be an auxiliary grid in radial direction satisfying the conditions of Lemma~\ref{lem:grid}, then, by \eqref{energy_bound} and \eqref{ineq:KE}, we have \begin{align}\label{e:I/E_1} \frac{I}{ E_{(\varepsilon,\gamma)} } &< \displaystyle \sum_{i=1}^K \int_{ \Omega_{(r_{i-1},r_i)} } \frac{2^{p-1} c^{2}}{ K E_{(r_{i-1},r_i)}}\, \kappa \left| \left( 2r^2 \widetilde{F} \right)^{p/2} - \left( 2r^2 F \right)^{p/2} \right|\, \mathrm{d} r \mathrm{d} \theta \nonumber \\ & \leq 2^{p-1} c^3 \kappa \sum_{i=1}^K \frac{1}{K \tau_i} \left( \frac{r_{i-1}}{r_i} \right)^{1-p} \int_{ \Omega_{(r_{i-1},r_i)} } \left| \left( 2r^2 \widetilde{F} \right)^{p/2} - \left( 2r^2 F \right)^{p/2} \right| \mathrm{d} r \mathrm{d} \theta . \end{align} Since hypotheses (H1), (H2) and Lemma~\ref{lem:Liguo} implies that both $r^2F$ and $r^2\widetilde{F}$ are bounded, by the H\"{o}lder inequality, we have \begin{align*} &\int_{ \Omega_{(r_{i-1},r_i)} } \left| \left( 2r^2 \widetilde{F} \right)^{p/2} - \left( 2r^2 F \right)^{p/2} \right| \mathrm{d} r \mathrm{d} \theta = \int_{ \Omega_{(r_{i-1},r_i)} } \frac{p}{2}| \vartheta_2^{p/2-1} | \left| 2r^2 \widetilde{F} - 2r^2 F \right| \mathrm{d} r \mathrm{d} \theta \\ &\leq c \int_{ \Omega_{(r_{i-1},r_i)} } \left| 2r^2 \widetilde{F} - 2r^2 F \right| \mathrm{d} r \mathrm{d} \theta \leq c \sqrt{\tau_i} \| 2r^2 \widetilde{F} - 2r^2 F \|_{L^2( \Omega_{(r_{i-1},r_i)} )} , \end{align*} where $\vartheta_2$ is between $2r^2 \widetilde{F}$ and $2r^2 F$, and thus $|\vartheta_2^{p/2-1}|$ is also bounded. Substituting this into \eqref{e:I/E_1} and applying the H\"{o}lder inequality, we have, by \eqref{ineq:rtau} and \eqref{e:2r2F_error}, \begin{align}\label{e:I/E_2} \frac{I}{ E_{(\varepsilon,\gamma)} } &\leq c \sum_{i=1}^K \frac{1}{K \sqrt{\tau_i} } \left( \frac{r_i}{r_{i-1}} \right)^{p-1} \cdot \| 2r^2 \widetilde{F} - 2r^2 F \|_{L^2( \Omega_{(r_{i-1},r_i)} )} \nonumber \\ &\leq c \left[ \sum_{i=1}^K \frac{1}{K^2} \cdot \frac{1}{\tau_i} \left( \frac{r_i}{r_{i-1}} \right)^{2p-2} \right]^{1/2} \cdot \left[ \sum_{i=1}^K \| 2r^2 \widetilde{F} - 2r^2 F \|_{L^2( \Omega_{(r_{i-1},r_i)} )}^2 \right]^{1/2} \nonumber \\ &\leq c \| 2r^2 \widetilde{F} - 2r^2 F \|_{L^2( \Omega_{(\varepsilon,\gamma)} )} \leq c \| 2r^2 \widetilde{F} - 2r^2 F \|_{\omega,\Omega'} \leq c \left(M^{2-l}+N^{1-l} \right) . \end{align} The proof is completed by combining\eqref{e:energy_error_bounds}, \eqref{e:II} and \eqref{e:I/E_2}. \end{proof} Notice that $\mathcal{B}_+^{NM} \subset \mathcal{B}_+$, the result of Theorem~\ref{thm:E_tilde} allows us to obtain the following elastic energy error estimate for the discrete cavitation solution. \begin{thm} \label{thm:E_NM} Let a cavitation solution $(P,Q)$ satisfy the hypotheses (H1), (H2) and be a global energy minimizer of $E(\cdot,\cdot)$ in $\mathcal{B}_+$. Let $(P^{NM},Q^{NM})$ be a global energy minimizer of $E(\cdot,\cdot)$ in $\mathcal{B}_+^{NM}$. Then, for $M$, $N$ sufficiently large, there exists a constant $C>0$ such that \begin{equation} \label{e:E^NM} E(P,Q) \le E(P^{NM},Q^{NM}) \le E(P,Q) + C \left( M^{2-l} + N^{1-l}\right). \end{equation} \end{thm} \begin{proof} The first inequality is a direct consequence of $\mathcal{B}_+^{NM} \subset \mathcal{B}_+$. By \eqref{det_error} and for $M$, $N$ sufficiently large, we have $(I^{NM}P,I^{NM}Q) \in \mathcal{B}_+^{NM}$. Then, the second inequality follows from Theorem~\ref{thm:E_tilde} and $E(P^{NM},Q^{NM}) \le E(I^{NM}P, I^{NM}Q)$. \end{proof} Let $(P,Q)\in \mathcal{B}_+$ be a global energy minimizer of $E(\cdot,\cdot)$ in $\mathcal{B}_+$, let $(I^{NM}P,I^{NM}Q)\in \mathcal{B}_+^{NM}$ be its interpolation functions. Let $(P^{NM},Q^{NM})\in \mathcal{B}_+^{NM}$ be global energy minimizers of $E(\cdot,\cdot)$ in $\mathcal{B}_+^{NM}$. Denote $\mathbf{\bar{u}}$, $\mathbf{\bar{u}}^{NM}$ and $\mathbf{U}^{NM}$ as the corresponding functions on $\Omega_{(\varepsilon,\gamma)}$ defined by $(P,Q)$, $(I^{NM}P,I^{NM}Q)$ and $(P^{NM},Q^{NM})$ respectively via the coordinates transformations \eqref{coordinate:polar} and \eqref{coordinate:rho_phi}. Then, \eqref{e:E^NM} can be rewritten as \begin{equation}\label{e:E(u)NM} E(\mathbf{\bar{u}}) = \inf_{\mathbf{v}\in \mathcal{A}_{\varepsilon}} E(\mathbf{v}) \le E(\mathbf{U}^{NM}) \le E(\mathbf{\bar{u}}^{NM}) \le E(\mathbf{\bar{u}}) + C\left( M^{2-l} + N^{1-l}\right). \end{equation} According to Theorem~4.9 in \cite{Su2015} and its proof, for a conforming discrete approximation method of the cavitation problem, the inequality \eqref{e:E(u)NM} implies the convergence of the discrete cavitation solutions. Thus, we have the following convergence theorem. For the convenience of the readers, we sketch its proof below. \begin{thm} Let a cavitation solution $(P,Q)$ satisfy the hypotheses (H1), (H2) and be a global energy minimizer of $E(\cdot,\cdot)$ in $\mathcal{B}_+$. Let $(P^{NM},Q^{NM})$ be global energy minimizers of $E(\cdot,\cdot)$ in $\mathcal{B}_+^{NM}$. Let $\mathbf{U}^{NM}$ correspond to $(P^{NM},Q^{NM})$ under the transformations \eqref{coordinate:polar} and \eqref{coordinate:rho_phi}. Then, there exists a subsequence, still denoted as $\{\mathbf{U}^{NM}\}$, and a function $\mathbf{u} \in \mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$, such that $\mathbf{U}^{NM} \rightarrow \mathbf{u}$ in $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$ and $\mathbf{u}$ is a global energy minimizer of $E(\cdot)$ in $\mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$. \end{thm} \begin{proof} Since $h>0$ is a convex function satisfying the growth conditions \eqref{note:h}, $1<p<2$ and $\mathbf{U}^{NM}$ satisfies the Direchlet boundary condition, by \eqref{e:E(u)NM} and the De La Vall\'{e}e Poussin theorem \cite{Poussin}, we conclude that $\{|\nabla \mathbf{U}^{NM}|^p\}_{N,M \rightarrow \infty}$ and $\{\det \nabla \mathbf{U}^{NM}\}_{N,M \rightarrow \infty}$ are equi-integrable. As a consequence, there exists a subsequence, still denoted as $\{\mathbf{U}^{NM}\}$, a function $\mathbf{u} \in W^{1,p}(\Omega_{(\varepsilon,\gamma)})$ and a function $\zeta \in L^1(\Omega_{(\varepsilon,\gamma)})$ such that \begin{equation}\label{e:Uweakconv} \mathbf{U}^{NM} \rightharpoonup \mathbf{u} ~\text{in}~ W^{1,p}(\Omega_{(\varepsilon,\gamma)}), \quad \mathbf{U}^{NM} \rightarrow \mathbf{u} ~a.e., \quad \det \nabla \mathbf{U}^{NM} \rightharpoonup \zeta ~\text{in}~ L^1(\Omega_{(\varepsilon,\gamma)}). \end{equation} Hence, by $\det \nabla \mathbf{U}^{NM} >0$, $a.e.$, we have $\zeta \geq 0$, $a.e.$. We conclude that $\zeta >0$, $a.e.$. Suppose otherwise, {\it i.e.} $\zeta =0$ on a set $S$ with positive measure, then there exists a subsequence, still denoted as $\{\mathbf{U}^{NM}\}$, such that $\int_S |\det \nabla \mathbf{U}^{NM}| \mathrm{d} \mathbf{x} \rightarrow 0$ and $\det \nabla \mathbf{U}^{NM} \rightarrow 0$, $a.e.$ on the set $S$, which, by \eqref{note:h}, implies $h(\det \nabla \mathbf{U}^{NM}) \rightarrow \infty$, $a.e.$ on the set $S$. Thus, by the Fatou lemma, we have and $E(\mathbf{U}^{NM}) \rightarrow \infty$, which contradicts to $\displaystyle \varlimsup_{N,M\rightarrow \infty} E(\mathbf{U}^{NM}) <\infty$. Thanks to Theorem~3 in \cite{Henao2010} and Theorem~3 in \cite{Henao2011}, as a consequence of \eqref{e:Uweakconv}, $\zeta >0$, $a.e.$ and the continuity of $\mathbf{U}^{NM}$, we have $\zeta = \det \nabla \mathbf{u}$, $a.e.$ and $\mathbf{u}$ is one-to-one a.e.. In addition, it is easily verified that $\mathbf{u}|_{\partial \Omega} = \mathbf{u}_0$. Hence $\mathbf{u} \in \mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$. On the other hand, since $\displaystyle E(\mathbf{u}) \leq \varliminf_{N,M\rightarrow \infty} E(\mathbf{U}^{NM})$, due to the weakly lower semi-continuity of $E(\cdot)$ on $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$ (see the theorem 5.4 in \cite{Ball1981}), we conclude from $\mathbf{u} \in \mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$ and \eqref{e:E(u)NM} that $\mathbf{u}$ is a global minimizer of $E(\cdot)$ in $\mathcal{A}_{(\varepsilon,\gamma)}( \mathbf{u}_0 )$ and $\displaystyle E(\mathbf{u}) = \lim_{N,M\rightarrow \infty} E(\mathbf{U}^{NM})$. Since $h$ is a convex function, it follows from $\det \nabla \mathbf{U}^{NM} \rightharpoonup \det \nabla \mathbf{u}$ in $L^1(\Omega_{(\varepsilon,\gamma)})$ that \begin{align*} &E(\mathbf{u}) - \kappa \int_{\Omega_{(\varepsilon,\gamma)}} |\nabla \mathbf{u}|^p \mathrm{d} \mathbf{x} = \int_{\Omega_{(\varepsilon,\gamma)}} h(\det \nabla \mathbf{u}) \mathrm{d} \mathbf{x} \leq \varliminf_{N,M\rightarrow \infty} \int_{\Omega_{(\varepsilon,\gamma)}} h(\det \nabla \mathbf{U}^{NM}) \mathrm{d} \mathbf{x} \\ &\quad = \varliminf_{N,M\rightarrow \infty} \left( E(\mathbf{U}^{NM}) - \kappa \int_{\Omega_{(\varepsilon,\gamma)}} |\nabla \mathbf{U}^{NM}|^p \mathrm{d} \mathbf{x} \right) = E(\mathbf{u}) - \kappa \varlimsup_{N,M\rightarrow \infty} \int_{\Omega_{(\varepsilon,\gamma)}} |\nabla \mathbf{U}^{NM}|^p \mathrm{d} \mathbf{x}, \end{align*} which leads to $\displaystyle \varlimsup_{N,M\rightarrow \infty} \norm{\mathbf{U}^{NM}}_{W^{1,p}(\Omega_{(\varepsilon,\gamma)})} \leq \norm{\mathbf{u}}_{W^{1,p}(\Omega_{(\varepsilon,\gamma)})}$. Thus, by the uniform convexity of $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$ (see \cite{Adams1975}) and $\mathbf{U}^{NM} \rightharpoonup \mathbf{u}$ in $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$, it follows from proposition 3.30 in \cite{Brezis1983} that $\mathbf{U}^{NM} \rightarrow \mathbf{u}$ in $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$. This completes the proof. \end{proof} \section{Numerical Experiments and Results} In our numerical experiments, the stored energy density function $W(\cdot)$ is taken of the form \eqref{note:W} with \begin{align} p=\frac32, \quad \kappa =\frac23, \quad h(t)= 2^{-1/4} \left( \frac{(t-1)^2}{2}+\frac{1}{t} \right). \end{align} The reference configuration is $\Omega_{(\varepsilon,\gamma)} = \mathbb{B}_\gamma(\mathbf{0}) \setminus \overline{\mathbb{B}_\varepsilon(\mathbf{0})}$, $(0<\varepsilon \ll \gamma \leq 1)$. We consider $\mathbf{u}_0(\mathbf{x})= \lambda \mathbf{x},~ \mathbf{x} \in \partial \mathbb{B}_\gamma( \mathbf{0})$, $\lambda >1$ in the radially-symmetric case and $\mathbf{u}_0(\mathbf{x}) = \left[ \lambda_1 x_1, \lambda_2 x_2 \right]^T$, $\mathbf{x} \in \partial \mathbb{B}_\gamma( \mathbf{0})$, $\lambda_1,\lambda_2 >1$ in the non-radially-symmetric case. By \eqref{det_error} and the hypotheses (H1), (H2), we expect to have $(I^{NM}P,I^{NM}Q) \in \mathcal{B}_+^{NM}$ for sufficiently large $N$ and $M$. Before proceeding to the numerical experiments, we first check in Table~\ref{tab:incompress_det} the orientation preservation condition $D(I^{NM}P,I^{NM}Q)>0$ for the exact cavitation solution $(P,Q)$ in the radially-symmetric case for incompressible elastic materials, since in such a case the cavity solutions have a simple explicit form $R(r)=\sqrt{\lambda^2+r^2-\gamma^2}$ in the polar coordinate systems. By $D(I^{NM}P,I^{NM}Q) = \frac{\rho_r}{r} \cdot I^{NM} P \cdot (I^{NM} P)_\rho$, we only need to check whether $(I^{NM} P)_\rho >0$ is satisfied. In fact, whenever $(I^{NM} P)_\rho >0$ is satisfied, we have $D(I^{NM}P,I^{NM}Q)\approx 1$. The data shown in Table~\ref{tab:incompress_det} suggest that the orientation preservation condition $D>0$ should not impose much real additional restrictions on the choice of $N$ and $M$ in practical computations. \begin{table}[H] \centering \footnotesize \renewcommand{1.0}{1.0}{ \caption{\footnotesize The minima of $(I^{NM} P)_\rho$ in $\Omega'$ in various cases (independent of $N$) } \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{~~}ccrrrrrr} \toprule \raisebox{-2.00ex}[0cm][0cm]{ $\lambda \times \gamma$ } & \raisebox{-2.00ex}[0cm][0cm]{ $(\varepsilon,\gamma)$ } & \multicolumn{6}{c}{ $M$ } \\ \cline{3-8} & & 4~~~ & 6~~~ & 8~~~ & 10~~~ & 12~~~ & 14~~~ \\ \midrule \multirow{3}{*}{ 2 } & $(10^{-2},1)$ & 2.66e-3 & 2.86e-3 & 2.86e-3 & 2.86e-3 & 2.86e-3 & 2.86e-3 \\[-1mm] & $(10^{-3},1) $ & 8.57e-5 & 2.92e-4 & 2.88e-4 & 2.88e-4 & 2.88e-4 & 2.88e-4 \\[-1mm] & $(10^{-4},1) $ & -1.75e-4 & 3.27e-5 & 2.88e-5 & 2.89e-5 & 2.89e-5 & 2.89e-5 \\[1mm] \multirow{3}{*}{1.25} & $(10^{-3},10^{-1})$ & 3.97e-5 & 3.97e-5 & 3.97e-5 & 3.97e-5 & 3.97e-5 & 3.97e-5 \\[-1mm] & $(10^{-4},10^{-2})$ & 3.96e-7 & 3.96e-7 & 3.96e-7 & 3.96e-7 & 3.96e-7 & 3.96e-7 \\[-1mm] & $(10^{-5},10^{-3}) $ & 3.96e-9 & 3.96e-9 & 3.96e-9 & 3.96e-9 & 3.96e-9 & 3.96e-9 \\ \bottomrule \end{tabular*} \label{tab:incompress_det} } \end{table} Next, we investigate the effect of the number of quadrature points $N',M'$ used in \eqref{EL:final_D}. Figure \ref{fig:numer_integ} shows the convergence behavior of the errors on the cavity radius for various $N'/N$ (with $M=32$ and $M'=8M$ fixed) and $M'/M$ (with $N=16$ and $N'=2N$ fixed), where for the non-radially-symmetric case $\Omega_{(\varepsilon,\gamma)}=\Omega_{(10^{-4},1)}$ with $\lambda_1 = 2.4$ and $\lambda_2 = 2$, and for the radially-symmetric case $\Omega_{(\varepsilon,\gamma)}=\Omega_{(10^{-2},1)}$ with $\lambda = 2$. To balance the accuracy and computational cost, we set in our numerical experiments below $N' = 2N$ and $M' = 8M$. \begin{figure} \caption{Effect of $N'/N$, $M'/M$ on the cavity radius errors.} \label{fig:numer_integ} \end{figure} \subsection{Radially-Symmetric Case} In the radially-symmetric case with $\mathbf{u}_0(\mathbf{x})=\lambda \mathbf{x}$, $\lambda >1$, the cavitation solution $\mathbf{u}$ can be written in polar coordinates systems as \begin{equation*} R = s(r),~~ \Theta=\theta, \quad \forall~ (r,\theta) \in [\varepsilon,\gamma]\times [0,2\pi], \end{equation*} where $s(r)$ satisfies $s(\varepsilon)>0$ and $s(\gamma) = \lambda \cdot \gamma$. Since in theory the numerical solution is independent of the circumferential DOF (degree of freedom) $N$, we fix $N=16$ and examine the effect of the radial DOF $M$ on the numerical performance of our method. In comparison, high precision numerical solutions to the equivalent 1-dimensional ODE boundary value problems \cite{Sivaloganathan2009}, obtained by the {\it ode15s} routine in MATLAB with the tolerance $10^{-16}$, are taken as the exact solutions. For the standard case of $\gamma=1$, $\lambda = 2$, and $\varepsilon= 10^{-3}$, $10^{-4}$, the convergence behavior of our numerical cavitation solutions $\mathbf{U}^{NM}$ is shown in Figure~\ref{fig:error_ode_3}, \ref{fig:error_ode_4}, where $L^2_{\omega}$ and $W^{1,p}$ represent $L^2_{\omega}(\Omega')$-norm and $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$-semi-norm respectively. For $\gamma=1$ and $\varepsilon = 10^{-2},10^{-3},10^{-4}$, we show in Figure \ref{fig:lambda_crit_symmetry} the numerical results obtained with $M=32$ on the $\lambda$-$R^{NM}(\varepsilon)$ ({\it i.e.} the expansion rate on the outer boundary against the cavity radius on the inner boundary) graph. Figure~\ref{fig:lambda_crit_sivaloganathan} shows, for $\varepsilon=10^{-4}$, the convergence of our numerical results to that of the 1-dimensional ODE solution. \begin{figure} \caption{The convergence behavior of radially-symmetric $\mathbf{U}^{NM}$ with $N=16$ fixed.} \label{fig:error_ode_3} \label{fig:error_ode_4} \end{figure} \begin{figure} \caption{$\lambda$-$R^{NM}(\varepsilon)$, $N=16$, $M=32$.} \label{fig:lambda_crit_symmetry} \caption{$\lambda$-$R^{NM}(10^{-4})$, $N=16$.} \label{fig:lambda_crit_sivaloganathan} \end{figure} \begin{figure} \caption{The convergence behavior on $\Omega_{(\varepsilon,\gamma)}$ with small $\gamma$.} \label{fig:subdomain} \end{figure} To explore the potential of the method in coupling with a domain decomposition method, especially when combining with a finite element method in a multi-defects problem, we examine the convergence behavior of our algorithm on a small neighbourhood of the defect. Taking $\Omega_{(10^{-3},10^{-1})}$, $\Omega_{(10^{-4},10^{-2})}$ and $\Omega_{(10^{-5},10^{-3})}$ as the reference configurations and setting $\lambda \cdot \gamma=1.25$, we show in Figure~\ref{fig:subdomain} the energy error and cavity radius error as a function of $M$ (with $N=16$ fixed), where it is clearly seen that high precision numerical results can be obtained with rather small $M$. \subsection{Non-radially Symmetric Case} For the non-radially symmetric case, we consider the circular ring reference configuration $\Omega_{(\varepsilon,\gamma)}$ with oval boundary stretch $\mathbf{u}_0 (\mathbf{x}) = \left[ \lambda_1 x_1, \lambda_2 x_2 \right]^T$, $\lambda_1$, $\lambda_2>1$. Assuming that the error of the numerical solution $\mathbf{U}^{NM}$ satisfies \begin{align} \label{eq:error} q^{NM} \approx q^{\infty} + c_1 N^{-\nu_1} + c_2 M^{-\nu_2}, \end{align} where $q^{\infty}$ and $q^{NM}$ represent the exact and numerical results of a specific quantity, such as the elastic energy, semi-major axis and semi-minor axis etc., and $c_1$, $c_2$, $v_1$, $v_2$ are the corresponding parameters to be determined by the least squares data fitting. For $\Omega_{(\varepsilon,\gamma)}=\Omega_{(10^{-4},1)}$, $\lambda_1 = 2.4$ and $\lambda_2 = 2$, we show in Figure~\ref{fig:nonsymmetry_N} the errors between $\mathbf{U}^N$ and $\mathbf{U}^{1.5N}$ with $M=32$ fixed, and in Figure~\ref{fig:nonsymmetry_M} the errors between $\mathbf{U}^M$ and $\mathbf{U}^{1.25 M}$ with $N=16$ fixed, where $L^2_{\omega}$ and $W^{1,p}$ represent $L^2_{\omega}(\Omega')$-norm and $W^{1,p}(\Omega_{(\varepsilon,\gamma)})$-semi-norm respectively. The regressed quantities and parameters are shown in Table~\ref{tab:fit}. \begin{figure} \caption{The convergence behavior of the non-radially-symmetric $\mathbf{U}^{NM}$.} \label{fig:nonsymmetry_N} \label{fig:nonsymmetry_M} \end{figure} \begin{table}[H] \centering \footnotesize \renewcommand{1.0}{1.0}{ \caption{The regressed quantities and parameters for the non-radially-symmetric case.} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{~~}crrrrr} \toprule $q$ & $c_1$ & $c_2$ & $\nu_1$ & $\nu_2$ & $q^{\infty}$ \\ \midrule energy & 1.45e+24 & -3.10e+6 & 25 & 6.1 & 22.85959048 \\[-1mm] semi-major axis & -2.02e+14 & -1.37e-1 & 17 & 2.1 & 1.67481624 \\[-1mm] semi-minor axis & -1.61e+14 & -8.42e-2 & 16 & 2.1 & 1.42872097 \\ \bottomrule \end{tabular*} \label{tab:fit} } \end{table} As a comparison, we show in Figure~\ref{fig:error_twoM} the corresponding errors obtained in the same way for the radially-symmetric case with $\lambda=2$, and show in Table~\ref{tab:fit_s} the regressed quantities and parameters. It is clearly seen that the regressed formula \eqref{eq:error} produces quite sharp numerical results in the radially-symmetric case. \begin{figure} \caption{The convergence behavior of the radially-symmetric $\mathbf{U}^{NM}$.} \label{fig:error_twoM} \end{figure} \begin{table}[H] \centering \footnotesize \renewcommand{1.0}{1.0}{ \caption{The regressed quantities and parameters for the radially-symmetric case.} \begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}@{~~}ccrrrr} \toprule $\varepsilon$ & $q$ & $c_2$ & $\nu_2$ & $q^{\infty}$ & ODE solution \\ \midrule \multirow{2}{*}{ $10^{-3}$ } & energy & -1.38e+8 & 9.5 & 18.61960091 & 18.61960090 \\[-1mm] & cavity radius & -3.18e-2 & 2.0 & 1.26772534 & 1.26772534 \\[-1mm] \multirow{2}{*}{ $10^{-4}$ } & energy & -2.22e+6 & 6.1 & 18.87582778 & 18.87582146 \\[-1mm] & cavity radius & -1.49e-1 & 2.1 & 1.25228561 & 1.25228643 \\[-1mm] \bottomrule \end{tabular*} \label{tab:fit_s} } \end{table} To see how well the regressed formula \eqref{eq:error} fits the data, we show in Figure~\ref{fig:fit_error_radius} the errors on the cavity dimensions, {\it i.e.} the cavity radius in the radially-symmetric case and the cavity major and minor axes in the non-radially-symmetric case, and in \ref{fig:fit_error_energy} the errors on the elastic energy, between the corresponding quantities produced by the numerical solution $\mathbf{U}^{NM}$ and the regressed formula \eqref{eq:error} respectively. In particular, compare also to Figure~\ref{fig:error_ode_4}, it is clearly seen that the regressed formula \eqref{eq:error} is highly accurate and reliable. \begin{figure} \caption{Errors on the key quantities produced by $\mathbf{U}^{NM}$ and the regressed data.} \label{fig:fit_error_radius} \label{fig:fit_error_energy} \label{fig:fit_error} \end{figure} To examine how the axial ratio of the oval stretch affect the critical displacement, we show in Figure~\ref{fig:lambda_crit_nonsymmetry} the semi-major and semi-minor axes of the numerical cavity formed as functions of $\lambda_1$ for $\lambda_1 / \lambda_2 = 1.2,1.3,1.4$, with $\Omega_{(\varepsilon,\gamma)}=\Omega_{(10^{-4},1)}$, $N=16$ and $M=32$, where it is obviously seen that both are monotonously increasing functions. \begin{figure} \caption{The semi-major and semi-minor axes of the numerical cavity formed.} \label{fig:lambda_crit_nonsymmetry} \end{figure} Taking $\Omega_{(\varepsilon,\gamma)}=\Omega_{(10^{-3},10^{-1})}$, $\Omega_{(10^{-4},10^{-2})}$, $\Omega_{(10^{-5},10^{-3})}$ as the reference configuration respectively, setting the oval boundary data $\lambda_1 \gamma=1.67$ and $\lambda_2 \gamma=1.42$ (see Table~\ref{tab:fit}), for $N=16$ fixed, the errors of the corresponding non-radially-symmetric numerical solutions $\mathbf{U}^{NM}$ are shown in with Figure~\ref{fig:subdomain_nonsym}, where the numerical solution with $M=32$ is taken as the exact solution. It is clearly seen that high precision numerical results can also be obtained in a neighborhood of a defect with rather small $M$ in non-radially-symmetric case. \begin{figure} \caption{The convergence behavior on $\Omega_{(\varepsilon,\gamma)}$ with small $\gamma$.} \label{fig:subdomain_nonsym} \end{figure} \addcontentsline{toc}{chapter}{\bibname} \end{document}

\begin{document} \title{Photon-mediated entanglement scheme between a ZnO semiconductor defect and a trapped Yb ion} \author{Jennifer F. Lilieholm} \email[]{[email protected]} \affiliation{Department of Physics, University of Washington, Seattle, Washington 98195, USA} \author{Vasilis Niaouris} \affiliation{Department of Physics, University of Washington, Seattle, Washington 98195, USA} \author{Alexander Kato} \affiliation{Department of Physics, University of Washington, Seattle, Washington 98195, USA} \author{Kai-Mei C. Fu} \affiliation{Department of Physics, University of Washington, Seattle, Washington 98195, USA} \affiliation{Department of Electrical and Computer Engineering, University of Washington, Seattle, Washington 98195, USA} \author{Boris B. Blinov} \affiliation{Department of Physics, University of Washington, Seattle, Washington 98195, USA} \date{\today} \begin{abstract} We propose an optical scheme to generate an entangled state between a trapped ion and a solid state donor qubit through which-path erasure of identical photons emitted from the two systems. The proposed scheme leverages the similar transition frequencies between In donor bound excitons in ZnO and the $^2P\textsubscript{1/2}$ to $^2S\textsubscript{1/2}$ transition in Yb\textsuperscript{+}. The lifetime of the relevant ionic state is longer than that of the ZnO system by a factor of 6, leading to a mismatch in the temporal profiles of emitted photons. A detuned cavity-assisted Raman scheme weakly excites the donor with a shaped laser pulse to generate photons with 0.99 temporal overlap to the Yb\textsuperscript{+} emission and partially shift the emission of the defect toward the Yb\textsuperscript{+} transition. The remaining photon shift is accomplished via the dc Stark effect. We show that an entanglement rate of 21\,kHz and entanglement fidelity of 94\% can be attained using a weak excitation scheme with reasonable parameters. \end{abstract} \pacs{} \maketitle Hybrid quantum systems offer the opportunity to combine the benefits of different qubit types while avoiding some of their pitfalls. Task-dependent qubit selection allows the usage of long-lived qubits for memory and qubits with rapid gate speeds for operations. For optical systems, a photon bus can be used to remotely link these systems via photon-heralded entanglement. To successfully generate entanglement, the two different qubit systems must emit identical photons, requiring spectro-temporal engineering of at least one qubit's photon wavepacket. While significant progress has been made toward efficient quantum-frequency conversion~\cite{Siverns2019naw,zaske2012vqf,bock2018hfe,rutz2017qfc}, post-emission temporal photon pulse-shaping~\cite{keller2004cgs, fan2019ssi,karpinski2017bmq} techniques for the narrow-band photons from both trapped ions and solid-state defects is an outstanding challenge. We have identified two disparate, complementary qubit systems in which high-fidelity photon-mediated entanglement should be possible by direct control over the photon emission process. Trapped ions are a well-studied qubit system with high operational fidelities\cite{Ballance2016} and long coherence times~\cite{wang2017sqq}, but relatively slow initialization and gate speeds~\cite{linke2017ect}. Electron spins in semiconductors have rapid initialization and gate speeds~\cite{warburton2013ssi, fu2008ucd,he2019tqg}, but have shorter coherence times. A hybrid system consisting of ions and electrons bound to donor defects would have the ability to use ions for quantum memory and defects for gate operations, producing a system more rapid and reliable than either qubit alone. Yb\textsuperscript{+} and the ZnO donor were chosen as the target systems for their shared transition near 369\,nm: the $^2P\textsubscript{1/2}$ to $^2S\textsubscript{1/2}$ transition in \textsuperscript{171}Yb\textsuperscript{+} and the In neutral donor bound exciton (D$\textsuperscript{0}$X) to neutral donor (D$\textsuperscript{0}$) transition in ZnO (Fig.\,\ref{trans}). In:ZnO is analogous in structure to the better-known P:Si qubit system~\cite{kane1998sbn}, however ZnO is a direct band gap semiconductor enabling efficient donor coupling to photons. While the two transition frequencies are quite close ($\delta$ = 0.36~THz), the excited state lifetimes differ by a factor of 6 resulting in a large temporal mismatch. Prior semiconductor spin - trapped ion entanglement schemes addressed similar temporal mismatch by using coherent scattering~\cite{waks2009phe} or sacrificing fidelity~\cite{meyer2015dpc}. Here we demonstrate that pulse shaping can be a powerful tool to attain high-fidelity entanglement and show that an entanglement rate of 21\,kHz and fidelity of 94\% is feasible. \begin{figure}\label{trans} \end{figure} A heralded entanglement scheme based on weak excitation, single-photon detection and which-path erasure can be used to entangle the two systems, similar to the proposal by Cabrillo \textit{et al.}~\cite{cabrillo1999ces} Fig.\,\ref{trans} depicts the relevant energy levels and excitation/decay pathways for the donor and ion. Here D\textsuperscript{0} system is in the Voigt (B$\perp \hat k$) geometry but the Faraday geometry could also be utilized. The donor is coupled to an optical cavity detuned by $\Delta$ from the D\textsuperscript{0}X-D\textsuperscript{0} transition. The diagram of the experiment is shown in Fig.\,\ref{ent}. The Yb\textsuperscript{+} and In donor are first initialized using optical pumping to $\ket{F=0,m_F=0}$ and $\ket{m_s=-1/2}$, respectively, producing the initial state $|\Psi\rangle_i = |0\rangle_{\textrm{Yb}}\otimes|0\rangle_{\textrm{In}}\otimes |\textrm{vac}\rangle \equiv |0;0;\textrm{vac}\rangle$. Next, each system is excited to $\ket{e}$\textsubscript{In} or $\ket{e}$\textsubscript{Yb}, using resonant or near-resonant pulsed excitation. Here, we assume the weak excitation limit (excitation probability $p_{1,x}<10\%$, $x=\{\text{Yb,In}\}$). \begin{figure} \caption{A schematic for remote entanglement of a trapped ion qubit (left) and donor qubit in ZnO (right). A transfer cavity phase-locks the two 369\,nm excitation lasers. The two acousto-optic modulators (AOM) are synchronized and programmed to output the calculated pulse shapes for their respective qubits. Photons collected from the two qubits interfere on the central beam splitter (BS) via inputs A and B. Successful entanglement is heralded by the detection of a single photon by photodetectors (PD) at outputs C, D. For the trapped ion, a B-field pointing along the direction of fluorescence collection breaks the degeneracy between the $\ket{F=1}$ states and defines the quantization axis. For the donor, the B-field points either parallel to the direction of emitted photons (Faraday geometry) or perpendicular to it (Voigt geometry).} \label{ent} \end{figure} The state of the ZnO donor and ion is now given by \begin{equation} \begin{split} \ket{\Psi}_{c} = \be_1 \ket{0;0;\text{vac}} & + \be_4 \ket{1;1;\gyb,\gin} + \\ + \be_2 \ket{0;1;\gin} & + \be_3 \ket{1;0;\gyb}, \end{split} \end{equation} where the emitted photons on paths A and B of Fig.~\ref{ent} $\ket{\gyb}=\sum_\omega \xi\uyb[,\omega] \ryb \vac$ and $\ket{\gin}=\sum_{\omega^{\prime}} \xi\uin[,\omega^{\prime}] \rin \vac$ are given by a sum over all modes $\omega$ ($\omega^\prime$) with coefficients $\xi\uyb[,\omega]$ ($\xi\uin[,\omega^\prime]$) and creation operators $\ryb$ ($\rin$). The coefficients $\beta$ emerge from the excitation ($p_{1,x}$) probabilities of the two systems, the phase gained from excitation laser phases ($\phi_{x,L}$), and the distance travelled by the collected photon ($\phi_{x,d}$): \begin{equation} \begin{split} \beta_1 =\sqrt{(1-p\uybn[1,])(1-p\uinn[1,])}e^{i(\phi\uyb[,L]+\phi\uin[,L])}\\%\al\uyb[,0] \al\uin[,0] \beta_2 = \sqrt{p\uinn[1,](1-p\uybn[1,])}e^{i(\phi\uyb[,L]+\phi\uin[,d])}\\%\al\uyb[,0] \al\uin[,1] \\ \beta_3=\sqrt{p\uybn[1,](1-p\uinn[1,])}e^{i(\phi\uyb[,d]+\phi\uin[,L])}\\%\al\uyb[,1] \al\uin[,0] \qquad \beta_4=\sqrt{p\uinn[1,]p\uybn[1,]}e^{i(\phi\uyb[,d]+\phi\uin[,d])}\\%\al\uyb[,1] \al\uin[,1], \end{split} \end{equation} By phase locking the laser pulses, we can ignore $\phi_{x,L}$. Collected photons from both systems interfere on the beamsplitter, which erases which-path information. Entanglement is heralded by the detection of a single photon at one of the two photodetectors. With the appropriate choice for $p\uybn[1,]$, $p\uinn[1,]$, and the collection efficiency from each system (supplemental material), photon detection in path D projects the ion-donor qubits onto the renormalized entangled state \begin{equation} \begin{split} \ket{\Psi} = \frac{1}{\sqrt{2}}\left( \ket{0;1;\gin}-i e^{i\Delta\phi} \ket{1;0;\gyb} \right), \end{split} \end{equation} where $\Delta\phi$ is determined by the optical path length difference. Similar expression can be derived for detector C. Tracing over all photon modes, we get the reduced Yb$^+ -$ In density matrix \begin{equation} \begin{split} \rho^{\text{Yb,In}} & = \frac{1}{2} \ket{0;1} \bra{0;1} + \frac{1}{2} \ket{1;0} \bra{1;0} +\\ & + \frac{1}{2} \left(i e^{i\Delta\phi} \ovybin \ket{0;1} \bra{1;0} +c.c.\right), \end{split} \end{equation} where $\ovybin = \sum_{\tilde{\omega}} \xybt^* \xint$ is the overlap of the photons from the Yb\textsuperscript{+} and ZnO systems. Factors which affect the entanglement fidelity are photon overlap, false identification of both-system excitation as a single-system excitation, and atomic recoil from the ion interacting with the excitation laser. Accounting for these sources of error, the final fidelity is: \begin{equation} F=\frac{1}{2+c_{1}^{2}}[1+F\utxt{dyn}\text{Re}(\ovybin)] \label{eq:fidelity} \end{equation} where c\textsubscript{1} depends on the excitation probabilities and detection efficiencies of both systems (supplemental material) and $F\textsubscript{dyn}$ is related to the photon recoil~\cite{cabrillo1999ces}. Motion of the trapped ion due to photon recoil during the absorption/emission process can shift the frequency of the photon and reduce fidelity of the entangled state. Note that for the ZnO donor, absorption/emission are recoilless due to the Mössbauer effect. For a Doppler-cooled $^{171}$Yb$^{+}$ in a 1~MHz trap in geometry where the ion is excited by a laser pulse parallel to the light collection direction, the expected $F\utxt{dyn}$ is 96\%~\cite{cabrillo1999ces}. In addition, uncertainty in both $\phi_{x,L}$ and $\phi_{x,d}$ leading to an undesired phase factor $e^{i\epsilon}$ between the terms in Eq. ~3 can further degrade the fidelity according to $\text{Re}(\ovybin) \rightarrow \text{Re}(e^{i\epsilon} \ovybin).$ Other factors that may further decrease the fidelity include photodetector dark counts, background luminescence from ZnO, and D$\textsuperscript{0}$X spectral diffusion~\cite{Humphreys2018,slodivcka2013aae}. Photon collection efficiency primarily affects the protocol’s probability of success. For trapped ions, light collection is challenging due to the high-vacuum environment and the need to isolate ions from decoherence-inducing surfaces. Typical light collection efficiency is 2-4\% utilizing off-the-shelf long working distance microscope objectives~\cite{blinov04nat}, while optics based on in-vacuum lenses~\cite{araneda2018prl} and custom high-NA objectives~\cite{stephenson2020hrh} are capable of collecting up to 10\% of the emitted photons. Further enhancement is possible by integrating a metallic parabolic mirror as an RF electrode of the ion trap~\cite{chou2017nsi}. Ions are trapped at the focus of the mirror, so that the emitted photons are collimated upon reflection from the mirror with an expected 32\% overall coupling efficiency into a single-mode optical fiber. As we show below, the parabolic mirror trap also provides a mechanism for polarization filtering. Longer term, integrated-photonics platforms may provide a path toward high-NA collection from scalable arrays of ions~\cite{bruzewicz2019tiq}. For the donor, a photonic cavity can be fabricated in ZnO to enhance collection efficiency. As shown in Fig.\,\ref{QV}, cavities which satisfy high cooperativity $C={g^2}/{\kappa\Gamma\uin}$ (here $g$ is the donor-cavity coupling strength, $\kappa$ is the cavity decay rate and $\Gamma\uin$ is the spontaneous decay rate) in the “bad cavity” limit necessary for the pulse-shaping procedure described below, lie in a band of readily achievable $Q/V$ ratios with today’s nanophotonic fabrication techniques (here $Q$ is the quality factor and $V$ is the mode volume of the cavity). Due to intrinsic band-edge absorption, the high quality factor region in Fig.\,\ref{QV} may not be achievable at D\textsuperscript{0}X-D\textsuperscript{0} transition~\cite{nur2019spc}, thus low mode volume cavities with moderate quality factors should be targeted. While nanophotonic fabrication in ZnO is relatively immature compared to other quantum defect host crystals, small mode volume ZnO nanowire cavities have enabled UV lasers~\cite{huang2001rtu} and ZnO cavities fabricated by focused ion beam milling~\cite{chang2016zob}, a method that has been used to achieve high cooperativity in rare-earth doped systems~\cite{zhong2018oas}, exhibit quality factors up to 1000. In the limit that the cavity photon loss rate $\kappa$ is dominated by coupling to the output mode, over 50\% collection efficiency into a waveguide for planar geometry cavities~\cite{arcari2014nuc} or into an objective lens for nanowire cavities~\cite{senellart2017hps} is possible. \begin{figure} \caption{ZnO cavity parameter space ($\kappa$, $g$, $C$) satisfying the photon pulse-shaping requirements in terms of the quality factor $Q$ and the mode volume $V$. The green area corresponds to $C\geq1$ and $g\leq\kappa$, and the blue area corresponds to $C\geq10$ and $\sqrt{10}g\leq\kappa$. } \label{QV} \end{figure} As shown in Eq.\,\ref{eq:fidelity}, for high fidelity entanglement, the frequency, polarization, and temporal shape of the photons emitted by the two systems must be matched to maximize $\text{Re}(\ovybin)$. The type of donor used affects the amount of frequency shift required to match the emission frequency of Yb\textsuperscript{+}. Of the three primary donor candidates, Al, Ga and In, the In D\textsuperscript{0}X transition is closest to the Yb$^+$ transition, $v\uin=v\uyb + 0.36$~THz~\cite{meyer2004bed}, where $v\uin$ and $v\uyb$ are the values of the $\ket{0}\rightarrow \ket{e}$ transitions with zero magnetic field, and in the absence a DC Stark shift. The donor will be integrated in an optical cavity detuned from the relevant transition by $\sim$200\,GHz. The remaining frequency shift will be attained via the DC Stark effect. Electric field tuning in a similar quantum dot trion system has shown that several meV of tuning is possible~\cite{bennett2010gse}. Decay from $\ket{\text{e}}$\textsubscript{Yb} ($^2P_{1/2}$ $\ket{F=1, m_F=0}$) can occur along three different channels, producing either a $\sigma^{\pm}$ Raman photon or a $\pi$ Rayleigh photon (see Fig.\ref{trans}). A pure polarization state is required for polarization matching with the photon emitted by the ZnO donor. While the use of a high-NA collection optic increases the photon collection efficiency, it can pose problems for polarization purity. However, the parabolic mirror can be utilized to filter out the undesired $\pi$ polarized photons when the optical axis is oriented along the quantization axis defined by the applied magnetic field~\cite{kim2011ecs}. In this geometry, the $\pi$-polarized photons reflected off the mirror have a radial polarization pattern, which completely destructively interferes when focused into a single-mode optical fiber. The $\sigma$-polarized photons, on the other hand, have an elliptical polarization upon reflection from the mirror. The eccentricity increases with radial distance from the center, with perfectly circular polarization at the center of the reflected beam and linear polarization at the edge. The linear component is filtered out by destructive interference in the optical fiber. \begin{figure} \caption{ZnO energy-level system used in pulse-shaping calculations. The kets represent the In:ZnO state and the associated photon number. } \label{pl} \end{figure} \begin{figure*} \caption{(a) Excitation pulse and temporal wavefunction of the emitted photon for the ZnO system. The parameters used are $\Delta=2\pi\times$(200\,GHz), $\sigma_{1}=8.9$\,ns, $\sigma_{2}=16$\,ns, $\tau=35.8$\,ns, $t_{h}=0.85$\,ns, $\Omega_{max}=2\pi\times$(2.9\,GHz), $\theta_1=2\pi\times$(6.9\, MHz), $\theta_0=2\pi\times(-0.15)$, $g=2\pi\times(15$\,GHz), and $\kappa=2\pi\times$(60\,GHz). (b) Excitation pulse and temporal wavefunction of emitted photon for the Yb\textsuperscript{+} system with $\sigma_1=7.0$\,ns, $\sigma_2=6.4$\,ns, $\tau=28$\,ns, $t_{h}=3.9$\,ns, $\Omega_{max}=2\pi\times$(8.1\,MHz), $\theta_1=0$\, GHz, and $\theta_0=2\pi\times(0.50)$. (c) Imaginary parts of both wavefunctions, leading to $\text{Re}(\ovybin)\simeq0.99$.} \label{pulses} \end{figure*} In the Voigt geometry, with the applied magnetic field perpendicular to the crystal axis, the branching ratio between the ZnO donor Raman transitions $\ket{\text{e}}\uin \rightarrow \ket{0}\uin$ and $\ket{\text{e}}\uin \rightarrow \ket{1}\uin$ is approximately 1:1~\cite{linpeng2018cps,wagner2009gvb}. For a cavity with large $V$ and high $Q$ (e.g. ring resonator~\cite{liu2018uhq}), the cavity resonance will be narrower than the Zeeman splitting of D$^0$, allowing for selective coupling of the desired Raman transition. For high $V$, the size of the cavity is large compared to the excitation beam diameter, so polarization selection can be attained by selectively exciting a small area of the cavity, where only one dipole moment is coupled to the cavity mode. For cavities with low $Q$ and $V$, polarization and frequency selection can be achieved via cross polarization~\cite{meyer2015dpc}, waveguide excitation~\cite{huber2020ffs} and spectral filtering. Matching the temporal profiles of the emitted photons poses a greater challenge. The $^2$P\textsubscript{1/2} Yb\textsuperscript{+} state lifetime is 8.1\,ns~\cite{olmschenk2009ml6}, while that of D\textsuperscript{0}X state in ZnO is only 1.4\,ns~\cite{wagner2011bez}. Post-emission pulse shaping~\cite{wright2017ssq,baek2008tsh} is not feasible because the ZnO and Yb photons are too narrow band for these dispersive methods. Instead, the photons emitted by the ZnO donor can be pulse-shaped at their creation~\cite{law1997dgb} by modulating the intensity of the excitation pulse. The ZnO cavity is constructed with parameters within the ``bad cavity" regime ($\kappa \gg g^2/ \kappa \gg \Gamma\uin$)~\cite{law1997dgb}. The large cavity decay rate ensures that we are not in the strong coupling regime, so the donor excitation follows the optical pulse, while the high cooperativity ensures that the donor decays via Raman emission into the cavity. While it is possible to obtain an analytic expression for the ideal excitation pulse shape for maximum photon overlap~\cite{vasilev2010spm}, in this work we limit ourselves to experimentally attractive Gaussian pulses and performed numerical simulations to determine photon temporal overlap, given the practical cavity considerations discussed above. The donor defect is modeled as a three level system with initial state $\ket{0}\uin$ (Fig. \ref{pl}) connected to the excited state $\ket{\text{e}}\uin$ by an excitation pulse of Rabi frequency $\Omega\uin(t)$ and detuning $\Delta$. We neglect the effect of the other excited state level. The cavity is coupled to the $\ket{e;0}\leftrightarrow\ket{1;1}$ transition with detuning $\Delta$ and coupling strength $g$. Photons from this transition have a spontaneous radiative decay rate of $\Gamma_{\text{In}}$. Photons escape the cavity at the cavity decay rate $\kappa$. The equations of motion for the population amplitudes are~\cite{law1997dgb,vasilev2010spm} \begin{equation} \label{3leveleqz} i \frac{d}{dt}a\uin(t)= \frac{1}{2}\begin{pmatrix} 0 & \Omega\textsubscript{In}(t) & 0\\ \Omega\textsuperscript{*}\textsubscript{In}(t) & 2\Delta - i \Gamma_{\text{In}} & 2g \\ 0 & 2g & -i\kappa \end{pmatrix} a\uin(t), \end{equation} where $a\uin(t)=[a\uinn[0,](t), a\uinn[\text{e},](t), a\uinn[1,](t)]^T$. The Yb\textsuperscript{+} is modeled in a similar manner but without a cavity. The ground state $\ket{0}\uyb$ is coupled to the excited state $\ket{\text{e}}\uyb$ by the Rabi pulse $\Omega\uyb(t)$. Decay from the excited state occurs with the rate $\Gamma_{\text{Yb}}$. The equations of motion are: \begin{equation} \label{3leveleqy} i \frac{d}{dt} \begin{pmatrix} a\textsubscript{0,Yb}(t) \\ a\textsubscript{e,Yb}(t) \\ \end{pmatrix}= \frac{1}{2}\begin{pmatrix} 0 & \Omega\textsubscript{Yb}(t)\\ \Omega\textsuperscript{*}\textsubscript{Yb}(t) & - i \Gamma_{\text{Yb}}\\ \end{pmatrix} \begin{pmatrix} a\textsubscript{0,Yb}(t) \\ a\textsubscript{e,Yb}(t) \\ \end{pmatrix} \end{equation} The emission rates of the photons from the ZnO and Yb\textsuperscript{+} systems are $\kappa|a\uinn[1,](t)|^2$ and $\Gamma\uyb|a\uybn[e,](t)|^2$, respectively~\cite{law1997dgb}, with temporal wavefunctions given by normalizing the population amplitudes $a\textsubscript{1,In} (t)\rightarrow A\textsubscript{1,In}(t)$ and $a\textsubscript{e,Yb}(t)\rightarrow A\textsubscript{e,Yb}(t)$. By controlling the Rabi frequencies $\Omega\uin (t)$ and $\Omega\uyb (t)$, it is possible to engineer the real component of the photon overlap $\int_{-\infty}^{\infty}A^{*}\uybn[e,](t)A\uinn[1,](t)dt=\ovybin$ to \textasciitilde 0.99 for practical experimental parameters using the control pulses shown in Fig.\,\ref{pulses}. The optimized pulse is restricted to a Gaussian pulse shape with adjustable rise time $\sigma_{1}$, fall time $\sigma_{2}$, time to pulse max $\tau$, hold time $t_{h}$, maximum pulse height $\Omega_{max}$, and phase factor e\textsuperscript{$i \alpha(t)$} where $\alpha(t)=\theta_0+\theta_1 t$ describes a linear time-dependent phase. Setting either pulse to achieve a desired excitation probability $p_{1,x}$, we iteratively sweep the pulse parameters for the other system to obtain local maxima in the overlap. The probability of successful entanglement is \begin{equation} P_{\text{succ}}=[p\uybn[1,]p\uybn[2,](1-p\uinn[1,]) +p\uinn[1,]p\uinn[2,](1-p\uybn[1,])]\eta \end{equation} where $p_{2,x}$ is the collection efficiency from each system, and $\eta$ is the quantum efficiency of the detector, which can be as high as \textasciitilde 80\% using superconducting nanowire single photon detectors (SNSPD's)~\cite{Crain2019} for photons at 369\,nm. With a parabolic mirror ion trap, collection efficiency for Yb\textsuperscript{+} systems is 32\%; the ZnO system is set to 34\% collection efficiency to match the coefficients in Eq.~2 to create a maximally entangled state. Excitation probabilities depend on the pulse shaping requirements, and need to be kept low (<10\%) to minimize error. For good fidelity while still maintaining a reasonably high success probability, we use excitation probabilities around 5\%. Each experimental run begins with \textasciitilde1\,\textmu s of optical pumping, followed by the \textasciitilde 10\,ns excitation pulse. If a single photon is detected, then the state readout is performed, taking \textasciitilde10\,\textmu s and limited by the ion~\cite{Crain2019}. We find a success probability of \textasciitilde2.7\%, leading to an entanglement generation rate of 21\,kHz. Practically, this rate will be further decreased by the interferometer phase stabilization and defect frequency stabilization steps~\cite{Humphreys2018}. With all experiments using this type of protocol, there is a tradeoff between success probability and fidelity~\cite{slodivcka2013aae,Humphreys2018}. One can always increase the success probability by increasing the excitation probability, but this degrades the fidelity according to Eq \ref{eq:fidelity}. Further, in order to be useful, the entanglement rate needs to be comparable to the rate of decoherence. While the demonstrated coherence time for trapped ytterbium ions is long~\cite{Wang2017} (10 minutes), the spin echo time T\textsubscript{2} of ensemble donor bound excitons in ZnO is only 50\,\textmu s. However, the fundamental limit of T\textsubscript{2} is the longitudinal spin relaxation time T\textsubscript{1} which exceeds 100~ms~\cite{linpeng2018cps} and may allow for improvement through chemical and isotope purification~\cite{tribollet2009ten}. In summary, a ZnO donor defect qubit and a single trapped Yb\textsuperscript{+} ion can be remotely entangled via a photonic link at 369\,nm. Pulse shaping techniques can be used to alter the temporal profile of the photon emitted by the donor to attain the temporal wavefunction overlap of 0.99 with the photon emitted by the trapped ion, leading to an entangled state fidelity of 94\% with realistic parameters. \section*{Supplementary Material} See supplemental material for a derivation of the fidelity expression (Eq. \ref{eq:fidelity}). \section*{AIP Publishing Data Sharing Policy} The data that supports the findings of this study are available within the article. \section{Supplemental Material} \subsection{Maximally entangled state and fidelity} We first define the state$\ket{\Psi_{1}}$ of $\text{Yb}^{+}$ and the state $\ket{\Psi_{2}}$ of the In donor. We begin by optically pumping both systems into the ground state \begin{subequations} \begin{align} \ket{\Psi_{1}}& = \ket{0}\uyb \tag{S1a}\\ \ket{\Psi_{2}}& = \ket{0}\uin. \tag{S1b} \end{align} \end{subequations} We now apply an excitation pulse to both species with $p_{1,x}\ll 1$ so that the probability of both systems being excited during the same experimental run is small. The states of both systems are given by: \begin{equation} \ket{\Psi_{1}}=\sqrt{p\uybn[1,]}e^{i\phi_{D_1}}\ket{1}\ket{\zeta\uyb}+\sqrt{1-p\uybn[1,]}e^{i\phi_{L_1}}\ket{0}\vac \tag{S2} \end{equation} \begin{equation} \ket{\Psi_{2}}=\sqrt{p\uinn[1,]}e^{i\phi_{D_2}}\ket{1}\ket{\zeta\uin}+\sqrt{1-p\uinn[1,]}e^{i\phi_{L_2}}\ket{0}\vac \tag{S3} \end{equation} where $\phi_{L_x}$ denotes the phase of the laser at species x, $\phi_{D_x}$ denotes the phase of the emitted photon after travelling a distance $D_{x}$, $\vac$ is the vacuum state, and $\ket{\zeta\uyb}=\sum_\omega \xi\uyb[,\omega] \ryb \vac$ and $\ket{\zeta\uin}=\sum_{\omega^{\prime}} \xi\uin[,\omega^{\prime}] \rin \vac$ are the temporal wavefunctions of emitted photons from each system. The temporal wavefunctions are given by a sum over all modes $\omega$ ($\omega^\prime$) with coefficients $\xi\uyb[,\omega]$ ($\xi\uin[,\omega^\prime]$) and raising operators $\ryb$ ($\rin$). We phase lock the laser systems to set $\phi_{L_1}=\phi_{L_2}=0$. Assuming we collect a single photon with efficiency $p_{2,x}$ from either system, we obtain the state (not normalized) \begin{equation} \begin{split} \ket{\Psi_{1,2}}= \sqrt{p\uybn[1,](1-p\uinn[1,])p\uybn[2,]}\ket{1,0}\sum_\omega \xi\uyb[,\omega] \ryb \vac \\ +\sqrt{p\uinn[1,](1-p\uybn[1,])p\uinn[2,]}e^{i\Delta\phi}\ket{0,1}\sum_{\omega^{'}}^{} \xi\uin[,\omega^{'}] \rin \vac \end{split} \tag{S4} \end{equation} where $\Delta\phi=\phi_{D_2}-\phi_{D_1}$ is the difference in optical path length between the two qubit systems and we have dropped the terms $\ket{1,1}$ and $\ket{0,0}$, since they will eventually be projected out upon the detection of a single photon. Here, we note that to obtain a maximally entangled state we want to set \begin{equation} p\uybn[1,](1-p\uinn[1,])p\uybn[2,]=p\uinn[1,](1-p\uybn[1,])p\uinn[2,]. \tag{S5} \end{equation} Since we use $p_{1,x}$ to achieve good temporal overlap, and typically $p\uybn[2,]<p\uinn[2,]$, this is accomplished by lowering $p\uinn[2,]$. Now, at the beamsplitter we choose the transformation $\ryb \rightarrow (\cyb + i \dyb)/\sqrt{2},\; \rin \rightarrow (\din + i \cin)/\sqrt{2}$ , where $\ryb$, $\rin$ are the raising operators of the respective paths A and B and $\cyb$, $\dyb$ are the raising operators in paths C and D, as depicted in Fig, 2 of main text. We also have that $i(\phi_{D_2}-\phi_{D_1})=i\Delta\phi$ where $\Delta\phi$ is the difference in phase between photons traversed from each system. To account for the reflection of one of the two paths in the beamsplitter, we set a phase difference of $\frac{\pi}{2}$ between $\ket{1,0}$ and $\ket{0,1}$ states. We then obtain the entangled state upon detection of a single photon \begin{equation} \begin{split} \ket{\Psi_{1,2}}&=\frac{1}{\sqrt{2}} [ \ket{1,0}\sum_\omega \xi\uyb[,\omega] \frac{\cyb+i\dyb}{\sqrt{2}} \vac \\ & -ie^{i\Delta\phi}\ket{0,1}\sum_{\omega^{'}}^{} \xi\uin[,\omega^{'}] \frac{\din+i\cin}{\sqrt{2}} \vac ] \end{split} \tag{S6} \end{equation} The density matrix can then be computed. Let us first assume the photon was detected on path D, and not on path C. Tracing over photon states in the path D, and over all photon frequencies $\omega$, we obtain \begin{equation} \begin{split} \rho^{\text{Yb,In,D}}=\frac{1}{4}& [ \sum_\omega \xi^{*}\uyb[,\omega]\xi\uyb[,\omega] \ket{1,0}\bra{1,0} \\ + &\sum_{\omega}\xi^{*}\uyb[,\omega]\xi\uyb[,\omega] \ket{0,1}\bra{0,1}\\ -ie^{i\Delta\phi}&\sum_\omega \xi^{*}\uyb[,\omega]\xi\uin[,\omega] \ket{0,1}\bra{1,0} \\ +ie^{-i\Delta\phi} & \sum_{\omega}^{} \xi^{*}\uin[,\omega]\xi\uyb[,\omega] \ket{1,0}\bra{0,1} ] \end{split} \tag{S7} \end{equation} The same matrix can be found for the path C. Summing the density matrices we then find the complete density matrix including paths C and D \begin{equation} \begin{split} \rho^{\text{Yb,In}}=\frac{1}{2}( \ket{1,0}\bra{1,0}+ \ket{0,1}\bra{0,1}\\ -ie^{i\Delta\phi}\ovybin \ket{0,1}\bra{1,0}\\ +ie^{-i\Delta\phi}\ovinyb \ket{1,0}\bra{0,1}) \end{split} \tag{S8} \end{equation} where we have used the relations $\ovinin=\ovybyb=1$ and $\ovybin=\ovinyb^{*}=\sum_{\omega}^{} \xi^{*}\uin[,\omega]\xi\uyb[,\omega]$. Finally, we compute the fidelity using the target state $\ket{\Psi_\text{ent}}=\frac{1}{\sqrt{2}}(\ket{1,0}-ie^{i\Delta\phi}\ket{0,1})$ \begin{equation} F=\bra{\Psi_\text{ent}} \rho^{\text{Yb,In}} \ket{\Psi_\text{ent}}=\frac{1}{2}\Big[1+\text{Re}(\ovybin)\Big] \tag{S9} \end{equation} \subsection{Double Excitations} Here we will derive the parameter $c_1$ of Eq.~5 of the main text. In an experimental set-up, when a photon is detected on either detector, there is a non-zero probability that both qubits were excited but only one was detected. This probability is given by \begin{equation} \begin{split} p_\text{double} = & [p\uybn[1,]p\uinn[1,]p\uybn[2,](1-p\uinn[2,])] \\ + & [p\uybn[1,]p\uinn[1,]p\uinn[2,](1-p\uybn[2,])] \end{split} \tag{S10} \end{equation} where the two terms come from the probability of detecting one photon from either qubit that has decayed from its excited state. There is a phase of $\pi/2$ between these two photons as with the state in Eq. S6, and an additional phase factor determined by the total optical path length of the In system $\phi_{D_2}=ikD_2$. Following through the same process, we arrive at an entangled state \begin{equation} \begin{split} \ket{\Psi_{1,2}}=\frac{1}{\sqrt{2+c_{1}^2}} \Bigg[ \ket{1,0}\sum_\omega \xi\uyb[,\omega] \frac{\cyb+i\dyb}{\sqrt{2}} \vac\\ -i e^{i\Delta\phi}\ket{0,1}\sum_{\omega^{'}}^{} \xi\uin[,\omega^{'}] \frac{\din+i\cin}{\sqrt{2}} \vac ]\\ +c_{1}\ket{1,1} \Big[ \sum_\omega \xi\uyb[,\omega] \frac{\cyb+i\dyb}{\sqrt{2}} \vac\\-ie^{i(\phi_{D_2})}\sum_{\omega^{'}}^{} \xi\uin[,\omega^{'}] \frac{\din+i\cin}{\sqrt{2}} \vac \Big]\Bigg] \end{split} \tag{S11} \end{equation} \hspace{-8pt}where we have that \begin{equation} \begin{split} c_{1} = & \frac{\sqrt{p_{double}}}{\sqrt{p\uybn[1,](1-p\uinn[1,])p\uybn[2,]}}\\ = & \frac{\sqrt{p\uinn[1,](p\uybn[2,](1-p\uinn[2,])+p\uinn[2,](1-p\uybn[2,]))}}{\sqrt{(1-p\uinn[1,])p\uybn[2,]}}. \end{split} \tag{S12} \end{equation} Since the target state has no $\ket{1,1}$ component, when we calculate the fidelity, we obtain the same result as before, with the only modification being the prefactor $\frac{1}{\sqrt{2}}\rightarrow\frac{1}{\sqrt{2+c_{1}^{2}}}$. Including the effect of $F_{dyn}$ from Cabrillo \textit{et al.}\cite{cabrillo1999ces} on photon distinguishibility, we then obtain the fidelity equation found in the main text: \begin{equation} F=\frac{1}{2+c_{1}^{2}}(1+F\utxt{dyn}\text{Re}(\ovybin). \tag{S13} \end{equation} \end{document}

\begin{document} \title{Extracting work from quantum systems} \author{Paul Skrzypczyk}\affiliation{Department of Applied Mathematics and Theoretical Physics$\text{,}$ University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom} \author{Anthony J.~Short}\affiliation{H. H. Wills Physics Laboratory, University of Bristol$\text{,}$ Tyndall Avenue, Bristol, BS8 1TL, United Kingdom} \author{Sandu Popescu} \affiliation{H. H. Wills Physics Laboratory, University of Bristol$\text{,}$ Tyndall Avenue, Bristol, BS8 1TL, United Kingdom} \begin{abstract} We consider the task of extracting work from quantum systems in the resource theory perspective of thermodynamics, where free states are arbitrary thermal states, and allowed operations are energy conserving unitary transformations. Taking as our work storage system a `weight' we prove the second law and then present simple protocols which extract \emph{average} work equal to the free energy change of the system -- the same amount as in classical thermodynamics. Crucially, for systems in `classical' states (mixtures of energy eigenstates) our protocol works on a single copy of the system. This is in sharp contrast to previous results, which showed that in case of \emph{almost-deterministic} work extraction, collective actions on multiple copies are necessary to extract the free energy. This establishes the fact that free energy is a meaningful notion even for individual systems in classical states. However, for non-classical states, where coherences between energy levels exist, we prove that collective actions are necessary, so long as no external sources of coherence are used. \end{abstract} \maketitle \section{Introduction} Thermodynamics forms part of the bedrock of our current understanding of the physical world. It has remained unchanged despite huge revolutions in physics, such as relativity and quantum theory, and few believe it will ever fail. Over time, it has been applied to situations well outside its original domain; from black holes \cite{Bek73,BarCarHaw73}, to quantum engines comprised of only a few qubits \cite{GevKos96,YouMahOba09,TonMah05,LinPopSkr10a}. A fundamental question is that of the applicability of thermodynamics to quantum systems \cite{GemMicMah04,GyfBer05}; it is this question that we wish to address in this paper. One of the most fundamental questions one would like to ask is how much work can be extracted from a system, given access to a thermal bath at temperature $T$. This question, for generic quantum systems and Hamiltonians has been investigated intensively, e.g. in \cite{ProLev76,AliHorHor04,HasIshDri10,TakHasDri10,EspVan11,Abe11,VerLac12} yet, until very recently only partial results have been obtained and it remained an open problem. Very recently however strong general results were obtained \cite{HorOpp11}, made possible by employing a new conceptual framework, namely that of ``resource theories". The advent of resource theories is one of the main paradigm shifts which occurred within quantum information theory; the main idea is to bring to the forefront the study of \emph{resources} needed to implement various tasks. The archetypal resource theory in this context is entanglement theory \cite{BenBerPop96,HorHorHor09}. However, numerous other examples have now been developed, including reference frames \cite{BarRudSpe07}, purity \cite{HorHorOpp03} and asymmetry \cite{MarSpe11}, which has culminated in a general formalised notion of a resource theory \cite{HorOpp12}. In \cite{BraHorOpp11} the authors introduced an appealing resource theory of thermodynamics, based on the class of allowed operations first studied in \cite{JanWocZei00}. Here we revisit and modify the resource theory framework of thermodynamics as originally put forward in \cite{BraHorOpp11}. In this new context we achieve two main results: First we prove the second law of thermodynamics. Second, as described in more detail below, we show that free energy is a relevant quantity for individual quantum systems. The latter result is particularly relevant following the results of \cite{HorOpp11} which call into question the role of free energy for individual quantum systems. Classical thermodynamics tells us that the total amount of work we are able to extract from a system is given by its change in \emph{free energy}, which was also supported by previous partial quantum results \cite{ProLev76,AliHorHor04,HasIshDri10,TakHasDri10,EspVan11,Abe11,VerLac12}. Yet in \cite{HorOpp11} it was shown that work equal to free energy can be extracted only if we {\it collectively} process many copies of the same system. When acting on each copy individually, the amount of work that can be extracted is generally significantly less than the free energy. These results therefore suggest that the free energy is not the relevant quantity for individual systems. Here we revisit the issue of work extraction and show that free energy is a significant quantity for individual systems. Our paradigm is similar to that of \cite{HorOpp11} but differs in an essential aspect. In \cite{HorOpp11} they considered {\it almost deterministic}\footnote{Specifically, they require that a deterministic amount of work is extracted, except with some small probability $\epsilon$ of failure} work extraction, from the `single-shot' viewpoint which has received much attention lately \cite{DahRenRie11,Abe11,RioAbeRen11,EglDahRen12,FaiDupOpp12}. Here, in contrast, we will consider {\it average} work extraction. We prove that there exist simple protocols which extract \emph{average} work equal to the free energy change of the system. Crucially, for systems in `classical' states (mixtures of energy eigenstates) our protocol works on a single copy of the system. This establishes the fact that free energy is a meaningful notion even for individual systems, when their states are classical. However, for non-classical states, where coherences between energy levels exist, we prove that collective actions are necessary, so long as no external sources of coherence are used. More precisely, we prove that when acting on an individual system the amount of average work that can be extracted is equal to the free energy of its decohered state -- the free energy due to coherences is `locked' and cannot be used. As more copies are collectively processed, some of this locked free energy is released -- we can extract the free energy of the decohered state of $n$ copies, which is larger than $n$ times the free energy of a single decohered system. In the limit of infinitely many copies, the average work extracted per copy reaches its entire free energy. Whether or not external coherent resources allow extraction of work equal to the free energy for individual systems, and how to account for such resources (given that they may provide free energy themselves even without the system) remains an open question. The paper is organised as follows. In Sec.~\ref{s:preliminaries} we go through the preliminaries, including a brief description of the resource theory of thermodynamics in Sec.~\ref{ss:resource theory}, work storage systems in Sec.~\ref{ss:work storage} and a proof that the second law holds in Sec.~\ref{s:second_law}. In Sec.~\ref{s:diagonal} we study work extraction from diagonal systems, beginning with a motivating example in Sec.~\ref{s:motivating example}, before moving onto a full qubit protocol in Sec.~\ref{ss:full protocol}. We discuss a special case in Sec.~\ref{ss:maxwell demon}, present a full protocol for arbitrary dimension systems in Sec.~\ref{ss:qudits} and prove optimality in Sec.~\ref{s:optimality}. In Sec.~\ref{s:coherence} we show how work can be extracted from coherence by acting collectively on multiple copies of the system. We finish by briefly discussing how the results of this paper can be understood from the perspective of `virtual temperatures' in Sec.~\ref{s:virtual}, before concluding in Sec.~\ref{s:conclusions}. \section{Preliminaries}\label{s:preliminaries} \subsection{The resource theory of thermodynamics}\label{ss:resource theory} The first ingredient in a resource theory is a set of free resources. By free we mean that we have an unlimited supply of this resource, since it is deemed easy to acquire. In the present context of thermodynamics this free resource is \emph{a thermal bath at temperature $T$}. More precisely, we assume that we have an unlimited supply of finite-dimensional systems, with any desired Hamiltonian $H$, in their thermal state $\tau_\beta(H)$, \begin{equation} \tau_\beta(H) = \tfrac{1}{\mathcal{Z}}e^{-\beta H} \end{equation} where $\beta = 1/T$ is the inverse temperature\footnote{We set $k_B=1$ throughout for convenience}, and $\mathcal{Z}~=~\textrm{tr}(e^{-\beta H})$ is the partition function. The second ingredient is a set of allowed operations, those which one has the ability to perform with ease. In the original resource framework these allowed operations were \emph{any energy conserving unitary transformation}. In particular, any unitary operation which commutes with the total (non-interacting) Hamiltonian of all the systems under consideration. This allows one to transform systems in a controlled way, but without admitting an external source of energy. However, we will also introduce an additional element into the resource framework, a \emph{work storage system}, whose sole purpose is to provide and store work for thermodynamic transformations. In the next section, we will place some additional constraints on the allowed operations, which prevent us from `cheating' by using work storage system for other purposes (e.g. by using it as a cold reservoir in a heat engine), and allow us to use the same work storage system for multiple thermodynamic protocols. Given these ingredients, any additional system which is not in thermal equilibrium with a bath at temperature $T$ is a useful resource. The question that remains is to quantify how useful different resource states are. \subsection{Work storage systems}\label{ss:work storage} One way to quantify the usefulness of different resource states is to find a transformation from every resource state to some \emph{fiducial} resource which allows for comparison in a well-defined and easy manner. In the resource theory of entanglement, for example, one can quantify the entanglement of a state by the optimal asymptotic number of maximally entangled states that can be created per resource state, or in other words the optimal rate at which states can be converted into maximally entangled states. Different states will have different optimal rates, and thus the rate quantifies how useful they are. We seek here an analogous fiducial resource for thermodynamics. Work is the desired output in this case, and different states can produce work at different rates. Because we wish to study a closed quantum system, with conserved total energy, we desire a system where we can store the work extracted from resource states. Similarly to entanglement theory, where the maximally entangled state is chosen for convenience, here we wish to find a work storage system which we deem to be most convenient. In previous work \cite{HorOpp11} a candidate was put forward -- raising a qubit deterministically from its ground state to it's excited state. This qubit was termed a \emph{wit}, short for \emph{work bit}. However, choosing the energy gap of the work bit requires advance knowledge of the work to be extracted, and this model does not translate well to non-deterministic work extraction, which we will be interested in here. Furthermore, we would prefer to be able to use a single work storage system as a `battery' capable of gaining and expending work in multiple thermodynamic processes. The alternative work storage system we will consider here is a suspended \emph{weight}, which is raised or lowered when work is done on or by it. In particular, we will consider a quantum system whose height is given by the position operator $\hat{x}$, with Hamiltonian $H_w= m g \hat{x}$ representing its gravitational potential energy\footnote{Note that it is not strictly necessary for the height of the weight to be a continuous degree of freedom. In Appendix \ref{a:discrete work} we show that the weight can be replaced by a system with discrete equally-spaced energy levels. This introduces a small error in the results, but this can be made as small as desired by appropriately scaling the spacing of the weight.}. For simplicity, we choose $mg =1 \textrm{ J/m}$, such that the value of $\hat{x}$ directly denotes the work stored by the mass. Such a system has a long history of being used as a work storage system in classical thermodynamics \cite{LieYng99}. We will place two additional constraints on the allowed dynamics governing interactions with the weight:\renewcommand{(\roman{enumi})}{(\roman{enumi})} \begin{enumerate} \item The average amount of work extracted in an allowed protocol must be independent of the initial state of the weight.\label{i:I} \item All allowed unitaries should commute with translation operations on the weight. This reflects the translational symmetry of the weight system, and the fact that only displacements in its height are important.\label{i:II} \end{enumerate} These constraints are intended to prevent us from `cheating' by using the weight for purposes other than as a work storage system (e.g. as a cold reservoir, or a source of coherence). For example if the weight starts in a pure state, it has zero entropy, so one needs to be careful to avoid unwittingly using this as a source of free energy, in addition to the system. Constraint (i) encapsulates this requirement. Constraint (ii) is a bit more subtle. On the one hand, it is very plausible that it is a consequence of constraint (i). We however have not been able to show this. On the other hand, even if it does not follow from (i), we would still like to assume it, as it seems wrong to use properties of the weight in the protocol, even if they do not affect the work extracted. Furthermore, constraint (i) ensures that we can use the same work storage system for multiple thermodynamics protocols (or on several copies of the same state) without having to worry how the initial state has been modified by earlier procedures. As we are free to choose any initial state of the weight due to constraint (i) above, for simplicity we will usually take the initial state of the weight to be a narrow normalised wavepacket $\ket{0}_w$ centred on the origin (i.e. $\bra{0} \hat{x} \ket{0} =0$). When the same wavepacket is raised above the origin by a distance $a$, we will denote the state by $\ket{a}_w=\Gamma_a \ket{0}_w$, where $\Gamma_a$ is the translation operator\footnote{The translation operator is $\Gamma_a=\exp(-i a \hat{p} / \hbar)$, where $\hat{p}$ is the usual momentum operator satisfying $[\hat{x}, \hat{p}] = i \hbar$. }. \section{The second law} \label{s:second_law} We now show that the second law of thermodynamics holds in our framework, by proving a probabilistic version of the second law -- that there is no protocol which extracts a positive quantity of average work from a thermal bath (i.e. that there is no way on average of turning heat into work) \cite{LadPreSho08}. To show this we will use proof by contradiction. Consider a thermal bath at temperature $T$ and the weight, i.e. there is no additional system out of thermal equilibrium. Let us first consider the energy changes during the protocol. Suppose that we are able to extract work from the bath and store it in the weight, $\Delta E_W > 0$. The average energy of the thermal bath must change by $\Delta E_B = - \Delta E_W $ due to energy conservation. Now consider the entropy changes during the same protocol (in particular, the changes in von Neumann entropy $S(\rho) = - \textrm{tr} (\rho \log \rho))$. As the bath and weight are initially uncorrelated, their initial entropy is simply the sum of their individual entropies. Unitary transformations conserve the total entropy, $\Delta S_{BW} = 0$. However, as correlations can form during the protocol, the sum of the final entropies of the bath and weight can be greater than the sum of their initial entropies (as the entropy is subadditive). This means that \begin{equation}\label{e:entropy eqn} \Delta S_B + \Delta S_W \geq \Delta S_{BW} = 0 \end{equation} Given an initial thermal state for the bath (with positive temperature), any change of the state which reduces its average energy must also reduce its entropy\footnote{This follows from the fact that the thermal state is the maximal entropy state with given average energy. For more details see Appendix \ref{a:second_law}}, $\Delta S_B < 0$. However, within our framework all allowed protocols are such that the average work extracted is independent of the initial state of the weight; we are therefore free to choose any initial state of the weight we like. Consider now that the initial state of the weight is a very wide wavepacket. We show in Appendix \ref{a:second_law} that the entropy change of the weight in this case can be made as small as desired, in particular we can make $\Delta S_W < |\Delta S_B|$. This would result in violating \eqref{e:entropy eqn}. Hence there is a contradiction, and thus there is no way to extract work from the bath. \section{Extracting work from diagonal systems}\label{s:diagonal} We will begin our exploration of work extraction by considering the special case of extracting work from systems whose states are diagonal in the energy eigenbasis. This represents systems which are essentially classical, since coherence plays no role, and states are simply mixtures of energy eigenstates. We will present a protocol which extracts work equal to the change in free energy of the system. In contrast to existing protocols, here our protocol will take one copy of the system, and extract average work equal to the free energy of the system. The protocol contains at its core a single elementary building block which is able to extract a vanishing amount of work blue, with maximal efficiency. The full protocol consists of many repetitions of this one step. We shall present first this building block in a motivating example and then show precisely how it is used to build the full protocol. In the main text the emphasis will be on the results of the protocol, with the more detailed analysis provided in the appendices. \subsection{A motivating example}\label{s:motivating example} Let us consider the task of performing the following transformation on a single qubit \begin{eqnarray}\label{e:rhoS} \rho_S &=& (1-p)\proj{0}{S} + p \proj{1}{S} \nonumber \\ &\downarrow \\ \rho_S' &=& (1-p+\delta \!\!p)\proj{0}{S} + (p-\delta \!\!p)\proj{1}{S} \nonumber \end{eqnarray} whose Hamiltonian is $H_S = E_S \proj{1}{S}$ and where $\delta \!\!p \ll 1$. That is, we wish to consider the small transformation from a diagonal state to a nearby diagonal state. The initial and final state differ in free energy, and hence the goal is to extract an amount of work as close as possible to this difference, namely \begin{align} \delta \!F_\beta &= F_\beta(\rho) - F_\beta(\rho') \nonumber \\ &= \left(\Exp{E}_{\rho} - T S(\rho)\right) - \left(\Exp{E}_{\rho'} - T S(\rho')\right) \nonumber \\ &\simeq \delta \!\!p \left(E_S - T S_c'(p)\right) \end{align} where $F_\beta(\sigma)=\Exp{E}_{\sigma} - TS(\sigma)$ is the free energy of $\sigma$ with respect to a bath at inverse temperature $\beta$, $\Exp{E}_\sigma = \textrm{tr}\left(H_S \sigma\right)$ is the average energy, $S_c(q)~=~-q \log q~-~(1-q)\log(1-q)$ is the (classical) binary entropy, $S_c'(q) = \frac{dS_c}{dq}$, and we have taken the first order expansion in $\delta \!\!p$ to arrive at the final line. We shall now see that to first order we can extract exactly this amount of work from the system. To do so let us bring in a thermal state from the bath -- the free resource at our disposal -- and choose its Hamiltonian such that the population in the excited state is $p-\delta \!\!p$. For thermal qubits the relation between excited state population $r$ and energy spacing $E_B$ is \begin{equation} E_B = T\log\left(\frac{1-r}{r}\right) = TS_c'(r) \end{equation} thus we will choose the Hamiltonian of the bath qubit to be $H_B = E_B\proj{1}{B}$ with \begin{align} E_B &= T S_c'(p-\delta \!\!p) \simeq T S_c'(p) - \delta \!\!p T S_c''(p) \end{align} Let us take the weight to be prepared initially in the state $\kets{0}{w}$ of zero average energy, and consider applying an energy conserving unitary transformation $U$ which interchanges \begin{equation}\label{e:U mot} \kets{0}{B}\kets{1}{S}\kets{x}{w} \leftrightarrow \kets{1}{B}\kets{0}{S}\kets{x+E_S-E_B}{w} \end{equation} for any $x$, whilst leaving all orthogonal states unchanged\footnote{$U$ can be expressed in terms of the translation operator $\Gamma_a$ as \begin{equation*} U=\ket{10}\bra{01} \otimes\Gamma_{\Delta } + \ket{01}\bra{10} \otimes\Gamma_{-\Delta } + \left(\ket{00}\bra{00} + \ket{11}\bra{11}\right)\otimes\openone, \end{equation*} where $\Delta = E_S -E_B$. Note that this commutes with the free Hamiltonian $H_S+H_B+H_w$.}. It is the change in average energy of the weight which is of interest. A straightforward calculation (given for completeness in the Appendix) shows that this is given by \begin{align} \delta \Exp{E}_w = \delta \!W &= \delta \!\!p(E_S - E_B) \simeq \delta \!\!p(E_S - T S_c'(p)) \end{align} which is equal to $\delta \!F_\beta$ to first order in $\delta \!\!p$. Thus we observe that if we wish to to make a change of state from $\rho$ to $\rho'$, a way to proceed is to bring in a qubit from the bath in the state $\tau_\beta(H_B) = \rho'$. Whenever $\rho$ is not the thermal state (given it's Hamiltonian $H_S$), we find that $H_B \neq H_S$, and therefore there is a difference in energy between the excited states of the two qubits. It is thus not possible to swap the state of the two qubits using an energy conserving unitary. However, by bringing in the weight we can perform the unitary \eqref{e:U mot} which is energy conserving, and, at the level of the reduced states of the system and bath, indeed performs a swap. The system therefore ends up in the desired state $\rho'$, and we find that the change in average energy of the weight, when the state of the system and bath are close, approaches exactly the change in free energy of the system. \subsection{Full qubit protocol}\label{ss:full protocol} The above insight can be used to build a full protocol which is able to extract an amount of work approaching the free energy change in transforming a system from a state $\rho$ to its thermal state $\tau_\beta(H_S)$. The basic idea is that, since in making a small change of state we have seen that the work extracted approaches the free energy change, we can divide a large change of state into many small changes and at each stage extract the desired amount of work. Therefore, let us fix an integer $N \gg 1$, and consider $N$ thermal qubits $\tau_\beta^{(k)}(H_B^{(k)})$, $k = 1,\ldots,N$, \begin{equation} \begin{split} \tau_\beta^{(k)}(H_B^{(k)}) &= (1-r^{(k)})\proj{0}{{B_k}} + r^{(k)}\proj{1}{{B_k}}\\ r^{(k)} &= p + \tfrac{k}{N}(p_{eq}-p) \end{split} \end{equation} where $p_{eq} = \frac{e^{-\beta E_s}}{1+e^{-\beta E_s}}$ is the excited-state probability in the thermal state $\tau_\beta(H_S)$. This collection of bath qubits have excited-state probabilities which vary linearly from that of the initial state of the system $\rho_S$ to that of the thermal state of the system $\tau_\beta(H_S)$, in increments of $\tfrac{(p_{eq}-p)}{N}$. We will apply, in turn, the sequence of unitary transformations $\{U^{(k)}\}$, where $U^{(k)}$ acts on bath qubit $k$, system and weight and swaps the states \begin{equation} \kets{0}{{B_k}}\kets{1}{S}\kets{x}{w} \leftrightarrow \kets{1}{{B_k}}\kets{0}{S}\kets{x+E_S-E_B^{(k)}}{w} \end{equation} whilst leaving all other orthogonal states unchanged. Assuming once again that the weight starts in the state $\kets{0}{w}$, it is the final state, and in particular the final average energy of the weight which is of interest, as the number of steps $N$ tends to infinity. In Appendix \ref{a:weight} we calculate explicitly this state in the asymptotic limit. The result is that the final reduced state in the infinite limit, denoted $\Omega_w^\infty$, is \begin{multline}\label{e:asymptotic ladder} \Omega_w^\infty = (1-p)\proj{T\log\left(\tfrac{1-p}{1-p_{eq}}\right)}{w} \\+ p\proj{T\log\left(\tfrac{p}{p_{eq}}\right)}{w} \end{multline} The average energy is thus \begin{align} \Exp{E}_w^\infty &= T\left(p \log\left(\tfrac{p}{p_{eq}}\right) + (1-p) \log\left(\tfrac{1-p}{1-p_{eq}}\right)\right) \nonumber \\ &= T D_c(p||p_{eq}) = F_\beta(\rho) - F_\beta\left(\tau_\beta(H_S)\right) \end{align} where $D_c(p||p_{eq})$ is the relative binary entropy. We observe that the average energy of the weight at the end of the protocol, and hence the average work extracted, approaches precisely the free energy difference in the asymptotic limit of infinitely many thermal qubits. Although this is promising, it should be noted that the final distribution of the weight does not have the form one may intuitively have asked for (i.e. a peak around a single value, at the free energy itself). Instead we find that the solution has perfect correlation between the initial state of the system and the final energy of the weight. It is worth noting that this interesting distribution coincides with the results found in \cite{Abe11} where, even though the details of the allowed transformations differ, it was shown that for classical systems, optimal work extraction protocols converge on this distribution in probability. Note also that in the large $N$ limit, although many thermal qubits are required, each of them is only slightly perturbed by the process. Finally, we consider the case of $n$ independent copies of the system -- i.e. the macroscopic limit. To do so we simply repeat the protocol above on each qubit separately, using a single weight to store the work. In each case we have seen that the weight is transformed from a single peak (at $\kets{0}{w}$ above) to two peaks. Since the protocol is linear, it follows immediately the the final distribution of the weight will be the $n$-fold convolution of the above distribution for a single qubit. From the central limit theorem one concludes that the final average energy of the weight is \begin{equation} \Exp{E}_w = n \left(F_\beta(\rho) - F_\beta(\tau_\beta(H_S))\right) \end{equation} with fluctuation of the order $\sqrt{n}$. For large $n$ these fluctuations will be insignificant compared to the average, and we obtain an almost deterministic quantity of work, with the amount of work extracted per copy of the system equal to the free energy difference. The results presented here are completely consistent with those presented in \cite{BraHorOpp11}. It should be noted however the stark difference in which the two protocols achieve the same goal. In \cite{BraHorOpp11} the analysis first considers all $n$ systems together and uses ideas from information theory to compress information within large typical subspaces. Here, on the other hand, each qubit is manipulated separately and nowhere is the idea of a typical subspace ever introduced, nevertheless they both achieve the same end result. \subsection{Special case: isothermal expansion}\label{ss:maxwell demon} The protocol outlined in the previous subsection is the general protocol which allows for the extraction of work from an arbitrary diagonal state of a qubit, for any given Hamiltonian. There is one particular choice of state that will be particularly important in later sections. As such, we shall briefly look at it in more detail. Consider a qubit for which the Hamiltonian vanishes ($E_S = 0$) and for which all the probability is initially in the state $\kets{0}{S}$ ($p = 0$). Since the Hamiltonian vanishes, the system lives in a degenerate space, and since it is pure we have maximal information about the state of the system. This information can be used to extract work from the heat bath. Since all states of the system have equal energy, it is clear that the system provides only information, whilst it is the bath which provides the energy. For a degenerate system the equilibrium state $\tau_\beta(0) = \tfrac{1}{2}\openone$, thus after applying the protocol from the previous section, in the asymptotic limit of infinitely many bath qubits, the final state of the weight is given by \begin{equation} \Omega_w^\infty = \proj{T\log 2}{w} \end{equation} corresponding to the extraction of an amount of work $W = T\log 2$ from the system. This is an analogous process to the isothermal expansion of a single-molecule gas from one side of a box into the full volume, as discussed in Szilard's well-known treatment of Maxwell's demon \cite{Szi29,LefRex03}. It should be noted that contrary to the previous subsection, where the final state of the weight was found to consist of two peaks, in this special case of a pure initial state it consists of only a single peak. Thus it is not necessary to go to the many-copy regime to extract a deterministic amount of work in this case. \subsection{Extracting work from diagonal quantum states of arbitrary dimension }\label{ss:qudits} To conclude the analysis of diagonal systems it is now necessary to move beyond qubits, to quantum systems of arbitrary finite dimension $d$, and arbitrary Hamiltonians. Our strategy is to connect a pair of energy levels in the system to a sequence of thermal qubits in the bath, shifting the occupation probability between these levels as in the qubit case (Sec.~\ref{ss:full protocol}). We then repeat this process for different pairs of levels until we have the desired final state. As the amount of work extracted in each step does not depend on the initial state of the weight (as proven in Appendix \ref{a:work_independence}), we can consider each step separately and simply sum the total work extracted. An important difference from the qubit case is that the occupation probability of the two levels in the system that are coupled to the bath at any given stage need not sum to unity. For example consider a qutrit (or larger system) in the state \begin{equation} \rho_S = (1-p-q)\proj{0}{S}+q\proj{1}{S} + p\proj{2}{S}, \end{equation} where we wish to shift a small amount $\delta \!\!p$ of probability from $\kets{0}{S}$ to $\kets{2}{S}$. To do this, we take a thermal qubit with excitation probability $(p+\delta \!\!p)/(1-q)$, so that the \emph{ratio} of occupation probabilities in the bath qubit is the same as in that in the desired final state of the system. We then apply the unitary \begin{equation}\label{e:U mot2} \kets{0}{B}\kets{2}{S}\kets{x}{w} \leftrightarrow \kets{1}{B}\kets{0}{S}\kets{x+E_2-E_B}{w} \end{equation} where $E_2$ is the energy of the state $\kets{2}{S}$, and all orthogonal states are left unchanged. We show in Appendix \ref{a:qudits} that the change in the average energy of the weight during this process is equal to the free energy change of the system, to first order in $\delta \!\!p$. From this it is straightforward to extend the qubit protocol to arbitrary diagonal states. For sufficiently large $N$ we can choose a sequence of $N+1$ states for the system where two adjacent states differ in occupation probability between a pair of levels by $\mathcal{O}(1/N)$, with the first and last states equal to the initial state of the system and its thermal state respectively\footnote{For example, we could first shift probability from all energy levels with higher probability in $\rho_S$ than in $\tau_\beta(H_S)$ to the $\kets{0}{S}$ state, then move probability from $\kets{0}{S}$ to the remaining levels, using $N/(d-1)$ steps for each pair of levels.}. We then pick the Hamiltonian for $N$ thermal qubits such that unitaries of the form \eqref{e:U mot2} take the reduced state of the system from from one state in the sequence to the next. This is done by taking the thermal qubits to have populations equal to those of the pair of levels of interest of the next state in the sequence, after renormalising. In the limit $N \rightarrow \infty$ the average amount of work put into the weight equals the free energy change of the system, regardless of the precise choice of path. Interestingly, an equivalent protocol would work for transitions between any two diagonal states, $\rho$ and $\sigma$, extracting an average amount of work equal to $F_\beta(\rho) - F_\beta(\sigma)$. This implies that creating a diagonal state $\rho$ from a thermal state costs the same average amount of work as can be extracted from $\rho$ when it is thermalised. In this sense thermodynamic transitions between diagonal states are reversible \footnote{However, note that if a state is thermalised and then recreated using our protocol, the fluctuations in the position of the weight will increase.}. This differs from the results of \cite{HorOpp11}, who show that such transitions are irreversible when considering (almost) deterministic work extraction, rather than average work. \subsection{Proof of optimality} \label{s:optimality} We now show that the above protocol is optimal. In particular, we will argue that the maximum free energy that can be extracted from a transition between diagonal states $\rho$ and $\sigma$ is $F_\beta(\rho) - F_\beta(\sigma)$, using a modification of our proof of the second law (in Section \ref{s:second_law}). Suppose that we have a protocol which extracts average work $F_\beta(\rho) - F_\beta(\sigma) + \delta$ (where $\delta > 0$) when the system is transformed from $\rho$ and $\sigma$. We can then use the above protocol to return the state from $\sigma$ to $\rho$ in a finite number of steps, extracting work $F_\beta(\sigma) - F_\beta(\rho) - \epsilon$, where we choose the number of thermal qubits such that $ \epsilon$ is in the range $0 < \epsilon < \delta/2$. The net effect is that a positive average work $\geq \delta/2$ is extracted into the weight, and the system begins and ends the combined procedure in the same state $\rho$. Considering total energy conservation for the combined protocol, we note that the average energy of the bath must have decreased by at least $\delta/2$, which means that the bath's entropy must also have decreased by a finite amount. However, the system does not change in entropy, and the change in entropy of the weight can be made as small as desired by choosing its initial state to be a sufficiently broad pure state (see Appendix \ref{a:work_independence}). Hence entropy conservation cannot be satisfied by our hypothetical protocol, and thus no protocol exists which extracts more average work than the free energy change. \section{Extracting work from coherence}\label{s:coherence} So far all of our discussions have involved only states which are diagonal in the energy eigenbasis and the protocol presented involved shifting probability only between these energy eigenstates. We can of course consider more general states, and ask the question of how to extract the maximum amount of work from such states. \subsection{Motivating example} As a motivating example, let us consider a system with Hamiltonian $H_S = E_S\proj{1}{S}$ prepared initially in the pure state \begin{equation}\label{e:pure} \kets{\psi}{S} = \sqrt{1-p_{eq}}\kets{0}{S} + \sqrt{p_{eq}}\kets{1}{S} \end{equation} This state has the same average energy as the thermal state $\tau_\beta(H_S)$, however, since this state is pure, it has zero entropy and hence its free energy differs from that of the thermal state. Thus one would like to extract work from this state, with the limit on extractable work given by \begin{equation} \label{e:opt work} W \leq F(\psi) - F(\tau_\beta(H_S)) = TS_c(p_{eq}) \end{equation} Previously we were able to extract work from single copies of diagonal states. However, we show in the next section that it is impossible to extract work from a single copy of $\kets{\psi}{S}$ given the assumptions of the framework. This is similar to the analysis of \cite{HorOpp11} where they also found that the state \eqref{e:pure} is useless for (almost) deterministic work extraction. However, it is only a single copy which is a useless resource; two copies can be used to extract work, although the amount is less than that given by (\ref{e:opt work}). The important point is the way in which work is extracted, which will suggest a different way in which one can understand an asymptotic protocol (in the number of system qubits) able to extract the optimal amount of work. Consider two copies of the state $\kets{\psi}{S}$, which written out in full is given by \begin{multline} \kets{\psi}{S}^{\otimes 2} = (1-p_{eq})\kets{0}{S}\kets{0}{S} + \sqrt{p_{eq}(1-p_{eq})}\Big(\kets{0}{S}\kets{1}{S}\\+\kets{1}{S}\kets{0}{S}\Big) + p_{eq}\kets{1}{S}\kets{1}{S} \end{multline} Let us consider that we now dephase this state in the energy eigenbasis, so called collective-dephasing. We can achieve this in two ways, either by allowing the state to evolve under its Hamiltonian freely for some unknown duration of time, or more systematically by applying controlled-$Z$ operations between maximally mixed (therefore degenerate) bath qubits and the system. Note that both of these are energy conserving unitaries. In either case, at the end of the dephasing the state $\rho$ is \begin{multline} \rho = (1-p_{eq})^2 \kets{0}{S}\kets{0}{S} \bras{0}{S}\bras{0}{S} + 2p_{eq}(1-p_{eq})\kets{\psi^+}{S}\bras{\psi^+}{S} \\+ p_{eq}^2\kets{1}{S}\kets{1}{S}\bras{1}{S}\bras{1}{S} \end{multline} where $\kets{\psi^+}{S} = \tfrac{1}{\sqrt{2}}\left(\kets{0}{S}\kets{1}{S}+\kets{1}{S}\kets{0}{S}\right)$. We note that since $\kets{0}{S}\kets{1}{S}$ and $\kets{1}{S}\kets{0}{S}$ have the same energy, the definite phase between them is preserved. The important point is that when considering the combined system we find that we have a two-dimensional degenerate subspace and within this subspace we are in the pure state $\kets{\psi^+}{S}$, and \emph{not} the orthogonal state $\kets{\psi^-}{S} \equiv \tfrac{1}{\sqrt{2}}\left(\kets{0}{S}\kets{1}{S}-\kets{1}{S}\kets{0}{S}\right)$. Thus we have knowledge that we are with certainty in one of two orthogonal states and we can use this information to extract work from a thermal bath by allowing it to `isothermally expand' into the full subspace. Randomising these two states brings the pair of qubits to their thermal state. We can thus apply the analogue of the protocol in Sec.~\ref{ss:maxwell demon}, coupling only the two states inside the degenerate subspace to the thermal bath, and it follows immediately that the final average energy of the weight, in the asymptotic limit, is \begin{equation} \Exp{E}_w = 2p_{eq}(1-p_{eq})T\log 2 \end{equation} Except for the limiting case $p_{eq} = 0$, corresponding to the unphysical condition $E_S \to \infty$, this value is strictly less than the change in free energy of two system qubits, \begin{equation} 2p_{eq}(1-p_{eq})T\log 2 < 2TS_c(p_{eq}) \end{equation} This inequality is due to the fact that the collective-dephasing changed the entropy of the system but extracted no work in the process. One may wondering therefore why this operation is considered in the first place. In Appendix \ref{a:coherence} we show that we can apply the same analysis, including the collective-dephasing, to $n$ copies of $\kets{\psi}{S}$ and extract, in the asymptotic limit $n \to \infty$, work equal to the free energy change $n T S_c(p_{eq})$. This is possible since, as we also show, collective-dephasing increases the entropy of $n$ systems by an amount which grows as $\log(n+1)$, which becomes negligible in the asymptotic limit. We thus see that coherence between states of different energy in fact plays no thermodynamic role asymptotically in work extraction\footnote{ In the other direction, that of creating non-thermal states from a supply of work, in \cite{BraHorOpp11} it was shown that a phase reference (e.g. a large coherent state) is consumed at a sub-linear rate in this process.}. Altogether this highlights an essential difference between diagonal and non-diagonal states; When $E_S \neq 0$, the state of multiple qubits in a diagonal state will be maximally mixed inside each energy eigenspace. However, for non-diagonal states, these subspaces take on non-trivial states, and work can be extracted from them by allowing them to `isothermally expand' into the full subspace. \subsection{General work extraction protocol} We now present a general strategy to extract work from $n$ copies of an arbitrary state $\rho$, of arbitrary dimension $d$. It is helpful to divide the protocol into three steps: \begin{enumerate} \item Collectively dephase the state $\rho^{\otimes n}$ in the joint energy eigenbasis. \label{i:dephase} \item Allow the resulting state to `isothermally expand' inside each energy eigenspace, until it equals $\omega^{\otimes n}$, where $\omega$ is the state obtained by dephasing $\rho$ in its energy eigenbasis. \label{i:expand} \item Individually transform each copy of $\omega$ into $\tau_\beta(H_S)$. \label{i:transform} \end{enumerate} It follows from the results of Sec.~\ref{ss:qudits} that steps (\ref{i:expand}) and (\ref{i:transform}) can be performed whilst extracting an average amount of work equal to the free energy change. The free energy change in the first step is not captured as work, but is simply lost. However, we show in the Appendix \ref{a:coherence} that the amount of work lost in this step is at most $T (d-1) \log(n+1)$. The average amount of work extracted per copy of the original system is therefore \begin{equation} \delta \!W \geq F_\beta(\rho) - F_\beta(\tau_\beta(H_S)) - \frac{T (d-1)}{n} \log(n+1). \end{equation} In the limit as $n\rightarrow \infty$, we therefore extract an average work per copy of $\rho$ equal to its free energy change, as desired. If we only have a single copy of the system, the amount of work we can extract from $\rho$ is given by the free energy change of the decohered state, $F_\beta(\omega) - F_\beta(\tau_\beta(H_S))$. This can be obtained by first dephasing $\rho$ into $\omega$ and then using the protocol in Sec.~\ref{ss:qudits}. We now show that this is optimal, following the approach given in \cite{HorOpp11}. Firstly, note that as we require the average work extracted by a protocol to be independent of the initial state of the weight, we are free to choose that state however we like -- here we choose it to be a very narrow wavepacket centred on zero. Now consider a decomposition of the total state space into subspaces, each of which has total energy (of the system, bath and weight) close to one of the energy eigenvalues of the system and bath\footnote{In particular the $i^{\textrm{th}}$ subspace corresponds to the total energy $E$ lying in the range $\frac{E_i-E_{i-1}}{2} \leq E \leq \frac{E_{i+1}-E_{i}}{2}$, where $E_i$ are the energy eigenvalues of the system and bath. Furthermore we choose the width of the weight's initial wavepacket to be narrower than the smallest subspace.}. Note that any work extraction protocol can be followed by a transformation which decoheres the state with respect to these total energy subspaces, without affecting the average work extracted. However, this decohering operation commutes with the unitaries used in the protocol, so we can move it to the beginning of the protocol without changing the work extracted. At the beginning of the protocol, this operation has the sole effect of decohering the system in its energy eigenbasis (changing $\rho$ to $\omega$). Hence the protocol extracts the same amount of work as it would have if it had operated on $\omega$. \section{Perspective using `virtual temperatures'}\label{s:virtual} Consider the model of the smallest possible heat engines \cite{YouMahOba09,BruLinPop11}, consisting of two qubits, one with energy spacing $E_1$ in thermal contact with a bath at temperature $T_1$, and a second with energy spacing $E_2$ in contact with a bath at $T_2$. As a composite system this pair of qubits is not in a global thermal state. Nevertheless, let us assign a `virtual temperature' to each transition via the identification \cite{BruLinPop11} \begin{equation} \tfrac{p(e)}{p(g)} = e^{-\beta_v (E_e-E_g)} \end{equation} where $p(e)$ and $p(g)$ are the probability of being in the `excited' state (the state of higher energy) and ground state (the state of lower energy) respectively, with $E_e$ and $E_g$ being their energy eigenvalues, and $\beta_v = 1/T_v$ is the `inverse virtual temperature'. For the case in hand, the virtual inverse temperature for the states $\kets{0}{1}\kets{1}{2}$ and $\kets{1}{1}\kets{0}{S}$ is \begin{equation} \beta_v = \frac{S_c'(p_1)-S_c'(p_2)}{E_1-E_2} = \frac{E_1 \beta_1 - E_2 \beta_2}{E_1-E_2} \end{equation} where we have denoted the probability for qubit $i$ to be in the excited state by $p_i$, and we have used the relation $S_c'(p) = \beta E$ for a thermal state. The motivation for introducing the concept of virtual temperatures is that for the smallest machines it gives a concise characterisation of their behaviour; the function of the machine is determined by the virtual temperature -- refrigerator, heat pump or heat engine, if colder, hotter, or negative temperature respectively; Carnot efficiency is approached when the virtual temperature approaches the external temperature for the refrigerator and pump, and when approaching minus infinite temperature for the heat engine. It is the final machine mentioned, the heat engine, which is relevant here, since this is the machine which produces work. We can think of the system qubit and bath qubit together to be a two-qubit heat engine and that the operating point of the protocol is the Carnot point of the engine. Let us return to the motivating example of Section \ref{s:motivating example}. Our system qubit had spacing $E_S$ with excited state probability $p$, whilst the bath qubit had spacing $E_B$ with excited state probability $q$. Let us rename the state $\kets{1}{B}\kets{0}{S} = \kets{0}{v}$ and $\kets{0}{B}\kets{1}{S} = \kets{1}{v}$. With this renaming, the interaction \eqref{e:U mot} becomes \begin{equation} \kets{1}{v}\kets{x}{w} \leftrightarrow \kets{0}{v}\kets{x+E_v}{w} \end{equation} where $E_v = E_S - E_B$ is the spacing of the virtual qubit. Thus from this perspective it is explicit that the unitary is the one which exchanges energy from the virtual qubit into the weight, analogous to the Hamiltonian of the smallest heat engine \cite{BruLinPop11}. The operating point of the protocol was to choose $E_B$ such that $q = p-\delta \!\!p$. In this regime, to first order, the inverse virtual temperature becomes \begin{equation} \beta_v \simeq \delta \!\!p\frac{ S_c''(p)}{E_S-TS_c'(p)} \end{equation} where $S_c''(p) = -\tfrac{1}{1-p}-\tfrac{1}{p} \leq 0$. Thus the inverse virtual temperature becomes vanishingly small and negative as $\delta \!\!p \to 0$, which is exactly the Carnot limit of the small heat engine. The picture that one obtains from this perspective is that the reason why any state which is not a thermal state at temperature $T$ is a resource is because the \emph{composite system of resource state + thermal state} contain non-trivial virtual temperatures. One can extract work from a system approaching its free energy when the only exchanges between the composite system and weight involve virtual qubits whose inverse temperatures become negatively infinite. Finally, we conclude by noting the protocol for systems with many energy levels becomes easier to explain using this perspective. A non-thermal state will have transitions (i.e. pairs of levels) at many different temperatures, and each pair needs to be bought to the same temperature in the thermalisation process. In the protocol the bath qubits interact only with a single pair of levels at a time. To approach reversibility it must be the case that any systems which interact only differ in temperature by an infinitesimal amount, but this is exactly the condition enforced by choosing the renormalised probability of the pair to be close to the bath qubit populations. Finally, the unitary which is applied is the basic thermalising unitary and the energy exchanged in the limit is precisely the free energy. \section{Conclusions}\label{s:conclusions} To summarise, in this paper we have analysed the fundamental laws of thermodynamics using the recently proposed concept of resource theories. We proposed an alternative resource theory which differs in certain aspects from the original one. In this conceptual framework we have proved a number of results. First we proved the second law of thermodynamics. Second we proved that for systems that are diagonal in the energy eigenbasis, but not at equilibrium with the thermal bath, we can extract work equal to the free energy by processing each system \emph{individually}, hence showing the free energy is a relevant quantity for individual systems. Third, we proved that when systems are not diagonal in the energy eigenbasis, i.e. they have coherences between different energy eigenstates, it is not possible to extract work equal to the free energy by processing them individually. Work equal to the free energy can however be extracted if we allow for collective processing of such systems (in the asymptotic limit of many copies). A number of important questions open immediately. First of all, whether the assumptions which underlie our resource theory framework are too restrictive. In particular, this may be the reason why we cannot extract the entire free energy by processing single systems with coherences between different energy levels. Indeed, given our assumptions we are very limited in how we can access and manipulate the relative phase between two different energy levels, hence we cannot interfere them in a controlled way. As such, for all practical purposes the different energy levels behave as if they were decohered; it is precisely the free energy of this decohered state that is the maximum we can extract in our framework. To be able to interfere different energy levels, i.e. to control their relative phase, we need a ``frame of reference" for it. One way to look at this is that we need some other resource with coherent superpositions between energy levels \footnote{Note that the weight cannot be used for this purpose because of our assumption that the work extracted must be independent of its initial state}. Another way to express this is that we essentially need a clock: the phase between two energy levels varies in time, so a ``frame of reference" for it is essentially a clock. When collectively processing many copies the different systems act effectively as reference systems for each other. For individual processing however we essentially need to add an external clock. In our present framework we did not include such a clock since we were not able to guarantee that one cannot unwittingly cheat by using the clock itself as a source of free energy. It is conceivable however that a more careful formulation of the resource theory would permit integrating a clock in a consistent and controlled manner. In this case, it may be possible that work equal to the free energy can be extracted from all quantum states by only processing individual copies. Second, while in the above we asked if we can relax some of the constraints imposed in the resource theory, for example by including a clock, we must also enquire whether we may indeed need to strengthen some of the constraints. In particular, one may be concerned that we have used unitary transformations as opposed to Hamiltonian evolution. Unitaries imply controlled evolution, i.e. turning on and off interactions at precise time intervals. One may wonder whether this does not require some free energy to achieve. If this is the case, then it is only an illusion that we have extracted work equal to the free energy from the system, since we had to pay to implement the process. Although such a detailed analysis is missing, the fact that all of the results obtained within the resource theory framework so far are consistent with our expectations is an implicit indication that the assumptions are indeed justified. To conclude, the resource theory framework seems to be a natural and powerful way to approach thermodynamics. It has already delivered significant results and we believe that further investigation along these lines will lead to a much deeper understanding of the foundations of thermodynamics. \begin{acknowledgments} SP and PS acknowledge support from the European project (Integrated Project ``Q-ESSENCE"). SP acknowledges support from the ERC (Advanced Grant ``NLST''), and the Templeton Foundation. AJS acknowledges support from the Royal Society. PS is grateful to Lidia del Rio, Johan {\AA}berg, Joe Renes, Philippe Faist, Renato Renner, Oscar Dahlsten and Luis Masannes for interesting discussions. \end{acknowledgments} \input{WorkExtraction_bibliography.bbl} \appendix \section*{APPENDICES} \section{Independence of the work on the initial state of the weight} \label{a:work_independence} In each step of our protocols in which work is extracted, the state of a thermal qubit is swapped with a pair of energy levels in the system, with a corresponding translation of the weight in order to conserve total energy. Consequently, each work-extraction step can be represented by a unitary transformation of the form \begin{equation} U = \sum_{ij} \Pi_{ij} \ket{i}\bra{j} \otimes{\Gamma_{E_j - E_i}} \end{equation} where the states $\{ \ket{i} \}$ are an orthonormal basis of energy eigenstates for the system and the relevant portion of the bath, such that $(H_B + H_S) \ket{i}= E_i \ket{i}$, $\Gamma_a$ is the translation operator on the weight, and $\Pi_{ij}$ is a permutation matrix. Furthermore, apart from the initial decohering of non-classical states (discussed in Section \ref{s:coherence}), which is entirely independent from the weight, the entire protocol can be cast in this form. It is easy to see that all such unitaries commute with translations on the weight. We now show that the average work extracted by a unitary of this form does not depend on the initial state of the weight (even if it is initially correlated with the state of the system). Let us denote the initial state of the system, bath and weight by the density operator $\rho$. The average work extracted is given by \begin{eqnarray} \Exp{W} &=& \textrm{tr} \left( H_W U \rho U^{\dag} \right) - \textrm{tr} \left( H_W \rho \right), \nonumber \\ &=& \textrm{tr} \left( \left(U^{\dag} H_W U - H_W \right) \rho \right) \label{eq:workindependence} \end{eqnarray} where $H_W = \openone_{SB} \otimes \hat{x}_w $ is the Hamiltonian of the weight (defined for convenience as an operator on the entire system). Now note that \begin{eqnarray} U^{\dag} H_W U &=& \sum_{ijkl} \Pi_{kl}\Pi_{ij} \ket{l}\braket{k}{i}\bra{j} \otimes \left(\Gamma_{E_k - E_l} \hat{x} \Gamma_{E_j - E_i} \right) \nonumber \\ &=& \sum_{ijl} \Pi_{il}\Pi_{ij} \ket{l}\bra{j} \otimes \left(\Gamma_{E_i - E_l} \hat{x} \Gamma_{E_j - E_i} \right) \nonumber \\ &=& \sum_{ij} \Pi_{ij} \ket{j}\bra{j} \otimes \left( \Gamma_{E_i - E_j} \hat{x} \Gamma_{E_j - E_i} \right) \nonumber \\ &=& \sum_{ij} \Pi_{ij} \ket{j}\bra{j} \otimes \left(\hat{x} + \left(E_j - E_i\right) \openone_w \right) \nonumber \\ &=& H_W + \sum_{ij} \Pi_{ij} \left(E_j - E_i\right) \ket{j}\bra{j} \otimes \openone_w \end{eqnarray} where in the third line we have used the fact that for permutation matrices $\Pi_{il} \Pi_{ij}= \Pi_{ij} \delta_{jl}$ (where $\delta_{jl}$ is the Kronecker delta function), and in the fourth line we have used the fact that $\Gamma_{-a} \hat{x} \Gamma_{a} = \hat{x} + a \openone_w$. Inserting this expression in (\ref{eq:workindependence}), we obtain \begin{equation} \Exp{W} = \textrm{tr}_{SB} \left( \Big(\sum_{ij} \Pi_{ij} \left(E_j - E_i\right) \ket{j}\bra{j}\Big) \rho_{SB} \right) \end{equation} where $\rho_{SB} = \textrm{tr}_W (\rho)$ is the reduced density matrix of the system and bath. Hence the average amount of work extracted is independent of the initial state of the weight as desired. Note that the most general energy-conserving unitary transformations which commute with all translations on the weight have the form \begin{equation} \label{e:general_unitary} U = \sum_{ij} u_{ij} \ket{i}\bra{j} \otimes{\Gamma_{E_j - E_i}} \end{equation} where $u_{ij}$ is an arbitrary unitary matrix. However, for such unitaries the work extracted would generally depend on the initial state of the weight. \section{Proof of second law of thermodynamics} \label{a:second_law} In this appendix, we provide further details for our proof of the second law in Section \ref{a:second_law}. We first argue that any reduction in the average energy of an initially thermal state (with positive temperature) must also yield a reduction in its entropy. We will only need to use the fact that a thermal state is the maximal entropy state with a given average energy. If a system starts in a thermal state and is transformed to a final state with fixed average energy, the entropy change $\Delta S_B$ will be maximised when the final state is also thermal. In the case where the average energy decreases, and the initial state has positive temperature, the final state will be a thermal state of lower temperature, and thus lower entropy. We now show that the entropy change of the weight in any particular protocol may be made as small as desired by choosing its initial state to be a sufficiently broad pure state (e.g. a state $\ket{\psi_L}$ with wavefunction $\psi_L(x)$ equal to $1/\sqrt{2L}$ for $|x|\leq L$, and 0 otherwise, where $L \gg 1$). First note that the requirement that transformations are translationally invariant on the weight means that they have the form given in (\ref{e:general_unitary}). When applied to a thermal state of the bath, the effect of the protocol on the weight is to transform it into a finite mixture of translations of the initial state $\rho_w = \sum_{ij} p_{ij} \Gamma_{E_{i} -E_{j}} \proj{\psi_L}{w} \Gamma_{E_{j} -E_{i}}$, where $p_{ij}$ are probabilities and $i$ and $j$ label energy eigenstates (of energies $E_i$ and $E_j$) of the bath. If the thermal systems used in the protocol have total dimension $d$, and the initial state of the weight is pure, then the final state of the weight lies in a $d^2$-dimensional subspace spanned by the states $ \Gamma_{E_{i} -E_{j}} \ket{\psi_L}{w}$ for $i,j \in \{ 1,2, \ldots d\}\}$. As the final state of the weight lives in a finite dimensional subspace, its entropy can be shown to be continuous due to Fannes' inequality \cite{NieChu00}. \begin{equation} |S(\rho_w) - S(\proj{\psi_L}{w})| \leq D \log \left( \frac{d^2}{D} \right) \end{equation} where $ D$ is the trace distance between $\rho_W$ and $\proj{\psi_L}{w}$. We can make $ D$ as small as we like by taking sufficiently large $L$ and therefore make the entropy change as small as we like. For the case in hand, a straightforward calculation shows that $D \leq \sqrt{\frac{a}{L}}$, where $a = \max_{ij}|E_i-E_j|$ is the maximum energy gap of the system-bath Hamiltonian. Note that all of the above arguments apply equally well if we consider a system in addition to the bath which is initially in a mixture of energy eigenstates, as considered in Section \ref{s:optimality}. \section{Motivating example details}\label{a:motivating example} In this appendix we shall provide more details into the example of Section \ref{s:motivating example}, where a state undergoes a transformation and in the process supplies an amount of work $W$. Consider the state \begin{equation} \rho_S = (1-p)\proj{0}{S} + p \proj{1}{S} \end{equation} with Hamiltonian $H_S = E_S \proj{1}{S}$, along with a thermal qubit \begin{equation} \tau_\beta(H_B) = (1-r)\proj{0}{B} + r\proj{1}{B} \end{equation} with Hamiltonian $H_B = E_B\kets{1}{B}$, where $E_B = T S_c'(r)$. Let us bring in a weight in the state $\kets{0}{w}$, and apply the unitary transformation $U$, which interchanges \begin{equation} \kets{0}{B}\kets{1}{S}\kets{x}{w} \leftrightarrow \kets{1}{B}\kets{0}{S}\kets{x+E_S-E_B}{w} \end{equation} whilst leaving all other orthogonal states unchanged. Applying this unitary to the combined system we find that \begin{multline}\label{e:U example} \sigma_{BSw} = U\tau_\beta(H_B)\otimes \rho_S \otimes \proj{0}{w}U^\dagger \\ = rp\kets{1}{B}\kets{1}{S}\kets{0}{w}\bras{1}{B}\bras{1}{S}\bras{0}{w} \\ + r(1-p)\kets{0}{B}\kets{1}{S}\kets{E_B-E_S}{w}\bras{0}{B}\bras{1}{S}\bras{E_B-E_S}{w} \\ + (1-r)p\kets{1}{B}\kets{0}{S}\kets{E_S-E_B}{w}\bras{1}{B}\bras{0}{S}\bras{E_S-E_B}{w} \\ +(1-r)(1-p)\kets{0}{B}\kets{0}{S}\kets{0}{w}\bras{0}{B}\bras{0}{S}\bras{0}{w} \end{multline} From this one can finds that the final reduced state of the system and bath qubit are $\rho' = \textrm{tr}_{Bw} \sigma =\tau_\beta(H_B)$ and $\tau'~=~\textrm{tr}_{Sw}\sigma = \rho_S$ respectively. Thus we see that at the level of reduced states these two systems have swapped states. The change in free energy of the system $\Delta F_\beta$ is \begin{align} \Delta F_\beta &= F_\beta(\rho) - F_\beta(\tau_\beta(H_B)) \nonumber \\ &= (p E_S - T S_c(p)) - (r E_S - T S_c(r)) \nonumber \\ &= (p-r)E_S -T(S_c(p)-S_c(r)) \end{align} Moving on to the weight its final reduced state $\Omega_w'$ is \begin{multline}\label{e:Lambda L} \Omega_w' = (rp+(1-r)(1-p))\proj{0}{w}\\+ r(1-p)\proj{E_B-E_S}{w}\\ + (1-r)p\proj{E_S-E_B}{w} \end{multline} The change in average energy of the weight $\Delta \Exp{E}_w$ is thus \begin{align} \Delta \Exp{E}_w &= \textrm{tr}(H_w \Omega_w) - \textrm{tr}(H_w \proj{0}{w}) \nonumber \\ &= (p-r)(E_S - E_B) \nonumber \\ &= (p-r)(E_S - TS_c(r)) \end{align} Now, let us take $r=p-\delta \!\!p$ and expand to second order in $\delta \!\!p$ both $\Delta F_\beta$ and $\Delta \Exp{E}_w$, \begin{align} \Delta F_\beta &= \delta \!\!p(E_S - T S_c'(p)) + \delta \!\!p^2 \tfrac{T S_c''(p)}{2} + \mathcal{O}(\delta \!\!p^3) \\ \Delta \Exp{E}_w &= \delta \!\!p(E_S - T S_c'(p)) + \delta \!\!p^2 T S_c''(p) + \mathcal{O}(\delta \!\!p^3) \end{align} The change in free energy and the change in average energy of the weight thus coincide to first order in $\delta \!\!p$ and differ only by a factor of 2 in the second order term. Since $S_c''(p) < 0$, we see immediately that $\Delta \Exp{E}_w \leq \Delta F_\beta$, so that the change in free energy is always larger than the change in average energy of the weight, as we would expect. \section{Calculating the average energy and spread}\label{a:weight} In this appendix we will show how to arrive at equation \eqref{e:asymptotic ladder} for the asymptotic state of the ladder. To do so let us consider first those instances where the system qubits starts in state $\kets{0}{S}$. Given this we shall then calculate the expected energy of the weight and its variance in the limit of large $N$ to first order, from which we shall be able to conclude that it approaches a definitive value in the asymptotic limit at the desired energy. A analogous method can also be employed assuming the system starts instead in the state $\kets{1}{S}$, with the desired result also obtained. By calculating the first few stages of the protocol and then by induction it is easy to show that the expected energy of the weight at the end of the protocol is \begin{equation}\label{e:Ew0} \Exp{E}_w^{[0]} = -p_{eq} E_S + p E_B^{(1)} + \sum_{k=1}^N (q^{(k)} - q^{(k-1)})E_B^{(k)} \end{equation} where the superscript $[0]$ is a reminder that this is the case for the system starting in the state $\kets{0}{S}$, and as defined in the main text we have \begin{align} q^{(k)}& = p + \tfrac{k}{N}(p_{eq}-p)& E_B^{(k)} &= \tfrac{1}{\beta}\log\left(\tfrac{1-q^{(k)}}{q^{(k)}}\right) \end{align} By expanding the logarithms in powers of $\tfrac{k}{N}$ and denoting $a = (p_{eq}-p)$, after some basic manipulations this can be re-written as \begin{multline} \Exp{E}_w^{[0]} = -p_{eq} E_S + p E_B^{(1)} + \tfrac{a}{\beta}\log\left(\tfrac{1-p}{p}\right) \\ + \frac{a}{\beta N}\sum_{k=1}^N \sum_{n=1}^{\infty} \left(-\frac{1}{n}\left(\frac{ak}{(1-p)N}\right)^n +\frac{1}{n}\left(\frac{-ak}{pN}\right)^n \right) \end{multline} The advantage of performing this expansion is that we can now interchange the order of summation and note that the only term which depends upon $k$ is the term $k^n$ which is common to both terms in the sum. We can thus use the expansion \begin{equation} \sum_{k=1}^N k^n= \frac{N^{n+1}}{n+1} + \frac{N^n}{2} + \mathcal{O}\left(N^{n-1}\right) \end{equation} to write \begin{multline} \Exp{E}_w^{[0]} = -p_{eq} E_S + p E_B^{(1)} + \tfrac{a}{\beta}\log\left(\tfrac{1-p}{p}\right) \\ + \frac{1}{\beta} \sum_{n=1}^{\infty} \left(\frac{a^{n+1}}{n}\left(\frac{1}{(-p)^n}-\frac{1}{(1-p)^n} \right)\left(\frac{1}{n+1} + \frac{1}{2N}\right)\right) \\+ \mathcal{O}\left(\tfrac{1}{N^2}\right) \end{multline} This sum can now be evaluated explicitly, and after re-expressing $E_S$ in terms of $S_c'(p_{eq})$, expanding $E_B^{(1)}$ to first order in powers of $\tfrac{1}{N}$, and an amount of manipulation we arrive at \begin{multline} \Exp{E}_w^{[0]} = \tfrac{1}{\beta}\log\left(\tfrac{1-p}{1-p_{eq}}\right) + \tfrac{1}{\beta}(p_{eq}-p)\Big(\tfrac{1}{2}\big(S_c'(p_{eq})\\-S_c'(p)\big) -\tfrac{1}{1-p}\Big)\tfrac{1}{N} + \mathcal{O}\left(\tfrac{1}{N^2}\right) \end{multline} thus demonstrating that the weight is peaked around the point expected when the system starts off in the state $\kets{0}{S}$ with an offset which vanishes in the asymptotic limit. Considering the case now when the system starts off in the state $\kets{1}{S}$ and again calculating the first few stages of the protocol and then by induction it can be shown that \begin{equation} \Exp{E}_w^{[1]} = \Exp{E}_w^{[0]} + E_S - E_B^{(1)} \end{equation} Thus by re-expressing $E_S$ in terms of $S_c'(p_{eq})$ and expanding $E_B^{(1)}$ to first order it is straightforward given the previous results to show that \begin{multline} \Exp{E}_w^{[1]} = \tfrac{1}{\beta}\log\left(\tfrac{p}{p_{eq}}\right) + \tfrac{1}{\beta}(p_{eq}-p)\Big(\tfrac{1}{2}\big(S_c'(p_{eq})\\-S_c'(p)\big) +\tfrac{1}{p}\Big)\tfrac{1}{N} + \mathcal{O}\left(\tfrac{1}{N^2}\right) \end{multline} which again gives the desired result. Let us move on now to a calculation of the variance $\Delta E_w^{2[i]} = \Exp{E^2}_w^{[i]} - \Exp{E}_w^{2[i]}$ of each of the two packets, where for simplicity we will neglect the finite width of the initial wavepacket. Again by induction it can be shown that the spread of both packets is identical (and hence we shall drop the superscript $[i]$) and equal to \begin{equation} \Delta E_w^2 = \sum_{k=1}^N q^{(k)}(1-q^{(k)})(E_B^{(k+1)}-E_B^{(k)}) \end{equation} Let us consider the $k^{th}$ term in this sum, multiplied by $N$. Defining $x = \tfrac{k}{N}$ and expanding this summand in terms of $\tfrac{1}{N}$ we find \begin{multline} q^{(k)}(1-q^{(k)})(E_B^{(k+1)}-E_B^{(k)}) \\= \frac{T^2(p_{eq}-p)^2}{(p+x(p_{eq}-p))(1-p-x(p_{eq}-p))N^2} + \mathcal{O}\left(\tfrac{1}{N^3}\right) \end{multline} Thus for large $N$ the leading order contribution to this sum becomes equal to \begin{equation} \Delta E_w^2 \simeq \frac{1}{N}\int_0^1 \frac{T^2(p_{eq}-p)^2 dx}{(p+x(p_{eq}-p))(1-p-x(p_{eq}-p))} \end{equation} Evaluating this integral explicitly we find finally that \begin{equation} \Delta E_w^2 = \frac{T^2 (p_{eq}-p)(S_c'(p_{eq}-S_c'(p))}{N} + \mathcal{O}\left(\frac{1}{N^2}\right) \end{equation} Thus we see that each peak has a spread of the order of $1/\sqrt{N}$ which vanishes in the asymptotic limit $N\to\infty$. Therefore in this limit we can conclude that the distribution of the ladder approaches a two-peaked distribution, as given by equation \eqref{e:asymptotic ladder}. \section{Extracting work from diagonal quantum systems of arbitrary dimension}\label{a:qudits} In this appendix we wish to show that the average work extracted when moving the occupation probability between energy levels is still equal to the free energy in a $d$-dimensional system. It will suffice to consider a qutrit, and moving probability $\delta \!\!p$ from the ground state $\kets{0}{S}$ to the second excited state $\kets{2}{S}$. Let us take our system to be initially in the state \begin{equation} \rho_S = (1-p-q)\proj{0}{S}+q\proj{1}{S} + p\proj{2}{S}, \end{equation} and thus to end in the state \begin{equation} \rho'_S = (1-p-q-\delta \!\!p)\proj{0}{S}+q\proj{1}{S}+(p+\delta \!\!p)\proj{2}{S}. \end{equation} where the system Hamiltonian is \begin{equation} H_S = E_1\proj{1}{S} + E_2\proj{2}{S}. \end{equation} To achieve this, we append a bath qubit with energy level spacing $E_B$ and ground state population $r$ and apply the unitary transformation which interchanges only the states \begin{equation} \kets{1}{B}\kets{0}{S}\kets{x}{w} \leftrightarrow \kets{0}{B}\kets{2}{S}\kets{x+E_B - E_2}{w} \end{equation} This induces on the system and weight the transformation \begin{multline}\label{e:trit 1} \rho_S \otimes \proj{0}{w} \xrightarrow{} (1-p-q)(1-r)\kets{0}{S}\kets{0}{w}\bras{0}{S}\bras{0}{w} \\ +q\kets{1}{S}\kets{0}{w}\bras{1}{S}\bras{0}{w} +pr\kets{2}{S}\kets{0}{w}\bras{2}{S}\bras{0}{w} \\ +p(1-r)\kets{0}{S}\kets{E_{2}-E_B}{w}\bras{0}{S}\bras{E_{2}-E_B}{w} \\ +(1-p-q)r\kets{2}{S}\kets{E_B-E_{2}}{w}\bras{2}{S}\bras{E_B-E_{2}}{w} \end{multline} where we now carry out the final trace over the bath qubit for brevity of presentation. Thus in order that the final reduced state of the system is $\rho'_S$ we find that the ground state population of the bath must be chosen to be \begin{equation} r = \frac{p+\delta \!\!p}{1-q} \end{equation} The denominator is the total probability to be in either of the states $\kets{0}{S}$ or $\kets{1}{S}$, and hence this expression can be seen as a renormalised version of the qubit case, where the total probability to be in either state always sums to unity. We see that it is the ratio of probabilities which must be close to the ratio of populations of the thermal qubits, which is equivalent to saying that they must be close in temperature (see Sec.~\ref{s:virtual}). The change in energy of the weight is given by \begin{eqnarray} \delta \!W &=& (p-(1-q)r)(E_2-E_B) \nonumber \\ &=& -\delta \!\!p(E_2-E_B) \end{eqnarray} thus using the fact that $E_B = \tfrac{1}{\beta}S_c'(r)$ and expanding to first order in $\delta \!\!p$ we find that the work extracted from the system is given by \begin{equation} \delta \!W \simeq -\delta \!\!p\left(E_2 -\tfrac{1}{\beta}S_c'\!\!\left(\tfrac{p}{1-q}\right)\right) \end{equation} which a straightforward calculation verifies is equal to the change in free energy $F(\rho_S)~-~F(\rho_S')$ to first order in $\delta \!\!p$. If we do a sequence of $N$ such transitions between the states $\rho$ and $\sigma$, in each of which $\delta \!\!p = \mathcal{O}(1/N)$, then we will obtain $\delta \!W = F(\rho_S)~-~F(\rho_f) + \mathcal{O}(1/N)$. In the asymptotic limit in which the number of thermal qubits $N$ goes to infinity, we will therefore extract average free energy equal to the free energy change. \section{Work extraction from coherence}\label{a:coherence} In this appendix we will present the full details for how work can be extracted from coherence. We shall first consider the special case presented in the main text for $n$ systems, and then move on to the general case. \subsection{Motivating example} Consider $n$ copies of $\kets{\psi}{S}$, where \begin{equation} \kets{\psi}{S} = \sqrt{1-p_{eq}}\kets{0}{S} + \sqrt{p_{eq}}\kets{1}{S} \end{equation} There will be $n+1$ energy eigenspaces with energies $k E_S$, $k = 0,\ldots, n$. The dimension of the eigenspace with energy $k E_S$ will be $\binom{n}{k}$. After collective-dephasing in each eigenspace the system will be a pure superposition over all states, and the probability to be in each eigenspace will be $\binom{n}{k}p_{eq}^k (1-p_{eq})^{(n-k)}$. We then cause the state to `isothermally expand' in each degenerate subspace, taking the state from the initial pure state to the final maximally mixed state and in the process extracting $T\log d$ energy from the bath, where $d$ is the dimension of the subspace. In total, the final average energy of the ladder will be \begin{equation} \Exp{E}_w = \sum_{k=0}^n \binom{n}{k}p_{eq}^k (1-p_{eq})^{(n-k)} T\log \binom{n}{k} \end{equation} In the asymptotic limit, as $n\to \infty$ it is straightforward to show that \begin{equation} \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^n \binom{n}{k}p_{eq}^k (1-p_{eq})^{(n-k)}T\log \binom{n}{k} =TH(p_{eq}) \end{equation} since in the limit the only contribution to the sum comes from $k = np_{eq}$. We thus see that asymptotically the expected amount of work is extracted. There are a couple of comments which are worth mentioning. First we note that due to concentration effects, we know that almost all weight lies in those subspaces with energy $n p_{eq} \pm \mathcal{O}(\sqrt{n})$. We could therefore ignore those subspaces which lie outside this range and obtain, to leading order, the same amount of work. The results obtain however are independent of the concentration effects. Second we recall that the protocol starts with collective-dephasing in the energy eigenbasis. It might appear that this operation, which changes the entropy of the system, throws away work. However, asymptotically we obtain to first order the optimal amount of work, and so we conclude that the amount thrown away must scale sub-linearly in $n$. In fact the amount of entropy thrown away is only $\mathcal{O}(\log n)$, the entropy of the binomial distribution, since the dephasing leaves the system binomially distributed amongst energy eigenspaces which have orthogonal support. This shows that asymptotically the coherence between states of different energy are thermodynamically insignificant. \subsection{General work extraction protocol} Finally, for the case of arbitrary states of arbitrary dimension $d$, we note that the same conclusion can also be drawn. First we need to show that the operation of collective-dephasing never increases the entropy of a collection of systems too much in the asymptotic limit. Let us denote the projectors onto the energy eigenspaces by $\Pi_k$, and the total number of different energies by $K$. After the operation of collective-dephasing the state becomes block diagonal in the energy eigenbasis, namely \begin{equation} \rho^{\otimes n} \mapsto \rho' = \sum_{k=0}^{K-1} \Pi_k \rho^{\otimes n} \Pi_k \end{equation} We are interested in the difference in entropy $S(\rho^{\otimes n}) - S(\rho')$, i.e. the entropy generated by this operation. To do so, let us first consider the coherent version of collective-dephasing. Let us introduce an ancillary system $A$ with $K$ states $\kets{z}{A}$ prepared initially in the state $\kets{0}{A}$. Let us also consider the unitary operation $V$ \begin{equation} V = \sum_{k,z} \Pi_k \otimes \kets{z+k \mod K}{A}\bra{z} \end{equation} Applied to the state $\rho_{SA} = \rho^{\otimes n}\otimes \proj{0}{A}$ this performs the transformation \begin{equation} V \left( \rho^{\otimes n}\otimes \proj{0}{A} \right) V^\dagger = \rho_{SA}' = \sum_{k,k'} \Pi_k \rho^{\otimes n}\Pi_{k'} \otimes \kets{k}{A}\bra{k'} \end{equation} It is straightforward to see that $\textrm{tr}_A \rho'_{SA} = \rho'$ whilst \begin{equation} \textrm{tr}_S \rho_{SA}' \equiv \rho_A' = \sum_k \textrm{tr}\left(\Pi_k \rho^{\otimes n}\right) \proj{k}{A} \label{e:rhoA'} \end{equation} Considering now the entropies, we have $S(\rho_{SA}')~=~S(\rho^{\otimes n})~=~n S(\rho)$, since the two states differ by the appended pure ancilla and the joint unitary transformation. Thus, from the Araki-Lieb inequality $S(\rho_{SA}) \geq |S(\rho_S) - S(\rho_A)|$ we have \begin{equation}\label{e:Araki-Lieb} n S(\rho) \geq |S(\rho') - S(\rho_A')| \end{equation} Next, note that we can rewrite $\rho'$ in the form \begin{equation} \rho' = \sum_{k} \textrm{tr}(\Pi_k \rho^{\otimes n}) \frac{ \Pi_k \rho^{\otimes n}\Pi_{k}}{ \textrm{tr}(\Pi_k \rho^{\otimes n}) } \end{equation} where the sum is restricted to those $k$ for which $\textrm{tr}(\Pi_k \rho^{\otimes n})~\neq 0$. Comparing this to (\ref{e:rhoA'}), we note that $S(\rho') \geq S(\rho_A')$, as a mixture of orthogonal mixed states always has higher entropy than the same mixture of orthogonal pure states. Hence the absolute value can be omitted on the right-hand-side of equation \eqref{e:Araki-Lieb} and after rearranging we get \begin{equation} S(\rho_A') \geq S(\rho') - n S(\rho) \end{equation} Next, note that $S(\rho'_A) \leq \log K$, since in the worst case $\rho'_A$ is maximally mixed over $K$ states. Hence we find \begin{equation} \log K \geq S(\rho') - n S(\rho) \end{equation} The number of different energy levels $K$ is upper bounded by the number of ways of distributing $n$ indistinguishable particles between $d$ distinguishable energy levels, which is given by $\binom{n+d-1}{n}$. Thus \begin{equation} S(\rho') - n S(\rho) \leq \log \binom{n+d-1}{n} \leq (d-1)\log (n+1) \end{equation} This demonstrates that the operation of collective-dephasing on $n$ qubits can only increase the entropy by a logarithmic amount in $n$. The increase \emph{per qubit} is thus of the order $\tfrac{d-1}{n}\log (n+1)$, which vanishes in the asymptotic limit. This result does not follow from any concentration of measure, but simply by the fact that the the number of energy eigenvalues of $n$ non-interacting systems is logarithmic in $n$. \section{Discrete work storage}\label{a:discrete work} In this appendix we will show that the energy level spacing of the weight need only become continuous in the asymptotic limit and that for any finite number of bath qubits $N$ it is sufficient to consider the levels to have a minimum spacing $\mathcal{E}$ which scales as $1/N^2$. Let us begin by recalling how the protocol works. We pick a collection of $N$ bath qubits which have excited state probabilities $q^{(k)}$ given by \begin{equation} q^{(k)} = p + \tfrac{k}{N}(p_{eq}-p) \end{equation} and therefore excited state energies $E_B^{(k)} = T S_c'(q^{(k)})$. At each stage of the protocol the weight either gains or loses energy $E_S - E_B^{(k)}$. Let us now assume instead that the of being a continuous degree of freedom that the weight has a discrete (but infinite) set of states $\kets{k}{w}$ and Hamiltonian \begin{equation} H_w = \sum_{k = -\infty}^{\infty} k \mathcal{E} \proj{k}{w} \end{equation} Accordingly, we shall adjust the energy levels of the bath qubits to \begin{equation} \tilde{E}_B^{(k)} = E_B^{(k)} + \varepsilon^{(k)} \end{equation} such that \begin{equation} E_S-\tilde{E}_B^{(k)}= m^{(k)} \mathcal{E} \end{equation} where the $m^{(k)}$ are integers, and we take the smallest possible $\varepsilon^{(k)}$ possible such that this condition is met, meaning that $|\varepsilon^{(k)}| < \mathcal{E}$. To proceed, we first note that \eqref{e:Ew0} can alternatively be re-written as \begin{align} \Exp{E}_w^{[0]} &= -p_{eq} E_S + p E_B^{(1)} + \sum_{k=1}^N (q^{(k)} - q^{(k-1)})E_B^{(k)} \nonumber \\ &= \sum_{k=0}^{N-1} q^{(k)} (E_B^{(k)} - E_B^{(k+1)}) \end{align} Thus upon making the replacement $E_B^{(k)} \to \tilde{E}_B^{(k)}$ for the case of a discrete weight, the average energy, conditional on the qubit starting in the ground state becomes \begin{align} \Exp{\tilde{E}}_w^{[0]} &= \sum_{k=0}^{N-1} q^{(k)} (\tilde{E}_B^{(k)} - \tilde{E}_B^{(k+1)}) \nonumber \\ &= \sum_{k=0}^{N-1} q^{(k)} (E_B^{(k)}\! -\! E_B^{(k+1)}) + \sum_{k=0}^{N-1} q^{(k)} (\varepsilon^{(k)}\! -\! \varepsilon^{(k+1)}) \nonumber \\ &= \Exp{E}_w^{[0]} + \sum_{k=0}^{N-1} q^{(k)} (\varepsilon^{(k)} - \varepsilon^{(k+1)}) \end{align} It is the final term in the last line which is the error introduced by considering a discrete weight, and thus it is the magnitude of this term which is of interest. This can easily be bounded as follows \begin{align} \left|\sum_{k=0}^{N-1} q^{(k)} (\varepsilon^{(k)} - \varepsilon^{(k+1)})\right| &\leq \sum_{k=0}^{N-1} q^{(k)} \left|\varepsilon^{(k)} - \varepsilon^{(k+1)}\right| \nonumber \\ &\leq \sum_{k=0}^{N-1} q^{(k)} 2\mathcal{E}\nonumber \\ &= \mathcal{E}\left((p+p_{eq})N + (p-p_{eq})\right) \end{align} Therefore as long as the product $\mathcal{E}N \to 0$ as $N\to \infty$, then this term will asymptotically vanish. Choosing $\mathcal{E} = \epsilon/N^2$ would ensure that this correction is $\mathcal{O}(1/N)$ and thus the same order as the leading order correction to $\Exp{E}_w^{[0]}$ for finite $N$. An identical analysis holds also for $\Exp{E}_w^{[1]}$, the average energy of the weight conditional on the system starting in the state $\kets{1}{S}$, and therefore we conclude that the results more generally do not rely crucially on the continuous nature of the weight. \end{document}

\begin{document} \begin{abstract} We explore the optimality of the constants making valid the recently established Little Grothendieck inequality for JB$^*$-triples and JB$^*$-algebras. In our main result we prove that for each bounded linear operator $T$ from a JB$^*$-algebra $B$ into a complex Hilbert space $H$ and $\varepsilon>0$, there is a norm-one functional $\varphi\in B^*$ such that $$\norm{Tx}\le(\sqrt{2}+\varepsilon)\norm{T}\norm{x}_\varphi\quad\mbox{ for }x\in B.$$ The constant appearing in this theorem improves the best value known up to date (even for C$^*$-algebras). We also present an easy example witnessing that the constant cannot be strictly smaller than $\sqrt2$, hence our main theorem is `asymptotically optimal'. For type I JBW$^*$-algebras we establish a canonical decomposition of normal functionals which may be used to prove the main result in this special case and also seems to be of an independent interest. As a tool we prove a measurable version of the Schmidt representation of compact operators on a Hilbert space. \end{abstract} \title{On optimality of constants in the Little Grothendieck Theorem} \section{Introduction} We investigate the optimal values of the constant in the Little Grothendieck theorem for JB$^*$-algebra. The story begins in 1956 when Grothendieck \cite{grothendieck1956resume} proved his famous theorem on factorization of bilinear forms on spaces of continuous functions through Hilbert spaces. A weaker form of this result, called Little Grothendieck Theorem, can be formulated as a canonical factorization of bounded linear operators from spaces of continuous functions into a Hilbert space. It was also proved by Grothendieck \cite{grothendieck1956resume} (see also \cite[Theorem 5.2]{pisier2012grothendieck}) and reads as follows. \begin{thma}\label{T-C(K)} There is a universal constant $k$ such that for any bounded linear operator $T:C(K)\to H$, where $K$ is a compact space and $H$ is a Hilbert space, there is a Radon probability measure $\mu$ on $K$ such that $$\norm{Tf}\le k\norm{T} \left(\int \abs{f}^2\,\mbox{\rm d}\mu\right)^{\frac12}\quad\mbox{ for }f\in C(K).$$ Moreover, the optimal value of $k$ is $\frac2{\sqrt\pi}$ in the complex case and $\sqrt{\frac\pi2}$ in the real case. \end{thma} The Grothendieck theorem was later extended to the case of C$^*$-algebras by Pisier \cite{pisier1978grothendieck} and Haagerup \cite{haagerup1985grothendieck}. Its `little version' reads as follows. Henceforth, all Hilbert spaces considered in this note will be over the complex field. \begin{thma}\label{T-C*alg} Let $A$ be a C$^*$-algebra, $H$ a Hilbert space and $T:A\to H$ a bounded linear operator. Then there are two states $\varphi_1,\varphi_2\in A^*$ such that $$\norm{Tx}\le\norm{T}\left(\varphi_1(x^*x)+\varphi_2(xx^*)\right)^{\frac12}\quad \mbox{ for }x\in A.$$ Moreover, the constant $1$ on the right-hand side is optimal. \end{thma} The positive part of the previous theorem is due to Haagerup \cite{haagerup1985grothendieck}, the optimality result was proved by Haagerup and Itoh in \cite{haagerup-itoh} (see also \cite[Section 11]{pisier2012grothendieck}). Let us recall that a \emph{state} on a C$^*$-algebra is a positive functional of norm one, hence in the case of a complex $C(K)$ space (which is a commutative C$^*$-algebra), a state is just a functional represented by a probability measure. Hence, as a consequence of Theorem~\ref{T-C*alg} we get a weaker version of the complex version of Theorem~\ref{T-C(K)} with $k\le\sqrt{2}$. Let us point out that Theorem~\ref{T-C*alg} is specific for (noncommutative) C$^*$-algebras due to the asymmetric role played there by the products $xx^*$ and $x^*x$. To formulate its symmetric version recall that the Jordan product on a C$^*$-algebra $A$ is defined by $$x\circ y=\frac12(xy+yx)\quad\mbox{ for }x,y\in A.$$ Using this notation we may formulate the following consequence of Theorem~\ref{T-C*alg}. \begin{thma}\label{T:C*alg-sym} Let $A$ be a C$^*$-algebra, $H$ a Hilbert space and $T:A\to H$ a bounded linear operator. Then there is a state $\varphi\in A^*$ such that $$\norm{Tx}\le2\norm{T}\varphi(x\circ x^*)^{\frac12}\quad \mbox{ for }x\in A.$$ \end{thma} To deduce Theorem~\ref{T:C*alg-sym} from Theorem~\ref{T-C*alg} it is enough to take $\varphi=\frac12(\varphi_1+\varphi_2)$ and to use positivity of the elements $xx^*$ and $x^*x$. However, in this case the question on optimality of the constant remains open. \begin{ques} Is the constant $2$ in Theorem~\ref{T:C*alg-sym} optimal? \end{ques} It is easy to show that the constant should be at least $\sqrt2$ (see Example~\ref{ex:alg vs triple} below) and, to the best of our knowledge, no counterexample is known showing that $\sqrt{2}$ is not enough. A further generalization of the Grothendieck theorem, to the setting of JB$^*$-triples (see Section~\ref{sec: notation} for basic definitions and properties), was suggested by Barton and Friedman \cite{barton1987grothendieck}. However, their proof contained a gap found later by Peralta and Rodr\'{\i}guez Palacios \cite{peralta2001little,peralta2001grothendieck} who proved a weaker variant of the theorem. A correct proof was recently provided by the authors in \cite{HKPP-BF}. The `little versions' of these results are summarized in the following theorem. \begin{thma}\label{T:triples} Let $E$ be a JB$^*$-triple, $H$ a Hilbert space and $T:E\to H$ a bounded linear operator. \begin{enumerate}[$(1)$] \item If $T^{**}$ attains its norm, there is a norm-one functional $\varphi\in E^*$ such that $$\norm{Tx}\le\sqrt{2}\norm{T}\norm{x}_\varphi\quad\mbox{ for }x\in E.$$ \item Given $\varepsilon>0$, there are norm-one functionals $\varphi_1,\varphi_2\in E^*$ such that $$\norm{Tx}\le(\sqrt{2}+\varepsilon)\norm{T}\left(\norm{x}^2_{\varphi_1}+\varepsilon\norm{x}^2_{\varphi_2}\right)^{\frac12}\quad\mbox{ for }x\in E.$$ \item Given $\varepsilon>0$, there is a norm-one functional $\varphi\in E^*$ such that $$\norm{Tx}\le(2+\varepsilon)\norm{T}\norm{x}_\varphi\quad\mbox{ for }x\in E.$$ \end{enumerate} \end{thma} The pre-hilbertian seminorms $\norm{\cdot}_\varphi$ used in the statement are defined in Subsection~\ref{subsec:seminorms} below. Let us comment the history and the differences of the three versions. It was claimed in \cite[Theorem 1.3]{barton1987grothendieck} that assertion $(1)$ holds without the additional assumption on attaining the norm, because the authors assumed this assumption is satisfied automatically. In \cite{peralta2001little} and \cite[Example 1 and Theorem 3]{peralta2001grothendieck} it was pointed out that this is not the case and assertion $(2)$ was proved using a variational principle from \cite{Poliquin-Zizler}. In \cite[Lemma 3]{peralta2001grothendieck} also assertion $(1)$ was formulated. Note that in $(2)$ not only the constant $\sqrt2$ is replaced by a slightly larger one, but also the pre-hilbertian seminorm on the right-hand side is perturbed. This perturbation was recently avoided in \cite[Theorem 6.2]{HKPP-BF}, at the cost of squaring the constant. Further, although the proof from \cite{barton1987grothendieck} was not correct, up to now there is no counterexample to the statement itself. In particular, the following question remains open. \begin{ques} What is the optimal constant in assertion $(3)$ of Theorem~\ref{T:triples}? In particular, does assertion $(1)$ of the mentioned theorem hold without assuming the norm-attainment? \end{ques} The main result of this note is the following partial answer. \begin{thm}\label{t constant >sqrt2 in LG for JBstar algebras} Let $B$ be a JB$^*$-algebra, $H$ a Hilbert space and $T:B\to H$ a bounded linear operator. Given $\varepsilon>0$, there is a norm-one functional $\varphi\in B^*$ such that $$\norm{Tx}\le(\sqrt{2}+\varepsilon)\norm{T}\norm{x}_\varphi\quad\mbox{ for }x\in B.$$ In particular, this holds if $B$ is a C$^*$-algebra. \end{thm} Note that JB$^*$-algebras form a subclass of JB$^*$-triples and can be viewed as a generalization of C$^*$-algebras (see the next section). We further remark that the previous theorem is `asymptotically optimal' as the constant cannot be strictly smaller than $\sqrt2$ by Example~\ref{ex:Tx=xxi} below. The paper is organized as follows. Section~\ref{sec: notation} contains basic background on JB$^*$-triples and JB$^*$-algebras. In Section~\ref{sec:majorizing} we formulate the basic strategy of the proof using majorization results for pre-hilbertian seminorms. In Section~\ref{sec:type I} we deal with a large subclass of JBW$^*$-algebras (finite ones and those of type I). The main result of this section is Proposition~\ref{P:type I approx} which provides a canonical decomposition of normal functionals on the just commented JBW$^*$-algebras. This statement may be used to prove the main result in this special case and, moreover, it seems to be of an independent interest. As a tool we further establish a measurable version of Schmidt decomposition of compact operators (see Theorem~\ref{T:measurable Schmidt}). In Section~\ref{sec:JW*} we address Jordan subalgebras of von Neumann algebras. Section~\ref{sec:proofs} contains the synthesis of the previous sections, the proof of the main result and some consequences. In particular, we show that Theorem~\ref{T-C*alg} (with the precise constant) follows easily from Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}. Section~\ref{sec:problems} contains several examples witnessing optimality of some results and related open problems. In Section~\ref{sec:triples} we discuss the possibility of extending our results to general JB$^*$-triples. \section{Basic facts on JB$^*$-triples and JB$^*$-algebras}\label{sec: notation} It is known that in most cases, like in $B(H)$, the hermitian part of a C$^*$-algebra $A$ need not be a subalgebra of $A$ because it is not necessarily closed for the associative product. This instability can be avoided, at the cost of loosing associativity, by replacing the associative product $a b$ in $A$ with the \emph{Jordan product} defined by \begin{equation}\label{eq special Jordan product} a\circ b := \frac12 (a b + ba). \end{equation} This may be seen as an inspiration for the following abstract definitions. A real or complex \emph{Jordan algebra} is a non-necessarily associative algebra $B$ over $\mathbb{R}$ or $\mathbb{C}$ whose multiplication (denoted by $\circ$) satisfies the identities:\begin{equation}\label{eq Jordan axioms} x\circ y = y \circ x \hbox{ (commutative law) and } ( x\circ y )\circ x^2 = x \circ ( y \circ x^2 ) \hbox{ (Jordan identity)} \end{equation} for all $x,y\in B$, where $x^2 = x\circ x$. Jordan algebras were the mathematical structures designed by the theoretical physicist P.~Jordan to formalize the notion of an algebra of observables in quantum mechanics in 1933. The term ``Jordan algebra'' was introduced by A.A.~Albert in the 1940s. Promoted by the pioneering works of I.~Kaplanski, E.M.~Alfsen, F.W.~Shultz, H.~Hanche-Olsen, E.~St\"{o}rmer, J.D.M.~Wright and M.A.~Youngson, JB$^*$- and JBW$^*$-algebras are Jordan extensions of C$^*$- and von Neumann algebras. A {\em JB$^*$-algebra} is a complex Jordan algebra $(B,\circ)$ equipped with a complete norm $\|\cdot\|$ and an involution $*$ satisfying the following axioms: \begin{enumerate}[$(a)$] \item $\norm{x\circ y}\le\norm{x}\norm{y}$ for $x,y\in B$; \item $\norm{U_{x} (x^*)}=\norm{x}^3$ for $x\in B$ ({\em a Gelfand-Naimark type axiom}), \end{enumerate} where $U_{x} (y) = 2(x\circ y)\circ x-x^2\circ y$ ($x,y\in B$). These axioms guarantee that the involution of every JB$^*$-algebra is an isometry (see \cite[Lemma 4]{youngson1978vidav} or \cite[Proposition 3.3.13]{Cabrera-Rodriguez-vol1}). JB$^*$-algebras were also called \emph{Jordan C$^*$-algebras} by I. Kaplansky and other authors at the early stages of the theory. Every C$^*$-algebra is a JB$^*$-algebra with its original norm and involution and the Jordan product defined in \eqref{eq special Jordan product}. Actually, every norm closed self-adjoint Jordan subalgebra of a C$^*$-algebra is a JB$^*$-algebra. Those JB$^*$-algebras obtained as JB$^*$-subalgebras of C$^*$-algebras are called \emph{JC$^*$-algebras}. There exist JB$^*$-algebras which are \emph{exceptional} in the sense that they cannot be identified with a JB$^*$-subalgebra of a C$^*$-algebra, this is the case of the JB$^*$-algebra $H_3(\mathbb O)$ of all $3\times 3$-hermitian matrices with entries in the algebra $\mathbb O$ of complex octonions (see, for example, \cite[\S 7.2]{hanche1984jordan}, \cite[\S 6.1 and 7.1]{Cabrera-Rodriguez-vol2} or \cite[\S 6.2 and 6.3]{Finite}). A JBW$^*$-algebra (respectively, a JW$^*$-algebra) is a JB$^*$-algebra (respectively, a JC$^*$-algebra) which is also a dual Banach space. JB$^*$-algebras are intrinsically connected with another mathematical object deeply studied in the literature. A \emph{JB-algebra} is a real Jordan algebra $J$ equipped with a complete norm satisfying \begin{equation}\label{eq axioms of JB-algebras} \|a^{2}\|=\|a\|^{2}, \hbox{ and } \|a^{2}\|\leq \|a^{2}+b^{2}\|\ \hbox{ for all } a,b\in J. \end{equation} In a celebrated lecture in St.\ Andrews in 1976 I. Kaplansky suggested the definition of JB$^*$-algebra and pointed out that the self-adjoint part $B_{sa}=\{x\in B : x^* =x\}$ of a JB$^*$-algebra is always a JB-algebra. One year later, J.D.M. Wright contributed one of the most influential results in the theory of JB$^*$-algebras by proving that the complexification of every JB-algebra is a JB$^*$-algebra (see \cite{Wright1977}). A \emph{JC-algebra} (respectively, a \emph{JW-algebra}) is a norm-closed (respectively, a weak$^*$-closed) real Jordan subalgebra of the hermitian part of a C$^*$-algebra (respectively, of a von Neumann algebra). Suppose $B$ is a unital JB$^*$-algebra. The smallest norm-closed real Jordan subalgebra $C(a)$ of $B_{sa}$ containing a self-adjoint element $a$ in $B$ and $1$ is associative. According to the usual notation, the \emph{spectrum of $a$} in $B$, denoted by $Sp(a)$, is the the set of all real $\lambda$ such that $a - \lambda 1$ does not have an inverse in $C(a)$ (cf. \cite[3.2.3]{hanche1984jordan}). If $B$ is not unital we consider the unitization of $B$ to define the spectrum of a self-adjoint element. It is known that the JB$^*$-subalgebra of $B$ generated by a single self-adjoint element $a\in B$ and the unit is isometrically JB$^*$-isomorphic to the commutative C$^*$-algebra $C(Sp(a))$, of all complex-valued continuous functions on $Sp(a)$ (see \cite[3.2.4. The spectral theorem]{hanche1984jordan}). An element $a\in B$ is called positive if $a=a^*$ and $Sp(a)\subseteq \mathbb{R}_{0}^+$ (cf. \cite[3.3.3]{hanche1984jordan}). Although there exist exceptional JB$^*$-algebras which cannot be embedded as JB$^*$-subalgebras of C$^*$-algebras, the JB$^*$-subalgebra of a JB$^*$-algebra $B$ generated by two hermitian elements (and the unit element) is a JC$^*$-algebra (compare Macdonald's and Shirshov-Cohn's theorems \cite[Theorems 2.4.13 and 2.4.14]{hanche1984jordan}, \cite[Corollary 2.2]{Wright1977} or \cite[Proposition 3.4.6]{Cabrera-Rodriguez-vol1}). Consequently, for each $x\in B$, the element $x\circ x^*$ is positive in $B$. We refer to the references \cite{hanche1984jordan,Cabrera-Rodriguez-vol1} and \cite{Cabrera-Rodriguez-vol2} for the basic background, notions and results on JB$^*$-algebras. C$^*$- and JB$^*$-algebras have been extensively employed as a framework for studying bounded symmetric domains in complex Banach spaces of infinite dimension, as an alternative notion to simply connected domains. The open unit ball of every C$^*$-algebra is a bounded symmetric domain (see \cite{harris1974bounded}) and the open unit balls of (unital) JB$^*$-algebras are, up to a biholomorphic mapping, those bounded symmetric domains which have a realization as a tube domain, i.e. an upper half-plane (cf. \cite{BraKaUp78}). These examples do not exhaust all possible bounded symmetric domains in arbitrary complex Banach spaces, a strictly wider class of Banach spaces is actually required. The most conclusive result was obtained by W. Kaup who proved in 1983 that every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB$^*$-triple \cite{kaup1983riemann}. A complex Banach space $E$ belongs to the class of {\em JB$^*$-triples} if it admits a triple product (i.e., a continuous mapping) $\J{\cdot}{\cdot}{\cdot}:E^3\to E$ which is symmetric and bilinear in the outer variables and conjugate linear in the middle variable and satisfies the next algebraic and geometric axioms: \begin{enumerate}[(JB$^*$-1)] \item $\J xy{\J abc}=\J{\J xya}bc-\J a{\J yxb}c+\J ab{\J xyc}$ for any $x,y,a,b,c\in E$ ({\em Jordan identity}); \item For any $a\in E$ the operator $L(a,a):x\mapsto \J aax$ is hermitian with non-negative spectrum; \item $\norm{\J xxx}=\norm{x}^3$ for all $x\in E$ ({\em a Gelfand-Naimark type axiom}). \end{enumerate} C$^*$-algebras and JB$^*$-algebras belong to the wide list of examples of JB$^*$-triples when they are equipped with the triple products given by \begin{equation}\label{eq triple product JCstar and JBstar} \{a,b,c\} =\frac12 ( a b^* c + c b^* a),\hbox{ and } \{a,b,c\} = (a\circ b^*) \circ c +(c\circ b^*) \circ a - (a\circ c) \circ b^*, \end{equation} respectively (see \cite[Theorem 3.3]{BraKaUp78} or \cite[Theorem 4.1.45]{Cabrera-Rodriguez-vol1}). The first triple product in \eqref{eq triple product JCstar and JBstar} induces a structure of JB$^*$-triple on every closed subspace of the space $B(H,K),$ of all bounded linear operators between complex Hilbert spaces $H$ and $K$, which is closed under this triple product. In particular, $B(H,K)$ and every complex Hilbert space are JB$^*$-triples with their canonical norms and the first triple product given in \eqref{eq triple product JCstar and JBstar}. In a JB$^*$-triple $E$ the triple product is contractive, that is, $$\|\{x,y,z\}\|\le\|x\| \|y\| \|z\| \quad\mbox{ for all } x,y,z \mbox{ in } E$$ (cf. \cite[Corollary 3]{Friedman-Russo-GN} or \cite[Corollary 7.1.7]{Cabrera-Rodriguez-vol2}, \cite[P.\ 215]{chubook}). A linear bijection between JB$^*$-triples is a triple isomorphism if and only if it is an isometry (cf. \cite[Proposition 5.5]{kaup1983riemann} or \cite[Theorems 3.1.7, 3.1.20]{chubook}). Thus, a complex Banach space admits a unique triple product under which it is a JB$^*$-triple. A JBW$^*$-triple is a JB$^*$-triple which is also a dual space. Every JBW$^*$-triple admits a unique (in the isometric sense) predual and its triple product is separately weak$^*$-continuous (see \cite{BaTi}, \cite[Theorems 5.7.20, 5.7.38]{Cabrera-Rodriguez-vol2}). Each idempotent $e$ in a Banach algebra $A$ produces a Peirce decomposition of $A$ as a sum of eigenspaces of the left and right multiplication operators by the idempotent $e$. A.A. Albert extended the classical Peirce decomposition to the setting of Jordan algebras in the middle of the last century. The notion of idempotent might mean nothing in a general JB$^*$-triple. The appropriate alternative is the concept of tripotent. An element $e$ in a JB$^*$-triple $E$ is a \emph{tripotent} if $\{e,e,e\}=e$. It is worth mentioning that when a C$^*$-algebra $A$ is regarded as a JB$^*$-triple with respect to the first triple product given in \eqref{eq triple product JCstar and JBstar}, an element $e\in A$ is a tripotent if and only if it is a partial isometry (i.e., $e e^*$, or equivalently $e^* e$, is a projection) in $A$. In case we fix a tripotent $e$ in a JB$^*$-triple $E$, the classical Peirce decomposition for associative and Jordan algebras extends to a \emph{Peirce decomposition} of $E$ associated with the eigenspaces of the mapping $L(e,e)$, whose eigenvalues are all contained in the set $\{0,\frac12,1\}$. For $j\in\{0,1,2\}$, the (linear) projection $P_{j} (e)$ of $E$ onto the eigenspace, $E_j(e)$, of $L(e, e)$ corresponding to the eigenvalue $\frac{j}{2},$ admits a concrete expression in terms of the triple product as follows: $$\begin{aligned} P_2 (e) &= L(e,e)(2 L(e,e) -id_{E})=Q(e)^2, \nonumber \\ P_1 (e) &= 4 L(e,e)(id_{E}-L(e,e)) =2\left(L(e,e)-Q(e)^2\right), \hbox{ and } \nonumber \\ P_0 (e) &= (id_{E}-L(e,e)) (id_{E}-2 L(e,e)), \end{aligned}$$ where $Q(e)(x)=\J exe$ for $x\in E$. The projection $P_{j} (e)$ is known as the \emph{Peirce}-$j$ \emph{projection} associated with $e$. Peirce projections are all contractive (cf. \cite[Corollary 1.2]{Friedman-Russo}), and the JB$^*$-triple $E$ decomposes as the direct sum $$E= E_{2} (e) \oplus E_{1} (e)\oplus E_0 (e),$$ which is termed the \emph{Peirce decomposition} of $E$ relative to $e$ (see \cite{Friedman-Russo}, \cite[Definition 1.2.37]{chubook} or \cite[Subsection 4.2.2]{Cabrera-Rodriguez-vol1} and \cite[Section 5.7]{Cabrera-Rodriguez-vol2} for more details). In the particular case in which $e$ is a tripotent (i.e. a partial isometry) in a C$^*$-algebra $A$ with initial projection $p_i= e^* e$ and final projection $p_f= e e^*$, the subspaces in the Peirce decomposition are precisely $$A_2(e) =p_f A p_i, \, A_1(e) = p_f A (1-p_i)\oplus (1-p_f) A p_i,\, A_0(e) = (1-p_f) A (1-p_i). $$ A tripotent $e$ in a JB$^*$-triple $E$ is called \emph{complete} if $E_0 (e) =\{0\}$. We shall say that $e$ is \emph{unitary} if $E= E_2(e)$, or equivalently, if $\{e,e,x\}={x}$ for all $x\in E$. Obviously, every unitary is a complete tripotent, but the converse implication is not always true; consider for example a non-surjective isometry $e$ in $B(H)$. A non-zero tripotent $e$ satisfying $E_2(e) = \mathbb{C} e$ is called \emph{minimal}. Note that in a unital JB$^*$ algebra there is another definition of a unitary element (cf.\ \cite[Definition 4.2.25]{Cabrera-Rodriguez-vol1}). However, it is equivalent to the above-defined notion as witnessed by the following fact (where condition $(3)$ is the mentioned alternative definition). We will work solely with the notion of unitary tripotent defined above (i.e., with condition $(1)$ from the fact below) but we include these equivalences for the sake of completeness. \begin{fact} Let $B$ be a unital JB$^*$-algebra and let $u\in B$. The following assertions are equivalent. \begin{enumerate}[$(1)$] \item $u$ is a unitary tripotent, i.e., $u$ is a tripotent with $B_2(u)=B$. \item $u$ is a tripotent and $u\circ u^*=1$. \item $u\circ u^*=1$ and $u^2\circ u^*=u$, i.e., $u^*$ is the Jordan inverse of $u$. \end{enumerate} \end{fact} \begin{proof} The equivalence $(1)\Leftrightarrow(3)$ is proved in \cite[Proposition 4.3]{BraKaUp78} (see also \cite[Theorem 4.2.28]{Cabrera-Rodriguez-vol1}). To prove the equivalence $(1)\Leftrightarrow(2)$ observe that assertion $(2)$ means that $1=\J uu1$, i.e., $1\in B_2(u)$. It remains to use \cite[Proposition 6.6]{hamhalter2019mwnc}. \end{proof} Complete tripotents in a JB$^*$-triple $E$ can be geometrically characterized since a norm-one element $e$ in $E$ is a complete tripotent if and only if it is an extreme point of its closed unit ball (cf. \cite[Lemma 4.1]{BraKaUp78}, \cite[Proposition 3.5]{kaup1977jordan} or \cite[Theorem 4.2.34]{Cabrera-Rodriguez-vol1}). Consequently, every JBW$^*$-triple contains an abundant collection of complete tripotents. Given a unitary element $u$ in a JB$^*$-triple $E$, the latter becomes a unital JB$^*$-algebra with Jordan product and involution defined by \begin{equation}\label{eq circ-star} x\circ_{u} y=\J xuy\mbox{ and }x^{*_u}=\J uxu\qquad\mbox{for }x,y\in E, \end{equation} see \cite[Theorem 4.1.55]{Cabrera-Rodriguez-vol1}. We even know that $u$ is the unit of this JB$^*$-algebra (i.e., $u\circ_u x=x$ for $x\in E$). Each tripotent $e$ in a JB$^*$-triple $E$ is a unitary in the JB$^*$-subtriple $E_2(e)$, and thus $(E_2(e),\circ_e,*_{e})$ is a unital JB$^*$-algebra. Therefore, since the triple product is uniquely determined by the structure of a JB$^*$-algebra, unital JB$^*$-algebras are in one-to-one correspondence with those JB$^*$-triples admitting a unitary element. A linear subspace $I$ of a JB$^*$-triple $E$ is called a \emph{triple ideal} or simply an \emph{ideal} of $E$ if $\J IEE\subset I$ and $\J EIE\subset I$ (see \cite{horn1987ideal}). Let $I, J$ be two ideals of $E$. We shall say that $I$ and $J$ are orthogonal if $I\cap J =\{0\}$ (and consequently $\{I,J,E\} = \{J,I,E\}=\{0\}$). It is known that every weak$^*$-closed ideal $I$ of a JBW$^*$-triple $M$ is orthogonally complemented, that is, there exists another weak$^*$-closed ideal $J$ of $M$ which is orthogonal to $I$ and $M = I \oplus^{\infty} J$ (see \cite[Theorem 4.2$(4)$ and Lemmata 4.3 and 4.4]{horn1987ideal}). For each weak$^*$-closed ideal $I$ of $M$, we shall denote by $P_{I}$ the natural projection of $M$ onto $I$.\label{eq ideals in JBW} Let us observe that, in this case $P_{I}$ is always a weak$^*$-continuous triple homomorphism. \subsection{Positive functionals and prehilbertian seminorms}\label{subsec:seminorms} As in the case of C$^*$-algebras, a functional $\phi$ in the dual space, $B^*$, of a JB$^*$-algebra $B$ is called positive if it maps positive elements to non-negative real numbers. We shall frequently apply that a functional $\phi$ in the dual space of a unital JB$^*$-algebra $B$ is positive if and only if $\|\phi\| = \phi (1)$ (cf. \cite[Lemma 1.2.2]{hanche1984jordan}). The same conclusion holds for functionals in the predual of a JBW$^*$-algebra. A positive normal functional $\varphi$ in the predual of a JBW$^*$-algebra $B$ is called \emph{faithful} if $\varphi (a) = 0$ for $a\geq 0$ in $B$ implies $a=0$. If $\phi$ is a positive functional in the dual of a C$^*$-algebra $A,$ and $1$ denotes the unit element in $A^{**}$, the mapping $$(a,b)\mapsto \phi \left(\frac{ a b^* + b^* a}{2} \right)= \phi \{a,b,1\} \ \ (a,b\in A)$$ is a positive semi-definite sesquilinear form on $A\times A$, whose associated prehilbertian seminorm is denoted by $\|x \|_{\phi} = (\phi \{x,x,1\})^{1/2}$. If we consider a positive functional $\phi$ in the dual of a JB$^*$-algebra $B$, the associated prehilbertian seminorm is defined by $\|x \|_{\phi}^2 = \phi \{x,x,1\} =\phi (x\circ x^*),$ where $1$ stands for the unit in $B^{**}$. The lacking of a local order or positive cone in a general JB$^*$-triple, and hence the lacking of positive functionals makes a bit more complicated the definition of appropriate prehilbertian seminorms. Namely, let $\varphi$ be a functional in the predual of JBW$^*$-triple $M$ and let $z$ be a norm-one element in $M$ satisfying $\varphi (z) = \|\varphi\|$. Proposition 1.2 in \cite{barton1987grothendieck} proves that the mapping $M\times M\to \mathbb{C}$, $(x,y)\mapsto \varphi\{x,y,z\}$ is a positive semi-definite sesquilinear form on $M$ which does not depend on the choice of the element $z$ (that is, $\varphi\{x,y,z\} = \varphi\{x,y,\tilde{z}\}$ for every $x,y\in M$ and every $\tilde{z}\in M$ with $\|\tilde{z}\|=1$ and $\varphi(\tilde{z})=\norm{\varphi}$, see \cite[Proposition 5.10.60]{Cabrera-Rodriguez-vol2}). The associated prehilbertian seminorm is denoted by $\norm{x}_{\varphi} =(\varphi\{x,x,z\})^{1/2}$ ($x\in M$). Since the triple product of every JB$^*$-triple is contractive it follows that \begin{equation}\label{eq seminorm inequality} \|x\|_\varphi\le\sqrt{\|\varphi\|}\|x\| \hbox{ for all } x\in M.\end{equation} If $\varphi$ is a non-zero functional in the dual $E^*$ of a JB$^*$-triple $E$, and we regard $E^*$ as the predual of the JBW$^*$-triple $E^{**}$, the prehilbertian seminorm $\norm{\cdot}_{\varphi}$ on $E^{**}$ acts on $E$ by mere restriction. \subsection{Comparison theory of projections and tripotents} Two projections $p,q$ in a C$^*$-algebra $A$ (respectively, in a JB$^*$-algebra $B$) are said to be orthogonal ($p\perp q$ in short) if $ p q = 0$ (respectively, $p\circ q =0$). The relation ``being orthogonal'' can be used to define a natural partial ordering on the set of projections in $A$ (respectively, in $B$) defined by $ p\leq q$ if $q-p$ is a projection and $q-p \perp p$. We write $p < q$ if $p\leq q$ and $p\neq q$. Two tripotents $e$ and $u$ in a JB$^*$-triple $E$ are called \emph{orthogonal} ($e\perp u$ in short) if $\{e,e,u\} = 0$ (equivalently, $u\in M_0 (e)$). It is known that $e\perp u$ if and only if any of the following equivalent reformulations holds: \begin{enumerate}[$(1)$] \item $e\in E_0(u)$. \item $E_2(u)\subset E_0(e)$. \item $L(u,e)=0$. \item $L(e,u)=0$. \item Both $u+e$ and $u-e$ are tripotents. \item $\{u,u,e\} =0$. \end{enumerate} For proofs see \cite[Lemma 3.9]{loos1977bounded}, \cite[Proposition 6.7]{hamhalter2019mwnc} or \cite[Lemma 2.1]{Finite}. The induced partial order defined by this orthogonality on the set of tripotents is given by $e\leq u$ if $u-e$ is a tripotent with $u-e \perp e$. Let $\varphi$ be a non-zero functional in the predual of a JBW$^*$-triple $M$. By Proposition 2 in \cite{Friedman-Russo} (or \cite[Proposition 5.10.57]{Cabrera-Rodriguez-vol2}) there exists a unique tripotent $s(\varphi)\in M$, called the \emph{support tripotent} of $\varphi$, such that $\varphi=\varphi\circ P_2(s(\varphi))$, and $\varphi|_{M_2(s(\varphi))}$ is a faithful positive functional on the JBW$^*$-algebra $M_2(s(\varphi))$. In particular, $\norm{x}_{\varphi}^2=\varphi\{x,x,s(\varphi)\}$ for all $x\in M$. The support tripotent of a non-zero functional $\varphi$ in the predual of a JBW$^*$-triple $M$ is also the smallest tripotent in $M$ at which $\varphi$ attains its norm, that is, \begin{equation}\label{eq minimality of the support tripotent} \hbox{$\varphi (u) =\|\varphi\|$ for some tripotent $u\in M$} \Rightarrow s(\varphi)\leq u. \end{equation} Namely, the element $P_2(s(\varphi))(u)$ lies in the unit ball of $M_2(s(\varphi))$ because $P_2(s(\varphi))$ is contractive. Since $\varphi = \varphi\circ P_2(s(\varphi))$ and $\varphi|_{M_2(s(\varphi))}$ is a faithful functional on the JBW$^*$-algebra $M_2(s(\varphi))$, we deduce that $P_2(s(\varphi)) (u) = s(\varphi)$. It follows from \cite[Lemma 1.6 or Corollary 1.7]{Friedman-Russo} that $s(\varphi)\leq u.$ Actually the previous arguments prove \begin{equation}\label{eq minimality of the support tripotent for elements} \hbox{$\varphi (a) =\|\varphi\|$ for some element $a\in M$ with $\|a\|=1$} \Rightarrow a= s(\varphi)+P_0 (s(\varphi)) (a). \end{equation} Two projections $p$ and $q$ in a von Neumann algebra $W$ are called \emph{(Murray-von Neumann) equivalent} (written $p\sim q$) if there is a partial isometry $e\in W$ whose initial projection is $p$ and whose final projection is $q$. This Murray-von Neumann equivalence is employed to classify projections and von Neumann algebras in terms of their properties. For example a projection $p$ in $W$ is said to be \emph{finite} if there is no projection $q< p$ that is equivalent to $p$. For example, all finite-dimensional projections in $B(H)$ are finite, but the identity operator on $H$ is not finite when $H$ is an infinite-dimensional complex Hilbert space. The von Neumann algebra $W$ is called \emph{finite} if its unit element is a finite projection. The set of all finite projections in the sense of Murray-von Neumann in $W$ forms a (modular) sublattice of the set of all projections in $W$ (see e.g. \cite[Theorem V.1.37]{Tak}). We recall that a projection $p$ in $W$ is {\em infinite} if it is not finite, and {\em properly infinite} if $p\ne 0$ and $zp$ is infinite whenever $z$ is a central projection such that $zp\ne0$ (cf. \cite[Definition V.1.15]{Tak}). In the setting of JBW$^*$-algebras the notion of finiteness was replaced by the concept of modularity, and the Murray-von Neumann equivalence by the relation ``being equivalent by symmetries'', that is, two projections $p,q$ in a JBW$^*$-algebra $N$ are called equivalent (by symmetries) (denoted by $p\stackrel{s}{\sim} q$) if there is a finite set $s_l, s_2 ,\ldots, s_n$ of self-adjoint symmetries (i.e. $s_j =1-2p_j$ for certain projections $p_j$) such that $Q(s_1)\cdots Q(s_n) (p) =q$, where $Q(s_j) (x) =\{s_j,x,s_j\} = 2 (s_j \circ x) \circ s_j - s_j^2 \circ x$ for all $x\in N$ (cf. \cite[\S 10]{Topping1965}, \cite[5.1.4]{hanche1984jordan}, \cite[\S 3]{AlfsenShultzGeometry2003} or \cite[\S 7.1]{Finite}). Unlike Murray-von Neumann equivalence, $p \stackrel{s}{\sim} q$ in $N$ implies $1-p \stackrel{s}{\sim} 1-q$. When $M$ is a von Neumann algebra regarded as a JBW$^*$-algebra, and $p,q$ are projections in $M$, $p\stackrel{s}{\sim}q$ if and only if $p$ and $q$ are unitarily equivalent, i.e. there exists a unitary $u\in M$ such that $u p u^* = q$ (see \cite[Proposition 6.56]{AlfsenShultzStateSpace2001}). In particular, $p\stackrel{s}{\sim} q$ implies $p\sim q$. In a recent contribution we study the notion of finiteness in JBW$^*$-algebras and JBW$^*$-triples from a geometric point of view. In the setting of von Neumann algebras, the results by H. Choda, Y. Kijima, and Y. Nakagami assert that a von Neumann algebra $W$ is finite if and only if all the extreme points of its closed unit ball are unitary (see \cite[Theorem 2]{ChodaKijimaNak69} or \cite[Proof of Theorem 4]{mil}). Therefore, a projection $p$ in $W$ is finite if and only if every extreme point of the closed unit ball of $pWp$ is a unitary in the latter von Neumann algebra. This is the motivation for the notion of finiteness introduced in \cite{Finite}. According to the just quoted reference, a tripotent $e$ in a JBW$^*$-triple $M$ is called \begin{enumerate}[$\bullet$] \item {\em finite} if any tripotent $u\in M_2(e)$ which is complete in $M_2(e)$ is already unitary in $M_2(e)$; \item {\em infinite} if it is not finite; \item {\em properly infinite} if $e\ne 0$ and for each weak$^*$-closed ideal $I$ of $M$ the tripotent $P_I (e)$ is infinite whenever it is nonzero. \end{enumerate} If any tripotent in $M$ is finite, we say that $M$ itself is {\em finite}. Finite-dimensional JBW$^*$-triples are always finite \cite[Proposition 3.4]{Finite}. The JBW$^*$-triple $M$ is said to be {\em infinite} if it is not finite. Finally, $M$ is {\em properly infinite} if each nonzero weak$^*$-closed ideal of $M$ is infinite. Every JBW$^*$-triple decomposes as an orthogonal sum of weak$^*$-closed ideals $M_1,$ $M_2$, $M_3$ and $M_4,$ where $M_1$ is a finite JBW$^*$-algebra, $M_2$ is either a trivial space or a properly infinite JBW$^*$-algebra, $M_3$ is a finite JBW$^*$-triple with no nonzero direct summand isomorphic to a JBW$^*$-algebra, and $M_4$ is either a trivial space or $M_4=qV_4$, where $V_4$ is a von Neumann algebra, $q\in V_4$ is a properly infinite projection such that $qV_4$ has no direct summand isomorphic to a JBW$^*$-algebra; we further know that $M_4$ is properly infinite in case that it is not zero (see \cite[Theorem 7.1]{Finite} where a more detailed description is presented). This decomposition applies in the particular case in which $M$ is a JBW$^*$-algebra with the appropriate modifications and simplifications on the summands to avoid those which are not JBW$^*$-algebras. In a von Neumann algebra $W$ the two notions of finiteness coincide for projections (see \cite[Lemma 3.2$(a)$]{Finite}). Every modular projection in a JBW$^*$-algebra is a finite tripotent in the sense above, but the reciprocal is not always true (cf. \cite[Lemma 7.12 and Remark 7.13]{Finite}). Finite JBW$^*$-triples enjoy formidable properties. For example, for each finite tripotent $u$ in a JBW$^*$-algebra $M$ there is a unitary element $e\in M$ with $u \leq e$ (cf. \cite[Proposition 7.5]{Finite}). More details and properties can be found in \cite{Finite}. A projection $p$ in a von Neumann algebra $W$ is called \emph{abelian} if the subalgebra $pW p$ is abelian (see \cite[Definition V.1.15]{Tak}). The von Neumann algebra $W$ is said to be of \emph{type I} or \emph{discrete} if every nonzero (central) projection contains a nonzero abelian subprojection \cite[Definition V.1.17]{Tak}. In the previous definition the word central can be relaxed (see, for example, \cite[Corollary 4.20]{stra-zsi}). A tripotent $e$ in a JB$^*$-triple is said to be \emph{abelian} if the JB$^*$-algebra $E_2(u)$ is associative, or equivalently, $(E_2(u),\circ_u,*_u)$ is a unital abelian C$^*$-algebra. Obviously, any minimal tripotent is abelian. We further know that every abelian tripotent is finite \cite[Lemma 3.2$(e)$]{Finite}. According to \cite{horn1987classification,horn1988classification} and \cite{horn1987ideal}, a JBW$^*$-triple $M$ is said to be of \emph{type $I$} (respectively, \emph{continuous}) if it coincides with the weak$^*$ closure of the span of all its abelian tripotents (respectively, it contains no non-zero abelian tripotents). Every JBW$^*$-triple can be written as the orthogonal sum of two weak$^*$-closed ideals $M_1$ and $M_2$ such that $M_1$ is of type $I$ and $M_2$ is continuous (any of these summands might be trivial). G. Horn and E. Neher established in \cite{horn1987classification,horn1988classification} structure results describing type $I$ and continuous JBW$^*$-triples. Concretely, every JBW$^*$-triple of type $I$ may be represented in the form \begin{equation}\label{eq:representation of type I JBW* triples} \bigoplus_{j\in J}^{\ell_\infty}A_j\overline{\otimes}C_j, \end{equation} where the $A_j$'s are abelian von Neumann algebras and the $C_j$'s are Cartan factors (the concrete definitions will be presented below in Section \ref{sec:type I}, the reader can also consult \cite{loos1977bounded, kaup1981klassifikation, kaup1997real} for details). To reassure the reader we shall simply note that every Cartan factor $C$ is a JBW$^*$-triple. In the case in which $C$ is a JW$^*$-subtriple of some $B(H)$ and $A$ is an abelian von Neumann algebra, the symbol $A\overline{\otimes}C$ denotes the weak$^*$-closure of the algebraic tensor product $A{\otimes}C$ in the von Neumann tensor product $A\overline{\otimes} B(H)$ (see \cite[Section IV.1]{Tak} and \cite[\S 1]{horn1987classification}). In the remaining cases $C$ is finite-dimensional and $A\overline{\otimes} C$ will stand for the completed injective tensor product (see \cite[Chapter 3]{ryan}). \section{Majorizing certain seminorms}\label{sec:majorizing} The main result will be proved using its dual version. The starting point is the following dual version of Theorem~\ref{T:triples}$(2)$. \begin{thm}[{\cite[Theorem 3]{peralta2001grothendieck}}]\label{T:triples-dual} Let $M$ be a JBW$^*$-triple, $H$ a Hilbert space and $T:M\to H$ a weak$^*$-to-weak continuous linear operator. Given $\varepsilon>0$, there are norm-one functionals $\varphi_1,\varphi_2\in M_*$ such that $$\norm{Tx}\le(\sqrt{2}+\varepsilon)\norm{T}\left(\norm{x}^2_{\varphi_1}+\varepsilon\norm{x}^2_{\varphi_2}\right)^{\frac12}\quad\mbox{ for }x\in M.$$ \end{thm} We continue by recalling two results from \cite{HKPP-BF}. The first one is essentially the main result and easily implies Theorem~\ref{T:triples}$(3)$. The second one was used to prove one of the particular cases and we will use it several times as well. \begin{prop}[{\cite[Theorem 2.4]{HKPP-BF}}]\label{P:2-1BF} Let $M$ be a JBW$^*$-triple. Then given any two functionals $\varphi_1,\varphi_2$ in $M_*$, there exists a norm-one functional $\psi\in M_*$ such that $$\norm{x}_{\varphi_1}^2+\norm{x}_{\varphi_2}^2\le 2(\norm{\varphi_1}+\norm{\varphi_2})\cdot \norm{x}_\psi^2$$ for all $x\in M.$ \end{prop} \begin{lemma}[{\cite[Proposition 3.2]{HKPP-BF}}]\label{L:rotation} Let $M$ be a JBW$^*$-triple and let $\varphi\in M_*$. Assume that $p\in M$ is a tripotent such that $s(\varphi)\in M_2(p)$. Then there exists a functional $\tilde{\varphi}\in M_*$ such that {$\norm{\tilde{\varphi}}=\norm{\varphi}$}, $s(\tilde{\varphi})\le p$ and $\norm{x}_\varphi\le\sqrt{2}\norm{x}_{\tilde{\varphi}}$ for all $x\in M$. \end{lemma} The key step to prove our main result is the following proposition which says that for JBW$^*$-algebras a stronger version of Proposition~\ref{P:2-1BF} is achievable. \begin{prop}\label{P:majorize 1+2+epsilon} Let $M$ be a JBW$^*$-algebra. Then given any two functionals $\varphi_1,\varphi_2$ in $M_*$ and $\varepsilon>0$, there exists a norm-one functional $\psi\in M_*$ such that $$\norm{x}_{\varphi_1}^2+\norm{x}_{\varphi_2}^2\le (\norm{\varphi_1}+2\norm{\varphi_2}+\varepsilon) \norm{x}_\psi^2 \mbox{ for }x\in M.$$ \end{prop} Using this proposition we will easily deduce the main result in Section~\ref{sec:proofs} below. Proposition \ref{P:majorize 1+2+epsilon} will be proved using the following result. \begin{prop}\label{P:key decomposition alternative} Let $M$ be a JBW$^*$-algebra, $\varphi\in M_*$ and $\varepsilon>0$. Then there are a functional $\tilde\varphi\in M_*$ and a unitary element $w\in M$ such that $$\norm{\tilde\varphi}\le\norm{\varphi}, \quad s(\tilde\varphi)\le w \quad\mbox{ and } \norm{\cdot}^2_\varphi\le(1+\varepsilon)\norm{\cdot}^2_{\tilde\varphi}.$$ \end{prop} This proposition will be proved at the beginning of Section~\ref{sec:proofs} using the results from Sections~\ref{sec:type I} and~\ref{sec:JW*}. Let us now show that it implies Proposition~\ref{P:majorize 1+2+epsilon}. \begin{proof}[Proof of Proposition~\ref{P:majorize 1+2+epsilon} from Proposition~\ref{P:key decomposition alternative}.] Assume that $M$ is a JBW$^*$-algebra, $\varphi_1,\varphi_2\in M_*$ and $\varepsilon>0$. Let $\tilde\varphi_1\in M_*$ and $w\in M$ correspond to $\varphi_1$ and $\frac{\varepsilon}{\norm{\varphi_1}}$ by Proposition~\ref{P:key decomposition alternative}. Since $w$ is unitary, we have $M_2(w)=M$, hence we may apply Lemma~\ref{L:rotation} to get $\psi_2\in M_*$ such that $$s(\psi_{2})\le w, \ \norm{\psi_{2}}\le\norm{\varphi_{2}},\ \norm{\cdot}_{\varphi_{2 }}\le\sqrt{2}\norm{\cdot}_{\psi_{2}}. $$ Then $$\begin{aligned} \norm{\cdot}_{\varphi_1}^2+\norm{\cdot}_{\varphi_2}^2&\le \left(1+\frac{\varepsilon}{\norm{\varphi_1}}\right)\norm{\cdot}_{\tilde\varphi_{1}}^2+\norm{\cdot}_{\varphi_2}^2\le \left(1+\frac{\varepsilon}{\norm{\varphi_1}}\right)\norm{\cdot}_{\tilde\varphi_{1}}^2+2\norm{\cdot}_{\psi_2}^2\\ &=\norm{\cdot}_{\left(1+\frac{\varepsilon}{\norm{\varphi_1}}\right)\tilde\varphi_{1}+2\psi_2}^2 =\left(\left(1+\frac{\varepsilon}{\norm{\varphi_1}}\right)\norm{\tilde\varphi_{1}}+2\norm{\psi_2}\right)\norm{\cdot}_\psi^2, \end{aligned} $$ where $$\psi=\frac{(1+\frac{\varepsilon}{\norm{\varphi_1}})\tilde\varphi_{1}+2\psi_2}{(1+\frac{\varepsilon}{\norm{\varphi_1}})\norm{\tilde\varphi_{1}}+2\norm{\psi_2}}.$$ (Note that the first equality follows from the fact that the support tripotents of both functionals are below $w$.) Since the functionals $\tilde\varphi_{1}$ and $\psi_2$ attain their norms at $w$, we deduce that $\norm{\psi}=1$. It remains to observe that $$\left(1+\frac{\varepsilon}{\norm{\varphi_1}}\right)\norm{\tilde\varphi_{1}}+2\norm{\psi_2}\le \norm{\varphi_1}+\varepsilon+2\norm{\varphi_2}.$$ \end{proof} \section{Finite or type I JBW$^*$-algebras}\label{sec:type I} The aim of this section is to prove a stronger version of Proposition~\ref{P:key decomposition alternative} for a large subclass of JBW$^*$-algebras (see Proposition \ref{P:type I approx}). We follow the notation from \cite{Finite} recalled in Section \ref{sec: notation}. Since in a finite JBW$^*$-algebra any tripotent is majorized by a unitary one (cf. \cite[Lemma 3.2(d)]{Finite}), we get the following observation. \begin{obs}\label{obs:finite JBW* algebras} Let $M$ be a finite JBW$^*$-algebra. Then Proposition~\ref{P:key decomposition alternative} holds for $M$ in a very strong version -- one can take $\tilde\varphi=\varphi$ and $\varepsilon =0$. \end{obs} There is a larger class of JBW$^*$-algebras for which we get a stronger and canonical version of Proposition~\ref{P:key decomposition alternative}. The concrete result appears in the content of the following proposition. The exact relationship with Proposition~\ref{P:key decomposition alternative} will be explained in Remark \ref{Rem} (1) below. We first recall that, in the setting of JBW$^*$-triples, two normal functionals $\varphi$ and $\psi$ in the predual of a JBW$^*$-triple $M$ are called (\emph{algebraically}) \emph{orthogonal} (written $\varphi\perp \psi$) if their support tripotents are orthogonal in $M$---that is, $s(\varphi)\perp s(\psi)$ (cf. \cite{FriRu87,EdRu01}). It is shown in \cite[Lemma\ 2.3]{FriRu87} (see also \cite[Theorem 5.4]{EdRu01}) that $\varphi,\psi\in M_*$ are orthogonal if and only if they are ``geometrically'' \emph{$L$-orthogonal} in $M_*$ i.e., $\|\varphi \pm \psi\| = \|\varphi\| + \|\psi\|$. In particular $\norm{\cdot}_{\varphi+\psi}^2=\norm{\cdot}_\varphi^2 + \norm{\cdot}_\psi^2$ if $\varphi$ and $\psi$ are orthogonal because in this case $\varphi$, $\psi$ and $\varphi+\psi$ attain their respective norms at $s(\varphi)+s(\psi)$. \begin{prop}\label{P:type I approx} Let $M$ be a JBW$^*$-algebra which is triple-isomorphic to a direct sum $M_1\oplus^{\ell_\infty}M_2$, where $M_1$ is a finite JBW$^*$-algebra and $M_2$ is a type I JBW$^*$-algebra. Let $\varphi\in M_*$ be arbitrary. Then for each $\varepsilon>0$ there are two functionals $\varphi_1,\varphi_2\in M_*$ such that \begin{enumerate}[$(i)$] \item $\varphi=\varphi_1+\varphi_2$; \item $\varphi_1\perp\varphi_2$; \item $\norm{\varphi_2}<\varepsilon$; \item $s(\varphi_1)$ is a finite tripotent in $M$. \end{enumerate} \end{prop} The rest of this section is devoted to prove Proposition~\ref{P:type I approx}. To this end we will use the following decomposition result which was essentially established in \cite{Finite}. Let us note that the concrete definition of a type 2 Cartan factor can be found in the next subsection. \begin{prop}\label{P:type I decomposition} Let $M$ be a JBW$^*$-algebra which is triple-isomorphic to a direct sum $M_1\oplus^{\ell_\infty}M_2$, where $M_1$ is a finite JBW$^*$-algebra and $M_2$ is a type I JBW$^*$-algebra. Then $M$ is triple-isomorphic to a JBW$^*$-algebra of the form $$N\oplus^{\ell_\infty}\left(\bigoplus_{j\in J}L^\infty(\mu_j)\overline{\otimes}C_j\right)\oplus^{\ell_\infty}\left(\bigoplus_{\lambda\in \Lambda}L^\infty(\nu_\lambda)\overline{\otimes}B(H_\lambda)\right),$$ where \begin{itemize} \item $N$ is a finite JBW$^*$-algebra; \item $J$ and $\Lambda$ are (possibly empty) sets; \item $\mu_j$'s and $\nu_\lambda$'s are probability measures; \item $C_j$ is an infinite-dimensional type 2 Cartan factor for each $j\in J$; \item $H_\lambda$ is an infinite-dimensional Hilbert space for each $\lambda\in\Lambda$. \end{itemize} \end{prop} \begin{proof} By \cite[Theorem 7.1]{Finite} $M$ is triple-isomorphic to $N\oplus^{\ell_\infty} N_1$, where $N$ is a finite JBW$^*$-algebra and $N_1$ is (either trivial or) a properly infinite JBW$^*$-algebra. By the same theorem $N_1$ is triple-isomorphic to $$\left(\bigoplus_{j\in J}L^\infty(\mu_j)\overline{\otimes}C_j\right)\oplus^{\ell_\infty}N_2,$$ where the first summand has the above-mentioned form and $N_2$ is (either trivial or) a properly infinite von Neumann algebra. Since by the assumptions $N_2$ is clearly of type I, we may conclude using \cite[Theorem V.1.27]{Tak}. \end{proof} We observe that the validity of Proposition~\ref{P:type I approx} is preserved by $\ell_\infty$-sums, so it is enough to prove it for the individual summands from Propostion~\ref{P:type I decomposition}. For the finite JBW$^*$-algebra $N$ we may use Observation~\ref{obs:finite JBW* algebras}. We will prove the desired conclusion for the summands $L^\infty(\mu_j)\overline{\otimes}C_j$. For the remaining summands an easier version of the same proof works as we will explain below. \subsection{The case of type 2 Cartan factors}\label{subsec:C2} Let us start by recalling the definition of type 2 Cartan factors. Let $H$ be a Hilbert space with a fixed orthonormal basis $(e_\gamma)_{\gamma\in \Gamma}$. Then $H$ is canonically represented as $\ell^2(\Gamma)$. For $\xi\in H$ let $\overline{\xi}$ be the coordinatewise complex conjugate of $\xi$. Further, for $x\in B(H)$ we denote by $x^t$ the operator defined by $$x^t\xi=\overline{x^*\overline{\xi}},\qquad\xi\in H.$$ Then $x^t$ is the transpose of $x$ with respect to the fixed orthonormal basis, i.e., $$\ip{x^t e_\gamma}{e_\delta}=\ip{x e_\delta}{e_\gamma}\mbox{ for }\gamma,\delta\in\Gamma$$ (see, e.g., \cite[Section 5.3]{Finite} for the easy computation). Then $$B(H)_s=\{x\in B(H);\, x^t=x\}\mbox{ and }B(H)_a=\{x\in B(H);\, x^t=-x\}$$ are the so-called \emph{Cartan factors} of \emph{type 3} and \emph{2}, respectively. They are formed by operators with symmetric (antisymmetric, respectively) `representing matrices' with respect to the fixed orthonormal basis. We will deal with the second case, i.e., with `antisymmetric operators'. So, assume that $H$ has infinite dimension (or, equivalently, $\Gamma$ is an infinite set). Let $M=B(H)_a$. Define $\pi:B(H)\to M$ by $\pi(x)=\frac12(x-x^t)$. Then $\pi$ is a norm-one projection which is moreover weak$^*$-to-weak$^*$ continuous. Hence $\pi_*:M_*\to B(H)_*$ defined by $\pi_*(\varphi)=\varphi\circ\pi$ is an isometric injection. Moreover $$\begin{aligned}\pi_*(M_*)&=\{\varphi\in B(H)_*;\, \varphi(x^t)=-\varphi(x)\mbox{ for }x\in B(H)\}\\&=\{\varphi\in B(H)_*;\, \varphi|_{B(H)_s}=0\}.\end{aligned}$$ Recall that $B(H)_*$ is isometric to the space of nuclear operators $N(H)$ via the trace duality (cf. \cite[Theorem II.1.8]{Tak}). Moreover, any $y\in N(H)$ is represented in the form $$y=\sum_{k\geq 1} \lambda_k\ip{\cdot}{\eta_k}\xi_k$$ where $(\xi_k)$ and $(\eta_k)$ are orthonormal sequences in $H$ and the $\lambda_k$ are positive numbers with $\displaystyle\sum_{k\geq 1}\lambda_k=\norm{y}_N$. Then clearly $$y^*=\sum_{k\geq 1} \lambda_k\ip{\cdot}{\xi_k}\eta_k,$$ hence for any $\xi\in H$ we have $$y^t\xi=\overline{y^*\overline{\xi}} =\overline{\sum_{k\geq 1} {\lambda_k} \ip{\overline{\xi}}{\xi_k}\eta_k} =\sum_{k\geq 1}\lambda_k\ip{\xi}{\overline{\xi_k}}\overline{\eta_k},$$ thus $$y^t=\sum_{k\geq 1} \lambda_k\ip{\cdot}{\overline{\xi_k}}\overline{\eta_k}.$$ In particular \begin{equation}\label{eq transpose preserves traces} \tr{y^t}=\sum_{k\geq 1} \lambda_k \ip{\overline{\eta_k}}{\overline{\xi_k}}=\sum_{k\geq 1} \lambda_k\ip{\xi_k}{\eta_k}=\tr{y}. \end{equation} Hence, given $\varphi\in B(H)_*$ represented by $y\in N(H)$, the functional $\varphi^t(x)=\varphi(x^t)$, $x\in B(H)$ is represented by $y^t$. Indeed, $$\varphi^t(x)=\varphi(x^t)=\tr{x^ty}=\tr{y^tx}=\tr{xy^t}\mbox{ for }x\in B(H).$$ It follows that $$\pi_*M_*=\{\varphi\in B(H)_*;\, \varphi\mbox{ is represented by an antisymmetric nuclear operator}\}.$$ \begin{proof}[Proof of Proposition~\ref{P:type I approx} for $M=B(H)_a$] Fix $\varphi\in M_*$ of norm one and $\varepsilon>0$. Let $u=s(\varphi)\in M$. Set $\tilde\varphi=\pi_*\varphi$. Fix $y\in N(H)$ representing $\tilde\varphi$. Then $$y=\sum_{k\geq 1} \lambda_k\ip{\cdot}{\eta_k}\xi_k$$ where $(\xi_k)$ and $(\eta_k)$ are orthonormal sequences in $H$ and the $\lambda_k$ are strictly positive numbers with $\displaystyle \sum_{k\geq 1}\lambda_k=1$. Observe that $$s(\tilde\varphi)=\sum_{k\geq 1}\ip{\cdot}{\xi_k}\eta_k.$$ Moreover, since $y$ is antisymmetric, we deduce that $s(\tilde\varphi)$ is also antisymmetric. Indeed, by the above we have $$y=-y^t=-\sum_{k\geq 1} \lambda_k\ip{\cdot}{\overline{\xi_k}}\overline{\eta_k}.$$ Hence $$s(\tilde\varphi)=-\sum_{k\geq 1}\ip{\cdot}{\overline{\eta_k}}\overline{\xi_k}=-s(\tilde\varphi)^t.$$ For $\delta>0$ set $$y_\delta=\sum_{\lambda_k\ge\delta} \lambda_k\ip{\cdot}{\eta_k}\xi_k.$$ Then $y_\delta$ is a finite rank operator and $$y_\delta^t=\sum_{\lambda_k\ge\delta}\lambda_k\ip{\cdot}{\overline{\xi_k}}\overline{\eta_k}.$$ By uniqueness of the nuclear representation (the sequence $(\lambda_k)$ is unique and for any fixed $\lambda>0$ the linear spans of those $\eta_k$, resp. $\xi_k$, for which $\lambda_k=\lambda$ are uniquely determined) we deduce that $y_\delta$ is antisymmetric and hence its support tripotent $$u_\delta=\sum_{\lambda_k\ge\delta}\ip{\cdot}{\xi_k}\eta_k$$ is antisymmetric as well. Fix $\delta>0$ such that $\sum_{\lambda_k<\delta}\lambda_k<\varepsilon$. Then $\norm{y-y_\delta}_N<\varepsilon$. Let $\tilde\varphi_1$ be the functional represented by $y_\delta$ and $\tilde\varphi_2=\tilde\varphi-\tilde\varphi_1$ (i.e., the functional represented by $y-y_\delta$). Since $y_\delta$ is antisymmetric, both $\tilde\varphi_1$ and $\tilde\varphi_2$ belong to $\pi_*M_*$. Moreover, $s(\tilde\varphi_1)=u_\delta$ and $s(\tilde\varphi_2)=u-u_\delta$. Since $u_\delta\perp u-u_\delta$, we deduce that $\tilde{\varphi}_1\perp\tilde{\varphi}_2$. Further, $u_\delta$ is a finite tripotent, being a finite rank partial isometry. Since we are in $\pi_*M_*$, we have functionals $\varphi_1,\varphi_2\in M_*$ such that $\tilde{\varphi}_j=\pi_*\varphi_j$. It is now clear that they provide the sought decomposition of $\varphi$. \end{proof} We have settled the case of $B(H)_a$. Note that for $M=B(H)$ the same proof works -- we just do not use the mapping $\pi$ and are not obliged to check the antisymmetry. The proof was done using the Schmidt decomposition of nuclear operators. To prove the result for the tensor product we will use a measurable version of Schmidt decomposition established in the following subsection. \subsection{Measurable version of Schmidt decomposition} In this subsection we are going to prove the following result (note that $K(H)$ denotes the C$^*$-algebra of compact operators on $H$). \begin{thm}\label{T:measurable Schmidt} Let $H$ be a Hilbert space. Then there are sequences $(\lambda_n)_{n=0}^\infty$ and $(\uu_n)_{n=0}^\infty$ of mappings such that the following properties are fulfilled for $n\in\mathbb N$ and $x\in K(H)$: \begin{enumerate}[$(a)$] \item $\lambda_n:K(H)\to[0,\infty)$ is a lower-semicontinuous mapping; \item $\lambda_{n+1}(x)<\lambda_n(x)$ whenever $x\in K(H)$ and $\lambda_n(x)>0$; \item $\uu_n:K(H)\to K(H)$ is a Borel measurable mapping; \item $\uu_n(x)$ is a finite rank partial isometry on $H$; \item $\uu_n(x)=0$ whenever $\lambda_n(x)=0$; \item The partial isometries $\uu_k(x)$, $k\in\mathbb N\cup\{0\}$, are pairwise orthogonal; \item $x=\sum_{n=0}^\infty\lambda_n(x)\uu_n(x),$ where the series converges in the operator norm. \end{enumerate} \end{thm} Let us point out that the Borel measurability in this theorem and in the lemmata used in the proof is considered with respect to the norm topology. However, if $X$ is a separable Banach space, it is well known and easy to see that any norm open set is weakly $F_\sigma$, hence the norm Borel sets coincide with the weak Borel sets (cf. \cite[pages 74 and 75]{Kuo75}). This applies in particular to $H$, $K(H)$ and $K(H)\times H$ where $H$ is a separable Hilbert space. The proof will be done in several steps contained in the following lemmata. \begin{lemma}\label{L:measurability of singular numbers} Let $H$ be a Hilbert space (not necessarily separable). For $x\in K(H)$ let $(\alpha_n(x))$ be the sequence of its singular numbers. Moreover, let $(\lambda_n(x))$ be the strictly decreasing version of $(\alpha_n(x))$ (recall that the sequence $(\alpha_n(x))$ itself is non-increasing), completed by zeros if necessary. I.e., $$\lambda_n(x)=\begin{cases} \alpha_k(x) & \mbox{ if }\card \{\alpha_0(x),\alpha_1(x),\dots,\alpha_k(x)\}=n+1\\ 0 & \mbox{ if such $k$ does not exist.} \end{cases}$$ Then the following assertions are valid for each $n\in\mathbb N\cup\{0\}$. \begin{enumerate}[$(i)$] \item $\alpha_n$ is a $1$-Lipschitz function on $K(H)$; \item $\lambda_n$ is a lower semicontinuous function on $K(H)$, in particular it is Borel measurable and of the first Baire class. \end{enumerate} \end{lemma} \begin{proof} $(i)$ This is proved in \cite[Corollary VI.1.6]{Gohberg90} and easily follows from the following well-known formula for singular numbers $$\alpha_n(x)=\operatorname{dist} \Big(x, \Big\{y\in K(H);\, \dim yH\le n\Big\}\Big), \quad x\in K(H), n\in\mathbb N\cup\{0\}$$ (cf. \cite[Theorem VI.1.5]{Gohberg90}). $(ii)$ Clearly $\lambda_n\ge0$. Moreover, for each $c>0$ we have $\lambda_n(x)>c$ if and only if $$\exists\, c_0>c_1>\dots>c_n>c_{n+1}=c,\, \exists\, k_0,k_1,\dots k_n\in \mathbb N\, \hbox{ such that }$$ $$ \alpha_{k_j} (x)\in (c_{j+1},c_j)\ \ \forall j\in\{0,\dots,n\}.$$ Since the functions $\alpha_k$ are continuous by $(i)$, $\{x;\, \lambda_n(x)>c\}$ is open. Now the lower semicontinuity easily follows. Finally, any lower semicontinuous function on a metric space is clearly $F_\sigma$-measurable, hence Borel measurable and also of the first Baire class (cf. \cite[Corollary 3.8(a)]{LMZ}). \end{proof} \begin{lemma}\label{L:measurable projections} Let $H$ be a Hilbert space. For any $x\in K(H)_+$ and $n\in\mathbb N\cup\{0\}$ let $p_n(x)$ be the projection onto the eigenspace with respect to the eigenvalue $\lambda_n(x)$ provided $\lambda_n(x)>0$ and $p_n(x)=0$ otherwise. Then the mapping $p_n$ is Borel measurable. \end{lemma} \begin{proof} We start by proving that the mapping $p_0$ is Borel measurable. For $x\in K(H)_+\setminus\{0\}$ we set $$\psi(x)=\frac{x-\lambda_0(x)\cdot I}{2(\lambda_0(x)-\lambda_1(x))}+I.$$ Then the mapping $\psi:K(H)_+\setminus\{0\}\to B(H)_{sa}$ is Borel measurable (by Lemma~\ref{L:measurability of singular numbers}$(ii)$, note that for $x\in K(H)_+\setminus\{0\}$ we have $\lambda_0(x)>\lambda_1(x)$). Moreover, since $$x=\sum_{n\ge0} \lambda_n(x)p_n(x),$$ by the Hilbert-Schmidt theorem, we deduce that $$\psi(x)=p_0(x)+\sum_{n\ge1} \frac{\lambda_0(x)-2\lambda_1(x)+\lambda_n(x)}{2(\lambda_0(x)-\lambda_1(x))}p_n(x) +\frac{\lambda_0(x)-2\lambda_1(x)}{2(\lambda_0(x)-\lambda_1(x))}(I-\sum_{n\ge0}p_n(x)),$$ hence the spectrum of $\psi(x)$ is $$\sigma(\psi(x))=\left\{1,\tfrac{\lambda_0(x)-2\lambda_1(x)}{2(\lambda_0(x)-\lambda_1(x))}\right\}\cup\left\{\tfrac{\lambda_0(x)-2\lambda_1(x)+\lambda_n(x)}{2(\lambda_0(x)-\lambda_1(x))};\, n\ge 1\right\}\subset\{1\}\cup(-\infty,\tfrac12].$$ It follows that $p_0(x)=f(\psi(x))$ whenever $f$ is a continuous function on $\mathbb R$ with $f=0$ on $(-\infty,\frac12]$ and $f(1)=1$. Since the mapping $y\mapsto f(y)$ is continuous on $B(H)_{sa}$ by \cite[Proposition I.4.10]{Tak}, we deduce that $p_0$ is a Borel measurable mapping. Further, for $n\in\mathbb N$ we have $$p_n(x)=\begin{cases}0,& \hbox{if } \lambda_n(x)=0,\\ p_0\left(x-\sum_{k=0}^{n-1}\lambda_k(x)p_k(x)\right),& \hbox{if } \lambda_n(x)>0,\end{cases}$$ hence by the obvious induction we see that $p_n$ is Borel measurable as well. \end{proof} \begin{proof}[Proof of Theorem~\ref{T:measurable Schmidt}] Fix any $x\in K(H)$. Let $x=u(x)\abs{x}$ be the polar decomposition. By the Hilbert-Schmidt theorem we have $$\abs{x}=\sum_n \lambda_n(x) \, p_n(\abs{x})$$ (note that $\lambda_n(x)=\lambda_n(\abs{x})$). Hence $$x=\sum_n \lambda_n(x) u(x) p_n(\abs{x})=\sum_n \lambda_n(x) u_n(x),$$ where $u_n(x)=u(x)p_n(\abs{x})$ are mutually orthogonal partial isometries (of finite rank). The mappings $\lambda_n$ are lower semicontinuous by Lemma~\ref{L:measurability of singular numbers}. Further, the assignment $x\mapsto\abs{x}=\sqrt{x^*x}$ is continuous by the properties of the functional calculus. Indeed, the mapping $x\mapsto x^*x$ is obviously continuous and the mapping $y\mapsto\sqrt{y}$ is continuous on the positive cone of $K(H)$ by \cite[Proposition I.4.10]{Tak}. Hence, we can deduce from Lemma~\ref{L:measurable projections} that the assignments $x\mapsto p_n(\abs{x})$ are Borel measurable. Since $u_n(x)=0$ whenever $\lambda_n(x)=0$ and $u_n(x)=\frac1{\lambda_n(x)}xp_n(\abs{x})$ if $\lambda_n(x)>0$, it easily follows that the mapping $u_n$ is Borel measurable. \end{proof} \begin{prop}\label{P:measurable nuclear rep} Let $H$ be a separable Hilbert space. Consider the mappings $\lambda_n$ and $u_n$ provided by Theorem~\ref{T:measurable Schmidt} restricted to $N(H)$. Then $\lambda_n$ and $u_n$ are Borel measurable also with respect to the nuclear norm. Moreover, the series from assertion $(g)$ converges absolutely in the nuclear norm and, moreover, $$\norm{x}=\sum_{n=0}\lambda_n\norm{u_n(x)}$$ where the norm is the nuclear one. \end{prop} \begin{proof} The Borel measurability of $\lambda_n$ and $u_n$ follows from the continuity of the canonical inclusion of $N(H)$ into $K(H)$ together with Theorem~\ref{T:measurable Schmidt}. The rest follows from the Schmidt representation of nuclear operators. \end{proof} \subsection{Proof of Proposition~\ref{P:type I approx}} Let us adopt the notation from Subsection~\ref{subsec:C2}. Moreover, let $\mu$ be a probability measure and $A=L^\infty(\mu)$. Set $W=A\overline{\otimes}B(H)$. Then $W$ is a von Neumann algebra canonically represented in $B(L^2(\mu,H))$ (for a detailed description see e.g. \cite[Section 5.3]{Finite}). Moreover, on $L^2(\mu,H)$ we have a canonical conjugation (the pointwise one -- recall that $H=\ell^2(\Gamma)$ is equipped with the coordinatewise conjugation). Therefore we have a natural transpose of any $x\in W$ defined by $$x^t(\f)=\overline{x^*(\overline{\f})}, \qquad \f\in L^2(\mu,H).$$ Then we have a canonical identification $$M=A\overline{\otimes}B(H)_a=W_a=\{x\in W;\, x^t=-x\}.$$ Similarly as in Subsection~\ref{subsec:C2} we denote by $\pi$ the canonical projection of $W$ onto $M$, i.e., $x\mapsto\frac12(x-x^t)$. Recall that, by \cite[Theorem IV.7.17]{Tak}, $W_*=L^1(\mu,N(H))$ (the Lebesgue-Bochner space). Since $\pi$ is a weak$^*$-weak$^*$ continuous norm-one projection, we have an isometric embedding $\pi_*:M_*\to W_*$ defined by $\pi_*\omega=\omega\circ \pi$. Moreover, clearly $$\pi_*(M_*)=\{\omega\in W_*;\, \omega^t=-\omega\}.$$ \begin{lemma} Assume that $\g\in L^1(\mu,N(H))=W_*$. Then the following assertions hold. \begin{enumerate}[$(i)$] \item $\g^*(\omega)=(\g(\omega))^*$ $\mu$-a.e., \item $\g^t(\omega)=(\g(\omega))^t$ $\mu$-a.e. \end{enumerate} \end{lemma} \begin{proof} Let us start by explaining the meaning. On the left-hand side we consider the involution and transpose applied to $\g$ as to a functional on $W$, while on the right-hand side these operations are applied to the nuclear operators $\g(\omega)$. Observe that it is enough to prove the equality for $\g=\chi_E y$ (where $E$ is a measurable set and $y\in N(H))$ as functions of this form are linearly dense in $L^1(\mu,N(H))$, i.e., we want to prove $$(\chi_E y)^*=\chi_E y^*\mbox{ and }(\chi_E y)^t=\chi_E y^t.$$ It is clear that the elements on the right-hand side belong to $L^1(\mu,N(H))=W_*$, so the equality may be proved as equality of functionals. Since these functionals are linear and weak$^*$-continuous on $W$, it is enough to prove the equality on the generators $f\otimes x$, $f\in L^\infty(\mu)$, $x\in B(H)$. So, fix such $f$ and $x$ and recall that $$(f\otimes x)^*=\overline{f}\otimes x^*\mbox{ and }(f\otimes x)^t=f\otimes x^t.$$ Indeed, the first equality follows from the very definition of the von Neumann tensor product, the second one is proved in the computation before Lemma 5.10 in \cite{Finite}. Hence we have $$\begin{aligned} \ip{(\chi_E y)^*}{f\otimes x}&=\overline{\ip{\chi_Ey}{\overline{f}\otimes x^*}}= \overline{\int_E\overline{f}\,\mbox{\rm d}\mu \cdot\tr{yx^*}} =\int_E f\,\mbox{\rm d}\mu \cdot\tr{(yx^*)^*}\\&=\int_E f\,\mbox{\rm d}\mu \cdot\tr{xy^*}=\int_E f\,\mbox{\rm d}\mu \cdot\tr{y^*x}=\ip{\chi_E y^*}{f\otimes x}\end{aligned}$$ and, similarly, by \eqref{eq transpose preserves traces}, we get $$\begin{aligned} \ip{(\chi_E y)^t}{f\otimes x}&=\ip{\chi_Ey}{f\otimes x^t}= \int_E f\,\mbox{\rm d}\mu \cdot\tr{yx^t} =\int_E f\,\mbox{\rm d}\mu \cdot\tr{(yx^t)^t}\\&=\int_E f\,\mbox{\rm d}\mu \cdot\tr{xy^t}=\int_E f\,\mbox{\rm d}\mu \cdot\tr{y^tx}=\ip{\chi_E y^t}{f\otimes x}.\end{aligned}$$ \end{proof} It easily follows that $$\pi_*(M_*)=L^1(\mu,N(H)_a).$$ \begin{lemma}\label{L:sepred} Let $\g\in L^1(\mu,N(H))=W_*$. Then the following assertions hold. \begin{enumerate}[$(i)$] \item $\ip{f\otimes x}{\g}=\int f(\omega)\tr{x\g(\omega)}\,\mbox{\rm d}\mu(\omega)$ for $f\in L^\infty(\mu)$ and $x\in B(H)$. \item There is a projection $p\in B(H)$ with separable range such that $p\g(\omega)p=\g(\omega)$ $\mu$-a.e. In this case we have $(1\otimes p)\g(1\otimes p)=\g$, i.e., $$\ip{T}{\g}=\ip{(1\otimes p)T(1\otimes p)}{\g}\mbox{ for }T\in W.$$ \end{enumerate} \end{lemma} \begin{proof} $(i)$ Fix $f\in L^\infty(\mu)$ and $x\in B(H)$. Consider both the left hand side and the right hand side as functionals depending on $\g$. Since both functionals are linear and continuous on $L^1(\mu,N(H))$, it is enough to prove the equality for $\g=\chi_E y$ where $E$ is a measurable set and $y\in N(H)$. In this case we have $$\ip{f\otimes x}{\chi_E y}=\int_E f\,\mbox{\rm d}\mu \tr{xy},$$ so the equality holds. $(ii)$ Note that $\g$ is essentially separably-valued, so there is a separable subspace $Y\subset N(H)$ such that $\g(\omega)\in Y$ $\mu$-a.e. Since for any $y\in N(H)$ there is a projection $q$ with separable range with $qyq=y$ (due the the Schmidt representation), the existence of $p$ easily follows. To prove the last equality it is enough to verify it for the generators $T=f\otimes x$ and this easily follows from $(i)$. \end{proof} \begin{prop}\label{P:L1 measurable repr} Let $\g\in L^1(\mu,N(H))$. Then there are a separable subspace $H_0\subset H$, a sequence $(\zeta_n)$ of nonnegative measurable functions and a sequence $(\uu_n)$ of measurable mappings with values in $K(H_0)$ such that the following holds for each $\omega$: \begin{enumerate}[$(a)$] \item $\zeta_{n+1}(\omega)<\zeta_n(\omega)$ whenever $\zeta_n(\omega)>0$; \item $\uu_n(\omega)$ is a finite rank partial isometry on $H_0$; \item $\uu_n(\omega)=0$ whenever $\zeta_n(x)=0$; \item the partial isometries $\uu_k(\omega)$, $k\in\mathbb N\cup\{0\}$, are pairwise orthogonal; \item $\g=\sum_{n=0}^\infty\zeta_n\uu_n$ where the series converges absolutely almost everywhere and also in the norm of $L^1(\mu,N(H))$. \end{enumerate} \end{prop} \begin{proof} Let $p\in B(H)$ be a projection with separable range provided by Lemma \ref{L:sepred}$(ii)$ and set $H_0=pH$. Let $(\lambda_n)$ and $(u_n)$ be the mappings provided by Theorem~\ref{T:measurable Schmidt}. Let $\uu_n(\omega)=u_n(\g(\omega))$ and $\zeta_n(\omega)=\lambda_n(\g(\omega))$. Then these functions are measurable due to measurability of $\g$ and Proposition~\ref{P:measurable nuclear rep}. Assertions $(a)-(d)$ now follow from Theorem~\ref{T:measurable Schmidt}. By Proposition~\ref{P:measurable nuclear rep} we get the first statement of $(e)$ and, moreover, $$\sum_n\norm{\zeta_n(\omega)\uu_n(\omega)}=\norm{\g(\omega)}\ \mu\mbox{-a.e.},$$ hence the convergence holds also in the norm of $L^1(\mu,N(H))$, by the Lebesgue dominated convergence theorem for Bochner integral. \end{proof} Set $$W_0=\{\f:\Omega\to B(H);\, \f\mbox{ is bounded, measurable and has separable range} \}.$$ By a measurable function we mean a strongly measurable one, i.e., an almost everywhere limit of simple functions. However, note that weak measurability is equivalent in this case by Pettis measurability theorem as we consider only functions with separable range. Then $W_0$ is clearly a C$^*$-algebra when equipped with the pointwise operation and supremum norm. We remark that the following lemma seems to be close to the results of \cite[Section IV.7]{Tak}. However, it is not clear how to apply these results in our situation, so we give the proofs. \begin{lemma} For $\f\in W_0$ and $\h\in L^2(\mu,H)$ define the function $T_{\f}\h$ by the formula $$T_{\f}\h (\omega)= \f(\omega)(\h(\omega)),\quad \omega\in\Omega.$$ \begin{enumerate}[$(i)$] \item For each $\f\in W_0$ the mapping $T_{\f}$ is a bounded linear operator on $L^2(\mu,H)$ which belongs to $W$ and satisfies $\norm{T_{\f}}\le\norm{\f}_\infty$. \item If $\f\in W_0$ and $\g\in W_*=L^1(\mu,N(H))$, then $$\ip{T_{\f}}{\g}=\int\tr{\f(\omega)\g(\omega)}\,\mbox{\rm d}\mu(\omega).$$ \item $T_{\f}$ is a partial isometry (a projection) in $W$ whenever $\f(\omega)$ is a partial isometry (a projection) $\mu$-a.e. \item If $\g\in L^1(\mu,N(H))$ is represented as in Proposition~\ref{P:L1 measurable repr}$(e)$, then $s(\g)\le \sum_n T_{\uu_n^*}$ where series converges in the SOT topology in $W$. \end{enumerate} \end{lemma} \begin{proof} $(i)$ It is clear that the mapping $\h\mapsto T_{\f}\h$ is a linear mapping assigning to each $H$-valued function another $H$-valued function. Moreover, $$\norm{T_{\f}\h(\omega)}=\norm{\f(\omega)(\h(\omega))}\le\norm{\f(\omega)}\norm{\h(\omega)}\le\norm{\f}_\infty\norm{\h(\omega)}.$$ In particular, if a sequence $(\h_n)$ converges almost everywhere to a function $\h$, then $(T_{\f}\h_n)$ converges almost everywhere to $T_{\f}\h$. It follows that $T_{\f}$ is well defined on $L^2(\mu,H)$ (in the sense that if $\h_1=\h_2$ a.e., then $T_{\f}\h_1=T_{\f}\h_2$ a.e.). The next step is to observe that $T_{\f}\h$ is measurable whenever $\h$ is measurable. This is easy for simple functions. Further, any measurable function is an a.e. limit of a sequence of simple functions, hence the measurability follows by the above. Further, it follows from the above inequality that $\norm{T_{\f}\h}_2\le \norm{\f}_\infty\norm{\h}_2$, thus $\norm{T_{\f}}\le\norm{\f}_\infty$. Finally, by \cite[Lemma 5.12]{Finite} we get that $T_{\f}\in W$. $(ii)$ Let us first show that $\f\g\in L^1(\mu,N(H))$ whenever $\f\in W_0$ and $\g\in L^1(\mu,N(H))$. By the obvious inequalities the only thing to be proved is measurability of this mapping. This is easy if $\g$ is a simple function. The general case follows from the facts that any measurable function is an a.e. limit of simple functions and that measurability is preserved by a.e. limits of sequences. It remains to prove the equality. Since the functions from $W_0$ are separably valued, countably valued functions are dense in $W_0$. So, it is enough to prove the equality for countably valued functions. To this end let $$\f=\sum_{k\in\mathbb N}\chi_{E_k}x_k,$$ where $(E_k)$ is a disjoint sequence of measurable sets and $(x_k)$ is a bounded sequence in $B(H)$. For any $\h\in L^2(\mu,H)$ we have $$T_{\f}\h(\omega)=\sum_{k\in\mathbb N}\chi_{E_k}(\omega)x_k(\h(\omega)),\qquad \omega\in \Omega.$$ Since $T_{\f}\h\in L^2(\mu,H)$ by $(i)$ and the sets $E_k$ are pairwise disjoint, we deduce that $$T_{\f}\h=\sum_{k\in\mathbb N}T_{\chi_{E_k}x_k}\h,$$ where the series converges in $L^2(\mu,H)$. Since this holds for any $\h\in L^2(\mu,H)$, we deduce that $$T_{\f}=\sum_{k\in\mathbb N}T_{\chi_{E_k}x_k}$$ unconditionally in the SOT topology, hence also in the weak$^*$ topology of $W$. Thus, for any $\g\in W_*=L^1(\mu,N(H))$ we get $$\ip{T_{\f}}{\g}=\sum_{k\in\mathbb N}\ip{T_{\chi_{E_k}x_k}}{\g}=\sum_{k\in\mathbb N}\int_{E_k}\tr{x_k\g(\omega)}\,\mbox{\rm d}\mu(\omega)=\int\tr{\f(\omega)\g(\omega)}\,\mbox{\rm d}\mu(\omega), $$ where in the second equality we used Lemma~\ref{L:sepred}$(i)$. $(iii)$ This is obvious as the mapping $\f\mapsto T_{\f}$ is clearly a $*$-homomorphism of $W_0$ into $W$. $(iv)$ First observe that the mappings $\uu_n^*$ belong to $W_0$. Indeed, by Proposition~\ref{P:L1 measurable repr} the mapping $\uu_n$ is measurable and has separable range (as $K(H_0)$ is separable). Moreover, $\norm{\uu_n}_\infty\le1$ for each $n\in\mathbb N$. These properties are shared by $\uu_n^*$, hence $\uu_n^*\in W_0$. By $(iii)$ we deduce that $T_{\uu_n^*}$ is a partial isometry for any $n\in\mathbb N$. Moreover, these partial isometries are pairwise orthogonal (cf. property $(d)$ from Proposition~\ref{P:L1 measurable repr}), hence $U=\sum_n T_{\uu_n^*}$ is a well-defined partial isometry in $W$. Moreover, by taking $\g$ as in Proposition~\ref{P:L1 measurable repr}$(e)$, we have $$\begin{aligned} \ip{U}{\g}&=\sum_{n=0}^\infty\ip{T_{\uu_n^*}}{\g}=\sum_{n=0}^\infty\int \tr{\uu_n^*(\omega)\g(\omega)}\,\mbox{\rm d}\mu\omega\\&= \sum_{n=0}^\infty\int \zeta_n(\omega)\tr{\uu_n^*(\omega)\uu_n(\omega)}\,\mbox{\rm d}\mu(\omega) \\&=\int\sum_{n=0}^\infty \zeta_n(\omega)\tr{\uu_n^*(\omega)\uu_n(\omega)}\,\mbox{\rm d}\mu(\omega)=\int \norm{\g(\omega)}\,\mbox{\rm d}\mu(\omega)=\norm{\g},\end{aligned}$$ thus $s(\g)\le U$. \end{proof} \begin{proof}[Proof of Proposition~\ref{P:type I approx} for $A\overline{\otimes}B(H)_a$] Fix any $\g\in M_*=L^1(\mu,N(H)_a)$ and $\varepsilon>0$. Fix its representation from Proposition~\ref{P:L1 measurable repr}. Fix $N\in\mathbb N$ such that $$\norm{\sum_{n>N}\zeta_n\uu_n}<\varepsilon.$$ This is possible by the convergence established in Proposition~\ref{P:L1 measurable repr}. Note that $$-\g=\g^t=\sum_{n=1}^\infty\zeta_n\uu_n^t,$$ hence $\uu_n^t=-\uu_n^t$. (Note that the representation from Proposition~\ref{P:L1 measurable repr} is unique due to the uniqueness of the Hilbert-Schmidt representation). Let $$\g_1=\sum_{n=1}^N\zeta_n\uu_n.$$ Then $\g_1\in M_*$ as $\g_1^t=-\g_1$. Further, let $$\vv=\sum_{n=1}^N \uu_n.$$ We have $\g-\g_1\perp \g_1$ as $$s(\g_1)\le T_{\vv^*} \mbox{ and }s(\g-\g_1)\le \sum_{n>N}T_{\uu_n^*}$$ and the two tripotents on the right-hand sides are orthogonal. Moreover, $T_{\vv^*}$ is a finite tripotent in $M$ by \cite[Proposition 5.31(i) and Lemma 5.16(ii)]{Finite}. This completes the proof. \end{proof} \begin{proof}[Proof of Proposition~\ref{P:type I approx} for $A\overline{\otimes}B(H)$] The proof is an easier version of the previous case. Fix $\g\in W_*=L^1(\mu,N(H))$ and $\varepsilon>0$. In the same way we find $N$ and define $\g_1$ and $\vv$. We omit the considerations of the transpose and antisymmetry. Finally, $T_{\vv^*}$ is a finite tripotent in $W$ by \cite[Proposition 4.7 and Lemma 5.16(ii)]{Finite}. \end{proof} \section{JW$^*$-algebras}\label{sec:JW*} The aim of this section is to prove the following proposition which will be used to prove Proposition~\ref{P:key decomposition alternative}. \begin{prop}\label{P:key decomposition} Let $M$ be a JBW$^*$-algebra, $\varphi\in M_*$ and $\varepsilon>0$. Then there are functionals $\varphi_1,\varphi_2\in M_*$ and a unitary element $w\in M$ satisfying the following conditions. \begin{enumerate}[$(i)$] \item\label{it:key decomposition i} $\norm{\varphi_1}\le\norm{\varphi}$; \item $\norm{\varphi_2}<\varepsilon$; \item $s(\varphi_1)\le w$; \item\label{it:key decomposition iv} $\norm{\cdot}_\varphi^2\le\norm{\cdot}_{\varphi_1}^2+\norm{\cdot}_{\varphi_2}^2$. \end{enumerate} \end{prop} The proof will be done at the end of the section with the help of several lemmata. We focus mainly on JW$^*$-algebras, i.e., on weak$^*$-closed Jordan $^*$-subalgebras of von Neumann algebras. To this end we recall some notation (cf. \cite[Section III.2]{Tak}). Let $A$ be a C$^*$-algebra and let $\phi\in A^*$. Then we define functionals $a\phi$ and $\phi a$ by \begin{equation} a\phi(x)=\phi(xa) \quad\mbox{ and }\quad \phi a(x)=\phi(ax) \quad\mbox{for } x\in A. \end{equation} Note that $a\phi,\phi a\in A^*$ and $\norm{a\phi}\le\norm{a}\norm{\phi}$, $\norm{\phi a}\le\norm{a}\norm{\phi}$. We recall the natural isometric involution $\phi\mapsto\phi^*$ defined by $\phi^*(x)=\overline{\phi(x^*)}$. Then clearly $(a\phi)^*=\phi^*a^*$, $(\phi a)^*=a^*\phi^*$. If $W$ is a von Neumann algebra and if $\phi\in W_*$, $a\in W$ then $a\phi, \phi a\in W_*$. Further, given $\phi\in W_*$ we set $\abs{\phi}=s(\phi)\phi$ where $s(\phi)\in W$ is the support tripotent of $\phi$. Then $\phi=s(\varphi)^*\abs{\phi}$ is the polar decomposition of $\phi$ (cf. \cite[Section III.4]{Tak}). More generally, if $a\in W$ is a norm-one element on which $\phi$ attains its norm then we have $\betr{\phi}=a\phi$, $\phi=a^*\betr{\phi}$, $\betr{\phi^*}=\phi a$ (cf. \eqref{eq minimality of the support tripotent for elements}). Note that $\betr{\phi}=\betr{\phi}^*$ since $\betr{\phi}$ is positive. All this is stable by small perturbations as witnessed by the following lemma. \begin{lemma}[{\cite[Lemma 3.3]{pfitzner-jot}}]\label{l perturbation of functionals} Let $A$ be a C$^*$-algebra, $\phi$ a functional on $A$ and $a,b$ in the unit ball of $A$. Then \begin{eqnarray} \norm{\phi-a^*\betr{\phi}\;} &\leq& ({2\norm{\phi}})^{1/2}\,\,\betr{\,\norm{\phi} - \phi(a)}^{1/2} \label{glA3_1}\\ \norm{\betr{\phi}-a\phi} &\leq& ({2\norm{\phi}})^{1/2}\,\,\betr{\,\norm{\phi} - \phi(a)}^{1/2} \label{glA3_2}\\ \norm{\betr{\phi^*}-\phi a} &\leq& ({2\norm{\phi}})^{1/2}\,\,\betr{\,\norm{\phi} - \phi(a)}^{1/2}. \label{glA3_3} \end{eqnarray} \end{lemma}\noindent (As to \eqref{glA3_3}, which is not stated explicitly in \cite[Lemma 3.3]{pfitzner-jot}, note that it follows easily from \eqref{glA3_2} by $\norm{\betr{\phi^*}-\phi a}=\norm{\betr{\phi^*}-a^*\phi^*}\le(2\norm{\phi^*})^{1/2}\,\,\betr{\norm{\phi^*} - \phi^*(a^*)}^{1/2}=({2\norm{\phi}})^{1/2}\,\,\betr{\,\norm{\phi} - \phi(a)}^{1/2}$.) There is another way to obtain positive functionals: We can write $\phi=\phi_1-\phi_2+i(\phi_3-\phi_4)$ with positive $\phi_k\in W_*$ ($k=1, 2, 3, 4$) such that $\norm{\phi_k-\phi_{k+1}}=\norm{\phi_k}+\norm{\phi_{k+1}}\le\norm{\phi}$, $k=1, 3$ (cf. \cite[Theorem III.4.2]{Tak}). Then we set $$[\phi]=\frac12\sum_{k=1}^4\phi_k=\frac12(\abs{\phi_1-\phi_2}+\abs{\phi_3-\phi_4})$$ and obtain that $[\phi]\in W_*$ is positive, $\norm{[\phi]}\le\norm{\phi}$ and $\betr{\phi(a)}\le2[\phi](a)$ for all positive $a\in W$. Finally, let us remark that if $A$ is a C$^*$-algebra, then $A^{**}$ is a von Neumann algebra and $A^*=(A^{**})_*$, thus $\abs{\phi}$ and $[\phi]$ make sense also for continuous functionals on a C$^*$-algebra. \begin{lemma}\label{l norm-one functional close enough to states at a unitary Cstar} Let $W$ be von Neumann algebra, let $w\in W$ be a unitary element and $\delta\in(0,1)$. Let $\phi\in W_*$ be a norm-one functional such that $\phi (w) >1-\delta$ (in particular, $\phi(w)\in\mathbb R$). Then $\psi := w^* |\phi|$ is a norm-one element of $W_*$ satisfying $\psi(w) =1$ and $\|\phi -\psi\| < \sqrt{2\delta}$. \end{lemma} \begin{proof} On the one hand we have that $\|\psi \| \le \| |\phi|\| = \|\phi\|=1$. On the other hand, since $\psi (w) = (w^* |\phi|) (w) = |\phi| (w w^*) = |\phi| (1) = \| |\phi|\| = 1$ we deduce that $\norm{\psi}=1$. Applying \eqref{glA3_1} of Lemma \ref{l perturbation of functionals} we obtain $$ \|\phi - w^* |\phi| \| \leq \sqrt{2 } \betr{ 1 - \phi(w)}^{1/2} \leq \sqrt{2 \delta},$$ which finishes the proof. \end{proof} We continue by extending the previous lemma to JW$^*$-algebras. \begin{lemma}\label{l norm-one functional close enough to states at a unitary} Let $M$ be a JW$^*$-algebra, $w\in M$ a unitary element and $\delta\in(0,1)$. Let $\phi\in M_*$ be a norm-one functional such that $\phi (w) >1-\delta$ (in particular, $\phi(w)\in\mathbb R$). Then there exists a norm-one functional $\psi\in M_*$ satisfying $\psi(w) =1$ and $\|\phi -\psi\| < \sqrt{2\delta}$. \end{lemma} \begin{proof} Let us assume that $M$ is a JW$^*$-subalgebra of a von Neumann algebra $W$. Let $1$ denote the unit of $M$. Then $1$ is a projection in $W$, thus, up to replacing $W$ by $1W1$, we may assume that $M$ contains the unit of $W$. We observe that $w$, being a unitary element in $M$, is unitary in $W$. Let $\tilde{\phi}\in W_*$ be a norm-preserving extension of $\phi$ provided by \cite[Theorem]{Bun01}. By hypothesis, $1-\delta < {\phi} (w)= \tilde{\phi} (w) \leq \|{\phi}\| = \|\tilde{\phi}\|=1$. Now, applying Lemma \ref{l norm-one functional close enough to states at a unitary Cstar} to $W$, $\tilde{\phi}\in W_*$ and the unitary $w$, we find a norm-one functional $\tilde{\psi}\in W_*$ satisfying $\tilde{\psi} (w) =1$ and $\|\tilde{\phi} -\tilde{\psi}\| < \sqrt{2\delta}$. Since $w\in M$ and $1= \tilde{\psi} (w)$, the functional $\psi = \tilde{\psi}|_{M}$ has norm-one, $\psi (w) =1$ and clearly $\|{\phi} -{\psi}\| < \sqrt{2\delta}$. \end{proof} \begin{lemma}\label{l ad-hoc 1 Jordan} Let $M$ be a JW$^*$-algebra, let $\phi\in M_*$ and $\delta>0$. Suppose $a_1, a_2$ are two norm-one elements in $M$ such that $$ \betr{\norm{\phi}-\phi(a_k)}<\delta\norm{\phi} \hbox{ for } k=1,2.$$ Then there is a positive functional $\omega\in M_*$ satisfying $\norm{\omega}\le2\sqrt{2\delta}\norm{\phi}$ and $$\betr{\phi\J xx{a_1-a_2}}\le4\norm{x}_\omega^2 \hbox{ for all } x\in M.$$ \end{lemma} \begin{proof} Similarly as in the proof of Lemma \ref{l norm-one functional close enough to states at a unitary} we may assume that $M$ is a JW$^*$-subalgebra of a von Neumann algebra $W$ containing the unit of $W$. Let $\tilde{\phi}\in W_*$ be a norm-preserving normal extension of $\phi$ (see \cite[Theorem]{Bun01}). Working in $W_*$ we set $\tilde\psi_l=a_1\tilde\phi-a_2\tilde\phi$ and $\tilde\psi_r=\tilde\phi a_1-\tilde\phi a_2$. By \eqref{glA3_2} of Lemma \ref{l perturbation of functionals} we have $\norm{\betr{\tilde\phi}-a_k\tilde\phi}\le\sqrt{2\delta}\norm{\tilde\phi}$ ($k=1,2$) hence $\norm{\tilde\psi_l}\le2\sqrt{2\delta}\norm{\tilde\phi}$. Likewise we get $\norm{\tilde\psi_r}\le2\sqrt{2\delta}\norm{\tilde\phi}$ with \eqref{glA3_3} of Lemma \ref{l perturbation of functionals}. Set $\tilde\omega=([\tilde\psi_l]+[\tilde\psi_r])/2$. Then $\norm{\tilde\omega}\le2\sqrt{2\delta}\norm{\tilde\phi}$ and $$\begin{aligned} \betr{\tilde\phi\J xx{a_1-a_2}}&=\frac12\betr{\tilde\psi_l(xx^*)+\tilde\psi_r(x^*x)}\le[\tilde\psi_l](xx^*)+[\tilde\psi_r](x^*x)\\ &\le([\tilde\psi_l]+[\tilde\psi_r])(xx^*+x^*x)=4\tilde\omega(\J xx1)=4\norm{x}_{\tilde\omega}^2. \end{aligned}$$ It remains to set $\omega=\tilde\omega|_M$. \end{proof} \begin{lemma}\label{l ad-hoc 2} Let $M$ be a JW$^*$-algebra, $\phi\in M_*$ and let $a$ be a norm-one element of $M$. Then there is a positive functional $\omega\in M_*$ such that $$\norm{\omega}\le\norm{\phi}\quad\mbox{ and }\quad\forall x\in W:\betr{\phi\J xxa}\le4\norm{x}_\omega^2.$$ \end{lemma} \begin{proof} The proof resembles the preceding one of Lemma \ref{l ad-hoc 1 Jordan}. Assume that $M$ is a JW$^*$-subalgebra of a von Neumann algebra $W$ and $1_W\in M$. Let $\tilde\phi\in W_*$ be a norm-preserving extension of $\phi$ (see \cite[Theorem]{Bun01}). Set $\tilde\psi_l=a\tilde\phi$ and $\tilde\psi_r=\tilde\phi a$. Then $\norm{\tilde\psi_l}\le\norm{a}\norm{\tilde\phi}=\norm{\tilde\phi}$ and, similarly, $\norm{\tilde\psi_r}\le\norm{\tilde\phi}$. Set $\tilde\omega=([\tilde\psi_l]+[\tilde\psi_r])/2$. Then $\norm{\tilde\omega}\le\norm{\tilde\phi}$ and $$\begin{aligned} \betr{\tilde\phi\J xxa}&=\frac12\betr{\tilde\psi_l(xx^*)+\tilde\psi_r(x^*x)}\le[\tilde\psi_l](xx^*)+[\tilde\psi_r](x^*x)\\ &\le([\tilde\psi_l]+[\tilde\psi_r])(xx^*+x^*x)=4\tilde\omega(\J xx1)=4\norm{x}_{\tilde\omega}^2. \end{aligned}$$ Finally, we may set $\omega=\tilde{\omega}|_M$. \end{proof} \begin{proof}[Proof of Proposition~\ref{P:key decomposition}] It follows from \cite[Theorem 7.1]{Finite} that any JBW$^*$-algebra $M$ can be represented by $M_1\oplus^{\ell_\infty} M_2$ where $M_1$ is a finite JBW$^*$-algebra and $M_2$ is a JW$^*$-algebra. The validity of Proposition~\ref{P:key decomposition} for finite JBW$^*$-algebras follows immediately from Observation~\ref{obs:finite JBW* algebras}. Since the validity of Proposition~\ref{P:key decomposition} is clearly preserved by $\ell_\infty$-sums, it remains to prove it for JW$^*$-algebras. So, assume that $M$ is a JW$^*$-algebra and $\varphi\in M_*$. By homogeneity we may assume $\norm{\varphi}=1$. Fix $\varepsilon>0$. Choose $\delta>0$ such that $12\sqrt{2\delta}<\varepsilon$. By the Wright-Youngson extension of the Russo-Dye theorem, the convex hull of all unitary elements in $M$ is norm dense in the closed unit ball of $M$ (see \cite[Theorem 2.3]{WrightYoungson77} or \cite[Fact 4.2.39]{Cabrera-Rodriguez-vol1}). We can therefore find a unitary element $w$ such that $\varphi (w) > 1-\delta$. By Lemma~\ref{l norm-one functional close enough to states at a unitary} there exists a norm-one functional $\psi\in M_*$ satisfying $\psi(w)=1$ and $\|\varphi -\psi\| < \sqrt{2\delta}.$ Set $u=s(\varphi)$. For $x\in M$ we then have $$\begin{aligned} \norm{x}_\varphi^2 &= \varphi \J xxu = \psi \J xxw +(\varphi-\psi)\J xxw + \varphi \J xx{u-w}. \end{aligned}$$ Applying Lemma~\ref{l ad-hoc 2} to $\varphi-\psi$ and $w$ we find a positive functional $\omega_1\in M_*$ with $\norm{\omega_1}\le\norm{\varphi-\psi}<\sqrt{2\delta}$ such that $$\abs{(\varphi-\psi)\J xxw}\le 4\norm{x}_{\omega_1}^2\mbox{ for }x\in M.$$ Applying Lemma~\ref{l ad-hoc 1 Jordan} to the functional $\varphi$ and the pair $w,u\in M$ we get a positive functional $\omega_2\in M_*$ with $\norm{\omega_2}\le2\sqrt{2\delta}$ such that $$\abs{\varphi \J xx{u-w}}\le 4\norm{x}_{\omega_2}^2\mbox{ for }x\in M.$$ Hence we have for each $x\in M$ $$\norm{x}_\varphi^2\le \norm{x}_\psi^2+4(\norm{x}_{\omega_1}^2+\norm{x}_{\omega_2}^2)=\norm{x}_\psi^2+\norm{x}_{4(\omega_1+\omega_2)}^2,$$ where we used that $\omega_1$ and $\omega_2$ are positive functionals. Since $s(\psi)\le w$ (just have in mind that $\psi(w)=1$ and \eqref{eq minimality of the support tripotent}), $w$ is unitary and $$\norm{4(\omega_1+\omega_2)}<12\sqrt{2\delta},$$ it is enough to set $\varphi_1=\psi$ and $\varphi_2=4(\omega_1+\omega_2)$. \end{proof} \begin{remark}\label{Rem} (1) Note that by \cite[Proposition 7.5]{Finite} any finite tripotent in a JBW$^*$-algebra is majorized by a unitary element, hence Proposition~\ref{P:type I approx} is indeed a stronger version of Proposition~\ref{P:key decomposition} in the special case in which the JBW$^*$-algebra $M$ is a direct sum of a finite JBW$^*$-algebra and a type $I$ JBW$^*$-algebra. (For (\ref{it:key decomposition i}) and (\ref{it:key decomposition iv}) of Proposition~\ref{P:key decomposition} see the remarks before the statement of Proposition~\ref{P:type I approx}.) Further, as will be seen at the beginning of the next section, Proposition~\ref{P:key decomposition} is the main ingredient for proving Proposition~\ref{P:key decomposition alternative}. (2) There is an alternative way of proving Proposition~\ref{P:key decomposition}. It follows from \cite[Theorem 7.1]{Finite} that any JBW$^*$-algebra $M$ can be represented by $M_1\oplus^{\ell_\infty} M_2\oplus^{\ell^\infty} M_3$ where $M_1$ is a finite JBW$^*$-algebra, $M_2$ is a type I JBW$^*$-algebra and $M_3$ is a von Neumann algebra. So, we can conclude using Proposition~\ref{P:type I approx} and giving the above argument only for von Neumann algebras (which is slightly easier). \end{remark} \section{Proofs of the main results}\label{sec:proofs} We start by proving Proposition~\ref{P:key decomposition alternative}. \begin{proof}[Proof of Proposition~\ref{P:key decomposition alternative}.] Let $M$ be a JBW$^*$-algebra, $\varphi\in M_*$ and $\varepsilon>0$. By homogeneity we may assume that $\norm{\varphi}=1$. Let $\varphi_1,\varphi_2$ and $w$ correspond to $\varphi$ and $\frac\varepsilon2$ by Proposition~\ref{P:key decomposition}. Since $w$ is unitary, we have $M_2(w)=M$, hence we may apply Lemma~\ref{L:rotation} to get $\psi_2\in M_*$ such that $$s(\psi_{2})\le w, \ \norm{\psi_{2}}\le\norm{\varphi_{2}},\ \norm{\cdot}_{\varphi_{2 }}\le\sqrt{2}\norm{\cdot}_{\psi_{2}}. $$ Then $$\begin{aligned} \norm{\cdot}_\varphi^2&\le\norm{\cdot}_{\varphi_1}^2+\norm{\cdot}_{\varphi_2}^2\le \norm{\cdot}_{\varphi_{1}}^2+2\norm{\cdot}_{\psi_2}^2=\norm{\cdot}_{\varphi_{1}+2\psi_2}^2 =(\norm{\varphi_{1}}+2\norm{\psi_2})\norm{\cdot}_\psi^2, \end{aligned} $$ where $$\psi=\frac{\varphi_{1}+2\psi_2}{\norm{\varphi_{1}}+2\norm{\psi_2}}.$$ (Note that the first equality follows from the fact that the support tripotents of both functionals are below $w$.) Since the functionals $\varphi_{1}$ and $\psi_2$ attain their norms at $w$, we deduce that $\norm{\psi}=1$. It remains to observe that $$\norm{\varphi_{1}}+2\norm{\psi_2}\le \norm{\varphi}+2\norm{\varphi_2}\le1+\varepsilon.$$ This completes the proof. \end{proof} Having proved Proposition~\ref{P:key decomposition alternative}, we know that Proposition~\ref{P:majorize 1+2+epsilon} is valid as well. Using it and Theorem~\ref{T:triples-dual} we get the following theorem. \begin{thm}\label{T:JBW*-algebras} Let $M$ be a JBW$^*$-algebra, let $H$ be a Hilbert space and let $T:M\to H$ be a weak$^*$-to-weak continuous linear operator. Given $\varepsilon>0$, there is a norm-one functional $\varphi\in M_*$ such that $$\norm{Tx}\le(\sqrt2+\varepsilon)\norm{T}\norm{x}_\varphi\mbox{ for }x\in M.$$ \end{thm} Now we get the main result by the standard dualization. \begin{proof}[Proof of Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}] Let $T:B\to H$ be a bounded linear operator from a JB$^*$-algebra into a Hilbert space. Let $\varepsilon>0$. Since Hilbert spaces are reflexive, the second adjoint operator $T^{**}$ maps $B^{**}$ into $H$ and it is weak$^*$-to-weak continuous. Further, $B^{**}$ is a JBW$^*$-algebra (cf. \cite[Theorem 4.4.3]{hanche1984jordan} and \cite{Wright1977} or \cite[Proposition 5.7.10]{Cabrera-Rodriguez-vol2} and \cite[Theorems 4.1.45 and 4.1.55]{Cabrera-Rodriguez-vol1}), so Theorem~\ref{T:JBW*-algebras} provides the respective functional $\varphi\in (B^{**})_*=B^*$. \end{proof} We further note that for JB$^*$-algebras we have two different forms of the Little Grothendieck theorem -- a triple version (the just proved Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}) and an algebraic version (an analogue of Theorem~\ref{T:C*alg-sym}). The difference is that the first form provides just a norm-one functional while the second one provides a state, i.e., a positive norm-one functional. Let us now show that the algebraic version may be proved from the triple version. \begin{thm}\label{T:algebraic version dual} Let $M$ be a JBW$^*$-algebra, let $H$ be a Hilbert space and let $T:M\to H$ be a weak$^*$-to-weak continuous linear operator. Given $\varepsilon>0$, there is a state $\varphi\in M_*$ such that $$\norm{Tx}\le(2+\varepsilon)\norm{T}\varphi(x\circ x^*)^{1/2}\mbox{ for }x\in M.$$ \end{thm} \begin{proof} By Theorem~\ref{T:JBW*-algebras} there is a norm-one functional $\psi\in M_*$ such that $$\norm{Tx}\le (\sqrt2+\frac{\varepsilon}{\sqrt2})\norm{T}\norm{x}_\psi\mbox{ for }x\in M.$$ Since $M$ is unital and $M_2(1)=M$, Lemma~\ref{L:rotation} yields a norm-one functional $\varphi\in M_*$ with $s(\varphi)\le1$ and $\norm{\cdot}_\psi\le\sqrt2\norm{\cdot}_\varphi$. Then $\varphi$ is a state (note that $\varphi(1)=1$) and $$\norm{Tx}\le(2+\varepsilon)\norm{T}\norm{x}_\varphi\mbox{ for }x\in M.$$ It remains to observe that $$\norm{x}_\varphi=\sqrt{\varphi\J xx1}=\sqrt{\varphi(x\circ x^*)}$$ for $x\in M$. \end{proof} \begin{thm}\label{T:algebraic version non-dual} Let $B$ be a JB$^*$-algebra, let $H$ be a Hilbert space and let $T:B\to H$ be a bounded linear operator. Then there is a state $\varphi\in B^*$ such that $$\norm{Tx}\le2\norm{T}\varphi(x\circ x^*)^{1/2}\mbox{ for }x\in B.$$ \end{thm} \begin{proof} Since $B^{**}$ is a JBW$^*$-algebra, $T^{**}$ maps $B^{**}$ into $H$ and $T^{**}$ is weak$^*$-to-weak continuous, by Theorem~\ref{T:algebraic version dual} we get a sequence $(\varphi_n)$ of states on $B$ such that $$\norm{Tx}\le(2+\frac1n)\norm{T}\varphi_n(x\circ x^*)^{1/2}\mbox{ for }x\in B\mbox{ and } n\in\mathbb N.$$ Let $\tilde\varphi$ be a weak$^*$-cluster point of the sequence $(\varphi_n)$. Then $\tilde\varphi$ is positive, $\norm{\tilde\varphi}\le 1$ and $$\norm{Tx}\le2\norm{T}\tilde\varphi(x\circ x^*)^{1/2}\mbox{ for }x\in B.$$ Now we can clearly replace $\tilde\varphi$ by a state. Indeed, if $\tilde\varphi\ne0$, we take $\varphi=\frac{\tilde\varphi}{\norm{\tilde\varphi}}$. If $\tilde\varphi=0$, then $T=0$ and hence $\varphi$ may be any state. (Note that in case $B$ is unital, $\tilde\varphi$ is already a state.) \end{proof} We finish this section by showing that our main result easily implies Theorem~\ref{T-C*alg}. \begin{proof}[Proof of Theorem~\ref{T-C*alg} from Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}] Let $A$ be a C$^*$-algebra, let $H$ be a Hilbert space and let $T:A\to H$ be a bounded linear operator. By Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras} there is a sequence $(\psi_n)$ of norm-one functionals in $A^*$ such that $$\norm{Tx}\le (\sqrt{2}+\frac1n)\norm{T}\norm{x}_{\psi_n}\mbox{ for }x\in A \mbox{ and }n\in\mathbb N.$$ Recall that $A^{**}$ is a von Neumann algebra. Set $u_n=s(\psi_n)\in A^{**}$. Then $$\norm{x}_{\psi_n}^2=\psi_n\J xx{u_n}=\frac12(\psi_n(xx^*u_n)+\psi_n(u_nx^*x))=\frac12(u_n\psi_n(xx^*)+\psi_n u_n(x^*x)) $$ for $x\in A$. Moreover, $\varphi_{1,n}=u_n\psi_n$ and $\varphi_{2,n}=\psi_n u_n$ are states on $A$ (note that $\varphi_{1,n}=\abs{\psi_n}$ and $\varphi_{2,n}=\abs{\psi_n^*}$) such that $${\norm{Tx}\le (\sqrt{2}+\frac1n)\norm{T}\cdot\frac1{\sqrt{2}}(\varphi_{1,n}(xx^*)+\varphi_{2,n}(x^*x))^{1/2}\mbox{ for }x\in A \mbox{ and }n\in\mathbb N.}$$ Let $(\varphi_1,\varphi_2)$ be a weak$^*$-cluster point of the sequence $((\varphi_{1,n},\varphi_{2,n}))_n$ in $B_{A^*}\times B_{A^*}$. Then $\varphi_1,\varphi_2$ are positive functionals of norm at most one such that $$\norm{Tx}\le \|T\| (\varphi_{1}(xx^*)+\varphi_{2}(x^*x))^{1/2}\mbox{ for }x\in A.$$ Similarly as above we may replace $\varphi_1$ and $\varphi_2$ by states. \end{proof} \section{Examples and problems}\label{sec:problems} \begin{ques} Do Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras} and Theorem~\ref{T:JBW*-algebras} hold with the constant $\sqrt2$ instead of $\sqrt2+\varepsilon$? \end{ques} We remark that these theorems do not hold with a constant strictly smaller than $\sqrt{2}$. Indeed, assume that Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras} holds with a constant $K$. Then Theorem~\ref{T-C*alg} holds with constant $\frac{K}{\sqrt{2}}$ (see the proof of the relationship of these two theorems in Section~\ref{sec:proofs}). But the best constant for Theorem~\ref{T-C*alg} is $1$ due to \cite{haagerup-itoh}. Since the example in \cite{haagerup-itoh} uses a rather involved combinatorial construction, we provide an easier example showing that the constant in Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras} has to be at least $\sqrt{2}$. \begin{example2}\label{ex:Tx=xxi} Let $H$ be an infinite-dimensional Hilbert space. Let $A=K(H)$ be the C$^*$-algebra of compact operators. Fix an arbitrary unit vector $\xi\in H$ and define $T:A\to H$ by $Tx=x\xi$ for $x\in A$. It is clear that $\norm{T}=\norm{\xi}=1$. Fix an arbitrary norm-one functional $\varphi\in A^*$. We are going to prove that \begin{equation} \sup \left\{\frac{\norm{Tx}}{\norm{T} \norm{x}_\varphi};\, x\in A, \norm{x}_\varphi\neq0\right\}\ge\sqrt{2}.\label{eq Example} \end{equation} Recall that $K(H)^*$ is identified with $N(H)$, the space of nuclear operators on $H$ equipped with the nuclear norm, and $K(H)^{**}$ is identified with $B(H)$, the von Neumann algebra of all bounded linear operators on $H$. Using the trace duality we deduce that there is a nuclear operator $z$ on $H$ such that $\tr{\abs{z}}=\norm{z}_N=1$ and $\varphi(x)=\tr{zx}$ for $x\in A$. Consider the polar decomposition $z=u\abs{z}$ in $B(H)$. Then $\abs{z}=u^*z$, hence $s(\varphi)\le u^*$. (Note that $\varphi(u^*)=\tr{zu^*}=\tr{u^*z}=\tr{\abs{z}}=1$, hence $s(\varphi)\le u^*$ by \eqref{eq minimality of the support tripotent}. The converse inequality holds as well, but it is not important.) It follows that for each $x\in A$ we have $$\begin{aligned}\norm{x}_\varphi^2&=\varphi(\J xx{u^*})=\frac12\varphi(xx^*u^*+u^*x^*x)= \frac12\tr{xx^*u^*z+u^*x^*xz}\\&=\frac12(\tr{xx^*\abs{z}}+\tr{u^*x^*xz})\end{aligned}$$ If $\eta\in H$ is a unit vector, we define the operator $$y_\eta(\zeta)=\ip{\zeta}{\xi}\eta,\qquad\zeta\in H.$$ Then $y_\eta\in A$, $\norm{y_\eta}=1$ and $\norm{Ty_\eta}=1$. Moreover, $$y_\eta^*(\zeta)=\ip{\zeta}{\eta}\xi,$$ hence $$y_\eta y_\eta^*(\zeta)=\ip{\zeta}{\eta}\eta\mbox{ and }y_\eta^*y_\eta(\zeta)=\ip{\zeta}{\xi}\xi.$$ Thus $$\norm{y_\eta}_\varphi^2=\frac12(\tr{\abs{z}y_\eta y_\eta^*}+\tr{zu^* y_\eta^*y_\eta})=\frac12(\ip{\abs{z}\eta}{\eta}+\ip{zu^*\xi}{\xi})\le\frac12(1+\ip{\abs{z}\eta}{\eta}).$$ It follows that $$\inf\{\norm{x}_\varphi^2;\, x\in A, \norm{Tx}=1\}\le \frac 12 \inf\{1+\ip{\abs{z}\eta}{\eta};\, \norm{\eta}=1\}=\frac12+\frac12\min \sigma(\abs{z}),$$ where the last equality follows from \cite[Theorem 15.35]{fabianetal2011}. Now, $z$ is a nuclear operator of norm one. Thus $0\in\sigma(\abs{z})$ as $H$ has infinite dimension. Hence $$\inf\{\norm{x}_\varphi;\, x\in A,\norm{Tx}=1\}\le\frac1{\sqrt2},$$ which yields inequality \eqref{eq Example}. \qed\end{example2} \begin{remark} If $H$ is a finite-dimensional Hilbert space, the construction from Example~\ref{ex:Tx=xxi} could be done as well. In this case $A=K(H)=B(H)$ can be identified with the algebra of $n\times n$ matrices where $n=\dim H$. In this case $\sigma(\abs{z})$ need not contain $0$, but at least one of the eigenvalues of $\abs{z}$ is at most $\frac1n$. So, we get a lower bound $\sqrt{\frac{2n}{n+2}}$ for the constant in Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}. \end{remark} Next we address the optimality of the algebraic version of the Little Grothendieck theorem. \begin{ques} What is the optimal constant in Theorem~\ref{T:C*alg-sym}, Theorem~\ref{T:algebraic version dual} and Theorem~\ref{T:algebraic version non-dual}? In particular, do these theorems hold with the constant $\sqrt{2}$? \end{ques} Note that the constant cannot be smaller than $\sqrt2$ due to Example~\ref{ex:Tx=xxi}. The following example shows that Example~\ref{ex:Tx=xxi} cannot yield a greater lower bound. \begin{example2} Let $H$, $A$, $\xi$ and $T$ be as in Example~\ref{ex:Tx=xxi}. Let $u\in A^{**}=B(H)$ be any unitary element. Then $$\varphi_u(x)=\ip{x\xi}{u\xi},\qquad x\in A$$ defines a norm-one functional in $A^*$ such that $s(\varphi_u)\le u$ and, moreover, $$\norm{Tx}\le\sqrt2 \norm{x}_{\varphi_u}\mbox{ for }x\in A.$$ Indeed, it is clear that $\norm{\varphi_u}\le 1$. Since $\varphi_u(u)=1$, necessarily $\norm{\varphi_u}=1$ and $s(\varphi)\le u$. Moreover, for $x\in A$ we have $$\begin{aligned} \norm{x}_{\varphi_u}^2&=\varphi_u\J xxu=\frac12\varphi_u(xx^*u+ux^*x) =\frac12(\ip{xx^*u\xi}{u\xi}+\ip{ux^*x\xi}{u\xi}) \\&=\frac12(\norm{x^*u\xi}^2+\norm{x\xi}^2)\ge\frac12\norm{x\xi}^2=\frac12\norm{Tx}^2.\end{aligned}$$ This completes the proof. \qed\end{example2} We continue by recalling the example of \cite{haagerup-itoh} showing optimality of Theorem~\ref{T-C*alg} and explaining that it does not show optimality neither of Theorem~\ref{T:C*alg-sym} nor of Theorem~\ref{T:algebraic version non-dual}. An important tool to investigate optimality of constants in Theorem~\ref{T-C*alg} is the following characterization. \begin{prop}[{\cite[Proposition 23.5]{pisier2012grothendieck}}]\label{p equivalent formulation C*} Let $A$ be a C$^*$-algebra, $H$ a Hilbert space, $T:A\rightarrow H$ a bounded linear map and $K$ a positive number. Then the following two assertions are equivalent. \begin{enumerate}[(i)] \item\label{it equiv formul2 C*} There are states $\varphi_1, \varphi_2$ on $A$ such that \begin{eqnarray} \norm{Tx}\le K\norm{T}(\varphi_1(x^*x)+\varphi_2(xx^*))^{1/2} \quad \mbox{for } x\in A. \end{eqnarray} \item\label{it equiv formul1 C*} For any finite sequence $(x_j)$ in $A$ we have \begin{eqnarray} \left(\sum_j\norm{Tx_j}^2\right)^{1/2}\le K\norm{T}\left(\Norm{\sum_j x_j^*x_j}+\Norm{\sum_j x_jx_j^*}\right)^{1/2}. \label{eq equiv formul1 C*} \end{eqnarray} \end{enumerate} \end{prop} The following proposition is a complete analogue of the preceding one and can be used to study optimality of Theorem~\ref{T:algebraic version non-dual}. We have not found it explicitly formulated in the literature, but its proof is completely analogous to the proof of Proposition~\ref{p equivalent formulation C*} given in \cite{pisier2012grothendieck}. \begin{prop}\label{p equivalent formulation JB*-algebra} Let $A$ be a unital JB$^*$-algebra, $H$ a Hilbert space, $T:A\rightarrow H$ a bounded linear map and $K$ a positive number. Then the following two assertions are equivalent. \begin{enumerate}[(i)] \item\label{it equiv formul2 JB*} There is a state $\varphi$ on $A$ such that \begin{eqnarray} \norm{Tx}\le K\norm{T}\varphi(x^*\circ x)^{1/2} \quad \mbox{for } x\in A. \end{eqnarray} \item\label{it equiv formul1 JB*} For any finite sequence $(x_j)$ in $A$ we have \begin{eqnarray} \left(\sum_j\norm{Tx_j}^2\right)^{1/2}\le K\norm{T}\Norm{\sum_j x_j^*\circ x_j}^{1/2}. \label{eq equiv formul1 JB*} \end{eqnarray} \end{enumerate} \end{prop} We recall the example originated in \cite{haagerup-itoh} and formulated and proved in this setting in \cite{pisier2012grothendieck}. \begin{example}[{\cite[Lemma 11.2]{pisier2012grothendieck}}]\label{example} Consider an integer $n\ge1$. Let $N=2n+1$ and $d=\begin{pmatrix}2n+1\\n\end{pmatrix}=\begin{pmatrix}2n+1\\n+1\end{pmatrix}$. Let $\tau_d$ denote the normalized trace on the space $M_d$ of $d\times d$ (complex) matrices. There are $x_1, \ldots, x_N$ in $M_d$ such that $\tau_d(x_i^*x_j)=1$ if $i=j$ and $=0$ otherwise, satisfying \begin{eqnarray} \sum_j x_j^*x_j=\sum_j x_jx_j^*=NI \label{eq1 example} \end{eqnarray} and moreover such that, with $a_n=(n+1)/(2n+1)$, \begin{eqnarray} \forall \alpha=(\alpha_i)\in\mathbb C^N,\quad \Norm{\sum _j\alpha_jx_j}_{(M_d)^*}=d\sqrt{a_n}\left(\sum_j\betr{\alpha_j}^2\right)^{1/2}. \label{eq2 example} \end{eqnarray} \end{example} In the following example we show that the previous one yields the optimality of Theorem~\ref{T-C*alg} but does not help to find the optimal constant for Theorem~\ref{T:C*alg-sym} or Theorem~\ref{T:algebraic version non-dual}. The first part is proved already in \cite{haagerup-itoh} (cf. \cite[Section 11]{pisier2012grothendieck}) but we include the proof for the sake of completeness and, further, in order to compare it with the second part. \begin{example2} Fix $n\ge1$. With the notation of Example \ref{example} define $T:M_d\to \ell_2^N$ by $$T(x)=(\tau_d(x_j^*x))_{j=1}^N,\qquad x\in M_d.$$ Let $(\eta_j)_{j=1}^N$ be the canonical orthonormal basis of $\ell_2^N$. Then the dual mapping $T^*:\ell_2^N\to M_d^*$ fulfils $$\ip{T^*(\eta_j)}{x}=\ip{\eta_j}{T(x)}=\tau_d(x_j^*x)=\frac1d\tr{x_j^*x}\mbox{ for }x\in M_d,$$ thus $T^*(\eta_j)=\frac1dx_j^*$ (we use the trace duality). Then \eqref{eq2 example} shows that $$\norm{T^*(\alpha)}=\frac1d\norm{\sum_{j=1}^N\alpha_jx_j^*}_{(M_d)^*}=\sqrt{a_n}\norm{\alpha}\mbox{ for }\alpha\in\ell_2^N.$$ In particular, $\frac1{\sqrt{a_n}}T^*$ is an isometric embedding, thus $\frac1{\sqrt{a_n}}T$ is a quotient mapping. Hence, $\norm{T}=\sqrt{a_n}$. Further, $T(x_j)=\eta_j$ for $j=1,\dots,N$, so $$\sum_{j=1}^N \norm{T(x_j)}^2=N$$ and $$\norm{\sum_{j=1}^N x_j^*x_j}+\norm{\sum_{j=1}^N x_jx_j^*}=2\norm{NI}=2N.$$ Thus due to Proposition~\ref{p equivalent formulation C*} the optimal value of the constant in Theorem~\ref{T-C*alg} is bounded below by $$\frac{1}{\sqrt{2a_n}}=\sqrt{\frac{2n+1}{2n+2}}\to 1.$$ On the other hand, $$\norm{\sum_{j=1}^N x_j^*\circ x_j}=\norm{NI}=N,$$ thus Proposition~\ref{p equivalent formulation JB*-algebra} yields that the optimal value of the constant in Theorem~\ref{T:C*alg-sym} is bounded below by $$\frac1{\sqrt{a_n}}=\sqrt{\frac{2n+1}{n+1}}\to \sqrt2,$$ so it gives nothing better than Example~\ref{ex:Tx=xxi}. In fact, this operator $T$ satisfies Theorem~\ref{T:C*alg-sym} with constant $\frac1{\sqrt{a_n}}\le\sqrt{2}$. To see this observe that $(x_j)_{j=1}^N$ is an orthonormal system in $M_d$ equipped with the normalized Hilbert-Schmidt inner product. Hence, any $x\in M_d$ can be expressed as $$x=y+\sum_{j=1}^N\alpha_jx_j,$$ where $\alpha_j$ are scalars and $y\in\{x_1,\dots,x_N\}^{\perp_{HS}}$. Then $T(x)=(\alpha_j)_{j=1}^N$ and $$\tau_d(x^*\circ x)=\tau_d(x^*x)=\tau_d(y^*y)+\sum_{j=1}^N\abs{\alpha_j}^2\ge \sum_{j=1}^N\abs{\alpha_j}^2=\norm{T(x)}^2.$$ Hence $$\norm{T(x)}\le \tau_d(x^*\circ x)^{1/2}=\frac1{\sqrt{a_n}}\norm{T} \tau_d(x^*\circ x)^{1/2}.$$ Since $\tau_d$ is a state, the proof is complete. \end{example2} We continue by an example showing that there is a real difference between the triple and algebraic versions of the Little Grothendieck theorem. \begin{example}\label{ex:alg vs triple} \ \begin{enumerate}[$(a)$] \item Let $M$ be any JBW$^*$-triple and let $\varphi\in M_*$ be a norm-one functional. Then $$\abs{\varphi(x)}\le\norm{x}_\varphi,\mbox{ for all }x\in M,$$ hence $\varphi:M\to\mathbb C$ satisfies Theorem~\ref{T:triples}$(3)$ with constant one. \item Let $M_2$ be the algebra of $2\times 2$ matrices. Then there is a norm-one functional $\varphi:M_2\to \mathbb C$ not satisfying Theorem~\ref{T:C*alg-sym} with constant smaller than $\sqrt{2}$. \item In particular, the constant $\sqrt{2}$ in Lemma~\ref{L:rotation} is optimal. \end{enumerate} \end{example} \begin{proof} $(a)$ The desired inequality was already stated in \cite[comments before Definition 3.1]{barton1990bounded}. Let us give some details. We set $e=s(\varphi)$. Then $$\abs{\varphi(x)}=\abs{\varphi(P_2(e)x)}=\abs{\varphi(\J{P_2(e)x}ee}\le\norm{P_2(e)x}_\varphi\norm{e}_\varphi=\norm{P_2(e)x}_\varphi.$$ Moreover, $$\begin{aligned}\norm{x}_\varphi^2&=\varphi\J xxe=\varphi(P_2(e)\J xxe)\\&=\varphi(\J{P_2(e)x}{P_2(e)x}e+\J{P_1(e)x}{P_1(e)x}e)=\norm{P_2(e)x}_\varphi^2+\norm{P_1(e)x}_\varphi^2\\&\ge\norm{P_2(e)x}_\varphi^2.\end{aligned}$$ $(b)$ Each $a\in M_2$ can be represented as $a=(a_{ij})_{i,j=1,2}$. Define $\varphi:M_2\to\mathbb C$ by $$\varphi(a)=a_{12},\quad a\in M_2.$$ { It is clear that $\norm{\varphi}=1$ and that $\varphi(s)=1$ where $$s=\begin{pmatrix} 0 & 1\\ 0& 0\end{pmatrix}.$$ Let $\psi$ be any state on $M_2$. Then $$\norm{s}_\psi^2=\psi (\J ss{\mathbf{1}})=\frac12\psi(s s^*+s^*s)=\frac12\psi(\mathbf{1})=\frac12.$$ Thus $\varphi(s)=\sqrt{2}\norm{s}_\psi$ for any state $\psi$ on $A=M_2$, which completes the proof. $(c)$ This follows from $(b)$ (consider $p=\mathbf{1}$).} \end{proof} \section{Notes and problems on general JB$^*$-triples}\label{sec:triples} The main result, Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras}, is formulated and proved for JB$^*$-algebras. The assumption that we deal with a JB$^*$-algebra, not with a general JB$^*$-triple, was strongly used in the proof. Indeed, the key step was to prove the dual version for JBW$^*$-algebras, Theorem~\ref{T:JBW*-algebras}, and we substantially used the existence of unitary elements. So, the following problem remains open. \begin{ques} Is Theorem~\ref{t constant >sqrt2 in LG for JBstar algebras} valid for general JB$^*$-triples? \end{ques} We do not know how to attack this question. However, there are some easy partial results. Moreover, some of our achievements may be easily extended to JBW$^*$-triples. In this section we collect such results. The first example shows that for some JB$^*$-triples the optimal constant in the Little Grothendieck Theorem is easily seen to be $\sqrt2$. This is shown by completely elementary methods. \begin{example2} Let $H$ be a Hilbert space considered as the triple $B(\mathbb C,H)$ (i.e., a type 1 Cartan factor). That is, the triple product is given by $$\J xyz=\frac12(\ip xyz+\ip zyx),\quad x,y,z\in H.$$ The dual coincides with the predual and it is isometric to $H$. Let $y\in H^*$ be a norm-one element, i.e. we consider it as the functional $\ip{\cdot}{y}$. Then $s(y)=y$. So, for $x\in H$ we have $$\norm{x}_y^2=\ip{\J xxy}{y}= \frac12\ip{\ip xxy+\ip yxx}{y}=\frac12(\norm{x}^2+\abs{\ip xy}^2)\ge\frac12\norm{x}^2.$$ Hence, if $K$ is another Hilbert space and $T:H\to K$ is a bounded linear operator, then for any norm-one $y\in H^*$ we have $$\norm{Tx}\le\norm{T}\norm{x}\le\sqrt{2}\norm{T}\norm{x}_y,$$ so we have the Little Grothendieck theorem with constant $\sqrt2$. Moreover, the constant $\sqrt2$ is optimal in this case as soon as $\dim H\ge2$. Indeed, let $T:H\to H$ be the identity. Given any norm-one element $y\in H$, we may find a norm-one element $x\in H$ with $x\perp y$. The above computation shows that $\norm{x}=\sqrt{2}\norm{x}_y$. \end{example2} Another case, nontrivial but well known, is covered by the following example. \begin{example2} Assume that $E$ is a finite-dimensional JB$^*$-triple. Then $E$ is reflexive and, moreover, any bounded linear operator $T:E\to H$ (where $H$ is a Hilbert space) attains its norm. Hence $E$ satisfies the Little Grothendieck theorem with constant $\sqrt{2}$ by Theorem~\ref{T:triples}$(1)$. \end{example2} We continue by checking which methods used in the present paper easily work for general triples. \begin{obs}\label{obs:type I approx triples} Proposition~\ref{P:type I approx} holds for corresponding JBW$^*$-triples as well. \end{obs} \begin{proof} It is clear that it is enough to prove it separately for finite JBW$^*$-triples and for type I JBW$^*$-triples. The case of finite JBW$^*$-triples is trivial (one can take $\varphi_2=0$). So, let $M$ be a JBW$^*$-triple of type I, $\varphi\in M_*$ and $\varepsilon>0$. Set $e=s(\varphi)$. Then $M_2(e)$ is a type I JBW$^*$-algebra (see \cite[comments on pages 61-62 or Theorem 4.2]{BuPe02}) and $\varphi|_{M_2(e)}\in M_2(e)_*$. Apply Proposition~\ref{P:type I approx} to $M_2(e)$ and $\varphi|_{M_2(e)}$ to get $\varphi_1$ and $\varphi_2$. The pair of functionals $\varphi_1\circ P_2(e)$ and $\varphi_2\circ P_2(e)$ completes the proof. \end{proof} Observe that the validity of Proposition~\ref{P:type I approx} for finite JBW$^*$-triples is trivial but useless if we have no unitary element. However, the `type I part' may be used at least in some cases. \begin{prop}\label{P:B(H,K)} Let $M=L^\infty(\mu)\overline{\otimes}B(H,K)$, where $H$ and $K$ are infinite-di\-men\-sional Hilbert spaces. Then Proposition~\ref{P:majorize 1+2+epsilon} holds for $M$. \end{prop} \begin{proof} Let us start by showing that Peirce-2 subspaces of tripotents in $M$ are upwards directed by inclusion. To this end first observe that $M=pV$, where $V$ is a von Neumann algebra and $p\in V$ is a properly infinite projection. This is explained for example in \cite[p. 43]{hamhalter2019mwnc}. Now assume that $u_1,u_2\in pV$ are two tripotents (i.e., partial isometries in $V$ with final projections below $p$). By \cite[Lemma 9.8(c)]{hamhalter2019mwnc} there are projections $q_1,q_2\in V$ such that $q_j\ge p_i(u_j)$ and $q_j\sim p$ for $j=1,2$. Further, by \cite[Lemma 9.8(a)]{hamhalter2019mwnc} we have $q_1\vee q_2\sim p$, so there is a partial isometry $u\in V$ with $p_i(u)=q_1\vee q_2$ and $p_f(u)=p$. Then $u\in pV=M$ and $M_2(u)\supset M_2(u_1)\cup M_2(u_2)$. Now we proceed with the proof of the statement itself. Let $\varphi_1,\varphi_2\in M_*$ and $\varepsilon>0$. Note that $M$ is of type I, hence we may apply Observation~\ref{obs:type I approx triples} to get the respective decomposition $\varphi_1=\varphi_{11}+\varphi_{12}$. Let $u\in M$ be a tripotent such that $M_2(u)$ contains $s(\varphi_{11}),s(\varphi_{12}),s(\varphi_2)$. Such a $u$ exists as Peirce-2 subspaces of tripotents in $M$ are upwards directed by inclusion as explained above. We can find a unitary $v\in M_2(u)$ with $s(\varphi_{11})\le v$ (recall that $s(\varphi_{11})$ is a finite tripotent and use \cite[Proposition 7.5]{Finite}). We conclude by applying Lemma~\ref{L:rotation}. \end{proof} Combining the previous proposition with Theorem~\ref{T:triples-dual} we get the following. \begin{cor} Let $M=L^\infty(\mu)\overline{\otimes}B(H,K)$, where $H$ and $K$ are infinite-di\-men\-sio\-nal Hilbert spaces. Then Theorem~\ref{T:JBW*-algebras} holds for $M$. \end{cor} We finish by pointing out main problems concerning JBW$^*$-triples. \begin{ques} Assume that $M$ is a JBW$^*$-triple of one of the following forms: \begin{itemize} \item $M=L^\infty(\mu,C)$, where $\mu$ is a probability measure and $C$ is a finite-di\-men\-sio\-nal JB$^*$-triple without unitary element. \item $M=pV$, where $V$ is a von Neumann algebra and $p$ is a purely infinite projection. \item $M=pV$, where $V$ is a von Neumann algebra and $p$ is a finite projection. \end{itemize} Is Theorem~\ref{T:JBW*-algebras} valid for $M$? \end{ques} Note that these three cases correspond to the three cases distinguished in \cite{HKPP-BF}. We conjecture that the second case may be proved by adapting the results of Section~\ref{sec:JW*} (but we do not see an easy way) and that the third case is the most difficult one (similarly as in \cite{HKPP-BF}). \begin{remark}{\rm Haagerup applied in \cite{haagerup1985grothendieck} ultrapower techniques to relax some of the extra hypotheses assumed by Pisier in the first approach to a Grothendieck inequality for C$^*$-algebras. We should include a few words justifying that Haagerup's techniques are not effective in the setting of JB$^*$-triples. Indeed, while a cluster point (in a reasonable sense) of states of a unital C$^*$-algebra is a state, a cluster point of norm-one functionals may be even zero. It is true for weak (weak$^*$) limits and also for ultrapowers. The ultrapower, $E_{\mathcal{U}},$ of a JB$^*$-triple, $E$, with respect to an ultrafilter $\mathcal{U}$, is again a JB$^*$-triple with respect to the natural extension of the triple product (see \cite[Corollary 10]{Dineen86}), and $E$ can be regarded as a JB$^*$-subtriple of $E_{\mathcal{U}}$ via the inclusion of elements as constant sequences. Given a norm one functional $\widetilde{\varphi}\in E_{\mathcal{U}}^*$ the restriction $\varphi = \widetilde{\varphi}|_{E}$ belongs to $E^*$ however we cannot guarantee that $\|x\|_{\widetilde{\varphi}} = \|[x]_{\mathcal{U}}\|_{\widetilde{\varphi}}$ is bounded by a multiple of $\|x\|_{{\varphi}}$. Let us observe that both prehilbertian seminorms coincide on elements of $E$ when the latter is a unital C$^*$-algebra and $\widetilde{\varphi}$ is a state on $E.$ }\end{remark} \textbf{Acknowledgements} A.M. Peralta partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, the IMAG–Mar{\'i}a de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033, and by Junta de Andaluc\'{\i}a grants FQM375 and A-FQM-242-UGR18. We would like to thank the referees for their carefully reading of our manuscript and their constructive comments. \def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$} \end{document}

\begin{document} \title{Saturated simple and $k$-simple topological graphs} \author{ Jan Kyn\v cl\thanks{Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics, Malostransk\'e n\'am.~25, 118~00~ Praha 1, Czech Republic; and Alfr\'ed R\'enyi Institute of Mathematics, Budapest, Hungary. Email: \texttt{[email protected]}. Supported by the GraDR EUROGIGA GA\v{C}R project No. GIG/11/E023, by the grant SVV-2013-267313 (Discrete Models and Algorithms) and by ERC Advanced Research Grant no 267165 (DISCONV).} \and J\'anos Pach\thanks{Ecole Polytechnique F\'ed\'erale de Lausanne and Alfr\'ed R\'enyi Institute of Mathematics, Budapest. Email: \texttt{[email protected]}. Research partially supported by Swiss National Science Foundation Grants 200021-137574 and 200020-144531, by Hungarian Science Foundation Grant OTKA NN 102029 under the EuroGIGA programs ComPoSe and GraDR, and by NSF grant CCF-08-30272.} \and Rado\v{s} Radoi\v{c}i\'{c}\thanks{Department of Mathematics, Baruch College, City University of New York, NY, USA. Email: \texttt{[email protected]}. Part of the research by this author was done at the Alfr\'ed R\'enyi Institute of Mathematics, partially supported by Hungarian Science Foundation Grant OTKA T 046246.} \and G\'eza T\'oth\thanks{Alfr\'ed R\'enyi Institute of Mathematics, Budapest, Hungary. Email: \texttt{[email protected]}. Partially supported by Hungarian Science Foundation Grants OTKA K 83767 and NN 102029.}} \maketitle \begin{abstract} A {\em simple topological graph} $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called {\em saturated\/} if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for {\em $k$-simple topological graphs}, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered. \end{abstract} \section{Introduction} Saturation problems in graph theory have been studied at length, ever since the paper of Erd\H{o}s, Hajnal, and Moon~\cite{EHM64}. Given a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph, but the addition of any edge joining two non-adjacent vertices of $G$ creates a copy of $H$. The saturation number of $H$, $\mbox{sat}(n,H)$, is the minimum number of edges in an $H$-saturated graph on $n$ vertices. The saturation number for complete graphs was determined in~\cite{EHM64}. A systematic study by K\'{a}szonyi and Tuza~\cite{KT86} found the best known general upper bound for $\mbox{sat}(n,H)$ in terms of the independence number of $H$. The saturation number is now known, often precisely, for many graphs; for these results and related problems in graph theory we refer the reader to the thorough survey of J. Faudree, R. Faudree, and Schmitt~\cite{FFS11}. It is worth noting that $\mbox{sat}(n,H) = O(n)$, quite unlike the Tur\'{a}n function $\mbox{ex}(n,H)$, which is often superlinear. In this paper, we study a saturation problem for {\em drawings\/} of graphs. In a drawing of a simple undirected graph $G$ in the plane, every vertex is represented by a point, and every edge is represented by a curve between the points that correspond to its endpoints. If it does not lead to confusion, these points and curves are also called {\em vertices\/} and {\em edges}. We assume that in a drawing no edge passes through a vertex and no two edges are tangent to each other. A graph, together with its drawing, is called a {\em simple topological graph\/} if any two edges have at most one point in common, which is either their common endpoint or a proper crossing. In general, for any positive integer $k$, it is called a {\em $k$-simple topological graph\/} if any two edges have at most $k$ points in common. We also assume that in a $k$-simple topological graph no edge crosses itself. Obviously, a $1$-simple topological graph is a simple topological graph. Our motivation partly comes from the following problem: At least how many pairwise disjoint edges can one find in every simple topological graph with $n$ vertices and $m$ edges~\cite{PST03}? (Note that the simplicity condition is essential here, as there are complete topological graphs on $n$ vertices and no {\em two\/} disjoint edges, in which every pair of edges intersect at most twice~\cite{PT10}.) For {\em complete\/} simple topological graphs, i.e., when $m={n\choose 2}$, Pach and T\'{o}th conjectured (\cite{BMP05}, page 398) that one can always find $\Omega(n^{\delta})$ disjoint edges for a suitable constant $\delta > 0$. This was shown by Suk \cite{S13} with $\delta=1/3$; see \cite{FR13} for an alternative proof. Recently, Ruiz-Vargas \cite{R13} has improved this bound to $\Omega\left(\sqrt{{n}/{\log{n}}}\right)$. Unfortunately, all known proofs break down for {\em non-complete\/} simple topological graphs. For {\em dense\/} graphs, i.e., when $m\ge \varepsilon n^2$ for some $\varepsilon>0$, Fox and Sudakov \cite{FS09} established the existence of $\Omega(\log^{1+\gamma}{n})$ pairwise disjoint edges, with $\gamma \approx 1/50$. However, if $m\ll n^2$, the best known lower bound, due to Pach and T\'oth \cite{PT10}, is only $\Omega\left((\log{m}-\log{n})/\log{\log{n}}\right)$. We know a great deal about the structure of complete simple topological graphs, but in the non-complete case our knowledge is rather lacunary. We may try to extend a simple topological graph to a complete one by adding extra edges and then explore the structural information we have for complete graphs. The results in the present note suggest that this approach is not likely to succeed: there exist very sparse simple topological graphs to which no edge can be added without violating the simplicity condition. A $k$-simple, non-complete topological graph is {\em saturated\/} if no further edge can be added so that the resulting drawing is still a $k$-simple topological graph. In other words, if we connect any two non-adjacent vertices by a curve, it will have at least $k+1$ common points with one of the existing edges. Consider the simple topological graph $G_1$ with eight vertices, depicted in Figure~\ref{kampok}. It is easy to verify that the vertices $x$ and $y$ cannot be joined by a new edge so that the resulting topological graph remains simple. Indeed, every edge of $G_1$ is incident either to $x$ or to $y$, and any curve joining $x$ and $y$ must cross at least one edge. On the other hand, $G_1$ can be extended to a (saturated) simple topological graph in which every pair of vertices except $x$ and $y$ are connected by an edge. \begin{figure} \caption{A topological graph $G_1$: the edge $\{x, y\}$ cannot be added.} \label{kampok} \end{figure} Another example was found independently by Kyn\v{c}l~\cite[Fig.~9]{K13}: The simple topological graph $G_2$ depicted in Figure~\ref{kynclpelda} has only six vertices, from which $x$ and $y$ cannot be joined by an edge without intersecting one of the original edges at least twice. Again, $G_2$ can be extended to a simple topological graph in which every pair of vertices except $x$ and $y$ are connected by an edge. \begin{figure} \caption{A topological graph $G_2$: the edge $\{x, y\}$ cannot be added.} \label{kynclpelda} \end{figure} In view of the fact that the graphs shown in Figures~\ref{kampok} and~\ref{kynclpelda} can be extended to nearly complete simple topological graphs, it is a natural question to ask whether every saturated simple topological graph with $n$ vertices must have $\Omega(n^2)$ edges. It is not obvious at all, whether there exist saturated non-complete $k$-simple topological graphs for some $k>1$. Our next theorem shows that there are such graphs, for every $k$, moreover, they may have only a linear number of edges. \begin{theorem}\label{linearbound} For any positive integers $k$ and $n\ge 4$, let $s_k(n)$ be the minimum number of edges that a saturated $k$-simple topological graph on $n$ vertices can have. Then \begin{enumerate} \item[(i)] we have $$1.5n\le s_1(n)\le 17.5n,$$ \item[(ii)] for $k>1$ we have $$n\le s_k(n)\le 16n.$$ \end{enumerate} \end{theorem} For our best upper bounds see Table~\ref{table1}. \begin{table} \begin{center} {\footnotesize \begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|c} $k$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & $\ge 11$ \\ \hline {\mbox{upper bound}} & $17.5n$ & $16n$ & $14.5n$ & $13.5n$ & $13n$ & $9.5n$ & $10n$ & $9.5n$ & $7n$ & $9.5n$ & $7n$ \end{tabular} \caption{Upper bounds on the minimum number of edges in saturated $k$-simple topological graphs.} } \label{table1} \end{center} \end{table} For any positive integers $k$ and $l$, $k < l$, a topological graph $G$ {\em together with\/} a pair of non-adjacent vertices $\{u, v\}$ is called a {\em $(k,l)$-construction\/} if $G$ is $k$-simple and any curve joining $u$ and $v$ has at least $l$ points in common with at least one edge of $G$. Using this terminology, every saturated non-complete $k$-simple topological graph together with any pair of non-adjacent vertices is a $(k,k+1)$-construction. \begin{theorem}\label{klconstruction} For every $k>0$, \begin{enumerate} \item[(i)] There exists a $(k, 2k)$-construction, \item[(ii)] There is no $(k, l)$-construction with $l > 2k$. \end{enumerate} \end{theorem} For any positive integers $k$ and $l$, $k < l$, a {\em non-complete\/} topological graph $G$ is called {\em $(k, l)$-saturated\/} if $G$ is $k$-simple and any curve joining {\em any pair\/} of non-adjacent vertices has at least $l$ points in common with at least one edge of $G$. Obviously, every saturated $k$-simple topological graph is $(k, k+1)$-saturated. Clearly, every $(k, l)$-saturated topological graph, together with any pair of its non-adjacent vertices, is a $(k, l)$-construction. However, for $l>k+1$, the existence of a $(k, l)$-construction does not necessarily imply the existence of a $(k, l)$-saturated topological graph. The best we could prove is the following. \begin{theorem}\label{3k/2saturated} For any $k>0$, there exists a $(k, \lceil 3k/2\rceil)$-saturated topological graph. \end{theorem} In the proof of Theorem~\ref{klconstruction} we obtain a set of six curves, any two of which cross at most once, and two points, such that any curve connecting them has to cross one of the six curves at least twice (see Figure~\ref{nembovitheto1}). On the contrary, it follows from Levi's enlargement lemma~\cite{L26} that if the curves have to be {\em bi-infinite}, that is, two-way unbounded, then there is no such construction. A {\em pseudoline arrangement\/} is a set of bi-infinite curves such that any two of them cross exactly once. By Levi's lemma, for any two points not on the same line, the arrangement can be extended by a pseudoline through these two points. A {\em $k$-pseudoline arrangement\/} is a set of bi-infinite curves such that any two of them cross at most $k$ times. A {\em $k$-pseudocircle arrangement\/} is a set of closed curves such that any two of them cross at most $k$ times. Elements of pseudoline arrangements and $k$-pseudoline arrangements are called {\em pseudolines}. Note that for even $k$, $k$-pseudoline arrangements can be considered a special case of $k$-pseudocircle arrangements. Snoeyink and Hershberger \cite{SH91} generalized Levi's lemma to $2$-pseudoline arrangements and $2$-pseudocircle arrangements as follows. They proved that for every $2$-pseudocircle arrangement and three points, not all on the same pseudocircle, the arrangement can be extended by a closed curve through these three points so that it remains a $2$-pseudocircle arrangement. They also showed that for $k\ge 3$, an analogous statement with $k$-pseudoline arrangements and $k+1$ given points is false. A $k$-pseudoline arrangement is {\em $(p,l)$-forcing} if there is a set $A$ of $p$ points such that every bi-infinite curve through the points of $A$ crosses one of the pseudolines at least $l$ times. Snoeyink and Hershberger~\cite{SH91} found $(k+1,k+1)$-forcing $k$-pseudoline arrangements for $k\ge 3$. We generalize their result as follows. \begin{theorem}\label{pseudoline} \begin{itemize} \item[(i)] For every $k\ge 1$, there is a $(3,5\lceil(k-7)/4\rceil)$-forcing $k$-pseudoline arrangement. \item[(ii)] For every $k\ge 1$, there is a $(k,\Omega(k\log k))$-forcing $k$-pseudoline arrangement. \end{itemize} \end{theorem} In Section 2 we define tools necessary for our constructions. In Section 2.1 we define {\em spirals\/} and use them in Lemma~\ref{lemma_weakklconstruction} to prove the existence of a $\left( k, \lceil 7(k-1)/6\rceil \right)$-construction (for $k\ge 8$). Although Lemma~\ref{lemma_weakklconstruction} is a very weak version of Theorem~\ref{klconstruction}, its proof is a relatively simple construction, which serves as the basis of all our further constructions. In Section 2.2 we define another tool, {\em forcing blocks}, and as an illustration, we prove Lemma~\ref{2k-1construction}, which is an improvement of Lemma~\ref{lemma_weakklconstruction}, yet still weaker than Theorem~\ref{klconstruction}. Finally, in Section 2.3, we define the remaining necessary tools, {\em grid blocks} and {\em double-$k$-forcing blocks}, and use them to prove Theorem~\ref{klconstruction} (i). In Section 3 we prove Theorem~\ref{klconstruction} (ii). Our proof is self-contained and independent of the tools developed in Section 2. In Section 4 we prove the upper bounds in Theorem~\ref{linearbound}, by giving five different constructions; the first one is for $k=1$, and it is essentially different from the other four, which use spirals, grid blocks, and forcing blocks described previously in Section 2. In Section 5 we prove the lower bounds in Theorem~\ref{linearbound}. Our proof is self-contained and independent of the remaining sections. In Section 6, we prove Theorem~\ref{3k/2saturated}. Our construction uses grid blocks and double-forcing blocks described in Section 2. In Section 7, we prove Theorem~\ref{pseudoline}. Our constructions use spirals from Section 2. We finish the paper with some remarks and open problems. \section{Building blocks for $(k, l)$-constructions} With the exception of the proof of Theorem~\ref{linearbound} (i), we construct drawings on a vertical cylinder, which can be transformed into a planar drawing. The cylinder will be represented by an axis-parallel rectangle whose left and right vertical sides are identified. Curves on the cylinder are also represented in the axis-parallel rectangle, where they can ``jump'' between the left and the right sides. Edges will be drawn as $y$-monotone curves. Drawings will be constructed from {\em blocks}. Each block is a horizontal ``slice'' of the cylinder, represented again by an axis-parallel rectangle, say, $R$, whose left and right vertical sides are identified. A {\em cable} in a block is a group of intervals of edges that go very close to each other but do not cross in the block. A cable is represented by a single curve which goes very close to each edge in the cable. A curve or a cable in a block $B$ whose endpoints are on the top and bottom boundary of $R$ is called a {\em transversal\/} of $B$. For any curves or cables $a$ and $b$, let ${\mbox {\sc cr}}(a,b)$ denote the number of crossings between $a$ and $b$. \subsection{\texorpdfstring{Spirals and a $\left(k, \lceil 7(k-1)/6\rceil \right)$-construction}{Spirals and a $(k, 7(k-1)/6)$-construction}} \begin{figure} \caption{A $4$-spiral formed by $\alpha$ and $\beta$. One of the four regions is shaded. The arrows represent the two edges of the rectangle that are glued together to form a cylinder.} \label{figure_spiral} \end{figure} Let $B$ be a block on the cylinder, represented by the unit square $R$ with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, the two vertical sides are identified. Let $\alpha$ be a straight line segment from $(0.5,0)$ to $(0.5,1)$, and let $\beta$ be represented as the union of $m$ straight line segments, $b_1, b_2, \ldots , b_m$, where $b_i$ is the segment from $(0, (i-1)/m)$ to $(1,i/m)$. See Figure~\ref{figure_spiral}. Cables $a$ and $b$ in $B$ form an {\em $m$-spiral\/} if there is a homeomorphism of $B$ that takes $a$ to $\alpha$ and $b$ to $\beta$ and maps the lower boundary of $B$ to itself. Clearly, such cables $a$ and $b$ are transversals of $B$ and intersect exactly $m$ times. \begin{observation}\label{obs_spiral} Suppose that $a$ and $b$ form an $m$-spiral in a block $B$. Then every transversal of $B$ crosses $a$ and $b$ together at least $m-1$ times. \end{observation} \begin{proof} Let $\kappa$ be a transversal of $B$. Extend $B$ to a two-way infinite cylinder, $B'$. Cables $a$ and $b$ together divide the cylinder $B'$ into $m$ regions, say, $B_1, B_2, \ldots, B_{m}$, from bottom to top. One endpoint of $\kappa$ is in $B_1$, the other one is in $B_{m}$. It is easy to see that $B_i$ and $B_j$ have a common boundary if and only if $|i-j|=1$. Therefore, to go from $B_1$ to $B_m$, $\kappa$ has to cross at least $m-1$ boundaries. \end{proof} Using spirals, we are able to prove the following weak version of Theorem~\ref{klconstruction} (i). \begin{lemma}\label{lemma_weakklconstruction} For $k\ge 8$ there exists a $(k, l)$-construction with $l=\lceil 7(k-1)/6\rceil >k$. \end{lemma} \begin{proof} The construction consists of $7$ consecutive blocks, $X,\allowbreak A,\allowbreak B,\allowbreak C,\allowbreak D,\allowbreak E,\allowbreak Y$, in this order (say, from bottom to top). First we define six independent edges, $\alpha_1,\allowbreak \alpha_2,\allowbreak \beta_1,\allowbreak \beta_2,\allowbreak \gamma_1,\allowbreak \gamma_2$, and two isolated vertices, $x$ and $y$. Put $x$ in $X$ and $y$ in $Y$. The edges $\alpha_1$ and $\alpha_2$ are in the blocks $A$ and $B$, both have one endpoint on the boundary of $X$ and $A$ and one on the boundary of $B$ and $C$. Edges $\beta_1$ and $\beta_2$ are in $B$, $C$ and $D$, both have one endpoint on the boundary of $A$ and $B$ and one on the boundary of $D$ and $E$. Edges $\gamma_1$ and $\gamma_2$ are in $D$ and $E$, both have one endpoint on the boundary of $C$ and $D$ and one on the boundary of $E$ and $Y$. The edges $\alpha_1$ and $\alpha_2$ form a $k$-spiral in $A$ and a cable in $B$. The edges $\beta_1$ and $\beta_2$ form another cable in $B$, and these two cables form a $k$-spiral. Further, $\beta_1$ and $\beta_2$ form a $k$-spiral in $C$ and a cable in $D$. The edges $\gamma_1$ and $\gamma_2$ form another cable in $D$, and these two cables form a $k$-spiral. Finally, $\gamma_1$ and $\gamma_2$ form a $k$-spiral in $E$. We show that every curve $\kappa$ from $x$ to $y$ crosses one of the curves $\alpha_1,\allowbreak \alpha_2,\allowbreak \beta_1,\allowbreak \beta_2,\allowbreak \gamma_1,\allowbreak \gamma_2$ at least $7(k-1)/6$ times. Let $\kappa$ be a fixed curve from $x$ to $y$. For every $\chi\in\{\alpha_1, \alpha_2, \beta_1, \beta_2, \gamma_1, \gamma_2\}$ and $Z\in\{A, B, C, D, E\}$, let $Z(\chi)$ denote the number of intersections of $\chi$ with $\kappa$ in $Z$. (That is, $A(\alpha_1)$ is the number of intersections between $\alpha_1$ and $\kappa$ in $A$.) By Observation~\ref{obs_spiral}, we have \begin{align*} A(\alpha_1)+A(\alpha_2)&\ge k-1,\\ B(\alpha_i)+B(\beta_j)&\ge k-1, \ \ 1\le i, j\le 2,\\ C(\beta_1)+C(\beta_2)&\ge k-1,\\ D(\beta_i)+D(\gamma_j)&\ge k-1, \ \ 1\le i, j\le 2,\\ E(\gamma_1)+E(\gamma_2)&\ge k-1.\\ \end{align*} Therefore, we can assume without loss of generality that $A(\alpha_1)\ge (k-1)/2$, $C(\beta_1)\ge (k-1)/2$ and $E(\gamma_1)\ge (k-1)/2$. It follows that $${\mbox {\sc cr}}(\alpha_1, \kappa)+ {\mbox {\sc cr}}(\beta_1, \kappa)+ {\mbox {\sc cr}}(\gamma_1, \kappa)$$ $$=A(\alpha_1)+B(\alpha_1)+ B(\beta_1)+C(\beta_1)+D(\beta_1)+ D(\gamma_1)+E(\gamma_1)\ge 7(k-1)/2.$$ Consequently, at least one of ${\mbox {\sc cr}}(\alpha_1, \kappa)$, ${\mbox {\sc cr}}(\beta_1, \kappa)$, ${\mbox {\sc cr}}(\gamma_1, \kappa)$ is at least $7(k-1)/6>k$, since $k\ge 8$. \end{proof} \begin{figure} \caption{Left: a crossing-forcing configuration formed by curves $\alpha,\beta,\gamma,\delta$. Right: cables in the subblock $C_2$.} \label{figure_XX} \end{figure} \subsection{Forcing blocks and a $(k, 2k-1)$-construction} We define another type of block, which is built from several subblocks. Let $B$ be a block on the cylinder represented by the unit square $R$ with vertices $(0,0)$, $(1,0)$, $(0,1)$, $(1,1)$, where the two vertical sides are identified. See Figure~\ref{figure_XX}, left. Let $\alpha$ be a straight line segment from $(0.25,0)$ to $(0.25,1)$, $\beta$ a straight line segment from $(0.75,0)$ to $(0.75,1)$, $\delta$ a straight line segment from $(0,0)$ to $(1,1)$, and let $\gamma$ be represented as the union of the segment from $(0.5,0)$ to $(1,0.5)$ and the segment from $(0,0.5)$ to $(0.5, 1)$. Cables $a$, $b$, $c$ and $d$ in a block $B$ form a {\em crossing-forcing configuration\/} if there is a homeomorphism of $B$ that takes $a$ to $\alpha$, $b$ to $\beta$, $c$ to $\gamma$, and $d$ to $\delta$, and maps the lower boundary of $B$ to itself. Clearly, in this case any two cables intersect at most once. \begin{observation}\label{obs_XX} Suppose that cables $a$, $b$, $c$ and $d$ form a crossing-forcing configuration in a block $B$. Then every transversal of $B$ crosses at least one of $a$, $b$, $c$, or $d$. \qed \end{observation} Fix $k>0$. Now we define the {\em $k$-forcing block} $B_k$ of $4^k$ edges, $a_1, a_2, \ldots , a_m$, $m=4^k$. We build $B_k$ from $k$ subblocks $C_1, C_2, \ldots, C_k$, arranged from top to bottom in this order. In $C_1$, divide our edges into four equal subsets, each form a cable, and these four cables form a crossing-forcing configuration in $C_1$. In general, suppose that for some $i$, $1\le i<k$, $C_i$ contains $4^i$ cables $c_1, c_2, \ldots , c_{4^i}$ and each of them contains $4^{k-i}$ edges. For each cable $c_j$ of $C_i$, divide the corresponding set of edges into four equal subsets, each of them form a cable in $C_{i+1}$, and let these four cables form a crossing-forcing configuration in $C_{i+1}$. It is possible to draw the cables so that any two of them intersect at most once in $C_{i+1}$ and so that for every two edges $e \in c_j$ and $f \in c_{j'}$, $j<j'$, the edges $e$ and $f$ intersect the top and the bottom boundary of $C_{i+1}$ in the same order. See Figure~\ref{figure_XX}, right. Clearly, $C_{i+1}$ contains $4^{i+1}$ cables and each of them contains $4^{k-i-1}$ edges. The resulting block $B_k=\cup_{i=1}^k C_i$ is called a {\em $k$-forcing block\/} of edges $a_1, a_2, \ldots ,\allowbreak a_m$, where $m=4^k$. The next lemma explains the name. \begin{lemma}\label{kforcinglemma} Suppose that $B_k=\cup_{i=1}^kC_i$ is a $k$-forcing block of edges $a_1,\allowbreak a_2,\allowbreak \ldots,\allowbreak a_m,\allowbreak m=4^k$. Then every transversal of $B_k$ intersects at least one of $a_1,\allowbreak a_2,\allowbreak \ldots,\allowbreak a_m$ at least $k$ times. \end{lemma} \begin{proof} We prove the statement by induction on $k$. For $k=1$ the statement is equivalent to Observation~\ref{obs_XX}. Suppose that the statement has been proved for $k-1$, and let $\kappa$ be a transversal of $B_k$. By Observation~\ref{obs_XX}, $\kappa$ crosses at least one of the cables in $C_1$. Consider now only the $4^{k-1}$ edges that belong to that cable. These edges form a {\em $(k-1)$-forcing block\/} in $B'_{k-1}=\cup_{i=2}^kC_i$. Therefore, by the induction hypothesis, $\kappa$ crosses one of the edges at least $k-1$ times in $B'_{k-1}$. It also crosses this edge in $C_1$, so we are done. \end{proof} Now we prove a statement which is slightly weaker than Theorem~\ref{klconstruction} (i), but much stronger than Lemma~\ref{lemma_weakklconstruction}. \begin{lemma}\label{2k-1construction} For every $k>0$, there exists a $(k, 2k-1)$-construction. \end{lemma} \begin{proof} The construction consists of $2k+1$ consecutive blocks, $X,\allowbreak F_1,\allowbreak S_1,\allowbreak F_2,\allowbreak S_2,\allowbreak \ldots,\allowbreak S_{k-1},\allowbreak F_k,\allowbreak Y$, in this order (from bottom to top). Let $m=4^k$. We define $km$ independent edges $\alpha_i^j$, $1\le i\le k$, $1\le j\le m$, and two isolated vertices $x$ and $y$ as follows. Put $x$ in $X$ and $y$ in $Y$. \begin{itemize} \item The edges $\alpha_1^j$, $1\le j\le m$, are in the blocks $F_1$ and $S_1$, \item for every $i$, $1<i<k$, the edges $\alpha_i^j$, $1\le j\le m$, are in the blocks $S_{i-1}$, $F_i$ and $S_i$, \item the edges $\alpha_k^j$, $1\le j\le m$, are in the blocks $S_{k-1}$ and $F_k$. \end{itemize} For every $i$, $1\le i\le k$, the edges $\alpha_i^j$, $1\le j\le m$, form a $k$-forcing block in $F_i$. For every $i$, $1\le i\le k-1$, the edges $\alpha_i^j$, $1\le j\le m$, form one cable in $S_i$, the edges $\alpha_{i+1}^j$, $1\le j\le m$, form another cable in $S_i$, and these two cables form a $k$-spiral in $S_i$. Let $\kappa$ be a fixed curve from $x$ to $y$. We show that $\kappa$ crosses one of the curves $\alpha_i^j$ at least $2k-1$ times. For every $i$, $1\le i\le k$, by Lemma~\ref{kforcinglemma}, there is a $j$, $1\le j\le m$, such that $\alpha_i^j$ and $\kappa$ cross at least $k$ times in $F_i$. Denote this $\alpha_i^j$ by $\alpha_i$. For every $Z\in\{F_1, F_2, \ldots , F_k, S_1, S_2, \ldots , S_{k-1}\}$ and $1\le i\le k$, let $Z(\alpha_i)$ denote the number of intersections of $\alpha_i$ with $\kappa$ in the block $Z$. By the choice of $\alpha_i$, for every $i$, $1\le i\le k$, we have $$F_i(\alpha_i)\ge k.$$ By Observation~\ref{obs_spiral}, for every $i$, $1\le i\le k-1$, we have $$S_i(\alpha_i)+S_i(\alpha_{i+1})\ge k-1.$$ Summing up, \begin{align*} \sum_{i=1}^k{\mbox {\sc cr}}(\kappa, \alpha_i) &=\sum_{i=1}^kF_i(\alpha_i)+\sum_{i=1}^{k-1}\left(S_i(\alpha_i)+S_i(\alpha_{i+1})\right)\\ &\ge k^2+(k-1)^2=(2k-2)k+1. \end{align*} Therefore, for some $i$, ${\mbox {\sc cr}}(\kappa, \alpha_i)\ge 2k-1$. \end{proof} \subsection{Grid blocks, Double-forcing blocks, and a proof of Theorem~\ref{klconstruction} (i)} \subsubsection*{Grid blocks.} Let $m, k>0$. An {\em $(m,1)$-grid block}, $G(m,1)$ contains two groups, $G'$ and $G''$, of cables (or edges). Both $G'$ and $G''$ contain $m$ cables. Refer to Figure~\ref{figure_block_Cij}. The cables of $G'$ form $m$ parallel segments in $G(m,1)$. The cables of $G''$ are also parallel in $G(m,1)$ but make exactly one twist around the cylinder, intersecting every cable of $G'$ exactly once. Moreover, the cables from $G'$ and $G''$ intersect both the upper and lower boundary alternately. An {\em $(m,k)$-grid block} $G(m,k)$ consists of $k$ identical subblocks $G(m,1)$ stacked on top of each other. Observe that $G(2,1)$ is a crossing-forcing configuration and that $G(1, k)$ is a $k$-spiral. So grid blocks are common generalizations of the spirals and crossing-forcing blocks. \begin{figure} \caption{A $(5,1)$-grid block $G(5,1)$.} \label{figure_block_Cij} \end{figure} The following observation generalizes Observation~\ref{obs_spiral} and is easily shown by induction. \begin{observation}\label{obs_grid} Every transversal of the grid block $G(m,k)$ has at least $mk-1$ crossings with the cables in $G(m,k)$. \qed \end{observation} \subsubsection*{Double-$k$-forcing blocks.} Let $k>0$. A {\em double-$k$-forcing block} $D_k$ contains two groups of cables (edges), say, $D'$ and $D''$. \begin{itemize} \item Each of $D'$ and $D''$ contains $4^k$ cables forming a $k$-forcing block in $B$. \item The cables of $D'$ and $D''$ are consecutive along the upper and lower boundaries (but they are ordered differently on the two boundaries). \item Any two cables in $D_k$ intersect at most $k$ times. \end{itemize} We can construct double-$k$-forcing blocks from subblocks in the same way as $k$-forcing blocks. For $k=1$, the construction is shown on Figure~\ref{figure_hypersubblock}. Suppose we already have $D_i$. We add a subblock $C'_{i+1}$ to the bottom of $D_i$ as follows. We divide each cable into four subcables and let these cables form a crossing-forcing configuration in $C'_{i+1}$, so that any two cables cross at most once in $C'_{i+1}$ and the subcables of the same cable are consecutive along the upper and lower boundary of $C'_{i+1}$. \begin{figure} \caption{A double-$1$-forcing block.} \label{figure_hypersubblock} \end{figure} The following is a direct consequence of Lemma~\ref{kforcinglemma}. \begin{observation}\label{obs_doublekforcing} Suppose that $D_k$ is a double-$k$-forcing block with two groups of cables $D'$ and $D''$. Then for any transversal $\kappa$ of $D_k$, there are cables $\alpha'\in D'$ and $\alpha''\in D''$ that both cross $\kappa$ at least $k$ times. \end{observation} Now, we are ready to prove Theorem~\ref{klconstruction} (i). The construction consists of $4k+1$ consecutive blocks, $X,\allowbreak D_1,\allowbreak G_1,\allowbreak D_2,\allowbreak G_2,\allowbreak \ldots,\allowbreak G_{2k-1},\allowbreak D_{2k},\allowbreak Y$, in this order (from bottom to top). Let $m=2\cdot 4^k$. We define $2km$ independent edges $\alpha_i^j$ and $\beta_i^j$, $1\le i\le k$, $1\le j\le m$, and two isolated vertices $x$ and $y$ as follows. Put $x$ in $X$ and $y$ in $Y$. \begin{itemize} \item The edges $\alpha_1^j$ and $\beta_1^j$, $1\le j\le m$, are in $D_1$ and $G_1$. \item For every $i$, $1<i<2k$, the edges $\alpha_i^j$ and $\beta_i^j$, $1\le j\le m$, are in $G_{i-1}$, $D_i$ and $G_i$. \item The edges $\alpha_{2k}^j$ and $\beta_{2k}^j$, $1\le j\le m$, are in $G_{2k-1}$ and $D_{2k}$. \end{itemize} For every $i$, $1\le i\le 2k$, $D_i$ is a double-$k$-forcing block, and the edges $\alpha_i^j$, $1\le j\le m$, and $\beta_i^j$, $1\le j\le m$, form its two groups $D'$ and $D''$. For every $i$, $1\le i\le 2k-1$, $G_i$ is a $(2,k)$-grid block $G(2,k)$ with groups of cables $G'$ and $G''$. The edges $\alpha_i^j$ form a cable $G'_1$, the edges $\beta_i^j$ form a cable $G'_2$, the edges $\alpha_{i+1}^j$ form a cable $G''_1$, and the edges $\beta_{i+1}^j$ form a cable $G''_2$. Cables $G'_1$ and $G'_2$ form the group $G'$, and cables $G''_1$ and $G''_2$ form the group $G''$. Let $\kappa$ be a fixed curve from $x$ to $y$. We show that $\kappa$ crosses one of the curves $\alpha_i^j$ or $\beta_i^j$ at least $2k$ times. For every $i$, $1\le i\le 2k$, by Observation~\ref{obs_doublekforcing}, there is a $j$, $1\le j\le m$, such that $\alpha_i^j$ and $\kappa$ cross at least $k$ times in $D_i$. For simplicity, denote this $\alpha_i^j$ by $\alpha_i$. Similarly, there is a $j'$, $1\le j'\le m$, such that $\beta_i^{j'}$ and $\kappa$ cross at least $k$ times in $D_i$. Denote $\beta_i^{j'}$ by $\beta_i$. For every $Z\in\{D_1,\allowbreak D_2,\allowbreak \ldots ,\allowbreak D_{2k},\allowbreak G_1,\allowbreak G_2,\allowbreak \ldots ,\allowbreak G_{2k-1}\}$ and $\chi\in\{\alpha_1,\allowbreak \ldots ,\allowbreak \alpha_{2k},\allowbreak \beta_1,\allowbreak \ldots ,\allowbreak \beta_{2k}\}$, let $Z(\chi)$ denote the number of intersections of $\chi$ with $\kappa$ in $Z$. By the choice of $\alpha_i$ and $\beta_i$, for every $i$, $1\le i\le 2k$, we have $$D_i(\alpha_i), \ D_i(\beta_i)\ge k.$$ By Observation~\ref{obs_grid}, for every $i$, $1\le i\le 2k-1$, we have $$G_i(\alpha_i)+G_i(\beta_i)+G_i(\alpha_{i+1})+G_i(\beta_{i+1})\ge 2k-1.$$ Summing up, \begin{align*} &\sum_{i=1}^{2k}\left({\mbox {\sc cr}}(\kappa, \alpha_i)+{\mbox {\sc cr}}(\kappa, \beta_i)\right)\\ =&\sum_{i=1}^{2k}\left(D_i(\alpha_i)+D_i(\beta_i)\right)+ \sum_{i=1}^{2k-1}\left(G_i(\alpha_i)+G_i(\alpha_{i+1})+G_i(\beta_i)+G_i(\beta_{i+1})\right)\\ \ge &4k^2+(2k-1)^2=4k(2k-1)+1. \end{align*} Therefore, for some $i$, ${\mbox {\sc cr}}(\kappa, \alpha_i)\ge 2k$ or ${\mbox {\sc cr}}(\kappa, \beta_i)\ge 2k$. \qed \section{Proof of Theorem~\ref{klconstruction} (ii)} Let $G$ be a non-complete $k$-simple topological graph, and let $u$ and $v$ be two non-adjacent vertices of $G$. We prove that $u$ and $v$ can be connected by a curve that has at most $2k$ points in common with any edge of $G$. Place a new vertex at each crossing of $G$ and subdivide the edges accordingly. Let $G'$ denote the resulting topological (multi)graph. Clearly, there is no loss of generality in assuming that $G'$ is connected. Choose an arbitrary path $\alpha$ in $G'$ connecting $u$ and $v$. We distinguish two types of vertices on $\alpha$. A vertex $x$ of $G'$ that lies on $\alpha$ is called a {\em passing vertex\/} if the two edges of $\alpha$ incident to $x$ belong to the same edge of $G$. A vertex $x$ of $G'$ that lies on $\alpha$ is a {\em turning vertex\/} if it is not a passing vertex, that is, if the two edges of $\alpha$ meeting at $x$ belong to distinct edges of $G$. Assign to $\alpha$ a unique {\em code}, denoted by $c(\alpha)$, as follows. Suppose that $\alpha$ contains $r$ turning vertices for some $r\ge 0$. These vertices divide $\alpha$ into $r+1$ intervals, $I^{\alpha}_1, I^{\alpha}_2, \ldots ,I^{\alpha}_{r+1}$, ordered from $u$ to $v$. Set $p^{\alpha}_0=r$ and for any $i$, $1\le i\le r+1$, let $p^{\alpha}_i$ denote the number of passing vertices on $I^{\alpha}_i$. Let $c(\alpha)=(p^{\alpha}_0, p^{\alpha}_1, p^{\alpha}_2, \ldots , p^{\alpha}_{r+1})$; see Figure~\ref{codes}. \begin{figure} \caption{A $(u,v)$-path $\alpha$ (in bold) with $c(\alpha)=(6,0,0,1,1,0,1,0)$ and its turning vertices $t_i$.} \label{codes} \end{figure} Order the codes of all $(u,v)$-paths lexicographically: if $\alpha$ and $\beta$ are two $(u,v)$-paths in $G'$, with codes $c(\alpha)=(p^{\alpha}_0=r, p^{\alpha}_1, p^{\alpha}_2, \ldots , p^{\alpha}_{r+1})$ and $c(\beta)=(p^{\beta}_0=s, p^{\beta}_1, p^{\beta}_2, \ldots , p^{\beta}_{s+1})$, respectively, then let $c(\alpha)\prec_{\mbox{\scriptsize{lex}}} c(\beta)$ if and only if $c(\alpha)\neq c(\beta)$ and for the smallest index $i$ such that $p_i\neq q_i$ we have $p_i<q_i$. Finally, define a {\em partial ordering} $\prec$ on the set of all the $(u,v)$-paths in $G'$: for any two $(u,v)$-paths, $\alpha$ and $\beta$, let $\alpha\prec\beta$ if and only if $c(\alpha)\prec_{\mbox{\scriptsize{lex}}} c(\beta)$. Let $\gamma$ be a {\em minimal\/} element with respect to $\prec$. Suppose that $\gamma$ has $r$ turning vertices, $t_1, t_2, \ldots , t_r$, $r\ge 0$, which divide $\gamma$ into intervals $I^{\gamma}_1, I^{\gamma}_2, \ldots , I^{\gamma}_{r+1}$, ordered from $u$ to $v$. Consider the intervals as half-closed, that is, for every $i$, $0\le i \le r$, let $t_i$ belong to $I^{\gamma}_{i+1}$. Next we establish some simple properties of the intersections of $\gamma$ with the edges of $G$. \begin{lemma}\label{consecutive} Let $e$ be an edge of $G$ that has only finitely many points in common with $\gamma$. Then all of these points belong to two consecutive intervals of $\gamma$. \end{lemma} \begin{proof} Suppose for contradiction that $e$ has nonempty intersection with at least two non-consecutive intervals of $\gamma$. Let $x$ (and $y$) denote the crossing of $e$ and $\gamma$, closest to (respectively, farthest from) $u$ along $\gamma$. Let $x$ belong to $I^{\gamma}_i$ and let $y$ belong to $I^{\gamma}_j$, where $i<j-1$. Let $\gamma'$ be another $(u,v)$-path, which is identical to $\gamma$ from $u$ to $x$, identical to $e$ from $x$ to $y$, and finally identical to $\gamma$ from $y$ to $v$; see Figure~\ref{modify}. If $i<j-2$, then it is evident that $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$, since $\gamma'$ has fewer turning vertices than $\gamma$. If $i=j-2$, then $\gamma$ and $\gamma'$ have the same number of turning vertices, but $I^{\gamma'}_i$ contains fewer passing vertices than $I^{\gamma}_i$ (hence $p^{\gamma'}_i < p^{\gamma}_i$), and we have $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$. In both cases we obtain that $\gamma'\prec\gamma$, contradicting the minimality of $\gamma$. \end{proof} \begin{figure} \caption{Two $(u,v)$-paths $\gamma$ and $\gamma'$ (both in bold) in the proof of Lemma~\ref{consecutive}.} \label{modify} \end{figure} \begin{lemma}\label{2k-2k-1} Let $e$ be an edge of $G$ that has only finitely many points in common with $\gamma$. \begin{itemize} \item[(i)] If none of the common points is a vertex of $e$, then $e$ crosses $\gamma$ at most $2k$ times. \item[(ii)] If one of the common points is a vertex of $e$, then $e$ crosses $\gamma$ at most $2k-1$ times. \end{itemize} \end{lemma} \begin{proof} First, suppose that no vertex of $e$ lies on $\gamma$. By Lemma~\ref{consecutive}, $e$ crosses at most two consecutive intervals of $\gamma$. Each interval is a part of some edge of $G$ and hence crosses $e$ at most $k$ times. This proves (i). Suppose next that one of the vertices of $e$ lies on $\gamma$. Observe that such a vertex must be a turning vertex of $\gamma$, say $t_i$. Again, by Lemma~\ref{consecutive}, $e$ crosses at most two consecutive intervals of $\gamma$. Each interval is a part of some edge of $G$. Moreover, one of them has a common endpoint with $e$. Therefore, $e$ crosses one of the intervals at most $k$ times and the other at most $k-1$ times. This proves (ii). \end{proof} Note that no edge $e$ of $G$ that has only finitely many points in common with $\gamma$ can have both of its endpoints on $\gamma$. Otherwise, both endpoints must be turning vertices of $\gamma$, say $t_i$ and $t_j$ for some $i < j$. Since the underlying abstract graph $G$ is simple (that is, $G$ has no multiple edges), the edge of $G$ that contains $I^{\gamma}_{i+1}$ must be different from the edge that contains $I^{\gamma}_j$. Hence, there is at least one turning vertex between $t_i$ and $t_j$ on $\gamma$. Now consider another $(u,v)$-path $\gamma'$ that is identical to $\gamma$ from $u$ to $t_i$, identical to $e$ from $t_i$ to $t_j$, and finally identical to $\gamma$ from $t_j$ to $v$. The turning vertices $t_i$ and $t_{j}$ of $\gamma$ are also turning vertices on $\gamma'$. Since the turning vertices of $\gamma$ that lie between $t_{i}$ and $t_j$ are not among the turning vertices of $\gamma'$, $\gamma'$ has fewer turning vertices than $\gamma$. Therefore, we have $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$, contradicting the minimality of $\gamma$. \begin{lemma}\label{int} Let $e$ be an edge of $G$ that contains an interval $I^{\gamma}_i$ of $\gamma$. Then $e$ and $\gamma$ have at most $k$ points in common outside of $I^{\gamma}_i$. Furthermore, one of these points is $t_i$, the endpoint of $I^{\gamma}_i$. \end{lemma} \begin{proof} Since $I^{\gamma}_i$ and $I^{\gamma}_{i+1}$ are separated by $t_i$, and $I^{\gamma}_i$ is contained in $e$, it follows that $e$ cannot contain $I^{\gamma}_{i+1}$. Similarly, $e$ cannot contain $I^{\gamma}_{i-1}$. If $e$ has a point $p$ in $I^{\gamma}_j$ with $j<i$, consider another $(u,v)$-path $\gamma'$ that is identical to $\gamma$ from $u$ to $p$, identical to $e$ from $p$ to $t_{i-1}$, and finally identical to $\gamma$ from $t_{i-1}$ to $v$; see Figure~\ref{int_flips}. If $j<i-1$, the turning vertices $t_j$ and $t_{i-1}$ of $\gamma$ are not among the turning vertices of $\gamma'$. Although $p$ was a passing vertex of $\gamma$ and is now a turning vertex of $\gamma'$, still $\gamma'$ has fewer turning vertices than $\gamma$. Therefore, $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$. If $j = i-1$, the turning vertex $t_j$ of $\gamma$ is not a turning vertex of $\gamma'$. Again, $p$ was a passing vertex of $\gamma$ and is now a turning vertex of $\gamma'$. So, $\gamma$ and $\gamma'$ have the same number of turning vertices. Since $p$ is not a passing vertex of $\gamma'$, $I^{\gamma'}_{i-1}$ has fewer passing vertices than $I^{\gamma}_{i-1}$ (hence $p^{\gamma'}_{i-1} < p^{\gamma}_{i-1}$), and we have that $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$. In all of the above cases, we obtain that $\gamma'\prec\gamma$, contradicting the minimality of $\gamma$. \begin{figure} \caption{Two $u,v$-paths $\gamma$ and $\gamma'$ (both in bold) in the proof of Lemma~\ref{int}; $j<i-1$.} \label{int_flips} \end{figure} Similarly, if $e$ has a point $p$ in $I^{\gamma}_j$ with $j>i+1$, consider another $(u,v)$-path $\gamma'$ that is identical to $\gamma$ from $u$ to $t_i$, identical to $e$ from $t_i$ to $p$, and finally identical to $\gamma$ from $p$ to $v$. The turning vertices $t_i$ and $t_{j-1}$ of $\gamma$ are not among the turning vertices of $\gamma'$. Although $p$ was a passing vertex of $\gamma$ and is a turning vertex of $\gamma'$, still $\gamma'$ has fewer turning vertices than $\gamma$. Therefore, $c(\gamma')\prec_{\mbox{\scriptsize{lex}}} c(\gamma)$, contradicting the minimality of $\gamma$. Note that the case $j=i+1$ cannot be settled in the same way as the previous cases, since the number of passing vertices on $e$ between $t_i$ and $p$ may not be smaller than the number of passing vertices on $\gamma$ between $t_i$ and $p$. Nevertheless, we can conclude that no interval of $\gamma$ other than $I^{\gamma}_i$ is contained in $e$. Furthermore, the only interval of $\gamma$ other than $I^{\gamma}_i$ that can share some points with $e$ is $I^{\gamma}_{i+1}$. Let $f$ be the edge of $G$ that contains $I^{\gamma}_{i+1}$. Since $e$ and $f$ have at most $k$ points in common, $e$ and $I^{\gamma}_{i+1}$ can have at most $k$ points in common, too. The point $t_{i}$, the common endpoint of $I^{\gamma}_i$ and $I^{\gamma}_{i+1}$, is one of these points. \end{proof} Now we are in a position to complete the proof of Theorem~\ref{klconstruction} (ii). Join $u$ and $v$ by a curve $\beta$ that runs very close to $\gamma$. We claim that any edge $e$ of $G$ has at most $2k$ points in common with $\beta$. If $e$ has only finitely many points in common with $\gamma$ and none of them is a vertex of $e$, then every crossing between $e$ and $\beta$ corresponds to a crossing between $e$ and $\gamma$. Therefore, by Lemma~\ref{2k-2k-1}(i), $e$ and $\beta$ cross each other at most $2k$ times. If $e$ has only finitely many points in common with $\gamma$, but one of them is a vertex of $e$, then each crossing between $e$ and $\beta$ corresponds to a crossing between $e$ and $\gamma$, and there may be an additional crossing near the vertex of $e$ on $\gamma$. Again, by Lemma~\ref{2k-2k-1}(ii), there are at most $2k$ crossings between $e$ and $\beta$. Finally, if $e$ contains a whole interval $I^{\gamma}_i$ of $\gamma$, then each crossing between $e$ and $\beta$ corresponds to a crossing between $e$ and $\gamma$, or to a vertex of $e$ on $\gamma$. There may be an additional crossing near the endpoint $t_{i}$ of $I^{\gamma}_i$. Thus, there are at most $k+1$ crossings. \qed \section{Proof of Theorem~\ref{linearbound}: Upper Bounds} The construction for $k=1$ is essentially different from the constructions for $k>1$. For $k>1$, all constructions are variations of the constructions used in the proofs of Theorem~\ref{klconstruction}, Lemma~\ref{lemma_weakklconstruction} and Lemma \ref{2k-1construction}, but they give different bounds for different values of $k$. Table~\ref{table1} shows our best upper bounds for different values of $k$. \paragraph{First construction.} This construction is for $k=1$. First we need to modify the graph $G_1$ on Figure~\ref{kampok}. Consider the edges of $G_1$ incident to $x$, and modify them in a small neighborhood of $x$ so that the resulting edges have distinct endpoints, they pairwise cross each other, and their union encloses a {\em region} $X$ (i.e., a connected component $X$ of the complement of the union of the edges) which contains $x$. Analogously, modify the other three edges of $G_1$ in a small neighborhood of $y$. Let $Y$ be the region that contains $y$ and is enclosed by the modified edges. The resulting simple topological graph $G$ has $12$ vertices and $6$ edges; see Figure~\ref{nembovitheto1}. The points $x,y\in V(G_1)$ do not belong to $V(G)$. \begin{figure} \caption{A topological graph $G$: the edge $\{x, y\}$ cannot be added.} \label{nembovitheto1} \end{figure} \begin{lemma}\label{2cr} Let $x$ and $y$ be any pair of points belonging to the regions $X$ and $Y$ in $G$, respectively. Then any curve joining $x$ and $y$ will meet at least one of the edges of $G$ at least twice. \end{lemma} \begin{proof} We prove the claim by contradiction. Let $a_1, a_2, a_3$, $b_1, b_2, b_3$ denote the edges of $G$. They divide the plane into eight regions, $X$, $Y$, $A_1, A_2, A_3$, $B_1, B_2, B_3$; see Figure~\ref{nembovitheto1}. Suppose there exists an oriented curve from $x$ to $y$ that crosses every edge of $G$ at most once. Let $\gamma$ be such a curve with the smallest number of crossings with the edges of $G$. Let $c_1, c_2, \ldots , c_{m-1}$ be the crossings between $\gamma$ and the edges of $G$, ordered according to the orientation of $\gamma$. They divide $\gamma$ into intervals $I_1, I_2, \ldots , I_m$, ordered again according to the orientation of $\gamma$. The first interval $I_1$ lies in $X$, and the last one, $I_m$, lies in $Y$. Observe that no other interval can belong to $X$ or to $Y$, because in this case we could simplify $\gamma$ and obtain a curve with a smaller number of crossings. By symmetry, we can assume that the first crossing, $c_1$, is a crossing between $\gamma$ and $a_1$. Then $I_2$ belongs to $A_1$. The following property holds. {\sl Property $\mathcal P$: If for some $j\ge 2$, the interval $I_j$ belongs to $A_i$ (or $B_i$), then one of the points $c_1, c_2, \ldots , c_{j-1}$ is a crossing between $\gamma$ and the edge $a_i$ (or $b_i$, respectively).} We prove Property $\mathcal P$ by induction on $j$. Clearly, the property holds for $j = 2$. Assume that $I_{j-1}$ is in $A_i$ (or $B_i$) and one of $c_1, c_2, \ldots , c_{j-2}$ is a crossing between $\gamma$ and $a_i$ (or $b_i$). For simplicity, assume that $I_{j-1}$ belongs to the region $A_1$ and that one of the points $c_1, c_2, \ldots , c_{j-2}$ is a crossing between $\gamma$ and $a_1$; the other cases are analogous. Since $c_{j-1}$ cannot belong to $a_1$, it must be a crossing between $\gamma$ and either $a_2$ or $b_2$. In the first case, $I_j$ belongs to $A_2$, in the second to $B_2$. In either case, Property $\mathcal P$ is preserved. Now, we can complete the proof of Lemma~\ref{2cr}. Consider the interval $I_{m-1}$. Since $I_m$ lies in $Y$, for some $i$, the interval $I_{m-1}$ must lie in $B_i$. Suppose for simplicity that $I_{m-1}$ lies in $B_1$. By Property $\mathcal P$ (with $j = m-1$, $m\ge 3$), one of the points $c_1, c_2, \ldots , c_{m-2}$ must be a crossing between $\gamma$ and $b_1$. However, using that $I_m$ is in $Y$, $c_{m-1}$ must be another crossing between $\gamma$ and $b_1$. Thus, $\gamma$ crosses $b_1$ twice, which is a contradiction. \end{proof} Now, we return to the proof of the upper bound in Theorem~\ref{linearbound}. Modify the drawing of $G$ in Figure~\ref{nembovitheto1} so that the region $Y$ becomes unbounded, and let $H$ be the resulting topological graph. Denote by $Y$ the {\em outer region\/} of $H$ and by $X$ the {\em inner region\/} of $H$; see Figure~\ref{nembovitheto-gyurus}. \begin{figure} \caption{A topological graph $H$, a modification of $G$.} \label{nembovitheto-gyurus} \end{figure} For every $n\ge 1$, construct a saturated simple topological graph $F_n$, as follows. Let $k=\lfloor{n/12}\rfloor$. Take a disjoint union of $k$ scaled and translated copies of $H$, denoted by $H^1,\allowbreak H^2,\allowbreak \ldots,\allowbreak H^k$, such that for any $i$, $1<i\le k$, the copy $H^i$ lies entirely in the inner region of $H^{i-1}$; see Figure~\ref{soknembovitheto}. For $1\le i\le k$, let $V_i$ be the vertex set of $H^i$. Finally, place $n-12k$ additional vertices in the inner region of $H^k$, and let $V_{k+1}$ denote the set of these vertices. Obviously, we have $|V_{k+1}| < 12$. \begin{figure} \caption{A saturated simple topological graph $F_n$. } \label{soknembovitheto} \end{figure} Add to this topological graph all possible missing edges one by one, in an arbitrary order, as long as it remains simple. We end up with a saturated simple topological graph $F_n$ with $n$ vertices. Observe that for every $i$ and $j$ with $1\le i<j-1<k$, $V_i$ lies in the outer region of $H^{i+1}$, while $V_j$ is in the inner region of $H^{i+1}$. By Lemma~\ref{2cr} (applied with $G = H^{i+1}$, $x\in V_j$, $y\in V_i$), no edge of $F_n$ runs between $V_i$ and $V_j$. Hence, every vertex in $V_i$ can be adjacent to at most $35$ other vertices; namely, to the elements of $V_{i-1}\cup V_i\cup V_{i+1}$. Therefore, $F_n$ is a saturated simple topological graph with $n$ vertices and at most $17.5n$ edges. \paragraph{Second construction.} This construction is used for all odd $k\ge 5$ and all even $k\ge 12$. Suppose for simplicity that $n$ is divisible by $3$ and let $m=n/3$. The construction consists of $2m+3$ consecutive blocks, $B_0,\allowbreak A_0,\allowbreak B_1,\allowbreak A_1,\allowbreak \ldots,\allowbreak B_{m},\allowbreak A_{m},\allowbreak B_{m+1}$, in this order, from bottom to top. See Figure~\ref{figure_saturated_first_second}, left. \begin{figure} \caption{Three consecutive blocks in the constructions of saturated $k$-simple topological graphs. Left: the second construction for $k=7$. Right: the third construction for $k=6$.} \label{figure_saturated_first_second} \end{figure} For every $i$, $1\le i\le m$, let $u_i$ be a vertex on the common boundary of $B_{i-1}$ and $A_{i-1}$ and let $v_i$ and $w_i$ be vertices on the common boundary of $A_{i}$ and $B_{i+1}$. Let $\alpha_i$ be an edge connecting $u_i$ and $v_i$ and let $\beta_i$ be an edge connecting $u_i$ and $w_i$. The pair $(\alpha_i, \beta_i)$ is called the {\em $i$th bundle}. The edges $\alpha_i$ and $\beta_i$ form a $(k-1)$-spiral in $B_i$. For $1< i\le m$, the edges $\alpha_i$ and $\beta_i$ form a cable in $A_{i-1}$, the edges $\alpha_{i-1}$ and $\beta_{i-1}$ also form a cable in $A_{i-1}$, and these two cables form a $k$-spiral in $A_{i-1}$. The resulting $k$-simple topological graph $G$ has $n$ vertices and $2n/3$ edges. Add to $G$ all possible missing edges one by one, as long as the drawing remains $k$-simple. We obtain a saturated $k$-simple topological graph $H$. Note that $H$ is not uniquely determined by $G$, not even as an abstract graph. Suppose that $k\ge 11$. Just like in the proof of Lemma~\ref{lemma_weakklconstruction}, we can prove that any curve from $A_i$ to $A_{i+3}$ has to cross one of the curves $\alpha_{i+1}$, $\beta_{i+1}$, $\alpha_{i+2}$, $\beta_{i+2}$, $\alpha_{i+3}$, $\beta_{i+3}$ at least $(k-2)/2+2(k-1)/3>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can possibly be connected only to other vertices from the five blocks $A_{i-2},\allowbreak A_{i-1},\allowbreak A_i,\allowbreak A_{i+1},\allowbreak A_{i+2}$. Since every block $A_j$ has at most three vertices, the maximum degree in $H$ is at most $5\cdot3-1=14$ and thus $H$ has at most $7n$ edges. For $k \ge 9$ odd and for every $j$, $1\le j\le 3$, every curve $\kappa$ from $A_i$ to $A_{i+3}$ has to cross one of the two curves $\alpha_{i+j},\allowbreak \beta_{i+j}$ at least $(k-1)/2$ times in $B_{i+j}$. Let $\gamma_{i+j}$ be this curve. Now, for every $j$, $1\le j\le 2$, the curve $\kappa$ crosses $\gamma_{i+j}$ and $\gamma_{i+j+1}$ together at least $k-1$ times in $A_{i+j}$. It follows that $\kappa$ crosses one of the curves $\gamma_{i+1},\allowbreak \gamma_{i+2},\allowbreak \gamma_{i+3}$ at least $(1/2 + 2/3)\cdot (k-1)>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can be connected only to other vertices from the five blocks $A_{i-2},\allowbreak A_{i-1},\allowbreak \dots,\allowbreak A_{i+2}$. The maximum degree in $H$ is thus at most $5\cdot 3 - 1 =14$ and $H$ has at most $7n$ edges. Similarly, for $k \ge 7$ odd, for every curve $\kappa$ from $A_i$ to $A_{i+4}$, there are four curves $\gamma_{i+1},\allowbreak \gamma_{i+2},\allowbreak \gamma_{i+3},\allowbreak \gamma_{i+4}$ such that $\kappa$ crosses one of them at least $(1/2 + 3/4)\cdot (k-1)>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can be connected only to vertices from the seven blocks $A_{i-3},\allowbreak A_{i-2},\allowbreak \dots,\allowbreak A_{i+3}$. The maximum degree in $H$ is thus at most $7\cdot 3 - 1 =20$ and $H$ has at most $10n$ edges. For $k \ge 5$ odd, for every curve $\kappa$ from $A_i$ to $A_{i+5}$, there are five curves, $\gamma_{i+1},\allowbreak \gamma_{i+2},\allowbreak \dots,\allowbreak \gamma_{i+5}$ such that $\kappa$ crosses one of them at least $(1/2 + 4/5)\cdot (k-1)>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can be connected only to vertices from the nine blocks $A_{i-4},\allowbreak A_{i-3},\allowbreak \dots,\allowbreak A_{i+4}$. The maximum degree in $H$ is thus at most $9\cdot 3 - 1 =26$ and $H$ has at most $13n$ edges. \paragraph{Third construction.} This construction is used for $k\in\{4,\allowbreak 6,\allowbreak 8,\allowbreak 10\}$. It is a modification of the second construction, where the edges of the $i$th bundle, $\alpha_i$ and $\beta_i$, do not have common endpoints, so they form a matching rather then a path, and they form a $k$-spiral in $B_i$. See Figure~\ref{figure_saturated_first_second}, right. For $k \ge 6$ even and for every $j$, $1\le j\le 3$, every curve $\kappa$ from $A_i$ to $A_{i+3}$ has to cross one of the two curves $\alpha_{i+j},\allowbreak \beta_{i+j}$ at least $k/2$ times in $B_{i+j}$. Let $\gamma_{i+j}$ be this curve. Now, for every $j$, $1\le j\le 2$, the curve $\kappa$ crosses $\gamma_{i+j}$ and $\gamma_{i+j+1}$ together at least $k-1$ times in $A_{i+j}$. It follows that $\kappa$ crosses one of the curves $\gamma_{i+1},\allowbreak \gamma_{i+2},\allowbreak \gamma_{i+3}$ at least $k/2 + 2(k-1)/3>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can be connected only to other vertices from the five blocks $A_{i-2},\allowbreak A_{i-1},\allowbreak \dots,\allowbreak A_{i+2}$. Since every block $A_j$ now has at most four vertices, the maximum degree in $H$ is at most $5\cdot 4 - 1 =19$ and $H$ has at most $9.5n$ edges. Similarly for $k \ge 4$ even, for every curve $\kappa$ from $A_i$ to $A_{i+4}$, there are four curves $\gamma_{i+1},\allowbreak \gamma_{i+2},\allowbreak \gamma_{i+3},\allowbreak \gamma_{i+4}$ such that $\kappa$ crosses one of them at least $k/2 + 3(k-1)/4>k$ times. Therefore, in $H$, a vertex from the block $A_i$ can be connected only to other vertices from the seven blocks $A_{i-3},\allowbreak A_{i-2},\allowbreak \dots,\allowbreak A_{i+3}$. The maximum degree in $H$ is thus at most $7\cdot 4 - 1 =27$ and $H$ has at most $13.5n$ edges. \paragraph{Fourth construction.} This construction is for $k=2$. First we present a weaker but simpler version. It is a modification of the previous constructions. Here each bundle contains $16$ independent edges. The edges of the $i$th bundle form a $2$-forcing block in $B_i$. In $A_i$, the edges of the $i$th bundle form a cable, the edges of the $(i+1)$st bundle form another cable, and these two cables form a $2$-spiral. Let $\kappa$ be a curve from $A_i$ to $A_{i+2}$. Just like in the previous arguments, using Observation~\ref{obs_spiral} and Lemma~\ref{kforcinglemma}, it is not hard to see that $\kappa$ has to cross an edge more than twice. Therefore, in $H$, a vertex from $A_i$ can be connected only to vertices from $A_{i-1},\allowbreak A_{i},\allowbreak A_{i+1}$. Every block $A_j$ has at most $32$ vertices, so the maximum degree in $H$ is at most $95$, therefore, $H$ has at most $47.5n$ edges. The best construction we have is very similar. To obtain it, in each bundle we identify some of the endpoints of the edges, and we also modify the order of the edges along the bottom boundary of $B_i$; see Figure~\ref{figure_saturated_for_k_4_2}, left. Then every block $A_i$ has at most $11$ vertices, so the maximum degree in $H$ is at most $3\cdot 11-1=32$ and $H$ has at most $16n$ edges. \begin{figure} \caption{A bundle in the construction of a saturated $2$-simple (left) and a saturated $3$-simple (right) topological graph.} \label{figure_saturated_for_k_4_2} \end{figure} \paragraph{Fifth construction.} This construction is for $k=3$. First we present a weaker but simpler version. It is again a modification of the previous constructions. Here each bundle contains four independent edges. The edges of the $i$th bundle form a grid block $G(2,3)$ in $B_i$. In $A_i$, the edges of the $i$th bundle form a cable, the edges of the $(i+1)$st bundle form another cable, and these two cables form a $3$-spiral. Let $\kappa$ be a curve from $A_i$ to $A_{i+3}$. Just like in the previous arguments, using Observations~\ref{obs_grid} and~\ref{obs_spiral}, it is not hard to see that $\kappa$ has to cross an edge more than three times. Therefore, in $H$, a vertex from $A_i$ can be connected only to vertices from $A_{i-2},\allowbreak A_{i-1},\allowbreak \dots,\allowbreak A_{i+2}$. Every block $A_j$ has at most $8$ vertices, so $H$ has at most $19.5n$ edges. To obtain our best construction, in each bundle we identify some endpoints of the edges; see Figure~\ref{figure_saturated_for_k_4_2}, right. Then every block $A_i$ has at most $6$ vertices, so the maximum degree in $H$ is at most $5\cdot 6-1=29$ and $H$ has at most $14.5n$ edges. A modification of this construction works for $k=1$, and it gives the same upper bound, $17.5n$, as the first construction. This concludes the proof of the upper bounds. \qed \section{Proof of Theorem~\ref{linearbound}: Lower Bounds} A vertex of a (topological) graph is {\em isolated\/} if its degree is zero. A triangle in a (topological) graph is called {\em isolated\/} if its vertices are incident to no edges other than the edges of the triangle. \begin{lemma}\label{triangle} A saturated simple topological graph on at least four vertices contains no isolated triangle. \end{lemma} \begin{proof} Let $G$ be a saturated simple topological graph with at least four vertices, and suppose for contradiction that $G$ has an isolated triangle $T$ with vertices $x$, $y$ and $z$. By definition, the edges of $T$ do not cross one another. If all vertices other than $x,y,z$ are isolated, it is trivial to add a new edge without crossings. Hence we may assume that $G$ has an edge not contained in $T$. We distinguish two cases. {\bf Case 1.} The edges of $T$ cross no other edges. The edges of $G$ divide the plane into regions. Let $R$ denote a region bounded by the edges of $T$ and at least one other nontrivial curve $\omega$. Let $e = \{u,v\}$ be an edge that contributes to $\omega$, and let $p$ be a point on $e$ that belongs to the boundary of $R$; see Figure~\ref{trianglecases}, left. Choose a point $p'$ inside of $R$, very close to $p$. Let $\beta$ be a curve running inside $R$ that connects a vertex of $T$, say $x$, to $p'$. Let $\beta'$ be a curve joining $p'$ and $u$, and running very close to the edge $e$. Adjoining $\beta$ and $\beta'$ at $p'$, we obtain a curve $\gamma$ connecting $x$ and $u$, two previously non-adjacent vertices of $G$. The curve $\gamma$ crosses neither an edge of $T$ or an edge of $G$ incident to $u$. Since $\beta$ is crossing-free, all crossings between $\gamma$ and the edges of $G$ must lie on $\beta'$ and, hence, must correspond to crossings along the edge $e$. Therefore, every edge of $G$ can cross $\gamma$ at most once. Consequently, $\gamma$ can be added to $G$ as an extra edge so that the topological graph remains simple. This contradicts the assumption that $G$ was saturated. {\bf Case 2.} At least one edge of $T$ participates in a crossing. Assume without loss of generality that $e=\{x,y\}$ is crossed by another edge of $G$. Let $p$ denote the crossing on $e$ {\em closest\/} to $x$, and suppose that $p$ is a crossing between $e$ and another edge $f=\{u,v\}$; see Figure~\ref{trianglecases}, right. The point $p$ divides $f$ into two parts. At least one of them, say, $up$, does not cross the edge $\{x,z\}$ of $T$. The edges $e$ and $f$ divide a small neighborhood of $p$ into four parts. Choose a point $p'$ in the part bounded by $up$ and $xp$. Let $\beta$ be a curve connecting $x$ and $p'$, running very close to $e$. Let $\beta'$ be a curve between $p'$ and $u$, running very close to $f$. Adjoining $\beta$ and $\beta'$ at $p'$ we obtain a curve $\gamma$ connecting $x$ and $u$, two vertices that were not adjacent in $G$. Just like in the previous case, add $\gamma$ to $G$ as an extra edge. The curve $\gamma$ crosses no edge incident to $x$ or $u$. Since the portion $xp$ of $e$ is crossing-free, $\beta$ must be crossing-free, too. Therefore, all possible crossings between $\gamma$ and the edges of $G$ must lie on $\beta'$ and, hence, correspond to crossings along $f$. Thus, every edge of $G$ crosses $\gamma$ at most once, contradicting our assumption that $G$ was saturated. \end{proof} \begin{figure} \caption{Case 1 and Case 2 of Lemma~\ref{triangle}.} \label{trianglecases} \end{figure} \begin{lemma}\label{isol+deg1} For any $k>0$, a saturated $k$-simple topological graph on at least three vertices contains \item[(i)] no isolated vertex, \item[(ii)] no vertex of degree one. \end{lemma} The proof of Lemma~\ref{isol+deg1} is very similar to the proof of Lemma~\ref{triangle}, but much easier. We omit the details. The lower bound in Theorem~\ref{linearbound} (ii) now follows directly. In a saturated $k$-simple topological graph on $n$ vertices, every vertex has degree at least two, therefore, it has at least $n$ edges. We are left with the proof of the lower bound of part (i). It follows immediately from the statement below. \begin{lemma}\label{deg3} In every saturated simple topological graph with at least four vertices, every vertex has degree at least 3. \end{lemma} \begin{proof} We prove the claim by contradiction. Let $G$ be a saturated simple topological graph, and let $x$ be a vertex of degree two in $G$. (By Lemma~\ref{isol+deg1}, the degree of $x$ cannot be $0$ or $1$.) Let $y$ and $z$ denote the neighbors of $x$. By definition, the edges $\{x,y\}$ and $\{x, z\}$ do not cross. We distinguish two cases. {\bf Case 1.} The edges $\{x,y\}$ and $\{x, z\}$ cross no other edges. By Lemma~\ref{triangle} and~\ref{isol+deg1}, $y$ and $z$ both have degree at least two, and $x$, $y$ and $z$ do not span an isolated triangle. Hence, at least one of the vertices $y$ and $z$, say, $y$, has a neighbor $w$ different from $x$ and $z$. Let $\gamma$ be a curve connecting $x$ to $w$ that runs very close to the edge $\{x,y\}$ from $x$ to a point in a small neighborhood of $y$, and from that point all the way to $w$ very close to the edge $\{y,w\}$. We can assume that $\gamma$ does not cross $\{x,y\}$ and $\{y,w\}$. Add $\gamma$ to $G$ as an extra edge. Clearly, $\gamma$ crosses no edge incident to $x$ or $w$, and crosses no edge of $G$ twice. This contradicts the assumption that $G$ was saturated. {\bf Case 2.} At least one of the edges $\{x,y\}$ and $\{x, z\}$ participates in a crossing. Assume without loss of generality that $e = \{x,y\}$ is crossed by another edge of $G$. Let $p$ be the crossing on $e$ {\em closest\/} to $x$, and suppose that the other edge passing through $p$ is $f=\{u,v\}$. The point $p$ divides $f$ into two pieces, at least one of which, say, $up$, has no point in common with the edge $\{x,z\}$. Let $\gamma$ be a curve connecting $x$ and $u$, following $e$ very closely from $x$ to a point in a small neighborhood of $p$, and from that point following $f$ all the way to $u$. We can assume that $\gamma$ does not cross $e$ and $f$. Add $\gamma$ to $G$ as an extra edge. It is again easy to see that this new edge meets no original edge of $G$ more than once, and again, this contradicts the assumption that $G$ was saturated. \end{proof} \section{Proof of Theorem~\ref{3k/2saturated}} We start with a piece of the construction we used in the proof of Theorem~\ref{klconstruction} (i). Then we add some edges so that it remains a $k$-simple topological graph, and we show that it is $(k, \lceil 3k/2\rceil)$-saturated. Let $D_1$, $G$, $D_2$ be three consecutive blocks, say, from bottom to top, and let $m=4^k$. We define $4m$ independent edges $\alpha_i^j$, $\beta_i^j$, $1\le i\le 2$, $1\le j\le m$. The edges $\alpha_1^j$ and $\beta_1^j$, $1\le j\le m$, are in $D_1$ and $G$, with endpoints on the lower boundary of $D_1$ and the upper boundary of $G$. Denote the sets of these vertices by $V_0$ and $V_2$, respectively. The edges $\alpha_2^j$ and $\beta_2^j$, $1\le j\le m$, are in $G$ and $D_2$, with endpoints on the lower boundary of $G$ and the upper boundary of $D_2$. Denote the sets of these vertices by $V_1$ and $V_3$, respectively. For $i=1, 2$, the block $D_i$ is a double-$k$-forcing block, the edges $\alpha_i^j$, $1\le j\le m$, and $\beta_i^j$, $1\le j\le m$, form its two groups $D'$ and $D''$. The block $G$ is a $(2,k)$-grid block $G(2,k)$ with groups of cables $G'$ and $G''$. The edges $\alpha_1^j$ form a cable $G'_1$, the edges $\beta_1^j$ form a cable $G'_2$, the edges $\alpha_{2}^j$ form a cable $G''_1$, and the edges $\beta_{2}^j$ form a cable $G''_2$. The cables $G'_1$ and $G'_2$ form the group $G'$, and the cables $G''_1$ and $G''_2$ form the group $G''$ in $G$. Let $T$ denote the resulting topological graph. Let $v_0\in V_0$ and $v_3\in V_3$ be arbitrary vertices, and let $\kappa$ be a curve connecting $v_0$ and $v_3$. By Observation~\ref{obs_doublekforcing} there are edges $\alpha_1=\alpha_1^j$ and $\beta_1=\beta_1^{j'}$ that both cross $\kappa$ at least $k$ times in $D_1$. Similarly, there are edges $\alpha_2=\alpha_2^l$ and $\beta_2=\beta_2^{l'}$ that both cross $\kappa$ at least $k$ times in $D_2$. Since $\alpha_1, \allowbreak \alpha_2, \allowbreak \beta_1, \allowbreak \beta_2$ form a $(2,k)$-grid block in $G$, by Observation~\ref{obs_grid}, $\kappa$ crosses them in $G$ together at least $2k-1$ times. Therefore, $\kappa$ crosses one of the curves at least $\lceil(6k-1)/4\rceil =\lceil 3k/2\rceil$ times. Now we show that any two vertices $v_i\in V_i$ and $v_i\in V_j$ with $|i-j|\le 2$, can be connected so that we still have a $k$-simple topological graph. We only sketch the argument. By definition, block $D_1$ is divided into $k$ subblocks, $C'_1, C'_2, \ldots, C'_k$, from top to bottom. Let $v_0\in V_0$, $v_2\in V_2$, let $\alpha$ be the edge of $T$ incident with $v_0$ and let $\beta$ be the edge of $T$ incident with $v_2$. We can assume that $\alpha\neq\beta$, otherwise we are done. Draw a curve $\kappa$ from $v_2$ very close to $\beta$ all the way in $G$, and then in the subblocks $C'_1, C'_2,\ldots, C'_{k-1}$. In the last subblock, $C'_k$, connect $\kappa$ to $v_0$ so that it crosses all edges at most once in $C'_k$. A straightforward but slightly technical argument shows that it is possible. For example, we can draw the cables in $C'_k$ as in Figure~\ref{figure_saturated_jump}, and then draw $\kappa$ as the shortest line with positive slope. Repeat this procedure for all pairs $v_0\in V_0$, $v_2\in V_2$. The resulting topological graph is still $k$-simple. We can add similarly all edges between vertices $v_1\in V_1$ and $v_3\in V_3$. We can connect all the remaining pairs, $v_i\in V_i$ and $v_j\in V_j$, $|i-j|\le 1$, in a similar, but simpler way. We obtain a $(k, \lceil 3k/2\rceil)$-saturated topological graph. \qed \begin{figure} \caption{Adding edges to the subblock $C_k$.} \label{figure_saturated_jump} \end{figure} \section{Proof of Theorem~\ref{pseudoline}} Let $\alpha$ and $\beta$ be two simple closed curves in the plane. Suppose that $\alpha$ contains $\beta$ in its interior. The region between $\alpha$ and $\beta$ is called an {\em annulus}. It is homeomorphic to the cylindrical surface, so we can transform {\em blocks\/} onto the annulus. The region outside $\alpha$ is called the {\em outer exterior\/} of the annulus. Similarly, the region inside $\beta$ is called the {\em inner exterior\/} of the annulus. It is enough to prove the following statement; Theorem~\ref{pseudoline} easily follows. \begin{theorem}\label{19} \begin{enumerate} \item[(i)] For $m\ge 2$ and $k=4m$, there is a $(3,5k/4-5)$-forcing $k$-pseudoline arrangement. \item[(ii)] For $m\ge 3$ and $k=2^m$, there is a $(k,(k/2-2)\cdot (\log_2k +1))$-forcing $k$-pseudoline arrangement. \end{enumerate} \end{theorem} \begin{figure} \caption{The arrangement from the proof of Theorem~\ref{19} (i).} \label{figure_k_pseudorays} \end{figure} \begin{proof} (i) Refer to Figure~\ref{figure_k_pseudorays}. First we construct an arrangement of one-way infinite curves. Let $x_1$, $x_2$ and $x_3$ be three distinct points in the plane. Let $A_1$, $A_2$ and $A_3$ be three disjoint annuli such that they contain each other in their outer exteriors, and for $i=1, 2, 3$, $A_i$ contains $x_i$ in its inner exterior. Let $B$ be an annulus that contains both $A_1$ and $A_2$ in its inner exterior, and $A_3$ in its outer exterior. Finally, let $C$ be an annulus that contains both $A_3$ and $B$ in its inner exterior. Now we define six one-way infinite curves, $\gamma_i^j$, for $i=1, 2, 3$, $j=1, 2$. For any fixed $i$, $i=1, 2, 3$, let $\gamma_i^1$ and $\gamma_i^2$ start very close to $x_i$ and form a $k/4$-spiral in $A_i$. In the outer exterior of $A_i$, let $\gamma_i^1$ and $\gamma_i^2$ form a cable $\gamma_i$. Let $\gamma_1$ and $\gamma_2$ form a $k/4$-spiral in $B$. In the outer exterior of $B$, let $\gamma_1$ and $\gamma_2$ form a cable $\gamma$. Finally, let $\gamma$ and $\gamma_3$ form a $k/4$-spiral in $C$. In the outer exterior of $C$ all six curves go to infinity. Now replace each one-way infinite curve $\gamma_i^j$ by two one-way infinite curves with the same endpoint, so that they go very close to each other. Each of these pairs of curves form a bi-infinite curve $\Gamma_i^j$, and any two intersect at most $k$ times. For the rest of the proof we call them {\em pseudolines}. Let $\rho$ be a bi-infinite curve containing $x_1$, $x_2$ and $x_3$. Then $\rho$ contains at least two transversals of each of $A_1$, $A_2$, $A_3$, $B$ and $C$. This means, by Observation~\ref{obs_spiral}, that in each of $A_1$, $A_2$ and $A_3$, there is a pseudoline, say, $\Gamma_1^1$, $\Gamma_2^1$ and $\Gamma_3^1$, respectively, that crosses $\rho$ at least $k/2-2$ times. Moreover, $\rho$ crosses one of the two cables in $B$ at least $k/4-1$ times, which implies that $\rho$ crosses one of the pseudolines $\Gamma_1^1$ or $\Gamma_2^1$, say, $\Gamma_1^1$, at least $k/2-2$ times in $B$. Finally, $\rho$ crosses the two cables in $C$ together at least $k/2-2$ times. Hence, in $C$, the curve $\rho$ has at least $k/8 - 1/2$ crossings with $\gamma$, or at least $3(k/8 - 1/2)$ crossings with $\gamma_3$. In the first case $\rho$ crosses $\Gamma_1^1$ at least $5k/4-5$ times, in the second case it crosses $\Gamma_3^1$ at least $5k/4-5$ times. (ii) For the second part of the theorem, we iterate the construction from the proof of part (i) $m$ times. Let $P(k,0)$ be the following arrangement. Take a point $x$ in the plane and an annulus $A$ around it (that is, $x$ is in the inner exterior of $A$). Let $\gamma^1$ and $\gamma^2$ be two one-way infinite curves, both starting near $x$ and forming a $k/4$-spiral in $A$. Suppose that we have already defined an arrangement $P(k,i)$ containing $2^i$ points and $2^{i+1}$ one-way infinite curves. Take two disjoint copies of $P(k,i)$, and an annulus $B$ that contains all annuli of both copies in its internal exterior. Merge all curves of each copy of $P(k,i)$ into a cable and let the two cables form a $k/4$-spiral in $B$. The resulting arrangement is $P(k,i+1)$. Once the arrangement $P(k,m)$ is constructed, take two copies of each curve in $P(k,m)$ and join their endpoints to form a bi-infinite curve, thus obtaining a $k$-pseudoline arrangement $P'(k,m)$. Let $X_m$ be the set of $2^m$ points in the centers of the innermost annuli of $P'(k,m)$. By induction, every bi-infinite curve containing all the points of $X_m$ crosses some pseudoline of $P'(k,m)$ at least $(m+1)(k/2-2)$ times. \end{proof} \section{Concluding Remarks} Our lower bound in Theorem~\ref{linearbound} for $k>1$ is weaker than for $k=1$. The reason is that for $k>1$, we could not prove that a saturated $k$-simple topological graph cannot contain an isolated triangle. The main difficulty is that for $k>1$, a triangle can cross itself, and our proof for Lemma~\ref{triangle} does not work in this case. \begin{problem} \begin{enumerate} \item[(i)] Is there a saturated $k$-simple topological graph, for some $k\ge 2$, that contains an isolated triangle? \item[(ii)] Is there a {\em disconnected\/} saturated $k$-simple topological graph, for some $k$? \end{enumerate} \end{problem} Problem 1 (ii) is open for every $k\ge 1$. It follows from Theorem~\ref{klconstruction} (ii) that there is no $(k,l)$-saturated graph with $l>2k$. By Theorem~\ref{3k/2saturated}, there is a $(k,l)$-saturated graph if $l\le \lceil 3k/2\rceil$. \begin{problem} Is there a $(k,l)$-saturated graph with $k\ge 2$ and $l>\lceil 3k/2\rceil$? \end{problem} In Theorem~\ref{pseudoline} we have shown that for sufficiently large $k$, there is a $(3, k+1)$-forcing arrangement of $k$-pseudolines. On the other hand, it is easy to see that there are no $(1, k+1)$-forcing arrangements of $k$-pseudolines. \begin{problem} Is there a $(2, k+1)$-forcing arrangement of $k$-pseudolines for some $k\ge 3$? \end{problem} We assumed that in a $k$-simple topological graph, no edge can cross itself. For any $k$, a graph drawn in the plane is called a {\em $k$-complicated topological graph\/} if any two edges have at most $k$ points in common, and an edge is allowed to cross itself, at most $k$ times. Somewhat surprisingly, for saturated $k$-complicated topological graphs we cannot even prove Lemma~\ref{isol+deg1} part (ii). We can only prove that a saturated $k$-complicated topological graph does not have isolated vertices. Therefore, the best lower bound we have for the minimum number of edges of a saturated $k$-complicated topological graph is $c_k(n)\ge n/2$. On the other hand, for $k\ge 6$, using self-crossings, we can improve our upper bound constructions from the proof of Theorem~\ref{linearbound} to obtain that $c_k(n)\le 5n/2$. We sketch the construction here. Suppose that $n$ is even, $k\ge 6$, and let $m=n/2$. The construction consists of $2m+1$ consecutive blocks, $A_0,B_1,\allowbreak A_1,\allowbreak B_2,\allowbreak \ldots,\allowbreak B_{m},\allowbreak A_{m}$, in this order, from bottom to top. For every $i$, $1\le i\le m$, let $u_i$ be a vertex on the lower boundary of $A_{i-1}$ and let $v_i$ be a vertex on the lower boundary of $B_i$. Let $\alpha_i$ be an edge joining $u_i$ and $v_i$. The block $B_i$ is a $k$-spiral, and both of its cables are formed by $\alpha_i$. For $2\le i\le m$, the block $A_i$ is a $3$-spiral, one cable is formed by $\alpha_i$, the other one is formed by a folded curve $\alpha_{i-1}$ (that is, two intervals of $\alpha_{i-1}$). Any curve $\kappa$ from $A_{i-2}$ to $A_{i}$ has to cross $\alpha_{i-1}$ at least $k-1$ times in $B_{i-1}$, and $\alpha_i$ at least $k-1$ times in $B_i$. In $A_{i-1}$, the curve $\kappa$ also crosses one of the curves $\alpha_i$ or $\alpha_{i-1}$ at least twice, since $\alpha_{i-1}$ is folded in $A_{i-1}$. It follows that when we extend this graph to a saturated $k$-complicated topological graph, each vertex has degree at most five. Note that using $\lfloor k/2\rfloor$-spirals in place of the $3$-spirals, we obtain a $(k,l)$-saturated $k$-complicated topological graph with $l=5k/3 - O(1)$. \end{document}

\begin{document} \input amssym.def \input amssym.tex \title{Natural Internal Forcing Schemata extending ZFC. A Crack in the Armor surrounding $CH?$} \author{Garvin Melles\thanks{Would like to thank Ehud Hrushovski for supporting him with funds from NSF Grant DMS 8959511} \\Hebrew University of Jerusalem} \newtheorem{theorem}{Theorem}[section] \newtheorem{defi}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{coro}[theorem]{Corollary} \newtheorem{conj}{Conjecture} \newcommand{{\sc proof} \hspace{0.1in}}{{\sc proof} \hspace{0.1in}} \newcommand{\iopp}{\stackrel{|}{\smile}} \newcommand{\niopp}{\not\!\!{\stackrel{|}{\smile}}} \newcommand{\noindep}[1]{\mathop{\niopp}\limits_{\textstyle{#1}}} \newcommand{\indep}[1]{\mathop{\iopp}\limits_{\textstyle{#1}}} \newcommand{\subseteq}{\subseteq} \mathsurround=.1cm \maketitle Mathematicians are one over on the physicists in that they already have a unified theory of mathematics, namely set theory. Unfortunately the plethora of independence results since the invention of forcing has taken away some of the luster of set theory in the eyes of many mathematicians. Will man's knowledge of mathematical truth be forever limited to those theorems derivable from the standard axioms of set theory, $ZFC?$ This author does not think so, and in fact he feels there is a schema concerning non-constructible sets which is a very natural candidate for being considered as part of the axioms of set theory. To understand the motivation why, let us take a very short look back at the history of the development of mathematics. Mathematics began with the study of mathematical objects very physical and concrete in nature and has progressed to the study of things completely imaginary and abstract. Most mathematicians now accept these objects as as mathematically legitimate as any of their more concrete counterparts. It is enough that these objects are consistently imaginable, i.e., exist in the world of set theory. Applying the same intuition to set theory itself, we should accept as sets as many that we can whose existence are consistent with $ZFC.$ Of course this is only a vague notion, but knowledge of set theory so far, namely of the existence of $L$ provides a good starting point. What sets can we consistently imagine beyond $L?$ Since by forcing one can prove the consistency of $ZFC$ with the existence of non-constructible sets and as $L$ is absolute, with these forcing extensions of $L$ you have consistently imagined more sets in a way which satisfies the vague notion mentioned above. The problem is which forcing extensions should you consider as part of the universe? But there is no problem, because if you prove the consistency of the existence of some $L$ generic subset of a partially ordered set $P\in L$ with $ZFC,$ then $P$ must be describable and we can easily prove the consistency of $ZFC$ with the existence of $L$ generic subsets of $P$ for every $P$ definible in $L.$ Namely, the axiom schema $IFS_L$ (For internal forcing schema over $L\!$) defined below is consistent with $ZFC.$ \begin{defi} $IFS_L$ is the axiom schema which says for every formula $\phi(x),$ if $L\models$ there is a unique partial order $P$ such that $\phi(P),$ then there is a $L$ generic subset of $P$ in the universe $V.$ \end{defi} \noindent $IFS_L$ is a natural closure condition on a universe of set theory. Given a class model of $ZFC$ which has no inner class model of the form $L[G]$ for some partial order $P$ definable in $L,$ we can (by forcing) consistently imagine expanding the model to include such a class. Conversely, no class model of $ZFC+IFS_L$ can be contained in a class model of $ZFC$ which does not satisfy $IFS_L.$ \begin{theorem} If there is a sequence $\langle M_n\mid n<\omega\rangle$ of transitive models with $M_n\models ZFC_n$ where $ZFC=\bigcup\limits_{n\in\omega}ZFC_n,$ then $Con(ZFC+IFS_L)$ \end{theorem} {\sc proof} \hspace{0.1in} By the compactness theorem and forcing. \begin{theorem}\label{IFSLCBLG} If $V$ is a model of $ZFC,$ then $V\models IFS_L$ if and only if $V\models$ every set definable in $L$ is countable. \end{theorem} {\sc proof} \hspace{0.1in} Certainly if every set definable in $L$ is countable, then every partially ordered set definable in $L$ is countable, so therefore is the set of dense subsets of $P$ in $L$ countable and so $P$ has generic subsets over $L$ in the universe. In the other direction, if $s$ is a set definable in $L,$ then so is the partially ordered set consisting of maps from distinct finite subsets of $s$ to distinct finite subsets of $\omega,$ so a $L$ generic subset over the partial ordering is a witness to $\big|s\big|=\omega.$ \noindent Perhaps $IFS_L$ is not surprising since $ZFC+0^{\#}\ \vdash\ IFS_L.$ But the same reasoning as led to $IFS_L$ leads to the following stronger schema, $IFS_{Ab\,L[r]}$ (For internal forcing schema for absolute class models of $ZFC$ constructible over an absolutely definable real) which implies that if $0^{\#}$ exists, then all sets definable in $L[0^{\#}]$ are countable. \begin{defi} A subset $r$ of $\omega$ is said to be absolutely definable if for some $\Pi_1$ formula $\theta(x),$ \begin{enumerate} \item $V\models \theta(r)$ \item $ZFC\vdash \exists\,x\theta(x)\ \rightarrow\ \exists!\,x\theta(x)$ \end{enumerate} \end{defi} \begin{defi} $IFS_{Ab\,L[r]}$ is the axiom schema of set theory which says if $r$ is an absolutely definable real then all definable elements of $L[r]$ are countable (equivalently, every partial order $P$ definable in $L[r]$ has an $L[r]$ generic subset.) \end{defi} \noindent The following theorem is a formal justification of $IFS_{Ab\,L[r]}.$ \begin{theorem}\label{CONIFSr} Suppose $V$ is a countable transitive model of $ZFC$ and let $\big\{\theta_i(x)\mid i<\omega\big\}$ be the list of all formulas defining absolute reals such that $V\models \bigwedge\limits_{i<\omega}\exists\,x\theta_i(x).$ Suppose that the supremum of the ordinals definable in $V$ is in $V.$ Then there is a countable transitive extension $V'$ of $V$ with the same ordinals such that $$V'\models ZFC+IFS_{Ab\,L[r]}+\bigwedge\limits_{i<\omega}\exists\,x\theta_i(x)$$ \end{theorem} {\sc proof} \hspace{0.1in} Let $\alpha^*$ be the sup of all the ordinals definable in $L.$ Let $P$ be the set of finite partial one to one functions from $\alpha^*$ to $\omega.$ Let $V'=V[G]$ where $G$ is a $V$ generic subset of $P.$ To finish the proof it is enough to prove the following claim. \noindent Claim: If $\psi(x)$ defines a real in $M[G]$ then it is in $M.$\\ {\sc proof} \hspace{0.1in} Since $P$ is separative, if $p\in P$ and $\pi$ is an automorphism of $P,$ then for every formula $\varphi(v_1,\ldots,v_n)$ and names $x_1,\ldots,x_n$ $$*\ \ \ \ p\Vdash \varphi(x_1,\ldots,x_n)\ \hbox{ iff }\ \pi p\Vdash \varphi(\pi x_1,\ldots,\pi x_n)$$ Let $\varphi(x)=\exists\,Y(\psi(Y)\ \wedge\ x\in Y).$ Let $n\in \omega.$ If for no $p\in P$ does $p\Vdash \big|\big|\varphi(\check n)\big|\big|$ then $\big|\big|\varphi(\check n)\big|\big|=0.$ So let $p\in P$ such that $p\Vdash \big|\big|\varphi(\check n)\big|\big|.$ By $*$ if $\pi$ is an automorphism of $P$ then $\pi p\Vdash \big|\big|\varphi(\check n)\big|\big|.$ Let $\pi$ be a permutation of $\omega.$ $\pi$ induces a permutation of $P$ by letting for $p\in P,$ $dom\,\pi p=dom\,p$ and letting $\pi p(\alpha)=\pi(p(\alpha)).$ By letting $\pi$ vary over the permutations of $\omega$ it follows that $\big|\big|\varphi(\check n)\big|\big|=1.$ Let $\dot r$ be the name with domain $\big\{\check n\mid n<\omega\big\}$ and such that $$\dot r(\check n)=\big|\big|\varphi(\check n)\big|\big|$$ $i_G(\dot r)=r,$ but then $r=\big\{n\mid \big|\big|\varphi(\check n)\big|\big|=1\big\}$ which means it is in $M.$ \begin{coro} $ZFC+IFS_{Ab\,L[r]}+$'there are no absolutely definable non-constructible reals' is consistent. (Relative to the assumption of a countable transitive model of $L$ with its definable ordinals having a supremum in the model.) \end{coro} \noindent Since classes of the form $L[r]$ are absolute if $r$ is an absolutely definable real, they provide reference points from which to measure the size of the universe. We can extend the schema $IFS_{Ab\,L[r]}$ by exploiting the similarity between a class such as $L({\Bbb R})$ and a class of the form $L[r]$ where $r$ is an absolutely definable real. We can argue that if $P$ is a partial order definable in $L({\Bbb R}),$ and if a $V$ generic subset of $P$ cannot add any reals to $V,$ then an $L({\Bbb R})$ generic subset of $P$ should exist in $V.$ $L({\Bbb R})$ is concrete in the sense the interpretation of $L({\Bbb R})$ is absolute in any class model containing ${\Bbb R}$, and thereby like classes of the form $L[r]$ where $r$ is an absolutely definable real, $L({\Bbb R})$ provides a reference point from which to measure the size of the universe. This leads to the following natural strengthening of $IFS_L$ and $IFS_{Ab\,L[r]}.$ \begin{defi} $x\in V$ is said to be weakly absolutely definable of the form $V_{\alpha}$ if for some formula $\psi(v)$ which provably defines an ordinal and which is provably $\Delta_1$ from $ZF,$ $$V\models \exists!\alpha\Big(\psi(\alpha)\ \wedge\ \forall y(y\in x\ \leftrightarrow\ \rho(x)\leq\alpha)\Big)$$ Let $\theta(x)$ denote $\exists!\alpha\Big(\psi(\alpha)\ \wedge\ \forall y(y\in x\ \leftrightarrow\ \rho(x)\leq\alpha)\Big)$ and let $ZF_{\theta}$ be a finite part of $ZF$ which proves $\psi(v)$ is $\Delta_1$ and proves $\psi(v)$ defines an ordinal. $\theta(x)$ is said to define a weakly absolutely definable set of the form $V_{\alpha}.$ ($\rho(x)$ denotes the foundation rank.) \end{defi} \begin{defi} $IFS_{W\!Ab\,L(V_{\alpha})}$ is the axiom schema of set theory which says for every weakly absolutely definable set of the form $V_{\alpha}$ for every partial order $P$ definable in $L(V_{\alpha}),$ if $$\big|\big|V_{\alpha}^{V[G]}=V_{\alpha}^{V}\big|\big|^{(r.o.P)^V}=1$$ then there exists an $L(V_{\alpha})$ generic subset $G$ of $P.$ \end{defi} \begin{theorem} If there is a sequence $\langle M_n\mid n<\omega\rangle$ of transitive models with $M_n\models ZFC_n$ where $ZFC=\bigcup\limits_{n\in\omega}ZFC_n$ then $Con(ZFC+IFS_{W\!Ab\,L(V_{\alpha})})$ \end{theorem} {\sc proof} \hspace{0.1in} Let $\langle\theta_i\mid i<n\rangle$ be a list of formulas defining weakly absolute sets of the form $V_{\alpha}.$ Let $\big\{\varphi_{ij}(x)\mid i<n, j<m,\big\}$ be a set of formulas. It is enough to show the consistency with $ZFC$ of $$\bigwedge\limits_{i<n,j<m}\exists V_{\alpha_i}\Big[\theta_i(V_{\alpha_i}) \ \wedge\ \exists! P_{ij}(L(V_{\alpha})\models \varphi_{ij}(P_{ij}))\ \longrightarrow$$ $$\exists G(G\subseteq P_{ij}\ \wedge\ G\hbox{ is }L(V_{\alpha_i})\hbox{ generic}\Big]$$ Let $M$ be a countable transitive model of enough of $ZFC$ (including $\mathop{\wedge}\limits_{i<n}ZF_{\theta_i}.\!$) Let $\langle\alpha_0,\ldots,\alpha_{n-1}\rangle$ be the increasing sequence of ordinals such that $$M\models \theta_i(V_{\alpha_i})$$ for $i<n.$ We define by induction on $(i,j)\in n\times m$ sets $G_{ij}.$ Suppose $P_{ij}$ is a partial order definable in $L(V_{\alpha_i}^{M[\{G_{h,l}|h\leq i,l<j\}]})$ by $\varphi_{ij}(x)$ and there exists a $M[\{G_{h,l}|h\leq i,l<j\}]$ generic subset of $P_{ij}$ not increasing $$V_{\alpha_i}^{M[\{G_{h,l}|h\leq i,l<j\}]}$$ Then let $G_{ij}$ be such a $M[\{G_{h,l}|h\leq i,l<j\}]$ subset of $P_{ij}.$ (If not, let $G_{ij}=\emptyset.$) Let $$N=M[\{G_{ij}|i<n,j<m\}]$$ $N$ has the property that if $P_{ij}$ is a partial order definable in $L(V_{\alpha_i})$ by $\varphi_{ij}(x)$ and $G$ is an $N$ generic subset of $P_{ij}$ such that $$\big|\big|V_{\alpha_i}^{N[G]}=V_{\alpha_i}^N\big|\big|^{(r.o.P)^N}=1$$ then an $L(V_{\alpha_i}^{N})$ generic subset of $P_{ij}$ exists in $N.$ \begin{theorem} If $\langle M_n\mid n<\omega\rangle$ is a sequence of transitive models with $M_n\models ZFC_n$ where $ZFC=\bigcup\limits_{n\in\omega}ZFC_n,$ then $Con(ZFC+IFS_{W\!Ab\,L[V_{\alpha}]}+IFS_{Ab\,L[r]})$ \end{theorem} {\sc proof} \hspace{0.1in} Same as the last theorem except we start with a model of enough of $ZFC+IFS_{Ab\,L[r]}.$ \begin{theorem}\label{Jech} $V[G]$ has no functions $f:\kappa\rightarrow\kappa$ not in the ground model if and only if $r.o.P$ is $(\kappa,\kappa)\!$-distributive. \end{theorem} {\sc proof} \hspace{0.1in} See [Jech1]. \begin{coro} $IFS_{W\!Ab\,L(V_{\alpha})}$ is equivalent to the axiom schema of set theory which says for every weakly absolutely definable set of the form $V_{\alpha},$ for every partial order $P$ definable in $L(V_{\alpha}),$ if $$(r.o.P)^V \hbox{ is }(\kappa,\kappa)\hbox{-distributive}$$ for each $\kappa$ such that for some $\beta<\alpha,\ \kappa\leq \big|V_{\beta}\big|,$ then there exists an $L(V_{\alpha})$ generic subset $G$ of $P.$ \end{coro} \begin{theorem} $ZFC+IFS_{W\!Ab\ L(V_{\alpha})}\vdash\ CH$ \end{theorem} {\sc proof} \hspace{0.1in} Let $P=$ the set of bijections from countable ordinals into ${\Bbb R}.$ Since $P$ is $\sigma$ closed, $\omega_1=\omega_1^{L({\Bbb R})},$ and $P$ is a definable element of $L({\Bbb R}),$ there is an $L({\Bbb R})$ generic subset of $P$ in $V.$ If $\alpha$ is an ordinal less than $\omega_1$ and $r$ is a real, let $D_{\alpha}=\big\{p\in P\mid \alpha\in dom\,p\big\}$ and $D_r=\big\{p\in P\mid r\in ran\,p\big\}.$ For each $\alpha<\omega_1,$ $G\cap D_{\alpha}\neq\emptyset$ and for each $r\in {\Bbb R},$ $G\cap D_r\neq\emptyset,$ so $\bigcup G$ is a bijection from $\omega_1$ to ${\Bbb R}.$ Perhaps the following is a better illustration of the kind of result obtainable from $ZFC+IFS_{W\!Ab\ L(V_{\alpha})}.$ \begin{defi} A Ramsey ultrafilter on $\omega$ is an Ultrafilter on $\omega$ such that every coloring of $\omega$ with two colors has a homogenous set in the ultrafilter. \end{defi} \begin{theorem} $ZFC+IFS_{W\!Ab\ L(V_{\alpha})}\vdash $ there is a Ramsey ultrafilter on $\omega.$ \end{theorem} {\sc proof} \hspace{0.1in} Let $P$ be the partial order $(P(\omega),\subseteq^*)$ where $P(\omega)$ is the power set of $\omega$ and $a\subseteq^* b$ means $a$ is a subset of $b$ except for finitely many elements. $P$ is definable is $L({\Bbb R})$ and is $\omega$ closed. The generic object is an Ramsey ultrafilter over $L({\Bbb R}),$ and since all colorings of $\omega$ are in $L({\Bbb R}),$ it is a Ramsey ultrafilter over $V.$ \noindent One can argue that $IFS_{W\!Ab\ L(V_{\alpha})}$ is not a natural axiom since among the definable sets $X$ with the property that $L(X)$ is absolute when not increasing $X,$ why should you choose only those of the form $V_{\alpha}?$ But it is natural in the sense it is a way of forcing the universe as large as possible with respect to the existence of generics by first fixing the height of the models under consideration and then by fixing more and more of their widths. In any case we should consider the strengthenings of $IFS_{W\!Ab\,L(V_{\alpha})}$ defined below. \begin{defi} $x\in V$ is said to be weakly absolutely definable if for some formula $\psi(x)$ which is provably $\Delta_1$ from $ZF,$ $$V\models \forall y(y\in x\ \leftrightarrow\ \psi(y))$$ \end{defi} \begin{defi} $IFS$ is the axiom schema of set theory which says for every weakly absolutely definable set $X,$ for every partial order $P$ definable in $L(X),$ if $$\big|\big|X^{V[G]}=X^{V}\big|\big|^{(r.o.P)^V}=1$$ then there exists an $L(X)$ generic subset $G$ of $P.$ \end{defi} \noindent If $X$ is an weakly absolutely definable set and $P$ is a partial ordering definable in $L(X)$ such that $$\big|\big|X^{V[G]}=X^{V}\big|\big|^{(r.o.P)^V}=1$$ and if there is no $L(X)$ generic subset of $P$ in $V,$ we say that $V$ has a gap. $IFS$ says there are no gaps. The intuition that such gaps should not occur in $V$ leads to the following: \begin{conj} $ZFC+IFS$ is consistent. \end{conj} \noindent If $ZFC+IFS$ is consistent, then this means that it is consistent that the universe is complete with respect to the natural yardstick classes, (the classes of the form $L(X)$ where $X$ is weakly absolutely definable.) In my view, confirming the consistency of $ZFC+IFS$ would be strong evidence that the universe of set theory conforms to the axioms of $IFS.$ One reason for this opinion is that there is no apriori reason for the consistency of $ZFC+IFS,$ so if $ZFC+IFS$ is consistent, it seems that confirmimg its consistency would involve some deep mathematics implying $IFS$ should be taken seriously. \section{Formalizing the arguments in favor of $IFS_L$ and the other schemata} \noindent In this section we try to formalize the vague notion that $IFS_L$ is a natural closure condition on the universe, and that gaps in general are esthetically undesirable. For simplicity we concentrate on $IFS_L.$ \begin{defi} Let $T$ be a recursive theory in the language of set theory extending $ZFC.$ Let $P$ be a unary predicate. If $\varphi$ is a formula of set theory then $\varphi^*$ is $\varphi$ with all its quantifiers restricted to $P,$ i.e., if $\exists x$ occurs in $\varphi$ then it is replaced by $\exists x(P(x)\,\wedge\ldots)$ and $\forall x$ is replaced by $\forall x(P(x)\ \rightarrow\ldots).$ The theory majorizing $T,$ $T',$ is the recursive theory in the language $\big\{\varepsilon, P(x)\big\}$ such that \begin{enumerate} \item $\varphi\in T\ \rightarrow\ \varphi^*\in T'$ \item $P(x)\ is\ transitive\ \in T'$ \item $\forall x(x\in ORD\ \rightarrow\ P(x))\in T'$ \item $ZFC\subseteq T'$ \end{enumerate} \noindent If $\theta(x)=\forall y(y\in x\leftrightarrow \psi(x))$ is a formula defining a weakly absolutely definable set then the theory majorizing $T$ with respect to $\theta(x)$ is $T'$ plus all the axioms of the form $$\Big(\varphi_1\ \wedge\ \ldots\ \wedge\ \varphi_n\ \rightarrow\ \big(\psi(y)\leftrightarrow \exists z\psi_0(y,z)\leftrightarrow\forall z\psi_1(y,z)\big)\Big)\ \longrightarrow\ \Big(\forall y(\psi(y)\ \rightarrow\ P(y))\Big)$$ where $\varphi_1,\ldots,\varphi_n\in ZF$ and $\psi_0(y,z)$ and $\psi_1(y,z)$ are $\Delta_0$ formulas. \end{defi} \begin{theorem} Let $T$ be a recursive extension of $ZFC.$ Let $T=\bigcup\limits_{n\in\omega}T_n$ where for some recursive function $F,$ for each $n,$ $F(n)=T_n,$ a finite subset of $T$ and the $T_n$ are increasing. If there is a sequence $\langle M_n\mid n<\omega\rangle$ of countable transitive models such that $$M_n\models T_n$$ then $T'+IFS_L$ ($T'$ is the theory majorizing $T\!$) is consistent and there is a sequence $\langle N_n\mid n\in\omega\rangle$ of countable transitive models such that $$N_n\models T_n'$$ where $T'=\bigcup\limits_{n\in\omega}T'_n$ and for some recursive function $H,$ for each $n\in\omega,$ $H(n)=T'_n$ a finite subset of $T'.$ \end{theorem} {\sc proof} \hspace{0.1in} Let $IFS_L=\bigcup\limits_{n\in\omega}(IFS_L)_n$ where for each $n\in\omega,\ (IFS_L)_n$ is finite. We can find a subsequence $\langle N_n\mid n\in\omega\rangle$ of the $\langle M_n\mid n\in\omega\rangle$ and $N_n$-generic sets $G_n$ such that $N_n[G_n]\models (IFS_L)_n,\ N_n\models T_n.$ Let $N_n[G_n]^*$ be the model in the language $\big\{\varepsilon,P(x)\big\}$ obtained by letting the interpretation of $P(x)$ to be $N_n.$ Let $D$ be an ultrafilter on $\omega.$ Then $$\prod N_n[G_n]^*/D$$ is a model for $T'+IFS_L.$ \begin{defi} A theory extending $ZFC$ is $\omega-\!$complete if whenever $\varphi(x)$ is a formula of set theory and if for each natural number $n,$ $$T\vdash\varphi(n)$$ then $T\vdash \forall n\in\omega\varphi(n).$ \end{defi} \begin{theorem} Let $T$ be a recursive extension of $ZFC$ and suppose it has a consistent, complete and $\omega-\!$complete extension $T^*.$ Then $T'+IFS_L$ is consistent. \end{theorem} {\sc proof} \hspace{0.1in} By reflection in $T^*,$ by its $\omega-\!$completeness and by the axiom of choice in $T^*,$ $$T^*\vdash\exists\langle N_n\mid n\in\omega\rangle$$ with the $\langle N_n\mid n\in\omega\rangle$ having the same properties as in the previous theorem. As in the previous theorem since $ZFC\subset T,$ $T^*\vdash Con(T'+IFS_L).$ Since $T^*$ is $\omega-\!$complete, (by the omitting types theorem) it has an model $M$ with the standard set of integers. Since $M\models T^*,$ $$M\models Con(T'+IFS_L)$$ and as $Con(T'+IFS_L)$ is an arithmetical statement, it must really be true. \noindent Certainly if the hypothesis of the theorem fails, then $T$ cannot be a suitable axiom system for set theory. \begin{defi} If $\theta(x)$ is a formula defining an weakly absolutely definable set, then $IFS\restriction\theta(x)$ is $IFS$ restricted to the set defined by $\theta(x),$ i.e., it says for all partial orders $P$ definable in $L(X)$ were $X$ is defined by $\theta(x)$ such that $$\big|\big|X^{V[G]}=X^{V}\big|\big|^{(r.o.P)^V}=1$$ there is an $L(X)$ generic subset of $P.$ \end{defi} \begin{theorem} Let $\theta(x)$ be a formula defining an weakly absolutely definable set. Let $T$ be a recursive extension of $ZFC$ and suppose it has a consistent, complete and $\omega-\!$complete extension $T^*.$ Then $T_{\theta(x)}'+IFS\restriction\theta(x)$ is consistent. \end{theorem} {\sc proof} \hspace{0.1in} Same as above. \begin{theorem} Let $\theta(x)$ be a formula defining an weakly absolutely definable set. Let $T$ be a recursive extension of $ZFC+IFS\restriction\theta(x)$ and suppose $T'_{\theta(x)}$ majorizes $T$ with respect to $\theta(x).$ Then $T_{\theta(x)}'\vdash IFS\restriction\theta(x).$ \end{theorem} {\sc proof} \hspace{0.1in} Working in $T'$ the generics in the inner model are still generic over $L(X)$ since the inner model is a transitive class containing all the ordinals. \noindent The theorems in this section are meant as the formalization of the notion that we can 'consistently imagine' a class model of $ZFC$ not satisfying $IFS_L$ as being contained in a larger class satisfying $ZFC+IFS_L,$ and that models of $ZFC$ not satisfying $IFS$ have a gap. \section{Conclusion} \noindent These axiom schemata lead to many questions, among them \begin{enumerate} \item Are there models of $IFS$ or $IFS_{W\!Ab\,L[V_{\alpha}]}$ which are forcing extensions of $L$? \item Are there similar natural schema's making the universe large, but contradicting $IFS$ or $IFS_{W\!Ab\,L[V_{\alpha}]}?$ \item What are the consequences for ordinary mathematics of these axioms? \end{enumerate} \noindent The conventional view of the history of set theory says that Godel in 1938 proved that the consistency of $ZF$ implies the consistency of $ZFC$ and of $ZFC+GCH,$ and that Cohen with the invention of forcing proved that $Con(ZF)$ implies $Con(ZF+\neg AC)$ and $Con(ZFC+\neg GCH)$ but from the point of view of $IFS_L$ a better way to state the history would be to say that Godel discovered $L$ and Cohen proved there are many generic sets over $L.$ I think confirming the consistency of $IFS$ with $ZFC$ would be a vindication of the idea that generics over partial orders definable in $L(X)$ with $X$ an weakly absolutely definable set exist, and thereby put a crack in the armor surrounding the continuim hypothesis as $ZFC+IFS\restriction{\Bbb R}\vdash CH.$ On the other hand, if $ZFC+IFS$ is not consistent, it would show the universe must have some gaps, i.e., incomplete with respect to some concrete set, an esthetically unpleasing result. It is ironic that although mathematics and especially mathematical logic is an art noted for its precise and formalized reasoning, it seems that in order to solve problems at the frontiers of logic's foundations we must tackle questions of an esthetic nature. \pagebreak \begin{center} REFERENCES \end{center} \noindent 1. [CK] C. C. Chang and J. Keisler, {\em Model Theory}, North Holland Publishing Co. \noindent 2. [Jech1] T. Jech, {\em Multiple Forcing}, Cambridge University Press. \noindent 3. [Jech2] T. Jech, {\em Set Theory}, Academic Press. \end{document}

\begin{document} \title[Pontryagin space structure in RKHS's]{Pontryagin space structure in reproducing kernel Hilbert spaces over $*$--semigroups} \author[F.H. Szafraniec, M. Wojtylak]{ Franciszek Hugon Szafraniec \and Micha\l{} Wojtylak } \address{F. H. Szafraniec, Instytut Matematyki, Uniwersytet Jagiello\'nski, ul. \L ojasiewicza 6, 30 348 Krak\'ow, Poland M. Wojtylak, Instytut Matematyki, Uniwersytet Jagiello\'nski, ul. \L ojasiewicza 6, 30 348 Krak\'ow, Poland\\ VU University Amsterdam, Department of Mathematics, Faculty of Exact Sciences, De Boelelaan 1081 a, 1081 HV Amsterdam\\ } \email{[email protected]} \email{[email protected]} \thanks{The first author was supported by the MNiSzW grant N201 026 32/1350. He also would like to acknowledge an assistance of the EU Sixth Framework Programme for the Transfer of Knowledge ``Operator theory methods for differential equations'' (TODEQ) \# MTKD-CT-2005-030042.} \subjclass[2000]{primary: 43A35 \and 46C20 \and 47B32} \keywords{$*$-semigroup \and shift operator \and Pontryagin space \and fundamental symmetry} \begin{abstract} The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of $*$-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries. \end{abstract} \maketitle \section*{Introduction} There are two ways of looking at $*$--semigroups and positive definite functions defined on them. The first consists in intense analysis of the algebraic structure of a semigroup so as to establish conditions on it, which ensure prescribed properties to hold for \underbar{any} positive definite functions. One of the properties frequently considered is representing \underbar{each} of the positive definite functions as moments of a positive measure. This attitude has been successfully undertaken by Bisgaard resulting in a considerable number of papers, see in particular \cite{biss-scmat,biss-two,biss-sep,biss-ext,biss-CC,biss-fact} and references therein, some of them we are going to exploit here. The other way is to determine \underbar{which} of the positive definite functions posses desired properties; a typical example within this category is to detect multidimensional moment sequences. In the present paper we are going to develop the first thread putting forward the following problem. Impose necessary and sufficient conditions on a distinguished element $u$ of a $*$--semigroup $S$ to generate a Pontryagin fundamental symmetry of \underbar{any} reproducing kernel Hilbert space over the semigroup in question; the precise formulation is exposed as ({\tt P}), p. \pageref{a}. Surprisingly, our problem has found a simple algebraic solution. In the context of $*$-separative commutative semigroups the condition on $u$ is: $u=u^*$, $u+u=0$ and $u+s\neq s$ for only a finite number of $s\in S$ (see Proposition \ref{krein}, Theorem \ref{main}). Let us mention that $*$-semigroups and positive definite functions on them have been originated by Sz.-Nagy in his famous Appendix \cite{nagy}. He uses the reproducing kernel Hilbert space factorization to prove his "general dilation theorem" for operator valued functions. Since then the RKHS technique has been used from time to time for proving results with dilation flavour behind the screen. We are going to follow suit here. \section{Shift operators connected with positive definite functions -- formulation of the problem} By a $*$-semigroup we understand a commutative semigroup with an involution, not necessarily having the neutral element. The involution is always denoted by the symbol ``$*$'' and the semigroup operation is always written in an additive way. In the case when the $*$--semigroup $S$ has a neutral element $0$ we say that $\phi:S\to\mathbb{C}$ is {\it positive definite} (we write $\phi\in\Pp S$) if for every $N\in\mathbb{N}:=\set{0,1,\dots}$ and every $s_0,\dots, s_N\in S$, $\xi_0,\dots, \xi_N\in\mathbb{C}$ we have $\sum_{i,j=0}^N\xi_i\bar\xi_j \phi(s_j^*+s_i)\geq 0$. With such $\phi$ we link a reproducing kernel Hilbert space $\mathcal{H}^\phi\subseteq \mathbb{C}^S$ with a reproducing kernel defined by $K^\phi(s,t):=\phi(t^*+s)$, $t,s\in S$ (see e.g. ~\cite[p.81]{BChR}). For $s\in S$ we set $K^\phi_s:=K(\cdot,s)$ ($s\in S$), it is known that $K^\phi_s\in\mathcal{H}^\phi$ and $f(s)=\seq{f,K^\phi_s}$ for every $f\in\mathcal{H}^\phi$. The set $\mathcal{D}^\phi:=\textrm{lin}\set{K^\phi_s:s\in S}$ is dense in $\mathcal{H}^\phi$. For an element $u\in S$ we define {\it the shift operator} (sometimes called also the translation operator) $ A(u,\phi):\mathcal{D}^\phi\to\mathcal{D}^\phi$, by $$ A(u,\phi)\left(\sum_j\xi_jK_{s_j}^\phi\right):=\sum_j\xi_jK_{s_j+u}^\phi. $$ It can be shown that $A( u,\phi)$ is well defined, linear and closable (see e.g. \cite[Proposition, p.253]{szaf-bdd}, \cite[p.90]{BChR}). Our main object will be the closure of the above operator, denoted below by $u_\phi$. It is a matter of simple verification that $\mathcal{D}^\phi$ is contained in the domain of $(u_\phi)^*$ (the adjoint of the operator $u_\phi$) and $(u_\phi)^*f=(u^*)_\phi f$ for $f\in\mathcal{D}^\phi$. Under the above circumstances we can state our problem as follows. \begin{enumerate} \label{a} \item[({\tt P})] Provide necessary and sufficient conditions on the element $u\in S$ for the operator $u_\phi$ to be a fundamental symmetry of a Pontryagin space, i.e. to satisfy $$ u_\phi=u_\phi^*,\quad u_\phi^2=I_{\mathcal{H}^\phi},\quad \dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty $$ \underbar{for every} $\phi\in\Pp S$. \end{enumerate} Obviously, the condition $u_\phi=u_\phi^*$ together with $u_\phi^2=I_{\mathcal{H}^\phi}$ imply that $u_\phi$ must be bounded on $\mathcal{H}$. We continue with some basic definitions and notations concerning $*$-semigroups. Let $S$ and $T$ be $*$-semigroups, a mapping $\chi:S\to T$ is called $*$-{\it homomorphism} if $\chi(s+t)=\chi(s)+\chi(t)$ for all $s,t\in S$, $\chi(s^*)=(\chi(s))^*$ for all $s\in S$. A {\it character} on $S$ is a nonzero $*$-homomorphism $\chi:S\to\mathbb{C}$ where the latter set is understood as a semigroup with multiplication as the operation and the conjugation as involution. It is obvious that if $S$ has a neutral element $0$ then $\chi(0)=1$ for all $\chi\in S^*$. Let $\mathcal{A}(S^*)$ be the least $\sigma$-algebra of subsets of $S^*$ rendering measurable all the functions \begin{equation}\label{shat} \hat s:S^*\ni\chi\mapsto\chi(s)\in\mathbb{C},\quad s\in S. \end{equation} If $\mu$ is a positive measure on $S^*$ such that all the functions listed in (\ref{shat}) are square-integrable we define a function $\mathcal{L}(\mu):S\to\mathbb{C}$ by $\mathcal{L}(\mu)(s):=\int_{S^*}\sigma(s)d\mu(\sigma)$ ($s\in S$). We call $\phi:S\to\mathbb{C}$ a {\it moment function} ($\phi\in\M S $) if $\phi=\mathcal{L}(\mu)$ for some measure $\mu$ on $S^*$. It is easy to verify that $\M S \subseteq\Pp S$, if the latter inclusion is an equality then we call $S$ {\it semiperfect}. For examples of semiperfect and non-semiperfect $*$-semigroups and more general concepts of semiperfectness see ~\cite{biss-scmat}. We call $S$ {\it $*$-separative} if the characters separate points in $S$. The {\it greatest $*$-homomorphic $*$-separative image of $S$} is the semigroup $S/_\sim$, where the equivalence relation $\sim$ on $S$ is defined by the condition that $s\sim t$ if and only if $\sigma(s)=\sigma(t)$ for all $\sigma\in S^*$; addition and involution in $S/_\sim$ are those that make the quotient mapping a $*$-homomorphism. The elements of $S/_\sim$ will be denoted as equivalence classes $[s]$ ($s\in S$). If $S$ has a zero, then we will use the symbol $0$ for the neutral element of both $S$ and $S/_\sim$, instead of using $[0]$. Since every character on $S$ generates a character on $S/_\sim$, the latter semigroup is in fact $*$-separative. The following simple proposition gives answer to the question when the operator $ u_\phi$ defined above is a fundamental symmetry of a Krein space, i.e. when $( u_\phi)^2= u_\phi$ and $ u_\phi=( u_\phi)^*$. \begin{prop}\label{krein} Let $S$ be a commutative $*$-semigroup with zero. For each element $u\in S$ the following conditions are equivalent: \begin{itemize} \item[(i)]{$[2u]= 0$ and $[u]=[u^*]$;} \item[(ii)]{for every $\phi\in\M S $ we have $(u_\phi)^2= I_{\mathcal{H}^\phi}$ and $(u_\phi)^*= u_\phi$;} \item[(iii)]{for every $\phi\in\Pp S$ we have $(u_\phi)^2 = I_{\mathcal{H}^\phi}$ and $(u_\phi)^*= u_\phi$;} \end{itemize} Moreover, {\rm (i)} implies that $ u_\phi$ is a bounded, selfadjoint operator on $\mathcal{H}^\phi$ for every $\phi\in\Pp S$. \end{prop} \begin{proof} (i)$ \Rightarrow$(iii) Suppose that (i) holds and let $\phi\in\Pp S$. For every $\sigma\in S^*$ we have $\sigma(u)=\sigma(u^*)$ and $\sigma(2u)=\sigma(0)=1$. Consequently, for every $\sigma\in S^*$ and every $s\in S$ \begin{equation}\label{sigma2} \sigma(s+u)=\sigma(s+u^*),\quad\sigma(t+2u)=\sigma(t),\quad \sigma(s^*+u^*+u+s)=\sigma(s^*+s). \end{equation} By \cite[Thm.2]{biss-sep} we have \begin{equation}\label{sigma3-1} \phi(s+u)=\phi(s+u^*),\quad s\in S, \end{equation} \begin{equation}\label{sigma3-2} \phi(s+2u)=\phi(s),\quad s\in S, \end{equation} \begin{equation}\label{sigma3-3} \phi(s^*+u^*+u+s)=\phi(s^*+s),\quad s\in S, \end{equation} The last of these three equalities, together with \cite[Cor.1]{szaf-bdd}, implies that the operator $u_\phi$ is in $\bold{B}(\mathcal{H}^\phi)$. It is also selfadjoint, since for $f=\sum_i \xi_i K_{s_i}^\phi\in\mathcal{D}^\phi$ we have $$ \seq{u_\phi f,f}= \sum_{i,j}\xi_i\bar\xi_j\seq{K_{s_i+u}^\phi,K_{s_j}^\phi}=\sum_{i,j}\xi_i\bar\xi_j\phi(s_i+u+s_j^*) \stackrel{(\ref{sigma3-1})}= $$ $$ \sum_{i,j}\xi_i\bar\xi_j\phi(s_i+u^*+s_j^*)= \seq{f, u_\phi f}. $$ The fact that $(u_\phi)^2=I_\mathcal{H}^\phi$ can be obtained similarly as selfadjointness of $u_\phi$, with the use of (\ref{sigma3-2}) instead of (\ref{sigma3-1}). This finishes the proof of (iii) and of the `Moreover' part of the proposition. The implication (iii)$ \Rightarrow$(ii) is trivial. The proof (ii)$ \Rightarrow$(i) goes by contraposition. Suppose first that $[2u]\neq 0$, i.e. there exists a character $\sigma$ such that $\sigma(u)^2=\sigma(2u)\neq\sigma(0)=1$. We put $\phi:=\mathcal{L}(\delta_\sigma)$, where $\delta_\sigma$ stands for the Dirack measure on $S^*$ concentrated in $\sigma$. Let $\seq{\,\cdot\,,-}$ denote the scalar product on $\mathcal{H}^\phi$. Observe that $$ \seq{K_0^\phi,K_0^\phi}=\phi(0)=\sigma(0)\neq\sigma(2u)=\phi(2u)=\seq{K_{2u}^\phi,K_0^\phi}=\seq{( u_\phi)^2K_0^\phi,K_0^\phi}. $$ In consequence, $I_{\mathcal{H}^\phi}\neq ( u_\phi)^2$. Similarly, if $\tau(u)\neq\tau(u^*)$ for some $\tau\in S^*$ then for $\psi:=\mathcal{L}(\delta_\tau)$ the operator $u_\psi$ is not symmetric in $\mathcal{H}^\psi$. \end{proof} Let us consider now a situation when $S$ and $T$ are $*$-semigroups with zeros and $h$ is a $*$-homomorphism from $S$ into $T$ satisfying $h(0)=0$. Note that if an element $u\in S$ is such that $[2u]=0$, $[u]=[u^*]$ then $[2h(u)]=0$, $[h(u)]=[h(u)^*]$. This comes from the fact that for every character $\sigma$ on $T$ the function $\sigma\circ h$ is a character on $S$. Observe also that $\phi\circ h\in\Pp S$ for every $\phi\in\Pp T$. \begin{prop}\label{ST} Assume that $S$, $T$ and $h$ are as above and that $h$ is additionally onto. Let $u\in S$ be such that $[2u]=0$, $[u]=[u^*]$ and let $\phi\in\Pp T$. Then the operators $u_{\phi\circ h}$ in $\mathcal{H}^{\phi\circ h}$ and $h(u)_{\phi}$ in $\mathcal{H}^\phi$ are unitarily equivalent. \end{prop} \begin{proof} Let $\seq{\cdot,-}_\phi$ and $\seq{\cdot,-}_{\phi\circ h}$ denote the scalar products on $\mathcal{H}^\phi$ and $\mathcal{H}^{\phi\circ h}$ respectively. Since \begin{eqnarray*} \seq{\sum_{j=1}^N \xi_j K^{\phi\circ h} _{s_j} ,\sum_{j=1}^N \xi_j K^{\phi\circ h} _{s_j}}_{\!\!\phi\circ h} &=& \sum_{i,j=1}^N \xi_i\bar\xi_j (\phi\circ h)(s_i+s_j^*) \\ = \sum_{i,j=1}^N \xi_i\bar\xi_j \phi(h(s_i)+h(s_j^*))&=& \seq{\sum_{j=1}^N \xi_j K^{\phi} _{h(s_j)},\sum_{j=1}^N \xi_j K^{\phi} _{h(s_j)}}_{\!\!\phi}, \end{eqnarray*} the condition $ V( K^{\phi\circ h}_{s}):= K^{\phi}_{h(s)} $ ($s\in S$) properly defines an isometry between $\mathcal{H}^{\phi\circ h}$ and $\mathcal{H}^{\phi}$. Since $h$ is onto, the range of $V$ is dense in $\mathcal{H}^\phi$, and so $V$ is a unitary operator. To finish the proof we need to show that \begin{equation}\label{VuuV} h(u)_\phi Vf=V u_{\phi\circ h}f, \quad f\in\mathcal{H}^{\phi\circ h}. \end{equation} This can be easily verified for $f\in\mathcal{D}^{\phi\circ h}$. Since all the operators appearing in (\ref{VuuV}) are bounded, the proof is finished. \end{proof} Applying the above to the quotient semigroup $T=S/_\sim$ and $h$ as the quotient mapping, together with the fact from \cite{biss-fact} that every $\phi\in\Pp S$ ($\phi\in\M S$) is of the form $\phi=\psi\circ h$ for some $\psi\in\Pp{S/_\sim}$ ($\psi\in\MS/_\sim$) gives the following. \begin{cor}\label{Nowyjeszcze} Assume that $S$ is a $*$-semigroup with zero and $u\in S$ is such that $[2u]=0$, $[u]=[u^*]$. Then \begin{eqnarray*} \dim\ker(u_\phi+I)&<&+\infty \textrm{ for every }\phi\in\Pp S\,\, (\phi\in\M S)\\ \iff \dim\ker([u]_{\psi}+I)&<&+\infty\textrm{ for every }\psi\in\PpS/_\sim\,\, (\psi\in\MS/_\sim) \end{eqnarray*} \end{cor} \section{Examples} \begin{exm}\label{exm1} Consider the semigroup $S={\mathbb{Z}}_2\times\mathbb{N}$ with standard addition and the identical involution. As usually (cf. \cite{BChR}) we identify $S^*$ with $\set{-1,1}\times\mathbb{R}$, note that $S$ is $*$-separative. The only nonzero element satisfying $u=u^*$, $2u=0$ is $u=(1,0)$. Let $\phi$ be a positive definite mapping, we will now compute the eigenspaces of $ u_\phi$. Since $S$ is semiperfect (\cite{biss-two}), there exists a Borel measure $\mu$ on $S^*$ such that $\phi=\mathcal{L}(\mu)$. Having our interpretation of characters in mind we get $$ \phi(x,n)=\int_\mathbb{R} (-1)^xt^nd\mu_-(t)+\int_\mathbb{R} t^nd\mu_+ (t),\quad x\in{\mathbb{Z}}_2,\,n\in\mathbb{N}, $$ with $\mu_\pm:=\mu\rest{{\set{\pm1}}\times\mathbb{R}}$. Let us define the functions $f_{x,n}:S^*\to S^*$ ($x\in{\mathbb{Z}}_2$, $n\in\mathbb{N}$) by $$ f_{x,n}(\varepsilon,t):=\varepsilon^xt^x ,\quad \varepsilon\in\set{-1,1},\,t\in\mathbb{R},\,x\in{\mathbb{Z}}_2,\,n\in\mathbb{N}, $$ and note that they all are square integrable. By $\mathcal P^\mu$ we define the closure in $L^2(\mu)$ of the linear span of the functions $f_{x,n}$ ($x\in{\mathbb{Z}}_2$, $n\in\mathbb{N}$). The formula \begin{equation}\label{Vdef} V(K^\phi_{x,n}):=f_{x,n},\qquad x\in{\mathbb{Z}}_2,\,n\in\mathbb{N}, \end{equation} constitutes a unitary isomorphism between $\mathcal{H}^\phi$ and $\mathcal P^\mu$. The shift operator $u_\phi$ is unitary equivalent (via $V$) to the following operator $M$ $$ (M f)(\varepsilon,t):=\varepsilon f (\varepsilon,t),\qquad (\varepsilon,t)\in\set{-1,1}\times\mathbb{R},\, f\in\mathcal{P}^\mu. $$ It is not hard to see that $$ \ker(M\pm I)=\set{f\in \mathcal{P}^\mu: f(\pm1,\cdot)=0\,\,\, \, \mu_\pm\textrm{--a.e}}. $$ Note that $\dim\ker(u_\phi\pm I)=\dim\ker(M\pm I)$ and the latter is finite dimensional if and only if the support of $\mu_{\mp}$ is a finite set. In particular there exists a mapping $\phi\in\Pp S=\M S $ such that $\dim\ker( u_\phi+I)=\infty$. \end{exm} At this point we present a useful for our purposes construction. Let $S$ and $T$ be two disjoint $*$-semigroups (which only a formal restriction) and let $h:S\mapsto T$ be a $*$-homomorphism. We endow the set $S\cup T$ with the $*$-semigroup structure in the following way. The addition on $S\cup T$, denoted by the same symbol `$+$', is defined by $$ s+t:=\left\{\begin{array}{rcl} s+t & : & s,t\in S\textrm{ or } s,t\in T\\ s+h(t) & : & s\in S,\, t\in T\\ t+h(s) & : & t\in S,\, s\in T\\ \end{array}\right. $$ The involution on $S\cup T$ (still denoted by `$*$') is such that its restriction to both $S$ and $T$ is the original involution on $S$ and $T$, respectively. We denote the above constructed semigroup by $\mathrm{U}(S,T,h)$. The reader can easily check a general fact, that if $S$ and $T$ are $*$-separative then $\mathrm{U}(S,T,h)$ is $*$-separative as well. \begin{exm}\label{exm2} Consider a semigroup $S=\mathrm{U}({\mathbb{Z}}_2,\mathbb{N},h_0)$ where $h_0(x)=0$ ($x\in {\mathbb{Z}}_2$). The element $0_{{\mathbb{Z}}_2}$ is the neutral element of $S$. Take $u:=1_{{\mathbb{Z}}_2}$, clearly $u=u^*$ and $2u=0_{{\mathbb{Z}}_2}$. Let $\phi$ be any positive definite function on $S$ and suppose that $f\in\ker( u_\phi+I)$. This means that for $n\in\mathbb{N}$ $$ f(n)=f(n+u)=\seq{f,K^\phi_{n+u}}=\seq{f,u_\phi K^\phi_n}=\seq{u_\phi f, K^\phi_n}=-f(n), $$ hence $f\rest\mathbb{N}=0$. A similar calculation shows that $f(u)=-f(0_{{\mathbb{Z}}_2})$. Hence, the eigenspace $\ker( e_\phi+I)$ is spanned by the single function $$ f(s)=\left\{\begin{array}{rcl} 0 &:& s\in\mathbb{N}\\ 1 & : & s=0_{{\mathbb{Z}}_2}\\ -1 & : & s=1_{{\mathbb{Z}}_2} \end{array} \right. $$ if $f\in\mathcal{H}^\phi$ or is trivial otherwise. Resuming, $\dim\ker(u_\phi+I)\leq 1$ for all positive definite $\phi$. \end{exm} \section{Main result} \begin{thm}\label{main} Let $S$ be a commutative $*$-semigroup with zero and let $u\in S$ be such that $[2u]=0$ and $[u]=[u^*]$. Then the following conditions are equivalent: \begin{itemize} \item[(i)] the set $\set{[s]\in S/_\sim\colon [u+s]\neq[s]}$ is finite\,\footnote{\ Cf. Remark \ref{stupidstyle}.}; \item[(ii)]{ $\sigma(u)=-1$ for only finitely many $\sigma\in S^*$;} \item[(iii)]{$\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\M S $;} \item[(iv)]{$\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\Pp S$.} \end{itemize} Moreover, if {\rm (i)} holds then $\dim\ker(u_\phi+I_{\mathcal{H}^\phi})$ is less or equal to the half of the number elements of the set mentioned in {\rm (i)}. \end{thm} Let us stress here the solution of the main problem of the paper, which follows from Proposition~\ref{krein} and Theorem~\ref{main} : {\it the operator $ u_\phi$ is a fundamental symmetry of a Pontryagin space for every $\phi\in\Pp S$ $(\!\!$ equivalently: $\phi\in\M S$$)$ if and only if $[2u]=0$, $[u]=[u^*]$ and the set $\set{[s]\inS/_\sim:[u+s]\neq[s]}$ is finite.} \begin{rem}\label{stupidstyle} Condition (ii) is equivalent to \begin{itemize} \item[(ii')]{\it $\sigma([u])=-1$ for only finitely many characters $\sigma$ on $S/_\sim$,} \end{itemize} since the characters on $S$ and $S/_\sim$ are in one-to-one correspondence. This, together with Corollary \ref{Nowyjeszcze} (and the remarks above it) allows us to reduce the proof of Theorem \ref{main} to the case when $S$ is $*$-separative. However, note that the conditions (i)--(iv) are \underbar{not} equivalent to the following: \begin{itemize} \item[]{\it the set $\set{s\in S: u+s\neq s}$ is finite.} \end{itemize} Indeed, consider the following example. Let $S={\mathbb{Z}}_4\times\mathbb{N}$ with the natural operation $+$ and the identical involution. The greatest $*$-separative homomorphic image of $S$ is ($*$-isomorphic with) ${\mathbb{Z}}_2\times\mathbb{N}$ and the quotient homomorphism maps $u:=(2,0)$ to $(0,0)$. We have that $u+s\neq s$ for all $s\in S$ but the set mentioned in (i) is empty. It remains on open problem if the condition \begin{itemize} \item[]{\it the set $\set{s\in S: [u+s]\neq [s]}$ is finite.} \end{itemize} is equivalent to (i). \end{rem} Before the proof we introduce the notion of a $*$-archimedean component of a semigroup. We call a $*$-semigroup $H$ (not necessarily with 0) {\it $*$-archimedean} if for all $s,t\in H$ there exists $m\in\mathbb{N}\setminus\set0$ such that $m(s+s^*)\in t+ H$. An {\it $*$-archimedean component} of a $*$-semigroup $S$ is a maximal $*$-archimedean $*$-subsemigroup. Though $*$-archimedean component is $*$-semigroup for itself it is possible for it not to have the neutral element even if $S$ does. It can be shown that two elements $s,t$ belong to the same $*$-archimedean component of $S$ if and only if $m(s+s^*)\in t+S$ and $n(t+t^*)\in s+S$ for some $m,n\in\mathbb{N}\setminus\set0$. Furthermore, $S$ is the disjoint union of its $*$-archimedean components, see \cite[Section 4.3]{clipres} for the case of identical involution. The following Lemma was proven in \cite{biss-ext} (Lemma 2), the proof for an arbitrary involution requires minimal effort. \begin{lem}\label{extchar} If $H$ is a $*$-archimedean component of a $*$-semigroup $S$ then every character on $H$ is everywhere nonzero and extends to a character on $S$. \end{lem} If $H$ and $K$ are two $*$-archimedean components of $S$ then $H+K$ is contained in one single $*$-archimedean component of $S$. If $(S_i)_{i\in I}$ is the family of all $*$-archimedean components of $S$ then we define the operation $+$ on $I$ by: $$ i+j=k\textrm{ if and only if } S_i+S_j\subseteq S_k,\quad i,j,k\in I. $$ Since $S_i+S_i\subseteq S_i$ for all $i\in I$ we have that $i+i=i$. Therefore $I$ is a semilattice with the natural partial order given by the condition that $i\leq j$ if and only if $i+j=j$. The following easy lemma is left as an exercise for the reader. \begin{lem}\label{elat} Let $S$ be a $*$-semigroup with zero and let $(S_i)_{i\in I}$ be the family of all $*$-archimedean components $S$. If $u\in S$ be such that $2u=0$, then $u$ belongs to the same $*$-archimedean component as 0. In particular, $u+S_i\subseteq S_i$ for all $i\in I$. \end{lem} \begin{proof}[Proof of Theorem ~\ref{main}] As it was said in Remark \ref{stupidstyle} we may assume that $S$ is $*$-separative. To prove (i)$ \Rightarrow$(iv) let us put $$ U:=\set{ s\in S: u+ s= s} $$ and suppose that the set $ S\setminus U$ contains only a finite number $M$ of elements. Take $\phi\in\Pp S$. We show that $$ \dim\ker( u_\phi+I)\leq M/2, $$ this will also prove the last statement of Theorem \ref{main}. Let us fix an arbitrary $f\in\ker( u_\phi+I)$. We have $$ f( s+ u)=\seq{f,u_\phi K_{s}^\phi}=\seq{u_\phi f,K_s^\phi}=-f( s),\quad s\in S. $$ This means that $f\rest{ U}\equiv 0$. Observe that the relation $$ aRb \Longleftrightarrow (a=u+b\textrm{ or }a=b) $$ is an equivalence relation on $ S\setminus U$ and that the each equivalence class contains exactly two elements. Take any representees $s_1,\dots, s_{M/2}$ of the equivalence classes of $R$. It is clear, that $$ \ker( u_\phi+I)\subseteq\textrm{lin}\set{\delta_{s_i}-\delta_{s_i+u}:i=1,\dots, M/2}. $$ Consequently $\dim\ker( u_\phi+I)\leq M/2$.\\ (iv)$ \Rightarrow$(iii) is obvious. (iii)$ \Rightarrow$(ii) Suppose that $\set{\sigma_n:n\in\mathbb{N}}$ is an infinite set of characters satisfying $\sigma_n(u)=-1$ ($n\in\mathbb{N}$). We define a measure $\mu$ on $S^*$ by $\mu:=\sum_{n=0}^\infty 2^{-n} \delta_{\sigma_n}$ and we take a mapping $\phi=\mathcal{L}(\mu)$. Note that for every $N\in\mathbb{N}$ and for every $s_0,\dots, s_N\in S$, $\xi_0,\dots,\xi_N\in\mathbb{C}$ we have $$ \bigg|\sum_{j=0}^N\xi_j\sigma_n(s_j)\bigg|^2\leq 2^n\int_{S^*}\bigg|\sum_{j=0}^N\xi_j\sigma(s_j)\bigg|^2d\mu(\sigma)= 2^n\sum_{i,j=0}^N\xi_i\bar\xi_j\int_{S^*}\sigma(s_i+s_j^*)d\mu(\sigma)= $$ $$ =2^n\sum_{i,j=0}^N\xi_i\bar\xi_j\phi(s_i+s_j^*),\quad n\in\mathbb{N}. $$ By the RKHS Test (\cite{test} and also \cite{test2}) we get that $\sigma_n\in\mathcal{H}^\phi$ for $n=1,2,\dots$. Now observe that $$ u_\phi(\sigma_n)(s)=\sigma_n(u+s)=-\sigma_n(s),\quad s\in S,n=1,2\dots. $$ Therefore $\sigma_n\in\ker( u_\phi+I)$, $n=1,2,\dots$. It remains to show that the functions $\sigma_n$, $n\in\mathbb{N}$, are linearly independent. But this results from the well known fact that all characters are linearly independent\,\footnote{\ We can use the following argument: For every $s\in S$ the function $\hat s$ is a character on the dual semigroup $S^*$ and it is trivial that the family $(\hat s)_{s\in S}$ separates elements of $S^*$. Proposition 2 of \cite{lacunae} (see also \cite[Proposition 6.1.8]{BChR}) says that if $T$ is a semigroup and $C\subseteq T^*$ separates points, then the functions $\hat t\rest C$, $t\in T$ are linearly independent in $\mathbb{C}^C$. We use this result for $T=S^*$ and $C=\set{\hat s:s\in S}$, the functions $\hat\sigma\rest C$ can be identified with characters on $S$.}.\\ (ii)$ \Rightarrow$(i) Suppose that the number of elements $ s\in S$ satisfying $ u+ s\neq s$ is infinite. We show that there exists infinitely many characters $\sigma$ on $ S$ such that $\sigma( u)=-1$. Let $( S_i)_{i\in I}$ be the family of all $*$-archimedean components of $ S$. Set $$ I_0:=\set{i\in I: u+ s\neq s\textrm{ for some } s\in S_i}. $$ Our assumption implies that\,\footnote{\ Remark \ref{eitheror} shows that it is even equivalent to} either \begin{equation}\label{Case1} \textrm{ $ S_j$ is infinite for some $j\in I_0$} \end{equation} or \begin{equation}\label{Case2} I_0\textrm{ is infinite}. \end{equation} Let us first assume (\ref{Case1}). Take $s_0\in S_j$ such that $u+s_0\neq s_0$. By Lemma \ref{elat} we have $ u+ s_0\in S_j$. The $*$-semigroup $S_j$ is $*$-separative as a subsemigroup of a $*$-separative semigroup $S$. Therefore, there exists a character $\sigma_0$ on $ S_j$ such that $\sigma_0( u+ s_0)\neq\sigma_0( s_0). $ By Lemma \ref{extchar} $\sigma_0$ extends to some character $\tilde{\sigma_0}$ on $ S$. Since $ u= u^*$ and $2 u=0$ we have $\tilde{\sigma_0}( u)\in\set{-1,1}$. But $$ \tilde{\sigma_0}( u)\sigma_0( s_0)=\sigma_0( u+ s_0)\neq\sigma_0( s_0). $$ Hence, $\tilde{\sigma_0}( u)=-1$, which means that $\sigma_0( u+ s_0)=-\sigma_0( s_0)$. Denote by $A$ the set of all those characters $\sigma$ on $ S_j$ satisfying $\sigma( u+ s_0)=-\sigma( s_0)$. Since $\sigma_0$ is everywhere nonzero on $ S_j$ (Lemma \ref{extchar}), the mapping $$ S_j^*\ni\sigma\longmapsto \sigma_0\sigma\in S_j^* $$ is bijective. Moreover, it maps $A$ onto $ S_j^*\setminus A$. Since $ S_j$ is infinite and $*$-separative, $ S_j^*$ is infinite as well. Hence, $A$ is infinite. By Lemma \ref{extchar}, there is an infinite number of characters $\sigma$ on $ S$ satisfying $\sigma( u+ s_0)=-\sigma( s_0)$ and consequently $\sigma( u)=-1$.\\ Let us assume now (\ref{Case2}). For each $i\in I_0$ we take a character $\sigma_i$ on $ S$ satisfying $\sigma_{i}( u)=-1$, $\sigma_{i}( s)\neq 0$ for $s\in S_{i}$ (such a character exist by repeating the proof from the previous case). We also define a family of characters $\chi_{i}\in S^*$ (${i\in I_0}$) by $$ \chi_{i}( s)=\left\{\begin{array}{rl} 1 & \textrm{ if } s\in S_j\textrm{ and }j\leq i\\ 0 & \textrm{ otherwise} \end{array}\right.,\qquad s\in S,\,i\in I_0. $$ Finally, we put $\rho_i:=\sigma_i\chi_i$ ($i\in I_0$). By Lemma \ref{elat} we have that $ u$ is in the same $*$-archimedean component as 0, denote this component by $ S_{j_0}$. It is clear that $j_0\leq i$ for all $i\in I$, therefore $\chi_i( u)=1$ and $\rho_i( u)=-1$ for all $i\in I_0$. The only thing that lasts is to show that $\rho_i\neq\rho_j$ for $i\neq j$. If $i\neq j$ then, by symmetry, we can assume that $j\nleq i$. Thus $\chi_i\rest{ S_j}=0$ and $\rho_i\rest{ S_j}=0$. But $\chi_j\rest{ S_j}\equiv1$ and $\sigma_j$ is, by definition, everywhere nonzero on $ S_j$. Therefore $\rho_i\rest{ S_j}\equiv0\neq\rho_j\rest{ S_j}$.\\ \end{proof} \begin{rem}\label{eitheror} The alternative of (\ref{Case1}) and (\ref{Case2}) in the proof of (ii)$ \Rightarrow$(i) becomes more clear if we observe that $ u+ s\neq s$ for \underbar{all} $ s\in S_i$, provided that $i\leq j\in I_0$. Indeed, suppose that $ u+ s= s$ for some $ s\in S_i$, $ u+ s_0\neq s_0$ for some $ s_0\in S_j$ and $i\leq j$. This gives us $$ ( u+ s_0)+( s+ s_0)= u+ s+ s_0+ s_0= s_0+( s+ s_0). $$ By Lemma \ref{elat} we have $ u+ s_0\in S_j$. Moreover, $ s+ s_0\in S_j$ because $i\leq j$. The semigroup $ S_j$ is cancellative as a $*$-archimedean component of a $*$-separable group (see \cite[p.63]{biss-CC}). This gives us $ u+ s_0= s_0$, contradiction. The example below shows that both (\ref{Case1}) and (\ref{Case2}) are possible. \end{rem} \begin{exm} Let $S={\mathbb{Z}}_2\times\mathbb{N}$, with the natural addition on ${\mathbb{Z}}_2$ and $\mathbb{N}$ and the identical involution. The element $u=(1,0)$ is like in (\ref{Case1}). Let us now consider the semigroup $T={\mathbb{Z}}_2\times\mathbb{N}$, with the natural addition on ${\mathbb{Z}}_2$ and maximum as the operation on $\mathbb{N}$, the involution is again set to identity. It is easy to see that $T$ is $*$--separable. The element $u=(1,0)$ is such that (\ref{Case2}) is satisfied. This example shows one more thing. Namely, the condition \begin{itemize} \item[]{ \it $\dim\ker(u_\phi+I_{\mathcal{H}^\phi})<\infty$ for all $\phi\in\M S $ of compact support} \end{itemize} is \underbar{not} equivalent to any of the conditions of Theorem \ref{main}. Indeed, the characters on $T$ form a discrete, enumerable set. If the mapping $\phi\in\M S $ is compactly supported then it is finitely supported and consequently the space $\mathcal{H}^\phi$ is finite dimensional. Hence, $u=(1,0)$ satisfies the above condition, but does not satisfy (i). Note that in Proposition \ref{krein} restricting to compactly supported moment functions is possible because the function $\phi$ constructed in the proof (ii)$ \Rightarrow$(i) is supported by only one character. \end{exm} \section{Functions with a finite number of negative squares} The condition $\dim\ker( u_\phi+I_{\mathcal{H}^\phi})<\infty$ can be written also in the language of negative squares. Precisely speaking, by {\it the number of negative squares of a mapping } $\psi:S\to \mathbb{C}$ satisfying \begin{equation}\label{symmetric} \psi(s)=\overline{\psi(s^*)}, \qquad s\in S, \end{equation} we understand the maximum, taken over all numbers $N\in\mathbb{N}$ and all sequences $s_0,\dots, s_N$, of the number of negative eigenvalues of the symmetric matrix $\left(\psi(s_i+s_j^*)\right)_{i,j=0}^N$. Note that if $[u]=[u^*]$ and $\phi\in\Pp S$ then the mapping $\psi:=\phi(\cdot+u)$ satisfies (\ref{symmetric}). Indeed, take any character $\sigma$. Then $\sigma(s+u)=\sigma(s)\sigma(u)=\sigma(s)\sigma(u^*)=\sigma(s+u^*)$. By the result of \cite{biss-sep} we get $\phi(s+u)=\phi(s+u^*)$. Combining this with $\phi(t)=\overline{\phi(t^*)}$ ($t\in S$) \cite[4.1.6]{BChR} proofs the claim. \begin{prop}\label{(v)} Let $S$ be a $*$--semigroup with zero and let $u\in S$ be such that $[2u]=0$ and $[u]=[u^*]$ and let $\phi\in\Pp S$. Then the number of negative squares of the mapping $\phi(\cdot+u)$ equals $\dim\ker(u_\phi+I)$. Consequently, conditions {\rm (i)--(iv)} of Theorem \ref{main} are equivalent to each of the following: \begin{itemize} \item[(v)] the mapping $\phi(\cdot+u)$ has a finite number of negative squares for every $\phi\in\Pp S$; \item[(vi)] the mapping $\phi(\cdot+u)$ has a finite number of negative squares for every $\phi\in\M S $. \end{itemize} \end{prop} \begin{proof} (cf. \cite{berg,langerio,sasvari} for similar arguments) First let us assume that $\dim\ker(u_\phi+I)=m\in\mathbb{N}$. Consider the indefinite inner product space $(\mathcal{H}^\phi, \seq{u_\phi\,\cdot\,,\,\cdot\,})$. Since $\mathcal{D}^\phi$ is dense in $\mathcal{H}^\phi$ we can find elements $s_1,\dots, s_k\in S$ and vectors $\alpha^i=(\alpha^i_1,\dots, \alpha^i_k)\in\mathbb{C}^k$ ($i=1,\dots, m$) such that the elements \begin{equation}\label{fi} f^i:=\sum_{j=1}^k\alpha^i_j K^\phi_{s_j}\qquad (i=1,\dots, m) \end{equation} span an $m$-dimensional negative subspace (\cite[Theorem IX.1.4]{bognar}). Let \begin{equation}\label{A} A:=\left(\phi(s_j+s_{j'}^*+u)\right)_{j,j'=1}^k\in\mathbb{C}^{k\times k}. \end{equation} Note that for $i,l=1,\dots, m$ we have \begin{equation}\label{fitransf} \seq{A \alpha^l,\alpha^i}= \sum_{j,j'=1}^k\alpha^{l}_j\overline{\alpha^{i}_{j'}}K^\phi(s_j+u,s_{j'}) =\seq{u_\phi f^l,f^i}. \end{equation} Hence, the subspace $\textrm{lin}\set{\alpha^1,\dots,\alpha^k}$ is a negative subspace of the indefinite inner product space $(\mathbb{C}^m,\seq{A\cdot,\cdot})$. Since $f^1,\dots, f^m$ are linearly independent, the vectors $\alpha^1,\dots,\alpha^m$ are linearly independent as well. Therefore, the matrix $A$ has at least $m$ negative eigenvalues. Now let us assume that for some choice of $s_1,\dots, s_k\in S$ the matrix $A$ defined as in (\ref{A}) has $m$ negative eigenvalues. Then there exists linearly independent vectors $\alpha^i=(\alpha^i_1,\dots, \alpha^i_k)\in\mathbb{C}^k$ ($i=1,\dots, m$) such that \begin{equation} \seq{A \alpha^l,\alpha^i}=\delta_{il}\lambda_i\qquad i,l=1,\dots, m, \end{equation} with some $\lambda_1,\dots,\lambda_m<0$. We define $f_1,\dots, f_m$ as in (\ref{fi}) (with the new meaning of $s_1,\dots, s_m$). Using the calculation in (\ref{fitransf}) we get that the space $\textrm{lin}\set{f_1,\dots, f_m}$ is a negative subspace of $(\mathcal{H}^\phi, \seq{u_\phi\cdot,\cdot})$. We show now that $f_1,\dots, f_m$ are linearly independent. If $\sum_{i=1}^m \beta_i f_i=0$ for some $\beta_1,\dots,\beta_m\in\mathbb{C}$ then, by (\ref{fitransf}), $$ \seq{A \sum_{i=1}^m\beta_i\alpha^i,\sum_{i=1}^m\beta_i\alpha^i}=\seq{u_\phi \sum_{i=1}^m \beta_i f_i, \sum_{i=1}^m \beta_i f_i}=0. $$ But $A$ is strictly negative on $\textrm{lin}\set{\alpha^1,\dots,\alpha^m}$ and $\alpha_1,\dots,\alpha_m$ are linearly independent. Hence, $\beta_1=\cdots=\beta_m=0$. \end{proof} \section{More examples} The reader can easily check that Theorem ~\ref{main} can be applied to Examples ~\ref{exm1} and~\ref{exm2}. The next example concerns the estimation of the dimension of the eigenspace in Theorem ~\ref{main}. We will show that this dimension can be any number between $0$ and $M/2$, where $M$ is defined as in the proof of Theorem 2. \begin{exm} Let $S={\mathbb{Z}}_2^m$ with identical involution and let $u=(1,0,0\dts0)$. We have $M=2^m$. The dual semigroup $S^*$ can be identified with $\set{-1,1}^m$. There are $2^{m-1}$ characters $\sigma$ on $S$ satisfying $\sigma(u)=-1$ and $2^{m-1}$ characters $\sigma$ satisfying $\sigma(u)=1$. We denote those characters by $\sigma_1,\dots,\sigma_{2^{m-1}}$ and $\rho_1,\dots,\rho_{2^{m-1}}$, respectively. For fixed $k,l\in\set{0,\dots, 2^{m-1}}$ we put\,\footnote{\ We use the convention $\sum_{i=1}^0 a_i:=0$ } $\mu:=\sum_{i=1}^k \delta_{\sigma_i}+\sum_{j=1}^{l}\delta_{\rho_j}$ and $\phi:=\mathcal{L}(\mu)$. Since the support of the measure is consists of $k+l$ points, the space $\mathcal{H}^\phi$ is $k+l$ dimensional. To see this one can use the interpretation of $\mathcal{H}^\phi$ as $\mathcal P^\mu$, as in Example \ref{exm1}. Now let us observe that \begin{equation}\label{sigmaker} \sigma_1,\dots,\sigma_k\in\ker( u_\phi+I),\quad \rho_1,\dots,\rho_{l}\in\ker( u_\phi-I), \end{equation} by the same argument as in the proof of Theorem \ref{main} (iii)$ \Rightarrow$(ii). Since the characters are always linearly independent we get that $\dim\ker( u_\phi+I)\geq k$ and $\dim\ker( u_\phi-I)\geq l$. But the eigenspaces corresponding to $-1$ and $1$ are orthogonal, thus $\dim\ker( u_\phi+I)=k$ and $\dim\ker( u_\phi-I)=l$. Let us put $e=(0,1,0,\dots, 0)$ and take two numbers $l_1\in\set{0,\dots, 2^{m-1}}$ and $l_2\in\set{0,\dots, 2^{m-2}}$. Using the same technique we can construct a mapping $\phi\in\Pp S$ such that $\dim\ker(u_\phi+I)=l_1$ and $\dim\ker(e_\phi+I)=l_2$. \end{exm} In the following example there are three elements satisfying $2u=0$ and $u=u^*$, with three different upper bounds for the dimensions of the kernel. \begin{exm} Let us consider the semigroup $S=\mathrm{U}({\mathbb{Z}}_2^2,{\mathbb{Z}}_2,\pi)$ where $\pi(x,y)=x$ for $x,y\in{\mathbb{Z}}_2$. The involution on $S$ is identity. Note that $(1,0)+s\neq s$ for $s\in S$, but $(0,1)+s\neq s$ only for $s\in{\mathbb{Z}}_2^2$. Hence, the upper bounds for the dimensions of the kernels are three and two, respectively. The dimension of the kernel for $(0,0)$ is obviously zero. \end{exm} \begin{rem} Let us take two $*$-separative semigroups $S$ and $T$, both having neutral elements ($0_S$ and $0_T$ respectively) and a $*$-homomorphism $h:S\to T$ satisfying $h(0_S)=0_T$. Take an element $u\in T$ satisfying $2u=0_T$ and $u=u^*$. The $*$-semigroup $\mathrm{U}(S,T,h)$ has a zero, namely $0_S$. However, the element $u$, understood as an element of $\mathrm{U}(S,T,h)$, does not satisfy the condition $2u=0_{\mathrm{U}(S,T,h)}$. Nevertheless, we have $3u=u$ and $u^*=u$, which by \cite{szaf-bdd} guaranties boundedness (and hence selfadjointness) of $u_\phi$ for any $\phi\in\Pp{\mathrm{U}(S,T,h)}$. The indefinite inner product space $(\mathcal{H}^\phi,\seq{u_\phi\cdot,-})$ is then degenerate, i.e. $u_\phi$ has a non-trivial kernel. \end{rem} We could investigate, instead of positive definite mappings on $S$, the set of positive definite forms on $S$. Namely, we say that $\phi:S\times \mathcal{E}\times \mathcal{E}\to\mathbb{C}$ is a { form over ($S,\mathcal{E}$)} if for every $s\in S$ the mapping $\phi(s,\,\cdot\,,-)$ is a hermitian bilinear form on the linear space $\mathcal{E}$. We say that a form is positive definite if for every finite sequences $(s_k)_k\subseteq S$, $(f_k)_k\subseteq \mathcal{E}$ we have $\sum_{i,j} \phi(s_j^*+s_i;f_i,f_j))\geq 0$. For a positive definite form $\phi$ we can construct a Hilbert space $\mathcal{H}^\phi$ which together with the functions $K_{s,f}^\phi$ ($s\in S$, $f\in\mathcal E$) constitute a RKHS. Like in the case of $\mathcal{E}=\mathbb{C}$, cf. \cite{szaf-bdd} and also \cite{general}, we can define the (closed) shift operator associated with an element $u\in S$ by $u_\phi(K_{s,f}^\phi)=K_{s+u,f}^\phi$. The following example shows, that in this setting the equivalence in Theorem \ref{main} no longer holds. \begin{exm} Let $S={\mathbb{Z}}_2$ (with the identical involution) and let $\mathcal{E}$ be an \underbar{infinite} dimensional {Hilbert} space. Consider the following form $$ \phi(x,f,g)=\left\{\begin{array}{rcl} \seq{f,g}_\mathcal{E} & : & x=0\\ \seq{-f,g}_\mathcal{E} & : & x=1 \end{array}\right. . $$ Note that $$ \sum_{x,y=0,1} \phi(x+y,f_x,f_y)=\seq{f_0,f_0}+\seq{f_1,f_1}-2\Re\seq{f_1,f_0}=\norm{f_1-f_0}^2 $$ which is greater or equal to zero for any choice of $f_0,f_1\in\mathcal{H}$. The space $\mathcal{H}^\phi$ can be realized as $\mathcal{H}^\phi=\mathcal{E}$ so as $K_{0,f}^\phi=f$ and $K_{1,g}=-g$, $f,g\in \mathcal{E}$. Take $u=1$. The condition (i) of Theorem \ref{main} is satisfied because the semigroup is of finite cardinality. On the other hand $u_\phi K_{0,f}^\phi = K_{1,f}^\phi=-f$ for $f\in\mathcal{H}$. Hence, $\dim\ker(u_\phi+I)=\dim\mathcal{H}=\infty$. \end{exm} \section{Final remarks} Our work is connected in a way with many other papers and books. Let us mention some of them. \begin{itemize} \item The transformation $\phi\mapsto\phi(\cdot+u)$ has been investigated by Bisgaard. He showed in \cite{biss-fact} that it is always a sum of four positive definite mappings. \item Functions with finite number of negative spaces on topological groups has been considered in the book of Sasv\'ari \cite{sasvari}. \item In \cite{berg} sequences on $\mathbb{N}$ with a finite number of negative squares are considered. \item In \cite{BChR} the authors consider negative definite sequences, which is a subclass of mappings with a finite number of negative squares. \item Finally, in \cite{biss-cor} definitizing ideals are investigated. \end{itemize} \end{document}

\begin{document} \title{Mellin Analysis and Its Distance Concept Applications to Sampling Theory} \noindent {\small {\bf Abstract:} In this paper a notion of functional ``distance'' in the Mellin transform setting is introduced and a general representation formula is obtained for it. Also, a determination of the distance is given in terms of Lipschitz classes and Mellin-Sobolev spaces. Finally applications to approximate versions of certain basic relations valid for Mellin band-limited functions are studied in details.} \section{Introduction} A Mellin version of the Paley-Wiener theorem of Fourier analysis was introduced in \cite{BBMS}, using both complex and real approaches. Moreover, the structure of the set of Mellin band-limited functions (i.e. functions with compactly supported Mellin transform) was studied. It turns out that a Mellin band-limited function cannot at the same time be Fourier band-limited, and it is extendable as an analytic function to the Riemann surface of the (complex) logarithm. This makes the theory of Mellin band-limited functions very different from the Fourier band-limited ones since one has to extend the notion of the Bernstein spaces in a suitable way, involving Riemann surfaces (Mellin-Bernstein spaces). In the classical frame, Fourier band-limitedness is a very fundamental assumption in order to obtain certain basic formulae such as the Shannon sampling theorem, the Mellin reproducing kernel formula, the Boas differentiation formula, the Bernstein inequality, quadrature formulae and so on. When a function $f$ is no longer (Fourier) band-limited, certain approximate versions of the above formulae are available with a remainder which needs to be estimated in a suitable way. This was done in \cite{BSS1}, \cite{BSS2}, \cite{BSS3} in terms of an appropriate notion of ``distance'' of $f$ from the involved Bernstein space. In the Mellin transform setting an exponential version of the Shannon sampling theorem for Mellin band-limited functions was first introduced in a formal way in \cite{OSP}, \cite{BP} in order to study problems arising in optical physics. A precise mathematical version of the exponential sampling formula, also in the approximate sense, was given in \cite{BJ3}, \cite{BJ2}, employing a rigorous Mellin transform analysis, as developed in \cite{BJ0}, \cite{BJ1} (see also \cite{BBM2}). Furthermore, a Mellin version of the reproducing kernel formula, both for Mellin band-limited funtions and in an approximate sense, was proved in \cite{BBM0}. Therefore it is quite natural to study estimates of the error in the approximate versions of the exponential sampling theorem, the reproducing kernel formula, the Bernstein inequality and the Mellin-Boas differentiation formula using a new notion of ``Mellin distance'' of a function $f$ from a Mellin-Bernstein space. In the present paper, we introduce a notion of distance in the Mellin frame, and we prove certain basic representation theorems for it (Sec.~3). In Sec.~4 we give precise evaluations of the Mellin distance in some fundamental function spaces such as Lipschitz classes and Mellin-Sobolev spaces. In Sec.~5 we describe some important applications to the approximate exponential sampling thoerem, the Mellin reproducing kernel theorem and the Boas differentiation formula in the Mellin setting, employing Mellin derivatives. Moreover, the theory developed here enables one to obtain an interesting approximate version of the Bernstein inequality with an estimation of the remainder. The present approach may also be employed in order to study other basic relations valid for Mellin band-limited functions. \section{Notations and preliminary results} Let $C(\mathbb{R}^+)$\,and $C(\{c\} \times i\mathbb{R})$ be the spaces of all uniformly continuous and bounded functions defined on $\mathbb{R}^+$ and on the line $\{c\} \times i\mathbb{R}, c \in \mathbb{R},$ respectively, endowed with the usual sup-norm $\|\cdot\|_\infty,$ and let $C_0(\mathbb{R}^+)$ be the subspace of $C(\mathbb{R}^+)$ of functions $f$ satisfying $\lim_{x \rightarrow 0^+} f(x) = \lim_{x \rightarrow +\infty}f(x) = 0.$ For $1\leq p < +\infty,$ let $L^p= L^p(\mathbb{R}^+)$~ be the space of all the Lebesgue measurable and $p$-integrable complex-valued functions defined on $\mathbb{R}^+$ endowed with the usual norm $\|\cdot\|_p.$ Analogous notations hold for functions defined on $\mathbb{R}.$ For $p=1$ and $c \in \mathbb{R},$ let us consider the space $$X^1_c = \{ f: \mathbb{R}^+\rightarrow \mathbb{C}: f(\cdot) (\cdot)^{c-1}\in L^1(\mathbb{R}^+) \}$$ endowed with the norm $$ \| f\|_{X^1_c} := \|f(\cdot) (\cdot)^{c-1} \|_1 = \int_0^\infty |f(u)|u^{c-1} du.$$ More generally, let $X^p_c$ denote the space of all functions $f: \mathbb{R}^+\rightarrow \mathbb{C}$ such that $f(\cdot) (\cdot)^{c-1/p}\in L^p(\mathbb{R}^+)$ with $1<p< \infty.$ In an equivalent form, $X^p_c$ is the space of all functions $f$ such that $(\cdot)^c f(\cdot) \in L^p_\mu(\mathbb{R}^+),$ where $L^p_\mu= L^p_\mu(\mathbb{R}^+)$ denotes the Lebesgue space with respect to the (invariant) measure $\mu (A) = \int_A dt/t$ for any measurable set $A \subset \mathbb{R}^+.$ Finally, by $X^\infty_c$ we will denote the space of all functions $f:\mathbb{R}^+ \rightarrow \mathbb{C}$ such that $\|(\cdot) f(\cdot)\|_\infty = \sup_{x>0}|x^cf(x)| < +\infty.$ The Mellin transform of a function $f\in X^1_c$ is defined by (see e.g. \cite{MA}, \cite{BJ1}) $$ M_c[f](s) \equiv [f]^{\wedge}_{M_c} (s) = \int_0^\infty u^{s-1} f(u) du~~~(s=c+ it, t\in \mathbb{R}).$$ Basic properties of the Mellin transform are the following: $$M_c[af + bg](s) = a M_c[f](s) + bM_c[g](s)~~~(f,g \in X^1_c,~a,b \in \mathbb{R}),$$ $$|M_c[f](s)| \leq \|f\|_{X^1_c}~~(s = c+it).$$ The inverse Mellin transform $M^{-1}_c[g]$ of the function $g \in L^1(\{c\} \times i\mathbb{R})$ is defined by $$M^{-1}_c[g](x) := \frac{x^{-c}}{2 \pi}\int_{-\infty}^{+\infty} g(c+it) x^{-it}dt ~~~(x \in \mathbb{R^+}),$$ where by $L^p(\{c\} \times i\mathbb{R})$ for $p \geq 1,$ will mean the space of all functions $g:\{c\} \times i\mathbb{R} \rightarrow \mathbb{C}$ with $g(c +i\cdot) \in L^p(\mathbb{R}).$ For $p=2$ the Mellin transform $M_c^2$ of $f \in X^2_c$ is given by (see \cite{BJ2}) $$M_c^2[f](s) \equiv [f]^{\wedge}_{M_c^2} (s) = \limm_{\rho \rightarrow +\infty}~\int_{1/\rho}^\rho f(u) u^{s-1}du~~~(s=c+it),$$ in the sense that $$\lim_{\rho \rightarrow \infty}\bigg\|M_c^2[f](c+it) - \int_{1/\rho}^\rho f(u) u^{s-1}du\bigg\|_{L^{2}(\{c\}\times i\mathbb{R})} = 0.$$ In this instance, the Mellin transform is norm-preserving in the sense that (see \cite{BJ2}) $$\|g\|_{X^2_c} = \frac{1}{\sqrt{2\pi}}\|[g]^\wedge_{M_c}\|_{L^2(\{c\}\times i\mathbb{R})}.$$ More generally, using the Riesz-Thorin convexity theorem, one may introduce a definition of Mellin transform in $X^p_c$ with $p\in ]1,2[$ in an analogous way, i.e., $$\lim_{\rho \rightarrow \infty}\bigg\|M_c^p[f](c+it) - \int_{1/\rho}^\rho f(u) u^{s-1}du\bigg\|_{L^{p'}(\{c\}\times i\mathbb{R})} = 0,$$ where $p'$ denotes the conjugate exponent of $p.$ Analogously, the inverse Mellin transform $M_c^{-1, 2}$ of a function $g \in L^{2}(\{c\} \times i\mathbb{R})$ is defined as $$M_c^{-1,2}[f](s) = \limm_{\rho \rightarrow +\infty}~\int_{-\rho}^\rho g(c + iv) v^{-c-iv}dv,$$ in the sense that $$\lim_{\rho \rightarrow \infty}\bigg\|M_c^{-1,2}[g](c+iv) - \frac{1}{2\pi}\int_{-\rho}^\rho g(c + iv) v^{-c-iv}dv\bigg\|_{X^2_c} = 0.$$ In a similar way one can define the inverse Mellin transform $M_c^{-1,p}$ with $p \in {]}1,2{[}.$ In what follows, we will continue to denote the Mellin transform of a function $g\in L^p(\mathbb{R}^+)$ by $[g]^\wedge_{M_c}$ and we will consider essentially the cases $p=1$ and $p=2.$ The Mellin translation operator $\tau_h^c$ for $h \in \mathbb{R}^+,~c \in \mathbb{R}$ and $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ is defined by $$(\tau_h^c f)(x) := h^c f(hx)~~(x\in \mathbb{R}^+).$$ \noindent Setting $\tau_h:= \tau^0_h,$ then $(\tau_h^cf)(x) = h^c (\tau_hf)(x)$ and $\|\tau_h^c f\|_{X^1_c} = \|f\|_{X^1_c}.$ \vskip0,4cm For $1\leq p \leq 2$, denote by $B^p_{c,\sigma}$ the Bernstein space of all functions $f\in X^p_c\cap C(\mathbb{R}^+),$ $c \in \mathbb{R},$ which are Mellin band-limited to the interval $[-\sigma,\sigma],$ $\sigma \in \mathbb{R}^+,$ thus for which $[f]^\wedge_{M_c}(c+it) = 0$ for all $|t| > \sigma.$ We notice that, as in Fourier analysis, the inclusion $B^{p}_{c,\sigma} \subset B^q_{c,\sigma}$ holds for $1 \leq p<q \leq 2.$ \section{A notion of distance} For $q \in [1,+\infty]$, let $G_c^q$ be the linear space of all functions $f:\mathbb{R}^+ \rightarrow \mathbb{C}$ that have the representation $$f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi(v) x^{-c-iv}dv \qquad (x>0),$$ where $\varphi \in L^1(\mathbb{R}) \cap L^q(\mathbb{R}).$ The space $G_c^q$ will be endowed with the norm $$[\!\!|f|\!\!]_q := \|\varphi\|_{L^q(\mathbb{R})} = \left(\int_{-\infty}^\infty |\varphi(v)|^qdv\right)^{1/q}.$$ Note that this is really a norm. Indeed, $[\!\!|f|\!\!]_q = 0$ iff $f=0$ due to the existence and uniqueness of Mellin inversion (see \cite{BJ1}, \cite{BJ2}). The above norm induces the metric $$\mbox{\rm dist}_q(f,g) := [\!\!|f-g|\!\!]_q \quad \quad f,g \in G_c^q.$$ Note that in case $q=2$ we have $$\mbox{\rm dist}_2(f,g) = \sqrt{2\pi}\|f-g\|_{X^2_c},$$ i.e., our distance reduces to the ``Euclidean'' distance in $X^2_c,$ up to the factor $\sqrt{2\pi}.$\newline \noindent As a consequence of the Mellin inversion formula, functions $f$ for which $[f]^\wedge_{M_c}(c+i\cdot) \in L^1(\mathbb{R}) \cap L^q(\mathbb{R})$ belong to $G_c^q.$ For $p \in [1,2]$, the Mellin-Bernstein space $B_{c,\sigma}^p$ is a subspace of $G_c^q$ since the Mellin transform of $f \in B_{c,\sigma}^p$ has compact support as a function of $v \in \mathbb{R}$ and so it belongs to any $L^q(\mathbb{R}).$ For $f \in G_c^q$ we define $$\mbox{\rm dist}_q(f,B_{c,\sigma}^p) = \inf_{g \in B_{c,\sigma}^p}[\!\!|f-g|\!\!]_q\,.$$ \vskip0,4cm The following representation theorem holds: \newtheorem{Theorem}{Theorem} \begin{Theorem} \label{representation1} For any $f \in G_c^q$, we have $$\mbox{\rm dist}_q(f,B_{c,\sigma}^p) = \left(\int_{|v| \geq \sigma}|\varphi (v)|^qdv\right)^{1/q} \quad \quad (1\leq q<\infty),$$ and if $\varphi$ is also continuous, then $$\mbox{\rm dist}_\infty(f,B_{c,\sigma}^p) = \sup_{|v| \geq \sigma}|\varphi(v)|.$$ \end{Theorem} {\bf Proof}. Assume $q< \infty.$ Clearly, since $g \in B_{c, \sigma}^p$ implies $|[g]^\wedge_{M_c}(c+iv)| = 0$ for $|v| \geq \sigma,$ one has $$\mbox{\rm dist}_q(f,B_{c,\sigma}^p) = \left\{\inf_{g \in B_{c,\sigma}^p}\int_{|v|\leq \sigma}|\varphi(v) - [g]^\wedge_{M_c}(c+iv)|^qdv + \int_{|v|\geq \sigma}|\varphi(v)|^qdv\right\}^{1/q}.$$ Therefore we have to prove that $$I_{p,q}:= \inf_{g \in B_{c,\sigma}^p}\int_{|v| \leq \sigma} |\varphi(v) - [g]^\wedge_{M_c}(c+iv)|^q dv = 0.$$ For the given $\sigma >0,$ the space $C^\infty_c({]}-\sigma, \sigma{[})$, whose elements are all the infinitely differentiable functions with compact support in ${]}-\sigma, \sigma{[},$ is dense in $L^q({]}-\sigma, \sigma{[})$ for $1\leq q< \infty$ (see e.g. \cite{AD}). Thus, given $\varphi \in L^q(\mathbb{R})$ and $\varepsilon >0$, we can take a function $P \in C^\infty_c({]}-\sigma, \sigma{[})$ such that $$\|\varphi - P\|_{L^q({]}-\sigma, \sigma{[})} < \varepsilon.$$ Now we define $$g_\varepsilon (x) := \frac{x^{-c}}{2\pi}\int_{-\sigma}^\sigma P(v) x^{-iv}dv \quad (x>0).$$ Integrating $k$-times by parts, one can easily see that $x^cg_\varepsilon(x) = {\cal O}((\log x)^{-k})$ for $x \rightarrow +\infty$ and $x\rightarrow 0^+.$ This implies that $g_\varepsilon \in X^p_c$ for any $p\geq 1$, and $[g_\varepsilon]^\wedge_{M_c}(c+iv) = P(v)$ for $|v| \leq \sigma$ and $0$ otherwise. Thus $g_\varepsilon \in B^p_{c, \sigma}.$ Now we conclude that \begin{eqnarray*} I_{p,q}^{1/q} &\leq& \|\varphi - [g_\varepsilon]^\wedge_{M_c}(c+i\cdot)\|_{L^q(]-\sigma, \sigma[)} \\&=& \|\varphi - P\|_{L^q(]-\sigma, \sigma[)} < \varepsilon. \end{eqnarray*} Hence the assertion follows for $1\leq q < \infty.$ The case $q=\infty$ is treated in a different way. To this end, given $\varepsilon >0,$ there exists a twice continuously differentiable function $\psi$ on $\mathbb{R}$ such that $\sup_{v \in \mathbb{R}}|\varphi (v) - \psi (v)| < \varepsilon/2.$ For example, $\psi$ may be chosen as an appropriate spline function. For $\eta \in {]}0,\sigma{[},$ define \begin{eqnarray*} \psi_1(x):= \left\{\begin{array}{llll} 0 \quad &\mbox{if}\quad |v|\leq \sigma -\eta,\\[2ex] \frac{\psi(-\sigma)}{-\eta^3}(v+\sigma-\eta)^3 \quad &\mbox{if}\quad -\sigma \leq v \leq -\sigma +\eta,\\[2ex] \frac{\psi(\sigma)}{\eta^3}(v-\sigma + \eta)^3 \quad &\mbox{if}\quad \sigma - \eta \leq v \leq \sigma,\\[2ex] \psi(v) \quad &\mbox{if}\quad |v| \geq \sigma. \end{array} \right. \end{eqnarray*} Note that $\psi_1$ is continuous on $\mathbb{R}$ and $$\|\psi_1\|_{L^\infty(\mathbb{R})} = \sup_{|v| \geq \sigma -\eta}|\psi_1(v)| = \sup_{|v| \geq \sigma}|\psi (v)| \leq \sup_{|v|\geq \sigma}|\varphi(v)| + \frac{\varepsilon}{2}\,.$$ Next define $\psi_0(v):= \psi(v) - \psi_1(v)$ for $v \in \mathbb{R}.$ Then $\psi_0$ is continuous on $\mathbb{R},$ twice continuously differentiable on ${]}-\sigma, \sigma{[},$ it vanishes at $\pm \sigma$ and it has support on $[-\sigma, \sigma].$ With these properties, two integrations by parts show that $$g_\varepsilon(x):=\frac{x^{-c}}{2 \pi}\int_{-\sigma}^\sigma \psi_0(v)x^{-iv}dv \quad \quad (x>0)$$ defines a function $g_\varepsilon \in X^p_c \cap B^p_{c,\sigma}.$ Furthermore, the Mellin inversion formula yields that $[g_\varepsilon]^\wedge_{M_c}(c+i\cdot) = \psi_0(\cdot).$ Now we conclude that \begin{eqnarray*} \mbox{\rm dist}_\infty(f, B^p_{c,\sigma}) &= & \inf_{g \in B^p_{c, \sigma}}\|\varphi - [g]^\wedge_{M_c}(c+i\cdot)\|_{L^\infty(\mathbb{R})}\\&\leq& \|\varphi - [g_\varepsilon]^\wedge_{M_c}(c+i\cdot)\|_{L^\infty(\mathbb{R})}\\ &=&\|\varphi - \psi_0\|_{L^\infty(\mathbb{R})}\\ &\leq& \|\varphi - \psi\|_{L^\infty(\mathbb{R})} +\|\psi - \psi_0\|_{L^\infty(\mathbb{R})}\\ &\leq& \frac{\varepsilon}{2} + \|\psi_1\|_{L^\infty(\mathbb{R})}\\ &\leq& \varepsilon + \sup_{|v| \geq \sigma} |\varphi (v)|. \end{eqnarray*} This implies that $\mbox{\rm dist}_\infty(f, B^p_{c, \sigma}) \leq \sup_{|v| \geq \sigma}|\varphi (v)|.$ On the other hand, \begin{eqnarray*} \mbox{\rm dist}_\infty(f, B^p_{c, \sigma}) &=& \max\left\{\inf_{g \in B^p_{c, \sigma}} \sup_{|v| \leq \sigma}|\varphi (v) - [g]^\wedge_{M_c}(c+iv)|, \sup_{|v| \geq \sigma}|\varphi (v)|\right\}\\&\geq& \sup_{|v| \geq \sigma}|\varphi(v)|. \end{eqnarray*} Hence the formula stated in the theorem holds. $\Box$ \vskip0,4cm Next we will obtain a distance formula for Mellin derivatives. We define the first Mellin derivative (the Mellin differential operator of first order) by $$(\Theta_c^1f)(x) := \lim_{h\rightarrow 1}\frac{\tau^c_hf(x) - f(x)}{h-1} = \lim_{h\rightarrow 1}\frac{h^cf(hx) - f(x)}{h-1} \quad (x>0),$$ and the Mellin differential operator of order $r \in \mathbb{N}$ is defined iteratively by $\Theta^r_c:= \Theta_c^1(\Theta_c^{(r-1)});$ see \cite{BJ1}. We have the following \begin{Theorem}\label{derivative} Let $f\in G_c^q.$ If $v^r\varphi (v)$ belongs to $L^1(\mathbb{R})$ as a function of $v$ for some $r \in \mathbb{N},$ then $f$ has Mellin derivatives up to order $r$ in $C_0(\mathbb{R})$ and $$(\Theta^k_cf)(x) = \frac{(-i)^k}{2 \pi}\int_{-\infty}^{+\infty} v^k\varphi (v)x^{-c-iv}dv \quad \quad(k=0,1,\ldots r).$$ \end{Theorem} {\bf Proof.} Suppose that $r=1.$ For $h \neq 1$ we have \begin{eqnarray*} \frac{h^cf(hx) - f(x)}{h-1} &=& \frac{1}{2 \pi}\frac{1}{h-1}\bigg(\int_{-\infty}^{+\infty} \varphi(v) h^{-iv}x^{-c-iv}dv - \int_{-\infty}^{+\infty} \varphi(v) x^{-c-iv}dv\bigg) \\[1ex] &=& \frac{x^{-c}}{2 \pi}\int_{-\infty}^{+\infty} \varphi(v) x^{-iv}\frac{h^{-iv} - 1}{h - 1}dv. \end{eqnarray*} Now \begin{eqnarray*} \bigg|\frac{h^{-iv} - 1}{h - 1}\bigg|= \frac{2}{|h-1|}\bigg|\frac{e^{-\frac{iv \log h}{2}}-e^{\frac{iv \log h}{2}}}{2i}\bigg|= 2 \bigg|\frac{\sin (\frac{v \log h}{2})}{h-1}\bigg| \leq |v|. \end{eqnarray*} Since $$\lim_{h \rightarrow 1} \frac{h^{-iv} - 1}{h - 1} = -iv$$ and $v\varphi(v)$ is absolutely integrable, Lebesgue's theorem on dominated convergence gives $$(\Theta^1_c f)(x) = \frac{-i}{2\pi}\int_{-\infty}^{+\infty} v\varphi(v) x^{-c-iv}dv.$$ Moreover, using the Mellin inversion theorem (see \cite[Lemma~4, p.~349]{BJ1}, we have $\Theta^1_cf \in C_0(\mathbb{R}^+).$ The proof for general $r$ follows by mathematical induction. $\Box$ \vskip0,3cm Using Theorems \ref{representation1} and \ref{derivative}, we obtain immediately the distance of $\Theta^k_cf$ from the Bernstein space $B_{c,\sigma}^p$. \vskip0,4cm \noindent \newtheorem{Corollary}{Corollary} \begin{Corollary}\label{cor1} Let $f \in G^q_c$ with $v^k\varphi \in L^1(\mathbb{R}) \cap L^q(\mathbb{R}).$ Then for every $p \in [1,2]$, we have $$\mbox{\rm dist}_q(\Theta^k_cf,B_{c,\sigma}^p) = \left(\int_{|v| \geq \sigma}|v^k\varphi (v)|^qdv\right)^{1/q} \quad \quad (1\leq q<\infty),$$ and if $\varphi$ is continuous, then $$\mbox{\rm dist}_\infty(\Theta^k_cf,B_{c,\sigma}^p) = \sup_{|v| \geq \sigma}|v^k\varphi(v)|.$$ \end{Corollary} \vskip0,4cm \section{Estimation of the Mellin distance} In this section we will introduce certain basic ``intermediate'' function spaces between the spaces $B^p_{c,\sigma}$ and the space $G_c^q.$ We will consider mainly the cases $p=1$ and $p=2.$ In the following for $p\in [1,2],$ we will denote by $\mathcal{M}_c^p$ the space comprising all functions $f \in X_c^p \cap C(\mathbb{R})$ such that $[f]^\wedge_{M_c} \in L^1(\{c\}\times i \mathbb{R}).$ This space is contained in $G_c^q$ for suitable values of $q,$ namely for $q \in [1, p']$ with $p'$ being the conjugate exponent of $p.$ As for the classes $B^p_{c,\sigma},$ we have again the inclusion $\mathcal{M}^p_c \subset \mathcal{M}^q_c$ for $1 \leq p<q\leq 2.$ We begin with the definitions of differences of integer order and an appropriate modulus of smoothness. For a function $f \in X^p_c,$ $r \in \mathbb{N}$ and $h >0$, we define $$(\Delta_h^{r,c}f)(u) := \sum_{j=0}^r (-1)^{r-j}\left(\begin{array}{ll} r \\ j \end{array}\right) f(h^ju)h^{jc},$$ and for $\delta >0,$ $$\omega_{r}(f, \delta, X^p_c) := \sup_{|\log h|\leq \delta}\|\Delta_h^{r,c}f\|_{X^p_c}.$$ In particular for $p=1,$ $$\omega_{r}(f, \delta, X^1_c) := \sup_{|\log h|\leq \delta}\|\Delta_h^{r,c}f\|_{X^1_c}.$$ Among the basic properties of the above modulus of smoothness $\omega_{r}$ we list the following three: \begin{enumerate} \item $\omega_r(f, \cdot, X^p_c)$ is a non decreasing function on $\mathbb{R}^+;$ \item $\omega_r(f, \delta, X^p_c) \leq 2^r\|f\|_{X^p_c}$; \item for any positive $\lambda$ and $\delta$, one has $$\omega_r(f, \lambda \delta, X^p_c) \leq (1+\lambda)^r\omega_r(f,\delta,X^p_c).$$ \end{enumerate} We know that for functions $f \in X^1_c$ or $f \in X^2_c$ one has (see \cite{BJ1}, \cite{BJ0}, \cite{BBM}) \begin{eqnarray} [\Delta_h^{r,c}f]^\wedge_{M_c}(c+iv) = (h^{-iv} - 1)^r [f]^\wedge_{M_c}(c+iv) \qquad (v \in \mathbb{R}). \end{eqnarray} We have the following \begin{Theorem} \label{estimate1} If $f \in \mathcal {M}^1_c,$ then for any $q \in [1, \infty]$, \begin{eqnarray*} \mbox{\rm dist}_q(f, B^1_{c, \sigma}) \leq D \cdot \left\{\begin{array}{ll} \displaystyle\left\{\int_{\sigma}^\infty [\omega_r(f, v^{-1}, X^1_c)]^qdv\right\}^{1/q} &\quad (q< \infty),\\[2ex] \omega_r(f, \sigma^{-1}, X^1_c) &\quad (q=\infty), \end{array} \right. \end{eqnarray*} where $D$ is a constant depending on $r$ and $q$ only. \end{Theorem} {\bf Proof}. From (1), setting $h= e^{\pi/v}$, we have $$[\Delta_{h}^{r,c}f]^\wedge_{M_c}(c+iv) = (-2)^r [f]^\wedge_{M_c}(c+iv)$$ or $$[f]^\wedge_{M_c}(c+iv) = \frac{1}{(-2)^r}\int_0^\infty (\Delta^{r,c}_{h}f)(u) u^{c+iv-1}du \quad \quad (h= e^{\pi/v})$$ and so $$|[f]^\wedge_{M_c}(c+iv)| \leq \frac{1}{2^r}\int_0^\infty |(\Delta^{r,c}_{h}f)(u)|u^{c-1}du \leq \frac{1}{2^r}\omega_r(f, \frac{\pi}{|v|}, X^1_c).$$ Now using the properties of the modulus $\omega_r,$ we find that $$\omega_r(f, \frac{\pi}{|v|}, X^1_c) \leq (1+\pi)^r \omega_r(f, \frac{1}{|v|}, X^1_c).$$ Thus $$|[f]^\wedge_{M_c}(c+iv)| \leq \left(\frac{1+\pi}{2}\right)^r \omega_r(f, \frac{1}{|v|}, X^1_c).$$ In view of Theorem~\ref{representation1}, this implies the assertion for $q<\infty$. The case $q=\infty$ is obtained analogously. $\Box$ \vskip0,3cm \begin{Theorem} \label{estimate2} If $f \in \mathcal{M}^2_c$, then for any $q\in [1,2],$ $$\mbox{\rm dist}_q(f, B^2_{c,\sigma}) \leq D \left\{\int_{\sigma}^\infty [v^{-q/2}\omega_r(f, v^{-1}, X^2_c)]^qdv\right\}^{1/q},$$ where $D$ is a constant depending on $r$ and $q$ only. \end{Theorem} {\bf Proof}. First we consider the case $q=2.$ Then $$|[\Delta_h^{r,c}f]^\wedge_{M_c}(c+iv)| = 2^r |\sin ((v \log h)/2)| |[f]^\wedge_{M_c}(c+iv)|$$ and (see \cite[Lemma 2.6]{BJ2}) $$\|[\Delta_h^{r,c}f]^\wedge_{M_c}\|_{L^2(\{c\}\times i \mathbb{R})} = \sqrt{2\pi}\|\Delta_h^{r,c}f\|_{X^2_c} \leq \sqrt{2\pi}\omega_r(f, |\log h|, X^2_c).$$ Now let $h\geq 1.$ For $v \in [(2\log h)^{-1}, (\log h)^{-1}]$, one has $$\sin ((v\log h)/2) \geq \frac{1}{2\pi}\,,$$ and hence $$\int_{1/(2\log h)}^{1/\log h} |[f]^\wedge_{M_c}(c+iv)|^2 dv \leq (2\pi)^{2r}\int_0^\infty |\sin ((v\log h)/2)|^{2r}| |[f]^\wedge_{M_c}(c+iv)|^2dv.$$ Analogously we have $$\int_{-1/\log h}^{-1/(2\log h)} |[f]^\wedge_{M_c}(c+iv)|^2 dv \leq (2\pi)^{2r}\int_{-\infty}^0 |\sin ((v\log h)/2)|^{2r}| |[f]^\wedge_{M_c}(c+iv)|^2dv,$$ and so \begin{eqnarray*} \lefteqn{\int_{1/(2\log h) \leq|v| \leq 1/\log h}|[f]^\wedge_{M_c}(c+iv)|^2 dv} \qquad\qquad\quad\\ &\leq & (2\pi)^{2r}\int_{-\infty}^\infty |\sin ((v\log h)/2)|^{2r}| |[f]^\wedge_{M_c}(c+iv)|^2dv \\ &\leq& \pi^{2r}[\omega_r(f, \log h, X^2_c)]^2. \end{eqnarray*} Now, let $\sigma >0$ be fixed, set $\sigma_k:= \sigma 2^k$ with $k \in \mathbb{N}_0$, and define $h$ by $\log h = 1/\sigma_{k+1}.$ Then $$\int_{\sigma_k \leq |v| \leq \sigma_{k+1}}|[f]^\wedge_{M_c}(c+iv)|^2dv \leq \pi^{2r}[\omega_r(f, \sigma_{k+1}^{-1}, X^2_c)]^2,$$ and so summation over $k$ yields $$\int_{|v| \geq \sigma}|[f]^\wedge_{M_c}(c+iv)|^2dv \leq \pi^{2r} \sum_{k=0}^\infty [\omega_r(f, \sigma_{k+1}^{-1}, X^2_c)]^2.$$ Now since $\sigma_{k+1}- \sigma_k = \sigma_k,$ from the monotonicity of $\omega_r$ as a function of $\delta,$ one has $$\int_{\sigma_k}^{\sigma_{k+1}}v^{-1}[\omega_r(f, v^{-1}, X^2_c)]^2dv \geq \frac{\sigma_k}{\sigma_{k+1}}[\omega_r(f, \sigma^{-1}_{k+1}, X^2_c)]^2,$$ from which we deduce $$\sum_{k=0}^\infty [\omega_r(f, \sigma^{-1}_{k+1}, X^2_c)]^2 \leq 2\int_\sigma^\infty v^{-1}[\omega_r(f, v^{-1}, X^2_c)]^2dv.$$ This gives the assertion for $q=2.$ For $q\in [1, 2{[}$ one can proceed as in the proof of Proposition 13 in \cite{BSS2}, using H\"{o}lder's inequality. $\Box$ \subsection{Mellin-Lipschitz spaces} For $\alpha \in {]}0,r]$ we define the Lipschitz class by $$\mbox{Lip}_r(\alpha, X^p_c) := \{f \in X^p_c: \omega_r(f;\delta;X^p_c) = {\cal O}(\delta^\alpha), \delta \rightarrow 0^+\}.$$ \noindent As a consequence of Theorems \ref{estimate1} and \ref{estimate2}, we obtain the following corollary which determines the Mellin distance of a function $f \in \mbox{Lip}_r(\alpha, X^p_c)$ from $B^p_{c,\sigma}$ for $p=1,2.$ \begin{Corollary}\label{lip} If $f \in \mbox{\rm Lip}_r(\alpha, X^1_c\cap C(\mathbb{R}))$ for some $r\in \mathbb{N},~ r \geq 2$ and $1<\alpha \leq r,$ then $$\mbox{\rm dist}_1(f, B^{1}_{c, \sigma}) = {\cal O}(\sigma^{-\alpha +1})\quad \quad (\sigma \rightarrow +\infty).$$ Moreover, if $f \in \mathcal{M}^2_c \cap \mbox{\rm Lip}_r(\beta, X^2_c)$ with $r \in \mathbb{N},$ $q^{-1} - 2^{-1} < \beta \leq r,$ then $$\mbox{\rm dist}_q (f, B^2_{c, \sigma}) = {\cal O}(\sigma^{-\beta - 1/2 + 1/q})\quad \quad (\sigma \rightarrow +\infty).$$ \end{Corollary} The proof follows immediately from Theorems \ref{estimate1} and \ref{estimate2}. Note that, if $f \in \mbox{Lip}_r(\alpha, X^1_c\cap C(\mathbb{R})),$ then from the proof of Theorem \ref{estimate1} one has that $[f]^\wedge_{M_c} \in L^1(\{c\}\times i\mathbb{R});$ thus $f \in\mathcal{M}^1_c.$ For $q=2$ we obtain the estimate $$\mbox{\rm dist}_2 (f, B^2_{c, \sigma}) = {\cal O}(\sigma^{-\beta})\quad \quad (\sigma \rightarrow +\infty).$$ \subsection{Mellin-Sobolev spaces} Denote by $AC_{{\tt loc}}(\mathbb{R}^+)$ the space of all locally absolutely continuous functions on $\mathbb{R}^+$. The Mellin-Sobolev space $W_c^{r,p}(\mathbb{R}^+)$ is defined as the space of all functions $f \in X^p_c$ which are equivalent to a function $g \in C^{r-1}(\mathbb{R}^+)$ with $g^{(r-1)}\in AC_{{\tt loc}}(\mathbb{R}^+)$ such that $\Theta^r_cg \in X^p_c$ (see \cite{BJ1}, \cite{BJ2}, \cite{BBM}). For $p=1$ it is well known that for any $f \in W_c^{r,1}$ one has (see \cite{BJ1}) $$[\Theta^r_c]^\wedge_{M_c}(c+iv) = (-iv)^r[f]^\wedge_{M_c}(c+iv)\quad \quad (v \in \mathbb{R}).$$ The same result also holds for $1< p\leq 2,$ taking into account the general convolution theorem for Mellin transforms (\cite[Lemma~3.1]{BJ2} in case $p=2$, \cite[Lemma 2]{BKT}). By the above result, the Mellin-Sobolev space $W_c^{r,p}(\mathbb{R}^+)$ can be characterized as $$W_c^{r,p}(\mathbb{R}^+) = \{f \in X^p_c: (-iv)^r [f]^\wedge_{M_c}(c+iv) = [g]^\wedge_{M_c}(c+iv), ~g \in L^p(\{c\}\times i \mathbb{R})\}.$$ We have the following \begin{Theorem}\label{sobolev1} Let $f \in \mathcal{M}^1_c\cap W^{r,1}_c(\mathbb{R}^+).$ Then for $q \in [1,\infty]$ and $r >1/q,$ \begin{eqnarray*} \mbox{\rm dist}_q(f, B^1_{c,\sigma}) \leq D\|\Theta^r_cf\|_{X^1_c} \cdot \left\{\begin{array}{ll} \sigma^{-r +1/q}, &\quad q<\infty,\\ \sigma^{-r}, &\quad q= \infty, \end{array} \right. \end{eqnarray*} where $D$ is a constant depending on $r$ and $q$ only. If, in addition, $v[f]^\wedge_{M_c}(c+iv) \in L^1(\mathbb{R})$, then for $r> 1 + 1/q,$ \begin{eqnarray*} \mbox{\rm dist}_q(\Theta_c f, B^1_{c,\sigma}) \leq D'\|\Theta^r_cf\|_{X^1_c} \cdot \left\{\begin{array}{ll} \sigma^{-r +1+1/q}, & \quad q<\infty,\\ \sigma^{-r+1}, & \quad q= \infty, \end{array} \right. \end{eqnarray*} where $D'$ is again a constant depending on $r$ and $q$ only. \end{Theorem} {\bf Proof}. First we consider the case $q<\infty.$ The formula for the Mellin transform of a Mellin derivative yields $$[f]^\wedge_{M_c}(c+iv) = (-iv)^{-r}[\Theta^r_cf]^\wedge_{M_c}(c+iv) \qquad (v \in \mathbb{R} \setminus \{0\}).$$ Thus, from Theorem \ref{representation1} we obtain \begin{eqnarray*} \mbox{\rm dist}_q(f, B^1_{c,\sigma}) = \left\{\int_{|v| \geq \sigma}|v^{-r}[\Theta^r_cf(v)]^\wedge_{M_c}(c+iv)|^qdv\right\}^{1/q}. \end{eqnarray*} Since $\Theta^r_cf \in X^1_c,$ its Mellin transform is continuous and bounded on $\{c\} \times i\mathbb{R}$ (see \cite{BJ1}). Therefore \begin{eqnarray*} \mbox{\rm dist}_q(f, B^1_{c,\sigma}) &\leq& \|[\Theta^r_cf]^\wedge_{M_c}\|_{C(\{c\}\times i \mathbb{R})} \left\{2\int_{v \geq \sigma}v^{-rq}dv\right\}^{1/q} \\ &\leq& \|\Theta^r_cf\|_{X^1_c}\bigg(\frac{2}{rq-1}\bigg)^{1/q} \frac{1}{\sigma^{r -1/q}}, \end{eqnarray*} and hence the assertion for $q<\infty$ is proved with $D= (2/(rq-1))^{1/q}.$ For $q= \infty$ we use again Theorem \ref{representation1} and proceed analogously, obtaining $D=1.$ For the second part, note that under the assumptions on $f$ and $v [f]^\wedge_{M_c}(c+iv) \in L^1(\{c\}\times i \mathbb{R})$, we have $\Theta_cf \in {M}^1_c \cap W^{r-1, 1}_c(\mathbb{R}^+).$ Therefore we can apply the first part of the proof to the function $\Theta_cf,$ obtaining immediately the assertion with the constant $D'= (2/(rq-q-1))^{1/q}$ for $q< \infty$ and $D'= 1$ for $q = \infty.$ $\Box$ \vskip0,3cm Note that if $f\in \mathcal{M}^1_c\cap W^{r,1}(\mathbb{R}^+)$ satisfies the further condition that $[\Theta^r_cf]^\wedge_{M_c} \in L^q(\{c\} \times i\mathbb{R}),$ then one may write \begin{eqnarray*} \mbox{\rm dist}_q(f, B^1_{c,\sigma}) &=& \left\{\int_{|v| \geq \sigma}|v^{-r}[\Theta^r_cf(v)]^\wedge_{M_c}(c+iv)|^qdv\right\}^{1/q}\\[2ex] &\leq& \frac{1}{\sigma^r}\|[\Theta^r_cf]^\wedge_{M_c}\|_{L^q(\{c\} \times i\mathbb{R})}. \end{eqnarray*} Moreover, one has $$\mbox{\rm dist}_q(f, B^1_{c,\sigma}) = \mathcal{O}(\sigma^{-r}) \qquad (\sigma \rightarrow +\infty).$$ \vskip0,4cm For $p=2$ we have the following \begin{Theorem}\label{sobolev2} Let $f \in \mathcal{M}^2_c\cap W^{r,2}_c(\mathbb{R}^+).$ Then for $q \in [1,2]$, \begin{eqnarray*} \mbox{\rm dist}_q(f, B^2_{c,\sigma}) \leq D\|\Theta^r_cf\|_{X^2_c}~ \sigma^{-r -1/2 + 1/q}, \end{eqnarray*} where $D$ is a constant depending on $r$ and $q$ only. If, in addition, $v[f]^\wedge_{M_c}(c+iv) \in L^1(\mathbb{R})$, then for $r> 1 + 1/2 +1/q,$ \begin{eqnarray*} \mbox{\rm dist}_q(\Theta_c f, B^2_{c,\sigma}) \leq D'\|\Theta^r_cf\|_{X^2_c}~ \sigma^{-r +1/2+1/q}, \end{eqnarray*} where $D'$ is again a constant depending on $r$ and $q$ only. \end{Theorem} {\bf Proof}. As in the previous theorem, by the formula of Mellin transform in $X^2_c$ for derivatives we have $$\mbox{\rm dist}_q(f, B^2_{c,\sigma}) = \left\{\int_{|v| \geq \sigma}|v^{-r}[\Theta^r_c]^\wedge_{M_c}(c+iv)|^qdv\right\}^{1/q}.$$ For $q=2,$ using the property that the Mellin transform in $X^2_c$ is norm-preserving (see \cite[Lemma~2.6]{BJ2}), we have \begin{eqnarray*} \mbox{\rm dist}_q(f, B^2_{c,\sigma})&\leq& \frac{1}{\sigma^r}\left\{\int_{|v| \geq \sigma}|[\Theta^r_c]^\wedge_{M_c}(c+iv)|^2dv\right\}^{1/2} \\[1ex] &\leq& \frac{1}{\sigma^r}\|[\Theta^r_cf]^\wedge_{M_c}\|_{L^2(\{c\}\times i \mathbb{R})} = \sqrt{2\pi}\frac{1}{\sigma^r}\|\Theta^r_cf\|_{X^2_c}. \end{eqnarray*} Therefore the assertion follows for $q=2$ with the constant $D = (2\pi)^{-1/2}.$ For $q\in [1,2{[}$ one can use H\"{o}lder's inequality with $\mu = 2/(2-q),~\nu = 2/q$, obtaining \begin{eqnarray*} \mbox{\rm dist}_q(f, B^2_{c,\sigma})&\leq& \left\{2 \int_{\sigma}^\infty v^{-rq\mu}dv\right\}^{1/(q\mu)} \left\{\int_{|v|\geq \sigma}|[\Theta^r_cf]^\wedge_{M_c}(c+iv)|^{q\nu}dv\right\}^{1/(q\nu)}\\[1ex] &\leq& \frac{1}{\sigma^{r+1/2 -1/q}}\left\{\frac{4-2q}{(2r+1)q - 2}\right\}^{1/q - 1/2} \|[\Theta^r_cf]^\wedge_{M_c}\|_{L^2(\{c\}\times i \mathbb{R})} \\[1ex] &=& \sqrt{2\pi} \frac{1}{\sigma^{r+1/2 -1/q}}\left\{\frac{4-2q}{(2r+1)q - 2}\right\}^{1/q - 1/2} \|\Theta^r_cf\|_{X^2_c}. \end{eqnarray*} Thus the first inequality holds with $$D= \sqrt{2\pi}\left\{\frac{4-2q}{(2r+1)q - 2}\right\}^{1/q - 1/2}.$$ The second inequality follows by arguments similar to those in the proof of Theorem~\ref{sobolev1}. $\Box$ \vskip0,4cm As a consequence, under the assumptions of the first part of Theorem \ref{sobolev2}, we can obtain an asymptotic estimate of the form $$\mbox{\rm dist}_q(f, B^2_{c,\sigma}) = \mathcal{O}(\sigma^{-r-1/2+1/q}) \qquad (\sigma \rightarrow +\infty).$$ \section{Applications} In this section we will illustrate applications to various basic formulae such as the approximate exponential sampling theorem, the approximate reproducing kernel formula in the Mellin frame (see \cite{BJ3}, \cite{BBM0}), a generalized Boas differentiation formula and an extension of a Bernstein-type inequality. In the following for $c \in \mathbb{R}$, we denote by $\mbox{lin}_c$ the function $$\mbox{lin}_c(x) := \frac{x^{-c}}{2\pi i}\frac{x^{\pi i} -x^{-\pi i}}{\log x} = \frac{x^{-c}}{2\pi}\int_{-\pi}^\pi x^{-it}dt \qquad (x>0, \, x\neq 1)$$ with the continuous extension $\mbox{lin}_c(1) = 1.$ Thus $$\mbox{lin}_c(x) = x^{-c}\mbox{sinc}(\log x) \qquad (x>0).$$ Here, as usual, the ``sinc'' function is defined as $$\mbox{sinc}(t):= \frac{\sin (\pi t)}{\pi t}~ \mbox{for}~t\neq 0,\qquad \mbox{sinc}(0) = 1.$$ It is clear that $\mbox{lin}_c \not \in X_{\overline{c}}$ for any $\overline{c}.$ However, it belongs to the space $X^2_c$ and its Mellin transform in $X^2_c$-sense is given by $$[\mbox{\rm lin}_c]^\wedge_{M_c}(c+iv) = \chi_{[-\pi, \pi]},$$ where $\chi_A$ denotes the characteristic function of the set $A.$ \subsection{Approximate exponential sampling formula} For a function $f \in B^2_{c,\pi T}$ the following exponential sampling formula holds (see \cite{BJ3}, \cite{BJ2}): $$f(x) = \sum_{k \in \mathbb{Z}}f(e^{k/T})\mbox{\rm lin}_{c/T}(e^{-k}x^T) \qquad (x>0).$$ As an approximate version in the space $\mathcal{M}^2_c$ we have (see \cite[Theorem~5.5]{BJ2}): \newtheorem{Proposition}{Proposition} \begin{Proposition}\label{approsamp} Let $f \in \mathcal{M}_c^2.$ Then there holds the error estimate \begin{eqnarray*} \lefteqn{\bigg|f(x) - \sum_{k=-\infty}^{\infty} f(e^{k/T})\mbox{\rm lin}_{c/T}(e^{-k}x^T)\bigg|}\qquad\qquad\quad\\[2ex] &\leq& \frac{x^{-c}}{\pi}\int_{|t| > \pi T}| [f]^\wedge_{M_c}(c+it)| dt \qquad(x \in \mathbb{R}^+,~T >0). \end{eqnarray*} \end{Proposition} This estimate can now be given a ``metric interpretation''. By Theorem \ref{representation1}, the right-hand side may be expressed as $$\frac{x^{-c}}{\pi}\mbox{\rm dist}_1(f,B^2_{c, \pi T}).$$ Hence, introducing a remainder $(R_{\pi T} f)(x)$ by writing \begin{equation}\label{approx_samp} f(x)\,=\, \sum_{k\in\mathbb{Z}} f\left(e^{k/T}\right) \mbox{lin}_{c/T}\left(e^{-k}x^T\right) + \left(R_{\pi T}f\right)(x), \end{equation} we have by Proposition~\ref{approsamp} $$ |\left(R_{\pi T} f\right)(x)|\,\leq\, \frac{x^{-c}}{\pi}\,\mbox{dist}_1(f, B_{c,\pi T}^2) \qquad (x>0),$$ or equivalently, \begin{eqnarray}\label{exp_samp} \|R_{\pi T} f\|_{X_c^\infty}\,\leq\, \frac{1}{\pi}\,\mbox{dist}_1(f, B_{c,\pi T}^2). \end{eqnarray} This relation is a trivial equality when $f\in B_{c, \pi T}^2$. But equality can also occur when $f\not\in B_{c,\pi T}^2.$ Indeed, consider the function $$f(x)\,:=\, x^{-c} \mbox{sinc} (2T\log x -1).$$ By a straight forward calculation, we find that $$ [f]_{M_c}^\wedge(c+iv)\,=\, \frac{e^{iv/(2T)}}{2T}\,\mbox{rect} \left(\frac{v}{2T}\right),$$ where rect denotes the rectangle function defined by \begin{eqnarray*} \mbox{rect}(x)\,:=\left\{ \begin{array}{ccc} 1 & \hbox{ if } & |x|<\pi,\\ \frac{1}{2} & \hbox{ if } &|x|=\pi,\\ 0 & \hbox{ if } & |x|>\pi. \end{array} \right. \end{eqnarray*} Thus, $f\not\in B_{c,\pi T}^2$ and $$\mbox{dist}_1(f, B_{c,\pi T}^2)\,=\, \int_{|v|\geq\pi T} |[f]_{M_c}^\wedge(c+iv)| dv\,=\, \frac{1}{2T} \int_{|v|\geq \pi T} \mbox{rect} \left(\frac{v}{2T}\right) dv\,=\, \pi.$$ Furthermore, $$f(e^{k/T})\,=\, e^{-kc/T} \mbox{sinc} (2k-1)\,=0$$ for all $k\in\mathbb{Z}$. Therefore $(R_{\pi T}f)(x)=f(x)$, which shows that $$\|R_{\pi T}f\|_{X_c^\infty}\,=\, \sup_{x>0} |\mbox{sinc} (2T\log x -1)|\,=\,1,$$ and so equality occurs in (\ref{exp_samp}). \vskip0,3cm Now, employing Theorem \ref{representation1}, Corollary \ref{lip} and the results on Mellin-Sobolev spaces, one has the following theorem. \begin{Theorem}\label{expestimates} For the remainder of the approximate exponential sampling formula (\ref{approx_samp}), the following asymptotic estimates hold: \begin{enumerate} \item If $f\in \mbox{\rm Lip}_r(\alpha, X^1_c\cap C(\mathbb{R}^+)),$ $r\in \mathbb{N},\,r\geq 2,\,1<\alpha \leq r,$ then $$\|(R_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-\alpha +1})\qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^2_c \cap \mbox{\rm Lip}_r(\beta, X^2_c), \,r\in \mathbb{N}, \,1/2 < \beta \leq r,$ then $$\|(R_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-\beta +1/2})\qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^1_c\cap W^{r,1}_c(\mathbb{R}^+), \, r>1,$ then $$\|(R_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-r+1}) \qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^2_c \cap W^{r,2}_c(\mathbb{R}^+), \, r>1/2,$ then $$\|(R_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-r+1/2}) \qquad (T \rightarrow +\infty).$$ \end{enumerate} \end{Theorem} \subsection{Approximate Mellin reproducing kernel formula} Another interesting formula is the ``Mellin reproducing kernel formula'' for Mellin band-limited functions $f \in B^2_{c, \pi T}$. It reads as (see \cite[Theorems 4 and 5]{BBM0}) $$f(x) = T\int_0^\infty f(y)\mbox{\rm lin}_{c/T}\bigg(\left(\frac{x}{y}\right)^T\bigg)\frac{dy}{y} \qquad (x>0).$$ An approximate version was established in \cite[Theorem 6]{BBM0} for functions in the class $\mathcal{M}^1_c.$ In the same way we can state a version in $\mathcal{M}^2_c,$ as follows \begin{Proposition}\label{amrkf} Let $f\in \mathcal{M}_c^2.$ Then for $x \in \mathbb{R}^+,$ and $T>0,$ there holds \begin{equation}\label{rep_kernel} f(x) = T\int_0^\infty f(y)\mbox{\rm lin}_{c/T}\bigg(\left(\frac{x}{y}\right)^T\bigg)\frac{dy}{y} + (R^\ast_{\pi T}f)(x), \end{equation} where $$(R^\ast_{\pi T}f)(x) := \frac{x^{-c}}{2 \pi}\int_{|t|\geq \pi T} [f]^\wedge_{M_c}(c+it) x^{-it} dt.$$ Furthermore, we have the error estimate $$|(R^\ast_{\pi T}f)(x)| \leq \frac{x^{-c}}{2 \pi}\int_{|t|\geq \pi T} |[f]^\wedge_{M_c}(c+it)|dt.$$ \end{Proposition} {\bf Proof}. The proof is essentially the same as in \cite[Theorem 6]{BBM0}. Here for the sake of completeness we give some details. First, note that the convolution integral in (\ref{rep_kernel}) exists, using a H\"{o}lder-type inequality. Putting $G(x) := \mbox{lin}_{c/T}(x^T),$ its $X^2_c$~- Mellin transform is given by $[G]^\wedge_{M_c} = T^{-1}\chi_{[-\pi T, \pi T]}.$ Since $[f]^\wedge_{M_c}\in L^1(\{c\}\times i \mathbb{R}),$ using the Mellin inversion and the Fubini Theorem, one can easily obtain, with the same proof, an $X^2_c$~- extension of the Mellin-Parseval formula for convolutions (see \cite[Theorem 9]{BJ1} for functions in $X^1_c$), obtaining $$T\int_0^\infty f(y)\mbox{\rm lin}_{c/T}\bigg(\left(\frac{x}{y}\right)^T\bigg)\frac{dy}{y} = \frac{x^{-c}}{2\pi}\int_{-\infty}^{+\infty}[f]^\wedge_{M_c}(c+it) [G]^\wedge_{M_c}(c+it)x^{-it}dt.$$ Therefore, by the Mellin inversion formula we have \begin{eqnarray*} &&T\int_0^\infty f(y)\mbox{\rm lin}_{c/T}\bigg(\left(\frac{x}{y}\right)^T\bigg)\frac{dy}{y} = \frac{x^{-c}}{2\pi}\int_{-\pi T}^{\pi T}[f]^\wedge_{M_c}(c+it) x^{-it}dt \\ &=&f(x) - \frac{x^{-c}}{2\pi} \int_{|t| \geq \pi T}[f]^\wedge_{M_c}(c+it) x^{-it}dt \end{eqnarray*} that is the assertion. \vskip0,3cm As before, employing Theorem \ref{representation1}, one can express the error estimate in terms of the distance, i.e., $$|(R^\ast_{\pi T}f)(x)|\leq \frac{x^{-c}}{2\pi}\mbox{\rm dist}_1(f,B^2_{c, \pi T})\qquad (x>0),$$ or equivalently, \begin{equation}\label{ak_est} \|R^\ast_{\pi T} f\|_{X_c^\infty}\,\leq\, \frac{1}{2\pi}\,\mbox{dist}_1(f, B_{c,\pi T}^2). \end{equation} This is again a sharp inequality. Indeed, consider $f(x):=x^{-c} \hbox{sinc}(2T\log x)$. Then $f$ satisfies the hypotheses of Proposition~\ref{amrkf}. By a calculation we find that $\hbox{dist}_1(f, B^2_{c,\pi T})=\pi$ and $\|R^\ast_{\pi T} f\|_{X_c^\infty}=1/2.$ Hence equality occurs in (\ref{ak_est}). Using the estimates of the distance functional in Mellin-Lipschitz and Mellin-Sobolev spaces, we obtain the following results. \begin{Theorem}\label{rkfest} For the remainder of the approximate Mellin reproducing kernel formula (\ref{rep_kernel}), the following asymptotic estimates hold: \begin{enumerate} \item If $f\in \mbox{\rm Lip}_r(\alpha, X^1_c\cap C(\mathbb{R}^+)),$ $r\in \mathbb{N},\,r\geq 2,\,1<\alpha \leq r,$ then $$\|(R^\ast_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-\alpha +1})\qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^2_c \cap \mbox{\rm Lip}_r(\beta, X^2_c), \,r\in \mathbb{N}, \,1/2 < \beta \leq r,$ then $$\|(R^\ast_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-\beta +1/2})\qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^1_c\cap W^{r,1}_c(\mathbb{R}^+), \, r>1,$ then $$\|(R^\ast_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-r+1}) \qquad (T \rightarrow +\infty).$$ \item If $f \in \mathcal{M}^2_c \cap W^{r,2}_c(\mathbb{R}^+), \, r>1/2,$ then $$\|(R^\ast_{\pi T}f)\|_{X^\infty_c} = \mathcal{O}(T^{-r+1/2}) \qquad (T \rightarrow +\infty).$$ \end{enumerate} \end{Theorem} \subsection{A sampling formula for Mellin derivatives} In the context of Fourier analysis the following differentiation formula has been considered: \begin{eqnarray}\label{diff1} f'(x)\,=\, \frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2} f\left(x+ \frac{2k-1}{2T}\right) \qquad (x\in\mathbb{R}). \end{eqnarray} It holds for all entire functions of exponential type $\pi T$ which are bounded on the real line. In particular, it holds for trigonometric polynomials of degree at most $\lfloor \pi T\rfloor$, where $\lfloor \pi T\rfloor$ denotes the integral part of $\pi T,$ and in this case the series on the right-hand side can be reduced to a finite sum. The formula for trigonometric polynomials was discovered by Marcel Riesz \cite{RIE} in 1914. Its generalization (\ref{diff1}) is due to Boas \cite{BOA}. Some authors refer to (\ref{diff1}) as the {\it generalized Riesz interpolation formula}, others name it after Boas. Formula (\ref{diff1}) has several interesting applications. It provides a very short proof of Bernstein's inequality in $L^p(\mathbb{R})$ for all $p\in [1, \infty]$. Modified by introducing a Gaussian multiplier, it leads to a stable algorithm of high precision for numerical differentiation (see \cite{SCH2}). Furthermore, it has been extended to higher order derivatives (see \cite{BSS1}, \cite{SCH2}). The following theorem gives an analogue of (\ref{diff1}) for Mellin derivatives. \begin{Theorem}\label{d_thm1} For $f\in B_{c,\pi T}^\infty$ there holds \begin{eqnarray}\label{d_thm1.1} \Theta_cf(x)\,=\, \frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2} \,e^{(k-1/2)c/T} f\left(x e^{(k-1/2)/T}\right)\qquad (x\in\mathbb{R}^+). \end{eqnarray} \end{Theorem} {\bf Proof}. \, Formula (\ref{d_thm1.1}) could be deduced from (\ref{diff1}) by making use of the relationship between the Fourier transform and the Mellin transform. In the following we give an independent proof completely within Mellin analysis. First assume that, in addition, \begin{eqnarray} \label{d_thm1p1} x^c f(x)\,=\, \mathcal{O}\left(\frac{1}{|\log x|}\right) \qquad (x\rightarrow 0_+ \hbox{ and } x\rightarrow \infty). \end{eqnarray} Then the exponential sampling formula applies to $f$ and yields $$ f(x)\,=\, \sum_{k\in\mathbb{Z}} f\left(e^{k/T}\right) \mbox{lin}_{c/T}\left(e^{-k}x^T\right).$$ The series converges absolutely and uniformly on compact subsets of $\mathbb{R}^+$. When we apply the differentiation operator $\Theta_c$ with respect to $x$, we may interchange it with the summation on the right-hand side. Thus, \begin{eqnarray}\label{d_thm1p2} \Theta_c f(x)\,=\, \sum_{k\in\mathbb{Z}} f\left(e^{k/T}\right) \Theta_c \mbox{lin}_{c/T}\left(e^{-k}x^T\right). \end{eqnarray} By a calculation we find that $$\Theta_c\mbox{lin}_{c/T}\left(e^{-k}x^T\right)\,=\, T\, \frac{e^{kc/T}x^{-c} \cos(\log(e^{-\pi k}x^{\pi T})) - \mbox{lin}_{c/T}(e^{-k}x^T)}{\log(e^{-k}x^T)}\,.$$ The complicated cosine term disappears at $x=e^{1/(2T)}$. Then (\ref{d_thm1p2}) becomes \begin{eqnarray}\label{d_thm1p3} \Theta_cf(e^{1/(2T)})\,=\,\frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2}\, e^{(k-1/2)c/T} f\left(e^{k/T}\right). \end{eqnarray} In order to obtain the $\Theta_c$ derivative of $f$ at $x$, we consider the function $g\,:\, t\mapsto f(xe^{-1/(2T)}t).$ It satisfies the assumptions used for deducing (\ref{d_thm1p3}). Now, applying (\ref{d_thm1p3}) to $g$, we arrive at the desired formula (\ref{d_thm1.1}). We still have to get rid of the additional assumption (\ref{d_thm1p1}). If $f$ is any function in $B_{c,\pi T}^\infty$, then $$ f_\varepsilon(x)\,:=\,f(x^{1-\varepsilon}) \,x^{c\varepsilon(T-1)} \mbox{lin}_c(x^{\varepsilon T})$$ belongs to $B_{c,\pi T}^\infty$ for each $\varepsilon \in (0, 1)$ and it satisfies (\ref{d_thm1p1}). Applying (\ref{d_thm1.1}) to $f_\varepsilon$ and letting $\varepsilon \rightarrow 0_+$, we find that (\ref{d_thm1.1}) holds for $f$ as well. $\Box$ \vskip0,3cm We note that formula (\ref{d_thm1.1}) yields a very short proof for a Bernstein-type inequality for Mellin derivatives in $L^p$ norms for any $p\in [1, \infty].$ Indeed, by the triangular inequality for norms we have \begin{eqnarray}\label{Bernst1} \left\|\Theta_cf\right\|_{X_c^p}\,\leq\, \frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{1}{(2k-1)^2}\,e^{(k-1/2)c/T}\,\left\|f( \cdot \,e^{(k-1/2)/T})\right\|_{X_c^p}. \end{eqnarray} It is easily verified that for any positive $a$, there holds $$\left\|f( \cdot\, a)\right\|_{X_c^p}\,=\, a^{-c} \|f\|_{X_c^p}\,.$$ Furthermore, it is known that $$\sum_{k\in\mathbb{Z}} \frac{1}{(2k-1)^2}\,=\, \frac{\pi^2}{4}.$$ Thus, it follows from (\ref{Bernst1}) that \begin{eqnarray}\label{Bernst2} \|\Theta_cf\|_{X_c^p}\,\leq\, \pi T \|f\|_{X_c^p}\,. \end{eqnarray} Inequality (\ref{Bernst2}) in conjunction with Theorem \ref{derivative} and the Mellin inversion formula shows that if $f\in B_{c,\pi T}^p$ for some $p\in[1, \infty]$, then $\Theta_cf \in B_{c,\pi T}^p$ as well. If $f$ does not belong to $B_{c,\pi T}^\infty$ but the two sides of formula (\ref{d_thm1.1}) exist, we may say that (\ref{d_thm1.1}) holds with a remainder $(R^B_{\pi T} f)(x)$ defined as the deviation of the right-hand side from $\Theta_cf(x)$. We expect that $|(R^B_{\pi T}f)(x)|$ is small if $\Theta_cf$ is close to $B_{c,\pi T}^\infty.$ For the Mellin inversion class for $p=2$, $M_c^2$, we may state a precise result as follows. \begin{Theorem}\label{d_thm2} Let $f\in \mathcal{M}_c^2$ and suppose that $v [f]_{M_c}^\wedge(c+iv)$ is absolutely integrable on $\mathbb{R}$ with respect to $v$. Then, for any $T>0$ and $x\in\mathbb{R}^+$, we have \begin{eqnarray}\label{d_thm2.1} \Theta_cf(x)\,=\, \frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2} \,e^{(k-1/2)c/T} f\left(x e^{(k-1/2)/T}\right) + (R^B_{\pi T}f)(x), \end{eqnarray} where \begin{eqnarray}\label{d_thm2.2} (R^B_{\pi T}f)(x)\,=\, \frac{1}{2\pi i} \int_{|v|\geq\pi T} \left[v- \pi T\phi\left(\frac{v}{\pi T}\right)\right] [f]_{M_c}^\wedge(c+iv) x^{-c-iv}dv \end{eqnarray} with \begin{eqnarray}\label{d_thm2.3} \phi(v)\,:=\, \bigg|v+1 -4\left\lfloor\frac{v+3}{4}\right\rfloor \bigg|-1. \end{eqnarray} In particular, \begin{eqnarray}\label{d_thm2.4} |(R^B_{\pi T}f)(x)|\,\leq\, \frac{x^{-c}}{2\pi} \int_{|v|\geq \pi T} \left(|v|+\pi T\right)|[f]_{M_c}^\wedge(c+iv)|dv \end{eqnarray} and \begin{eqnarray}\label{d_thm2.5} \|R^B_{\pi T}f\|_{X_c^\infty}\,\leq\, \frac{1}{\pi} \mbox{\rm dist}_1\left(\Theta_cf, B_{c,\pi T}^2\right). \end{eqnarray} \end{Theorem} \begin{figure} \caption{ The graph of the function $\phi$. } \label{phi} \end{figure} {\bf Proof.}\, Define \begin{eqnarray}\label{d_thm2p1} f_1(x)\,:=\, \frac{1}{2\pi} \int_{|v|\geq \pi T} [f]_{M_c}^\wedge(c+iv) x^{-c-iv}dv. \end{eqnarray} Then $f-f_1\in B_{c,\pi T}^\infty$, and so (\ref{d_thm1.1}) applies. It yields that \begin{eqnarray}\label{d_thm2p2} (R^B_{\pi T}f)(x)\,=\, \Theta_c f_1(x) - \frac{4T}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2}\, e^{(k-1/2)c/T}\,f_1\left(xe^{(k-1/2)/T}\right). \end{eqnarray} We know from Theorem~\ref{derivative} that $$\Theta_c f_1(x)\,=\, \frac{1}{2\pi i} \int_{|v|\geq \pi T} v [f]_{M_c}^\wedge(c+iv) x^{-c-iv} dv.$$ Furthermore, by (\ref{d_thm2p1}), $$f_1\left(x e^{(k-1/2)/T}\right)\,=\, \frac{1}{2\pi} \int_{|v|\geq \pi T} [f]_{M_c}^\wedge(c+iv)\left( x e^{(k-1/2)/T}\right)^{-c-iv} dv.$$ Using these integral representations and interchanging summation and integration, which is allowed by Levi's theorem, we may rewrite (\ref{d_thm2p2}) as \begin{eqnarray}\label{d_thm2p3} (R^B_{\pi T} f)(x)\,=\, \frac{1}{2\pi i} \int_{|v|\geq \pi T} \bigg(v -\psi(v)\bigg) [f]_{M_c}^\wedge(c+iv) x^{-c-iv} dv, \end{eqnarray} where $$ \psi(v)\,:=\, \frac{4Ti}{\pi} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k+1}}{(2k-1)^2}\, e^{-i(k-1/2)v/T}.$$ Now, for $v\in\mathbb{R}$, consider the function $g_v\,:\, x \mapsto ix^{-iv}.$ We note that $g_v \in B_{0,\pi T}^\infty$ if $|v|\leq \pi T.$ Hence $g_v$ satisfies the hypotheses of Theorem~\ref{d_thm1} for $c=0$ and this restriction on $v$. Since $\Theta_0 g_v(1)=v$, we find by applying (\ref{d_thm1.1}) to $g_v$ with $c=0$ and $x=1$ that $\psi(v)= v$ for $v\in [-\pi T, \pi T].$ We also note that $\psi(v+ 2\pi T)=-\psi(v)$ and (consequently) $\psi(v+4\pi T) = \psi(v).$ Hence $\psi$ is a $4\pi T$-periodic function that is given on the interval $[-\pi t, 3\pi T]$ by \begin{eqnarray*} \psi(v)\, =\left\{ \begin{array}{cl} v & \hbox{ if } -\pi T\le v \le \pi T,\\ \\ 2\pi T -v & \hbox{ if } \pi T \leq v \leq 3\pi T. \end{array} \right. \end{eqnarray*} Thus, using the function $\phi$ defined in (\ref{d_thm2.3}), whose graph is shown in Fig.~\ref{phi}, we can express $\psi(v)$ as $\pi T\phi(v/(\pi T)).$ Hence (\ref{d_thm2p3}) implies (\ref{d_thm2.2}). Inequalities (\ref{d_thm2.4}) and (\ref{d_thm2.5}) are easily obtained by noting that $|\phi(v)|\leq 1$ for $v\in\mathbb{R}$ and by recalling Corollary~\ref{cor1}. $\Box$ \subsection{An extension of the Bernstein-type inequality} Just the same way as we deduced (\ref{Bernst2}) from (\ref{d_thm1.1}), we may use (\ref{d_thm2.1}) to obtain $$ \|\Theta_c f\|_{X_c^p}\,\leq\, \pi T \|f\|_{X_c^p} + \|R^B_{\pi T} f\|_{X_c^p}.$$ For $p=2$ we can profit from the isometry of the Mellin transform expressed by the formula $$ \|f\|_{X_c^2}\,=\, \frac{1}{\sqrt{2\pi}}\left(\int_\mathbb{R} |[f]_{M_c}^\wedge(c+iv)|^2dv \right)^{1/2},$$ obtaining the following theorem \begin{Theorem}\label{Bernapprox} Under the assumptions of Theorem \ref{d_thm2} we have $$\|\Theta_c f\|_{X_c^2} \,\leq\,\pi T \|f\|_{X_c^2} + \frac{1}{\sqrt{2\pi}} \,\mbox{\rm dist}_2(\Theta_c f, B_{c,\pi T}^2)$$ for any $T>0$. \end{Theorem} {\bf Proof}. With $f_1$ defined in (\ref{d_thm2p1}) and $f_0:=f-f_1$, we have \begin{eqnarray}\label{Bernst3} \|\Theta_c f\|_{X_c^2}\,\leq\, \|\Theta_c f_0\|_{X_c^2} + \|\Theta_c f_1\|_{X_c^2} \,\leq\, \pi T \|f_0\|_{X_c^2} +\|\Theta_c f_1\|_{X_c^2} \end{eqnarray} since (\ref{Bernst2}) applies to $f_0$. We are going to estimate the quantities on the right-hand side in terms of $ f$. Using the isometry of the Mellin transform, we find that \begin{align*} \|f\|_{X_c^2}^2 &=\, \frac{1}{2\pi} \int_\mathbb{R}|[f]_{M_c}^\wedge(c+iv)|^2 dv\\ &=\, \frac{1}{2\pi}\left[ \int_{|v|\leq \pi T}|[f]_{M_c}^\wedge(c+iv)|^2 dv + \int_{|v|\geq \pi T}|[f]_{M_c}^\wedge(c+iv)|^2 dv\right] \\ &=\, \|f_0\|_{X_c^2}^2 +\|f_1\|_{X_c^2}^2\,, \end{align*} which implies that $\|f_0\|_{X_c^2} \leq\|f\|_{X_c^2}.$ Next we note that \begin{align*} \|\Theta_c f_1\|_{X_c^2} &=\, \frac{1}{\sqrt{2\pi}} \left(\int_\mathbb{R} |\left[\Theta_cf_1\right]_{M_c}^\wedge(c+iv)|^2dv\right)^{1/2}\\ &=\,\frac{1}{\sqrt{2\pi}} \left(\int_\mathbb{R} |v\left[f_1\right]_{M_c}^\wedge(c+iv)|^2 dv\right)^{1/2}\\ &=\,\frac{1}{\sqrt{2\pi}} \left(\int_{|v|\geq\pi T}|v[f]_{M_c}^\wedge(c+iv)|^2 dv\right)^{1/2}\\ &=\, \frac{1}{\sqrt{2\pi}} \mbox{dist}_2(\Theta_c f, B_{c,\pi T}^2). \end{align*} Thus (\ref{Bernst3}) implies the assertion. $\Box$ \vskip0,4cm \noindent {\bf Aknowledgments}. Carlo Bardaro and Ilaria Mantellini have been partially supported by the ``Gruppo Nazionale per l'Analisi Matematica e Applicazioni (GNAMPA) of the ``Istituto Nazionale di Alta Matematica'' (INDAM) as well as by the Department of Mathematics and Computer Sciences of the University of Perugia. \flushright{{\footnotesize \today}} \end{document} \end{document}

\begin{document} \title{An Imputation-Consistency Algorithm for High-Dimensional Missing Data Problems and Beyond} \author{} \author{Faming Liang, Bochao Jia, Jingnan Xue, Qizhai Li, Ye Luo\thanks{ F. Liang is with Department of Statistics, Purdue University, West Lafayette, IN 47907, email: [email protected]; B. Jia is with Department of Biostatistics, University of Florida, Gainesville, FL 32611; J. Xue is with Department of Statistics, Texas A\&M University, College Station, TX 77843. Q. Li is with Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100864, China. Y. Luo is with Department of Economics, University of Florida, Gainesville, FL 32611. } } \maketitle \begin{abstract} Missing data are frequently encountered in high-dimensional problems, but they are usually difficult to deal with using standard algorithms, such as the expectation-maximization (EM) algorithm and its variants. To tackle this difficulty, some problem-specific algorithms have been developed in the literature, but there still lacks a general algorithm. This work is to fill the gap: we propose a general algorithm for high-dimensional missing data problems. The proposed algorithm works by iterating between an imputation step and a consistency step. At the imputation step, the missing data are imputed conditional on the observed data and the current estimate of parameters; and at the consistency step, a consistent estimate is found for the minimizer of a Kullback-Leibler divergence defined on the pseudo-complete data. For high dimensional problems, the consistent estimate can be found under sparsity constraints. The consistency of the averaged estimate for the true parameter can be established under quite general conditions. The proposed algorithm is illustrated using high-dimensional Gaussian graphical models, high-dimensional variable selection, and a random coefficient model. \underline{Keywords:} EM Algorithm; Gaussian Graphical Model; Gibbs Sampler; Random Coefficient Model; Variable Selection. \end{abstract} {\centering \section{Introduction}} Missing data are frequently encountered in both low and high-dimensional data, where low and high refer to that the number of variables is smaller or larger than the sample size, respectively. For example, the microarray data is usually considered as high-dimensional, where the number of genes can be much larger than the number of samples. Missing values can appear in microarray data due to various factors such as scratches on slides, spotting problems, experimental errors, etc. In some microarray experiments, missing values can occur for more than 90\% of the genes (Ouyang et al., 2004). Simply deleting the samples or genes for which missing values occur can lead to a significant loss of information of the data. How to deal with missing data has been a long-standing problem in statistics. For low-dimensional problems, the missing data can be dealt with using the EM algorithm (Dempster et al., 1977) or its variants. Let ${\boldsymbol X}^{{\rm obs}}=(X^{{\rm obs}}_1,X^{{\rm obs}}_2,\ldots,X^{{\rm obs}}_n)$ denote the observed incomplete data, where $n$ denotes the sample size. Let ${\boldsymbol X}^{{\rm mis}}=(X^{{\rm mis}}_1, X^{{\rm mis}}_2,\ldots, X^{{\rm mis}}_n)$ denote the missing data, and let ${\boldsymbol X}=({\boldsymbol X}^{{\rm obs}},{\boldsymbol X}^{{\rm mis}})$ denote the complete data. Let ${\boldsymbol \theta}$ denote the vector of unknown parameters, and let $f({\boldsymbol X}|{\boldsymbol \theta})$ denote the likelihood function of the complete data. Then the maximum likelihood estimate (MLE) of ${\boldsymbol \theta}$ can be determined by maximizing the marginal likelihood of the observed data, \[ f({\boldsymbol X}^{{\rm obs}}|{\boldsymbol \theta})=\int f({\boldsymbol X}^{{\rm obs}},{\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}) h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}, {\boldsymbol X}^{{\rm obs}}) d{\boldsymbol x}^{{\rm mis}}, \] where $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta},{\boldsymbol X}^{{\rm obs}})$ denotes the predictive density of the missing data. The EM algorithm seeks to maximize the marginal likelihood function by iterating between the following two steps: \begin{itemize} \item {\bf E-step}: {\it Calculate the expected value of the log-likelihood function with respect to the predictive distribution of the missing data given the current estimate ${\boldsymbol \theta}^{(t)}$, i.e., } \[ Q({\boldsymbol \theta}|{\boldsymbol \theta}^{(t)})=\int \log f({\boldsymbol X}^{{\rm obs}},{\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}) h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}^{(t)},{\boldsymbol X}^{{\rm obs}}) d{\boldsymbol x}^{{\rm mis}}. \] \item {\bf M-step}: {\it Find a value of ${\boldsymbol \theta}$ that maximizes the quantity $Q({\boldsymbol \theta}|{\boldsymbol \theta}^{(t)})$, i.e., set } \[ {\boldsymbol \theta}^{(t+1)}=\arg\max_{{\boldsymbol \theta}} Q({\boldsymbol \theta}|{\boldsymbol \theta}^{(t)}). \] \end{itemize} Dempster et al. (1977) showed that the marginal likelihood value increases with each iteration and, under fairly general conditions, it converges to a local or global maximum of the marginal likelihood. A rigorous study for the convergence is given by Wu (1983). Both the $E$ and $M$-steps of the algorithm can be rather complicated or even intractable. Meng and Rubin (1993) found that in many cases, the $M$-step is relatively simple when conditioned on some function of the parameters under estimation. Motivated by this observation, they introduced the expectation-conditional maximization (ECM) algorithm, which is to replace the M-step by a number of computationally simpler conditional maximization steps. Later, the EM algorithm was further speeded up by some other variants, such as the ECME algorithm (Liu and Rubin, 1994, He and Liu, 2012) and the PX-EM algorithm (Liu et al., 1998). When the E-step is analytically intractable, Wei and Tanner (1990) introduced the Monte Carlo EM algorithm, which is to simulate multiple missing values from the predictive distribution $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}^{(t)},{\boldsymbol X}^{{\rm obs}})$ at the $(t+1)th$ iteration, and then maximize the approximate conditional expectation of the complete-data log-likelihood \[ \widehat{Q}({\boldsymbol \theta}|{\boldsymbol \theta}^{(t)}) = \frac{1}{m} \sum_{j=1}^m \log f({\boldsymbol X}^{{\rm obs}},{\boldsymbol X}^{{\rm mis}}_j|{\boldsymbol \theta}), \] which converges to $ Q({\boldsymbol \theta}|{\boldsymbol \theta}^{(t)})$ as $m\to \infty$, where ${\boldsymbol X}^{{\rm mis}}_1,\ldots, {\boldsymbol X}^{{\rm mis}}_m$ denote the missing values simulated from $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol \theta}^{(t)}$, ${\boldsymbol X}^{{\rm obs}})$. When the dimension of ${\boldsymbol X}^{{\rm mis}}$ is high, the Monte Carlo approximation can be rather expensive. An alternative algorithm to deal with the intractable E-step is the stochastic EM (SEM) algorithm (Celeux and Diebolt, 1985). In this algorithm, the E-step is replaced by an imputation step, where the missing data are imputed with plausible values conditioned on the observed data and the current parameter estimate. At the M-step, the parameters are estimated by maximizing the likelihood function of the pseudo-complete data. Unlike the deterministic EM algorithm, the imputation-step and M-step of the SEM algorithm generate a Markov chain which converges to a stationary distribution whose mean is close to the MLE and whose variance reflects the information loss due to the missing data (Nielsen, 2000). Although EM and its variants work well for low-dimensional problems, see McLachlan and Krishnan (2008) for an overview, they essentially fail for high-dimensional problems. For the latter, the MLE can be non-unique or inconsistent. To address this issue, some problem-specific algorithms have been proposed, see e.g., misgLasso (St\"adler and B\"uhlmann, 2012), misPALasso (St\"adler et al., 2014), and matrix completion algorithms (Cai et al. 2010; Mazumder et al., 2010). MisgLasso is specifically designed for estimating Gaussian graphical models in presence of missing data. Similar to misgLasso, MisPALasso also deals with multivariate Gaussian data in presence of missing data. The matrix completion algorithm deals with large incomplete matrices, which is to learn a low-rank approximation for a large-scale matrix with missing entries. However, there still lacks a general algorithm for high-dimensional missing data problems. This work is to fill the gap: we propose a general algorithm for dealing with high-dimensional missing data problems. The proposed algorithm consists of two steps, an imputation step and a consistency step, and is therefore called an imputation-consistency (IC) algorithm. The imputation step is to impute the missing data with plausible values conditioned on the observed data and the current estimate of the parameters. The consistency step is to find a consistent estimate for the minimizer of a Kullback-Leibler divergence defined on the pseudo-complete data. For high dimensional problems, the consistent estimate is suggested to be found under sparsity constraints. Like the SEM algorithm, the IC algorithm generates a Markov chain which converges to a stationary distribution. Under mild conditions, we show that the mean of the stationary distribution converges to the true value of the parameters in probability as the sample size becomes large. For low-dimensional problems, the SEM algorithm can be viewed as a special case of the IC algorithm. The IC algorithm has strong implications for big data computing: Based on it, we propose a general strategy to improve Bayesian computation for big data. The IC algorithm also facilitates data integration from multiple sources, which plays an important role in big data analysis. A R package accompanying this paper is currently available at {\it http://www.stat.purdue.edu/$\sim$fmliang} and later will be distributed to the public via CRAN upon the acceptance of the paper. The remainder of this paper is organized as follows. Section 2 describes the IC algorithm with the theoretical development deferred to the Appendix. Section 3 applies the IC algorithm to high-dimensional Gaussian graphical models. Section 4 applies the IC algorithm to high-dimensional variable selection. Section 5 applies the IC algorithm to a random coefficient model and discusses its potential use for big data problems. Section 6 concludes the paper with a brief discussion. {\centering \section{The Imputation-Consistency Algorithm}} \subsection{The IC Algorithm} Let $X_{1},\dots,X_{n}$ denote a random sample drawn from the distribution $f(x|{\boldsymbol \theta})$ (also denoted by $f_{{\boldsymbol \theta}}(x)$ depending on convenience), where ${\boldsymbol \theta}$ is a vector of parameters. Let $X_i=(X^{{\rm obs}}_i,X^{{\rm mis}}_i)$, $i=1,\ldots, n$, where $X^{{\rm obs}}_i$ is observed and $X^{{\rm mis}}_i$ is missed. Let ${\boldsymbol X}=(X_1,\ldots, X_n)$, ${\boldsymbol X}^{{\rm obs}}=(X^{{\rm obs}}_1,\ldots,X^{{\rm obs}}_n)$ and ${\boldsymbol X}^{{\rm mis}}=(X^{{\rm mis}}_1,\ldots, X^{{\rm mis}}_n)$. To indicate the dependence of the dimension of ${\boldsymbol \theta}$ on the sample size $n$, we also write ${\boldsymbol \theta}$ as ${\boldsymbol \theta}_n$ and denote by ${\boldsymbol \theta}_n^{(t)}$ the estimate of ${\boldsymbol \theta}$ obtained at the $t^{th}$ iteration of the IC algorithm. The IC algorithm works by starting with an initial guess ${\boldsymbol \theta}_n^{(0)}$ and then iterating between the imputation and consistency steps: \begin{itemize} \item {\bf I-step}: {\it Draw ${\tilde{\boldsymbol X}}^{{\rm mis}}$ from the predictive distribution $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol X}^{{\rm obs}}, {\boldsymbol \theta}_n^{(t)})$ given ${\boldsymbol X}^{{\rm obs}}$ and the current estimate ${\boldsymbol \theta}_n^{(t)}$.} \item {\bf C-step}: {\it Based on the pseudo-complete data $\tilde{{\boldsymbol X}}=({\boldsymbol X}^{{\rm obs}},{\tilde{\boldsymbol X}}^{{\rm mis}})$, find an updated estimate ${\boldsymbol \theta}_n^{(t+1)}$ which forms a consistent estimate of \begin{equation} \label{Cequation} {\boldsymbol \theta}_*^{(t)}=\arg\max_{{\boldsymbol \theta}} E_{{\boldsymbol \theta}_n^{(t)}} \log f_{{\boldsymbol \theta}}(\tilde{{\boldsymbol x}}), \end{equation} where $E_{{\boldsymbol \theta}_n^{(t)}} \log f_{{\boldsymbol \theta}}(\tilde{{\boldsymbol x}}) = \int \log(f({\boldsymbol x}^{{\rm obs}}, {\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol \theta}))f({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}^*) h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_n^{(t)}) d{\boldsymbol x}^{{\rm obs}} d {\tilde{\boldsymbol x}}^{{\rm mis}}$, ${\boldsymbol \theta}^*$ denotes the true value of the parameters, and $f({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}^*)$ denotes the marginal density function of ${\boldsymbol x}^{{\rm obs}}$. } \end{itemize} To find a consistent estimate of ${\boldsymbol \theta}_*^{(t)}$, which is the minimizer of the Kullback-Leibler divergence from $f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta})$ to the joint density $f({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}^*) h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_n^{(t)})$, sparsity constraints can be imposed on ${\boldsymbol \theta}$ for high-dimensional problems. In general, we have two ways. The first way is via regularization methods. Corollary \ref{cor0A} in the Appendix shows that the regularization methods can be employed here to find consistent estimates for ${\boldsymbol \theta}_*^{(t)}$'s with appropriate penalty functions. For regularization methods, we recommend to use the same penalty functions as they would use if there are no missing data. The second way is via sure screening-based methods, which are to first reduce the space of ${\boldsymbol \theta}_*^{(t)}$ to a low-dimensional subspace and then find a consistent estimate of ${\boldsymbol \theta}_*^{(t)}$ in the low-dimensional subspace using a conventional statistical method, such as maximum likelihood, moment estimation or even regularization. In the Appendix, we point out that the sure screening-based methods can be viewed as a subclass of regularization methods, for which the solutions in the low-dimensional subspace receives a zero penalty and those outside the subspace receives a penalty of $\infty$. Such a binary-type penalty function satisfies the condition (C1) we imposed on regularization methods. Other than the regularization and sure screening-based methods, we justify in Corollary \ref{cor0} and Remark (R3) the use of general consistent estimation procedures in the IC algorithm, provided that the resulting estimates are accurate enough at each iteration $t$. For low-dimensional problems, the consistent estimator of ${\boldsymbol \theta}_*^{(t)}$ can be obtained by maximizing the pseudo-complete likelihood function. In this sense, the SEM algorithm can be viewed as a special case of the IC algorithm. It is easy to see that by simulating new independent missing values at each iteration, the sequence of estimates, $\{{\boldsymbol \theta}_n^{(t)} \}$, forms a time-homogeneous Markov chain. Also, the imputed values at different iterations form a Markov chain. The two Markov chains are interleaved and share many properties, such as irreducibility, aperiodicity and ergodicity. Refer to Nielsen (2000) for more discussions on this issue. In Theorem \ref{them1} and Theorem \ref{them2} of the Appendix, we prove that the Markov chain $\{{\boldsymbol \theta}_n^{(t)} \}$ has a stationary distribution and, furthermore, the mean of the stationary distribution forms a consistent estimate of ${\boldsymbol \theta}^*$. Like for other Markov chains, a good initial value will accelerate the convergence of the simulation. There are many different ways to specify initial values for the IC algorithm. In most examples of this paper, we started the simulation with an I-step, where all missing values are filled by the median of the variable. This method is simple and usually works when the missing rate is not high. However, when we perceive that such a constant filling method does not work well, we may start the simulation with a C-step. In this case, the initial estimate ${\boldsymbol \theta}_n^{(0)}$ may be obtained based on the complete samples (i.e., those without missing information) only. We note that many of the assumptions we made for proving the convergence of the IC algorithm are quite regular. For example, we assumed that $\log f_{{\boldsymbol \theta}}(\tilde{x})$ is a continuous function of ${\boldsymbol \theta}$ for each $\tilde{x} \in {\cal X}$ and a measurable function of $\tilde{x}$ for each ${\boldsymbol \theta}$. Since we aim to address the missing data issue for a wide range of problems and it is hard to specify the structure of each problem, we incorporate the assumptions about the parameters and problem structures into a metric entropy condition, see condition (A2). As discussed in Remark (R1) of the Appendix, this condition allows $p$ to grow with $n$ at a polynomial rate $O(n^\gamma)$ for some constant $0<\gamma<\infty$, and allows the number of nonzero elements in ${\boldsymbol \theta}$ to grow with $n$ at a rate of $O(n^{\alpha})$ for some $0<\alpha<1/2$. These rates seem a little more restrictive than the exponential rate, i.e., $\log(p)=n^{b}$ for some constant $0<b<1$, seeking for in the literature of high-dimensional regression. However, more or less, they are just some technical conditions. Moreover, our theory is more general and can be applied to many other problems. Note that the metric entropy condition has often been used in studying the minimax rate of estimation under the high-dimensional scenario, see e.g. Raskutti et al. (2011). Regarding conditions on missing data, we note that the IC algorithm essentially works with any missing data mechanism, as long as the predictive distribution $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol X}^{{\rm obs}}, {\boldsymbol \theta}_n^{(t)})$ is available, well behaved, and unchanged with the sample size $n$. Our current theory rules out the case that the missing data mechanism changes as the sample size increases, e.g., the missing rate increases due to increased wear and tear on measurement instruments or fatigue among data subjects measured later in the study. Our condition (A3) constrains the behavior of $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol X}^{{\rm obs}}, {\boldsymbol \theta}_n^{(t)})$ via some moment conditions on the log-likelihood function of the pseudo-complete data. It implies that a high missing rate may hurt the performance of the method. \subsection{An Extension of the IC Algorithm} Like the EM algorithm, the IC algorithm is attractive only when the consistent estimate of ${\boldsymbol \theta}_*^{(t)}$ can be easily obtained at each C-step. We found that for many problems, similar to the ECM algorithm (Meng and Rubin, 1993), the consistent estimate of ${\boldsymbol \theta}_*^{(t)}$ can be easily obtained with a number of conditional consistency steps. That is, we can partition the parameter ${\boldsymbol \theta}$ into a number of blocks and then find the consistent estimator for each block conditioned on the current estimates of other blocks. Note that for many problems, e.g., the examples studied in Sections 4 and 5, the partitioning of ${\boldsymbol \theta}$ is natural. Suppose that ${\boldsymbol \theta}=({\boldsymbol \theta}^{(1)}, \ldots, {\boldsymbol \theta}^{(k)})$ has been partitioned into $k$ blocks. The imputation-conditional consistency (ICC) algorithm can be described as follows: \begin{itemize} \item {\bf I-step}. Draw ${\tilde{\boldsymbol X}}^{{\rm mis}}$ from the conditional distribution $h({\boldsymbol x}^{{\rm mis}}|{\boldsymbol X}^{{\rm obs}}, {\boldsymbol \theta}_n^{(t,1)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})$ given ${\boldsymbol X}^{{\rm obs}}$ and the current estimate ${\boldsymbol \theta}_n^{(t)}=({\boldsymbol \theta}_n^{(t,1)},\ldots, {\boldsymbol \theta}_n^{(n,k)})$. \item {\bf CC-step}. Based on the pseudo-complete data $\tilde{{\boldsymbol X}}=({\boldsymbol X}^{{\rm obs}},{\tilde{\boldsymbol X}}^{{\rm mis}})$, do the following: \begin{itemize} \item[(1)] Conditioned on $({\boldsymbol \theta}_n^{(t,2)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})$, find ${\boldsymbol \theta}_n^{(t+1,1)}$ which forms a consistent estimate of \[ {\boldsymbol \theta}_*^{(t,1)}=\arg\max_{{\boldsymbol \theta}^{(t,1)'}} E_{{\boldsymbol \theta}_n^{(t,1)},\ldots, {\boldsymbol \theta}_n^{(t,k)}} \log f(\tilde{{\boldsymbol x}}| {\boldsymbol \theta}_n^{(t,1)'}, {\boldsymbol \theta}_n^{(t,2)}, \ldots, {\boldsymbol \theta}_n^{(t,k)} ), \] where the expectation is taken with respect to the joint distribution function of $\tilde{{\boldsymbol x}}=({\boldsymbol x}^{{\rm obs}},{\boldsymbol x}^{{\rm mis}})$ and the subscript of $E$ gives the current estimate of ${\boldsymbol \theta}$. \item[(2)] Conditioned on $({\boldsymbol \theta}_n^{(t+1,1)}, {\boldsymbol \theta}_n^{(t,3)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})$, find ${\boldsymbol \theta}_n^{(t+1,2)}$ which forms a consistent estimate of \[ {\boldsymbol \theta}_*^{(t,2)}=\arg\max_{{\boldsymbol \theta}^{(t,2)'}} E_{{\boldsymbol \theta}_n^{(t+1,1)}, {\boldsymbol \theta}_n^{(t,2)}, {\boldsymbol \theta}_n^{(t,3)}, \ldots, {\boldsymbol \theta}_n^{(t,k)}} \log f(\tilde{{\boldsymbol x}}| {\boldsymbol \theta}_n^{(t+1,1)}, {\boldsymbol \theta}_n^{(t,2)'}, {\boldsymbol \theta}_n^{(t,3)}, \ldots, {\boldsymbol \theta}_n^{(t,k)} ). \] \item[] $\ldots\ldots$ \item[(k)] Conditioned on $({\boldsymbol \theta}_n^{(t+1,1)}, \ldots, {\boldsymbol \theta}_n^{(t,k-1)})$, find ${\boldsymbol \theta}_n^{(t+1,k)}$ which forms a consistent estimate of \[ {\boldsymbol \theta}_*^{(t,k)}=\arg\max_{{\boldsymbol \theta}^{(t,k)'}} E_{{\boldsymbol \theta}_n^{(t+1,1)}, \ldots, {\boldsymbol \theta}_n^{(t+1,k-1)}, {\boldsymbol \theta}_n^{(t,k)}} \log f(\tilde{{\boldsymbol x}}| {\boldsymbol \theta}_n^{(t+1,1)}, \ldots, {\boldsymbol \theta}_n^{(t+1,k-1)}, {\boldsymbol \theta}_n^{(t,k)'} ). \] \end{itemize} \end{itemize} It is easy to see that the sequence $\{({\boldsymbol \theta}_n^{(t,1)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})\}$ forms a Markov chain. The convergence of the Markov chain can be studied under similar conditions as the IC algorithm. In Theorem \ref{them3} and Theorem \ref{them4} (see Appendix), we prove that the Markov chain $\{{\boldsymbol \theta}_n^{(t)} \}$ has a stationary distribution and the mean of the stationary distribution forms a consistent estimate of ${\boldsymbol \theta}^*$. {\centering \section{Learning High-Dimensional Gaussian Graphical Models in Presence of Missing Data}} Gaussian graphical models (GGMs) have often been used in learning gene regulatory networks from microarray data, see e.g., Dobra et al. (2004) and Friedman et al. (2008). As mentioned in the Introduction, missing values can appear in microarray data due to many factors. To deal with missing values in microarray data, many imputation methods, such as single value decomposition (SVD) imputation (Troyanskaya et al., 2001), least-square imputation (Bo et al., 2004), and Bayesian principal component analysis (BPCA) imputation (Oba et al., 2003), have been proposed. Since these methods impute the missing values independent of the models under consideration, they are often ineffective. Moreover, the statistical inference based on the ``one-time'' imputed data is potentially biased, because the uncertainty of the missing values cannot be properly accounted for. In this section, we apply the IC algorithm to handle missing values for microarray data. The IC algorithm iteratively impute missing values based on the updated parameter estimate. Therefore, it overcomes the weakness of the ``one-time'' imputation methods, and improves accuracy of statistical inference. Let ${\boldsymbol X}=({\boldsymbol x}_1, \ldots, {\boldsymbol x}_n)^T$ denote a microarray dataset of $n$ samples and $p$ genes, where ${\boldsymbol x}_i$ is assumed to follow a multivariate Gaussian distribution $N_p({\boldsymbol \mu}, {\boldsymbol \Sigma})$. According to the theory of GGMs, estimation of the GGM is equivalent to identify non-zero elements of the concentration matrix (i.e., the inverse of the covariance matrix ${\boldsymbol \Sigma}$) or to identify non-zero partial correlation coefficients for different pairs of genes. During the recent years, a couple of methods have been proposed to estimate high-dimensional GGMs, e.g., graphical Lasso (Yuan and Lin, 2007; Friedman et al., 2008), node-wise regression (Meinshausen and B\"uhlmann, 2006), and $\psi$-learning (Liang et al., 2015). However, none of the methods can be directly applied in presence of missing data. \subsection{The IC Algorithm} To apply the IC algorithm to learn GGMs in presence of missing data, we choose the $\psi$-learning algorithm as the consistent estimation procedure used in the C-step. For GGMs, ${\boldsymbol \theta}$ corresponds to the concentration matrix, which can be uniquely determined from the network structure using the algorithm given in Hastie et al. (2009, p.634). Under mild conditions, Liang et al. (2015) showed that the $\psi$-learning algorithm provides a consistent estimator for Gaussian graphical networks. Refer to the Supplementary Material for a brief review of the algorithm. As mentioned in the Appendix, the $\psi$-learning algorithm belongs to the class of sure-screening-based methods, which are to first reduce the dimension of the solution space via correlation screening and then conduct GGM estimation via covariance selection (Dempster, 1972) which, by nature, produces a maximum likelihood estimate. As mentioned in Section 2.1, such a sure-screening-based method can be used in IC simulations. Other than the $\psi$-learning algorithm, node-wise regression and graphical Lasso can also be used, which both belong to the class of regularization methods and are consistent in Gaussian graphical network estimation. The Gaussian graphical network specifies the dependence between different genes, according to which the missing values can be imputed. For convenience, we let $A=(a_{jk})$ denote the adjacency matrix of a Gaussian graphical network, where $a_{jk}=1$ if an edge exists between node $j$ and node $k$ and 0 otherwise. For microarray data, a node corresponds to a gene. Let $x_{ij}$ denote a missing entry, and let $\omega(j)=\{k: a_{jk}=1 \}$ denote the neighborhood of node $j$. According to the faithfulness property of GGMs, conditional on the neighboring genes in $\omega(j)$, gene $j$ is independent of all other genes. Therefore, $x_{ij}$ can be imputed conditional on the expression values of the neighboring genes. Mathematically, we have \begin{equation}\label{10} \left(\begin{array}{c} x_{ij}\\ {\boldsymbol x}_{i\omega}\\ \end{array}\right) \sim N\left(\left(\begin{array}{c} \mu_j \\ {\boldsymbol \mu}_{\omega} \\ \end{array}\right),\left(\begin{array}{cc} \sigma_j^2 & {\boldsymbol \Sigma}_{j \omega}\\ {\boldsymbol \Sigma}_{j\omega}^T & {\boldsymbol \Sigma}_{\omega\omega}\\ \end{array}\right)\right), \end{equation} where ${\boldsymbol x}_{i\omega}=\{x_{ik}: k\in \omega(j)\}$, and $\mu_j$, ${\boldsymbol \mu}_{\omega}$, $\sigma_j^2$, ${\boldsymbol \Sigma}_{j \omega}$ and ${\boldsymbol \Sigma}_{\omega\omega}$ denote the corresponding mean and variance components. The mean and variance of $x_{ij}$ conditional on ${\boldsymbol x}_{i\omega}$ is thus given by \begin{equation} \label{condeq1} \mu_{ij|\omega} = \mu_j+{\boldsymbol \Sigma}_{j\omega} {\boldsymbol \Sigma}_{\omega\omega}^{-1} ({\boldsymbol x}_{i\omega}-{\boldsymbol \mu}_{\omega}), \quad \sigma_{ij|\omega} = \sigma_j^2- {\boldsymbol \Sigma}_{j\omega} {\boldsymbol \Sigma}_{\omega\omega}^{-1} {\boldsymbol \Sigma}_{j\omega}^T. \end{equation} As shown in Liang et al. (2015), for each gene, the neighborhood size can be upper bounded by $\lceil n/log(n)\rceil$, where $\lceil z\rceil$ denotes the smallest integer not smaller than $z$. Hence, in practice, $\sigma_j^2$, ${\boldsymbol \Sigma}_{j\omega}$ and ${\boldsymbol \Sigma}_{\omega\omega}$ can be directly estimated from the data. Let ${\boldsymbol s}_j^2$, ${\boldsymbol S}_{j\omega}$, ${\boldsymbol S}_{\omega\omega}$, $\bar{x}_j$ and $\bar{{\boldsymbol x}}_{\omega}$ denote the respective sample estimates of $\sigma_j^2$, ${\boldsymbol \Sigma}_{j\omega}$, ${\boldsymbol \Sigma}_{\omega\omega}$, $\mu_j$ and ${\boldsymbol \mu}_{\omega}$. Then, at each iteration, $x_{ij}$ can be imputed by sampling from the distribution \begin{equation} \label{imputeq1} X_{ij|\omega} \sim N(\bar{x}_j+{\boldsymbol S}_{j\omega} {\boldsymbol S}_{\omega\omega}^{-1} ({\boldsymbol x}_{i\omega}-\bar{{\boldsymbol x}}_{\omega}), {\boldsymbol s}_j^2- {\boldsymbol S}_{j\omega} {\boldsymbol S}_{\omega\omega}^{-1} {\boldsymbol S}_{j\omega}^T). \end{equation} In this way, exact evaluation of the concentration matrix can be skipped. In summary, we have the following algorithm for learning GGMs in presence of missing data: {\it \begin{itemize} \item (Initialization) Fill each missing entry by the median of the corresponding variable, and then iterates between the C- and I-steps. \item (C-step) Apply the $\psi$-learning algorithm to learn the structure of the Gaussian graphical network. \item (I-step) Impute missing values according to (\ref{imputeq1}) based on the network learned in the C-step. \end{itemize} } This algorithm outputs a series of Gaussian graphical networks. To integrate/average these networks into a single network, we adopt the $\psi$-score averaging approach suggested by Liang et al. (2015). Let $(\psi_{ij}^{(t)})$ denote the $\psi$-scores at iteration $t$, where are obtained from $\psi$-partial correlation coefficients via Fisher's transformation. Let $\bar{\psi}_{ij}=\sum_{t=1}^T \psi_{ij}^{(t)}/T$, $i,j=1,2,\ldots,p$ and $i\ne j$, denote the averaged $\psi$-score for gene $i$ and gene $j$. Then the averaged network can be obtained by applying a multiple hypothesis approach to threshold the averaged $\psi$-scores; if an averaged $\psi$-score is greater than the threshold value, we set the corresponding element of the adjacency matrix to 1 and 0 otherwise. The multiple hypothesis test can be done using the method of Liang and Zhang (2008), which can be viewed as a generalized empirical Bayesian method (Efron, 2004). The significance level of the multiple hypothesis test can be specified in terms of Storey's $q$-value (Storey, 2002). In this paper, we set it to $0.05$. \subsection{A Simulated Example} We consider an autoregressive process of order two with the concentration matrix given by \begin{equation}\label{plugin} C_{i,j}=\left\{\begin{array}{ll} 0.5,&\textrm{if $\left| j-i \right|=1, i=2,...,(p-1),$}\\ 0.25,&\textrm{if $\left| j-i \right|=2, i=3,...,(p-2),$}\\ 1,&\textrm{if $i=j, i=1,...,p,$}\\ 0,&\textrm{otherwise.} \end{array}\right. \end{equation} This example has been used by multiple authors, e.g., Yuan and Lin (2007), Mazumder and Hastie (2012), and Liang et al. (2015) to illustrate different GGM methods. In this paper, we generated multiple datasets with $n=200$ and different values of $p$=100, 200, 300 and 400. For each combination of $(n,p)$, we generated 10 datasets independently; and for each dataset, we randomly deleted 10\% of the observations as missing values. To evaluate the performance of the IC algorithm, the precision-recall curves were drawn by varying the threshold value of $\psi$-scores, where the precision is the fraction of true edges among the retrieved edges, and the recall is the fraction of true edges that have been retrieved over the total amount of true edges. \begin{figure} \caption{ Precision-recall curves resulted from different imputation methods for one simulated dataset with $p=400$: ``True'' refers to the curve obtained with complete data; ``misgLasso'' refers to the curve produced by the misgLasso algorithm; ``IC-Ave'' and ``IC-Last'' refer to the curves obtained with the $\psi$-scores generated in the last IC iteration and averaged over last 20 IC iterations, respectively; and ``Median'', ``BPCA'' and ``RegTree'' refer to the curves obtained with missing values imputed by the median filling, BPCA and regression tree methods, respectively. } \label{PRcurveEx1} \end{figure} For each dataset, the IC algorithm was run for 50 iterations. Figure \ref{PRcurveEx1} shows the resulting precision-recall curves for one dataset with $p=400$. For comparison, Figure \ref{PRcurveEx1} also includes the precision-recall curves produced by misgLasso and those produced by the $\psi$-learning algorithm with missing values imputed by the median filling, BPCA(Oba et al., 2003), and regression tree (Buuren and Groothuis-Oudshoorn, 2011) methods. The regression tree method has been implemented in the R package {\it MICE} and was applied to this example under its default setting. The misgLasso algorithm is a combination of the gLasso and EM algorithms, which is to integrate out the missing data as in the EM algorithm (see e.g., St\"adler and B\"uhlmann, 2012) and then learn the GGM using the gLasso algorithm. The misgLasso algorithm has been implemented in the R package {\it spaceExt} (He, 2011). Refer to Figure 1 of the Supplementary Material for the curves with other values of $p$. \begin{table}[htbp] \tabcolsep=3pt\fontsize{7}{9} \begin{center} \caption{Average areas (over 10 datasets) under the Precision-Recall curves resulted from different imputation methods, where the number in the parentheses denotes the standard deviation of the average. } \label{AUC} \begin{tabular}{ccccccccc} \hline p& misgLasso & Median & BPCA & RegTree & IC-Last & IC-Ave & True\\ \hline 100 & 0.678(0.006)& 0.882(0.007)& 0.874(0.006)& 0.817(0.005)& 0.877(0.007)& 0.904(0.006)& 0.949(0.006) \\\hline 200 & 0.633(0.005)& 0.856(0.004)& 0.855(0.004)& 0.442(0.004)& 0.887(0.004)& 0.902(0.003)& 0.941(0.002) \\\hline 300 &0.599(0.004)& 0.830(0.003)& 0.833(0.003)& 0.574(0.003)& 0.869(0.003)& 0.901(0.002)& 0.936(0.002) \\\hline 400 & 0.580(0.003)& 0.824(0.003)& 0.824(0.003)& 0.620(0.003)& 0.868(0.003)& 0.900(0.002)& 0.932(0.001) \\\hline \end{tabular} \end{center} \end{table} Table \ref{AUC} compares the averaged areas (over 10 datasets) under the precision-recall curves produced by different methods. The comparison indicates that IC-Ave outperforms all others for this example. It is interesting to note that although IC-Last is also based on one-time imputation, it is much better than the median filling, regression tree and BPCA methods. This suggests that for microarray data, the model-based imputation method is potentially more accurate than other one-time imputation methods. The misgLasso algorithm does not work well for this example. This inferiority is not due to the EM algorithm, but due to the gLasso algorithm which does not work well for the example. This is consistent with Liang et al. (2015), where it is shown that the $\psi$-learning algorithm works much better than gLasso for the complete data version of this example. \subsection{Yeast Cell Expression Data} Gasch et al. (2000) explored genomic expression patterns in the yeast {\it Saccharomyces cerevisiae} responding to diverse environmental changes. The whole dataset has a missing rate of 3.01\% and is available at {\tt http://genome-www.stanford.edu/yeast-stress/}. Our numerical results for a subset of 1000 genes, reported in the Supplementary Material, indicate that the IC algorithm works reasonably well for this example with a few hub genes successfully identified, which are expected to play an important role for yeast cells in response to environmental changes. \section{High-Dimensional Variable Selection in Presence of Missing Data} This problem is also motivated by microarray data analysis, but the goal has been shifted to selection of genes relevant to a particular phenotype. To be more general, we let ${\boldsymbol Y}=(Y_1,\ldots,Y_n)^T$ denote the response vector for $n$ observations, and let ${\boldsymbol X}=(X_1,\ldots, X_n)^T$ denote the matrix of covariates, where each $X_i$ is a $p$-dimensional vector and $p$ can be much larger than $n$ (a.k.a. small-$n$-large-$p$). The response variable and covariates are linked through the regression, \begin{equation} \label{Lineareq} {\boldsymbol Y}=(\bm{1}_n, {\boldsymbol X}) {\boldsymbol \beta}+{\boldsymbol \epsilon}, \end{equation} where ${\boldsymbol \beta}=(\beta_0,\beta_1,\ldots,\beta_p)^T$ denotes the vector of regression coefficients, and ${\boldsymbol \epsilon} \sim N(0, \sigma_{\epsilon}^2 I_n)$ denotes the vector of random errors. Variable selection for the model (\ref{Lineareq}) with complete data has been extensively studied in the recent literature. Methods have been developed from both frequentist and Bayesian perspectives, see e.g., Tibshirani (1996), Johnson and Rossell (2012), and Song and Liang (2015a). For incomplete data, Garcia et al. (2010) proposed to conduct variable selection by maximizing the penalized likelihood function of the incomplete data. However, when $p$ is large and the covariates $X_i$'s are generally correlated, the incomplete data likelihood function can be intractable, rendering failure of their method. Zhao and Long (2013) showed through numerical studies that for the high dimensional data the standard multiple imputation approach performs poorly, while the imputation method based on Bayesian Lasso often works better. However, since Bayesian Lasso tends to over-shrink the non-zero regression coefficients, its consistency in variable selection is hard to be justified when $p$ is much greater than $n$ (Castillo et al., 2015). Quite recently, Long and Johnson (2015) proposed to combine Bayesian Lasso imputation and stability selection (Meinshausen and B\"uhlmann, 2010). Again, the consistency of this method is hard to be justified due to the inconsistency of Bayesian Lasso. \subsection{The ICC Algorithm} In what follows, we consider a general setting of the model (\ref{Lineareq}), where the covariates follow a multivariate Gaussian distribution ${\boldsymbol X} \sim N({\boldsymbol \mu}, {\boldsymbol \Sigma})$. Under this setting, the parameter vector ${\boldsymbol \theta}$ consists of three natural blocks ${\boldsymbol \beta}$, $\sigma_{\epsilon}^2$ and the concentration matrix ${\boldsymbol C}={\boldsymbol \Sigma}^{-1}$. Since $n$ has been assumed to be smaller than $p$, we further assume the sparsity for both the regression coefficients ${\boldsymbol \beta}$ and the concentration matrix ${\boldsymbol C}$. To apply the ICC algorithm to this problem, we choose the SIS-MCP algorithm as the consistent estimator of ${\boldsymbol \beta}$. That is, the variables are first subject to a sure independence screening procedure, and then the survived variables are selected using the MCP method (Zhang, 2010). This algorithm has been implemented in the R-package {\it SIS}. Given an estimates of ${\boldsymbol \beta}$, $\sigma_{\epsilon}^2$ can be estimated by $\hat{\sigma}_{\epsilon}^2=\sum_{i=1}^n \hat{\epsilon}_i^2/(n-|\hat{{\boldsymbol \beta}}|-1)$, where $\hat{\epsilon}_i$ denotes the residual of sample $i$, and $|\hat{{\boldsymbol \beta}}|$ denotes the number of nonzero elements included in the estimate $\hat{{\boldsymbol \beta}}$. Given the consistency of $\hat{{\boldsymbol \beta}}$, the consistency of $\hat{\sigma}_{\epsilon}^2$ is easy to be justified. To estimate the concentration matrix ${\boldsymbol C}$, we choose the $\psi$-learning algorithm. As mentioned previously, the $\psi$-learning algorithm provides a consistent estimate for the Gaussian graphical network, based on which a consistent estimate of the concentration matrix can be uniquely determined by the algorithm given in Hastie et al. (2009, p.634). Note that SIS-MCP does not make use of the dependency among the covariates. Given the structure of the ICC algorithm, some other variable selection algorithms which have made use of the dependency among the covariates, e.g., Yu and Liu (2016), can also be applied here. Next, we consider the imputation step. Suppose that the value of $x_{hk}$ is missed in ${\boldsymbol X}$. Section 4 of the Supplementary Material presents the conditional distributions of $X_{hk}$ given ${\boldsymbol Y}$ and the rest elements of ${\boldsymbol X}$ under different scenarios. Based on the conditional distributions, $x_{hk}$ can be easily imputed by sampling from the respective samplized conditional distributions. Here the samplized conditional distribution refers to the distribution with its population parameters replaced by their respective estimates calculated from samples. For example, $\beta_i$'s are replaced by their SIS-MCP estimates, $\sigma_{\epsilon}^2$ is replaced by $\hat{\sigma}_{\epsilon}^2$, etc. In summary, the ICC algorithm works as follows: {\it \begin{itemize} \item (Initialization) Fill each missing entry of ${\boldsymbol X}$ by the median of the corresponding variable, and then iterates between the CC- and I-steps. \item (CC-step) (i) Apply the SIS-MCP algorithm to estimate the regression coefficients ${\boldsymbol \beta}$; (ii) estimate $\sigma_{\epsilon}^2$ conditional on the estimate of ${\boldsymbol \beta}$; and (iii) apply the $\psi$-learning algorithm to learn the structure of the Gaussian graphical network. \item (I-step) Impute missing values according to the conditional distributions (given in the Supplemental Material) based on the regression model and network structure learned in the CC-step. \end{itemize} } \subsection{A Simulated Example} The datasets were simulated from the model (\ref{Lineareq}) with $n=100$ and $p$=200 and 500. The covariates ${\boldsymbol X}$ were generated under two settings: (i) the covariates are mutually independent, where ${\boldsymbol x}_i \sim N(0,2 I_n)$ for $i=1,\ldots,n$; and (ii) the covariates are generated according to the concentration matrix (\ref{plugin}). For both settings, we set $(\beta_0,\beta_1,\ldots,\beta_5)=(1,1,2,-1.5, -2.5, 5)$ and $\beta_6=\cdots=\beta_p=0$, and random error ${\boldsymbol \epsilon} \sim N(0, I_n)$. For each pair of $(n,p)$, we simulated 10 datasets independently. For each dataset, we considered two missing rates, randomly deleting 5\% and 10\% entries of ${\boldsymbol X}$ as missing values. The performance of different methods was measured using three criteria: \[ \mbox{err}_{{\boldsymbol \beta}}^2=\|\hat{{\boldsymbol \beta}}-{\boldsymbol \beta}\|^2, \quad \mbox{fsr}=\frac{|{\boldsymbol s} \backslash {\boldsymbol s}^*|}{|{\boldsymbol s}|}, \quad \mbox{nsr}=\frac{|{\boldsymbol s}^* \backslash {\boldsymbol s}|}{|{\boldsymbol s}^*|}, \] where $\|\cdot\|$ denotes the Euclidean norm, $\hat{{\boldsymbol \beta}}$ denotes the estimate of ${\boldsymbol \beta}$, ${\boldsymbol s}^*$ denotes the set of true covariates, and ${\boldsymbol s}$ denotes the set of selected covariates. The ICC algorithm was first applied to this example with the results summarized in Table \ref{RegressionTab1} and \ref{RegressionTab2}. For each dataset, the algorithm was run for 30 iterations. For variable selection, we kept only the variables appeared 5 or more times in the last 10 iterations. For estimation of ${\boldsymbol \beta}$, we averaged the estimates of ${\boldsymbol \beta}$ obtained in the last 10 iterations. For comparison, we also tried the one-time imputation methods, including median filling and BPCA. As explained previously, the median filling method is to fill each missing value by the median of the corresponding variable, and BPCA is to impute the missing values based on the principal component regression. Then the variables are selected using the SIS-MCP method. \begin{table}[htbp] \begin{center} \caption{Comparison of the ICC algorithm with the median filling and BPCA methods for high-dimensional variable selection with independent covariates. ''True'' denotes the results obtained by the MCP method from the complete data. The values in the table are obtained by averaging over 10 independent datasets with the standard deviation reported in the parentheses.} \label{RegressionTab1} \begin{tabular}{ccccccc} \hline $p$ &Missing Rate & & BPCA & Median & ICC & True \\ \hline \multirow{6}{2cm}{\centering $200$} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 0.257(0.267) & 0.262(0.261) & 0.042(0.041) & 0.046(0.048) \\ &5\%&fsr & 0.119(0.143)& 0.082(0.092) & 0(0) & 0(0) \\ && nsr & 0(0) & 0(0) & 0(0) & 0(0) \\ \cline{2-7} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 0.903(0.396) & 0.856(0.421) & 0.065(0.087) & 0.046(0.048) \\ &10\%&fsr & 0.310(0.159) & 0.308(0.178) & 0(0) & 0(0) \\ && nsr & 0(0) & 0(0) & 0(0) & 0(0) \\ \hline \multirow{6}{2cm}{\centering $500$} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 0.339(0.214) & 0.350(0.206) & 0.029(0.034) & 0.027(0.023) \\ &5\%&fsr & 0.249(0.225) & 0.266(0.237) & 0(0) & 0(0) \\ && nsr & 0(0) & 0(0) & 0(0) & 0(0) \\ \cline{2-7} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 1.532(1.071) & 1.354(0.895) & 0.044(0.022) & 0.027(0.023) \\ &10\%&fsr & 0.470(0.265) & 0.420(0.255) & 0(0) & 0(0) \\ && nsr & 0.033(0.070) & 0.017(0.053) & 0(0) & 0(0) \\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htbp] \begin{center} \caption{Comparison of the ICC algorithm with the median filling and BPCA methods for high-dimensional variable selection with dependent covariates. ''True'' denotes the results obtained by the MCP method from the complete data.} \label{RegressionTab2} \begin{tabular}{ccccccc} \hline $p$ &Missing Rate & & BPCA & Median & ICC & True \\ \hline \multirow{6}{2cm}{\centering $200$} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 0.580(0.413) & 0.548(0.140) & 0.118(0.097) & 0.071(0.050) \\ &5\%&fsr & 0.262(0.204)& 0.263(0.200) & 0(0) & 0(0) \\ && nsr & 0.017(0.052) & 0.017(0.052) & 0(0) & 0(0) \\ \cline{2-7} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 1.604(0.666) & 1.575(0.974) & 0.424(0.461) & 0.071(0.050) \\ &10\%&fsr & 0.247(0.229) & 0.273(0.238) & 0(0) & 0(0) \\ && nsr & 0.100(0.086) & 0.083(0.088) & 0.033(0.070) & 0(0) \\ \hline \multirow{6}{2cm}{\centering $500$} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 0.669(0.366) & 0.717(0.358) & 0.172(0.195) & 0.096(0.083) \\ &5\%&fsr & 0.262(0.202) & 0.289(0.236) & 0(0) & 0(0) \\ && nsr & 0.017(0.053) & 0.017(0.053) & 0(0) & 0(0)\\ \cline{2-7} && $\mbox{err}_{{\boldsymbol \beta}}^2$ & 2.752(2.306) & 2.896(2.601) & 0.578(0.587) & 0.096(0.083) \\ &10\%&fsr & 0.297(0.230) & 0.327(0.224) & 0(0) & 0(0) \\ && nsr & 0.133(0.070) & 0.133(0.070) & 0.050(0.081) & 0(0) \\ \hline \end{tabular} \end{center} \end{table} The comparison indicates that the ICC algorithm works extremely well for this example. For the case of independent covariates, its results are almost as good as those obtained from the complete data. In both cases, the ICC algorithm significantly outperforms the one-time imputation methods. \subsection{A Real Data Example} We analyzed one real gene expression dataset about Bardet-Biedl syndrome (Scheetz et al., 2006). The complete dataset contains 120 samples, where the expression level of the gene TRIM32 works as the response variable and the expression levels of 200 other genes work as the predictors. The dataset is available in the R package {\it flare}. We generated ten incomplete datasets from the complete one by randomly deleting 5\% observations. For each incomplete dataset, we ran the ICC algorithm for 30 iterations and averaged the estimates of ${\boldsymbol \beta}$ obtained in the last 10 iterations as the final estimate. For comparison, the median filling and BPCA methods were also applied to this example. Table \ref{eyetab} summarizes the estimation errors of $\hat{{\boldsymbol \beta}}$ (with respect to ${\boldsymbol \beta}_c$, the estimate of ${\boldsymbol \beta}$ from the complete data) produced by the three methods for ten incomplete datasets. \begin{table}[htbp] \begin{center} \caption{Estimation errors of $\hat{{\boldsymbol \beta}}$ (with respect to ${\boldsymbol \beta}_c$) produced by ICC, median filling and BPCA for the Bardet-Biedl syndrome example, where err$_{{\boldsymbol \beta}}^2$ is calculated by averaging $\|\hat{{\boldsymbol \beta}}-{\boldsymbol \beta}_c\|^2$ over ten incomplete datasets, and ``s.d.'' represents the standard deviation of err$_{{\boldsymbol \beta}}^2$. } \label{eyetab} \begin{tabular}{cccc} \hline Method & BPCA & Median & ICC \\ \hline err$_{{\boldsymbol \beta}}^2$ & 0.428 & 0.397 & 0.187 \\ s.d. & 0.091 & 0.086 & 0.040 \\ \hline \end{tabular} \end{center} \end{table} We have also explored the results of variable selection. The complete data model selects 5 variables: v.153, v.180, v.185, v.87 and v.200. For the ICC, median filling and BPCA models, we count the selection frequency of each variable for the ten incomplete datasets. For the ICC models, the top 5 variables in selection frequency are v.153, v.185, v.180, v.87 and v.200, which are the same (ignoring the order) as the complete data model. For the median filling models, the top 5 variables are v.153, v.185, v.62, v.200 and v.54. For the BPCA models, the top 5 variables are v.153, v.87, v.185, v.62 and v.200. Both the results of ${\boldsymbol \beta}$ estimation and variable selection indicate the superiority of the ICC algorithm over the one-time imputation methods. \section{A Random Coefficient Linear Model} To further illustrate the use of the ICC algorithm, we consider a random coefficient linear model. Such a model often arises, for instance, in recommendation systems where the customers rate different items, e.g., products or service. Specifically, we simulate the data from the following model \begin{equation} \label{randomcoefmodel} \begin{split} & y_{ij}={\boldsymbol x}_{ij}^T {\boldsymbol \beta}+{\boldsymbol z}_i^T {\boldsymbol \lambda}_i +{\boldsymbol w}_j^T {\boldsymbol \gamma}_j +e_{ij}, \\ & e_{ij}\sim N(0, \sigma^2), \quad {\boldsymbol \lambda}_i \sim N(0,\Lambda), \quad {\boldsymbol \gamma}_j \sim N(0,\Gamma), \\ \end{split} \end{equation} where $y_{ij}$ represents the response for customer $i$ on item $j$. Assuming that there are $I$ customers and each customer responds to $J$ items. Thus, the dataset consists of a total of $n=IJ$ observations. The vector ${\boldsymbol x}_{ij}$ represents the covariates that characterize the customers and items, e.g., how and how long the customer has purchased the item; ${\boldsymbol z}_i$ represents customer-specific covariates such as gender, education and demographics; and ${\boldsymbol w}_j$ represents item-specific covariates, e.g., the manufacturer and category of the item. The vector ${\boldsymbol \lambda}_i$ represents the customer-specific (random) coefficients and ${\boldsymbol \gamma}_j$ represents the item-specific (random) coefficients. This model can be easily extended to the case where each customer responds to only a subset of items. For this model, we treat the random coefficients ${\boldsymbol \lambda}_i$'s and ${\boldsymbol \gamma}_j$'s as missing data, and are interested in estimation of ${\boldsymbol \beta}$. For simplicity, we assume that ${\boldsymbol \beta}$ is low-dimensional, although the whole dataset can be big when $I$ and/or $J$ become large. Under this assumption, the ICC algorithm is essentially reduced to the stochastic EM algorithm for this example. Instead of using the ICC algorithm in this straightforward way, we propose to use it under the Bayesian framework. This extends the applications of the ICC algorithm to Bayesian computation. To conduct Bayesian analysis for the model, we assume the following semiconjugate priors: \begin{equation} \label{prioreq} {\boldsymbol \beta} \sim N({\boldsymbol \mu}_{{\boldsymbol \beta}}, \Sigma_{{\boldsymbol \beta}}), \quad \sigma^2 \sim IG(a,b), \quad \Lambda \sim IW(\rho_{\Lambda}, {\boldsymbol R}_{\Lambda}), \quad \Gamma \sim IW(\rho_{\Gamma}, {\boldsymbol R}_{\Gamma}), \end{equation} where $IG(\cdot,\cdot)$ denotes the inverted Gamma distribution, $IW(\cdot,\cdot)$ denotes the inverted Wishart distribution, and ${\boldsymbol \mu}_{{\boldsymbol \beta}}$, $\Sigma_{{\boldsymbol \beta}}$, $a$, $b$, $\rho_{\Lambda}$, ${\boldsymbol R}_{\Lambda}$, $\rho_{\Gamma}$, and ${\boldsymbol R}_{\Gamma}$ are hyperparameters to be specified by the user. Each of these priors is individually conjugate to the normal likelihood function, given the other parameters, although the joint prior is not conjugate. Given these priors, the full conditional posterior distributions are derived in Section 5 of the Supplementary Material. Since, under the low-dimensional setting, the mode of the full conditional posterior distribution provides a consistent estimator for the corresponding parameter, the ICC algorithm can work as follows: \begin{itemize} \item {\it (Initialization) Initialize $\lambda_i$'s, $\gamma_j$'s, and all parameters by some random numbers. } \item {\it (CC-step) Estimate the parameters ${\boldsymbol \beta}$, $\Lambda$, $\Gamma$, and $\sigma^2$ by the mode of their respective full conditional posterior distributions. } \item {\it (I-step) Impute the values of ${\boldsymbol \lambda}_i$'s and ${\boldsymbol \gamma}_j$'s according to their respective full conditional posterior distributions. } \end{itemize} Under the Bayesian framework, the ICC algorithm works in a similar way to the Gibbs sampler except that it replaces posterior samples of the parameters by their respective full conditional posterior modes. Also, in this case, it is reduced to a hybrid of data augmentation (Tanner \& Wong, 1987) and iterative conditional modes (Besag, 1974) by using imputation for the missing data and conditional modes for the parameters. However, the ICC algorithm offers more, whose consistency step allows it to conduct parameter estimation based on sub-samples only and this can create great savings in computation for big data problems. For high-dimensional problems, if the choice of prior distributions ensures posterior consistency, then the above algorithm can still be employed. Figure 3 of the supplementary material compares the sampling path and autocorrelations of the ICC and Gibbs samples. As expected, the comparison shows that the ICC algorithm can converge faster than the Gibbs sampler and, in addition, the samples generated by the ICC algorithm tend to have smaller variations than those by the Gibbs sampler. We are aware that the accuracy of the ICC estimates is achieved at the price that we scarify the variance information contained in the posterior samples. As pointed out by Nielsen (2000), the variance of the ICC samples reflects the information loss due to the missing data. However, for the random coefficient example, the variance information can be obtained from the full conditional posterior distributions (given in the Supplementary Material) by simply plugging the parameter estimates into their variances. This observation suggests a general strategy to improve simulations of the Gibbs sampler: At each iteration, we only need to draw samples for the components for which the posterior variance is not analytically available and also of interest to us, and the other components can be replaced by the mode of the respective full conditional posterior distributions. As aforementioned, the mode can be found with a subset of samples, which can be much cheaper than sampling from the full data conditional posterior. We expect that the proposed strategy can significantly facilitate Bayesian computation for big data problems. A further study of this proposed strategy will be reported elsewhere. \section{Discussion} In this paper, we have proposed the imputation-consistency algorithm, or the IC algorithm in short, as a general algorithm for dealing with high-dimensional missing data problems. Under quite general conditions, we show that the IC algorithm can lead to a consistent estimate for the parameters. We have also extended the IC algorithm to the case of multiple block parameters, which leads to the imputation-conditional consistency (ICC) algorithm. We illustrate the proposed algorithms using the high-dimensional Gaussian graphical models, high-dimensional variable selection, and a random coefficient model. Like the EM algorithm for low-dimensional data, we expect that the IC/ICC algorithm can have many applications for high-dimensional data. With the IC/ICC algorithm, many problems can be much simplified, e.g., variable selection for high-dimensional mixture regression (Khalili and Chen, 2007) and variable selection for high-dimensional mixed effect models (Fan and Li, 2012). For the former, the group index of each sample can be treated as missing data, and then the IC algorithm can be applied: The I-step is to assign the samples into different groups and the C-step is to conduct variable selection for each group separately. For the latter, at each iteration of the IC algorithm, it is reduced to variable selection for a high-dimensional regression with fixed designs given imputed random effects. To assess the convergence of IC simulations, we recommend the Gelman-Rubin statistic (Gelman and Rubin, 1992). Since the IC algorithm usually converges very fast, we do not recommend a long run. Our experience shows that 20 iterations have often been long enough to produce a stable parameter estimate. Due to the MCMC nature of IC simulations, we recommend the averaging method for parameter estimation, and allow a burn-in period before collecting the samples for averaging. In Section 3.2, we reported the results based on the last iteration is just to compare with other one-time imputation methods. Theoretically, the variation of the IC estimates collected at different iterations reflects the missing data information. Hence, in addition to the parameter estimates, one might report their variance over iterations. However, this information is often not of interest to us, we choose not to report them in the paper. Please be aware that when a large amount of noise was brought into the system through missing data, the IC/ICC algorithm may fail to work as implied by condition (A3). Also, please be aware that the IC/ICC algorithm targets consistency by design. When the sample size is small, the consistent estimators might not be adequately accurate. Our numerical experience shows that in this case, the IC/ICC algorithm might not significantly outperform one-time imputation methods, such as median filling and BPCA, as there is no much information to use for improving imputation during iterations. In general, when the sample size increases, the IC/ICC algorithm can significantly outperform one-time imputation methods. Regarding statistical inference for parameters, we note that, theoretically, it can be done in general scenarios, not limited to that the posterior distribution of the parameters has a closed form. For example, for high-dimensional regression, if the Lasso algorithm is employed as the consistent estimator in the IC algorithm, then the inference for ${\boldsymbol \theta}$, e.g., constructing confidence intervals for each component of ${\boldsymbol \theta}$, can be done using the de-sparsified method (Zhang and Zhang, 2014; van de Geer et al., 2014). Let $V({\boldsymbol x}^{{\rm obs}}, {\tilde{\boldsymbol x}}^{{\rm mis}}_t,{\boldsymbol \theta}_n^{(t)})$ denote an uncertainty assessment statistic obtained at iteration $t$. Assume that $V({\boldsymbol x}^{{\rm obs}}, {\boldsymbol x}^{{\rm mis}}, {\boldsymbol \theta})$ is a Lipschitz function with respect to ${\boldsymbol \theta}$ and it is integrable. Then, by Corollary \ref{corMCMC1} or Corollary \ref{corMCMC2} (depending on IC or ICC being used), we will be able to get an uncertainty assessment for ${\boldsymbol \theta}$ by averaging $V({\boldsymbol x}^{{\rm obs}}, {\tilde{\boldsymbol x}}^{{\rm mis}}_t,{\boldsymbol \theta}_n^{(t)})$ along the IC/ICC chain. However, how to get the uncertainty assessment statistic $V({\boldsymbol x}^{{\rm obs}}, {\tilde{\boldsymbol x}}^{{\rm mis}}_t,{\boldsymbol \theta}_n^{(t)})$ for general high-dimensional problems is beyond the scope of this paper. As a variant of the ICC algorithm, we note that the I-step can be replaced by an E-step if it is available. In this case, the C-step is to maximize the objective function \begin{equation} \label{nsfeq1} \max_{{\boldsymbol \theta}} E_{{\boldsymbol \theta}^*} Q({\boldsymbol x}^{{\rm obs}}, {\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}), \end{equation} where the Q-function is as defined in the EM algorithm, and $E_{{\boldsymbol \theta}^*}$ denotes expectation with respect to the true distribution $\pi({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}^*)$. Suppose that ${\boldsymbol \theta}$ has been partitioned into a few blocks ${\boldsymbol \theta}=(\theta^{(1)}, \ldots, \theta^{(k)})$. To solve (\ref{nsfeq1}), the ICC algorithm is reduced to a blockwise consistency algorithm, which is to iteratively find consistent estimates for the parameters of each block conditioned on the current estimates of the parameters of other blocks. The blockwise consistency algorithm is closely related to, but more flexible than, the coordinate ascent algorithm (Tseng, 2001; Tseng and Yun, 2009). The coordinate ascent algorithm is to iteratively find the exact maximizer for each block conditioned on the current estimates of the parameters of other blocks. Under appropriate conditions, such as contraction as in (A5$'$) and uniform consistency of ${\boldsymbol \theta}_n^{(t)}$'s as shown in Theorem \ref{them1new}, we will be able to show that the paths of the two algorithms will converge to the same point. This will be explored elsewhere. The ICC algorithm have strong implications for big data computing. Based on the ICC algorithm, we have proposed a general strategy to improve Bayesian computation under big data scenario; that is, we can replace posterior samples by posterior modes in Gibbs iterations to accelerate simulations, where the posterior modes can be calculated with a subset of samples. In addition, the IC/ICC algorithm facilitates data integration from multiple sources when missing data are present. With the IC/ICC algorithm, the problem of data integration for incomplete data is converted to a problem for complete data and thus many of the existing meta-analysis methods can be conveniently applied for inference. This is very important for big data analysis. \section*{Appendix} \paragraph{1. Proof of Consistency of ${\boldsymbol \theta}_n^{(t+1)}$.} Define $\Theta_n$ as the parameter space of ${\boldsymbol \theta}$, where the subscript $n$ indicates the dependence of the dimension of ${\boldsymbol \theta}$ on the sample size $n$. Without possible confusion, we will refer to $\Theta_n$ as $\Theta$. In addition, we let $\Theta_n^T=\{ \theta_n^{(1)}, \ldots, \theta_n^{(T)}\}$ denote a path of $\theta_n$ in the IC algorithm, which can be considered as an arbitrary subset of $\Theta_n$ with $T$ elements (replicates are allowed). Let $\tilde{x}=(x^{{\rm obs}},{\tilde{x}}^{{\rm mis}})$ and define \begin{equation} \label{Liangeq1} \begin{split} G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) & =E_{{\boldsymbol \theta}_n^{(t)}}\log f_{{\boldsymbol \theta}}(\tilde{x}) =\int \log (f_{{\boldsymbol \theta}}(\tilde{x}) ) f(x^{{\rm obs}}|{\boldsymbol \theta}^*) h({\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta}_n^{(t)}) d \tilde{x}, \\ \hat{G}_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) & =\frac{1}{n}\sum_{i=1}^n \log f(x^{{\rm obs}}_i,{\tilde{x}}^{{\rm mis}}_i|{\boldsymbol \theta}), \\ \tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) &=\frac{1}{n}\sum_{i=1}^n \int\log f(x^{{\rm obs}}_i,{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta})h({\tilde{x}}^{{\rm mis}}|x^{{\rm obs}}_i,{\boldsymbol \theta}_n^{(t)})d{\tilde{x}}^{{\rm mis}} := \frac{1}{n} \sum_{i=1}^n q(x^{{\rm obs}}_i). \end{split} \end{equation} Our first goal is to show the following uniform law of large numbers (ULLN) holds for any $T$ such that $\log T =o(n)$: \begin{equation} \label{Liangeq2} \sup_{{\boldsymbol \theta}_n^{(t)}\in \Theta_n^T}\sup_{{\boldsymbol \theta}\in \Theta_n}|\hat G_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) -G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) |\rightarrow_p 0, \end{equation} where $\rightarrow_p$ denotes convergence in probability. To achieve this goal, we need the following conditions: \begin{itemize} \item[(A1)] $\log f_{{\boldsymbol \theta}}(\tilde{x})$ is a continuous function of ${\boldsymbol \theta}$ for each $\tilde{x}\in \mathcal{X}$ and a measurable function of $\tilde{x}$ for each $\theta$. \item[(A2)] [Conditions for Glivenko-Cantelli theorem] \begin{itemize} \item[(a)] There exists a function $m_n(\tilde{x})$ such that $\sup_{{\boldsymbol \theta}\in \Theta_n,\tilde{x}\in \mathcal{X}}|\log f_{{\boldsymbol \theta}}(\tilde{x})|\leq m_n(\tilde{x})$. \item[(b)] Define $\tilde{m}_n(x^{{\rm obs}},{\boldsymbol \theta}_n^{(t)})=\int m_n(\tilde{x})$ $h({\tilde{x}}^{{\rm mis}}|x^{{\rm obs}},{\boldsymbol \theta}_n^{(t)})d {\tilde{x}}^{{\rm mis}}$. Assume that there exists $m^*_n(x^{{\rm obs}})$ such that $0\leq \tilde{m}_n(x^{{\rm obs}},{\boldsymbol \theta}_n^{(t)}) \leq m^*_n(x^{{\rm obs}})$ for all ${\boldsymbol \theta}_n^{(t)}$, $E[m^*_n(x^{{\rm obs}})]$ $<\infty$, and $\sup_{n\in \mathbb{Z}^+}E[m_n^*(x^{{\rm obs}})$ $1(m_n^*(x^{{\rm obs}})\geq \zeta)]\rightarrow 0$ as $\zeta\rightarrow \infty$. In addition, $\sup_{n\geq 1}\sup_{x\in \mathcal{X},{\boldsymbol \theta}\in \Theta_n}$ $ |\int m_n(\tilde{x})1(m_n(\tilde{x})>\zeta)h({\tilde{x}}^{{\rm mis}}|x,{\boldsymbol \theta}) d{\tilde{x}}^{{\rm mis}}|\rightarrow 0$ as $\zeta\rightarrow \infty$. \item[(c)] Define $\mathcal{F}_n=\{\int \log f(x^{{\rm obs}},{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta})h({\tilde{x}}^{{\rm mis}}|x^{{\rm obs}}$, ${\boldsymbol \theta}_n^{(t)})d{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta}, {\boldsymbol \theta}_n^{(t)}\in \Theta_n\}$, and $\mathcal{G}_{n,M}=\{q1(m^*_n(x^{{\rm obs}})$ $\leq M)|q\in \mathcal{F}_n\}$. Suppose that for every $\epsilon$ and $M>0$, the metric entropy $\log N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))=o_p^*(n)$, where $\mathbb{P}_n$ is the empirical measure of $x^{{\rm obs}}$, and $N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))$ is the covering number with respect to the $L_1(\mathbb{P})$-norm. \end{itemize} \item[(A3)] [Conditions for imputed data] Define $Z_{t,i}=\log f(x^{{\rm obs}}_i,{\tilde{x}}^{{\rm mis}}_i|{\boldsymbol \theta})-\int \log f(x^{{\rm obs}}_i,{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta}) h({\tilde{x}}^{{\rm mis}}| $ $x^{{\rm obs}}_i, {\boldsymbol \theta}_n^{(t)})$ $d{\tilde{x}}^{{\rm mis}}$. Suppose that for any ${\boldsymbol \theta}$ and ${\boldsymbol \theta}_n^{(t)} \in \Theta_n$, $E|Z_{t,i}|^m \leq m! M_b^{m-2} v_i/2$ for every $m\geq 2$ and some constants $M_b>0$ and $v_i=O(1)$. That is, $Z_{t,i}$'s are sub-exponential random variables. \end{itemize} \begin{theorem} \label{them0} Assume conditions (A1)--(A3), then (\ref{Liangeq2}) holds for any $T$ such that $\log T =o(n)$. \end{theorem} \begin{proof} By the definitions in (\ref{Liangeq1}), we have the decomposition \begin{equation} \label{decompeq} \hat G_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)})-G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) =\big\{\hat{G}_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)})-\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})\big\} +\big\{\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})-G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})\big\}, \end{equation} which consists of two terms, the first term comes from imputation of missing data, and the second term comes from the observed data. First, we show that the second term of (\ref{decompeq}) converges to 0 uniformly, following the proof of Theorem 2.4.3 of van der Vaart and Wellner (1996). By the symmetrization Lemma 2.3.1 of van der Vaart and Wellner (1996), measurability of the class $\mathcal{F}_n$, and Fubini's theorem, \[ \begin{split} & E^*\sup_{{\boldsymbol \theta},{\boldsymbol \theta}_n^{(t)}\in \Theta_n}|\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})-G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})| \leq 2 E_{x^{{\rm obs}}}E_\epsilon \sup_{q(x)\in \mathcal{F}_n}\|\frac{1}{n}\sum_{i=1}^n \epsilon_i q(x_i^{obs})\| \\ & \leq 2 E_{x^{{\rm obs}}}E_\epsilon \sup_{q(x)\in \mathcal{G}_{n,M}}\|\frac{1}{n}\sum_{i=1}^n \epsilon_i q(x_i^{obs})\| +2 E^*[m^*_n(x^{obs})1(m^*_n(x^{obs})>M)], \\ \end{split} \] where $\epsilon_i$ are i.i.d. Rademacher random variables with $P(\epsilon_i=+1)=P(\epsilon_i=-1)=1/2$, and $E^*$ denotes the outer expectation. By condition (A2), $2 E^*[m^*_n(x^{{\rm obs}})1(m^*_n(x^{{\rm obs}})>M)]\rightarrow 0$ for sufficiently large $M$. To prove convergence in mean, it suffices to show that the first term converges to zero for fixed $M$. Fix $x^{{\rm obs}}_1,...,x^{{\rm obs}}_n$, and let $\mathcal{H}$ be a $\epsilon$-net in $L_1(\mathbb{P}_n)$ over $\mathcal{G}_{n,M}$, then \[ E_\epsilon \sup_{q(x)\in \mathcal{G}_{n,M}}\|\frac{1}{n}\sum_{i=1}^n \epsilon_i q(x^{{\rm obs}}_i)\|\leq E_\epsilon \sup_{q(x)\in \mathcal{H}}\|\frac{1}{n}\sum_{i=1}^n \epsilon_i q(x^{{\rm obs}}_i)\|+\epsilon. \] The cardinality of $\mathcal{H}$ can be chosen equal to $N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n)$. Bound the $L_1$-norm on the right by the Orlicz-norm $\psi_2$ and use the maximal inequality (Lemma 2.2.2 of van der Vaart and Wellner (1996)) and Hoeffding's inequality, it can be shown that \begin{equation} \label{Liangeq5} E_\epsilon \sup_{q(x)\in \mathcal{G}_{n,M}}\|\frac{1}{n}\sum_{i=1}^n \epsilon_i q(x^{{\rm obs}}_i)\| \leq K \sqrt{1+\log N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))}\sqrt{6/n}M +\epsilon \rightarrow_{P^*} \epsilon, \end{equation} where $K$ is a constant, and $P^*$ denotes outer probability. It has been shown that the left side of (\ref{Liangeq5}) converges to zero in probability. Since it is bounded by $M$, its expectation with respect to $x^{{\rm obs}}_1,\ldots,x^{{\rm obs}}_n$ converges to zero by the dominated convergence theorem. This concludes the proof that $\sup_{{\boldsymbol \theta}_n^{(t)}\in \Theta_n}\sup_{{\boldsymbol \theta}\in \Theta_n}|\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) -G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})|\rightarrow_p 0$ in mean. Further, by Markov inequality, we conclude that \begin{equation} \label{Liangeq3} \sup_{{\boldsymbol \theta}_n^{(t)}\in \Theta_n}\sup_{{\boldsymbol \theta}\in \Theta_n}|\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) -G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})| \to_p 0. \end{equation} To establish the uniform convergence of the first term of (\ref{decompeq}), we fix $x^{{\rm obs}}_1,...,x^{{\rm obs}}_n$. By condition (A3), $n(\hat{G}_n({\boldsymbol \theta}|\tilde{x},{\boldsymbol \theta}_n^{(t)})-\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}))= Z_{t,1}+Z_{t,2}+\cdots+Z_{t,n}$. By Bernstein's inequality, \[ P(n|\hat{G}_n({\boldsymbol \theta}|\tilde{x},{\boldsymbol \theta}_n^{(t)})-\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})|>z) =P(|Z_{t,1}+Z_{t,2}+\cdots+Z_{t,n}|>z)\leq 2 \exp\left\{-\frac{1}{2} \frac{z^2}{v+M_b z}\right\}, \] for $v \geq v_1+\cdots +v_n$. By Lemma 2.2.10 of van der Vaart and Wellner (1996), for Orlicz norm $\psi_1$, we have \[ \begin{split} & \|\sup_{{\boldsymbol \theta}\in \Theta_n, t=1,2,\ldots,T} n \big\{\hat{G}_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)})-\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) \big\} \|_{\psi_1} \\ & \leq \epsilon + K(M_b \log (1+TN(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))) +\sqrt{v}\sqrt{\log(1+TN(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n)))}), \end{split} \] for a constant $K$ and any $\epsilon>0$. By condition (A2)-(c) and the condition $\log(T)=o(n)$, \[ \begin{split} & \|\sup_{{\boldsymbol \theta}\in \Theta_n, t=1,2,\ldots,T} \{ \hat{G}_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)})-\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)}) \} \|_{\psi_1} \\ & \leq \epsilon+ K(M_b \log (1+TN(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n)))/n +\sqrt{v/n}\sqrt{\log(1+TN(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n)))/n}) \rightarrow_{P^*} \epsilon. \end{split} \] Therefore, \begin{equation} \label{Liangeq4} \sup_{{\boldsymbol \theta}\in \Theta_n,t=1,2,...,T}|\hat G_n({\boldsymbol \theta}|\widetilde{x},{\boldsymbol \theta}_{n}^{(t)}) -\tilde{G}_n({\boldsymbol \theta}|{\boldsymbol \theta}_{n}^{(t)})|\rightarrow_{p} 0. \end{equation} The theorem can then be concluded by combining (\ref{Liangeq3}) and (\ref{Liangeq4}). \end{proof} \noindent {\bf Remark R1} {\it (On the metric entropy condition)} Assume that all elements in $\cup_{n\geq 1}\mathcal{F}_n$ are uniformly Lipschitz with respect to the $L_1$-norm. Then the metric entropy $\log N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))$ can be measured based on the parameter space $\Theta$. Since the functions in $\mathcal{G}_{n,M}$ are all bounded, the corresponding parameter space $\Theta_{n,M}$ can be contained in a $L_1$-ball by the continuity of $\log f(\tilde{x}|{\boldsymbol \theta})$ in ${\boldsymbol \theta}$. Further, we assume that the diameter of the $L_1$-ball or the space $\Theta_{n,M}$ grows at a rate of $O(n^{\alpha})$ for some $0 \leq \alpha<1/2$, then $\log N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n)) = O(n^{2\alpha} \log p)$ holds, which allows $p$ to grow at a polynomial rate of $O(n^{\gamma})$ for some constant $0<\gamma<\infty$. Note that the increased diameter accounts for the conventional assumption that the size of the true model grows with the sample size $n$. Refer to Vershynin (2015) for more discussions on this issue. Similar conditions on metric entropy have been used in the literature of high-dimensional statistics. For example, Raskutti et al. (2011) studied minimax rates of estimation for high-dimensional linear regression over $L_q$-balls. \noindent {\bf Remark R2} {\it (On the condition of $T$)}. Since the imputation step draws random data at each iteration $t$, there is no way to show uniform convergence of ${\boldsymbol \theta}_n^{(t+1)}$ to ${\boldsymbol \theta}_*^{(t)}$ over all possible ${\boldsymbol \theta}_n^{(t)} \in \Theta_n$. However, we are able to prove that the consistency results hold for any sequence of ${\boldsymbol \theta}_n^{(1)},...,{\boldsymbol \theta}_n^{(T)}$ with $T$ being not too large compared to $e^n$. This is enough for Theorems \ref{them1}-\ref{them4}. To justify this, we may consider the case that the dimension of ${\boldsymbol \theta}_n$ grows with $n$ at a rate of $p=O(n^{\gamma})$ for a constant $\gamma>0$, say, $\gamma=5$. Then it is easy to see that when $n>13$, the ratio $T/p$ has an order of \[ O(e^n/p)=O(e^{n-\gamma \log(n)}) \succ O(e^{0.1 n}) \succ O(p^{100}), \] which implies that essentially there is no constraint on the setting of $T$. Note that for MCMC simulations, the number of iterations is often set to a low-order polynomial of $p$ for a given set of observations. For any ${\boldsymbol \theta}_n^{(t)}\in \Theta_n^{T}$, we define ${{\boldsymbol \theta}}_n^{(t+1)}=\arg\max_{{\boldsymbol \theta}\in \Theta_n} \hat{G}_n({\boldsymbol \theta}|\widetilde{x},{\boldsymbol \theta}_n^{(t)})$ and ${\boldsymbol \theta}_*^{(t)}=\arg\max_{{\boldsymbol \theta}\in \Theta_n} G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$. We would like to establish the uniform consistency of ${{\boldsymbol \theta}}_n^{(t+1)}$ with respect to $t$, i.e., \begin{equation} \label{Faeq1} \sup_{t \in\{1,2,\ldots,T\}} \|{{\boldsymbol \theta}}_n^{(t+1)}-{\boldsymbol \theta}_*^{(t)}\| \to_p 0, \quad \mbox{as $n\to \infty$}. \end{equation} To achieve this goal, we assume the following condition: \begin{itemize} \item[(A4)] For each $t=1,2,\ldots, T$, $G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ has a unique maximum at ${\boldsymbol \theta}_*^{(t)}$; for any $\epsilon>0$, $\sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ exists, where $B_t(\epsilon)=\{{\boldsymbol \theta} \in \Theta_n: \|{\boldsymbol \theta}-{\boldsymbol \theta}_*^{(t)}\| < \epsilon\}$. Let $\delta_t=G_n({\boldsymbol \theta}_*^{(t)}|{\boldsymbol \theta}_n^{(t)})- \sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$, $\delta=\min_{t \in \{1,2,\ldots,T\}} \delta_t>0$. \end{itemize} Note that the existence of $\sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ can be easily satisfied if $\Theta_n$ is restricted to a compact set, which implies that $\Theta_n\setminus B_t(\epsilon)$ is also a compact set and thus the supremum is achievable. This condition can also be satisfied by assuming that $\Theta_n$ is convex and for each $t$, ${\boldsymbol \theta}_*^{(t)}$ is in the interior of $\Theta_n$ and $G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ is concave in ${\boldsymbol \theta}$. \begin{theorem} \label{them1new} Assume conditions (A1)-(A4) hold, then the maximum pseudo-complete data likelihood estimate ${{\boldsymbol \theta}}_n^{(t+1)}$ is uniformly consistent to ${\boldsymbol \theta}_*^{(t)}$ over $t=1,2,\ldots,T$, i.e. (\ref{Faeq1}) holds. \end{theorem} \begin{proof} Since both $\hat{G}_n({\boldsymbol \theta}|\widetilde{x},{\boldsymbol \theta}_n^{(t)})$ and $G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ are continuous in $\theta$ as implied by the continuity of $\log f_{{\boldsymbol \theta}}(\tilde{\boldsymbol{x}})$, the remaining part of the proof follows from Lemma \ref{lem00} by setting the penalty function $P_{\lambda_n}({\boldsymbol \theta})=0$ for all ${\boldsymbol \theta} \in \Theta_n$. \end{proof} \begin{lemma}\label{lem00} Consider a sequence of functions $Q_t({\boldsymbol \theta}, {\boldsymbol X}_n)$ for $t=1,2,\ldots,T$. Suppose that the following conditions are satisfied: (B1) For each $t$, $Q_t({\boldsymbol \theta},{\boldsymbol X}_n)$ is continuous in ${\boldsymbol \theta}$ and there exists a function $Q_t^*({\boldsymbol \theta})$, which is continuous in ${\boldsymbol \theta}$ and uniquely maximized at ${\boldsymbol \theta}_*^{(t)}$. (B2) For any $\epsilon>0$, $\sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} Q_t^*({\boldsymbol \theta})$ exists, where $B_t(\epsilon)=\{{\boldsymbol \theta}: \|{\boldsymbol \theta}-{\boldsymbol \theta}_*^{(t)}\| < \epsilon\}$; Let $\delta_t=Q_t^*({\boldsymbol \theta}_*^{(t)})- \sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} Q_t^*({\boldsymbol \theta})$, $\delta=\min_{t \in \{1,2,\ldots,T\}} \delta_t>0$. (B3) $\sup_{t\in\{1,2,\ldots,T\}} \sup_{{\boldsymbol \theta} \in \Theta_n} |Q_t({\boldsymbol \theta}, {\boldsymbol X}_n)-Q_t^*({\boldsymbol \theta})| \to_p 0$ as $n\to \infty$. (B4) The penalty function $P_{\lambda_n}({\boldsymbol \theta})$ is non-negative and converges to 0 uniformly over the set $\{{\boldsymbol \theta}_*^{(t)}: t=1,2,\ldots,T\}$ as $n\to \infty$, where $\lambda_n$ is a regularization parameter and its value can depend on the sample size $n$. Let ${\boldsymbol \theta}_n^{(t)}=\arg\max_{{\boldsymbol \theta}\in \Theta_n} \{ Q_t({\boldsymbol \theta}, {\boldsymbol X}_n)-P_{\lambda_n}({\boldsymbol \theta})\}$. Then the uniform convergence holds, i.e., $\sup_{t \in \{1,2,\ldots,T\}} \|{\boldsymbol \theta}_n^{(t)}- {\boldsymbol \theta}_*^{(t)}\|\to_p 0$. \end{lemma} \begin{proof} Consider two events (i) $\sup_{t \in \{1,2,\ldots,T\} } \sup_{{\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)} |Q_t({\boldsymbol \theta},{\boldsymbol X}_n)-Q_t^*({\boldsymbol \theta})| < \delta/2$, and (ii) $\sup_{t \in \{1,2,\ldots,T\} }$ $\sup_{{\boldsymbol \theta} \in B_t(\epsilon)} |Q_t({\boldsymbol \theta},{\boldsymbol X}_n)-Q_t^*({\boldsymbol \theta})| < \delta/2$. From event (i), we can deduce that for any $t \in \{1,2,\ldots, T\}$ and any ${\boldsymbol \theta} \in \Theta_n\setminus B_t(\epsilon)$, $Q_t({\boldsymbol \theta}, {\boldsymbol X}_n) < Q_t^*({\boldsymbol \theta})+\delta/2 \leq Q_t^*({\boldsymbol \theta}_*^{(t)}) -\delta_t +\delta/2 \leq Q_t^*({\boldsymbol \theta}_*^{(t)}) -\delta/2$. Therefore, $Q_t({\boldsymbol \theta}, {\boldsymbol X}_n) -P_{\lambda_n}({\boldsymbol \theta}) < Q_t^*({\boldsymbol \theta}_*^{(t)}) -\delta/2 -o(1)$ by condition (B4). From event (ii), we can deduce that for any $t \in \{1,2,\ldots, T\}$ and any ${\boldsymbol \theta} \in B_t(\epsilon)$, $Q_t({\boldsymbol \theta}, {\boldsymbol X}_n)> Q_t^*({\boldsymbol \theta}) -\delta/2$ and $Q_t({\boldsymbol \theta}_*^{(t)}, {\boldsymbol X}_n)> Q_t^*({\boldsymbol \theta}_*^{(t)}) -\delta/2$. Therefore, $Q_t({\boldsymbol \theta}_*^{(t)}, {\boldsymbol X}_n)-P_{\lambda_n}({\boldsymbol \theta}_*^{(t)}) > Q_t^*({\boldsymbol \theta}_*^{(t)}) -\delta/2- o(1)$ by condition (B4). If both events hold simultaneously, then we must have ${{\boldsymbol \theta}}_n^{(t)} \in B_t(\epsilon)$ for all $t \in \{1,2,\ldots, T\}$ as $n \to \infty$. By condition (B3), the probability that both events hold tends to 1. Therefore, \[ P(\mbox{${\boldsymbol \theta}_n^{(t)} \in B_t(\epsilon)$ for all $t=1,2,\ldots,T$}) \to 1, \] which concludes the lemma. \end{proof} Theorem \ref{them1new} establishes the consistency of ${{\boldsymbol \theta}}_n^{(t+1)}$ with respect to ${\boldsymbol \theta}_*^{(t)}$ for each $t=1,2,\ldots,T$. However, in the small-$n$-large-$p$ scenario, ${{\boldsymbol \theta}}_n^{(t+1)}$ is not well defined. For this reason, a sparsity constraint needs to be imposed on ${\boldsymbol \theta}$. For example, we can apply a regularization method to get an estimate of ${\boldsymbol \theta}_*^{(t)}$; that is, we can define \begin{equation} \label{mingeq5} {\boldsymbol \theta}_{n,p}^{(t+1)} =\arg\max_{{\boldsymbol \theta}\in \Theta_n}\left\{ \hat{G}_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) -P_{\lambda_n}({\boldsymbol \theta})\right\}, \end{equation} where the penalty function $P_{\lambda_n}({\boldsymbol \theta})$ constrains the sparsity of the solution. Assume that \begin{itemize} \item[(C1)] The penalty function $P_{\lambda_n}({\boldsymbol \theta})$ is non-negative, ensures the existence of ${\boldsymbol \theta}_{n,p}^{(t+1)}$ for all $n \in \mathbb{N}$ and $t=1,2,\ldots,T$, and converges to 0 uniformly over the set $\{{\boldsymbol \theta}_*^{(t)}: t=1,2\ldots,T\}$ as $n\to\infty$. \end{itemize} \begin{corollary} \label{cor0A} If the conditions (A1)-(A4) and (C1) hold, then the regularization estimator ${\boldsymbol \theta}_{n,p}^{(t+1)}$ in (\ref{mingeq5}) is uniformly consistent to ${\boldsymbol \theta}_*^{(t)}$ over $t=1,2,\ldots,T$, i.e., $\sup_{t \in\{1,2,\ldots,T\}} \|{\boldsymbol \theta}_{n,p}^{(t+1)}-{\boldsymbol \theta}_*^{(t)}\| \to_p 0$ as $n\to \infty$. \end{corollary} \begin{proof} It follows the proof of Lemma \ref{lem00} directly. \end{proof} Take the high-dimensional regression as example. If we allow $p$ to grow with $n$ at the rate $p=O(n^\gamma)$ for some constant $\gamma>0$, allow the size of ${\boldsymbol \beta}_*^{(t)}$ for all $t$ to grow with $n$ at the rate $O(n^{\alpha})$ for some constant $0<\alpha<1/2$, choose $\lambda_n=O(\sqrt{\log(p)/n})$, and set $P_{\lambda_n}({\boldsymbol \theta})=\lambda_n \sum_{i=1}^p c_{\lambda_n}(|\theta_i|)$, where $c_{\lambda_n}(\cdot)$ is set in the form of SCAD (Fan and Li, 2001) or MCP (Zhang, 2010) penalties, then the condition (C1) is satisfied. For both the SCAD and MCP penalties, $c_{\lambda_n}(|\theta_i|)=0$ if $\theta_i=0$ and bounded by a constant otherwise. Similarly, if the beta-min assumption holds, i.e., there exists a constant $\beta_{\min}>0$ such that $\min_{j \in S^*} |\beta_{*j}| \geq \beta_{\min}$, where $S^*=\{j: \beta_{*j} \ne 0\}$ denotes the index set of non-zero regression coefficients, then the reciprocal Lasso penalty (Song and Liang, 2015b) also satisfies (C1). Note that, if $\Theta=\mathbb{R}^p$, the Lasso penalty does not satisfy (C1) as which is unbounded. This explains why the Lasso estimate is unbiased even as $n\to\infty$. However, if $\Theta_n$ is restricted to a bounded space, then the Lasso penalty also satisfies (C1). Alternative to regularization methods, one may first restrict the space of ${\boldsymbol \theta}_*^{(t)}$ to some low-dimensional subspace through sure screening, and then find a consistent estimate in the subspace using a conventional statistical methods, such as maximum likelihood, moment estimation, or even regularization. Both the $\psi$-learning (Liang, Song and Qiu, 2015) sure independence screening (SIS) (Fan and Lv, 2008; Fan and Song, 2010) methods belong to this class. For $\psi$-learning, after correlation screening (based on the pseudo-complete data), the remaining network structure estimation procedure is essentially the same with the covariance selection method (Dempster, 1972) which, by nature, is a maximum likelihood estimation method. It is interesting to point out that the sure screening-based methods can be viewed as a special subclass of regularization methods, for which the solutions in the low-dimensional subspace receives a zero penalty, and those outside the subspace receives a penalty of $\infty$. It is easy to see that such a binary-type penalty function satisfies condition (C1). Both the regularization and sure screening-based methods are constructive. In what follows, we give a proof for the use of general consistent estimation procedures in the IC algorithm. Let ${\boldsymbol \theta}_{n,g}^{(t+1)}$ denote the estimate of ${\boldsymbol \theta}_*^{(t)}$ produced by such a general consistent estimation procedure at iteration $t+1$. Corollary \ref{cor0} shows that if ${\boldsymbol \theta}_{n,g}^{(t+1)}$ is accurate enough for each $t$ (pointwisely) and the log-likelihood function of the pseudo-complete data satisfies some moment conditions, then the estimation procedure can be used in the IC algorithm. Therefore, by its MLE nature in the subspace, the use of the $\psi$-learning algorithm in the IC algorithm can also be justified by Corollary \ref{cor0}. \begin{itemize} \item[(C2)] [Conditions for general consistent estimate ${\boldsymbol \theta}_{n,g}^{(t)}$] Assume that for each $t=1,2,\ldots, T$, ${\boldsymbol \theta}_{n,g}^{(t+1)}-{\boldsymbol \theta}_*^{(t)}=O_p(1/\sqrt{n})$ (pointwisely) and the Hessian matrix $\partial^2 G_n({\boldsymbol \theta}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)})/\partial {\boldsymbol \theta} \partial {\boldsymbol \theta}'$ is bounded in a neighborhood of ${\boldsymbol \theta}_*^{(t)}$; let $$Z_{t,i}'=\log f(x^{{\rm obs}}_i,{\tilde{x}}^{{\rm mis}}_i|{\boldsymbol \theta}_{n,g}^{(t+1)})-\int \log f(x^{{\rm obs}},{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta}_{n,g}^{(t+1)}) f(x^{{\rm obs}}|{\boldsymbol \theta}^*) h({\tilde{x}}^{{\rm mis}}|x^{{\rm obs}}_i, {\boldsymbol \theta}_n^{(t)})d{\tilde{x}}^{{\rm mis}} dx^{{\rm obs}},$$ then $E|Z_{t,i}'|^m \leq m! \tilde{M}_b^{m-2} \tilde{v}_i/2$ for every $m\geq 2$ and some constants $\tilde{M}_b>0$ and $\tilde{v}_i=O(1)$. \end{itemize} \begin{corollary} \label{cor0} Assume (A1)-(A4) and (C2). Then ${\boldsymbol \theta}_{n,g}^{(t+1)}$ is uniformly consistent to ${\boldsymbol \theta}_*^{(t)}$ over $t=1,2,\ldots,T$, i.e., $\sup_{t \in\{1,2,\ldots,T\}} \|{\boldsymbol \theta}_{n,g}^{(t+1)}-{\boldsymbol \theta}_*^{(t)} \| \to_p 0$ as $n\to \infty$. \end{corollary} \begin{proof} Applying Taylor expansion to $G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ at ${\boldsymbol \theta}_*^{(t)}$, we get $G_n({\boldsymbol \theta}_{n,g}^{(t+1)}|{\boldsymbol \theta}_n^{(t)}) - G_n({\boldsymbol \theta}_{*}^{(t)}|{\boldsymbol \theta}_n^{(t)}) =O_p(1/n)$, following from condition (C2) and condition (A4) that $G_n({\boldsymbol \theta}|{\boldsymbol \theta}_n^{(t)})$ is maximized at ${\boldsymbol \theta}_*^{(t)}$. Therefore, \[ \begin{split} n[\hat{G}_n({\boldsymbol \theta}_{n,g}^{(t+1)}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) - G_n({\boldsymbol \theta}_{*}^{(t)}|{\boldsymbol \theta}_n^{(t)})] &=Z_{t,1}'+\cdots+Z_{t,n}'+n[G_n({\boldsymbol \theta}_{n,g}^{(t+1)}|{\boldsymbol \theta}_n^{(t)})- G_n({\boldsymbol \theta}_{*}^{(t)}|{\boldsymbol \theta}_n^{(t)})]\\ & = Z_{t,1}'+\cdots+Z_{t,n}'+ \epsilon_n, \\ \end{split} \] where $\epsilon_n=O_p(1)$, and \begin{equation} \label{Goodeq3} P(n|\hat{G}_n({\boldsymbol \theta}_{n,g}^{(t+1)}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) - G_n({\boldsymbol \theta}_{*}^{(t)}|{\boldsymbol \theta}_n^{(t)})|>nz) \leq P(|Z_{t,1}'+\cdots+Z_{t,n}'|>nz-|\epsilon_n|). \end{equation} By Bernstein's inequality, \begin{equation} \label{Goodeq1} P(|Z_{t,1}'+\cdots+Z_{t,n}|>nz-|\epsilon_n|) \leq 2 \exp\left\{-\frac{1}{2} \frac{(z-|\epsilon_n|/n)^2}{ \tilde{v}'+\tilde{M}_b'(z-|\epsilon_n|/n)} \right\}, \end{equation} for $\tilde{v}' \geq (\tilde{v}_1+\cdots+\tilde{v}_n)/n^2$ and $\tilde{M}_b'=\tilde{M}_b/n$. Applying Taylor expansion to the right of (\ref{Goodeq1}) at $z$ and combining with (\ref{Goodeq3}) leads to \begin{equation} \label{Goodeq4} P(|\hat{G}_n({\boldsymbol \theta}_{n,g}^{(t+1)}|\tilde{\boldsymbol{x}},{\boldsymbol \theta}_n^{(t)}) - G_n({\boldsymbol \theta}_{*}^{(t)}|{\boldsymbol \theta}_n^{(t)})|>z) \leq K \exp\left\{-\frac{1}{2} \frac{z^2}{\tilde{v}'+\tilde{M}_b' z} \right\}, \end{equation} where $K=2+\frac{3}{\tilde{M}_b'}O_p(1/n)=2+\frac{3}{\tilde{M}_b}O_p(1)$, since the derivative $|d[z^2/(\tilde{v}'+\tilde{M}_b' z)]/dz| \leq 3/\tilde{M}_b'$. As in the proof of Theorem \ref{them0}, by applying Lemma 2.2.10 of van der Vaart and Wellner (1996), we can prove \begin{equation} \label{penneq0} \sup_{{\boldsymbol \theta}_n^{(t)} \in \Theta_n, t \in \{1,2,\ldots,T\}} \left|\hat{G}_n({\boldsymbol \theta}_{n,g}^{(t+1)}|\tilde{\boldsymbol{x}}, {\boldsymbol \theta}_n^{(t)})- G_n({\boldsymbol \theta}_*^{(t)} |{\boldsymbol \theta}_n^{(t)}) \right| \to_p 0. \end{equation} Note that, as implied by the proof of Lemma 2.2.10 of van der Vaart and Wellner (1996), (\ref{penneq0}) holds for a general constant $K$ in (\ref{Goodeq4}). Then, by condition (A4), we must have the uniform convergence that ${\boldsymbol \theta}_{n,g}^{(t+1)} \in B_t(\epsilon)$ for all $t$ as $n\to \infty$, where $B_t(\epsilon)$ is as defined in (A4). This statement can be proved by contradiction as follows: Assume ${\boldsymbol \theta}_{n,g}^{(i+1)} \notin B_{i}(\epsilon)$ for some $i\in\{1,2,\ldots,T\}$. By the uniform convergence established in Theorem \ref{them0}, $\left|\hat{G}_n({\boldsymbol \theta}_{n,g}^{(i+1)}|\tilde{\boldsymbol{x}}, {\boldsymbol \theta}_n^{(i)})- G_n( {\boldsymbol \theta}_{n,g}^{(i+1)} |{\boldsymbol \theta}_n^{(i)}) \right| =o_p(1)$. Further, by condition (A4) and the assumption ${\boldsymbol \theta}_{n,g}^{(i+1)} \notin B_{i}(\epsilon)$, \[ \begin{split} \left| \hat{G}_n({\boldsymbol \theta}_{n,g}^{(i+1)}|\tilde{\boldsymbol{x}}, {\boldsymbol \theta}_n^{(i)})- G_n({\boldsymbol \theta}_*^{(i)} |{\boldsymbol \theta}_n^{(i)}) \right| & \geq \left|G_n( {\boldsymbol \theta}_{n,g}^{(i+1)} |{\boldsymbol \theta}_n^{(i)}) - G_n({\boldsymbol \theta}_*^{(i)} |{\boldsymbol \theta}_n^{(i)}) \right| - \left|\hat{G}_n({\boldsymbol \theta}_{n,g}^{(i+1)}|\tilde{\boldsymbol{x}}, {\boldsymbol \theta}_n^{(i)})- G_n( {\boldsymbol \theta}_{n,g}^{(i+1)} |{\boldsymbol \theta}_n^{(i)}) \right| \\ & \geq \delta-o_p(1), \end{split} \] which contradicts with the uniform convergence established in (\ref{penneq0}). This concludes the proof. \end{proof} \noindent {\bf Remark R3} {\it (On the accuracy of ${\boldsymbol \theta}_{n,g}^{(t)}$'s)} Condition (C2) restricts the consistent estimates to those having a distance to the true parameter point of the order $O_p(1/\sqrt{n})$. Such condition can be satisfied by some estimation procedures in the low-dimensional subspace, e.g., maximum likelihood, for which both the variance and bias are often of the order $O(1/n)$ (Firth, 1993) and therefore the root mean squared error is of the order $O(1/\sqrt{n})$. To make the result of Corollary \ref{cor0} more general to include more estimation procedures, we can relax this order to ${\boldsymbol \theta}_{n,g}^{(t+1)}-{\boldsymbol \theta}_*^{(t)}=O_p(n^{-1/4})$, if we would like to relax the order of $T$ to $\log(T)=o(\sqrt{n})$ and the order of metric entropy to $\log N(\epsilon,\mathcal{G}_{n,M},L_1(\mathbb{P}_n))=o_p^*(\sqrt{n})$. As mentioned in remarks (R1) and (R2), both the order of $T$ and the order of metric entropy are technical conditions and relaxing them to the order of $O(\sqrt{n})$ will not restrict much the applications of the IC algorithm. The proof for this relaxation is straightforward, following the proof of Corollary \ref{cor0}. \paragraph{2. Proof of ergodicity of the Markov chain $\{{\boldsymbol \theta}_n^{(t)} \}$ } Although the IC algorithm is different from the stochastic EM algorithm in the ${\boldsymbol \theta}_n^{(t)}$-updating step, the Markov chains $\{{\boldsymbol \theta}_n^{(t)}\}$ induced by the two algorithms share some similar properties as well as similar proofs. The following two lemmas, Lemma \ref{lem1} and Lemma \ref{lem2}, can be proved in the same way as in Nielsen (2000), and thus the proofs are omitted. \begin{lemma} \label{lem1} The Markov chain $\{{\boldsymbol \theta}_n^{(t)}\}$ is irreducible and aperiodic. \end{lemma} \begin{lemma} \label{lem2} If (A1) holds, then the Markov chain $\{{\boldsymbol \theta}_n^{(t)} \}$ has the weak Feller property, and any compact subsets of $\Theta$ are small. \end{lemma} If (A1) holds and $\Theta_n$ is restricted to a compact set, then the Markov chain $\{{\boldsymbol \theta}_n^{(t)}\}$ is ergodic. Here we would like to establish the ergodicity of the Markov chain $\{{\boldsymbol \theta}_n^{(t)}\}$ under a more general scenario $\Theta_n=\mathbb{R}^p$. This can be done by verifying a drift condition. Similar to Nielsen (2000), we choose the negative log-likelihood function of the observed data as the drift function, motivated by the drift in the EM algorithm towards high-density areas. \begin{theorem} \label{them1} If (A1)--(A3) hold, then $\{{\boldsymbol \theta}_n^{(t)} \}$ is almost surely ergodic for sufficiently large $n$. \end{theorem} \begin{proof} Let $\upsilon({\boldsymbol \theta})=C-\frac{1}{n}\log f(x^{{\rm obs}}_{1},\dots,x^{{\rm obs}}_{n}|{\boldsymbol \theta})$, where $C$ denotes a constant such that $\upsilon({\boldsymbol \theta}) \geq 0$ for all ${\boldsymbol \theta} \in \Theta_n$. Since $\upsilon({\boldsymbol \theta})$ is nonnegative, it can be used to build the drift condition. Define \[ \begin{split} \Delta\upsilon({\boldsymbol \theta})&=E_h[\upsilon({\boldsymbol \theta}_{n}^{(t+1)})-\upsilon({\boldsymbol \theta}_{n}^{(t)})] =E_h[\frac{1}{n}\log f({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}_{n}^{(t)})-\frac{1}{n}\log f({\boldsymbol x}^{{\rm obs}}|{\boldsymbol \theta}_n^{(t+1)})] \\ &=E_h[\frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_n^{(t)})-\frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)})] -E_h[\frac{1}{n}\log h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t)})-\frac{1}{n}\log h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t+1)})] \\ &= (I) +(II), \end{split} \] where $E_h$ refers to the expectation with respect to the predictive distribution $h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_n^{(t)})$. First, we consider the negative of part (I), which can be decomposed as \[ \begin{split} -(I) &= E_h[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_n^{(t)})] \\ &= E_h[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) ] + E_h[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_n^{(t)}) ], \\ \end{split} \] where the function $M({\boldsymbol \theta})$ is defined by \begin{equation} \label{mapeq} M({\boldsymbol \theta})=\arg\max_{{\boldsymbol \theta}'} E_{{\boldsymbol \theta}} \log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}')=\arg\max_{{\boldsymbol \theta}'} \int f(x^{{\rm obs}},{\tilde{x}}^{{\rm mis}}|{\boldsymbol \theta}') f(x^{{\rm obs}}|{\boldsymbol \theta}^*) h({\tilde{x}}^{{\rm mis}}|x^{{\rm obs}},{\boldsymbol \theta})d{\tilde{x}}^{{\rm mis}} dx^{{\rm obs}}. \end{equation} By the ULLN established in Theorem \ref{them0}, we have \[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) \to_p G_n({\boldsymbol \theta}_n^{(t+1)}|{\boldsymbol \theta}_n^{(t)}) - G_n(M({\boldsymbol \theta}_n^{(t)})|{\boldsymbol \theta}_n^{(t)}). \] From Theorem \ref{them1new}, we have ${\boldsymbol \theta}_{n}^{(t+1)} - M({\boldsymbol \theta}_{n}^{(t)}) \to_p 0$. Further, by the continuity of $G_n({\boldsymbol \theta}|{\boldsymbol \theta}')$ with respect to ${\boldsymbol \theta}$, we have $G_n({\boldsymbol \theta}_n^{(t+1)}|{\boldsymbol \theta}_n^{(t)}) - G_n(M({\boldsymbol \theta}_n^{(t)})|{\boldsymbol \theta}_n^{(t)}) \to_p 0$ and thus $\frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) \to_p 0$. Then, by the boundedness of $\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta})$ (condition (A2)) and the dominated convergence theorem, \begin{equation} \label{proofeq11} E \left[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) \right] \to 0, \end{equation} where the expectation is with respect to the joint density function of $\tilde{{\boldsymbol x}}=({\boldsymbol x}^{{\rm obs}},{\tilde{\boldsymbol x}}^{{\rm mis}})$. Note that for any ${\boldsymbol \theta} \in \Theta_n$, we have \begin{equation} \label{proofeq14} E_h [ \frac{1}{n} \log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}) ] =\frac{1}{n} \sum_{i=1}^n E_h \log f(\tilde{x_i} |{\boldsymbol \theta}) \stackrel{\Delta}{=} \frac{1}{n} \sum_{i=1}^n g(x^{{\rm obs}}_i), \end{equation} where $g(x^{{\rm obs}}_i)$'s are mutually independent, but not necessarily identically distributed due to the presence of missing data. Then, by (\ref{proofeq11}), (A2) and Kolmogorov's SLLN, we have \begin{equation} \label{aseq0} E_h \left[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) \right ] \to 0, \quad \mbox{a.s.}, \end{equation} as $n \to \infty$. Therefore, there exists a constant $c>0$ and a large number $N$ such that \begin{equation} \label{proofeq12} -c< E_h\left[ \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_{n}^{(t+1)}) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) \right] < c, \quad \mbox{a.s.}, \end{equation} for any $n>N$ and any $t>0$. With a similar argument to (\ref{proofeq14}), by invoking Kolmogorov's SLLN, it can be shown that there exists a constant $\delta>0$ such that \begin{equation}\label{proofeq13} E_h[\frac{1}{n}\log f(\tilde{{\boldsymbol x}}| M({\boldsymbol \theta}_{n}^{(t)})) - \frac{1}{n}\log f(\tilde{{\boldsymbol x}}|{\boldsymbol \theta}_n^{(t)})] \to \delta, \quad \mbox{a.s.}, \end{equation} for any $t>0$ as $n \to \infty$. Combining (\ref{proofeq12}) and (\ref{proofeq13}), we have $-c-\delta < (I) < c$ holds almost surely for sufficiently large $n$. Next, by Jensen's inequality, we have \[ \begin{split} (II) &= E_h \left[\frac{1}{n}\log h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t+1)})-\frac{1}{n} \log h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t)}) \right ] \leq \frac{1}{n} \log E_h\left( \frac{ h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t+1)})}{ h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t)}) } \right) \\ & = \frac{1}{n} \log \int h({\tilde{\boldsymbol x}}^{{\rm mis}}|{\boldsymbol x}^{{\rm obs}},{\boldsymbol \theta}_{n}^{(t+1)}) d {\tilde{\boldsymbol x}}^{{\rm mis}} =0. \\ \end{split} \] Combining the results of (I) and (II), we have that $\Delta\upsilon({\boldsymbol \theta}) < c$ almost surely for all ${\boldsymbol \theta} \in \Theta_n$. Choose $b$ as a positive number less than $c+\delta$ and $D$ as a compact set including $\{{\boldsymbol \theta} \in \Theta_{n}: \Delta \upsilon(\boldsymbol{\theta}) \in [-b,c)\}$. In summary, we have \[ \Delta \upsilon(\boldsymbol{\theta})\leq \begin{cases} c, & {\boldsymbol \theta}\in D, \\ -b, & {\boldsymbol \theta}\in \Theta_n\setminus D, \\ \end{cases} \] almost surely. Hence, the strict drift condition $V_2$ (Meyn and Tweedie, 2009, p263) is almost surely satisfied. Since $({\boldsymbol \theta}_{n}^{(t)})_{t \in \mathbb{N}_{0}}$ also has weak Feller property (see Lemma \ref{lem2}), we can further conclude that an invariant probability measure $\pi$ almost surely exists for this Markov chain (Meyn and Tweedie, 2009, Theorem 12.3.4). Since $({\boldsymbol \theta}_{n}^{(t)})_{t \in \mathbb{N}_{0}}$ is irreducible (shown in Lemma \ref{lem1}), $D$ is a compact set and thus a small set (shown in Lemma \ref{lem2}), and the drift condition $V_2$ is stronger than the drift condition $V_1$ (Meyn and Tweedie, 2009, p189), we can show that $({\boldsymbol \theta}_{n}^{(t)})_{t \in \mathbb{N}_{0}}$ is Harris recurrent (Meyn and Tweedie, 2009, Theorem 9.1.8). Since $({\boldsymbol \theta}_{n}^{(t)})_{t \in \mathbb{N}_{0}}$ is irreducible and has an invariant probability measure $\pi$, it is also a positive chain (Meyn and Tweedie, 2009, p 230). Therefore, it is a positive Harris recurrent chain (Meyn and Tweedie, 2009, p 231). Finally, since $({\boldsymbol \theta}_{n}^{(t)})_{t \in \mathbb{N}_{0}}$ is aperiodic (shown in Lemma \ref{lem1}) and positive Harris recurrent, we can conclude that it is almost surely ergodic (Meyn and Tweedie, 2009, Theorem 13.3.3). \end{proof} \paragraph{3. Proof of consistency of the IC estimator} To prove the consistency of the IC estimator, we consider the mapping defined in (\ref{mapeq}). For the C-step, we have ${\boldsymbol \theta}_*^{(t)}=M({\boldsymbol \theta}_n^{(t)})$. Also, ${\boldsymbol \theta}^*$, the true value of ${\boldsymbol \theta}_n$, is a fixed point of the mapping. Further, to show that the mean of the stationary distribution of the Markov chain forms a consistent estimate of ${\boldsymbol \theta}^*$, we make the following assumption. \begin{itemize} \item[(A5)] The mapping $M({\boldsymbol \theta})$ is differentiable. Let $\lambda_n({\boldsymbol \theta})$ be the largest singular value of $\partial M({\boldsymbol \theta})/\partial {\boldsymbol \theta}$. There exists a number $\lambda^* <1$ such that $\lambda_n({\boldsymbol \theta}) \leq \lambda^*$ for all ${\boldsymbol \theta} \in \Theta_n$ for sufficiently large $n$ and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence. \end{itemize} \noindent {\bf Remark R4} {\it (On contraction mapping)} The condition (A5) directly implies \begin{equation} \label{mapeq2} \|M({\boldsymbol \theta}_n^{(t)})-{\boldsymbol \theta}^*\| = \|M({\boldsymbol \theta}_n^{(t)})-M({\boldsymbol \theta}^*)\| \leq \lambda^* \|{\boldsymbol \theta}_n^{(t)}-{\boldsymbol \theta}^*\|, \end{equation} that is, the mapping is a contraction. We note that a continuous application of the mapping, i.e., setting ${\boldsymbol \theta}_n^{(t+1)}={\boldsymbol \theta}_*^{(t)}=M({\boldsymbol \theta}_n^{(t)})$ for all $t$, leads to a monotone increase of the expectation $E_{{\boldsymbol \theta}_n^{(t)}} \log f_{{\boldsymbol \theta}}(\tilde{{\boldsymbol x}})$. Since $E_{{\boldsymbol \theta}_n^{(t)}} \log f_{{\boldsymbol \theta}}(\tilde{{\boldsymbol x}})$ attains its maximum at $E_{{\boldsymbol \theta}^*} \log f_{{\boldsymbol \theta}^*}(\tilde{{\boldsymbol x}})$, it is reasonable to assume that $M({\boldsymbol \theta}_n^{(t)})$ is closer to ${\boldsymbol \theta}^*$ than ${\boldsymbol \theta}_n^{(t)}$. This condition should hold for sufficiently large $n$, at which ${\boldsymbol \theta}_*^{(t)}$'s and ${\boldsymbol \theta}^*$ are all unique as assumed in condition (A4). We note that a similar contraction condition has been used in analysis of the SEM algorithm (Proposition 3, Nielsen, 2000). Some other conditions can potentially be specified based on the fixed-point theory (see e.g., Khamsi and Kirk, 2001). \begin{theorem} \label{them2} Assume (A1)-(A5) and $\sup_{n,t} E\|{\boldsymbol \theta}_n^{(t)}\| <\infty$. Then for sufficiently large $n$, sufficiently large $t$, and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence, $ \| {\boldsymbol \theta}_n^{(t)} - {\boldsymbol \theta}^* \|=o_p(1)$. Furthermore, the sample average of the Markov chain forms a consistent estimate of ${\boldsymbol \theta}^*$, i.e., $\| \frac{1}{T} \sum_{t=1}^T {\boldsymbol \theta}_n^{(t)} - {\boldsymbol \theta}^* \|=o_p(1)$, as $n \to \infty$ and $T\to \infty$. \end{theorem} \begin{proof} By Theorem \ref{them1}, the Markov chain $\{{\boldsymbol \theta}_n^{(t)}\}$ converges to a stationary distribution. For simplicity, we suppress the supscript $t$, let ${\boldsymbol \theta}_n$ denote the current sample, and let ${\boldsymbol \theta}_n'$ denote the next iteration sample. Therefore, $\|{\boldsymbol \theta}_n' - {\boldsymbol \theta}^*\| \leq \|{\boldsymbol \theta}_n' - M({\boldsymbol \theta}_n) \| + \| M({\boldsymbol \theta}_n) - {\boldsymbol \theta}^* \| \leq \|{\boldsymbol \theta}_n' - M({\boldsymbol \theta}_n) \| + \lambda^* \|{\boldsymbol \theta}_n-{\boldsymbol \theta}^*\|$, where the last inequality follows from (\ref{mapeq2}). Taking expectation on both sides leads to \begin{equation} \label{proofeq1} E \|{\boldsymbol \theta}_n' - {\boldsymbol \theta}^*\| \leq E \|{\boldsymbol \theta}_n' - M({\boldsymbol \theta}_n) \| + \lambda^* E \|{\boldsymbol \theta}_n-{\boldsymbol \theta}^*\| \leq \frac{1}{1-\lambda^*} E \|{\boldsymbol \theta}_n' - M({\boldsymbol \theta}_n) \| = \frac{1}{1-\lambda^*} o(1)=o(1), \end{equation} where the second inequality follows from the stationarity of the Markov chain, and the first equality follows from Theorem \ref{them1new} and the existence of $E \|{\boldsymbol \theta}_n\|$. Finally, by Markov's inequality, we conclude the consistency of ${\boldsymbol \theta}_n^{(t)}$ as an estimator of ${\boldsymbol \theta}^*$. By (\ref{proofeq1}), we have $\|E ({\boldsymbol \theta}_n) -{\boldsymbol \theta}^* \| \leq E \|{\boldsymbol \theta}_n-{\boldsymbol \theta}^*\| =o(1)$, which implies that the mean of the stationary distribution of $\{{\boldsymbol \theta}_n^{(t)} \}$ converges to ${\boldsymbol \theta}^*$ for sufficiently large $n$. Further, by the ergodicity of the Markov chain $\{{\boldsymbol \theta}_n^{(t)} \}$, we conclude the proof. \end{proof} \begin{corollary} \label{corMCMC1} Assume (A1)-(A5), $\sup_{n,t} E\|{\boldsymbol \theta}_n^{(t)}\| <\infty$, $h({\boldsymbol \theta})$ is a Lipschitz function on $\Theta_n$, and $\sup_{n,t}$ $E\|h({\boldsymbol \theta}_n^{(t)})\|<\infty$. Then for sufficiently large $n$, sufficiently large $t$, and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence, $ \| h({\boldsymbol \theta}_n^{(t)}) - h({\boldsymbol \theta}^*) \|=o_p(1)$. Furthermore, $\| \frac{1}{T} \sum_{t=1}^T h({\boldsymbol \theta}_n^{(t)}) - h({\boldsymbol \theta}^*) \|=o_p(1)$, as $n \to \infty$ and $T\to \infty$. \end{corollary} \begin{proof} The proof follows from the definition of Lipschitz function and the proof of Theorem \ref{them2}. \end{proof} \paragraph{4. Proof of ergodicity of the Markov chain for the ICC algorithm } \begin{theorem} \label{them3} If (A1)-(A3) hold, the Markov chain $\{({\boldsymbol \theta}_n^{(t,1)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})\}$ is almost surely ergodic for sufficiently large $n$. \end{theorem} This theorem can be proved in a similar way to Theorem \ref{them1} with the detail given in the Supplementary Material. \paragraph{5. Proof of consistency of the ICC estimator} \begin{itemize} \item[(A5$'$)] Let $M_i$ denote the mapping of the $i$th part of the CC-step, i.e., ${\boldsymbol \theta}_*^{(t,i)}=M_i({\boldsymbol \theta}_n^{(t+1,1)}, \ldots, {\boldsymbol \theta}_n^{(t+1,i-1)},$ ${\boldsymbol \theta}_n^{(t,i)}, \ldots, {\boldsymbol \theta}_n^{(t,k)})$. Let $M=M_k\circ M_{k-1}\circ \ldots\circ M_1$ denote the joint mapping of $M_1,\ldots, M_k$. Let $\lambda_n({\boldsymbol \theta})$ denote the largest singular value of $\partial M({\boldsymbol \theta})/\partial {\boldsymbol \theta}$. There exists a number $\lambda^*<1$ such that $\lambda_n({\boldsymbol \theta}) \leq \lambda^*$ for all ${\boldsymbol \theta} \in \Theta_n$, all sufficiently large $n$, and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence. \end{itemize} This condition is reasonable: It is easy to see that a continuous application of the mapping $M$, i.e., applying $M_i$'s in a circular manner, leads to a monotone increase of the function $E_{{\boldsymbol \theta}_n^{(t)}} \log f_{{\boldsymbol \theta}}(\tilde{{\boldsymbol x}})$. Similar to Theorem \ref{them2}, we can prove the following theorem with the detail given in the Supplementary Material. \begin{theorem} \label{them4} Assume (A1)-(A4), (A5$'$) and $\sup_{n,t} E|{\boldsymbol \theta}_n^{(t)}| <\infty$. Then for sufficiently large $n$, sufficiently large $t$, and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence, $\| {\boldsymbol \theta}_n^{(t)} - {\boldsymbol \theta}^* \|=o_p(1)$. Furthermore, the sample average of the Markov chain also forms a consistent estimate of ${\boldsymbol \theta}^*$, i.e., $ \| \frac{1}{T} \sum_{t=1}^T {\boldsymbol \theta}_n^{(t)} - {\boldsymbol \theta}^* \|=o_p(1)$, as $n\to \infty$ and $T\to \infty$. \end{theorem} \begin{corollary} \label{corMCMC2} Assume (A1)-(A4), (A5$'$), $\sup_{n,t} E\|{\boldsymbol \theta}_n^{(t)}\| <\infty$, $h({\boldsymbol \theta})$ is a Lipschitz function on $\Theta_n$, and $\sup_{n,t}$ $E\|h({\boldsymbol \theta}_n^{(t)})\|<\infty$. Then for sufficiently large $n$, sufficiently large $t$, and almost every ${\boldsymbol x}^{{\rm obs}}$-sequence, $ \| h({\boldsymbol \theta}_n^{(t)}) - h({\boldsymbol \theta}^*) \|=o_p(1)$. Furthermore, $\| \frac{1}{T} \sum_{t=1}^T h({\boldsymbol \theta}_n^{(t)}) - h({\boldsymbol \theta}^*) \|=o_p(1)$, as $n \to \infty$ and $T\to \infty$. \end{corollary} \begin{proof} The proof follows from the definition of Lipschitz function and the proof of Theorem \ref{them4}. \end{proof} \section*{References} \begin{description} \item[] Besag, J. (1974). Spatial Interaction and the Statistical Analysis of Lattice Systems (with discussion). {\it Journal of the Royal Statistical Society, Series B}, {\bf 36}(2), 192-225. \item[] Bo, T.H., Dysvik, B. and Jonassen, I. (2004). LSimpute: accurate estimation of missing values in microarray data with least square methods. {\it Nucleic Acids Research}, {\bf 32}:e34. \item[] Buuren, S. and Groothuis-Oudshoorn, K. (2011). mice: Multivariate imputation by chained equations in R. {\it Journal of statistical software}, {\bf 45}(3). \item[] Cai, J.-F., Cand\`es, E. and Shen, Z. (2010). 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\begin{document} { \begin{center} \Large\bf Devinatz's moment problem: a description of all solutions. \end{center} \begin{center} \bf S.M. Zagorodnyuk \end{center} \section{Introduction.} We shall study the following problem: to find a non-negative Borel measure $\mu$ in a strip $$ \Pi = \{ (x,\varphi):\ x\in \mathbb{R},\ -\pi\leq \varphi < \pi \}, $$ such that \begin{equation} \label{f1_1} \int_\Pi x^m e^{in\varphi} d\mu = s_{m,n},\qquad m\in \mathbb{Z}_+, n\in \mathbb{Z}, \end{equation} where $\{ s_{m,n} \}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$ is a given sequence of complex numbers. We shall refer to this problem as to {\bf the Devinatz moment problem}. \noindent A.~Devinatz was the first who introduced and studied this moment problem~\cite{cit_1000_D}. He obtained the necessary and sufficient conditions of solvability for the moment problem~(\ref{f1_1}) and gave a sufficient condition for the moment problem to be determinate~\cite[Theorem 4]{cit_1000_D}. \noindent Our aim here is threefold. Firstly, we present a new proof of the Devinatz solvability criterion. Secondly, we describe canonical solutions of the Devinatz moment problem (see the definition below). Finally, we describe all solutions of the Devinatz moment problem. We shall use an abstract operator approach~\cite{cit_1500_Z} and results of Godi\v{c}, Lucenko and Shtraus~\cite{cit_2000_GL},\cite[Theorem 1]{cit_3000_GP},\cite{cit_4000_S}. {\bf Notations. } As usual, we denote by $\mathbb{R},\mathbb{C},\mathbb{N},\mathbb{Z},\mathbb{Z}_+$ the sets of real numbers, complex numbers, positive integers, integers and non-negative integers, respectively. For a subset $S$ of the complex plane we denote by $\mathfrak{B}(S)$ the set of all Borel subsets of $S$. Everywhere in this paper, all Hilbert spaces are assumed to be separable. By $(\cdot,\cdot)_H$ and $\| \cdot \|_H$ we denote the scalar product and the norm in a Hilbert space $H$, respectively. The indices may be ommited in obvious cases. For a set $M$ in $H$, by $\overline{M}$ we mean the closure of $M$ in the norm $\| \cdot \|_H$. For $\{ x_k \}_{k\in T}$, $x_k\in H$, we write $\mathop{\rm Lin}\nolimits \{ x_k \}_{k\in T}$ for the set of linear combinations of vectors $\{ x_k \}_{k\in T}$ and $\mathop{\rm span}\nolimits \{ x_k \}_{k\in T} = \overline{ \mathop{\rm Lin}\nolimits \{ x_k \}_{k\in T} }$. Here $T := \mathbb{Z}_+ \times \mathbb{Z}$, i.e. $T$ consists of pairs $(m,n)$, $m\in \mathbb{Z}_+$, $n\in\mathbb{Z}$. The identity operator in $H$ is denoted by $E$. For an arbitrary linear operator $A$ in $H$, the operators $A^*$,$\overline{A}$,$A^{-1}$ mean its adjoint operator, its closure and its inverse (if they exist). By $D(A)$ and $R(A)$ we mean the domain and the range of the operator $A$. By $\sigma(A)$, $\rho(A)$ we denote the spectrum of $A$ and the resolvent set of $A$, respectively. We denote by $R_z (A)$ the resolvent function of $A$, $z\in \rho(A)$. The norm of a bounded operator $A$ is denoted by $\| A \|$. By $P^H_{H_1} = P_{H_1}$ we mean the operator of orthogonal projection in $H$ on a subspace $H_1$ in $H$. By $\mathbf{B}(H)$ we denote the set of all bounded operators in $H$. \section{Solvability.} Let a moment problem~(\ref{f1_1}) be given. Suppose that the moment problem has a solution $\mu$. Choose an arbitrary power-trigonometric polynomial $p(x,\varphi)$ of the following form: \begin{equation} \label{f1_2} \sum_{m=0}^\infty \sum_{n=-\infty}^\infty \alpha_{m,n} x^m e^{in\varphi},\qquad \alpha_{m,n}\in \mathbb{C}, \end{equation} where all but finite number of coefficients $\alpha_{m,n}$ are zeros. We can write $$ 0 \leq \int_\Pi |p(x,\varphi)|^2 d\mu = \int_\Pi \sum_{m=0}^\infty \sum_{n=-\infty}^\infty \alpha_{m,n} x^m e^{in\varphi} \overline{ \sum_{k=0}^\infty \sum_{l=-\infty}^\infty \alpha_{k,l} x^k e^{il\varphi} } d\mu $$ $$ = \sum_{m,n,k,l} \alpha_{m,n}\overline{\alpha_{k,l}} \int_\Pi x^{m+k} e^{i(n-l)\varphi} d\mu = \sum_{m,n,k,l} \alpha_{m,n}\overline{\alpha_{k,l}} s_{m+k,n-l}. $$ Thus, for arbitrary complex numbers $\alpha_{m,n}$ (where all but finite numbers are zeros) we have \begin{equation} \label{f2_1} \sum_{m,k=0}^\infty \sum_{n,l=-\infty}^\infty \alpha_{m,n}\overline{\alpha_{k,l}} s_{m+k,n-l} \geq 0. \end{equation} Let $T = \mathbb{Z}\times \mathbb{Z}_+$ and for $t,r\in T$, $t=(m,n)$, $r=(k,l)$, we set \begin{equation} \label{f2_2} K(t,r) = K((m,n),(k,l)) = s_{m+k,n-l}. \end{equation} Thus, for arbitrary elements $t_1,t_2,...,t_n$ of $T$ and arbitrary complex numbers $\alpha_1,\alpha_2,...,\alpha_n$, with $n\in \mathbb{N}$, the following inequality holds: \begin{equation} \label{f2_3} \sum_{i,j=1}^n K(t_i,t_j) \alpha_{i} \overline{\alpha_j} \geq 0. \end{equation} The latter means that $K(t,r)$ is a positive matrix in the sense of E.H.~Moore \cite[p.344]{cit_5000_A}. Suppose now that a Devinatz moment problem is given and conditions~(\ref{f2_1}) (or what is the same conditions~(\ref{f2_3})) hold. Let us show that the moment problem has a solution. We shall use the following important fact (e.g.~\cite[pp.361-363]{cit_6000_AG}). \begin{thm} \label{t2_1} Let $K = K(t,r)$ be a positive matrix on $T=\mathbb{Z}\times \mathbb{Z}_+$. Then there exist a separable Hilbert space $H$ with a scalar product $(\cdot,\cdot)$ and a sequence $\{ x_t \}_{t\in T}$ in $H$, such that \begin{equation} \label{f2_4} K(t,r) = (x_t,x_r),\qquad t,r\in T, \end{equation} and $\mathop{\rm span}\nolimits\{ x_t \}_{t\in T} = H$. \end{thm} {\bf Proof. } Consider an arbitrary infinite-dimensional linear vector space $V$ (for example, we can choose a space of complex sequences $(u_n)_{n\in \mathbb{N}}$, $u_n\in \mathbb{C}$). Let $X = \{ x_t \}_{t\in T}$ be an arbitrary infinite sequence of linear independent elements in $V$ which is indexed by elements of $T$. Set $L_X = \mathop{\rm Lin}\nolimits\{ x_t \}_{t\in T}$. Introduce the following functional: \begin{equation} \label{f2_5} [x,y] = \sum_{t,r\in T} K(t,r) a_t\overline{b_r}, \end{equation} for $x,y\in L_X$, $$ x=\sum_{t\in T} a_t x_t,\quad y=\sum_{r\in T} b_r x_r,\quad a_t,b_r\in \mathbb{C}. $$ Here all but finite number of indices $a_t,b_r$ are zeros. \noindent The set $L_X$ with $[\cdot,\cdot]$ will be a pre-Hilbert space. Factorizing and making the completion we obtain the required space $H$ (\cite[p. 10-11]{cit_7000_B}). $\Box$ By applying this theorem we get that there exist a Hilbert space $H$ and a sequence $\{ x_{m,n} \}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$, $x_{m,n}\in H$, such that \begin{equation} \label{f2_6} (x_{m,n}, x_{k,l})_H = K((m,n),(k,l)),\qquad m,k\in \mathbb{Z}_+,\ n,l\in \mathbb{Z}. \end{equation} Set $L = \mathop{\rm Lin}\nolimits\{ x_{m,n} \}_{(m,n)\in T}$. We introduce the following operators \begin{equation} \label{f2_7} A_0 x = \sum_{(m,n)\in T} \alpha_{m,n} x_{m+1,n}, \end{equation} \begin{equation} \label{f2_8} B_0 x = \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n+1}, \end{equation} where \begin{equation} \label{f2_9} x = \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n} \in L. \end{equation} We should show that these definitions are correct. Indeed, suppose that the element $x$ in~(\ref{f2_9}) has another representation: \begin{equation} \label{f2_10} x = \sum_{(k,l)\in T} \beta_{k,l} x_{k,l}. \end{equation} We can write $$ \left( \sum_{(m,n)\in T} \alpha_{m,n} x_{m+1,n}, x_{a,b} \right) = \sum_{(m,n)\in T} \alpha_{m,n} K((m+1,n),(a,b)) $$ $$= \sum_{(m,n)\in T} \alpha_{m,n} s_{m+1+a,n-b} = \sum_{(m,n)\in T} \alpha_{m,n} K((m,n),(a+1,b)) $$ $$ = \left(\sum_{(m,n)\in T} \alpha_{m,n} x_{m,n}, x_{a+1,b} \right) = (x,x_{a+1,b}), $$ for arbitrary $(a,b)\in T$. In the same manner we get $$ \left(\sum_{(k,l)\in T} \beta_{k,l} x_{k+1,l}, x_{a,b} \right) = (x,x_{a+1,b}). $$ Since $\mathop{\rm span}\nolimits\{ x_{a,b} \}_{(a,b)\in T} = H$, we get $$ \sum_{(m,n)\in T} \alpha_{m,n} x_{m+1,n} = \sum_{(k,l)\in T} \beta_{k,l} x_{k+1,l}. $$ Thus, the operator $A_0$ is defined correctly. \noindent We can write $$ \left\| \sum_{(m,n)\in T} (\alpha_{m,n}-\beta_{m,n}) x_{m,n+1} \right\|^2 $$ $$= \left( \sum_{(m,n)\in T} (\alpha_{m,n}-\beta_{m,n}) x_{m,n+1}, \sum_{(k,l)\in T} (\alpha_{k,l}-\beta_{k,l}) x_{k,l+1} \right) $$ $$ = \sum_{(m,n),(k,l)\in T} (\alpha_{m,n}-\beta_{m,n}) \overline{(\alpha_{k,l}-\beta_{k,l})} K((m,n+1),(k,l+1)) $$ $$= \sum_{(m,n),(k,l)\in T} (\alpha_{m,n}-\beta_{m,n}) \overline{(\alpha_{k,l}-\beta_{k,l})} K((m,n),(k,l)) $$ $$= \left( \sum_{(m,n)\in T} (\alpha_{m,n}-\beta_{m,n}) x_{m,n}, \sum_{(k,l)\in T} (\alpha_{k,l}-\beta_{k,l}) x_{k,l} \right) = 0. $$ Consequently, the operator $B_0$ is defined correctly, as well. Choose an arbitrary $y = \sum_{(a,b)\in T} \gamma_{a,b} x_{a,b} \in L$. We have $$ (A_0 x,y) = \sum_{m,n,a,b} \alpha_{m,n}\gamma_{a,b} (x_{m+1,n},x_{a,b}) = \sum_{m,n,a,b} \alpha_{m,n}\gamma_{a,b} K((m+1,n),(a,b)) $$ $$ = \sum_{m,n,a,b} \alpha_{m,n}\gamma_{a,b} K((m,n),(a+1,b)) = \sum_{m,n,a,b} \alpha_{m,n}\gamma_{a,b} (x_{m,n},x_{a+1,b}) = (x,A_0 y). $$ Thus, $A_0$ is a symmetric operator. Its closure we denote by $A$. On the other hand, we have $$ (B_0 x,B_0 y) = \sum_{m,n,a,b} \alpha_{m,n}\overline{\gamma_{a,b}} (x_{m,n+1},x_{a,b+1}) = \sum_{m,n,a,b} \alpha_{m,n}\overline{\gamma_{a,b}} K((m,n+1),(a,b+1)) $$ $$ = \sum_{m,n,a,b} \alpha_{m,n}\overline{\gamma_{a,b}} K((m,n),(a,b)) = \sum_{m,n,a,b} \alpha_{m,n}\overline{\gamma_{a,b}} (x_{m,n},x_{a,b}) = (x,y). $$ In particular, this means that $B_0$ is bounded. By continuity we extend $B_0$ to a bounded operator $B$ such that $$ (Bx,By) = (x,y),\qquad x,y\in H. $$ Since $R(B_0)=L$ and $B_0$ has a bounded inverse, we have $R(B)=H$. Thus, $B$ is a unitary operator in $H$. Notice that operators $A_0$ and $B_0$ commute. It is straightforward to check that $A$ and $B$ commute: \begin{equation} \label{f2_11} AB x = BA x,\qquad x\in D(A). \end{equation} Consider the following operator: \begin{equation} \label{f2_12} J_0 x = \sum_{(m,n)\in T} \overline{\alpha_{m,n}} x_{m,-n}, \end{equation} where \begin{equation} \label{f2_13} x = \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n} \in L. \end{equation} Let us check that this definition is correct. Consider another representation for $x$ as in~(\ref{f2_10}). Then $$ \left\| \sum_{(m,n)\in T} (\overline{\alpha_{m,n}} - \overline{\beta_{m,n}}) x_{m,-n} \right\|^2 $$ $$= \left( \sum_{(m,n)\in T} \overline{ (\alpha_{m,n}-\beta_{m,n}) } x_{m,-n}, \sum_{(k,l)\in T} \overline{ (\alpha_{k,l}-\beta_{k,l}) } x_{k,-l} \right) $$ $$ = \sum_{(m,n),(k,l)\in T} \overline{(\alpha_{m,n}-\beta_{m,n})} (\alpha_{k,l}-\beta_{k,l}) K((m,-n),(k,-l)) $$ $$= \overline{ \sum_{(m,n),(k,l)\in T} (\alpha_{m,n}-\beta_{m,n}) \overline{(\alpha_{k,l}-\beta_{k,l})} K((m,n),(k,l)) } $$ $$= \overline{ \left( \sum_{(m,n)\in T} (\alpha_{m,n}-\beta_{m,n}) x_{m,n}, \sum_{(k,l)\in T} (\alpha_{k,l}-\beta_{k,l}) x_{k,l} \right) } = 0. $$ Thus, the definition of $J_0$ is correct. For an arbitrary $y = \sum_{(a,b)\in T} \gamma_{a,b} x_{a,b} \in L$ we can write $$ (J_0 x,J_0 y) = \sum_{m,n,a,b} \overline{\alpha_{m,n}}\gamma_{a,b} (x_{m,-n},x_{a,-b}) = \sum_{m,n,a,b} \overline{\alpha_{m,n}}\gamma_{a,b} K((m,-n),(a,-b)) $$ $$ = \sum_{m,n,a,b} \overline{\alpha_{m,n}} \gamma_{a,b} K((a,b),(m,n)) = \sum_{m,n,a,b} \overline{\alpha_{m,n}}\gamma_{a,b} (x_{a,b},x_{m,n}) = (y,x). $$ In particular, this implies that $J_0$ is bounded. By continuity we extend $J_0$ to a bounded antilinear operator $J$ such that $$ (Jx,Jy) = (y,x),\qquad x,y\in H. $$ Moreover, we get $J^2 = E_H$. Consequently, $J$ is a conjugation in $H$ (\cite{cit_8000_S}). \noindent Notice that $J_0$ commutes with $A_0$. It is easy to check that \begin{equation} \label{f2_14} AJ x = JA x,\qquad x\in D(A). \end{equation} On the other hand, we have $J_0 B_0 = B_0^{-1} J_0$. By continuity we get \begin{equation} \label{f2_15} JB = B^{-1}J. \end{equation} Consider the Cayley transformation of the operator A: \begin{equation} \label{f2_16} V_A := (A+iE_H)(A-iE_H)^{-1}, \end{equation} and set \begin{equation} \label{f2_17} H_1 := \Delta_A(i),\ H_2 := H\ominus H_1,\ H_3:= \Delta_A(-i),\ H_4 := H\ominus H_3. \end{equation} \begin{prop} \label{p2_1} The operator $B$ reduces subspaces $H_i$, $1\leq i\leq 4$: \begin{equation} \label{f2_18} BH_i = H_i,\qquad 1\leq i\leq 4. \end{equation} Moreover, the following equality holds: \begin{equation} \label{f2_19} BV_Ax = V_ABx,\qquad x\in H_1. \end{equation} \end{prop} {\bf Proof. } Choose an arbitrary $x\in \Delta_A(z)$, $x=(A-zE_H)f_A$, $f_A\in D(A)$, $z\in \mathbb{C}\backslash \mathbb{R}$. By~(\ref{f2_11}) we get $$ Bx = BAf_A - zBf_A = ABf_A - zBf_A = (A-zE_H)Bf_A\in \Delta_A(z). $$ In particular, we have $BH_1\subseteq H_1$, $BH_3\subseteq H_3$. Notice that $B_0^{-1}A_0 = A_0 B_0^{-1}$. It is a straightforward calculation to check that \begin{equation} \label{f2_20} AB^{-1} x = B^{-1}A x,\qquad x\in D(A). \end{equation} Repeating the above argument with $B^{-1}$ instead of $B$ we get $B^{-1}H_1\subseteq H_1$, $B^{-1}H_3\subseteq H_3$, and therefore $H_1\subseteq BH_1$, $H_3\subseteq BH_3$. Consequently, the operator $B$ reduces subspaces $H_1$ and $H_3$. It follows directly that $B$ reduces $H_2$ and $H_4$, as well. \noindent Since $$ (A-iE_H) Bx = B(A-iE_H)x,\qquad x\in D(A), $$ for arbitrary $y\in H_1$, $y = (A-iE_H)x_A$, $x_A\in D(A)$, we have $$ (A-iE_H) B (A-iE_H)^{-1} y = B y; $$ $$ B (A-iE_H)^{-1} y = (A-iE_H)^{-1} B y,\qquad y\in H_1, $$ and~(\ref{f2_19}) follows. $\Box$ Our aim here is to construct a unitary operator $U$ in $H$, $U\supset V_A$, which commutes with $B$. Choose an arbitrary $x\in H$, $x= x_{H_1} + x_{H_2}$. For an operator $U$ of the required type by~Proposition~\ref{p2_1} we could write: $$ BU x = BV_Ax_{H_1} + BU x_{H_2} = V_ABx_{H_1} + BU x_{H_2}, $$ $$ UB x = UB x_{H_1} + UB x_{H_2} = V_ABx_{H_1} + UB x_{H_2}. $$ So, it is enough to find an isometric operator $U_{2,4}$ which maps $H_2$ onto $H_4$, and commutes with $B$: \begin{equation} \label{f2_21} B U_{2,4} x = U_{2,4}B x,\qquad x\in H_2. \end{equation} Moreover, all operators $U$ of the required type have the following form: \begin{equation} \label{f2_22} U = V_A \oplus U_{2,4}, \end{equation} where $U_{2,4}$ is an isometric operator which maps $H_2$ onto $H_4$, and commutes with $B$. \noindent We shall denote the operator $B$ restricted to $H_i$ by $B_{H_i}$, $1\leq i\leq 4$. Notice that \begin{equation} \label{f2_23} A^* J x= JA^* x,\qquad x\in D(A^*). \end{equation} Indeed, for arbitrary $f_A\in D(A)$ and $g_{A^*}\in D(A^*)$ we can write $$ \overline{ (Af_A,Jg_{A^*}) } = (JAf_A, g_{A^*}) = (AJf_A, g_{A^*}) = (Jf_A, A^*g_{A^*}) $$ $$ = \overline{ (f_A,JA^*g_{A^*}) }, $$ and~(\ref{f2_23}) follows. \noindent Choose an arbitrary $x\in H_2$. We have $$ A^* x = -i x, $$ and therefore $$ A^* Jx = JA^* x = ix. $$ Thus, we have $$ JH_2 \subseteq H_4. $$ In a similar manner we get $$ JH_4 \subseteq H_2, $$ and therefore \begin{equation} \label{f2_24} JH_2 = H_4,\quad JH_4 = H_2. \end{equation} By the Godi\v{c}-Lucenko Theorem (\cite{cit_2000_GL},\cite[Theorem 1]{cit_3000_GP}) we have a representation: \begin{equation} \label{f2_25} B_{H_2} = KL, \end{equation} where $K$ and $L$ are some conjugations in $H_2$. We set \begin{equation} \label{f2_26} U_{2,4} := JK. \end{equation} From~(\ref{f2_24}) it follows that $U_{2,4}$ maps isometrically $H_2$ onto $H_4$. Notice that \begin{equation} \label{f2_27} U_{2,4}^{-1} := KJ. \end{equation} Using relation~(\ref{f2_15}) we get $$ U_{2,4} B_{H_2} U_{2,4}^{-1} x = JK KL KJ x = J LK J x = J B_{H_2}^{-1} J x $$ $$ = JB^{-1}J x = B x = B_{H_4} x,\qquad x\in H_4. $$ Therefore relation~(\ref{f2_21}) is true. We define an operator $U$ by~(\ref{f2_22}) and define \begin{equation} \label{f2_28} A_U := i(U+E_H)(U-E_H)^{-1} = iE_H + 2i(U-E_H)^{-1}. \end{equation} The inverse Cayley transformation $A_U$ is correctly defined since $1$ is not in the point spectrum of $U$. Indeed, $V_A$ is the Cayley transformation of a symmetric operator while eigen subspaces $H_2$ and $H_4$ have the zero intersection. Let \begin{equation} \label{f2_29} A_U = \int_\mathbb{R} s dE(s),\quad B = \int_{ [-\pi,\pi) } e^{i\varphi} dF(\varphi), \end{equation} where $E(s)$ and $F(\varphi)$ are the spectral measures of $A_U$ and $B$, respectively. These measures are defined on $\mathfrak{B}(\mathbb{R})$ and $\mathfrak{B}([-\pi,\pi))$, respectively (\cite{cit_9000_BS}). Since $U$ and $B$ commute, we get that $E(s)$ and $F(\varphi)$ commute, as well. By induction argument we have $$ x_{m,n} = A^m x_{0,n},\qquad m\in \mathbb{Z}_+,\ n\in \mathbb{Z}, $$ and $$ x_{0,n} = B^n x_{0,0},\qquad n\in \mathbb{Z}. $$ Therefore we have \begin{equation} \label{f2_30} x_{m,n} = A^m B^n x_{0,0},\qquad m\in \mathbb{Z}_+,\ n\in \mathbb{Z}. \end{equation} We can write $$ x_{m,n} = \int_\mathbb{R} s^m dE(s) \int_{ [-\pi,\pi) } e^{in\varphi} dF(\varphi) x_{0,0} = \int_\Pi s^m e^{in\varphi} d(E\times F) x_{0,0}, $$ where $E\times F$ is the product spectral measure on $\mathfrak{B}(\Pi)$. Then \begin{equation} \label{f2_31} s_{m,n} = (x_{m,n},x_{0,0})_H = \int_\Pi s^m e^{in\varphi} d((E\times F) x_{0,0}, x_{0,0})_H,\quad (m,n)\in T. \end{equation} The measure $\mu := ((E\times F) x_{0,0}, x_{0,0})_H$ is a non-negative Borel measure on $\Pi$ and relation~(\ref{f2_31}) shows that $\mu$ is a solution of the Devinatz moment problem. Thus, we obtained a new proof of the following criterion. \begin{thm} \label{t2_2} Let a Devinatz moment problem~(\ref{f1_1}) be given. This problem has a solution if an only if conditions~(\ref{f2_1}) hold for arbitrary complex numbers $\alpha_{m,n}$ such that all but finite numbers are zeros. \end{thm} {\bf Remark. } The original proof of Devinatz used the theory of reproducing kernels Hilbert spaces (RKHS). In particular, he used properties of RKHS corresponding to the product of two positive matrices and an inner structure of a RKHS corresponding to the moment problem. We used an abstract approach with the Godi\v{c}-Lucenko Theorem and basic facts from the standard operator theory. \section{Canonical solutions. A set of all solutions.} Let a moment problem~(\ref{f1_1}) be given. Construct a Hilbert space $H$ and operators $A,B,J$ as in the previous Section. Let $\widetilde A\supseteq A$ be a self-adjoint extension of $A$ in a Hilbert space $\widetilde H\supseteq H$. Let $R_z(\widetilde A)$, $z\in \mathbb{C}\backslash \mathbb{R}$, be the resolvent function of $\widetilde A$, and $E_{\widetilde A}$ be its spectral measure. Recall that the function \begin{equation} \label{f3_1} \mathbf{R}_z(A) := P^{\widetilde H}_H R_z(\widetilde A),\qquad z\in \mathbb{C}\backslash \mathbb{R}, \end{equation} is said to be a generalized resolvent of $A$. The function \begin{equation} \label{f3_2} \mathbf{E}_A (\delta) := P^{\widetilde H}_H E_{\widetilde A} (\delta),\qquad \delta\in \mathfrak{B}(\mathbb{R}), \end{equation} is said to be a spectral measure of $A$. There exists a one-to-one correspondence between generalized resolvents and spectral measures established by the following relation~\cite{cit_6000_AG}: \begin{equation} \label{f3_3} (\mathbf{R}_z(A) x,y)_H = \int_{\mathbb{R}} \frac{1}{t-z} d(\mathbf{E}_A x,y)_H,\qquad x,y\in H. \end{equation} We shall reduce the Devinatz moment problem to a problem of finding of generalized resolvents of a certain class. \begin{thm} \label{t3_1} Let a Devinatz moment problem~(\ref{f1_1}) be given and conditions~(\ref{f2_1}) hold. Consider a Hilbert space $H$ and a sequence $\{ x_{m,n} \}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$, $x_{m,n}\in H$, such that relation~(\ref{f2_6}) holds where $K$ is defined by~(\ref{f2_2}). Consider operators $A_0$,$B_0$ defined by~(\ref{f2_7}),(\ref{f2_8}) on $L = \mathop{\rm Lin}\nolimits\{ x_{m,n} \}_{(m,n)\in T}$. Let $A=\overline{A_0}$, $B=\overline{B_0}$. Let $\mu$ be an arbitrary solution of the moment problem. Then it has the following form: \begin{equation} \label{f3_4} \mu (\delta)= ((\mathbf{E}\times F)(\delta) x_{0,0}, x_{0,0})_H,\qquad \delta\in \mathfrak{B}(\mathbb{R}), \end{equation} where $F$ is the spectral measure of $B$, $\mathbf{E}$ is a spectral measure of $A$ which commutes with $F$. By $((\mathbf{E}\times F)(\delta) x_{0,0}, x_{0,0})_H$ we mean the non-negative Borel measure on $\mathbb{R}$ which is obtained by the Lebesgue continuation procedure from the following non-negative measure on rectangules \begin{equation} \label{f3_5} ((\mathbf{E}\times F)(I_x\times I_\varphi)) x_{0,0}, x_{0,0})_H := ( \mathbf{E}(I_x) F(I_\varphi)) x_{0,0}, x_{0,0})_H, \end{equation} where $I_x\subset \mathbb{R}$, $I_\varphi\subseteq [-\pi,\pi)$ are arbitrary intervals. \noindent On the other hand, for an arbitrary spectral measure $\mathbf{E}$ of $A$ which commutes with the spectral measure $F$ of $B$, by relation~(\ref{f3_4}) it corresponds a solution of the moment problem~(\ref{f1_1}). \noindent Moreover, the correspondence between the spectral measures of $A$ which commute with the spectral meeasure of $B$ and solutions of the Devinatz moment problem is bijective. \end{thm} {\bf Remark. } The measure in~(\ref{f3_5}) is non-negative. Indeed, for arbitrary intervals $I_x\subset \mathbb{R}$, $I_\varphi\subseteq [-\pi,\pi)$, we can write $$ \left( \mathbf{E}(I_x) F(I_\varphi) x_{0,0}, x_{0,0} \right)_H = \left( F(I_\varphi) \mathbf{E}(I_x) F(I_\varphi) x_{0,0}, x_{0,0} \right)_H $$ $$ = \left( \mathbf{E}(I_x) F(I_\varphi) x_{0,0}, F(I_\varphi) x_{0,0} \right)_H = \left( \widehat E(I_x) F(I_\varphi) x_{0,0}, \widehat E(I_x) F(I_\varphi) x_{0,0} \right)_{\widehat H} \geq 0, $$ where $\widehat E$ is the spectral function of a self-adjoint extension $\widehat A\supseteq A$ in a Hilbert space $\widehat H\supseteq H$ such that $\mathbf{E} = P^{\widehat H}_H \widehat E$. The measure in~(\ref{f3_5}) is additive. If $I_\varphi = I_{1,\varphi}\cup I_{2,\varphi}$, $I_{1,\varphi}\cap I_{2,\varphi} = \emptyset$, then $$ \left( \mathbf{E}(I_x) F(I_\varphi) x_{0,0}, x_{0,0} \right)_H = \left( F( I_{1,\varphi}\cup I_{2,\varphi} )\mathbf{E}(I_x) x_{0,0}, x_{0,0} \right)_H $$ $$ = \left( F(I_{1,\varphi})\mathbf{E}(I_x) x_{0,0}, x_{0,0} \right)_H + \left( F(I_{2,\varphi})\mathbf{E}(I_x) x_{0,0}, x_{0,0} \right)_H. $$ The case $I_x = I_{1,x}\cup I_{2,x}$ is analogous. Moreover, repeating the standard arguments~\cite[Chapter 5, Theorem 2, p. 254-255]{cit_9500_KF} we conclude that the measure in~(\ref{f3_5}) is $\sigma$-additive. Thus, it posesses the (unique) Lebesgue continuation to a (finite) non-negative Borel measure on $\Pi$. {\bf Proof. } Consider a Hilbert space $H$ and operators $A$,$B$ as in the statement of the Theorem. Let $F$ be the spectral measure of $B$. Let $\mu$ be an arbitrary solution of the moment problem~(\ref{f1_1}). Consider the space $L^2_\mu$ of complex functions on $\Pi$ which are square integrable with respect to the measure $\mu$. The scalar product and the norm are given by $$ (f,g)_\mu = \int_\Pi f(x,\varphi) \overline{ g(x,\varphi) } d\mu,\quad \|f\|_\mu = \left( (f,f)_\mu \right)^{ \frac{1}{2} },\quad f,g\in L^2_\mu. $$ Consider the following operators: \begin{equation} \label{f3_6} A_\mu f(x,\varphi) = xf(x,\varphi),\qquad D(A_\mu) = \{ f\in L^2_\mu:\ xf(x,\varphi)\in L^2_\mu \}, \end{equation} \begin{equation} \label{f3_7} B_\mu f(x,\varphi) = e^{i\varphi} f(x,\varphi),\qquad D(B_\mu) = L^2_\mu. \end{equation} The operator $A_\mu$ is self-adjoint and the operator $B_\mu$ is unitary. Moreover, these operators commute and therefore the spectral measure $E_\mu$ of $A_\mu$ and the spectral measure $F_\mu$ of $B_\mu$ commute, as well. \noindent Let $p(x,\varphi)$ be a (power-trigonometric) polynomial of the form~(\ref{f1_1}) and $q(x,\varphi)$ be a (power-trigonometric) polynomial of the form~(\ref{f1_1}) with $\beta_{m,n}\in \mathbb{C}$ instead of $\alpha_{m,n}$. Then $$ (p,q)_\mu = \sum_{(m,n)\in T, (k,l)\in T} \alpha_{m,n}\overline{ \beta_{k,l} } \int_\Pi x^{m+k} e^{i(n-l)\varphi} d\mu $$ $$ = \sum_{(m,n)\in T, (k,l)\in T} \alpha_{m,n}\overline{ \beta_{k,l} } s_{m+k,n-l}, $$ On the other hand, we can write $$ \left( \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n}, \sum_{(k,l)\in T} \beta_{k,l} x_{k,l} \right)_H = \sum_{(m,n)\in T, (k,l)\in T} \alpha_{m,n}\overline{ \beta_{k,l} } (x_{m,n},x_{k,l})_H $$ $$ = \sum_{(m,n)\in T, (k,l)\in T} \alpha_{m,n}\overline{ \beta_{k,l} } K((m,n),(k,l)) = \sum_{(m,n)\in T, (k,l)\in T} \alpha_{m,n}\overline{ \beta_{k,l} } s_{m+k,n-l}. $$ Therefore \begin{equation} \label{f3_8} (p,q)_\mu = \left( \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n}, \sum_{(k,l)\in T} \beta_{k,l} x_{k,l} \right)_H. \end{equation} Consider thr following operator: \begin{equation} \label{f3_9} V[p] = \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n},\quad p=\sum_{(m,n)\in T} \alpha_{m,n} x^m e^{in\varphi}. \end{equation} Here by $[p]$ we mean the class of equivalence in $L^2_\mu$ defined by $p$. If two different polynomials $p$ and $q$ belong to the same class of equivalence then by~(\ref{f3_8}) we get $$ 0 = \| p-q \|_\mu^2 = (p-q,p-q)_\mu = \left( \sum_{(m,n)\in T} (\alpha_{m,n}-\beta_{m,n}) x_{m,n}, \sum_{(k,l)\in T} (\alpha_{k,l}-\beta_{k,l}) x_{k,l} \right) $$ $$ = \left\| \sum_{(m,n)\in T} \alpha_{m,n} x_{m,n} - \sum_{(m,n)\in T} \beta_{m,n} x_{m,n} \right\|_\mu^2. $$ Thus, the definition of $V$ is correct. It is not hard to see that $V$ maps a set of polynomials $P^2_{0,\mu}$ in $L^2_\mu$ on $L$. By continuity we extend $V$ to the isometric transformation from the closure of polynomials $P^2_\mu = \overline{P^2_{0,\mu}}$ onto $H$. \noindent Set $H_0 := L^2_\mu \ominus P^2_\mu$. Introduce the following operator: \begin{equation} \label{f3_10} U := V \oplus E_{H_0}, \end{equation} which maps isometrically $L^2_\mu$ onto $\widetilde H := H\oplus H_0$. Set \begin{equation} \label{f3_11} \widetilde A := UA_\mu U^{-1},\quad \widetilde B := UB_\mu U^{-1}. \end{equation} Notice that $$ \widetilde A x_{m,n} = UA_\mu U^{-1} x_{m,n} = UA_\mu x^m e^{in\varphi} = Ux^{m+1} e^{in\varphi} = x_{m+1,n}, $$ $$ \widetilde B x_{m,n} = UB_\mu U^{-1} x_{m,n} = UB_\mu x^m e^{in\varphi} = Ux^{m} e^{i(n+1)\varphi} = x_{m,n+1}. $$ Therefore $\widetilde A\supseteq A$ and $\widetilde B\supseteq B$. Let \begin{equation} \label{f3_12} \widetilde A = \int_\mathbb{R} s d\widetilde E(s),\quad \widetilde B = \int_{ [-\pi,\pi) } e^{i\varphi} d \widetilde F(\varphi), \end{equation} where $\widetilde E(s)$ and $\widetilde F(\varphi)$ are the spectral measures of $\widetilde A$ and $\widetilde B$, respectively. Repeating arguments after relation~(\ref{f2_29}) we obtain that \begin{equation} \label{f3_13} x_{m,n} = \widetilde A^m \widetilde B^n x_{0,0},\qquad m\in \mathbb{Z}_+,\ n\in \mathbb{Z}, \end{equation} \begin{equation} \label{f3_14} s_{m,n} = \int_\Pi s^m e^{in\varphi} d((\widetilde E\times \widetilde F) x_{0,0}, x_{0,0})_{\widetilde H},\quad (m,n)\in T, \end{equation} where $(\widetilde E\times \widetilde F)$ is the product measure of $\widetilde E$ and $\widetilde F$. Thus, the measure $\widetilde \mu := ((\widetilde E\times \widetilde F) x_{0,0}, x_{0,0})_{\widetilde H}$ is a solution of the Devinatz moment problem. \noindent Let $I_x\subset \mathbb{R}$, $I_\varphi\subseteq [-\pi,\pi)$ be arbitrary intervals. Then $$ \widetilde \mu (I_x \times I_\varphi) = ((\widetilde E\times \widetilde F) (I_x \times I_\varphi) x_{0,0}, x_{0,0})_{\widetilde H} $$ $$ = ( \widetilde E(I_x) \widetilde F(I_\varphi) x_{0,0}, x_{0,0})_{\widetilde H} = ( P^{\widetilde H}_H \widetilde E(I_x) \widetilde F(I_\varphi) x_{0,0}, x_{0,0})_{\widetilde H} $$ $$ = ( \mathbf{E}(I_x) F(I_\varphi) x_{0,0}, x_{0,0})_{H}, $$ where $\mathbf{E}$ is the correponding spectral function of $A$ and $F$ is the spectral function of $B$. Thus, the measure $\widetilde \mu$ has the form~(\ref{f3_4}) since the Lebesgue continuation is unique. \noindent Let us show that $\widetilde \mu = \mu$. Consider the following transformation: \begin{equation} \label{f3_15} S:\ (x,\varphi) \in \Pi \mapsto \left( \mathop{\rm Arg }\nolimits \frac{x-i}{x+i}, \varphi \right) \in \Pi_0, \end{equation} where $\Pi_0 = [-\pi,\pi) \times [-\pi,\pi)$ and $\mathop{\rm Arg }\nolimits e^{iy} = y\in [-\pi,\pi)$. By virtue of $V$ we define the following measures: \begin{equation} \label{f3_16} \mu_0 (VG) := \mu (G),\quad \widetilde\mu_0 (VG) := \widetilde\mu (G),\qquad G\in \mathfrak{B}(\Pi), \end{equation} It is not hard to see that $\mu_0$ and $\widetilde\mu_0$ are non-negative measures on $\mathfrak{B}(\Pi_0)$. Then \begin{equation} \label{f3_17} \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d\mu = \int_{\Pi_0} e^{im\psi} e^{in\varphi} d\mu_0, \end{equation} \begin{equation} \label{f3_18} \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d\widetilde\mu = \int_{\Pi_0} e^{im\psi} e^{in\varphi} d\widetilde\mu_0,\qquad m,n\in \mathbb{Z}; \end{equation} and $$ \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d\widetilde\mu = \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d((\widetilde E\times \widetilde F) x_{0,0}, x_{0,0})_{\widetilde H} $$ $$ = \left( \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d(\widetilde E\times \widetilde F) x_{0,0}, x_{0,0} \right)_{\widetilde H} $$ $$ = \left( \int_\mathbb{R} \left( \frac{x-i}{x+i} \right)^m d\widetilde E \int_{[-\pi,\pi)} e^{in\varphi} d\widetilde F x_{0,0}, x_{0,0} \right)_{\widetilde H} $$ $$ = \left( \left( (\widetilde A - iE_{\widetilde H})(\widetilde A + iE_{\widetilde H})^{-1} \right)^m \widetilde B^n x_{0,0}, x_{0,0} \right)_{\widetilde H} $$ $$ = \left( U^{-1}\left( (\widetilde A - iE_{\widetilde H})(\widetilde A + iE_{\widetilde H})^{-1} \right)^m \widetilde B^n U 1, U 1 \right)_\mu $$ $$ = \left( \left( (A_\mu - iE_{L^2_\mu})(A_\mu + iE_{L^2_\mu})^{-1} \right)^m B_\mu^n 1, 1 \right)_\mu $$ \begin{equation} \label{f3_19} = \int_\Pi \left( \frac{x-i}{x+i} \right)^m e^{in\varphi} d\mu,\qquad m,n\in \mathbb{Z}. \end{equation} By virtue of relations~(\ref{f3_17}),(\ref{f3_18}) and~(\ref{f3_19}) we get \begin{equation} \label{f3_20} \int_{\Pi_0} e^{im\psi} e^{in\varphi} d\mu_0 = \int_{\Pi_0} e^{im\psi} e^{in\varphi} d\widetilde\mu_0,\qquad m,n\in \mathbb{Z}. \end{equation} By the Weierstrass theorem we can approximate any continuous function by exponentials and therefore \begin{equation} \label{f3_21} \int_{\Pi_0} f(\psi) g(\varphi) d\mu_0 = \int_{\Pi_0} f(\psi) g(\varphi) d\widetilde\mu_0, \end{equation} for arbitrary continuous functions on $\Pi_0$. In particular, we have \begin{equation} \label{f3_22} \int_{\Pi_0} \psi^n \varphi^m d\mu_0 = \int_{\Pi_0} \psi^n \varphi^m d\widetilde\mu_0,\qquad n,m\in \mathbb{Z}_+. \end{equation} However, the two-dimensional Hausdorff moment problem is determinate (\cite{cit_10000_ST}) and therefore we get $\mu_0 = \widetilde\mu_0$ and $\mu=\mu_0$. Thus, we have proved that an arbitrary solution $\mu$ of the Devinatz moment problem can be represented in the form~(\ref{f3_4}). Let us check the second assertion of the Theorem. For an arbitrary spectral measure $\mathbf{E}$ of $A$ which commutes with the spectral measure $F$ of $B$, by relation~(\ref{f3_4}) we define a non-negative Borel measure $\mu$ on $\Pi$. Let us show that the measure $\mu$ is a solution of the moment problem~(\ref{f1_1}). \noindent Let $\widehat A$ be a self-adjoint extension of the operator $A$ in a Hilbert space $\widehat H\supseteq H$, such that $$ \mathbf{E} = P^{\widehat H}_H \widehat E, $$ where $\widehat E$ is the spectral measure of $\widehat A$. By~(\ref{f2_30}) we get $$ x_{m,n} = A^m B^n x_{0,0} = \widehat A^m B^n x_{0,0} = P^{\widehat H}_H \widehat A^m B^n x_{0,0} $$ $$ = P^{\widehat H}_H \left( \lim_{a\to +\infty} \int_{[-a,a)} x^m d\widehat E \right) \int_{[-\pi,\pi)} e^{in\varphi} dF x_{0,0} = \left( \lim_{a\to +\infty} \int_{[-a,a)} x^m d\mathbf{E} \right) $$ $$ * \int_{[-\pi,\pi)} e^{in\varphi} dF x_{0,0} = \left( \lim_{a\to +\infty} \left( \int_{[-a,a)} x^m d\mathbf{E} \int_{[-\pi,\pi)} e^{in\varphi} dF \right) \right) x_{0,0}, $$ \begin{equation} \label{f3_23} \qquad m\in \mathbb{Z}_+,\ n\in \mathbb{Z}, \end{equation} where the limits are understood in the weak operator topology. Then we choose arbitrary points $$ -a = x_0 < x_1 < ... < x_{N}=a; $$ \begin{equation} \label{f3_24} \max_{1\leq i\leq N}|x_{i}-x_{i-1}| =: d,\quad N\in \mathbb{N}; \end{equation} $$ -\pi = \varphi_0 < \varphi_1 < ... < \varphi_{M}=\pi; $$ \begin{equation} \label{f3_25} \max_{1\leq j\leq M}|\varphi_{j}-\varphi_{j-1}| =: r;\quad M\in \mathbb{N}. \end{equation} Set $$ C_a := \int_{[-a,a)} x^m d\mathbf{E} \int_{[-\pi,\pi)} e^{in\varphi} dF = \lim_{d\rightarrow 0} \sum_{i=1}^N x_{i-1}^m \mathbf{E}([x_{i-1},x_i)) $$ $$ * \lim_{r\rightarrow 0} \sum_{j=1}^M e^{in\varphi_{j-1}} F([\varphi_{j-1},\varphi_j)), $$ where the integral sums converge in the strong operator topology. Then $$ C_a = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N x_{i-1}^m \mathbf{E}([x_{i-1},x_i)) \sum_{j=1}^M e^{in\varphi_{j-1}} F([\varphi_{j-1},\varphi_j)) $$ $$ = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N \sum_{j=1}^M x_{i-1}^m e^{in\varphi_{j-1}} \mathbf{E}([x_{i-1},x_i)) F([\varphi_{j-1},\varphi_j)), $$ where the limits are understood in the strong operator topology. Then $$ (C_a x_{0,0}, x_{0,0})_H = \left( \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N \sum_{j=1}^M x_{i-1}^m e^{in\varphi_{j-1}} \mathbf{E}([x_{i-1},x_i)) F([\varphi_{j-1},\varphi_j)) x_{0,0}, x_{0,0} \right)_H $$ $$ = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N \sum_{j=1}^M x_{i-1}^m e^{in\varphi_{j-1}} \left( \mathbf{E}([x_{i-1},x_i)) F([\varphi_{j-1},\varphi_j)) x_{0,0}, x_{0,0} \right)_H $$ $$ = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N \sum_{j=1}^M x_{i-1}^m e^{in\varphi_{j-1}} \left( (\mathbf{E}\times F) ( [x_{i-1},x_i)\times [\varphi_{j-1},\varphi_j) ) x_{0,0}, x_{0,0} \right)_H $$ $$ = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \sum_{i=1}^N \sum_{j=1}^M x_{i-1}^m e^{in\varphi_{j-1}} \left( \mu ( [x_{i-1},x_i)\times [\varphi_{j-1},\varphi_j) ) x_{0,0}, x_{0,0} \right)_H. $$ Therefore $$ (C_a x_{0,0}, x_{0,0})_H = \lim_{d\rightarrow 0} \lim_{r\rightarrow 0} \int_{[-a,a)\times[-\pi,\pi)} f_{d,r} (x,\varphi) d\mu, $$ where $f_{d,r}$ is equal to $x_{i-1}^m e^{in\varphi_{j-1}}$ on the rectangular $[x_{i-1},x_i) \times [\varphi_{j-1},\varphi_j)$, $1\leq i\leq N$, $1\leq j\leq M$. \noindent If $r\rightarrow 0$, then the simple function $f_{d,r}$ converges uniformly to the function $f_d$ which is equal to $x_{i-1}^m e^{in\varphi}$ on the rectangular $[x_{i-1},x_i) \times [\varphi_{j-1},\varphi_j)$, $1\leq i\leq N$, $1\leq j\leq M$. Then $$ (C_a x_{0,0}, x_{0,0})_H = \lim_{d\rightarrow 0} \int_{[-a,a)\times[-\pi,\pi)} f_{d} (x,\varphi) d\mu. $$ If $d\rightarrow 0$, then the function $f_{d}$ converges uniformly to the function $x^m e^{in\varphi}$. Since $|f_d|\leq A^m$, by the Lebesgue theorem we get \begin{equation} \label{f3_26} (C_a x_{0,0}, x_{0,0})_H = \int_{[-a,a)\times[-\pi,\pi)} x^m e^{in\varphi} d\mu. \end{equation} By virtue of relations~(\ref{f3_23}) and~(\ref{f3_26}) we get $$ s_{m,n} = (x_{m,n},x_{0,0})_H = \lim_{a\to +\infty} (C_a x_{0,0},x_{0,0})_H $$ \begin{equation} \label{f3_27} = \lim_{a\to+\infty} \int_{[-a,a)\times[-\pi,\pi)} x^m e^{in\varphi} d\mu = \int_\Pi x^m e^{in\varphi} d\mu. \end{equation} Thus, the measure $\mu$ is a solution of the Devinatz moment problem. Let us prove the last assertion of the Theorem. Suppose to the contrary that two different spectral measures $\mathbf{E}_1$ and $\mathbf{E}_1$ of $A$ commute with the spectral measure $F$ of $B$ and produce by relation~(\ref{f3_4}) the same solution $\mu$ of the Devinatz moment problem. Choose an arbitrary $z\in \mathbb{C}\backslash \mathbb{R}$. Then $$ \int_\Pi \frac{x^m}{x-z} e^{in\varphi} d\mu = \int_\Pi \frac{x^m}{x-z} e^{in\varphi} ((\mathbf{E}_k\times F)(\delta) x_{0,0}, x_{0,0})_H $$ \begin{equation} \label{f3_28} = \lim_{a\to +\infty} \int_{[-a,a)\times [-\pi,\pi)} \frac{x^m}{x-z} e^{in\varphi} d((\mathbf{E}_k\times F)(\delta) x_{0,0}, x_{0,0})_H,\quad k=1,2. \end{equation} Consider arbitrary partitions of the type~(\ref{f3_24}),(\ref{f3_25}). Then $$ D_a := \int_{[-a,a)\times [-\pi,\pi)} \frac{x^m}{x-z} e^{in\varphi} d((\mathbf{E}_k\times F)(\delta) x_{0,0}, x_{0,0})_H $$ $$ = \lim_{d\to 0} \lim_{r\to 0} \int_{[-a,a)\times [-\pi,\pi)} g_{z;d,r}(x,\varphi) d((\mathbf{E}_k\times F)(\delta) x_{0,0}, x_{0,0})_H. $$ Here the function $g_{z;d,r}(x,\varphi)$ is equal to $\frac{x_{i-1}^m}{x_{i-1}-z} e^{in\varphi_{j-1}}$ on the rectangular $[x_{i-1},x_i) \times [\varphi_{j-1},\varphi_j)$, $1\leq i\leq N$, $1\leq j\leq M$. Then $$ D_a = \lim_{d\to 0} \lim_{r\to 0} \sum_{i=1}^N \sum_{j=1}^M \frac{ x_{i-1}^m }{ x_{i-1}-z } e^{in\varphi_{j-1}} \left( \mathbf{E}_k ([x_{i-1},x_i)) F([\varphi_{j-1},\varphi_j)) x_{0,0}, x_{0,0} \right)_H $$ $$ = \lim_{d\to 0} \lim_{r\to 0} \left( \sum_{i=1}^N \frac{ x_{i-1}^m }{ x_{i-1}-z } \mathbf{E}_k ([x_{i-1},x_i)) \sum_{j=1}^M e^{in\varphi_{j-1}} F([\varphi_{j-1},\varphi_j)) x_{0,0}, x_{0,0} \right)_H $$ $$ = \left( \int_{[-a,a)} \frac{ x^m }{ x-z } d\mathbf{E}_k \int_{[-\pi,\pi)} e^{in\varphi} dF x_{0,0}, x_{0,0} \right)_H. $$ Let $n = n_1+n_2$, $n_1,n_2\in \mathbb{Z}$. Then we can write: $$ D_a = \left( B^{n_1} \int_{[-a,a)} \frac{ x^m }{ x-z } d\mathbf{E}_k B^{n_2} x_{0,0}, x_{0,0} \right)_H $$ $$ = \left( \int_{[-a,a)} \frac{ x^m }{ x-z } d\mathbf{E}_k x_{0,n_2}, x_{0,-n_1} \right)_H. $$ By~(\ref{f3_28}) we get $$ \int_\Pi \frac{x^m}{x-z} e^{in\varphi} d\mu = \lim_{a\to +\infty} D_a = \lim_{a\to +\infty}\left( \int_{[-a,a)} \frac{ x^m }{ x-z } d \widehat{E}_k x_{0,n_2}, x_{0,-n_1} \right)_{\widehat H_k} $$ $$ = \left( \int_\mathbb{R} \frac{ x^m }{ x-z } d\widehat{E}_k x_{0,n_2}, x_{0,-n_1} \right)_{\widehat H_k} = \left( \widehat{A}^{m_2} R_z(\widehat{A}_k) \widehat{A}^{m_1} x_{0,n_2}, x_{0,-n_1} \right)_{\widehat H_k} $$ \begin{equation} \label{f3_29} = \left( R_z(\widehat{A}_k) x_{m_1,n_2}, x_{m_2,-n_1} \right)_H, \end{equation} where $m_1,m_2\in \mathbb{Z}_+:\ m_1+m_2 = m$, and $\widehat A_k$ is a self-adjoint extension of $A$ in a Hilbert space $\widehat H_k\supseteq H$ such that its spectral measure $\widehat E_k$ generates $\mathbf{E}_k$: $\mathbf{E}_k = P^{\widehat H_k}_H \widehat E_k$; $k=1,2$. \noindent Relation~(\ref{f3_29}) shows that the generalized resolvents corresponding to $\mathbf{E}_k$, $k=1,2$, coincide. That means that the spectral measures $\mathbf{E}_1$ and $\mathbf{E}_2$ coincide. We obtained a contradiction. This completes the proof. $\Box$ \begin{dfn} \label{d3_1} A solution $\mu$ of the Devinatz moment problem~(\ref{f1_1}) we shall call {\bf canonical} if it is generated by relation~(\ref{f3_4}) where $\mathbf{E}$ is an {\bf orthogonal} spectral measure of $A$ which commutes with the spectral measure of $B$. Orthogonal spectral measures are those measures which are the spectral measures of self-adjoint extensions of $A$ inside $H$. \end{dfn} Let a moment problem~(\ref{f1_1}) be given and conditions~(\ref{f2_1}) hold. Let us describe canonical solutions of the Devinatz moment problem. In the proof of Theorem~\ref{t2_2} we have constructed one canonical solution, see relation~(\ref{f2_31}). Let $\mu$ be an arbitrary canonical solution and $\mathbf{E}$ be the corresponding orthogonal spectral measure of $A$. Let $\widetilde A$ be the self-adjoint operator in $H$ which corresponds to $\mathbf{E}$. Consider the Cayley transformation of $\widetilde A$: \begin{equation} \label{f3_30} U_{\widetilde A} = (\widetilde A + iE_H)(\widetilde A - iE_H)^{-1} \supseteq V_A, \end{equation} where $V_A$ is defined by~(\ref{f2_16}). Since $\mathbf{E}$ commutes with the spectral measure $F$ of $B$, then $U_{\widetilde A}$ commutes with $B$. By relation~(\ref{f2_22}) the operator $U_{\widetilde A}$ have the following form: \begin{equation} \label{f3_31} U_{\widetilde A} = V_A \oplus \widetilde U_{2,4}, \end{equation} where $\widetilde U_{2,4}$ is an isometric operator which maps $H_2$ onto $H_4$, and commutes with $B$. Let the operator $U_{2,4}$ be defined by~(\ref{f2_26}). Then the following operator \begin{equation} \label{f3_32} U_2 = U_{2,4}^{-1} \widetilde U_{2,4}, \end{equation} is a unitary operator in $H_2$ which commutes with $B_{H_2}$. Denote by $\mathbf{S}(B;H_2)$ a set of all unitary operators in $H_2$ which commute with $B_{H_2}$. Choose an arbitrary operator $\widehat U_2\in \mathbf{S}(B;H_2)$. Define $\widehat U_{2,4}$ by the following relation: \begin{equation} \label{f3_33} \widehat U_{2,4} = U_{2,4} \widehat U_2. \end{equation} Notice that $\widehat U_{2,4}$ commutes with $B_{H_2}$. Then we define a unitary operator $U = V_A \oplus \widehat U_{2,4}$ and its Cayley transformation $\widehat A$ which commute with the operator $B$. Repeating arguments before~(\ref{f2_31}) we get a canonical solution of the Devinatz moment problem. \noindent Thus, all canonical solutions of the Devinatz moment problem are generated by operators $\widehat U_2\in \mathbf{S}(B;H_2)$. Notice that different operators $U',U''\in \mathbf{S}(B;H_2)$ produce different orthogonal spectral measures $\mathbf{E}',\mathbf{E}$. By Theorem~\ref{t3_1}, these spectral measures produce different solutions of the moment problem. Recall some definitions from~\cite{cit_9000_BS}. A pair $(Y,\mathfrak{A})$, where $Y$ is an arbitrary set and $\mathfrak{A}$ is a fixed $\sigma$-algebra of subsets of $A$ is said to be a {\it measurable space}. A triple $(Y,\mathfrak{A},\mu)$, where $(Y,\mathfrak{A})$ is a measurable space and $\mu$ is a measure on $\mathfrak{A}$ is said to be a {\it space with a measure}. Let $(Y,\mathfrak{A})$ be a measurable space, $\mathbf{H}$ be a Hilbert space and $\mathcal{P}=\mathcal{P}(\mathbf{H})$ be a set of all orthogonal projectors in $\mathbf{H}$. A countably additive mapping $E:\ \mathfrak{A}\rightarrow \mathcal{P}$, $E(Y) = E_{\mathbf{H}}$, is said to be a {\it spectral measure} in $\mathbf{H}$. A set $(Y,\mathfrak{A},H,E)$ is said to be a {\it space with a spectral measure}. By $S(Y,E)$ one means a set of all $E$-measurable $E$-a.e. finite complex-valued functions on $Y$. Let $(Y,\mathfrak{A},\mu)$ be a separable space with a $\sigma$-finite measure and to $\mu$-everyone $y\in Y$ it corresponds a Hilbert space $G(y)$. A function $N(y) = \dim G(y)$ is called the {\it dimension function}. It is supposed to be $\mu$-measurable. Let $\Omega$ be a set of vector-valued functions $g(y)$ with values in $G(y)$ which are defined $\mu$-everywhere and are measurable with respect to some base of measurability. A set of (classes of equivalence) of such functions with the finite norm \begin{equation} \label{f3_34} \| g \|^2_{\mathcal{H}} = \int |g(y)|^2_{G(y)} d\mu(y) <\infty \end{equation} form a Hilbert space $\mathcal{H}$ with the scalar product given by \begin{equation} \label{f3_35} ( g_1,g_2 )_{\mathcal{H}} = \int (g_1,g_2)_{G(y)} d\mu(y). \end{equation} The space $\mathcal{H}= \mathcal{H}_{\mu,N} = \int_Y \oplus G(y) d\mu(y)$ is said to be a {\it direct integral of Hilbert spaces}. Consider the following operator \begin{equation} \label{f3_36} \mathbf{X}(\delta) g = \chi_\delta g,\qquad g\in \mathcal{H},\ \delta\in \mathfrak{A}, \end{equation} where $\chi_\delta$ is the characteristic function of the set $\delta$. The operator $\mathbf{X}$ is a spectral measure in $\mathcal{H}$. Let $t(y)$ be a measurable operator-valued function with values in $\mathbf{B}(G(y))$ which is $\mu$-a.e. defined and $\mu-\sup \|t(y)\|_{G(y)} < \infty$. The operator \begin{equation} \label{f3_37} T:\ g(y) \mapsto t(y)g(y), \end{equation} is said to be {\it decomposable}. It is a bounded operator in $\mathcal{H}$ which commutes with $\mathbf{X}(\delta)$, $\forall\delta\in \mathfrak{A}$. Moreover, every bounded operator in $\mathcal{H}$ which commutes with $\mathbf{X}(\delta)$, $\forall\delta\in \mathfrak{A}$, is decomposable~\cite{cit_9000_BS}. In the case $t(y) = \varphi(y)E_{G(y)}$, where $\varphi\in S(Y,\mu)$, we set $T =: Q_\varphi$. The decomposable operator is unitary if and only if $\mu$-a.e. the operator $t(y)$ is unitary. Return to the study of canonical solutions. Consider the spectral measure $F_2$ of the operator $B_{H_2}$ in $H_2$. There exists an element $h\in H_2$ of the maximal type, i.e. the non-negative Borel measure \begin{equation} \label{f3_38} \mu(\delta) := (F_2(\delta)h,h),\qquad \delta\in \mathfrak{B}([-\pi,\pi)), \end{equation} has the maximal type between all such measures (generated by other elements of $H_2$). This type is said to be the {\it spectral type} of the measure $F_2$. Let $N_2$ be the multiplicity function of the measure $F_2$. Then there exists a unitary transformation $W$ of the space $H_2$ on $\mathcal{H}=\mathcal{H}_{\mu,N_2}$ such that \begin{equation} \label{f3_39} W B_{H_2} W^{-1} = Q_{e^{iy}},\qquad W F_2(\delta) W^{-1} = \mathbf{X}(\delta). \end{equation} Notice that $\widehat U_2\in \mathbf{S}(B;H_2)$ if and only if the operator \begin{equation} \label{f3_40} V_2 := W \widehat U_2 W^{-1}, \end{equation} is unitary and commutes with $\mathbf{X}(\delta)$, $\forall\delta\in \mathfrak{[-\pi,\pi)}$. The latter is equivalent to the condition that $V_2$ is decomposable and the values of the corresponding operator-valued function $t(y)$ are $\mu$-a.e. unitary operators. A set of all decomposable operators in $\mathcal{H}$ such that the values of the corresponding operator-valued function $t(y)$ are $\mu$-a.e. unitary operators we denote by $\mathbf{D}(B;H_2)$. \begin{thm} \label{t3_2} Let a Devinatz moment problem~(\ref{f1_1}) be given. In conditions of Theorem~\ref{t3_1} all canonical solutions of the moment problem have the form~(\ref{f3_4}) where the spectral measures $\mathbf{E}$ of the operator $A$ are constructed by operators from $\mathbf{D}(B;H_2)$. Namely, for an arbitrary $V_2\in \mathbf{D}(B;H_2)$ we set $U_2 = W^{-1} V_2 W$, $\widehat U_{2,4} = U_{2,4} \widehat U_2$, $U = V_A \oplus \widehat U_{2,4}$, $\widehat A = i(U+E_H)(U-E_H)^{-1}$, and then $\mathbf{E}$ is the spectral measure of $\widehat A$. \noindent Moreover, the correspondence between $\mathbf{D}(B;H_2)$ and a set of all canonical solutions of the Devinatz moment problem is bijective. \end{thm} {\bf Proof. } The proof follows directly from the previous considerations. $\Box$ Consider a Devinatz moment problem~(\ref{f1_1}) and suppose that conditions~(\ref{f2_1}) hold. Let us turn to a parameterization of all solutions of the moment problem. We shall use Theorem~\ref{t3_1}. Consider relation~(\ref{f3_4}). The spectral measure $\mathbf{E}$ commutes with the operator $B$. Choose an arbitrary $z\in \mathbb{C}\backslash \mathbb{R}$. By virtue of relation~(\ref{f3_3}) we can write: $$ (B\mathbf{R}_z(A) x,y)_H = (\mathbf{R}_z(A) x,B^*y)_H = \int_{\mathbb{R}} \frac{1}{t-z} d(\mathbf{E}(t) x,B^*y)_H $$ \begin{equation} \label{f3_41} \int_{\mathbb{R}} \frac{1}{t-z} d(B\mathbf{E}(t) x,y)_H = \int_{\mathbb{R}} \frac{1}{t-z} d(\mathbf{E}(t)B x,y)_H,\qquad x,y\in H; \end{equation} \begin{equation} \label{f3_42} (\mathbf{R}_z(A) Bx,y)_H = \int_{\mathbb{R}} \frac{1}{t-z} d(\mathbf{E}(t) Bx,y)_H,\qquad x,y\in H, \end{equation} where $\mathbf{R}_z(A)$ is the generalized resolvent which corresponds to $\mathbf{E}$. Therefore we get \begin{equation} \label{f3_43} \mathbf{R}_z(A) B = B \mathbf{R}_z(A),\qquad z\in \mathbb{C}\backslash \mathbb{R}. \end{equation} On the other hand, if relation~(\ref{f3_43}) holds, then \begin{equation} \label{f3_44} \int_{\mathbb{R}} \frac{1}{t-z} d(\mathbf{E} Bx,y)_H = \int_{\mathbb{R}} \frac{1}{t-z} d(B\mathbf{E} x,y)_H,\quad x,y\in H,\ z\in \mathbb{C}\backslash \mathbb{R}. \end{equation} By the Stieltjes inversion formula~\cite{cit_10000_ST}, we obtain that $\mathbf{E}$ commutes with $B$. \noindent We denote by $\mathbf{M}(A,B)$ a set of all generalized resolvents $\mathbf{R}_z(A)$ of $A$ which satisfy relation~(\ref{f3_43}). Recall some known facts from~\cite{cit_4000_S} which we shall need here. Let $K$ be a closed symmetric operator in a Hilbert space $\mathbf{H}$, with the domain $D(K)$, $\overline{D(K)} = \mathbf{H}$. Set $N_\lambda = N_\lambda(K) = \mathbf{H} \ominus \Delta_K(\lambda)$, $\lambda\in \mathbb{C}\backslash \mathbb{R}$. Consider an arbitrary bounded linear operator $C$, which maps $N_i$ into $N_{-i}$. For \begin{equation} \label{f3_45} g = f + C\psi - \psi,\qquad f\in D(K),\ \psi\in N_i, \end{equation} we set \begin{equation} \label{f3_46} K_C g = Kf + i C \psi + i \psi. \end{equation} Since an intersection of $D(K)$, $N_i$ and $N_{-i}$ consists only of the zero element, this definition is correct. Notice that $K_C$ is a part of the operator $K^*$. The operator $K_C$ is said to be a {\it quasiself-adjoint extension of the operator $K$, defined by the operator $K$}. The following theorem can be found in~\cite[Theorem 7]{cit_4000_S}: \begin{thm} \label{t3_3} Let $K$ be a closed symmetric operator in a Hilbert space $\mathbf{H}$ with the domain $D(K)$, $\overline{D(K)} = \mathbf{H}$. All generalized resolvents of the operator $K$ have the following form: \begin{equation} \label{f3_47} \mathbf R_\lambda (K) = \left\{ \begin{array}{cc} (K_{F(\lambda)} - \lambda E_\mathbf{H})^{-1}, & \mathop{\rm Im}\nolimits\lambda > 0\\ (K_{F^*(\overline{\lambda}) } - \lambda E_\mathbf{H})^{-1}, & \mathop{\rm Im}\nolimits\lambda < 0 \end{array}\right., \end{equation} where $F(\lambda)$ is an analytic in $\mathbb{C}_+$ operator-valued function, which values are contractions which map $N_i(A) = H_2$ into $N_{-i}(A) = H_4$ ($\| F(\lambda) \|\leq 1$), and $K_{F(\lambda)}$ is the quasiself-adjoint extension of $K$ defined by $F(\lambda)$. On the other hand, for any operator function $F(\lambda)$ having the above properties there corresponds by relation~(\ref{f3_47}) a generalized resolvent of $K$. \end{thm} Notice that the correspondence between all generalized resolvents and functions $F(\lambda)$ in Theorem~\ref{t3_3} is bijective~\cite{cit_4000_S}. Return to the study of the Devinatz moment problem. Let us describe the set $\mathbf{M}(A,B)$. Choose an arbitrary $\mathbf{R}_\lambda\in \mathbf{M}(A,B)$. By~(\ref{f3_47}) we get \begin{equation} \label{f3_48} \mathbf{R}_\lambda = (A_{F(\lambda)} - \lambda E_H)^{-1},\qquad \mathop{\rm Im}\nolimits\lambda > 0, \end{equation} where $F(\lambda)$ is an analytic in $\mathbb{C}_+$ operator-valued function, which values are contractions which map $H_2$ into $H_4$, and $A_{F(\lambda)}$ is the quasiself-adjoint extension of $A$ defined by $F(\lambda)$. Then $$ A_{F(\lambda)} = \mathbf{R}_\lambda^{-1} + \lambda E_H,\qquad \mathop{\rm Im}\nolimits\lambda > 0. $$ By virtue of relation~(\ref{f3_43}) we obtain \begin{equation} \label{f3_49} BA_{F(\lambda)} h = A_{F(\lambda)} B h,\qquad h\in D(A_{F(\lambda)}),\ \lambda\in \mathbb{C}_+. \end{equation} Consider the following operators \begin{equation} \label{f3_50} W_{\lambda} := (A_{F(\lambda)} + iE_H)(A_{F(\lambda)} - iE_H)^{-1} = E_H + 2i(A_{F(\lambda)} - iE_H)^{-1}, \end{equation} \begin{equation} \label{f3_51} V_A = (A +iE_H)(A - iE_H)^{-1} = E_H + 2i(A - iE_H)^{-1}, \end{equation} where $\lambda\in \mathbb{C}_+$. Notice that (\cite{cit_4000_S}) \begin{equation} \label{f3_52} W_{\lambda} = V_A \oplus F(\lambda),\qquad \lambda\in \mathbb{C}_+. \end{equation} The operator $(A_{F(\lambda)} - iE_H)^{-1}$ is defined on the whole $H$, see~\cite[p.79]{cit_4000_S}. By relation~(\ref{f3_49}) we obtain \begin{equation} \label{f3_53} B (A_{F(\lambda)} - iE_H)^{-1} h = (A_{F(\lambda)} - iE_H)^{-1} B h,\qquad h\in H,\ \lambda\in \mathbb{C}_+. \end{equation} Then \begin{equation} \label{f3_54} B W_\lambda = W_\lambda B,\qquad \lambda\in \mathbb{C}_+. \end{equation} Recall that by Proposition~\ref{p2_1} the operator $B$ reduces the subspaces $H_j$, $1\leq j\leq 4$, and $BV_A = V_A B$. If we choose an arbitrary $h\in H_2$ and apply relations~(\ref{f3_54}),(\ref{f3_52}), we get \begin{equation} \label{f3_55} B F(\lambda) = F(\lambda) B,\qquad \lambda\in \mathbb{C}_+. \end{equation} Denote by $\mathbf{F}(A,B)$ a set of all analytic in $\mathbb{C}_+$ operator-valued functions which values are contractions which map $H_2$ into $H_4$ and which satisfy relation~(\ref{f3_55}). Thus, for an arbitrary $\mathbf{R}_\lambda\in \mathbf{M}(A,B)$ the corresponding function $F(\lambda)\in \mathbf{F}(A,B)$. On the other hand, choose an arbitrary $F(\lambda)\in \mathbf{F}(A,B)$. Then we derive~(\ref{f3_54}) with $W_\lambda$ defined by~(\ref{f3_50}). Then we get~(\ref{f3_53}),(\ref{f3_49}) and therefore \begin{equation} \label{f3_56} B \mathbf{R}_\lambda = \mathbf{R}_\lambda B,\qquad \lambda\in \mathbb{C}_+. \end{equation} Calculating the conjugate operators for the both sides of the last equality we conclude that this relation holds for all $\lambda\in \mathbb{C}$. \noindent Consider the spectral measure $F_2$ of the operator $B_{H_2}$ in $H_2$. We have obtained relation~(\ref{f3_39}) which we shall use one more time. Notice that $F(\lambda)\in \mathbf{F}(A,B)$ if and only if the operator-valued function \begin{equation} \label{f3_57} G(\lambda) := W F(\lambda) U_{2,4}^{-1} W^{-1},\qquad \lambda\in \mathbb{C}_+, \end{equation} is analytic in $\mathbb{C}_+$ and has values which are contractions in $\mathcal{H}$ which commute with $\mathbf{X}(\delta)$, $\forall\delta\in \mathfrak{[-\pi,\pi)}$. This means that for an arbitrary $\lambda\in \mathbb{C}_+$ the operator $G(\lambda)$ is decomposable and the values of the corresponding operator-valued function $t(y)$ are $\mu$-a.e. contractions. A set of all decomposable operators in $\mathcal{H}$ such that the values of the corresponding operator-valued function $t(y)$ are $\mu$-a.e. contractions we denote by $\mathrm{T}(B;H_2)$. A set of all analytic in $\mathbb{C}_+$ operator-valued functions $G(\lambda)$ with values in $\mathrm{T}(B;H_2)$ we denote by $\mathbf{G}(A,B)$. \begin{thm} \label{t3_4} Let a Devinatz moment problem~(\ref{f1_1}) be given. In conditions of Theorem~\ref{t3_1} all solutions of the moment problem have the form~(\ref{f3_4}) where the spectral measures $\mathbf{E}$ of the operator $A$ are defined by the corresponding generalized resolvents $\mathbf{R}_\lambda$ which are constructed by the following relation: \begin{equation} \label{f3_58} \mathbf{R}_\lambda = (A_{F(\lambda)} - \lambda E_H)^{-1},\qquad \mathop{\rm Im}\nolimits\lambda > 0, \end{equation} where $F(\lambda) = W^{-1} G(\lambda) W U_{2,4}$, $G(\lambda)\in \mathbf{G}(A,B)$. \noindent Moreover, the correspondence between $\mathbf{G}(A,B)$ and a set of all solutions of the Devinatz moment problem is bijective. \end{thm} {\bf Proof. } The proof follows from the previous considerations. $\Box$ Consider an arbitrary non-negative Borel measure $\mu$ in the strip $\Pi$ which has all finite moments~(\ref{f1_1}). What can be said about the density of power-trigonometric polynomials~(\ref{f1_2}) in the corresponding space $L^2_\mu$? The measure $\mu$ is a solution of the corresponding moment problem~(\ref{f1_1}). Thus, $\mu$ admits a representation~(\ref{f3_4}) where $F$ is the spectral measure of $B$ and $\mathbf{E}$ is a spectral measure of $A$ which commutes with $F$ (the operators $A$ and $B$ in a Hilbert space $H$ are defined as above). Suppose that (power-trigonometric) polynomials are dense in $L^2_\mu$. Repeating arguments from the beginning of the Proof of Theorem~\ref{t3_1} we see that in our case $H_0 = \{ 0 \}$ and $\widetilde A$, $\widetilde B$ are operators in $H$. Moreover, we have $\mu = ((\widetilde E\times \widetilde F) x_{0,0}, x_{0,0})_{H}$, where $\widetilde E$ is the spectral measure of $\widetilde A$, $\widetilde F = F$. Consequently, $\mu$ is a canonical solution of the Devinatz moment problem. \noindent The converse assertion is more complicated and will be studied elsewhere. Sergey M. Zagorodnyuk School of Mathematics and Mekhanics Karazin Kharkiv National University Kharkiv, 61077 Ukraine \begin{center} \bf Devinatz's moment problem: a description of all solutions. \end{center} \begin{center} \bf S.M. Zagorodnyuk \end{center} In this paper we study Devinatz's moment problem: to find a non-negative Borel measure $\mu$ in a strip $\Pi = \{ (x,\varphi):\ x\in \mathbb{R},\ -\pi\leq \varphi < \pi \},$ such that $\int_\Pi x^m e^{in\varphi} d\mu = s_{m,n}$, $m\in \mathbb{Z}_+$, $n\in \mathbb{Z}$, where $\{ s_{m,n} \}_{m\in \mathbb{Z}_+, n\in \mathbb{Z}}$ is a given sequence of complex numbers. We present a new proof of the Devinatz solvability criterion for this moment problem. We obtained a parameterization of all solutions of Devinatz's moment problem. We used an abstract operator approach and results of Godi\v{c}, Lucenko and Shtraus. Key words: moment problem, measure, generalized resolvent. MSC 2000: 44A60, 30E05. } \end{document}

\begin{document} \title{Complete Test Sets And Their Approximations} \author{ \IEEEauthorblockN{Eugene Goldberg} \IEEEauthorblockN{\emph{[email protected]}}} \maketitle \begin{abstract} We use testing to check if a combinational circuit $N$ always evaluates to 0 (written as $N \equiv 0$). We call a set of tests proving $N \equiv 0$ a complete test set (CTS). The conventional point of view is that to prove $N \equiv 0$ one has to generate a \ti{trivial} CTS. It consists of all $2^{|X|}$ input assignments where $X$ is the set of input variables of $N$. We use the notion of a Stable Set of Assignments (SSA) to show that one can build a \ti{non-trivial} CTS consisting of less than $2^{|X|}$ tests. Given an unsatisfiable CNF formula $H(W)$, an SSA of $H$ is a set of assignments to $W$ that proves unsatisfiability of $H$. A trivial SSA is the set of all $2^{|W|}$ assignments to $W$. Importantly, real-life formulas can have non-trivial SSAs that are much smaller than $2^{|W|}$. In general, construction of even non-trivial CTSs is inefficient. We describe a much more efficient approach where tests are extracted from an SSA built for a ``projection'' of $N$ on a subset of variables of $N$. These tests can be viewed as an approximation of a CTS for $N$. We give experimental results and describe potential applications of this approach. \end{abstract} \section{Introduction} Testing is an important part of verification flows. For that reason, any progress in understanding testing and improving its quality is of great importance. In this paper, we consider the following problem. Given a single-output combinational circuit $N$, find a set of input assignments (tests) proving that $N$ evaluates to 0 for every test (written as $N \equiv 0$) or find a counterexample. We will call a set of input assignments proving $N \equiv 0$ a \ti{complete test set} (\ti{CTS})\footnote{\input{f7ootnote}}. We will call the set of all possible tests a \ti{trivial CTS}. Typically, one assumes that proving $N \equiv 0$ involves derivation of the trivial CTS, which is infeasible in practice. Thus, testing is used only for finding an input assignment refuting $N \equiv 0$. We present an approach for building a non-trivial CTS consisting only of a subset of all possible tests. In general, finding even a non-trivial CTS for a large circuit is impractical. We describe a much more efficient approach where an \ti{approximation} of a CTS is generated. The circuit $N$ above usually describes a property $\xi$ of a multi-output combinational circuit $M$, the latter being the \ti{real object of testing}. For instance, $\xi$ may state that $M$ never produces some output assignments. To differentiate CTSs and their approximations from conventional test sets verifying $M$ ``as a whole'', we will refer to the former as \ti{property-checking test sets}. Let $\Xi :=\s{\xi_1,\dots,\xi_k}$ be the set of properties of $M$ formulated by a designer. Assume that every property of $\Xi$ holds and $T_i$ is a test set generated to check property $\xi_i \in \Xi$. There are at least two reasons why applying $T_i$ to $M$ makes sense. First, if $\Xi$ is \ti{incomplete}\footnote{That is $M$ can be incorrect even if all properties of $\Xi$ hold.}, a test of $T_i$ can expose a bug, if any, breaking a property of $M$ that is not in $\Xi$. Second, if property $\xi_i$ is defined \ti{incorrectly}, a test of $T_i$ may expose a bug breaking the correct version of $\xi_i$. On the other hand, if $M$ produces proper output assignments for all tests of $T_1 \cup \dots \cup T_k$, one gets extra guarantee that $M$ is correct. In Section~\ref{sec:appl}, we list some other applications of property-checking test sets such as verification of design changes, hitting corner cases and testing sequential circuits. Let $N(X,Y,z)$ be a single-output combinational circuit where $X$ and $Y$ specify the sets of input and internal variables of $N$ respectively and $z$ specifies the output variable of $N$. Let $F_N(X,Y,z)$ be a formula defining the functionality of $N$ (see Section~\ref{sec:cts}). We will denote the set of variables of circuit $N$ (respectively formula $H$) as \V{N} (respectively \V{H}). Every assignment\footnote{\input{f1ootnote}} to \V{F_N} satisfying $F_N$ corresponds to a consistent assignment\footnote{\input{f2ootnote}} to \V{N} and vice versa. Then the problem of proving $N \equiv 0$ reduces to showing that formula $F_N \wedge z$ is unsatisfiable. From now on, we assume that all formulas mentioned in this paper are \ti{propositional}. Besides, we will assume that every formula is represented in CNF i.e. as a conjunction of disjunctions of literals. Our approach is based on the notion of a Stable Set of Assignments (SSA) introduced in~\cite{ssp}. Given formula $H(W)$, an SSA of $H$ is a set $P$ of assignments to variables of $W$ that have two properties. First, every assignment of $P$ falsifies $H$. Second, $P$ is a transitive closure of some neighborhood relation between assignments (see Section~\ref{sec:ssa}). The fact that $H$ has an SSA means that the former is unsatisfiable. Otherwise, an assignment satisfying $H$ is generated when building its SSA. If $H$ is unsatisfiable, the set of all $2^{|W|}$ assignments is always an SSA of $H$ . We will refer to it as \ti{trivial}. Importantly, a real-life formula $H$ can have a lot of SSAs whose size is much less than $2^{|W|}$. We will refer to them as \ti{non-trivial}. As we show in Section~\ref{sec:ssa}, the fact that $P$ is an SSA of $H$ is a \ti{structural} property of the latter. That is this property cannot be expressed in terms of the truth table of $H$ (as opposed to a \ti{semantic} property of $H$). For that reason, if $P$ is an SSA for $H$, it may not be an SSA for some other formula $H'$ that is logically equivalent to $H$. In other words, a structural property is \ti{formula-specific}. We show that a CTS for $N$ can be easily extracted from an SSA of formula $F_N \wedge z$. This makes a non-trivial CTS a structural property of circuit $N$ that cannot be expressed in terms of its truth table. Building an SSA for a large formula is inefficient. So, we present a procedure constructing a simpler formula $H(V)$ implied by $F_N \wedge z$ $($where $V \subseteq \V{F_N \wedge z})$ and building an SSA of $H$. The existence of such an SSA means that $H$ (and hence $F_N \wedge z$) is unsatisfiable. So, $N \equiv 0$ holds. A test set extracted from an SSA of $H$ can be viewed as a way to verify a ``projection'' of $N$ on variables of $V$. On the other hand, one can consider this set as an approximation of a CTS for $N$. We will refer to the procedure above as \mbox{$\mi{SemStr}$}\xspace (``\ti{Sem}antics and \ti{Str}ucture''). \mbox{$\mi{SemStr}$}\xspace combines semantic and structural derivations, hence the name. The semantic part of \mbox{$\mi{SemStr}$}\xspace is\footnote{\input{f6ootnote}} to derive $H$. Its structural part consists of constructing an SSA of $H$ thus proving that $H$ is unsatisfiable. The contribution of this paper is fourfold. First, we introduce the notion of non-trivial CTSs (Section~\ref{sec:cts}). Second, we present a method for efficient construction of property-checking tests that are approximations of CTSs (Sections~\ref{sec:algor} and~\ref{sec:app_cts}). Third, we describe applications of such tests (Section~\ref{sec:appl}). Fourth, we give experimental results showing the effectiveness of property-checking tests (Section~\ref{sec:exper}). \section{Stable Set Of Assignments} \label{sec:ssa} \subsection{Definitions} We will refer to a disjunction of literals as a \ti{clause}. Let \pnt{p}\, be an assignment to a set of variables $V$. Let \pnt{p}\, falsify a clause $C$. Denote by {\boldmath \nbhd{p}{C}} the set of assignments to $V$ satisfying $C$ that are at Hamming distance 1 from \pnt{p}. (Here \ti{Nbhd} stands for ``Neighborhood''). Thus, the number of assignments in \nbhd{p}{C} is equal to that of literals in $C$. Let \pnt{q}\, be another assignment to $V$ (that may be equal to \pnt{p}). Denote by {\boldmath \Nbh{q}{p}{C}} the subset of \nbhd{p}{C} consisting only of assignments that are farther away from \pnt{q} than \pnt{p} (in terms of the Hamming distance). \begin{example} Let $V=\s{v_1,v_2,v_3,v_4}$ and \pnt{p}=0110. We assume that the values are listed in \pnt{p} in the order the corresponding variables are numbered i.e. \mbox{$v_1=0$}, $v_2=1,v_3=1,v_4=0$. Let $C= v_1 \vee \overline{v_3}$. (Note that \pnt{p} falsifies $C$.) Then \nbhd{p}{C}=\s{\ppnt{p}{1},\ppnt{p}{2}} where \ppnt{p}{1} = 1110 and \ppnt{p}{2}=0100. Let \pnt{q} = 0000. Note that \ppnt{p}{2} is actually closer to \pnt{q} than \pnt{p}. So \Nbh{q}{p}{C}=\s{\ppnt{p}{1}}. \end{example} \begin{definition} \label{def:ac_fun} Let $H$ be a formula\footnote{\input{f3ootnote}} specified by a set of clauses \s{C_1,\dots,C_k}. Let $P$ = \s{\ppnt{p}{1},\dots,\ppnt{p}{m}} be a set of assignments to \V{H} such that every $\ppnt{p}{i} \in P$ falsifies $H$. Let \mbox{$\Phi$}\xspace denote a mapping $P \rightarrow H$ where \ac{\ppnt{p}{i}} is a clause $C$ of $H$ falsified by \ppnt{p}{i}. We will call \mbox{$\Phi$}\xspace an \tb{AC-mapping} where ``AC'' stands for ``Assignment-to-Clause''. We will denote the range of \mbox{$\Phi$}\xspace as \ac{P}. (So, a clause $C$ of $H$ is in \ac{P} iff there is an assignment $\ppnt{p}{i} \in P$ such that $C = \mbox{$\Phi$}\xspace(\ppnt{p}{i})$.) \end{definition} \begin{definition} \label{def:ssa} Let $H$ be a formula specified by a set of clauses \s{C_1,\dots,C_k}. Let $P$ = \s{\ppnt{p}{1},\dots,\ppnt{p}{m}} be a set of assignments to \V{H}. $P$ is called a \tb{Stable Set of Assignments}\footnote{\input{f5ootnote}} (\tb{SSA}) of $H$ with \tb{center} $\sub{p}{init} \in P$ if there is an AC-mapping \mbox{$\Phi$}\xspace such that for every $\ppnt{p}{i}\in P$, $\NNbhd{p}{p}{i}{C} \subseteq P$ holds where $C = \ac{\ppnt{p}{i}}$. \end{definition} \begin{example} \label{exmp:ssa} Let $H$ consist of four clauses: $C_1 = v_1 \vee v_2 \vee v_3$, $C_2 = \overline{v}_1$, $C_3 = \overline{v}_2$, $C_4 = \overline{v}_3$. Let $P =\s{\ppnt{p}{1},\ppnt{p}{2},\ppnt{p}{3},\ppnt{p}{4}}$ where $\ppnt{p}{1} = 000$, $\ppnt{p}{2} = 100$, $\ppnt{p}{3} = 010$, $\ppnt{p}{4}=001$. Let \mbox{$\Phi$}\xspace be an AC-mapping specified as $\ac{\ppnt{p}{i}} = C_i, i = 1,\dots,4$. Since $\ppnt{p}{i}$ falsifies $C_i$, $i=1,\dots,4$,~~\mbox{$\Phi$}\xspace is a correct AC-mapping. $P$ is an SSA of $H$ with respect to \mbox{$\Phi$}\xspace and center \sub{p}{init}=\ppnt{p}{1}. Indeed, \NNbhd{p}{p}{1}{C_1}=\s{\ppnt{p}{2},\ppnt{p}{3},\ppnt{p}{4}} where $C_1 = \ac{\ppnt{p}{1}}$ and \NNbhd{p}{p}{i}{C_i} = $\emptyset$, where $C_i = \ac{\ppnt{p}{i}}$, $i=2,3,4$. Thus, $\mi{Nbhd}(\sub{p}{init},\ppnt{p}{i},\ac{\ppnt{p}{i}}) \subseteq P$, $i=1,\dots,4$. \end{example} \subsection{SSAs and satisfiability of a formula} \label{ssec:ssa_sat} \begin{proposition} \label{prop:ssa} Formula $H$ is unsatisfiable iff it has an SSA. \end{proposition} The proof\footnote{The proof of Proposition~\ref{prop:ssa} presented in report~\cite{cmpl_tst} is inacurate.} is given Appendix~\ref{app:proofs}. A similar proposition is proved in~\cite{ssp} for ``uncentered'' SSAs (see Footnote~\ref{ftn:ssa}). \input{b0uild_path.fig} The set of all assignments to \V{H} forms the \ti{trivial} uncentered SSA of $H$. Example~\ref{exmp:ssa} shows a \ti{non-trivial} SSA. The fact that formula $H$ has a non-trivial SSA $P$ is its \ti{structural} property. That is one cannot check whether $P$ is an SSA of $H$ if only the truth table of $H$ is known. In particular, $P$ may not be an SSA of a formula $H'$ logically equivalent to $H$. \input{b1uild_ssa.fig} The relation between SSAs and satisfiability can be explained as follows. Suppose that formula $H$ is satisfiable. Let \sub{p}{init} be an arbitrary assignment to \V{H} and \pnt{s} be a satisfying assignment that is the closest to \sub{p}{init} in terms of the Hamming distance. Let $P$ be the set of all assignments to \V{H} that falsify $H$ and \mbox{$\Phi$}\xspace be an AC-mapping from $P$ to $H$. Then \pnt{s} can be reached from \sub{p}{init} by procedure \ti{BuildPath} shown in Figure~\ref{fig:bld_path}. It generates a sequence of assignments $\ppnt{p}{1},\dots,\ppnt{p}{i}$ where \ppnt{p}{1} = \sub{p}{init} and \ppnt{p}{i}=\pnt{s}. First, \ti{BuildPath} checks if current assignment \ppnt{p}{i} equals \pnt{s}. If so, then \pnt{s} has been reached. Otherwise, \ti{BuildPath} uses clause $C=\ac{\ppnt{p}{i}}$ to generate next assignment. Since \pnt{s} satisfies $C$, there is a variable $v \in \V{C}$ that is assigned differently in \ppnt{p}{i} and \pnt{s}. \ti{BuildPath} generates a new assignment \ppnt{p}{i+1} obtained from \ppnt{p}{i} by flipping the value of $v$. \ti{BuildPath} reaches \pnt{s} in $k$ steps where $k$ is the Hamming distance between \sub{p}{init} and \pnt{s}. Importantly, \ti{BuildPath} reaches \pnt{s} for \ti{any} AC-mapping. Let $P$ be an SSA of $H$ with respect to center \sub{p}{init} and AC-mapping \mbox{$\Phi$}\xspace. Then if \ti{BuildPath} starts with \sub{p}{init} and uses \mbox{$\Phi$}\xspace as an AC-mapping, it can reach only assignments of $P$. Since every assignment of $P$ falsifies $H$, no satisfying assignment can be reached. A procedure for generation of SSAs called \ti{BuildSSA} is shown in Figure~\ref{fig:bld_ssa}. It accepts formula $H$ and outputs either a satisfying assignment or an SSA of $H$, center \sub{p}{init} and AC-mapping \mbox{$\Phi$}\xspace. \ti{BuildSSA} maintains two sets of assignments denoted as $E$ and $Q$. Set $E$ contains the examined assignments i.e. those whose neighborhood is already explored. Set $Q$ specifies assignments that are queued to be examined. $Q$ is initialized with an assignment \sub{p}{init} and $E$ is originally empty. \ti{BuildSSA} updates $E$ and $Q$ in a \ti{while} loop. First, \ti{BuildSSA} picks an assignment \pnt{p} of $Q$ and checks if it satisfies $H$. If so, \pnt{p} is returned as a satisfying assignment. Otherwise, \ti{BuildSSA} removes \pnt{p}~\,from $Q$ and picks a clause $C$ of $H$ falsified by \pnt{p}. The assignments of $\Nbhd{p}{p}{C}$ that are not in $E$ are added to $Q$. After that, \pnt{p} is added to $E$ as an examined assignment, pair $(\pnt{p},C)$ is added to \mbox{$\Phi$}\xspace and a new iteration begins. If $Q$ is empty, $E$ is an SSA with center \sub{p}{init} and AC-mapping \mbox{$\Phi$}\xspace. \section{Complete Test Sets} \label{sec:cts} \input{m0iter.fig} Let $N(X,Y,z)$ be a single-output combinational circuit where $X$ and $Y$ specify the input and internal variables of $N$ respectively and $z$ specifies the output variable of $N$. Let $N$ consist of gates $G_1,\dots,G_k$. Then $N$ can be represented as $F_N = F_{G_1} \wedge \dots \wedge F_{G_k}$ where $F_{G_i},i=1,\dots,k$ is a CNF formula specifying the consistent assignments of gate $G_i$. Proving $N \equiv 0$ reduces to showing that formula $F_N \wedge z$ is unsatisfiable. \begin{example} \label{exmp:circ} Circuit $N$ shown in Figure~\ref{fig:miter} represents equivalence checking of expressions $(x_1 \vee x_2) \wedge x_3$ and $(x_1 \wedge x_3) \vee (x_2 \wedge x_3)$ specified by gates $G_1,G_2$ and $G_3,G_4,G_5$ respectively. Formula $F_N$ is equal to $F_{G_1} \wedge \dots \wedge F_{G_6}$ where, for instance, $F_{G_1} = C_1 \wedge C_2 \wedge C_3$, $C_1 = x_1 \vee x_2 \vee \overline{y}_1$, $C_2 = \overline{x}_1 \vee y_1$, $C_3 = \overline{x}_2 \vee y_1$. Every satisfying assignment to \V{F_{G_1}} corresponds to a consistent assignment to gate $G_1$ and vice versa. For instance, $(x_1=0,x_2=0,y_1=0)$ satisfies $F_{G_1}$ and is a consistent assignment to $G_1$ since the latter is an OR gate. Formula $F_N \wedge z$ is unsatisfiable due to functional equivalence of expressions $(x_1 \vee x_2) \wedge x_3$ and $(x_1 \wedge x_3) \vee (x_2 \wedge x_3)$. Thus, $N \equiv 0$. \end{example} Let \pnt{x} be a test i.e. an assignment to $X$. The set of assignments to \V{N} sharing the same assignment \pnt{x} to $X$ forms a cube of $2^{|Y|+1}$ assignments. $($Recall that $\V{N} = X \cup Y \cup \s{z}).$ Denote this set as \cube{x}. Only one assignment of \cube{x} specifies the correct execution trace produced by $N$ under \pnt{x}. All other assignments can be viewed as ``erroneous'' traces under test \pnt{x}. \input{s2em_str.fig} \begin{definition} \label{def:cts} Let $T$ be a set of tests \s{\ppnt{x}{1},\dots,\ppnt{x}{k}} where $k \leq 2^{|X|}$. We will say that $T$ is a \tb{Complete Test Set (CTS)} for $N$ if $\Cube{x}{1} \cup \dots \cup \Cube{x}{k}$ contains an SSA for formula $F_N \wedge z$. \end{definition} If $T$ satisfies Definition~\ref{def:cts}, set $\Cube{x}{1} \cup \dots \cup \Cube{x}{k}$ ``contains'' a proof that $N \equiv 0$ and so $T$ can be viewed as complete. If $k = 2^{|X|}$, $T$ is the \ti{trivial} CTS. In this case, $\Cube{x}{1} \cup \dots \cup \Cube{x}{k}$ contains the trivial SSA consisting of all assignments to \V{F_N \wedge z}. Given an SSA $P$ of $F_N \wedge z$, one can easily generate a CTS by extracting all different assignments to $X$ that are present in the assignments of $P$. \begin{example} Formula $F_N \wedge z$ of Example~\ref{exmp:circ} has an~SSA of 21 assignments to \V{F_N\!\wedge\!z}. They have only~5 different assignments to\,\,$X\!=\!\s{x_1,\!x_2,\!x_3}$. The set $\{101,\!100,\!011,\!010,\!000\}$ of\,\,those assignments\,is a CTS for $N$. \end{example} Definition~\ref{def:cts} is meant for circuits that are not ``too redundant''. Highly-redundant circuits are discussed in report \cite{cmpl_tst} and Appendix~\ref{app:red}. \section{\mbox{$\mi{SemStr}$}\xspace Procedure} \label{sec:algor} \subsection{Motivation} Building an SSA for a large formula is inefficient. So, constructing a CTS of $N$ from an SSA of $F_N \wedge z$ is impractical. To address this problem, we introduce a procedure called \mbox{$\mi{SemStr}$}\xspace (a short for ``Semantics and Structure''). Given formula $F_N \wedge z$ and a set of variables $V \subseteq \V{F_N \wedge z}$, \mbox{$\mi{SemStr}$}\xspace generates a simpler formula $H(V)$ implied by $F_N \wedge z$ at the same time trying to build an SSA for $H$. If \mbox{$\mi{SemStr}$}\xspace succeeds in constructing such an SSA, formula $H$ is unsatisfiable and so is $F_N \wedge z$. Then a set of tests $T$ is extracted from this SSA. As we show in Subsection~\ref{ssec:approx}, one can view $T$ as an approximation of a CTS for $N$ (if $X \subseteq V$) or an ``approximation of approximation'' of a CTS (if $X \not\subseteq V$). \begin{example} Consider the circuit $N$ of Figure~\ref{fig:miter} where $X=\s{x_1,x_2,x_3}$. Assume that $V = X$. Application of \mbox{$\mi{SemStr}$}\xspace to $F_N \wedge z$ produces $H(X)= (\overline{x}_1 \vee \overline{x}_3) \wedge (\overline{x}_2 \vee \overline{x}_3) \wedge (x_1 \vee x_2) \wedge x_3$. \mbox{$\mi{SemStr}$}\xspace also generates an SSA of $H$ of four assignments to $X$: \s{000,001,011,101} with center \sub{p}{init}=000. (We omit the AC-mapping here.) These assignments form an approximation of CTS for $N$. \end{example} \subsection{Description of \mbox{$\mi{SemStr}$}\xspace} The pseudocode of \mbox{$\mi{SemStr}$}\xspace is shown in Figure~\ref{fig:sem_str}. \mbox{$\mi{SemStr}$}\xspace accepts formula $G$ (in our case, $G := F_N \wedge z$) and a set of variables $V \subseteq \V{G}$. \mbox{$\mi{SemStr}$}\xspace outputs an assignment satisfying $G$ or formula $H(V)$ implied by $G$ and an SSA of $H$. Originally, the set of clauses $H$ is empty. $H$ is computed in a \ti{while} loop. First, \mbox{$\mi{SemStr}$}\xspace tries to build an SSA for the current formula $H$ by calling \ti{BuildSSA} (line 3). If $H$ is unsatisfiable, \ti{BuildSSA} computes an SSA $P$ returned by \mbox{$\mi{SemStr}$}\xspace (line 5). Otherwise, \ti{BuildSSA} returns an assignment \pnt{v} satisfying $H$. In this case, \mbox{$\mi{SemStr}$}\xspace calls procedure \ti{GenCls} to build a clause $C$ falsified by \pnt{v}. Clause $C$ is obtained by resolving clauses of $G$ on variables of $W$. (Hence $C$ is implied by $G$.) If \pnt{v} can be extended to an assignment \pnt{s} satisfying $G$, \mbox{$\mi{SemStr}$}\xspace terminates (lines 7-8). Otherwise, $C$ is added to $H$ and a new iteration begins. \input{g3en_clause.fig} Procedure \ti{GenCls} is shown in Figure~\ref{fig:gen_cls}. First, \ti{GenCls} generates formula \cof{G}{v} obtained from $G$ by discarding clauses satisfied by \pnt{v} and removing literals falsified by \pnt{v}. Then \ti{GenCls} checks if there is an assignment \pnt{s} satisfying \cof{G}{v}. If so, $\pnt{s} \cup \pnt{v}$ is returned as an assignment satisfying $G$. Otherwise, a proof $R$ of unsatisfiability of \cof{G}{v} is produced. Then \ti{GenCls} forms a set $V' \subseteq V$. A variable $w$ is in $V'$ iff a clause of \cof{G}{v} is used in proof $R$ and its parent clause from $G$ has a literal of $w$ falsified by \pnt{v}. Finally, clause $C$ is generated as a disjunction of literals of $V'$ falsified by \pnt{v}. By construction, clause $C$ is implied by $G$ and falsified by \pnt{v}. \section{Building Approximations Of CTS} \label{sec:app_cts} \subsection{Two kinds of approximations of CTSs} \label{ssec:approx} \input{g4en_tests.fig} As before, let $H(V)$ denote a formula implied by $F_N \wedge z$ that is generated by \mbox{$\mi{SemStr}$}\xspace and $P$ denote an SSA for $H$. Projections of $N$ can be of two kinds depending on whether $X \subseteq V$ holds. Let $X \subseteq V$ hold and $T$ be the test set extracted from $P$ as described in Section~\ref{sec:cts}. That is $T$ consists of all different assignments to $X$ present in the assignments of $P$. On one hand, using the reasoning of Section~\ref{sec:cts} one can show that $T$ is a CTS for projection of $N$ on $V$. On the other hand, since $H(V)$ is essentially an abstraction of $F_N \wedge z$, set $T$ is an approximation of a CTS for $N$. For that reason, we will refer to $T$ as a \tb{CTS\textsuperscript{a}} of $N$ where superscript ``a'' stands for ``approximation''. Now assume that $X \not\subseteq V$ holds. Generation of a test set $T$ from $P$ for this case is described in the next section. The set $T$ can be viewed as an approximation of a set $T'$ built for projection of $N$ on set $V \cup X$. Since $T'$ is a CTS\textsuperscript{a}\xspace for $N$, we will refer to $T$ as \tb{CTS\textsuperscript{aa}} where ``aa'' stands for ``approximation of approximation''. \subsection{Construction of CTS\textsuperscript{aa}\xspace} Consider extraction of a test set $T$ from SSA $P$ of formula $H(V)$ when $X \not\subseteq V$. Since $V$, in general, contains internal variables\footnote{If the special case $V \subset X$ holds, every assignment of $P$ can be easily turned into a test by assigning values to variables of $X \setminus V$ (e.g. randomly).} of $N$, translation of $P$ to a test set $T$ needs a special procedure \ti{GenTests} shown in Figure~\ref{fig:gen_tests}. For every assignment \pnt{v} of $P$, \ti{GenTests} checks if formula $F_N$ is satisfiable under assignment \pnt{v} (i.e. if there exists a test under which $N$ assigns \pnt{v} to $V$). If so, an assignment \pnt{x} to $X$ is extracted from the satisfying assignment and added to $T$ as a test. Otherwise, \ti{GenTests} runs a \ti{for} loop (lines 8-13) of $\mi{Tries}$ iterations. In every iteration, \ti{GenTests} relaxes $F_N$ by removing the clauses specifying a small subset of gates picked randomly. If the relaxed version of $F_N$ is satisfiable, a test is extracted from the satisfying assignment and added to $T$. \subsection{Finding a set of variables to project on} \label{ssec:int_cut} \input{i3nt_cut.fig} Intuitively, a good choice of the set $V$ to project $N$ on is a (small) coherent subset of variables of $N$ reflecting its structure and/or semantics. One obvious choice of $V$ is the set $X$ of input variables of $N$. In this section, we describe generation of a set $V$ whose variables form an internal cut of $N$ denoted as \ti{Cut}. Procedure \ti{GenCut} for generation of set \ti{Cut} consisting of \ti{Size} gates is shown in Figure~\ref{fig:int_cut}. Set $V$ is formed from output variables of the cut gates. The current cut is specified by $\mi{Gts} \cup \mi{Inps}$. Set \ti{Gts} is initialized with the output gate \Sub{G}{out} of circuit $N$ and \ti{Inps} is originally empty. \ti{GenCut} computes the \ti{depth} of every gate of \ti{Gts}. The depth of \Sub{G}{out} is set to 0. Set \ti{Gts} is processed in a \ti{while} loop (lines 5-15). In every iteration, a gate of the smallest depth is picked from \ti{Gts}. Then \ti{GenCut} removes gate $G$ from \ti{Gts} and examines the fan-in gates of $G$ (lines 9-15). Let $G'$ be a fan-in gate of $G$ that has not been seen yet and is not a primary input of $N$. Then the depth of $G'$ is set to that of $G$ plus 1 and $G'$ is added to \ti{Gts}. If $G'$ is a primary input of $N$ it is added to \ti{Inps}. \section{Applications Of Property-Checking Tests} \label{sec:appl} Given a multi-output circuit $M$, traditional testing is used to verify $M$ ``as a whole''. In this paper, we describe generation of a test set meant for checking a \ti{particular property} of $M$ specified by a single-output circuit $N$. In this section, we present some applications of property-checking test sets. \subsection{Testing properties specified by similar circuits} \label{ssec:des_changes} Let $N$ be a single-output circuit and $T$ be a test set generated when proving $N\equiv0$. Let $N^*$ be a circuit that is similar to $N$. (For instance, $N$ can specify a property of a circuit $M$ whereas $N^*$ specifies the same property after a modification of $M$.) Then one can use $T$ to verify if $N^* \equiv 0$. Since $T$ is generated for a similar circuit $N$, there is a good chance that it contains a counterexample to $N^* \equiv 0$, if any. (Of course, the fact that $N^*$ evaluates to 0 for all tests of $T$ does not mean that $N^* \equiv 0$ even if $T$ is a CTS for $N$). In Subsection~\ref{ssec:bug}, we give experimental evidence supporting the observation above. Assuming that $N \equiv 0$ was proved formally, checking if $N^* \equiv 0$ holds can be verified formally too. So applying tests of $T$ to $N^*$ can be viewed as a ``light'' verification procedure for exposing bugs. On the other hand, one can re-use test $T$ in situations where the necessity to apply a formal tool is overlooked or formal methods are not powerful enough. Let $N$ specify a property $\xi$ of a \ti{component} of a design $D$. Suppose that this component is modified under assumption that preserving $\xi$ is not necessary any more. By applying $T$ to $D$ one can invoke behaviors that break $\xi$ and expose a bug in $D$, if any, caused by ignoring $\xi$. If $D$ is a large design, finding such a bug by formal verification may not be possible. \subsection{Verification of corner cases} \label{ssec:corners} \input{s3ubcirc.fig} Let $K$ be a single-output subcircuit of circuit $M$ as shown in Figure~\ref{fig:subcirc}. For the sake of simplicity we consider here the case where the set $X_K$ of input variables of $K$ is a subset of the set $X$ of input variables of $M$. (The technique below can also be applied when input variables of $K$ are \ti{internal} variables of $M$.) Suppose $K$ evaluates, say, to value 0 much more frequently then to 1. Then one can view an input assignment of $M$ for which $K$ evaluates to 1 as specifying a ``corner case'' i.e. a rare event. Hitting such a corner case by a random test can be very hard. This issue can be addressed by using a coverage metric that \ti{requires} setting the value of $K$ to both 0 and 1. (The task of finding a test for which $K$ evaluates to 1 can be solved, for instance, by a SAT-solver.) The problem however is that hitting a corner case only once may be insufficient. One can increase the frequency of hitting the corner case above as follows. Let $N$ be a miter of circuits $K'$ and $K''$ (see Figure~\ref{fig:gen_miter}) i.e. a circuit that evaluates to 1 iff $K'$ and $K''$ are functionally inequivalent. Let $K'$ and $K''$ be two copies of circuit $K$. So $N \equiv 0$ holds. Let test set $T_K$ be extracted from an SSA built for a projection of $N$ on a set $V \subseteq \V{N}$. Set $T_K$ can be viewed as a result of ``squeezing'' the truth table of $K$. Since this truth table is dominated by input assignments for which $K$ evaluates to 0, this part of the truth table is \ti{reduced the most}. So, one can expect that the ratio of tests of $T_K$ for which $K$ evaluates to 1 is higher than in the truth table of $K$. In Subsection~\ref{ssec:ecorners}, we substantiate this intuition experimentally. One can easily extend an assignment \ppnt{x}{K} of $T_K$ to an assignment \pnt{x} to $X$ e.g. by randomly assigning values to the variables of $X \setminus X_K$. \subsection{Dealing with incomplete specifications} \label{ssec:incomp_spec} One can use property-checking tests to mitigate the problem of incomplete specifications. By running tests generated for an incomplete set of properties of $M$, one can expose bugs overlooked due to missing some properties. An important special case of this problem is as follows. Let $\xi$ be a property of $M$ that holds. Assume that the correctness of $M$ requires proving a slightly \ti{different} property $\xi'$ that is not true. By running a test set $T$ built for property $\xi$, one may expose a bug overlooked in formal verification due to proving $\xi$ instead of $\xi'$. In Subsection~\ref{ssec:missed_props}, we illustrate the idea above experimentally. \subsection{Testing sequential circuits} \label{ssec:seq_circ} There are a few ways to apply property-checking tests meant for combinational circuits to verification of \ti{sequential} circuits. Here is one of them based on bounded model checking~\cite{bmc}. Let $M$ be a sequential circuit and $\xi$ be a property of $M$. Let $N(X,Y,z)$ be a circuit such that $N \equiv 0$ holds iff $\xi$ is true for $k$ time frames. Circuit $N$ is obtained by unrolling $M$ $k$ times and adding logic specifying property $\xi$. Set $X$ consists of the subset $X'$ specifying the state variables of $M$ in the first time frame and subset $X''$ specifying the combinational input variables of $M$ in $k$ time frames. \input{g1en_miter.fig} Having constructed $N$, one can build CTSs, CTS\textsuperscript{a}\xspace{s} and CTS\textsuperscript{aa}\xspace{s} for testing property $\xi$ of $M$. The only difference here from the problem we have considered so far is as follows. Circuit $M$ starts in a state satisfying some formula $I(X')$ that specifies the initial states. So, one needs to check if $N \equiv 0$ holds only for the assignments to $X$ satisfying $I(X')$. A test here is an assignment $(\ppnt{x'}{1},\ppnt{x''}{1},\dots,\ppnt{x''}{k})$ where \ppnt{x'}{1} is an initial state and \ppnt{x''}{i}, $1 \leq i \leq k$ is an assignment to the combinational input variables of $i$-th time frame. Given a test, one can easily compute the corresponding sequence of states $(\ppnt{x'}{1},\dots,\ppnt{x'}{k})$ of $M$. In Subsection~\ref{ssec:missed_props}, we give an example of building an CTS\textsuperscript{aa}\xspace for a sequential circuit. \section{Experiments} \label{sec:exper} \input{g5en_pc_tests} In this section, we describe experiments with property-checking tests (PCT) generated by procedure \ti{GenPCT} shown in Figure~\ref{fig:gen_pct}. \ti{GenPCT} accepts a single-output circuit $N$ and outputs a set of tests $T$. (For the sake of simplicity, we assume here that $N \equiv 0$ holds.) \ti{GenPCT} starts with generating formula $F_N \wedge z$ and a set of variables $V \subseteq \V{F_N \wedge z}$. Then it calls \mbox{$\mi{SemStr}$}\xspace (see Fig.~\ref{fig:sem_str}) to compute an SSA $P$ of formula $H(V)$ describing a projection of circuit $N$ on $V$\!. If $H(V)$ does not depend on a variable $w \in V$, all assignments of $P$ have the same value of $w$. Procedure \ti{Diversify} randomizes the value of $w$ in the assignments of $P$. Finally, \ti{BldTests} uses $P$ to extract a test set for circuit $N$. If $X \subseteq V$ holds (where $X$ is the set of input variables of $N$), \ti{BldTests} outputs all the different assignments to $X$ present in assignments of $P$. Otherwise, \ti{BldTests} calls procedure \ti{GenTests} (see Fig.~\ref{fig:gen_tests}). If $V = \V{F_N \wedge z}$, then $H(V)$ is $F_N \wedge z$ itself and \ti{GenPCT} produces a CTS of $N$. Otherwise, according to definitions of Subsection~\ref{ssec:approx}, \ti{GenPCT} generates a CTS\textsuperscript{a}\xspace (if $X \subseteq V$) or CTS\textsuperscript{aa}\xspace (if $X \not\subseteq V$). In the following subsections, we describe results of four experiments. In the first three experiments we used circuits specifying next state functions of latches of HWMCC-10 benchmarks. (The motivation was to use realistic circuits.) In our implementation of \mbox{$\mi{SemStr}$}\xspace, as a SAT-solver, we used Minisat 2.0~\cite{minisat,minisat2.0}. We also employed Minisat to run simulation. To compute the output value of $N$ under test \pnt{x}, we added unit clauses specifying \pnt{x} to formula $F_N \wedge z$ and checked its satisfiability. \subsection{Comparing CTSs, CTS\textsuperscript{a}\xspace{s} and CTS\textsuperscript{aa}\xspace{s}} \label{ssec:cts} \input{c5ts.tbl} The objective of the first experiment was to give examples of circuits with non-trivial CTSs and compare the efficiency of computing CTSs, CTS\textsuperscript{a}\xspace{s} and CTS\textsuperscript{aa}\xspace{s}. In this experiment, $N$ was a miter specifying equivalence checking of circuits $M'$ and $M''$ (see Figure~\ref{fig:gen_miter}). $M''$ was obtained from $M'$ by optimizing the latter with ABC~\cite{abc}. The results of the first experiment are shown in Table~\ref{tbl:cts}. The first two columns specify an HWMCC-10 benchmark and its latch whose next state function was used as $M'$. The next two columns give the number of input variables and that of gates in the miter $N$. The following pair of columns describe computing a CTS for $N$. The first column of this pair gives the size of the SSA $P$ found by \ti{GenPCT} in thousands. The number of tests in the set $T$ extracted from $P$ is shown in the parentheses in thousands. The second column of this pair gives the run time of \ti{GenPCT} in seconds. The last four columns of Table~\ref{tbl:cts} describe results of computing test sets for a projection of $N$ on a set of variables $V$. The first column of this group shows if CTS\textsuperscript{a}\xspace or CTS\textsuperscript{aa}\xspace was computed whereas the next column gives the size of $V$. The third column of this group provides the size of SSA $P$ and the test set $T$ extracted from $P$ (in parentheses). Both sizes are given in thousands. The last column shows the run time of \ti{GenPCT}. For the first five examples, we used a projection of $N$ on $X$, thus constructing a CTS\textsuperscript{a}\xspace of $N$. For the last four examples we computed a projection of $N$ on an internal cut (see Subsection~\ref{ssec:int_cut}) thus generating a CTS\textsuperscript{aa}\xspace of $N$. \ti{GenPCT}\xspace was called with parameter $\mi{Tries}$ set to 5 (see Fig.~\ref{fig:gen_tests} and~\ref{fig:gen_pct}). For the first three examples, \ti{GenPCT}\xspace managed to build non-trivial CTSs that are smaller than $2^{|X|}$. For instance, the trivial CTS for example \ti{bob3} consists of $2^{14}$=16,384 tests, whereas \ti{GenPCT}\xspace found a CTS of 2,004 tests. (So, to prove $M'$ and $M''$ equivalent it suffices to run 2,004 out of 16,384 tests.) For the other examples, \ti{GenPCT}\xspace failed to build a non-trivial CTS due to exceeding the memory limit (1.5 Gbytes). On the other hand, \ti{GenPCT}\xspace built a CTS\textsuperscript{a}\xspace or CTS\textsuperscript{aa}\xspace for all nine examples of Table~\ref{tbl:cts}. Note, however, that CTS\textsuperscript{a}\xspace{s} give only a moderate improvement over CTSs. For the last four examples \ti{GenPCT}\xspace failed to compute an CTS\textsuperscript{a}\xspace of $N$ due to memory overflow whereas it had no problem computing an CTS\textsuperscript{aa}\xspace of $N$. So CTS\textsuperscript{aa}\xspace{s} can be computed efficiently even for large circuits. Further, we show that CTS\textsuperscript{aa}\xspace{s} are also very effective. \input{e7xper_bug_hunting} \input{e8xper_corner_cases} \input{e9xper_missed_props} \section{Background} As we mentioned earlier, traditional testing checks if a circuit $M$ is correct as a whole. This notion of correctness means satisfying a conjunction of \ti{many} properties of $M$. For this reason, one tries to spray tests uniformly in the space of all input assignments. To improve the effectiveness of testing, one can try to run many tests at once as it is done in symbolic simulation~\cite{SymbolSim}. To avoid generation of tests that for some reason should be or can be excluded, a set of constraints can be used~\cite{cnst_rand}. Another method of making testing more reliable is to generate tests exciting a particular set of events specified by a coverage metric~\cite{cov_metr}. Our approach is different from those above in that it is aimed at testing a particular property of $M$. The method of testing introduced in~\cite{bridging} is based on the idea that tests should be treated as a ``proof encoding'' rather than a sample of the search space. (The relation between tests and proofs have been also studied in software verification, e.g. in~\cite{UnitTests,godefroid,Beckman}). In this paper, we take a different point of view where testing becomes a \ti{part} of a formal proof namely the part that performs structural derivations. Reasoning about SAT in terms of random walks was pioneered in~\cite{rand_walk}. The centered SSAs we introduce in this paper bear some similarity to sets of assignments generated in de-randomization of Sch\"oning's algorithm~\cite{balls}. Typically, centered SSAs are much smaller than uncentered SSAs of~\cite{ssp}. The first version of \mbox{$\mi{SemStr}$}\xspace procedure is presented in report~\cite{cmpl_tst}. It has a much tighter integration between the structural part (computation of SSAs) and semantic part (derivation of formula $H$ implied by the original formula). The advantage of the new version of \mbox{$\mi{SemStr}$}\xspace described in this paper is twofold. First, it is much simpler than \mbox{$\mi{SemStr}$}\xspace of~\cite{cmpl_tst}. In particular, any resolution based SAT-solver that generates proofs can be used to implement the new \mbox{$\mi{SemStr}$}\xspace. Second, the simplicity of the new version makes it much easier to achieve the level of scalability where \mbox{$\mi{SemStr}$}\xspace becomes practical. \section{Conclusion} We consider the problem of finding a Complete Test Set (CTS) for a combinational circuit $N$ that is a test set proving $N \equiv 0$. We use the machinery of stable sets of assignments to derive non-trivial CTSs i.e. those that do not include all possible input assignments. Computing a CTS for a large circuit $N$ is inefficient. So, we present a procedure that generates a test set for a ``projection'' of $N$ on a subset $V$ of variables of $N$. Depending on the choice of $V$\!, this procedure generates a test set CTS\textsuperscript{a}\xspace that is an approximation of an CTS or a test set CTS\textsuperscript{aa}\xspace that is an approximation of CTS\textsuperscript{a}\xspace. We give experimental results showing that CTS\textsuperscript{aa}\xspace{s} can be efficiently computed even for large circuits and are effective in solving verification problems. \appendices \section{Proofs} \label{app:proofs} \setcounter{proposition}{0} \begin{proposition} Formula $H$ is unsatisfiable iff it has an SSA. \end{proposition} \begin{proof} \tb{If part.} Assume the contrary i.e. $P$ is an SSA of $H$ with center \sub{p}{init} and AC-mapping \mbox{$\Phi$}\xspace and $H$ is satisfiable. Let \pnt{s}~\,be an assignment satisfying $H$ that is the closest to \sub{p}{init} in terms of the Hamming distance. Then procedure \ti{BuildPath} (see Fig.~\ref{fig:bld_path}) can build a sequence of assignments $\ppnt{p}{1},\dots,\ppnt{p}{i}$ such that \begin{itemize} \item $i = \mi{Hamming\_distance}(\sub{p}{init},\pnt{s})+1$ \item $\ppnt{p}{1} = \sub{p}{init}$ and $\ppnt{p}{i} = \pnt{s}$ \end{itemize} By definition of \ti{BuildPath}, assignment \ppnt{p}{j+1} is closer to \pnt{s} and farther away from \sub{p}{init} than \ppnt{p}{j} where $1 \leq j \leq i-1$. This means that \ppnt{p}{j+1} is in $\mi{Nbhd}(\sub{p}{init},\ppnt{p}{j},C)$ where $C = \mbox{$\Phi$}\xspace(\ppnt{p}{j})$. In particular, \pnt{s} is in $\mi{Nbhd}(\sub{p}{init},\ppnt{p}{i-1},C)$ and so \pnt{s} is in $P$. However, by definition of an SSA, $P$ consists only of assignments falsifying $H$. Thus, we have a contradiction. \noindent\tb{Only if part}. Assume that formula $H$ is unsatisfiable. By applying \ti{BuildSSA} (see Fig.~\ref{fig:bld_ssa}) to $H$, one generates a set $P$ that is an SSA of $H$ with respect to some center \sub{p}{init} and AC-mapping \mbox{$\Phi$}\xspace. \end{proof} \section{CTSs And Circuit Redundancy} \label{app:red} Let $N \equiv 0$ hold. Let $R$ be a cut of circuit $N$. We will denote the circuit between this cut and the output of $N$ as $N_R$ (see Figure~\ref{fig:cut}). We will say that $N$ is \tb{non-redundant} if $N_R \not\equiv 0$ for any cut $R$ other than the cut specified by primary inputs of $N$. Note that if $N_{R} \not\equiv 0$ for some cut $R$, then $N_{R'} \not\equiv 0$ for \ti{every} cut $R'$ located above $R$. Definition~\ref{def:cts} of a CTS may not work well if $N$ is highly redundant. Assume, for instance, that $N_R \equiv 0$ holds for a cut $R$. This means that the clauses specifying gates of $N$ below $R$ (i.e. those that are not in $N_R$) are redundant in $F_N \wedge z$. Then one can build an SSA $P$ for $F_N \wedge z$ as follows. Let $P_R$ be an SSA for $F_{N_R} \wedge z$. Let \pnt{v} be an arbitrary assignment to the variables of $\V{N} \setminus \V{N_R}$. Then by adding \pnt{v} to every assignment of $P_R$ one obtains an SSA for $F_N \wedge z$. This means that for any test \pnt{x}, \cube{x} contains an SSA of $F_N \wedge z$. Therefore, according to Definition~\ref{def:cts}, circuit $N$ has a CTS consisting of just one test. \input{c2ut.fig} The problem above can be solved using the following observation. Let $T$ be a set of tests \s{\ppnt{x}{1},\dots,\ppnt{x}{k}} for $N$ where $k \leq 2^{|X|}$. Denote by $\vec{r}_i$ the assignment to the variables of cut $R$ produced by $N$ under input \ppnt{x}{i}. Let $T_R$ denote \s{\ppnt{r}{1},\dots, \ppnt{r}{k}}. Denote by $T^*_R$ the set of assignments to variables of $R$ that cannot be produced in $N$ by any input assignment. Now assume that $T$ is constructed so that $T_R \cup T^*_R$ is a CTS for circuit $N_R$. This does not change anything if $N_R$ is itself redundant (i.e. if $N_{R'} \equiv 0$ for some cut $R'$ that is closer to the output of $N$ than $R$). In this case, it is still sufficient to use $T$ of one test because $N_R$ has a CTS of one assignment (in terms of cut $R$). Assume however, that $N_R$ is non-redundant. In this case, there is no ``degenerate'' CTS for $N_R$ and $T$ has to contain at least $|T_R|$ tests. Assuming that $T^*_R$ alone is far from being a CTS for $N_R$, a CTS $T$ for $N$ will consist of many tests. So, one can modify the definition of CTS for a redundant circuit $N$ as follows. A test set $T$ is a CTS for $N$ if there is a cut $R$ such that \begin{itemize} \item circuit $N_R$ is non-redundant i.e. \begin{itemize} \item[$\bullet$] $N_R \equiv0$ holds \item[$\bullet$] $N_R' \not\equiv 0$ for every cut $R'$ above $R$ \end{itemize} \item set $T_R \cup T^*_R$ is a CTS for $N_R$. \end{itemize} just something to fill out the page just something to fill out the page \end{document}

\begin{document} \begin{center} {\huge{\bf On the conjugacy class of the Fibonacci \\[.3cm]dynamical system}} {\large{Michel Dekking (Delft University of Technology)\\ and\\ Mike Keane (Delft University of Technology and University of Leiden)}} {\large{{Version: August 16, 2016}}} \end{center} \section{Introduction}\label{sec:intro} We study the Fibonacci substitution $\varphi$ given by $$\varphi:\quad 0\rightarrow\,01,\;1\rightarrow 0.$$ The infinite Fibonacci word $w_{\rm F}$ is the unique one-sided sequence (to the right) which is a fixed point of $\varphi$: $$w_{\rm F}=0100101001\dots.$$ We also consider one of the two two-sided fixed points $x_{\rm F}$ of $\varphi^2$: $$x_{\rm F}=\dots01001001\!\cdot\!0100101001\dots.$$ The dynamical system generated by taking the orbit closure of $x_{\rm F}$ under the shift map $\sigma$ is denoted by $(X_\varphi,\sigma)$. The question we will be concerned with is: what are the substitutions $\eta$ which generate a symbolical dynamical system topologically isomorphic to the Fibonacci dynamical system? Here topologically isomorphic means that there exists a homeomorphism $\psi: X_\varphi\rightarrow X_\eta$, such that $\psi\sigma=\sigma\psi$, where we denote the shift on $X_\eta$ also by $\sigma$. In this case $(X_\eta, \sigma)$ is said to be conjugate to $(X_\varphi,\sigma)$. This question has been completely answered for the case of constant length substitutions in the paper \cite{CDK}. It is remarkable that there are only finitely many injective primitive substitutions of length $L$ which generate a system conjugate to a given substitution of length $L$. Here a substitution $\alpha$ is called \emph{injective} if $\alpha(a)\ne \alpha(b)$ for all letters $a$ and $b$ from the alphabet with $a\ne b$. When we extend to the class of all substitutions, replacing $L$ by the Perron-Frobenius eigenvalue of the incidence matrix of the substitution, then the conjugacy class can be infinite in general. See \cite{Dekking-TCS} for the case of the Thue-Morse substitution. In the present paper we will prove that there are infinitely many injective primitive substitutions with Perron-Frobenius eigenvalue $\Phi=(1+\sqrt{5})/2$ which generate a system conjugate to the Fibonacci system---see Theorem~\ref{th:inf}. In the non-constant length case some new phenomena appear. If one has an injective substitution $\alpha$ of constant length $L$, then all its powers $\alpha^n$ will also be injective. This is no longer true in the general case. For example, consider the injective substitution $\zeta$ on the alphabet $\{1,2,3,4,5\}$ given by $$\zeta: \qquad 1\rightarrow 12,\; 2\rightarrow 3,\; 3\rightarrow 45,\; 4\rightarrow 1,\; 5\rightarrow 23.$$ An application of Theorem~\ref{th:Nblock} followed by a partition reshaping (see Section~\ref{sec:reshaping}) shows that the system $(X_\zeta,\sigma)$ is conjugate to the Fibonacci system. However, the square of $\zeta$ is given by $$\zeta^2: \qquad 1\rightarrow 123,\; 2\rightarrow 45,\; 3\rightarrow 123,\; 4\rightarrow 12,\; 5\rightarrow 345, $$ which is \emph{not} injective. To deal with this undesirable phenomenon we introduce the following notion. A substitution $\alpha$ is called a \emph{full rank} substitution if its incidence matrix has full rank (non-zero determinant). This is a strengthening of injectivity, because obviously a substitution which is not injective can not have full rank. Moreover, if the substitution $\alpha$ has full rank, then all its powers $\alpha^n$ will also have full rank, and thus will be injective. Another phenomenon, which does not exist in the constant length case, is that non-primitive substitutions $\zeta$ may generate uniquely defined minimal systems conjugate to a given system. For example, consider the injective substitution $\zeta$ on the alphabet $\{1,2,3,4\}$ given by $$\zeta:\qquad 1\rightarrow 12,\quad 2\rightarrow 31,\quad 3\rightarrow 4,\quad 4\rightarrow 3. $$ With the partition reshaping technique from Section~\ref{sec:reshaping} one can show that the system $(X_\zeta,\sigma)$ is conjugate to the Fibonacci system (ignoring the system on two points generated by $\zeta$). In the remainder of this paper we concentrate on primitive substitutions. The structure of the paper is as follows. In Section~\ref{sec:Nblock} we show that all systems in the conjugacy class of the Fibonacci substitution can be obtained by letter-to-letter projections of the systems generated by so-called $N$-block substitutions. In Section~\ref{sec:C3} we give a very general characterization of symbolical dynamical systems in the Fibonacci conjugacy class, in the spirit of a similar result on the Toeplitz dynamical system in \cite{CKL08}. In Section~\ref{sec:reshaping} we introduce a tool which admits to turn non-injective substitutions into injective substitutions. This is used in Section~\ref{sec:C1} to show that the Fibonacci class has infinitely many primitive injective substitutions as members. In Section~\ref{sec:two} we quickly analyse the case of a 2-symbol alphabet. Sections \ref{sec:equi} and \ref{sec:mat} give properties of maximal equicontinuous factors and incidence matrices, which are used to analyse the 3-symbol case in Section \ref{sec:C2}. In the final Section \ref{sec:L2L} we show that the system obtained by doubling the 0's in the infinite Fibonacci word is conjugate to the Fibonacci dynamical system, but can not be generated by a substitution. \section{$N$-block systems and $N$-block substitutions}\label{sec:Nblock} For any $N$ the $N$-block substitution $\hat{\theta}_N$ of a substitution $\theta$ is defined on an alphabet of $p_\theta(N)$ symbols, where $p_\theta(\cdot)$ is the complexity function of the language ${\cal L}_\theta$ of $\theta$ (cf.\ \cite[p.~95]{Queff}). What is \emph{not} in \cite{Queff}, is that this $N$-block substitution generates the $N$-block presentation of the system $(X_\theta,\sigma)$. We denote the letters of the alphabet of the $N$-block presentation by $[a_1a_2\dots a_N]$, where $a_1a_2\dots a_N$ is an element from ${\cal L}_\theta^N$, the set of words of length $N$ in the language of $\theta$. The $N$-block presentation $(X^{[N]}_\theta,\sigma)$ emerges by applying an sliding block code $\Psi$ to the sequences of $X_\theta$, so $\Psi$ is the map \\[-.3cm] $$\Psi(a_1a_2\dots a_N)=[a_1a_2\dots a_N].$$ We denote by $\psi$ the induced map from $X_\theta$ to $X^{[N]}_\theta$: $$\psi(x)=\dots\Psi(x_{-N},\dots,x_{-1})\Psi(x_{-N+1},\dots,x_{0})\dots.$$ It is easy to see that $\psi$ is a conjugacy, where the inverse is $\pi_0$ induced by the 1-block map (also denoted $\pi_0$) given by $\pi_0([a_1a_2\dots a_N])=a_1$. The $N$-block substitution $\hat{\theta}_N$ is defined by requiring that for each word $a_1a_2\dots a_N$ the length of $\hat{\theta}_N([a_1a_2\dots a_N])$ is equal to the length $L_1$ of $\theta(a_1)$, and the letters of $\hat{\theta}_N([a_1a_2\dots a_N])$ are the $\Psi$-codings of the first $L_1$ consecutive $N$-blocks in $\theta(a_1a_2\dots a_N)$. \begin{theorem}\label{th:Nblock} Let $\hat{\theta}_N$ be the $N$-block substitution of a primitive substitution $\theta$. Let $(X^{[N]}_\theta,\sigma)$ be the $N$-block presentation of the system $(X_\theta,\sigma)$. Then the set $X^{[N]}_\theta$ equals $X_{\hat{\theta}_N}$. \end{theorem} {\em Proof:} Let $x$ be a fixed point of $\theta$, and let $y=\psi(x)$, where $\psi$ is the $N$-block conjugacy, with inverse $\pi_0$. The key equation is $\pi_0\,\hat{\theta}_N=\theta\,\pi_0$. This implies\\[-.3cm] $$\pi_0\,\hat{\theta}_N(y)=\theta\,\pi_0(y)=\theta\,\pi_0(\psi(x))=\theta(x)=x.$$ Applying $\psi$ on both sides gives $\hat{\theta}_N(y)=\psi(x)=y$, i.e., $y$ is a fixed point of $\hat{\theta}_N$. But then $X^{[N]}_\theta=X_{\hat{\theta}_N}$, by minimality of $X^{[N]}_\theta$. $\Box$ It is well known (see, e.g., \cite[p.~105]{Queff}) that $p_\varphi(N)=N+1$, so for the Fibonacci substitution $\varphi$ the $N$-block substitution $\hat{\varphi}_N$ is a substitution on an alphabet of $N+1$ symbols. We describe how one obtains $\hat{\varphi}_2$. We have ${\cal L}_\varphi^2=\{00, 01, 10\}$. Since 00 and 01 start with 0, and 10 with 1, we obtain $$\hat{\varphi}_2:\quad [00]\mapsto [01][10],\;[01]\mapsto [01][10],\; [10]\mapsto [00],$$ reading off the consecutive 2-blocks from $\varphi(00)=0101,\, \varphi(01)=010$ and $\varphi(10)=001$. It is useful to recode the alphabet $\{[00],[01],[10]\}$ to the standard alphabet $\{1,2,3\}$. We do this in the order in which they appear for the first time in the infinite Fibonacci word $w_{\rm F}$--- we call this the \emph{canonical coding}, and will use the same principle for all $N$. For $N=2$ this gives $[01]\rightarrow 1,\; [10]\rightarrow 2,\; [00]\rightarrow 3$. Still using the notation $\hat{\varphi}_2$ for the substitution on this new alphabet, we obtain $$\hat{\varphi}_2(1)=12 \quad \hat{\varphi}_2(2)=3, \quad \hat{\varphi}_2(3)=12.$$ In this way the substitution is in standard form (cf.~\cite{CDK} and \cite{Dekking-2016}). \section{The Fibonacci conjugacy class}\label{sec:C3} Let $F_n$ for $n=1,2,\dots$ be the Fibonacci numbers $$F_1=1,\, F_2=1,\, F_3=2,\, F_4=3,\, F_5=5, \dots.$$ \begin{theorem} Let $(Y,\sigma)$ be any subshift. Then $(Y,\sigma)$ is topologically conjugate to the Fibonacci system $(X_\varphi,\sigma)$ if and only if there exist $n\ge 3 $ and two words $B_0$ and $B_1$ of length $F_n$ and $F_{n-1}$, such that any $y$ from $Y$ is a concatenation of $B_0$ and $B_1$, and moreover, if\, $\cdots B_{x_{-1}} B_{x_0} B_{x_1}\cdots B_{x_k}\cdots$ is such a concatenation, then $x=(x_k)$ is a sequence from the Fibonacci system. \end{theorem} \noindent \emph{Proof:} First let us suppose that $(Y,\sigma)$ is topologically isomorphic to the Fibonacci system. By the Curtis-Hedlund-Lyndon theorem, there exists an integer $N$ such that $Y$ is obtained by a letter-to-letter projection $\pi$ from the $N$-block presentation $(X^{[N]}_\varphi, \sigma)$ of the Fibonacci system. Now if $B_0$ and $B_1$ are two decomposition blocks of sequences from $X^{[N]}_\varphi$ of length $F_n$ and $F_{n-1}$, then $\pi(B_0)$ and $\pi(B_1)$ are decomposition blocks of sequences from $Y$ with lengths $F_n$ and $F_{n-1}$, again satisfying the concatenation property. So it suffices to prove the result for $X^{[N]}_\varphi$. Note that we may suppose that the integers $N$ pass through an infinite subsequence; we will use $N=F_n$,where $n=3,4,\dots$. Useful to us are the \emph{singular words} $w_n$ introduced in \cite{WenWen}. The $w_n$ are the unique words of length $F_{n+1}$ having a different Parikh vector from all the other words of length $F_{n+1}$ from the language of $\varphi$. Here $w_1=1, w_2=00$, $w_3=101$, and for $n\ge4$ $$w_n=w_{n-2}w_{n-3}w_{n-2}.$$ The set of return words of $w_n$ has only two elements which are $u_n=w_nw_{n+1}$ and $v_n=w_nw_{n-1}$ (see page 108 in \cite{HuangWen}). The lengths of these words are $|u_n|=F_{n+3}$ and $|v_n|=F_{n+2}$. Let $w_n^-$ be $w_n$ with the last letter deleted. Define for $n\ge5$ $$B_0=\Psi(u_{n-3}w_{n-3}^-), \quad B_1=\Psi(v_{n-3}w_{n-3}^-),$$ where $\Psi$ is the $N$-block code from ${\cal L}_\varphi^N$ to ${\cal L}_{\varphi^{[N]}}$, with $N=F_{n-2}$. Then these blocks have the right lengths, and by Theorem 2.11 in \cite{HuangWen}, the two return words partition the infinite Fibonacci word $w_{\rm F}$ according to the infinite Fibonacci word---except for a prefix $r_{n,0}$: $$w_{\rm F}=r_{n,0}u_nv_nu_nu_nv_nu_n\dots.$$ By minimality this property carries over to all two-sided sequences in the Fibonacci dynamical system. For the converse, let $Y$ be a Fibonacci concatenation system as above. Let $C_0=\varphi^{n-2}(0)$ and $C_1=\varphi^{n-2}(1)$. We define a map $g$ from $(Y,\sigma)$ to a subshift of $\{0,1\}^{\mathbb{Z}}$ by $$g:\quad \cdots B_{x_{-1}} B_{x_0} B_{x_1}\cdots B_{x_k}\cdots\; \mapsto \; \cdots C_{x_{-1}} C_{x_0} C_{x_1}\cdots C_{x_k}\cdots,$$ respecting the position of the $0^{\rm th}$ coordinate. Since $|C_0|=|B_0|$ and $|C_1|=|B_1|$, $g$ commutes with the shift. Also, $g$ is obviously continuous. Moreover, since for any sequence $x$ in the Fibonacci system $\varphi^{n-2}(x)$ is again a sequence in the Fibonacci system, $g(Y)\subseteq X_\varphi$. So, by minimality, $(X_\varphi,\sigma)$ is a factor of $(Y,\sigma)$. Since $g$ is invertible, with continuous inverse, $(Y,\sigma)$ is in the conjugacy class of the Fibonacci system. $\Box$ \noindent {\bf Example}\; The case $(F_n,F_{n-1})=(13,8)$. Then $n=7$, so we have to consider the singular word $w_4=00100$ of length 5. \noindent The set of $5$-blocks is $\{01001,\,10010,\,00101,\,01010,\,10100,\,00100\}.$\\ These will be coded by the canonical coding $\Psi$ to the standard alphabet $\{1,2,3,4,5,6\}$. Note that $\Psi(w_4)=6$. Further, $w_3=101$ and $w_5=10100101$. So $u_n=0010010100101$ and $v_n=00100101$. Applying $\Psi$ gives the two decomposition blocks $B_0 = 6123451234512$ and $B_1 = 61234512$. \section{Reshaping substitutions}\label{sec:reshaping} We call a language preserving transformation of a substitution a reshaping. An example is the prefix-suffix change used in \cite{Dekking-TCS}. Here we consider a variation which we call a \emph{partition reshaping}. We give an example of this technique. Take the $N$-block representation of the Fibonacci system for $N=4$. All five 4-blocks occur consecutively at the beginning of the Fibonacci word $w_{\rm F}$ as $\{0100,\,1001,\,0010,\, 0101,\, 1010\}.$ The canonical coding to $\{1,2,3,4,5\}$ gives the 4-block substitution $\hat{\varphi}_4$: $$\hat{\varphi}_4:\qquad 1\rightarrow 12,\; 2\rightarrow 3,\; 3\rightarrow 45, \; 4\rightarrow 12,\; 5\rightarrow 3.$$ \noindent Its square is equal to $$\hat{\varphi}_4^2: \qquad 1\rightarrow 123,\; 2\rightarrow 45,\; 3\rightarrow 123,\; 4\rightarrow 123,\; 5\rightarrow 45. $$ Since the two blocks $B_0=123$ and $B_1=45$ have no letters in common this permits to do a partition reshaping. Symbolically this can be represented by \begin{table}[h!] \centering \caption{\small Partition reshaping.} \label{tab:table1} \begin{tabular}{ccccccccc}\\[.005cm] 1 & \; & 2 & 3 & & \qquad\qquad 4 & \; & 5 & \\ $\downarrow$ & \; & $\downarrow$ & $\downarrow$& & \qquad\qquad $\downarrow$ & \; & $\downarrow$ & \\ 1 & 2 & 3 & 4 & 5 & \qquad\qquad 1 & 2 & 3 & \\ 1 & \,\; 2 $\|$& 3 &\,\; 4 $\|$ & 5 & \qquad\qquad \,\; 1 $\|$ & 2 & \,\; 3 $\|$ & \end{tabular} \end{table} Here the third line gives the images $\hat{\varphi}_4(B_0)=\hat{\varphi}_4(123)=12345$ and $\hat{\varphi}_4(B_1)=\hat{\varphi}_4(45)=123$; the fourth line gives a \emph{another} partition of these two words in three, respectively two subwords from which the new substitution $\eta$ can be read of: $$\eta: \qquad 1\rightarrow 12,\; 2\rightarrow 34,\; 3\rightarrow 5 ,\; 4\rightarrow 1,\; 5\rightarrow 23. $$ What we gain is that the partition reshaped substitution $\eta$ generates the same language as $\hat{\varphi}_4$, but that $\eta$ is injective---it is even of full rank. \section{The Fibonacci class has infinite cardinality}\label{sec:C1} \begin{theorem}\label{th:inf} There are infinitely many primitive injective substitutions with Perron-Frobenius eigenvalue the golden mean that generate dynamical systems topologically isomorphic to the Fibonacci system. \end{theorem} We will explicitly construct infinitely many primitive injective substitutions whose systems are topologically conjugate to the Fibonacci system. The topological conjugacy will follow from the fact that the systems are $N$-block codings of the Fibonacci system, where $N$ will run through the numbers $F_n-1$. As an introduction we look at $n=5$, i.e., we consider the blocks of length $N=F_5-1=4$. With the canonical coding of the $N$-blocks we obtain the 4-block substitution $\hat{\varphi}_4$---see Section~\ref{sec:reshaping}: $$\hat{\varphi}_4:\qquad 1\rightarrow 12,\, 2\rightarrow 3,\, 3\rightarrow 45, \, 4\rightarrow 12,\,\ 5\rightarrow 3.$$ \noindent An \emph{interval} $I$ starting with $a\in A$ is a word of length $L$ of the form $$I=a,a+1,...,a+L-1.$$ \noindent Note that $\hat{\varphi}_4(123)=12345$, and $\hat{\varphi}_4(45)=123$, and these four words are intervals. This is a property that holds in general. First we need the fact that the first $F_n$ words of length $F_n-1$ in the fixed point of $\varphi$ are all different. This result is given by Theorem 2.8 in \cite{Chuan-Ho}. We code these $N+1$ words by the canonical coding to the letters $1,2,\dots,F_n$. We then have \begin{equation}\label{eq:Fib}\hat{\varphi}_N(12...F_{n-1})=12\dots F_{n}, \qquad \hat{\varphi}_N(F_{n-1}\!+1,\dots F_n)=12\dots F_{n-1}.\end{equation} This can be seen by noting that $\pi_0 \hat{\varphi}_N^n=\varphi^n \pi_0,$ for all $n$, and that the fixed point of $\varphi$ starts with $\varphi^{n-2}(0)\varphi^{n-3}(0)$. We continue for $n \ge 5$ with the construction of a substitution $\eta=\eta_n$ which is a partition reshaping of $\hat{\varphi}_N$. The $F_n$ letters in the alphabet $A^{[N]}$ are divided in three species, S, M and L (for Small, Medium and Large). $${\rm S}:={1,...,F_{n-3}}, \quad {\rm M}:={F_{n-3}+1,...,F_{n-1}},\quad {\rm L}:={F_{n-1}\!+1,...,F_n}.$$ Note that ${\rm Card}\, {\rm M}=F_{n-1}-F_{n-3}=F_{n-2}=F_{n}-F_{n-1}={\rm Card}\, {\rm L}.$ An important role is played by $a_{ \rm M}:=F_{n-3}+1$, the smallest letter in M, and $a_{ \rm L}:=F_{n-1}+1$, the smallest letter in L. For the letters in M (except for $a_{ \rm M})$ the rules are very simple: $$\eta(a)= a+F_{n-2}$$ (i.e., a single letter obtained by addition of the two integers). The first letter in M has the rule $$\eta(a_{ \rm M})=\eta(F_{n-3}\!+1)= F_{n-1}, F_{n-1}\!+1= F_{n-1},a_{ \rm L} .$$ The images of the letters in L are intervals of length 1 or 2, obtained from a partition of the word $12\dots F_{n-1}$. Their lengths are coming from $\varphi^{n-4}(0)$, rotated once (put the 0 in front at the back). This word is denoted $\rho(\varphi^{n-4 }(0))$. The choice of this word is somewhat arbitrary, other choices would work. The properties of $v:=\rho(\varphi^{n-4}(0))$ which matter to us are (V1) $\ell:=|v|=F_{n-2}$. (V2) $v_1=1$, $v_\ell=0$. (V3) $v$ does not contain any 11. \noindent Now the images of the letters in L are determined by $v$ according to the following rule: $|\eta(a_{ \rm L}+k-1)|=2-v_k$, for all $k=1,\dots,F_{n-2}$. Note that this implies in particular that for all $n\ge 5$ one has by property (V2) $$\eta(a_{\rm L})=\eta(F_{n-1}\!+1)= 1, \;\qquad \eta(F_n)= F_{n-1}-1,F_{n-1}.$$ The images of the letters in S are then obtained by choosing the lengths of the $\eta(a)$ in such a way that the largest common refinement of the induced partitions of the images of S and L is the singleton partition. \noindent{\bf Example} The case $n=7$, so $ F_n=13$, $ F_{n-1}=8$, and $ F_{n-2}=5$. \noindent Then ${\rm S}=\{1,2,3\},\, {\rm M}=\{4,5,6,7,8\},\, {\rm L}=\{9,10,11,12,13\}.$ \noindent Rules for M: \quad $4\rightarrow 89,\;5\rightarrow 10, \;6\rightarrow11, \;7\rightarrow 12, \;8\rightarrow13.$ Now $$\varphi^3(0)=01001\; \Rightarrow\; v= 10010\; \Rightarrow\; {\rm the\, partition\, is}\; 1|23|45|6|78.$$ This partition gives the following rules for L: $$9\rightarrow1,\; 10\rightarrow23,\; 11\rightarrow45,\; 12\rightarrow6,\; 13\rightarrow 78.$$ The induced partition for the images of the letters in S is $|12|34|567|8$, yielding rules $$1\rightarrow12,\; 2\rightarrow34,\; 3\rightarrow567.$$ \noindent In summary we obtain the substitution $\eta=\eta_7$ given by : \vspace*{-0.1cm} \begin{align*} {\rm S}: \begin{cases} 1& \rightarrow 1,2\\ 2& \rightarrow 3,4\\ 3& \rightarrow 5,6,7 \end{cases} \qquad {\rm M}: \begin{cases} 4& \rightarrow 8,9\\ 5& \rightarrow 10\\ 6& \rightarrow 11\\ 7& \rightarrow 12\\ 8& \rightarrow 13 \end{cases} \qquad {\rm L}: \begin{cases} \,\,9\!\!& \rightarrow 1\\ 10& \rightarrow 2,3\\ 11& \rightarrow 4,5\\ 12& \rightarrow 6\\ 13& \rightarrow 7,8. \end{cases} \end{align*} The substitution $\eta$ is primitive because you `can go' from the letter 1 to any letter and from any letter to the letter 1. This gives irreducibility; there is primitivity because periodicity is impossible by the first rule $1\rightarrow 1,2$. \noindent The substitution $\eta$ has full rank because any unit vector $$e_a=(0,\dots,0,1,0,\dots,0)$$ is a linear combination of rows of the incidence matrix $M_\eta$ of $\eta$. For $a\in {\rm L}\setminus\{9\}$ this combination is trivial, and for the other letters this is exactly forced by the choice of lengths in such a way that the largest common refinement of the induced partitions of the images of S and L is the singleton partition. In more detail: denote the $a^{\rm th}$ row of $M_\eta$ by $R_a$. Then $e_1=R_9$, and thus $e_2=R_1-R_9$, $e_3=R_{10}-e_2=R_{10}-R_1+R_9$, etc. The argument yielding the property of full rank will hold in general for all $n\ge5$. To prove primitivity for all $n$ we need some more details. \begin{proposition} The substitution $\eta=\eta_n$ is primitive for all $n \ge 5$.\end{proposition} \noindent \emph{Proof:} The proposition will be proved if we show that for all $a\in A$ the letter $a$ will occur in some iteration $\eta^k(1)$, and conversely, that for all $a\in A$ the letter $1$ will occur in some iteration $\eta^k(a)$. The first part is easy to see from the fact that $\eta(1)=1,2$ and that $\eta^2(1,\dots,F_{n-2})=1,\dots,F_n-1$, plus $\eta^2(a_{\rm M})=F_n,1$. For the second part, we show that A) for any $a\in$ M$\cup$L a letter from S will occur in $\eta^k(a)$ in $k\le$ Card M$\cup$L steps (see Lemma~\ref{lem:dec}) and B), that for any $a\in$ S the letter 1 will occur in $\eta^k(a)$ in $k\le$ 2Card $A$ steps (see Lemma~\ref{lem:occ1}). $\Box$ \begin{lemma}\label{lem:dec} Let $f:A\rightarrow A$ be the map that assigns the first letter of $\eta^2(a)$ to $a$. Then $f$ is strictly decreasing on L $\cup$ M$\backslash \{a_{\rm M}\}$.\end{lemma} \noindent \emph{Proof:} First we consider $f$ on ${\rm L}$. We have $$\eta^2(a_{\rm L}\dots F_n)=\eta(1,\dots, F_{n-1}-1, F_{n-1})=1\dots F_n.$$ Since $$\eta^2(F_n)=\eta( F_{n-1}-1,F_{n-1})=F_{n-1}-1+F_{n-2},F_{n-1}+F_{n-2}=F_n-1, F_n,$$ we obtain $f(F_n)=F_n-1<F_n$. This implies that also the previous letters in ${\rm L}$ are mapped by $f$ to a smaller letter. \noindent Next we consider $f$ on M$\backslash \{a_{\rm M}\}$. Here $$\eta^2(a_{\rm M}+1,\dots, F_{n-1})=\eta(a_{\rm L}+1,\dots, F_{n})=2,3,\dots, F_{n-1}.$$ Now $$\eta^2(F_{n-1})=\eta( F_{n})=F_{n-1}-1,F_{n-1}.$$ So we obtain $f(F_{n-1})=F_{n-1}-1<F_{n-1}$. This implies that also the previous letters in ${\rm M}$ are mapped by $f$ to a smaller letter. $\Box$ \begin{lemma}\label{lem:occ1} For all $a\in S$ there exists $k \le 2\,{\rm Card}\, A$ such that the letter 1 occurs in $\eta^{k}(a)$. \end{lemma} \noindent \emph{Proof:} The substitution $\eta^2$ maps intervals $I$ to intervals $\eta^2(I)$, provided $I$ does not contain $a_{\rm M}$ or $a_{\rm L}$. By construction, since the $\eta(b)$ for $b\in {\rm L}$ have length 1 or 2, the length of $\eta(a)$ for $a\in {\rm S}$ is 2 or 3, and so $\eta(a)$ contains a word $c, c+1$ for some $c\in A$. Since $\rho\varphi^{(n-4)}(0)$ does not contain two consecutive 1's (property (V3)), the image $\eta^2(c,c+1)$ has at least length 3. Since\footnote{This follows from the fact that any word in the language of $\eta$ occurs in some concatenation of the two words $12\dots F_{n}$ and $12\dots F_{n-1}$.} any word of length at least 3 in the language of $\eta$ contains an interval of length 2, the length increases by at least 1 if you apply $\eta^2$. It follows that for all $n\ge 5$ and all $a\in {\mathrm S}$ one has $|\eta^{2n+1}(a)| \ge n+2$. But then after less than ${\rm Card}\, A$ steps a letter $a_{\rm M}$ or a letter $a_{\rm L}$ must occur in $\eta^{2n+1}(a)$. This implies that the letter 1 occurs in $\eta^{2n+3}(a)$, since both $\eta^2(a_{\rm M})$ and $\eta^2(a_{\rm L})$ contain a 1. $\Box$ \section{The 2-symbol case}\label{sec:two} The eigenvalue group of the Fibonacci system is the rotation over the small golden mean $\gamma=(\sqrt{5}-1)/2$ on the unit circle, and any system topologically isomorphic to the Fibonacci system must have an incidence matrix with Perron Frobenius eigenvalue the golden mean or a power of the golden mean (cf.~\cite[Section 7.3.2]{Pytheas}). Thus, modulo a permutation of the symbols, on an alphabet of two symbols the incidence matrix with Perron-Frobenius eigenvalue the golden mean has to be $\left( \begin{smallmatrix} 1 \, 1\\ 1\, 0 \end{smallmatrix} \right).$ There are two substitutions with this incidence matrix: Fibonacci $\varphi$, and reverse Fibonacci ${\varphi_{\textsc{\tiny R}}}$, defined by $${\varphi_{\textsc{\tiny R}}}: \qquad 0\rightarrow\,10,\;1\rightarrow 0.$$ These two substitutions are essentially different, as they have different standard forms (see \cite{Dekking-2016} for the definition of standard form). However, it follows directly from Tan Bo's criterion in his paper \cite{Tan} that ${\varphi_{\textsc{\tiny R}}}$ and $\varphi$ have the same language\footnote{This follows also directly from the well-known formula ${\varphi_{\textsc{\tiny R}}}^{\!2n}(0)\,10=01\,\varphi^{2n}(0)$ for all $n\ge1$ (see \cite[p.17]{Berstel}).}, but then they also generate the same system. Conclusion: the conjugacy class of Fibonacci with Perron-Frobenius eigenvalue the golden mean restricted to two symbols consists of Fibonacci and reverse Fibonacci. \section{Maximal equicontinuous factors}\label{sec:equi} Let $T$ be the mapping from the unit circle $Z$ to itself defined by $Tz=z+\gamma \mod 1$, where $\gamma$ is the small golden mean. This, being an irrational rotation, is indeed an equicontinuous dynamical system -- the usual distance metric is an invariant metric under the mapping. The factor map from the Fibonacci dynamical system $(X_\varphi,\sigma)$ to $(Z,T)$ is given by requiring that the cylinder sets $\{x:x_0=0\}$ and $\{x:x_0=1\}$ are mapped to the intervals $[0,\gamma]$ and $[\gamma,1]$ respectively, and requiring equivariance. If we take any point of $Z$ not of the form $n\gamma \mod 1$ ($n$ any integer), then the corresponding sequence is unique. If, however, we use an element in the orbit of $\gamma$, then for this point there will be two codes, a ``left" one and a ``right" one. We want to understand more generally why two or more points map to a single point. Suppose $x$ and $y$ are two points of a system $(X,\sigma)$ that map to two points $x'$ and $y'$ in an equicontinuous factor. Then for any power of $T$ (the map of the factor system) the distance between $T^n(x')$ and $T^n(y')$ is just equal to the distance between $x'$ and $y'$. So $x$ and $y$ map to the same point $x'$ if either all $x_n$ and $y_n$ are equal for sufficiently large $n$, or all $x_n$ and $y_n$ are equal for sufficiently large $-n$. We say that $x$ and $y$ are respectively \emph{right asymptotic} or \emph{left asymptotic} A pair of letters $(b,a)$ is called a \emph{cyclic pair} of a substitution $\alpha$ if $ba$ is an element of the language of $\alpha$, and for some integer $m$ $$\alpha^m(b)=\dots b \quad{\rm and}\quad \alpha^m(a)=a\dots. $$ Such a pair gives an infinite sequence of words $\alpha^{mk}(ba)$ in the language of $\alpha$, which---if properly centered---converge to an infinite word which is a fixed point of $\alpha^m$. With a slight abuse of notation we denote this word by $\alpha^{\infty}(b)\cdot \alpha^{\infty}(a)$. For the Fibonacci substitution $\varphi$, $(0,0)$ and $(1,0)$ are cyclic pairs, and the two synchronized points $\varphi^\infty(0)\cdot\varphi^\infty(0)$ and $\varphi^\infty(1)\cdot\varphi^\infty(0)$, are right asymptotic so they map to the same point in the equicontinuous factor. Because of these considerations we now define $Z$-triples. Let $\eta$ be a primitive substitution. Call three points $x$, $y$, and $z$ in $X_\eta$ a $Z$-\emph{triple} if they are generated by three cyclic pairs of the form $(b,a),\, (b,d)$ and $(c,d)$, where $a,b,c,d \in A$. Then $x$, $y$, and $z$ are mapped to the same point in the maximal equicontinuous factor. \begin{theorem}\label{th:Zth} \; Let $(X_\eta,\sigma)$ be any substitution dynamical system topologically isomorphic to the Fibonacci dynamical system. Then there do not exist $Z$-triples in $X_\eta$. \end{theorem} \noindent \emph{Proof:} Since $(X_\eta,\sigma)$ is topologically isomorphic to $(X_\varphi,\sigma)$, its maximal equicontinuous factor is $(Z,T$), and the factor map is at most 2-to-1. Suppose $(b,a),\, (b,d)$ and $(c,d)$ gives a $Z$-triple $x,y,z$ in $X_\eta$. Noting that $$x=\eta^\infty(b)\cdot\eta^\infty(a), \quad y=\eta^\infty(b)\cdot \eta^\infty(d)$$ are left asymptotic, and $y=\eta^\infty(b)\cdot \eta^\infty(d)$ and $z=\eta^\infty(c)\cdot\eta^\infty(d)$ are right asymptotic, this would give a contradiction. $\Box$ \noindent {\bf Example} Let $\eta$ be the substitution given by $$\eta:\qquad 1\rightarrow 12,\, 2\rightarrow 34,\, 3\rightarrow 5 ,\, 4\rightarrow 1,\, 5\rightarrow 23. $$ Then $\eta$ generates a system that is topologically isomorphic to the Fibonacci system ($\eta$ is the substitution at the end of Section~\ref{sec:reshaping}). Quite remarkably, $\eta^6$ admits 5 fixed points generated by the cyclic pairs $(1, 2),\, (2, 3),\, (3, 1),\, (4, 5)$ and $(5, 1)$. Note however, that no three of these form a $Z$-triple. \section{Fibonacci matrices}\label{sec:mat} Let $\mathcal{F}_r$ be the set of all non-negative primitive $r\times r$ integer matrices, with Perron-Frobenius eigenvalue the golden mean $\Phi = (1+\sqrt{5})/2$.\\ We have seen already that $\mathcal{F}_2$ consists of the single matrix $\left( \begin{smallmatrix} 1 \, 1\\ 1\, 0 \end{smallmatrix} \right).$ \begin{theorem}\label{th:F3} The class $\mathcal{F}_3$ essentially consists of the three matrices \qquad $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right),\; \left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right),\; \left( \begin{smallmatrix} 0\, 1\, 0\\ 1 \,0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$ \end{theorem} Here essentially means that in each class of 6 matrices corresponding to the permutations of the $r=3$ symbols, one representing member has been chosen (actually corresponding to the smallest standard form of the substitutions having that matrix). \emph{Proof:} Let $M$ be a non-negative primitive $3\times 3$ integer matrix, with Perron-Frobenius eigenvalue the golden mean $\Phi = (1+\sqrt{5})/2$. We write\\ [-0.7cm] $$ M= \left( \begin{matrix} a\; b\; c\\ d\; e\; f\\ g\; h\; i \end{matrix} \right).$$ The characteristic polynomial of $M$ is $\chi_M(u)=u^3-Tu^2+Fu-D,$ where $T=a+e+i$ is the trace of $M$, and \begin{equation}\label{eq:FandD} F=ae+ai+ei-bd-cg-fh,\quad D=aei+bfg+cdh-afh-bdi-ceg.\quad \end{equation} Of course $D$ is the determinant of $M$. Since $\Phi$ is an eigenvalue of $M$, and we consider matrices over the integers, $u^2-u-1$ has to be a factor of $\chi_M$. Performing the division we obtain $$\chi_M(u)=\big(u-(T-1)\big)\big(u^2-u-1\big),$$ and requiring that the remainder vanishes, yields \begin{equation}\label{eq:DF} F=T-2,\quad D=1-T. \end{equation} Note that the third eigenvalue equals $\lambda_3=T-1$. From the Perron-Frobenius theorem follows that this has to be smaller than $\Phi$ in absolute value, and since it is an integer, only $\lambda_3=-1, 0, 1$ are possible. Thus there are only 3 possible values for the trace of $M$: $T=0,\, T=1$ and $T=2$. The smallest row sum of $M$ has to be smaller than the PF-eigenvalue $\Phi$ (well known property of primitive non-negative matrices). Therefore $M$ has to have one of the rows $(0,0,1)$, $(0,1,0)$ or $(0,0,1)$. Also, because of primitivity of $M$, the 1 in this row can not be on the diagonal. By performing permutation conjugacies of the matrix we may then assume that $M$ has the form $$ M= \left( \begin{matrix} 0\;\; 1\;\; 0\\ d\;\; e\;\; f\\ g\;\; h\;\; i \end{matrix} \right).$$ The equation \eqref{eq:FandD} combined with \eqref{eq:DF} then simplifies to \begin{equation}\label{eq:DF2} T-2=F=ei-d-fh, \quad 1-T=D=fg-di. \end{equation} \noindent{\bf Case $\mathbf{{\emph T}=0}$} \noindent In this case $e=i=0$, so \eqref{eq:DF2} simplifies to \begin{equation}\label{eq:T0F} -2=F=-d-fh, \quad 1=D=fg. \end{equation} Then $f=g=1$, and so $d+h=2$. This gives three possibilities leading to the matrices $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right),\; \left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right),\; \left( \begin{smallmatrix} 0\, 1\, 0\\ 2 \,0\, 1\\ 1\, 0\, 0 \end{smallmatrix} \right).$ \noindent Here the third matrix is permutation conjugate to the second one. \noindent{\bf Case $\mathbf{{\emph T}=1}$} \noindent In this case $e=1, i=0$, or $e=0, i=1$. \noindent First case: $e=1, i=0$. Now \eqref{eq:DF2} simplifies to \begin{equation}\label{eq:T1F} -1=F=-d-fh, \quad 0=D=fg. \end{equation} Then $g=0$, since $f=0$ is not possible because of primitivity. But $g=0$ also contradicts primitivity, as $d+fh=1$, gives either $d=0$ or $h=0$. \noindent Second case: $e=0, i=1$. Now \eqref{eq:DF2} simplifies to \begin{equation}\label{eq:T1F2} -1=F=-d-fh, \quad 0=D=fg-d. \end{equation} Then $d=0$ would imply that $f=h=1$. But, as $g>0$ because of primitivity, we get a contradiction with $fg=d=0$. On the other hand, if $d>0$, then $d=1$ and $f=0$ or $h=0$. But $fg=d=1$ gives $f=g=1$, so $h=0$, and we obtain the matrix $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$ \noindent{\bf Case $\mathbf{{\emph T}=2}$} \noindent In this case \eqref{eq:DF2} becomes \begin{equation}\label{eq:T1F} 0=F=ei-d-fh, \quad -1=D=fg-di. \end{equation} Since $ei=0$ would lead to $f=0$, which is not allowed by primitivity, what remains is $e=1,i=1$. Then, substituting $d=fg+1$ in the first equation gives $0=f(g+h)$. But both $f=0$ and $g=h=0$ contradict primitivity. Final conclusion: there are three matrices in $\mathcal{F}_3$, modulo permutation conjugacies. $\Box$ \noindent {\bf Remark} It is well-known that the PF-eigenvalue lies between the smallest and the largest row sum of the matrix. One might wonder whether this largest row sum is bounded for the class $\mathcal{F}=\cup_r\mathcal{F}_r$. Actually the class $\mathcal{F}_r$ contains matrices with some row sum equal to $r-1$ for all $r\ge 3$: take the matrix $M$ with $M_{1,j}=1$ for $j=2,\dots,r$, $M_{2,2}=1$ and $M_{i,{i+1}}=1$, for $i=2,\dots,r-1$, $M_{r,1}=1$ and all other entries 0. Now note that $(1, \Phi,...,\Phi)$ is a left eigenvector of $M$ with eigenvalue $\Phi$ (since $\Phi^2=1+\Phi$). Since the eigenvector has all entries positive, it must be a PF-eigenvector (well known property of primitive, non-negative matrices), and hence $M$ is in $\mathcal{F}_r$. \section{The 3-symbol case}\label{sec:C2} \begin{theorem} There are two primitive injective substitutions $\eta$ and $\zeta$ on a three letter alphabet $\{a,b,c\}$ that generate dynamical systems topologically isomorphic to the Fibonacci system. These are given\footnote{Standard forms: replace $a,b,c$ by $1,2,3$.} by\\[-.4cm] $$\eta(a)=b,\,\eta(b)=ca,\, \eta(c)=ba,\quad \zeta(a)=b,\,\zeta(b)=ac,\, \zeta(c)=ab.$$ \end{theorem} \noindent\emph{Proof:} The possible matrices for candidate substitutions are given in Theorem~\ref{th:F3}. Let us consider the first matrix $ \left( \begin{smallmatrix} 0\, 1\, 0\\ 1\, 0\, 1\\ 1\, 1\, 0 \end{smallmatrix} \right)$. \noindent There are four substitutions with this matrix as incidence matrix: \begin{align*} \eta_1: \; a& \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ba,& \eta_2: \; a \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ab,\\ \eta_3: \; a& \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ba,& \eta_4: \; a \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ab, \end{align*} Here $\eta_1=\eta$. To prove that the system of $\eta$ is conjugate to the Fibonacci system consider the letter-to-letter map $\pi$ given by $$\pi(a)=1,\quad \pi(b)=\pi(c)=0.$$ Then $\pi$ maps $X_\eta$ onto $X_\varphi$, because $\pi\eta=\varphi\pi$. Moreover, $\pi$ is a conjugacy, since if $x\ne y$ and $\pi(x)=\pi(y)$, then there is a $k$ such that $x_k=b$ and $y_k=c$. But the words of length 2 in the language of $\eta$ are $ab, ba, bc$ and $ca$, implying that $x_{k-1}=a$ and $y_{k-1}=b$, contradicting $\pi(x)=\pi(y)$. Since $\zeta$ is the time reversal of $\eta$, and we know already that the system of ${\varphi_{\textsc{\tiny R}}}$ is conjugate to the Fibonacci system, the system generated by $\eta_4=\zeta={\eta_{\textsc{\tiny R}}}$ is conjugate to the Fibonacci system. It remains to prove that $\eta_2$ and $\eta_3$ generate systems that are \emph{not} conjugate to the Fibonacci system. Again, since $\eta_3$ is the time reversal of $\eta_2$, it suffices to do this for $\eta_2$. The language of $\eta_2$ contains the words $ab, bb$ and $bc$. These words generate fixed points of $\eta_2^6$ in the usual way. But these three fixed points form a $Z$-triple, implying that the system of $\eta_2$ can not be topologically isomorphic to the Fibonacci system (see Theorem~\ref{th:Zth}). The next matrix we have to consider is $\left( \begin{smallmatrix} 0\, 1\, 0\\ 0\, 0 \,1 \\ 1\, 2\,0 \end{smallmatrix} \right).$ There are three substitutions with this matrix as incidence matrix: \begin{align*} \eta_1: \; a& \rightarrow b,\, b\rightarrow c,\,c\rightarrow abb,& \eta_2: \; a \rightarrow b,\, b\rightarrow c,\,c\rightarrow bab,\\ \eta_3: \; a& \rightarrow b,\, b\rightarrow c,\,c\rightarrow bba. \end{align*} Again, the system of $\eta_1$ contains a $Z$-triple generated by $ab, bb$ and $bc$. So this system is not conjugate to the Fibonacci system, and neither is the one generated by $\eta_3$ (time reversal of $\eta_1$). The system generated by $\eta_2$ behaves similarly to the Fibonacci system, \emph{but} is has an eigenvalue $-1$ (it has a two-point factor via the projection $a,c\rightarrow 0, b\rightarrow 1$.) Finally, we have to consider the matrix $\left( \begin{smallmatrix} 0\, 1\, 0\\ 1 \,0\, 1\\ 1\, 0\, 1 \end{smallmatrix} \right).$ There are four substitutions with this matrix as incidence matrix: \begin{align*} \eta_1: \; a& \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ac,& \eta_2: \; a \rightarrow b,\, b\rightarrow ac,\,c\rightarrow ca,\\ \eta_3: \; a& \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ac,& \eta_4: \; a \rightarrow b,\, b\rightarrow ca,\,c\rightarrow ca. \end{align*} Here $\eta_1$ and $\eta_4$ generate systems conjugate to the Fibonacci system, but the substitutions are not injective. The substitution $\eta_2$ has all 9 words of length 2 in its language, and all of these generate fixed points of $\eta_2^6$. So the system of $\eta_2$ is certainly not topologically isomorphic to the Fibonacci system. The proof is finished, since $\eta_3$ is the time reversal of $\eta_2$. $\Box$ \section{Letter-to-letter maps}\label{sec:L2L} By the Curtis-Hedlund-Lyndon theorem all members in the conjugacy class of the Fibonacci system can be obtained by applying letter-to-letter maps $\pi$ to $N$-block presentations $(X^{[N]},\sigma)$. Here we analyse the case $N=2$. The 2-block presentation of the Fibonacci system is generated by (see Section~\ref{sec:Nblock}) the 2-block substitution $$\hat{\varphi}_2(1)=12 \quad \hat{\varphi}_2(2)=3, \quad \hat{\varphi}_2(3)=12.$$ There are (modulo permutations of the symbols) three letter-to-letter maps from $\{1,2,3\}$ to $\{0,1\}$. Two of these project onto the Fibonacci system, as they are projections on the first respectively the second letter of the 2-blocks. The third is $\pi$ given by $$\pi(1)=0,\quad \pi(2)=0, \quad \pi(3)=1.$$ What is the system $(Y,\sigma)$ with $Y=\pi\big(X^{[2]}\big)$? First note that $(Y,\sigma)$ is conjugate to the Fibonacci system since $\pi$ is clearly invertible. Secondly, we note that the points in $Y$ can be obtained by doubling the 0's in the points of the Fibonacci system. This holds because $\pi(12)=00,\, \pi(3)=1$, but also $$\pi(\hat{\varphi}_2(12))=\pi(123)=001,\;\pi(\hat{\varphi}_2(3))=\pi(12)=00.$$ Thirdly, we claim that the system $(Y,\sigma)$ cannot be generated by a substitution. This follows from the fact that the sequences in $Y$ contain the word 0000, but no other fourth powers. This is implied by the $4^{\rm th}$ power free-ness of the Fibonacci word, proved in \cite{Karhumaki}. A fourth property is that the sequence $y^+$ obtained by doubling the 0's in $w_{\rm F}$, where $w_{\rm F}$ is the infinite Fibonacci word is given by $$y^+_n=[(n+2)\Phi]-[n\Phi]-[2\Phi], \qquad {\rm for\;} n\ge 1,$$ according to \cite{Wolfdieter}, and \cite{OEIS-Fib} (here $[\cdot]$ denotes the floor function). Finally we remark that Durand shows in the paper \cite{Durand} that the Fibonacci system is prime \emph{modulo topological isomorphism}, and ignoring finite factors and rotation factors. This implies that all the projections are automatically invertible, if the projected system is not finite. \end{document}

\begin{document} \title{Equidistribution estimates for eigenfunctions and eigenvalue bounds for random operators} \author{D. Borisov} \address{Institute of Mathematics USC RAS, Chernyshevskii str., 112,\\ Ufa, 450000, Russia\\ \& Bashkir State Pedagogical University, October rev. st., 3a, \\ Ufa, 450008, Russia,\\ E-mail: [email protected]\\ matem.anrb.ru \& www.bspu.ru} \author{M.~Tautenhahn} \address{Fakult\"at f\"ur Mathematik, Reichenhainer Str. 41,\\ Chemnitz, D-09126, Germany,\\ www.tu-chemnitz.de/\~{}mtau} \author{I.~Veseli\'c} \address{Fakult\"at f\"ur Mathematik, Reichenhainer Str. 41,\\ Chemnitz, D-09126, Germany,\\ www.tu-chemnitz.de/stochastik/} \begin{abstract} We discuss properties of $L^2$-eigenfunctions of Schr\"odinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones. \end{abstract} \keywords{scale-free unique continuation property, equidistribution property, observability estimate, uncertainty relation, Carleman estimate, Schr\"odinger operator, elliptic differential equation} \bodymatter \section{Introduction} In this note we present recent results in Harmonic Analysis for solutions of (time-independent) Schr{\"o}dinger equations and other partial differential equations. They are motivated by interest in techniques relevant for proving localization for random Schr{\"o}dinger operators. The mentioned Harmonic Analysis results which we present are a quantitative unique continuation principle and an equidistribution property for eigenfunctions, which is scale-uniform. These results, and variants thereof, go under various names, depending on the particular field of mathematics: They are called observability estimate, uncertainty relation, scale-free unique continuation principle, or local positive definiteness. The latter term signifies that a self-adjoint operator is (strictly) positive definite when restricted to a relevant subspace, while it is not so on the whole Hilbert space. For the purpose of motivation we discuss this property in the next section. \par The term \emph{localization} refers to the phenomenon, that quantum Hamiltonians describing the movement of electrons in certain disordered media exhibit pure point spectrum in appropriately specified energy regions. The corresponding eigenfunctions decay exponentially in space. The (time-dependent) wavepackets describing electrons stay localized essentially in a compact region of space for all times. Nota bene, all mentioned properties hold \emph{almost surely}. This is natural in the context of random operators. \par An important partial result for deriving localization are Wegner estimates. These are bounds on the expected number of eigenvalues in a bounded energy interval of a random Schr{\"o}dinger operator restricted to a box. \par The localization problem has been studied for other classes of random operators beyond those of Schr{\"o}dinger type. An example are random divergence type operators, see e.g.\ Refs.~\citenum{FigotinK-96} and \citenum{Stollmann-98}. This are partial differential operators with randomness in coefficients of higher order terms. In paricular, the second order term is no longer the Laplacian, but a variable coefficient operator. In this context one is again lead to consider the above mentioned questions of Harmonic Analysis for eigenfunctions of differential operators. In this note we present an exposition of recently published results, and an announcement of a quantitative unique continuation principle and an equidistribution estimate for eigenfunctions for a class of elliptic operators with variable coefficients. \subsection{Motivation: Moving and lifting of eigenvalues} \label{ss:motivation} Here we discuss some aspects of eigenvalue perturbation theory. It will provide an accessible explanation why one is interested in the results presented in Sections~\ref{sec:schroedinger} and \ref{sec:elliptic} below in the context of random Schr{\"o}dinger operators and elliptic differential operators, respectively. In fact, to illustrate the main questions it will be for the moment completely sufficient to restrict our attention to the finite dimensional situation, i.e.\ to perturbation theory for finite symmetric matrices. The focus will be on how (local) positive definiteness of the perturbation relates to lifting of eigenvalues. \par Let $A$ and $B$ be symmetric $n \times n$ matrices, with $B\geq b>0$ positive definite. The variational min-max principle for eigenvalues shows that for any $k \in \{1,\dots, n\}$ and $t\geq 0$ \begin{equation} \label{eq:positive_definite_perturbation} \lambda_k(A+tB) \geq \lambda_k(A) + b \, t \end{equation} where $\lambda_k(M)$ denotes the $k$th lowest eigenvalue, counting multiplicities, of a symmetric matrix $M$. Note that the dimension $n$ does not enter in the bound \eqref{eq:positive_definite_perturbation}. Without the positive definiteness assumption on $B$ this universal bound will fail, most blatantly if \begin{equation*} A =\begin{pmatrix} A_1 & \ 0 \\ 0 & A_2 \end{pmatrix} \quad \text{and} \quad B =\begin{pmatrix} \operatorname{Id} & \ 0 \\ 0 & -\operatorname{Id} \end{pmatrix} . \end{equation*} In this case, all eigenvalue $\lambda_k(A+tB)$ \emph{will move}, even with constant speed w.r.t.\ the variable $t$, albeit in different directions. If $B$ is singular, some eigenvalues may not move at all. However, for appropriate classes of symmetric matrices $A$, and of positive semidefinite matrices $B$, one may still aim to prove \begin{equation} \label{eq:positive_semidefinite_perturbation} \forall \, t\geq 0, k \in \{1,\dots, n\} \, \exists \, \kappa >0 \text{ such that } \lambda_k(A+tB) \geq \lambda_k(A) + \kappa t \end{equation} Note however, that $\kappa $ is now not a uniform bound but depends on \begin{itemlist} \item the class of symmetric matrices from which $A$ is chosen, \item the class of semidefinite matrices from which $B$ is chosen, \item the range from which the coupling $t$ is chosen, and \item the range from which the index $k \in \{1,\dots, n\}$ is chosen. \end{itemlist} In the case of random operators or matrices one in is interested in the situation where \begin{equation} \label{eq:multiparameter_family} A(\omega) =A_0+\sum_{j \in Q} \omega_j B_j =\Big(A_0+\sum_{j \in Q, j\neq 0} \omega_j B_j \Big)+ \omega_0 B_0 \end{equation} is a multi-parameter pencil. Here $Q$ is some subset of $\mathbb{Z}^d$ containing $0$. The real variables $\omega_j$ model random coupling constants determining the strength of the perturbation $B_j$ in each configuration $\omega=(\omega_j)_{j \in Q}$. Now, \eqref{eq:multiparameter_family} already suggest to write $A(\omega)$ as \[ A(\omega_0^\perp)+tB \quad\text{where} \quad t=\omega_0,\ B=B_0,\ \text{and} \ \omega_0^\perp=(\omega_j)_{j \in Q, j\neq0} . \] This highlights that if we consider $A(\omega)$ as a function of the single variable $t=\omega_0$, it is clearly a one-parameter family of operators, albeit the ``unperturbed part'' $A(\omega_0^\perp)$ of $A(\omega)=A(\omega_0^\perp)+tB$ is not a single operator, but varying over the ensemble $(A(\omega_0^\perp))_{\omega_0^\perp}$. To have a useful version of \eqref{eq:positive_semidefinite_perturbation} in this situation, the constant $\kappa $ needs to have a uniform lower bound $\inf_{A} \kappa $ where $A=A(\omega_0^\perp)$ varies over all matrices in the ensemble. \par In what follows we present rigorous results of the type \eqref{eq:positive_semidefinite_perturbation}, but where $A$ and $B$ are not finite matrices, but differential and multiplication operators. The relevant operators have all compact resolvent, ensuring that the entire spectrum consists of eigenvalues. \section{Equidistribution property of Schr\"odinger eigenfunctions} \label{sec:schroedinger} The following result is taken from Ref.~\citenum{Rojas-MolinaV-13}. It is an equidistribution estimate for Schr{\"o}dinger eigenfunctions, which is uniform w.r.t.\ the naturally arising length scales, and has strong implications for the spectral theory of random Schr\"odinger operators. \par We fix some notation. For $L>0$ we denote by $\Lambda_L = (-L/2 , L/2)^d$ a cube in $\mathbb{R}^d$. For $\delta>0$ the open ball centered at $x\in \mathbb{R}$ with radius $\delta$ is denoted by $B(x, \delta)$. For a sequence of points $(x_j)_j$ indexed by $j \in \mathbb{Z}^d$ we denote the collection of balls $\cup_{j \in \mathbb{Z}^d} B(x_j , \delta) $ by $S$ and its intersection with $\Lambda_L$ by $S_L$. We will be dealing with certain subspaces of the standard second order Sobolev space $W^{2,2}(\Lambda_L)$ on the cube. Let $\Delta$ be the $d$-dimensional Laplacian. Its restriction to the cube $\Lambda=\Lambda_L$ needs boundary conditions to be self-adjoint. The domain of the Dirichlet Laplacian will be denoted by $\mathcal{D}(\Delta_{\Lambda,0})$ and the domain of the Laplacian with periodic boundary conditions by $\mathcal{D}(\Delta_{\Lambda,\mathrm{per}})$. Let $V \colon \mathbb{R}^d\to \mathbb{R}$ be a bounded measurable function, and $H_L = (-\Delta + V)_{\Lambda_L} $ a Schr\"odinger operator on the cube $\Lambda_L$ with Dirichlet or periodic boundary conditions. The corresponding domains are still $\mathcal{D}(\Delta_{\Lambda,0})$ and $ \mathcal{D}(\Delta_{\Lambda,\mathrm{per}})$, respectively. Note that we denote a multiplication operator by the same symbol as the corresponding function. \par The following theorem was proven in Ref.~\citenum{Rojas-MolinaV-13}. \begin{theorem}[Scale-free unique continuation principle] \label{thm:RojasVeselic} Let $\delta, K_{V} > 0$. Then there exists $C_{\rm sfUC} \in (0,\infty)$ such that for all $L \in 2\mathbb{N}+1 $, all measurable $V : \mathbb{R}^d \to [-K_{V} , K_{V}]$, all real-valued $\psi \in \mathcal{D}(\Delta_{\Lambda,0}) \cup \mathcal{D}(\Delta_{\Lambda,\mathrm{per}})$ with $(-\Delta + V)\psi = 0$ almost everywhere on $\Lambda_L$, and all sequences $(x_j)_{j \in \mathbb{Z}^d} \subset \mathbb{R}^d$, such that for all $j \in \mathbb{Z}^d$ the ball $B(x_j , \delta) \subset \Lambda_1 + j$, we have \begin{equation} \label{eq:observability} \int_{S_L} \psi^2 \geq C_{\rm sfUC} \int_{\Lambda_L} \psi^2 . \end{equation} \end{theorem} \begin{figure} \caption{Examples of collections of balls $S_L$ within region $\Lambda_L\subset \mathbb{R}^2$.} \label{fig:equidistributed} \end{figure} The value of the result is not in the \emph{existence} of the constant $C_{\rm sfUC}$, but in the \emph{quantitative control} of the dependence of $C_{\rm sfUC}$ on parameters entering the model. The very formulation of the theorem states that $C_{\rm sfUC}$ is independent of the position of the balls $B(x_j,\delta)$ within $\Lambda_1 +j$, and independent of the scale $L\in2\mathbb{N} +1$. From the estimates given in Section~2 of Ref.~\citenum{Rojas-MolinaV-13} one infers that $C_{\rm sfUC}$ depends on the potential $V$ only through the norm $\lVert V \rVert_\infty$ (on an exponential scale), and it depends on the small radius $\delta>0$ polynomially, i.e.\ $C\gtrsim \delta^N$, for some $N\in\mathbb{N}$ which depends on the dimension on $d$ and $\lVert V \rVert_\infty$. \par The theorem states a property of functions in the kernel of the operator. It is easily applied to eigenfunctions corresponding to other eigenvalues since \[ H_L\psi=E\psi \Leftrightarrow (H_L-E)\psi=0 . \] As a consequence of the energy shift the constant $K_{V}$ has to be replaced with $K_{V-E}$, which may be larger than $K_{V}$. It may always be estimated by $K_{V-E}\leq K_V+|E|$. \par There is a very natural question supported by earlier results, which was spelled out in Ref.~\citenum{Rojas-MolinaV-13}, namely does the following generalisation of Theorem~\ref{thm:RojasVeselic} hold: Given $\delta >0$, $K\geq0$ and $E\in\mathbb{R}$ there is a constant $C>0$ such that for all measurable $ V\colon \mathbb{R}^d \rightarrow [-K,K] $, all $L \in 2\mathbb{N}+1$, and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in \mathbb{Z}^d$ we have \begin{equation} \label{eq:uncertainty} \chi_{(-\infty,E]} (H_L) \, W_L \, \chi_{(-\infty,E]} (H_L) \geq C~ \chi_{(-\infty,E]} (H_L) , \end{equation} where $W_L=\chi_{S_L}$ is the indicator function of $S_L$ and $\chi_{I} (H_L)$ denotes the spectral projector of $H_L$ onto the interval $I$. Here $C=C_{\delta, K, E}$ is determined by $\delta, K, E$ alone. \par Klein obtained a positive answer to the question for sufficiently short subintervals of $(-\infty,E]$. \begin{theorem}[Ref.~\citenum{Klein-13}] \label{thm:Klein-13} Let $d \in \mathbb{N}$, $E\in \mathbb{R}$, $\delta\in (0,1/2]$ and $V:\mathbb{R}^d \to \mathbb{R}$ be measurable and bounded. There is a constant $M_d>0$ such that if we set \[ \gamma = \frac{1}{2} \delta^{M_d \bigl(1 + (2\lVert V \rVert_\infty + E)^{2/3}\bigr)} , \] then for all energy intervals $I\subset (-\infty, E]$ with length bounded by $2\gamma$, all $L \in 2\mathbb{N}+1$, $L\geq 72 \sqrt{d}$ and all sequences $(x_j)_{j\in\mathbb{Z}^d} \subset \mathbb{R}^d$ with $B(x_j,\delta) \subset\Lambda_1 +j$ for all $j \in \mathbb{Z}^d$ \begin{equation} \chi_{I} (H_L) \, W_L \, \chi_{I} (H_L) \geq \gamma^2\chi_{I} (H_L) . \end{equation} \end{theorem} This does not answer the above posed question question completely due to the restriction $|I| \leq 2\gamma$. However, the result is sufficient for many questions in spectral theory of random Schr\"odinger operators. For a history of the questions discussed here and earlier results we refer to Ref.~\citenum{Rojas-MolinaV-13}. \subsection{Random Schr{\"o}dinger operators}\label{ss:rSo} Let ${\Lambda_L}$ be a cube of side $L\in2\mathbb{N}+1$, $(\Omega, \mathbb{P})$ a probability space, $V_0 \colon {\Lambda_L}\to \mathbb{R}$ a bounded, measurable deterministic potential, $V_\omega \colon {\Lambda_L}\to \mathbb{R}$ a bounded random potential and $H_{\omega,L}= (-\Delta + V_0+V_\omega)_{\Lambda_L}$ a random Schr\"odinger operator on $L^2({\Lambda_L})$ with Dirichlet or periodic boundary conditions. We assume that the random potential is of Delone-Anderson form \begin{equation*} V_\omega(x):= \sum_{j \in{\mathbb{Z}^d}} \ \omega_j u_j(x) . \end{equation*} The random variables $\omega_j, j\in {\mathbb{Z}^d},$ are independent with probability distributions $\mu_j$, such that for some $m>0$ an all $j\in {\mathbb{Z}^d}$ we have $\supp \mu_j \subset [-m, m]$. Fix $0 < \delta_- < \delta_+<\infty$ and $0 < C_- \leq C_+ <\infty$. The sequence of measurable functions $u_j \colon \mathbb{R}^d \to \mathbb{R}$, $j \in {\mathbb{Z}^d}$, is such that \begin{align*} \forall j \in {\mathbb{Z}^d}: \quad C_- \chi_{B(z_j,\delta_-)} \leq u_j \leq C_+ \chi_{B(z_j,\delta_+)}, \ \text{and} \ B(z_j,\delta_-) \subset \Lambda_1 + j . \end{align*} \subsection{Lifting of eigenvalues} \label{ss:lifting} Let $\lambda_k^L(\omega)$ denote the eigenvalues of $H_{\omega,L}$ enumerated in non-decreasing order and counting multiplicities and $\psi_k=\psi_k^L(\omega)$ the normalised eigenvectors corresponding to $\lambda_k^L(\omega)$. While we suppress the dependence of $\psi_k$ on $L$ and $\omega$ in the notation, it should be kept in mind. Then \[ \lambda_k^L(\omega) = \langle \psi_k, H_{\omega,L} \psi_k\rangle = \int_{\Lambda_L} \overline{\psi_k} ( H_{\omega,L} \psi_k ) . \] Define the vector $ e=(e_j)_{j\in{\mathbb{Z}^d}}$ by $e_j=1$ for $j\in{\mathbb{Z}^d}$. Consider the monotone shift of $V_\omega$ \[ V_{\omega+ {t} \cdot e} = \sum_{j \in{\mathbb{Z}^d}} (\omega_j+ {t} ) u_j \] and set $Q=Q_L= \Lambda_L \cap {\mathbb{Z}^d}$. By first order perturbation theory we have \[ \frac{\rm d}{{\rm d}{\tau}} \lambda_k^L(\omega+ {\tau} \cdot e) |_{\tau=t} = \langle \psi_k, \sum_{k \in Q} u_j \, \psi_k \rangle. \] Note that the right hand side depends on $t$ implicitly through the eigenfunction $\psi_k$. Let us fix some $E_0\in\mathbb{R}$ and restrict our attention only to those eigenvalues satisfying $\lambda_n^L(\omega) \leq E_0$. By Theorem~\ref{thm:RojasVeselic} there exists a constant $C_{\rm sfUC}$ depending on the energy $E_0$, $\delta_-$ and the overall supremum \begin{equation*} \label{eq:Vsupremum} \sup_{|s|\leq m} \ \sup_{|\omega_j|\leq m} \ \sup_{x\in\mathbb{R}^d} \big|V_{0}(x) +V_\omega(x) +s \sum_{j\in Q} u_j \big| \end{equation*} of the potential, such that \begin{equation*} \sum_{k \in Q} \langle \psi_k, u_j \, \psi_k \rangle \geq C_- \sum_{k \in Q}\langle \psi_k, \chi_{B(z_k,\delta_-)}\psi_k \rangle \geq C_-\cdot C_{\rm sfUC} =: \kappa . \end{equation*} Here we used that $\|\psi\|_{L^2 (\Lambda)}=1$. (Note that the quantity $\kappa$ depends a-priori on the model parameters.) Integrating the derivative gives \begin{align} \nonumber \lambda_k^L(\omega+ {t} \cdot e) &= \lambda_k^L(\omega) + \int_0^{t} \frac{\mathrm{d} \lambda_k^L(\omega+ \tau \cdot e) }{\mathrm{d} \tau}|_{\tau=s} \, \mathrm{d} s \\ & \geq \lambda_k^L(\omega) + \int_0^{t} \kappa \, \mathrm{d} s = \lambda_k^L(\omega) + t \kappa . \label{eq:lifting} \end{align} This is the lifting estimate for eigenvalues of random (Schr\"odinger) operators alluded to in \S \ref{ss:motivation}. It should be compared with \eqref{eq:positive_semidefinite_perturbation} there. Indeed, due to the uniform nature of the estimate in Theorem~\ref{thm:RojasVeselic} we have \begin{equation} \label{eq:uniform_kappa} \inf_{ L \in 2\mathbb{N}+1} \ \inf_{\omega \text{ s.t. } \forall \, k : |\omega_j|\leq m} \ \inf_{ |{t}|\leq m} \ \inf_{n \text{ s.t. } \lambda_n^L(\omega)\leq E_0} \kappa >0 . \end{equation} Thus eigenvalues lifting estimate is almost as uniform as \eqref{eq:positive_definite_perturbation}. A parameter, with respect to which the lifting estimate is \emph{not} uniform is the cut-off energy $E_0$. Indeed, if we add in \eqref{eq:uniform_kappa} an infimum over $E_0>0$ on the left hand side, it becomes zero, unless $\sum_k\chi_{B(z_k,\delta_-)}\geq 1$ almost everywhere on $\mathbb{R}^d$. \subsection{Wegner estimates} Here we present a Wegner estimate. Such estimates play an important role in the proof of localization via the multiscale analysis. The latter is an induction argument over increasing length scales. The Wegner bound is used to prove the induction step. \par Let $ s\colon [0,\infty) \to[0,1]$ be the global modulus of continuity of the family $\{\mu_j\}_{j\in {\mathbb{Z}^d}}$, that is, \begin{equation*} \label{definition-s-mu-epsilon} s(\epsilon):= \sup_{j \in {\mathbb{Z}^d}} \sup_{a \in \mathbb{R}} \, \mu_j\Big(\Big[a-\frac{\epsilon}{2},a+\frac{\epsilon}{2}\Big]\Big) \end{equation*} The main result of Ref.~\citenum{Rojas-MolinaV-13} on the model described in the last paragraph is a Wegner estimate which is valid for all compact energy intervals. \begin{theorem}[Ref.~\citenum{Rojas-MolinaV-13}] \label{t:Wegner} Let $H_{\omega,L}$ be a random Schr\"odinger operator as in \S \ref{ss:rSo}. Then for each $E_0\in \mathbb{R}$ there exists a constant $C_W$, such that for all $E\le E_0$, $\epsilon \le 1/3$, and all $L\in 2\mathbb{N}+1$ we have \begin{equation*} \label{eq:WE} \mathbb{E}\{{\mathop{\mathrm{Tr} \,}} [ \chi_{[E-\epsilon,E+\epsilon]}(H_{\omega, L}) ]\} \le C_W \ s(\epsilon) \, \lvert \ln \, \epsilon \rvert^d \ \lvert \Lambda_L \rvert . \end{equation*} \end{theorem} The Wegner constant $C_W$ depends only on $E_0$, $\|V_0\|_\infty$, $m$, $C_-$, $C_+$, $\delta_-$, and $\delta_+$. Klein\cite{Klein-13} obtains an improvement over this result based on his above quoted Theorem~\ref{thm:Klein-13}. There are many earlier, related Wegner estimates. For an overview we refer to Ref.~\citenum{Rojas-MolinaV-13}. \subsection{Comparison of local $L^2$-norms} An important step in the proof of Theorem~\ref{thm:RojasVeselic} is the following result which compares $L^2$-norms of the restrictions of a PDE-solution to two distinct subsets. In our applications the solution will be an eigenfunction of the Schr\"odinger operator. Various estimates of this type have been given in Refs.~\citenum{GerminetK-13}, \citenum{BourgainK-13} and \citenum{Rojas-MolinaV-13}. We quote here the version from the last mentioned paper. \begin{theorem} \label{thm:quantitative-UCP} Let $K, R, \beta\in [0, \infty), \delta \in (0,1]$. There exists a constant $C_{\rm qUC}=C_{\rm qUC}(d,K, R,\delta, \beta) >0$ such that, for any $G\subset \mathbb{R}^d$ open, any $\Theta\subset G$ measurable, satisfying the geometric conditions \[ \operatorname{diam} \Theta + \operatorname{dist} (0 , \Theta) \leq 2R \leq 2 \operatorname{dist} (0 , \Theta), \quad \delta < 4R, \quad B(0, 14R ) \subset G, \] and any measurable $V\colon G \to [-K,K]$ and real-valued $\psi\in W^{2,2}(G)$ satisfying the differential inequality \begin{equation*} \label{eq:subsolution} \lvert \Delta \psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.on } G \quad \text{ as well as } \quad \int_{G} \lvert \psi \rvert^2 \leq \beta \int_{\Theta} \lvert \psi \rvert^2 , \end{equation*} we have \begin{equation} \label{eq:aim} \int_{B(0,\delta)} \lvert \psi \rvert^2 \geq C_{\rm qUC} \int_{\Theta} \lvert \psi\rvert^2 . \end{equation} \end{theorem} \begin{figure} \caption{Assumptions in Theorem~\ref{thm:quantitative-UCP} on the geometric constellation of $G$, $\Theta$, and $B(0,\delta)$} \end{figure} \section{Equidistribution property eigenfunctions of second order elliptic operators}\label{sec:elliptic} \subsection{Notation} Let $\mathcal{L}$ be the second order partial differential operator \[ \mathcal{L} u = -\sum_{i,j=1}^d \partial_i \left( a^{ij} \partial_j u \right) \] acting on functions $u$ on $\mathbb{R}^d$. Here $\partial_i$ denotes the $i$th weak derivative. Moreover, we introduce the following assumption on the coefficient functions $a^{ij}$. \begin{assumption}\label{ass:elliptic+} Let $r,\vartheta_1 , \vartheta_2 > 0$. The operator $\mathcal{L} $ satisfies $A(r,\vartheta_1 , \vartheta_2)$, if and only if $a^{ij} = a^{ji}$ for all $i,j \in \{1,\ldots , d\}$ and for almost all $x,y \in B(0,r)$ and all $\xi \in \mathbb{R}^d$ we have \begin{equation*} \label{eq:elliptic} \vartheta_1^{-1} \lvert \xi \rvert^2 \leq \sum_{i,j=1}^d a^{ij} (x) \xi_i \xi_j \leq \vartheta_1 \lvert \xi \rvert^2 \quad\text{and}\quad \sum_{i,j=1}^d \lvert a^{ij} (x) - a^{ij} (y) \rvert \leq \vartheta_2 \lvert x-y \rvert . \end{equation*} \end{assumption} \subsection{A quantitative unique continuation principle} We first present an extension of the quantitative continuation principle, formulated for Schr\"odinger operators in Theorem~\ref{thm:quantitative-UCP}, to elliptic operators with variable coefficients. \begin{theorem}[Ref.~\citenum{BorisovTV}] \label{thm:qUC-elliptic} Let $R\in (0,\infty)$, $K_V, \beta \in [0,\infty)$ and $\delta \in (0, 4 R]$. There is an $\epsilon> 0$, such that if $ A(14R, 1+\epsilon, \epsilon)$ holds then there is a constant $C_{\rm qUC} > 0$, such that for any open $G\subset \mathbb{R}^d$ containing the origin and $\Theta \subset G$ measurable satisfying \[ \operatorname{diam} \Theta + \dist (0 , \Theta) \leq 2R \leq 2 \dist (0 , \Theta) \quad \text{and} \quad B(0,14R) \subset G, \] any measurable $V : G \to [-K_V , K_V]$ and real-valued $\psi \in W^{2,2} (G)$ satisfying the differential inequality \begin{equation*} \label{eq:psi} \lvert \mathcal{L} \psi \rvert \leq \lvert V\psi \rvert \quad \text{a.e.\ on $G$} \quad \text{as well as} \quad \frac{\lVert \psi \rVert_G^2}{\lVert \psi \rVert_\Theta^2} \leq \beta , \end{equation*} we have \begin{equation} \lVert \psi \rVert_{B(x,\delta)}^2 \geq C_{\rm qUC} \lVert \psi \rVert_{\Theta}^2 . \end{equation} \end{theorem} \subsection{Scale-free unique continuation principle} We move on to discuss the equidistribution property or scale-free unique continuation principle for eigenfunctions. The aim is to formulate an analog of Theorem~\ref{thm:RojasVeselic} for variable coefficient elliptic operators. As presented below, for the moment we have solved only the situation where the second order term is sufficiently close to the Laplacian. \par As before, we denote by $\Lambda_L$ a box of side $L\in \mathbb{N}$. By $V$ we indicate a bounded measurable potential on $\mathbb{R}^d$ taking values in $[-K_V,K_V]$, where $K_V$ is a positive constant. We restrict the operator $\mathcal{L} $ on $\Lambda_L(0)$ and add either periodic or Dirichlet boundary conditions. In the former case we denote such an operator by $\mathcal{L} _{L,0}$, and its domain $\mathcal{D}(\mathcal{L} _{L,0})$ is the subspace of $W^{2,2}(\Lambda_L)$ consisting of functions vanishing on $\partial \Lambda_L$. The notation for the operator with periodic boundary condition is $\mathcal{L} _{L,\mathrm{per}}$ and its domains $\mathcal{D}(\mathcal{L} _{L,\mathrm{per}})$ consists of the functions in $W^{2,2}(\Lambda_L)$ satisfying periodic boundary conditions. \begin{assumption}\label{ass:periodicCoefficients} For each pair $i,j$ the function $a^{ij}\colon \mathbb{R}^d \to \mathbb{R}$ is $\mathbb{Z}^d$-periodic. \par Assume that in the case of operator $\mathcal{L} _{L,0}$ its coefficients $a^{ij}$, $i\not= j$ vanish on the sides of box $\Lambda_L$, while the coefficients $a^{ii}$ satisfy periodic boundary conditions on the sides of box $\Lambda_L$. In the case of operator $\mathcal{L} _{L,\mathrm{per}}$ suppose that all its coefficients satisfy periodic boundary conditions on the sides of box $\Lambda_L$. \end{assumption} \begin{theorem}\label{thm:equidistribution-elliptic} Fix $K_V\in [0,+\infty)$, $\delta\in(0,1]$. Assume $A(\sqrt{d},1+\epsilon,\epsilon)$ with $\epsilon>0$ as in Theorem \ref{thm:qUC-elliptic} . Assume \ref{ass:periodicCoefficients}. \par Then there exists a constant $C_{sfUC}>0$ such that for any $L\in 2\mathbb{N}+1$, any sequence \begin{equation*}\label{d1.1} Z:=\{z_k\}_{k\in\mathbb{Z}^d} \ \text{ in }\ \mathbb{R}^d \quad \text{such that} \ B(z_k,\delta)\subset \Lambda_1(k) \text{ for each } k\in\mathbb{Z}^d, \end{equation*} any measurable $V: \Lambda_L\mapsto [-K_V,K_V]$ and any real-valued $\psi\in\mathcal{D}(\mathcal{L} _{L,0})$, respectively $\psi\in \mathcal{D}(\mathcal{L} _{L,\mathrm{per}})$ satisfying \begin{equation*}\label{d1.2} |\mathcal{L}\psi|\leqslant |V\psi|\quad \text{a.e.}\quad \Lambda_L \end{equation*} we have \begin{equation}\label{d1.3} \int\limits_{S_L} |\psi(x)|^2 dx=\sum\limits_{k\in Q_L} \|\psi\|_{L_2(B(z_k,\delta))}^2\geqslant C_{sfUC} \|\psi\|_{L_2(\Lambda_L)}^2, \end{equation} where $S_L:=S\cap\Lambda_L=\cup_{k\in Q_L} B(z_k,\delta)$, $Q_L=\Lambda_L\cap \mathbb{Z}^d$, and $S:=\cup_{k\in \mathbb{Z}^d} B(z_k,\delta)$. \end{theorem} As a \emph{Corollary} we obtain immediately an eigenvalue lifting estimate analogous to \eqref{eq:lifting}, where $\kappa$ is again uniform w.r.t.\ many parameters, as spelled out in subsection \ref{ss:lifting} explicitly. \par The proof of Theorem~\ref{thm:equidistribution-elliptic} is based on the strategy implemented in Ref.~\citenum{Rojas-MolinaV-13}. First one uses the conditions on the coefficients $a^{ij}$ described in Assumption \ref{ass:periodicCoefficients} to extend $\psi$ as well as the differential expression $\mathcal{L}$ to the whole of $\mathbb{R}^d$ while keeping the $W^{2,2}$-regularity and the differential inequality originally satisfied by $\psi$. Then one uses the comparison Theorem~\ref{thm:qUC-elliptic} for local $L^2$-norms. Note that now the condition concerning the minimal distance to the boundary of $G$ plays no role, since $\psi$ has been extended to the whole of $\mathbb{R}^d$. From this point the combinatorial and geometric arguments of Ref~\citenum{Rojas-MolinaV-13} take over. In fact, one can prove a abstract meta-theorem: Once the comparison of local $L^2$-norms of $\psi$ holds up to the boundary, an equidistribution property for $\psi$ follows. Interestingly, such an argument no longer uses the fact that $\psi$ is a solution of an differential equation or inequality. \section*{Acknowledgments} D.B. was partially supported by RFBR, the grant of the President of Russia for young scientists - doctors of science (MD-183.2014.1), and the fellowship of Dynasty foundation for young mathematicians. \par M.T. and I.V. have been partially supported by the DAAD and the Croatian Ministry of Science, Education and Sports through the PPP-grant ``Scale-uniform controllability of partial differential equations''. M.T. and I.V. have been partially supported by the DFG. \end{document}

\begin{document} \title{Isomorphism of Hilbert modules over stably finite C$^*$-algebras} \author{Nathanial P. Brown} \address{Department of Mathematics, Penn State University, State College, PA, 16802, USA} \email{[email protected]} \author{Alin Ciuperca} \address{Fields Institute, 222 College Street, Toronto, Ontario, Canada, M5T 3J1} \email{[email protected]} \keywords{$C^*$-algebras, Hilbert modules, Cuntz semigroup, compact} \subjclass[2000]{Primary 46L08, Secondary 46L80} \thanks{N.B. was partially supported by DMS-0554870; A.C. was partially supported by Fields Institute.} \begin{abstract} It is shown that if $A$ is a stably finite C$^*$-algebra and $E$ is a countably generated Hilbert $A$-module, then $E$ gives rise to a compact element of the Cuntz semigroup if and only if $E$ is algebraically finitely generated and projective. It follows that if $E$ and $F$ are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and $E$ is algebraically finitely generated and projective, then $E$ and $F$ are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C$^*$-algebra that are not isomorphic. \end{abstract} \maketitle \section{Introduction} In \cite{cowelliottiv} a new equivalence relation -- we'll call it \emph{CEI equivalence} -- on Hilbert modules was introduced. In general CEI equivalence is weaker than isomorphism, but it was shown that if $A$ has stable rank one, then it is the same as isomorphism (\cite[Theorem 3]{cowelliottiv}). Quite naturally, the authors wondered whether their result could be extended to the stably finite case. Unfortunately, it can't. In Section \ref{sec:counterexample}, we give examples of Hilbert modules over a stably finite C$^*$-algebra which are CEI-equivalent, but not isomorphic. On the other hand, we show in Section \ref{sec:main} that CEI equivalence amounts to isomorphism when restricted to ``compact" elements of the Cuntz semigroup, in the stably finite case. \noindent\textbf{Acknowledgments:} We thank George Elliott, Francesc Perera, Leonel Robert, Luis Santiago, Andrew Toms and Wilhelm Winter for valuable conversations on topics related to this work. \section{Definitions and Preliminaries} Throughout this note all C$^*$-algebras are assumed to be separable and all Hilbert modules are assumed to be right modules and countably generated. We will follow standard terminology and notation in the theory of Hilbert modules (see, for example, \cite{lance}). In particular, $\mathcal{K}$ denotes the compact operators on $\ell^2(\mathbb{N})$, while $\mathcal{K}(E)$ will denote the ``compact" operators on a Hilbert module $E$. For the reader's convenience, we recall a few definitions that are scattered throughout \cite{cowelliottiv}. \begin{defn} \label{defn:compactcontain} If $E \subset F$ are Hilbert $A$-modules, we say $E$ is \emph{compactly contained in} $F$ if there exists a self-adjoint $T \in \mathcal{K}(F)$ such that $T|_E = \operatorname{id}_E$. In this situation we write $E \subset \subset F$. \end{defn} Note that $E \subset \subset E$ if and only if $\mathcal{K}(E)$ is unital; it can be shown that this is also equivalent to $E$ being algebraically finitely generated and projective (in the purely algebraic category of right $A$-modules) -- see the proof of \cite[Corollary 5]{cowelliottiv} (this part of the proof did not require the assumption of stable rank one.). \begin{defn} We say a Hilbert $A$-module $E$ is \emph{CEI subequivalent} to another Hilbert $A$-module $F$ if every compactly contained submodule of $E$ is isomorphic to a compactly contained submodule of $F$. We say $E$ and $F$ are \emph{CEI equivalent} if they are CEI subequivalent to each other -- i.e., a third Hilbert $A$-module $X$ is isomorphic to a compactly contained submodule of $E$ if and only if $X$ is isomorphic to a compactly contained submodule of $F$. \end{defn} \begin{defn} We let $Cu(A)$ denote the set of Hilbert $A$-modules, modulo CEI equivalence. The class of a module $E$ in $Cu(A)$ will be denoted $[E]$. \end{defn} It turns out that $Cu(A)$ is an abelian semigroup with $[E] + [F] := [E\oplus F]$. (Note: it isn't even obvious that this is well defined!) Moreover $Cu(A)$ is partially ordered -- $[E] \leq [F] \Longleftrightarrow$ $E$ is CEI subequivalent to $F$ -- and every increasing sequence has a supremum (i.e., least upper bound). See \cite[Theorem 1]{cowelliottiv} for proofs of these facts. \begin{defn} An element $x \in Cu(A)$ is \emph{compact} (in the order-theoretic sense) if for every increasing sequence $\{ x_n \} \subset Cu(A)$ with $x \leq \sup_n x_n$ there exists $n_0 \in \mathbb{N}$ such that $x \leq x_{n_0}$. \end{defn} For a unital C$^*$-algebra $A$, \emph{stable finiteness} means that for every $n \in \mathbb{N}$, $M_n(A)$ contains no infinite projections. In the nonunital case there are competing definitions, but it seems most popular to say $A$ is stably finite if the unitization $\tilde{A}$ is stably finite, so this is the definition we will use. \section{Main Results} \label{sec:main} The proof of our first lemma is essentially contained in the proof of \cite[Corollary 5]{cowelliottiv}. \begin{lem} \label{lem:equality} Assume $E\subset \subset F$ is a compact inclusion of Hilbert $A$-modules. If $E \cong F$ then either $E = F$ or $A\otimes \mathcal{K}$ contains a scaling element (in the sense of \cite{BC}). If $A$ is stably finite, then $A\otimes \mathcal{K}$ cannot contain a scaling element; hence, in this case, $E \cong F$ if and only if $E = F$ \end{lem} \begin{proof} Assume $E$ is properly contained in $F$; we'll show $A\otimes \mathcal{K}$ contains a scaling element. Let $v\colon F \to E$ be an isomorphism and $T \in \mathcal{K}(F)$ be a positive operator such that $T|_E = \operatorname{id}_E$. As observed in \cite{cowelliottiv}, the map $vT$ is adjointable -- i.e.\ defines an element of $\mathcal{L}(F)$ -- and, in fact, is compact. (This assertion is readily checked whenever $T$ is a ``finite-rank" operator). Moreover, a calculation shows that $(vT)^*|_E = Tv^{-1}$. It is also worth noting that $T(vT) = vT$, since $T|_E = \operatorname{id}_E$ and $vT(F) \subset E$. The scaling element we are after is $x = vT$. Indeed, one checks that $x^* x = T^2$; hence, $(x^* x)(xx^*) = T^2(vT)(vT)^* = (vT)(vT)^* = xx^*$. Finally, we must see why $xx^* \neq x^* x$. But if $xx^* = x^* x$, then $T^2 = (vT)(vT)^*$ and thus $T^2(F) \subset vT(F) \subset E$. It follows that $T^2$ is a self-adjoint projection onto $E$ (since $T^2|_E = \operatorname{id}_E$, too), and hence $x = vT$ is a partial isometry whose support and range coincide with $E$. But this is impossible because $T = T^2$ (since $T \geq 0$), so $vT(F) \subsetneqq E$ (since $T(F) = E \subsetneqq F$). We've shown that if $E \subsetneqq F$, then $\mathcal{K}(F)$ contains a scaling element. But Kasparov's stabilization theorem provides us with an inclusion $\mathcal{K}(F) \subset A\otimes \mathcal{K}$, so the proof of the first part is complete. In the case that $A$ is stably finite, it is well known to the experts that $A\otimes \mathcal{K}$ cannot contain a scaling element. Indeed, if it did, then \cite[Corollary 4.4]{BC} implies that $M_n(A)$ contains a scaling element, for some $n \in \mathbb{N}$. But it was shown in \cite{BC} that the unitization $\widetilde{M_n(A)}$ would then have an infinite projection. However, there is a natural embedding $\widetilde{M_n(A)} \subset M_n(\tilde{A})$, which contradicts the assumption of stable finiteness. \end{proof} Note that the canonical Hilbert module $\ell^2(A)$ is isomorphic to lots of (non-compactly contained) proper submodules. \begin{prop} \label{prop} Let $E$ be a Hilbert $A$-module such that $[E]$ is compact in $Cu(A)$. Then either $E \subset \subset E$ or $A\otimes \mathcal{K}$ contains a scaling element. \end{prop} \begin{proof} Let $h \in \mathcal{K}(E)$ be strictly positive. If $0$ is an isolated point in the spectrum $\sigma(h)$, then functional calculus provides a projection $p \in \mathcal{K}(E)$ such that $p = \operatorname{id}_E$; so $E \subset \subset E$, in this case. If $0 \in \sigma(h)$ is not isolated, then, again using functional calculus, we can find $E_1 \subset \subset E_2 \subset \subset E_3 \cdots \subset \subset E$ such that $\cup_i E_i$ is dense in $E$ and $E_i \subsetneqq E_{i+1}$ for all $i \in \mathbb{N}$. Since $[E]$ is compact, there exists $i$ such that $[E_i] = [E]$. Since $E_{i+1} \subset \subset E$, $E_{i+1}$ is isomorphic to a compactly contained submodule of $E_i$ and this isomorphism restricted to $E_i$ maps onto a \emph{proper} submodule of $E_i$ (since $E_i \subsetneqq E_{i+1}$). Thus $E_i$ is isomorphic to a proper compactly contained submodule of itself. Hence, by Lemma \ref{lem:equality}, $A\otimes \mathcal{K}$ contains a scaling element. \end{proof} \begin{cor} Let $A$ be stably finite and $E$ be a Hilbert $A$-module. Then $[E] \in Cu(A)$ is compact if and only if $E \subset \subset E$. In particular, if $[E]$ is compact and $[E] \leq [F]$, then $E$ is isomorphic to a compactly contained submodule of $F$. \end{cor} \begin{proof} The ``only if" direction is immediate from the previous proposition. So assume $E \subset \subset E$ and let $[F_n] \in Cu(A)$ be an increasing sequence such that $[E] \leq [F] := \sup [F_n]$. By definition, $E$ is then isomorphic to a compactly contained submodule $E' \subset \subset F$. In the proof of \cite[Theorem 1]{cowelliottiv} it is shown that if $E' \subset \subset F$ and $[F] = \sup [F_n]$, then there is some $n \in \mathbb{N}$ such that $[E'] \leq [F_n]$. Since $[E] = [E']$, the proof is complete. \end{proof} \begin{cor} \label{cor:isom} Let $A$ be stably finite and $E, F$ be Hilbert $A$-modules. If $[E]= [F] \in Cu(A)$ is compact, then $E \cong F$. In particular, if $[E]= [F]$ and $E$ is algebraically finitely generated and projective, then $[E] \in Cu(A)$ is compact; hence, $E \cong F$. \end{cor} \begin{proof} Assume $[E] = [F]$ is compact. Then $E \subset \subset E$ and $F \subset \subset F$, by the previous corollary. Hence there exist isomorphisms $v\colon F \to F' \subset \subset E$ and $u\colon E \to E' \subset \subset F$. It follows that $F \cong u(v(F)) \subset \subset F$, which, by Lemma \ref{lem:equality}, implies that $u(v(F)) = F$. Hence $u$ is surjective, as desired. As mentioned after Definition \ref{defn:compactcontain}, if $E$ is algebraically finitely generated and projective, then $E \subset \subset E$, which implies $[E]$ is compact (as we've seen). \end{proof} In the appendix of \cite{cowelliottiv} it is shown that $Cu(A)$ is isomorphic to the classical Cuntz semigroup $W(A\otimes \mathcal{K})$. When $A$ is stable, the isomorphism $W(A) \to Cu(A)$ is very easy to describe: the Cuntz class of $a \in A_+$ is sent to $H_a := \overline{aA}$ (with its canonical Hilbert $A$-module structure). \begin{thm} \label{thm:main} Let $A$ be a stable, finite C$^*$-algebra, $a \in A_+$ and $H_a = \overline{aA}$. The following are equivalent: \begin{enumerate} \item $H_a$ is algebraically finitely generated and projective; \item $[H_a] \in Cu(A)$ is compact; \item $\sigma(a) \subset \{0\} \cup [\varepsilon, \infty)$ for some $\varepsilon > 0$; \item $\langle a \rangle = \langle p \rangle \in W(A)$ for some projection $p\in A$. \end{enumerate} \end{thm} \begin{proof} The implication $(1) \Longrightarrow (2)$ was explained above. $(2) \Longrightarrow (3)$: Let $a_\varepsilon = (a-\varepsilon)_+$. Then $H_{a_\varepsilon} \subset \subset H_a$ and $\cup_\varepsilon H_{a_\varepsilon}$ is dense in $H_a$. Since $[H_a] \in Cu(A)$ is compact, there exists $\varepsilon > 0$ such that $[H_a] = [H_{a_\varepsilon}]$. Corollary \ref{cor:isom} implies that $H_a \cong H_{a_\varepsilon}$; thus $H_a = H_{a_\varepsilon}$, by Lemma \ref{lem:equality}. It follows that $\sigma(a) \subset \{0\} \cup [\varepsilon, \infty)$, because otherwise functional calculus would provide a nonzero element $b \in C^*(a)$ such that $0 \leq b \leq a$ (so $b \in H_a$) and $a_\varepsilon b = 0$ (so $b \notin H_{a_\varepsilon}$), which would contradict the equality $H_a = H_{a_\varepsilon}$. $(3) \Longrightarrow (4)$ is a routine functional calculus exercise. $(4) \Longrightarrow (1)$: Assume $\langle a \rangle = \langle p \rangle \in W(A)$. Since $pA$ is singly generated and algebraically projective, Corollary \ref{cor:isom} implies $H_a$ is isomorphic to $pA$. \end{proof} The equivalence of $(3)$ and $(4)$ above generalizes Proposition 2.8 in \cite{PT}. \begin{cor} If $A$ is stably finite, then $A\otimes \mathcal{K}$ has no nonzero projections if and only if $Cu(A)$ contains no compact element. \end{cor} \section{A Counterexample} \label{sec:counterexample} Now let us show that if $A$ is stably finite and $E,F$ are Hilbert $A$-modules such that $[E] = [F]$, then it need not be true that $E$ and $F$ are isomorphic. Let $A = C_0(0,1] \otimes \mathcal{O}_3 \otimes \mathcal{K}$, where $\mathcal{O}_3$ is the Cuntz algebra with three generators. Voiculescu's homotopy invariance theorem (cf.\ \cite{dvv}) implies that $A$ is quasidiagonal, hence stably finite. Let $p, q \in \mathcal{O}_3 \otimes \mathcal{K}$ be two nonzero projections which are \emph{not} Murray-von Neumann equivalent. If $x \in C_0(0,1]$ denotes the function $t \mapsto t$, then we define $f_p = x \otimes p$ and $f_q = x \otimes q$ in $A$. Since $A$ is purely infinite in the sense of \cite{KR} and the ideals generated by $f_p$ and $f_q$ coincide, it follows that $[\overline{f_p A}] = [\overline{f_q A}] \in Cu(A)$. We claim that the modules $\overline{f_p A}$ and $\overline{f_q A}$ are not isomorphic. Indeed, if they were isomorphic, then we could find $v \in A$ such that $v^* v = f_p$ and $\overline{vv^*A} = \overline{f_q A}$. (See \cite[Lemma 3.4.2]{ciuperca}; if $T\colon \overline{f_p A}\to\overline{f_q A}$ is an isomorphism, then $v = T(f_p^{1/2})$ has the asserted properties.) Letting $\pi\colon A \to \mathcal{O}_3 \otimes \mathcal{K}$ be the quotient map corresponding to evaluation at $1 \in (0,1]$, it follows that $\pi(v)^* \pi(v) = p$ and $\overline{\pi(v) \pi(v)^* (\mathcal{O}_3 \otimes \mathcal{K})} = \overline{q (\mathcal{O}_3 \otimes \mathcal{K})}$. Since $\pi(v) \pi(v)^*$ is a projection whose associated hereditary subalgebra agrees with the hereditary subalgebra generated by $q$, it follows that $\pi(v) \pi(v)^* = q$ (since both projections are units for the same algebra). This contradicts the assumption that $p$ and $q$ are not Murray-von Neumann equivalent, so $\overline{f_p A}$ and $\overline{f_q A}$ cannot be isomorphic. \section{Questions and Related Results} If the following question has an affirmative answer, then the proof of \cite[Corollary 5]{cowelliottiv} would show that $A$ has real rank zero if and only if the compacts are ``dense" in $Cu(A)$. \begin{question} Can Corollary \ref{cor:isom} be extended to the ``closure" of the compact elements? That is, if $A$ is stably finite and $E$ and $F$ are Hilbert A-modules such that $[E]=[F] = \sup [C_n]$ for an increasing sequence of compact elements $[C_n]$, does it follow that $E\cong F$? \end{question} The next question was raised in \cite{cowelliottiv}, but we repeat it because the modules in Section \ref{sec:counterexample} are not counterexamples -- they mutually embed into each other. (To prove this, use the fact that $p$ is Murray-von Neumann equivalent to a subprojection of $q$, and vice versa.) \begin{question} Are there two Hilbert modules $E$ and $F$ such that $[E] = [F]$, but $F$ is not isomorphic to a submodule of $E$? \end{question} \begin{question} If $x \in Cu(A)$ is compact, is there a projection $p \in A\otimes \mathcal{K}$ such that $x = \langle p \rangle$? \end{question} Of course, in the stably finite case the results of Section \ref{sec:main} tell us that much more is true, but for general C$^*$-algebras we don't know the answer to this question. However, we can give an affirmative answer in some interesting cases, as demonstrated below. First, a definition. \begin{defn} An element $x \in Cu(A)$ will be called \emph{infinite} if $x+y=x$ for some non-zero $y\in Cu(A)$. Otherwise, $x$ will be called \emph{finite}. \end{defn} Note that $[\ell^2(A)] \in Cu(A)$ is always infinite. \begin{lem} \label{lem:unique} If $A$ is simple, then $[\ell^2(A)] \in Cu(A)$ is the unique infinite element. \end{lem} \begin{proof} Assume $[E] + [F] = [E]$ for some nonzero Hilbert $A$-module $F$. Adding $[F]$ to both sides, we see that $[E] + 2[F] = [E]$; repeating this, we have that $[E] + k[F] = [E]$ for all $k \in \mathbb{N}$. By uniqueness of suprema, it follows that $[E] + [\ell^2(F)] = [E]$ (cf.\ \cite[Theorem 1]{cowelliottiv}). Since $A$ is simple, $F$ is necessarily full and hence $\ell^2(F) \cong \ell^2(A)$ (\cite[Proposition 7.4]{lance}). Thus $$[E] = [E] + [\ell^2(F)] = [E \oplus \ell^2(A)] = [\ell^2(A)],$$ by Kasparov's stabilization theorem. \end{proof} In the proof of the following lemma, we use the operator inequality $$xbx^*+ y^*by\geq xby + y^*bx^*,$$ for any $b$ in $A^+$, and $x, y\in A$. (Which follows from the fact that $(x-y^*)b(x-y^*)^*\geq0$.) \begin{lem}\label{algsimple} Let $A$ be a stable algebraically simple C*-algebra. \begin{enumerate} \item For any non-zero $x\in Cu(A)$ there exists $n\in \mathbb{N}$ such that $nx=[A]$. \item There exists a projection $q\in A$ such that $[A]=[qA]$. In particular, $[A]$ is a compact element of the Cuntz semigroup $Cu(A)$. \end{enumerate} \end{lem} \begin{proof} It will be convenient to work in the original positive-element picture of the Cuntz semigroup. Our notation is by now standard (cf.\ \cite{PT}). Proof of (1): Let $x=[\overline{bA}]$ for some $0\neq b\in A^+$ and let $a\in A$ be a strictly positive element. (Stability implies that every right Hilbert $A$-module is isomorphic to a closed right ideal of $A$.) Since $A$ is algebraically simple, one can find $x_1,\ldots, x_n, y_1, \ldots, y_n \in A$ such that $a=\sum_{i=1}^k x_iby_i$. Thus, \begin{align*} a\sim 2a=a+a^* &=\sum_{i=1}^k (x_iby_i + y_i^*bx_i^*)\\ & \leq \sum_{i=1}^k (x_ibx_i^*+y_i^*by_i)\\ & \lesssim x_1bx_1^*\oplus y_1^*by_1 \oplus \cdots x_kbx_k^*\oplus y_k^*by_k\\ & \lesssim b\oplus b\oplus \cdots \oplus b, \end{align*} where the last sum has $n=2k$ summands. Since $A$ is stable, one can embed the Cuntz algebra $O_n$ in the multiplier algebra $M(A)$. This gives us isometries $s_1,\cdots, s_n\in M(A)$ with orthogonal ranges. Set $b_i'=s_ibs_i^*$ and note that $b_i'\sim b$ and $b_i'\perp b_j'$. Moreover, $a\lesssim b_1'+\cdots +b_n'\lesssim a$ (since $a$ is strictly positive, it Cuntz-dominates any element of $A$). Therefore, $\langle a \rangle = n\langle b\rangle = nx$, or equivalently, $[A] = nx$. Proof of (2): Since $A$ is stable and algebraically simple, \cite[Theorem 3.1]{BC} implies $A$ has a non-zero projection $p$. As above, we can find orthogonal projections $p_1,\ldots, p_n \in A$ such that $p_i\sim p$ and $\langle p_1+\cdots +p_n \rangle = n\langle p \rangle = [A]$. Defining $q = p_1+\cdots +p_n$, we are done. \end{proof} We'll also need a consequence of the work in Section \ref{sec:main}. \begin{prop} \label{stablyfinite} If $A$ is stable, $\langle a \rangle \in W(A) = Cu(A)$ is compact and $0 \in \sigma(a)$ is not an isolated point, then $A$ contains a scaling element and $\langle a \rangle$ is infinite. \end{prop} \begin{proof} Assume $A$ contains no scaling element. Since $\langle a \rangle$ is compact, Proposition \ref{prop} implies that $H_a \subset \subset H_a$. As in the proof of $(2) \Longrightarrow (3)$ in Theorem \ref{thm:main}, there exists $\varepsilon > 0$ such that $[H_a] = [H_{a_\varepsilon}]$ and hence $H_a$ is isomorphic to a compactly contained submodule $E$ of $H_{a_\varepsilon}$. Lemma \ref{lem:equality} implies $E = H_a$, so $H_{a_\varepsilon} = H_a$ too. As we've seen, this implies $\sigma(a) \subset \{0\} \cup [\varepsilon, \infty)$, contradicting our hypothesis; hence, $A$ contains a scaling element. To prove the second assertion, choose $\varepsilon > 0$ such that $[H_a] = [H_{a_\varepsilon}]$. Since $0 \in \sigma(a)$ is not isolated, we can find a nonzero positive function $f \in C_0(0,\|a\|]$ such that $f(t) = 0$ for all $t \geq \varepsilon$. Thus $f(a) + (a-\varepsilon)_+ \precsim a$ and $f(a) (a-\varepsilon)_+ = 0$. It follows that $$[H_{f(a)}] + [H_a] = [H_{f(a)}] + [H_{a_\varepsilon}] \leq [H_a]$$ and thus $[H_a]$ is infinite. \end{proof} \begin{thm} Let $x \in Cu(A)$ be compact. \begin{enumerate} \item If $A$ is simple, then there exists a projection $p \in A\otimes \mathcal{K}$ such that $x = \langle p \rangle$. \item If $x$ is finite, then there exists a projection $p \in A\otimes \mathcal{K}$ such that $x = \langle p \rangle$. \end{enumerate} \end{thm} \begin{proof} In both cases we may assume $A$ is stable. Proof of (1): Fix a nonzero positive element $a\in A$ such that $x = [H_a]$. If $0 \in \sigma(a)$ is an isolated point, then functional calculus provides us with a Cuntz equivalent projection, and we're done. Otherwise Proposition \ref{stablyfinite} tells us that $x$ is infinite and $A$ contains a scaling element. By simplicity and Lemma \ref{lem:unique}, we have that $x = [\ell^2(A)] = [A]$ (by stability). Moreover, the existence of a scaling element ensures that $A$ is algebraically simple (see \cite[Theorem 1.2]{BC}). Hence part (2) of Lemma \ref{algsimple} provides the desired projection. Proof of (2): Choose $a \in A_+$ such that $x = \langle a \rangle$. Since $x$ is finite, Proposition \ref{stablyfinite} implies $0 \in \sigma(a)$ is an isolated point, so we're done. \end{proof} \begin{rem} It is possible to improve part (2) of the theorem above. Namely, it is shown in \cite{ciuperca} that if $x \in Cu(A)$ is compact and there is no \emph{compact} element $y \in Cu(A)$ such that $x = x + y$, then there exists a projection $p \in A\otimes \mathcal{K}$ such that $x = \langle p \rangle$. \end{rem} \end{document}

\begin{document} \title{Differential Operators, Gauges, and Mixed Hodge Modules} \author{Christopher Dodd} \begin{abstract} The purpose of this paper is to develop a new theory of gauges in mixed characteristic. Namely, let $k$ be a perfect field of characteristic $p>0$ and $W(k)$ the $p$-typical Witt vectors. Making use of Berthelot's arithmetic differential operators, we define for a smooth formal scheme $\mathfrak{X}$ over $W(k)$, a new sheaf of algebras $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ which can be considered a higher dimensional analogue of the (commutative) Dieudonne ring. Modules over this sheaf of algebras can be considered the analogue (over $\mathfrak{X}$) of the gauges of Ekedahl and Fontain-Jannsen. We show that modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ admit all of the usual $\mathcal{D}$-module operations, and we prove a robust generalization of Mazur's theorem in this context. Finally, we show that an integral form of a mixed Hodge module of geometric origin admits, after a suitable $p$-adic completion, the structure of a module over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. This allows us to prove a version of Mazur's theorem for the intersection cohomology and the ordinary cohomology of an arbitrary quasiprojective variety defined over a number field. \end{abstract} \maketitle \tableofcontents{} \section{Introduction} In this work, we will develop the technology needed to state and prove \emph{Mazur's theorem for a mixed Hodge module}. In order to say what this means, we begin by recalling the original Mazur's theorem. Fix a perfect field $k$ of positive characteristic; let $W(k)$ denote the $p$-typical Witt vectors. Let $X$ be a smooth proper scheme over $k$. To $X$ is attached its crystalline cohomology groups $\mathbb{H}_{crys}^{i}(X)$, which are finite type $W(k)$-modules; the complex $\mathbb{H}_{crys}^{\cdot}(X)$ has the property that $\mathbb{H}_{crys}^{\cdot}(X)\otimes_{W(k)}^{L}k\tilde{\to}\mathbb{H}_{dR}^{\cdot}(X)$ (the de Rham cohomology of $X$ over $k$). Furthermore, if $\mathfrak{X}$ is a smooth, proper formal scheme over $W(k)$, whose special fibre is $X$, then there is a canonical isomorphism \[ \mathbb{H}_{crys}^{i}(X)\tilde{\to}\mathbb{H}_{dR}^{i}(\mathfrak{X}) \] for any $i$. In particular, the action of the Frobenius endomorphism on $X$ endows $\mathbb{H}_{dR}^{i}(\mathfrak{X})$ with an endomorphism $\Phi$ which is semilinear over the Witt-vector Frobenius $F$. It is known that $\Phi$ becomes an automorphism after inverting $p$; the ``shape'' of the map $\Phi$ is an interesting invariant of the pair $(\mathbb{H}_{crys}^{i}(X),\Phi)$. To make this precise, one attaches, to any $r\in\mathbb{Z}$, the submodule $(\mathbb{H}_{crys}^{i}(X))^{r}=\{m\in\mathbb{H}_{crys}^{i}(X)|\Phi(m)\in p^{r}\mathbb{H}_{crys}^{i}(X)\}$ (the equality takes place in $\mathbb{H}_{crys}^{i}(X)[p^{-1}]$). Thus we have a decreasing, exhaustive filtration, whose terms measure how far $\Phi$ is from being an isomorphism. On the other hand, the de Rham cohomology of $X$ comes with another filtration, the Hodge filtration, which comes from the Hodge to de Rham spectral sequence $E_{1}^{r,s}=\mathbb{H}^{s}(X,\Omega_{X}^{r})\Rightarrow\mathbb{H}_{dR}^{r+s}(X)$. Then we have the following remarkable \begin{thm} \label{thm:(Mazur)}(Mazur) Suppose that, for each $i$, the group $\mathbb{H}_{crys}^{i}(X)$ is $p$-torsion-free, and that the Hodge to de Rham spectral sequence of $X$ degenerates at $E_{1}$. Then the image of the filtration $(\mathbb{H}_{crys}^{i}(X))^{r}$ in $\mathbb{H}_{dR}^{i}(X)$ is the Hodge filtration. \end{thm} This is (the first half of) \cite{key-13}, theorem 3 (in fact, under slightly weaker hypotheses; compare \cite{key-14} corollary 3.3, and \cite{key-10}, theorem 8.26). The theorem also includes a similar description of the conjugate filtration (the filtration coming from second spectral sequence of hypercohomology) on $\mathbb{H}_{dR}^{i}(X)$; we will address this as part of the more general theorem 1.2 below. This result allowed Mazur to prove Katz's conjecture relating the slopes of $\Phi$ to the Hodge numbers of $X$. In the years following \cite{key-13}, it was realized that the theorem can be profitably rephrased in terms of certain additional structures on $\mathbb{H}_{crys}^{i}(X)$. Let $A$ be a commutative ring. Denote by $D(A)$ the commutative ring $A[f,v]/(fv-p)$; put a grading on this ring by placing $A$ in degree $0$, $f$ in degree $1$, and $v$ in degree $0$. Then a\emph{ gauge }(over \emph{$A$}) is a graded module over $D(A)$, ${\displaystyle M=\bigoplus_{i\in\mathbb{Z}}M^{i}}$. Set ${\displaystyle M^{\infty}:=M/(f-1)\tilde{=}\lim_{\to}M^{i}}$, and ${\displaystyle M^{-\infty}:=M/(v-1)\tilde{=}\lim_{\to}M^{-i}}$. One says that $M$ is an $F$-gauge if there is an isomorphism $F^{*}M^{\infty}\tilde{\to}M^{-\infty}$ (c.f. \cite{key-20}, definition 2.1, \cite{key-5}, chapter 1, or section 2.1 below). Then, in the above situation, one associates the $W(k)$- gauge \begin{equation} \mathbb{H}_{\mathcal{G}}^{i}(X):=\bigoplus_{r\in\mathbb{Z}}(\mathbb{H}_{crys}^{i}(X))^{r}\label{eq:Basic-Gauge-defn} \end{equation} where $f:(\mathbb{H}_{crys}^{i}(X))^{r}\to(\mathbb{H}_{crys}^{i}(X))^{r+1}$ acts by multiplication by $p$, and $v:(\mathbb{H}_{crys}^{i}(X))^{r}\to(\mathbb{H}_{crys}^{i}(X))^{r-1}$ acts as the inclusion. One has $\mathbb{H}_{\mathcal{G}}^{i}(X)^{\infty}\tilde{=}\mathbb{H}_{crys}^{i}(X)$, and the isomorphism\linebreak{} $F^{*}(\mathbb{H}_{crys}^{i}(X))^{\infty}\to(\mathbb{H}_{crys}^{i}(X))^{-\infty}$ comes from the action of $\Phi$. Remarkably, it turns out that there is a reasonable definition of $\mathbb{H}_{\mathcal{G}}^{i}(X)$ for any $X$, even without the assumption that each group $\mathbb{H}_{crys}^{i}(X)$ is $p$-torsion-free, or that the Hodge to de Rham spectral sequence degenerates at $E_{1}$. To state the result, note that for any gauge $M$ (over any $A$), $M^{-\infty}$ carries a decreasing filtration defined by $F^{i}(M^{-\infty})=\text{image}(M^{i}\to M^{\infty})$. Passing to derived categories, we obtain a functor $D(\mathcal{G}(D(A)))\to D((A,F)-\text{mod})$ (here $\mathcal{G}(D(A))$ is the category of gauges, and $D((A,F)-\text{mod})$ is the filtered derived category of $A$); we will denote this functor $M^{\cdot}\to M^{\cdot,-\infty}$. The analogous construction can be carried out for $+\infty$ as well using the increasing filtration $C^{i}(M^{\infty})=\text{image}(M^{i}\to M^{\infty})$. In particular, if $M^{\cdot}\in D(\mathcal{G}(D(A)))$, then each $H^{i}(M^{\cdot,-\infty})$ and each $H^{i}(M^{\cdot,\infty})$ is a filtered $A$-module. \begin{thm} \label{thm:=00005BFJ=00005D} For any smooth $X$ over $k$, there is a functorially attached complex of $W(k)$-gauges, $\mathbb{H}_{\mathcal{G}}^{\cdot}(X)$, such that $\mathbb{H}_{\mathcal{G}}^{i}(X)^{\infty}\tilde{=}\mathbb{H}_{crys}^{i}(X)$ for all $i$. Further, there is an $F$-semilinear isomorphism $H^{i}((\mathbb{H}_{\mathcal{G}}^{\cdot}(X)\otimes_{W(k)}^{L}k))^{-\infty}\tilde{\to}(\mathbb{H}_{dR}^{i}(X),F)$ and a linear isomorphism $H^{i}((\mathbb{H}_{\mathcal{G}}^{\cdot}(X)\otimes_{W(k)}^{L}k))^{\infty}\tilde{\to}(\mathbb{H}_{dR}^{i}(X),C)$, where $F$ and $C$ denote the Hodge and conjugate filtrations, respectively. When $\mathbb{H}_{crys}^{i}(X)$ is torsion-free for all $i$ and the Hodge to de Rham spectral sequence degenerates at $E_{1}$, then this functor agrees with the gauge constructed above in \eqref{Basic-Gauge-defn}. \end{thm} As far as I am aware, the first proof of this theorem appears in Ekedahl's book \cite{key-20}. This is also the first place that the above notion of gauge is defined; Ekedahl points out that Fontaine discovered the notion independantly. Ekedahl's proof relies on deep properties of the de Rham-Witt complex and on the results of the paper \cite{key-37}; in that paper, it is shown that there is attached to $X$ a complex inside another category $D^{b}(\mathcal{R}-\text{mod})$ where $\mathcal{R}$ is the so-called Raynaud ring; then, in definition 2.3.1 of \cite{key-20} Ekehahl constructs a functor from $D^{b}(\mathcal{R}-\text{mod})$ to the derived category $D^{b}(\mathcal{G}(D(A)))$; the composition of these two functors yeilds the construction of the theorem. Another, rather different proof of the theorem is given in \cite{key-5}, section 7. Now let us turn to $\mathcal{D}$-modules and Hodge modules. From at least the time of Laumon's work (\cite{key-19}), it has been understood that the filtered complex $\mathbb{H}_{dR}^{\cdot}(X)$ (with its Hodge filtration) can be understood as an object of filtered $\mathcal{D}$-module theory. To explain this, let $\mathcal{D}_{X}^{(0)}$ denote the level zero PD-differential operators on $X$. Then $\mathcal{D}_{X}^{(0)}$ acts on $\mathcal{O}_{X}$, and we have a canonical isomorphism \[ \int_{\varphi}\mathcal{O}_{X}[d_{X}]\tilde{\to}\mathbb{H}_{dR}^{\cdot}(X) \] where $\varphi$ denotes the map $X\to\text{Spec}(k)$, $d_{X}=\text{dim}(X)$, and ${\displaystyle \int_{\varphi}}$ is the push-forward for $\mathcal{D}_{X}^{(0)}$-modules. In addition, $\mathcal{D}_{X}^{(0)}$ comes equipped with a natural \emph{increasing }filtration, the symbol filtration. Laumon's work\footnote{Strictly speaking, Laumon works in characteristic zero. But the same formalism works for $\mathcal{D}_{X}^{(0)}$ in positive characteristic; I'll address this below in the paper} upgrades the push-forward functor to a functor from filtered $\mathcal{D}_{X}^{(0)}$-modules to filtered $k$-vector spaces; and we have that\emph{ \[ \int_{\varphi}\mathcal{O}_{X}[d_{X}]\tilde{\to}(\mathbb{H}_{dR}^{\cdot}(X),F') \] }where $F'$ is the Hodge filtration, suitably re-indexed to make it an increasing filtration. Furthermore, Laumon works in the relative setting; i.e., he constructs a filtered push-forward for any morphism $\varphi:X\to Y$ of smooth varieties. This leads to the question, of weather the construction of \thmref{=00005BFJ=00005D} can be understood in terms of some sort of upgrade of filtered $\mathcal{D}$-modules to a category of graded modules. The main body of this work shows that, at least when the schemes in question lift to smooth formal schemes over $W(k)$, the answer is yes\footnote{In fact, the answer is always yes. But we will adress the non-liftable case in future work}. To state the first result, recall that, in addition to the symbol filtration, the algebra $\mathcal{D}_{X}^{(0)}$ carries a decreasing filtration by two sided ideals, the conugate filtration, denoted $\{C^{i}(\mathcal{D}_{X}^{(0)})\}_{i\in\mathbb{Z}}$ (it was first defined in \cite{key-11}, section 3.4, c.f. also \defref{Hodge-and-Con} below). \begin{thm} \label{thm:D01}Let $\mathfrak{X}$ be a smooth formal scheme over $W(k)$. Then there is a locally noetherian sheaf of algebras $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ with the following properties: 1) ${\displaystyle \widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}=\bigoplus_{i}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),i}}$ is a graded $D(W(k))$-algebra, and $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/(v-1)\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$, while the sheaf $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/(f-1)$ has $p$-adic completion equal to $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. 2) Let $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/p:=\mathcal{D}_{X}^{(0,1)}$, a graded sheaf of $k$-algebras on $X$. The filtration $\text{im}(\mathcal{D}_{X}^{(0,1),i}\to\mathcal{D}_{X}^{(0)}\tilde{=}\mathcal{D}_{X}^{(0,1)}/(v-1))$ agrees with the conugate filtration on $\mathcal{D}_{X}^{(0)}$. 3) We have $\mathcal{D}_{X}^{(1)}=\mathcal{D}_{X}^{(0,1)}/(f-1)$. Consider the filtration $F^{i}(\mathcal{D}_{X}^{(1)})=\text{im}(\mathcal{D}_{X}^{(0,1),i}\to\mathcal{D}_{X}^{(0)}\tilde{=}\mathcal{D}_{X}^{(0,1)}/(f-1))$. Then filtered modules over $(\mathcal{D}_{X}^{(1)},F^{\cdot})$ are equivalent to filtered modules over $(\mathcal{D}_{X}^{(0)},F^{\cdot})$ (the symbol filtration on $\mathcal{D}_{X}^{(0)}$). \end{thm} This sheaf of algebras is constructed in \secref{The-Algebra} below; part $2)$ of the theorem is proved in \remref{Description-of-conjugate}, and part $3)$ is \thmref{Filtered-Frobenius}. This theorem shows that a graded module over $\mathcal{D}_{X}^{(0,1)}$ is a simultanious generalization of a conugate-filtered and a Hodge-filtered $\mathcal{D}_{X}^{(0)}$-module. The algebra $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ admits analogues of all of the usual $\mathcal{D}$-module operations; namely, tensor product, duality, left-right interchange, and well as push-forward and pull-back over arbitrary morphisms (between smooth formal schemes). By construction the sheaf $D(\mathcal{O}_{\mathfrak{X}})=\mathcal{O}_{\mathfrak{X}}[f,v]/(fv-p)$ carrries an action of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. Let $D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ denotes the derived category of graded $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-modules; then we have \begin{thm} For any morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ of smooth formal schemes we denote the pushforward by ${\displaystyle \int_{\varphi}:D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))\to D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))}$. If $\varphi$ is proper, then the pushforward takes $D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ to $D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. We have ${\displaystyle (\int_{\varphi}\mathcal{M})^{-\infty}\tilde{=}(\int_{\varphi}\mathcal{M}^{-\infty})}$, where the pushforward on the right is in the category of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules. In particular, if $\mathfrak{Y}$ is $\text{Specf}(W(k))$, then ${\displaystyle {\displaystyle \int_{\varphi}}D(\mathcal{O}_{\mathfrak{X}})}$ is a bounded complex of finite type gauges, and we have isomorphisms \[ ({\displaystyle \int_{\varphi}}D(\mathcal{O}_{\mathfrak{X}}))^{-\infty}[d_{X}]\tilde{=}\mathbb{H}_{dR}^{\cdot}(\mathfrak{X}) \] and \[ ({\displaystyle \int_{\varphi}}D(\mathcal{O}_{\mathfrak{X}}))^{\infty}[d_{X}]\tilde{=}F^{*}\mathbb{H}_{dR}^{\cdot}(\mathfrak{X}) \] where $F$ is the Witt-vector Frobenius. After passing to $k$ we obtain isomorphisms in the filtered derived category \[ ({\displaystyle \int_{\varphi}}D(\mathcal{O}_{\mathfrak{X}})\otimes_{W(k)}^{L}k)^{-\infty}[d_{X}]\tilde{=}(\mathbb{H}_{dR}^{\cdot}(X),C') \] (where $C'$ in the conjugate filtration, appropriately re-indexed to make it a decreasing filtration), and \[ ({\displaystyle \int_{\varphi}}D(\mathcal{O}_{\mathfrak{X}})\otimes_{W(k)}^{L}k)^{\infty}[d_{X}]\tilde{=}F^{*}(\mathbb{H}_{dR}^{\cdot}(X),F') \] where where $F'$ is the Hodge filtration, suitably re-indexed to make it an increasing filtration \end{thm} This theorem is proved in \secref{Push-Forward} below. In fact $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ has many more favorable properties which are developed extensively in this paper; including a well-behaved pull-back for arbitrary maps, an internal tensor product which satisfies the projection formula, and a relative duality theory; these are sections five through eight below. Simultaneously, we develop the analogous theory $\mathcal{D}_{X}^{(0,1)}$-modules; here $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/p$; technically, we do a little more than that, and develop the theory of $\mathcal{D}_{X}^{(0,1)}$-modules over smooth varieties which do not have to lift to $W(k)$. The two theories play off each other nicely- we often use reduction mod $p$ and various versions of Nakayama's lemma to reduce statements about $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ to statements about $\mathcal{D}_{X}^{(0,1)}$; on the other hand, there are always local lifts of a smooth variety over $k$, so local questions about $\mathcal{D}_{X}^{(0,1)}$ often reduce to questions about $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. There is also an interesting and rich theory over the truncated Witt vectors $W_{n}(k)$, but, given the length of this paper, we will undertake a detailed study of it in another work. We also have a comparison with the gauge constructed in \thmref{=00005BFJ=00005D}; however, we will defer the proof of this result to a later paper. That is because it seems best to prove it as a consequence of a more general comparison theorem between the category of gauges constructed here and the one constructed in \cite{key-5}; and this general statement is still a work in progress. It also seems that there is a close connection with the recent works of Drinfeld \cite{key-39} and Bhatt-Lurie \cite{key-40} via a kind of Koszul duality formalism; again, the details are a work in progress\footnote{The author has been discussing these topics with Bhargav Bhatt }. Now we discuss Mazur's theorem in the relative context. We begin with the \begin{defn} A module $\mathcal{M}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ is standard if if $\mathcal{M}^{-\infty}$ and $\mathcal{M}^{\infty}$ are $p$-torsion-free, each map $f_{\infty}:\mathcal{M}^{i}\to\mathcal{M}^{\infty}$ is injective; and, finally, there is a $j_{0}\in\mathbb{Z}$ so that \[ f_{\infty}(\mathcal{M}^{i+j_{0}})=\{m\in\mathcal{M}^{\infty}|p^{i}m\in f_{\infty}(\mathcal{M}^{j_{0}})\} \] for all $i\in\mathbb{Z}$. \end{defn} Note that, over $W(k)$, this is a generalization of the construction of the gauge in \eqref{Basic-Gauge-defn}; with the roles of $f$ and $v$ reversed (this is related to the re-indexing of the Hodge and conjugate filtrations; c.f. also \remref{basic-equiv} below). Thus a general version of Mazur's theorem will give conditions on a complex of gauges which ensure that each cohomology group is standard. In order to state such a theorem, we need to note that there is a notion of $F$-gauge in this context, or, to be more precise, a notion of $F^{-1}$-gauge: \begin{defn} (c.f. \defref{Gauge-Defn!}) Let $F^{*}:\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}-\text{mod}\to\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}-\text{mod}$ denote Berthelot's Frobenius pullback (c.f. \thmref{Berthelot-Frob} below for details). Then an $F^{-1}$-gauge over $\mathfrak{X}$ is an object of $\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ equipped with an isomorphism $F^{*}\mathcal{M}^{-\infty}\tilde{\to}\widehat{\mathcal{M}^{\infty}}$ (here $\widehat{?}$ denotes $p$-adic completion). There is also a version for complexes in $D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, namely, an $F^{-1}$-gauge in $D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$ is a complex $\mathcal{M}^{\cdot}$ equipped with an isomorphism $F^{*}\mathcal{M}^{\cdot,-\infty}\tilde{\to}\widehat{\mathcal{M}^{\cdot,\infty}}$ (here $\widehat{?}$ denotes the cohomolocial or derived completion, c.f. \defref{CC} and \propref{Basic-CC-facts} below). \end{defn} We denote by $D_{F^{-1}}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ the category of complexes for which there exists an isomorphism $F^{*}\mathcal{M}^{\cdot,-\infty}\tilde{\to}\widehat{\mathcal{M}^{\cdot,\infty}}$ as above. Then we have the following rather general version of Mazur's theorem: \begin{thm} (c.f. \thmref{F-Mazur}) Let $\mathcal{M}^{\cdot}\in D_{\text{coh},F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. Suppose that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{-\infty}$ is $p$-torsion-free for all $n$, and suppose that $\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free for all $n$. Then $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is standard for all $n$. \end{thm} Using the formalism of filtered $\mathcal{D}$-modules one verifies that the condition that $\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free for all $n$ is a generalization of the degeneration of the Hodge-to-de Rham spectral sequence. Therefore this theorem, along with the previous one, provide a robust generalization of Mazur's theorem, which allows much more general kinds of coefficients. The conditions of the theorem are satisfied in several important cases. Suppose $R$ is a finitely generated $\mathbb{Z}$-algebra, and suppose that $X_{R}$ is a smooth $R$ scheme, and let $\varphi:X_{R}\to Y_{R}$ be a proper map. Suppose that $(\mathcal{M}_{R},F)$ is a filtered coherent $\mathcal{D}_{X_{R}}^{(0)}$-module on $X_{R}$. If the associated complex filtered $\mathcal{D}$-module, $(\mathcal{M}_{\mathbb{C}},F)$ undergirds a mixed Hodge module, then by Saito's theory the Hodge-to-de Rham spectral sequence for ${\displaystyle \int_{\varphi}(\mathcal{M}_{\mathbb{C}},F)}$ degenerates at $E_{1}$. Thus the same is true over $R$, after possibly localizing. Further localization ensures that each ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}(\mathcal{M}_{R},F))}$ is flat over $R$. Now suppose we have a map $R\to W(k)$. Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ denote the formal completion of the base change to $W(k)$. Then the theorem applies if there exist a $p$-torsion-free gauge $\mathcal{N}$ over $\mathfrak{X}$ such that $\mathcal{N}^{-\infty}\tilde{=}\widehat{\mathcal{M}\otimes_{R}W(k)}$ and $F^{*}(\mathcal{M}_{k},F)\tilde{\to}\mathcal{N}^{\infty}/p$. By a direct construction, this happens for $\mathcal{M}=\mathcal{O}_{X}$ as well as $\mathcal{M}=j_{\star}\mathcal{O}_{U}$ and $\mathcal{M}=j_{!}\mathcal{O}_{U}$ (where $U\subset X$ is an open inclusion whose compliment is a normal crossings divisor, and $j_{\star}$ and $j_{!}$ denote the pushforwards in mixed Hodge module theory). Therefore, by the theorem itself, it happens when $(\mathcal{M}_{\mathbb{C}},F)$ is itself a Hodge module ``of geometric origin'' (c.f. \corref{Mazur-for-Hodge-1}). In this paper we give some brief applications of this to the case where $\mathcal{M}_{\mathbb{C}}$ is the local cohomology along some subcheme; but we expect that there are many more. Finally, let's mention that Hodge modules of geometric origin control both the intersection cohomology and singular cohomology of singular varieties over $\mathbb{C}$. So we can obtain \begin{thm} \label{thm:Mazur-for-IC-Intro}Let $X_{R}$ be a (possibly singular) quasiprojective variety over $R$. Then, after possibly localizing $R,$ there is a filtered complex of $R$-modules $I\mathbb{H}^{\cdot}(X_{R})$, whose base change to $\mathbb{C}$ yields $I\mathbb{H}^{\cdot}(X_{\mathbb{C}})$, with its Hodge filtration. Now suppose $R\to W(k)$ for some perfect field $k$. Then for each $i$, there is a standard gauge $\tilde{I\mathbb{H}}^{i}(X)_{W(k)}$ so that \[ \tilde{I\mathbb{H}}^{i}(X)_{W(k)}^{-\infty}\tilde{=}I\mathbb{H}^{\cdot}(X_{R})\otimes_{R}W(k) \] and so that \[ \tilde{I\mathbb{H}}^{i}(X)_{W(k)}^{\infty}\tilde{=}F^{*}(I\mathbb{H}^{\cdot}(X_{R})\otimes_{R}W(k)) \] Under this isomorphism, the Hodge filtration on $\tilde{I\mathbb{H}}^{i}(X)_{W(k)}^{\infty}/p$ agrees with the Frobenius pullback of the image of the Hodge filtration in $I\mathbb{H}^{\cdot}(X_{R})\otimes_{R}k$. The analogous statement holds for the ordinary cohomology of a quasiprojective variety $X_{R}$, with its Hodge filtration; as well as the compactly supported cohomology. \end{thm} This is proved in \corref{Mazur-for-IC} and \corref{Mazur-for-Ordinary}below. As in \cite{key-13} and \cite{key-38}, \cite{key-55} this result implies that the ``Newton polygon'' lies on or above the ``Hodge polygon'' for both the ordinary and the intersection cohomology of quasiprojective varieties, in the circumstances of the above theorem. We note here that the theorem gives an $F$-semilinear action on the groups $I\mathbb{H}^{\cdot}(X_{R})\otimes_{R}W(k)[p^{-1}]$, as well as the ordinary cohomology groups $\mathbb{H}^{\cdot}(X_{R})\otimes_{R}W(k)[p^{-1}]$, and the compactly supported cohomology as well. This action has already been constructed as a consequence of the formalism of rigid cohomology (c.f. \cite{key-80},\cite{key-81}). However, to my knowledge this ``integral'' version of the action has not been considered before. \subsection{Plan of the Paper} The first chapter has two sections. In the first, we quickly review the theory of gauges over $W(k)$, and in particular give the equivalence between $F$-guages and $F^{-1}$-guages in this context. In the second, we give a quick recollection of some generalities on graded modules, before reviewing and extending (to the case of graded modules) the very important technical notion of cohomological completeness (also known as derived completeness). The Nakayama lemma is key here, as the reduction mod $p$ will be one of our main technical tools for proving theorems. The next chapter introduces $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$, as well as its analogue $\mathcal{D}_{X}^{(0,1)}$ over a smooth $k$-variety $X$ (which does not have to lift to a smooth formal scheme), and performs some basic local calculations. In particular, we prove \corref{Local-coords-over-A=00005Bf,v=00005D}, which provides a local description of $\mathcal{D}_{X}^{(0,1)}$ which is analogous to the basic descriptions of differential operators ``in local coordinates'' that one finds in other contexts. In chapter $4$, we study the categories of graded modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ and $\mathcal{D}_{X}^{(0,1)}$, importing and generalizing some key results of \cite{key-5}. We prove the ``abstract'' version of Mazur's theorem (\thmref{Mazur!}) for a complex of gauges. Then we go on to introduce the notion of an $F^{-1}$-gauge over $X$ (and $\mathfrak{X}$), which makes fundamental use of Berthelot's Frobenius descent. We explain in \thmref{Filtered-Frobenius} how this Frobenius descent interacts with the natural filtrations coming from the grading on $\mathcal{D}_{X}^{(0,1)}$. Along the way, we look at the relationship between modules over $\mathcal{D}_{X}^{(0,1)}$ and modules over two important Rees algebras:$\mathcal{R}(\mathcal{D}_{X}^{(0)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, the Rees algebras of $\mathcal{D}_{X}^{(0)}$ with respect to the symbol and conjugate filtrations, respectively. Chapters $5$ through $8$ introduce and study the basic $\mathcal{D}$-module operations in this context: pullback, tensor product, left-right interchange, pushforward, and duality. Much of this is similar to the story for algebraic $\mathcal{D}$-modules (as covered in \cite{key-49}, for instance). For instance, even though $\mathcal{D}_{X}^{(0,1)}$ and $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ do not have finite homological dimension, we show that the pushforward, pullback, and duality functors do have finite homological dimension. As usual, the study of the pushforward (chapter $7$ below) is the most involved, and we spend some time exploring the relationship with the pushforwards for $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, respectively; these admit descriptions in terms of the more standard filtered pushforwards of $\mathcal{D}$-modules. Finally, in the last chapter we put everything together and prove Mazur's theorem for a Hodge module of geometric origin; this uses, essentially, all of the theory built in the previous sections. In addition to the applications explained in the introduction, we give some applications to the theory of the Hodge filtration on the local cohomology of a subcheme of a smooth complex variety. There is one appendix to the paper- in which we prove a technical result useful for constructing the gauge $j_{\star}(D(\mathcal{O}))$, the pushforward of the trivial gauge over a normal crossings divisor. \subsection{Notations and Conventions} Let us introduce some basic notations which are used throughout the paper. For any ring (or sheaf of rings) $\mathcal{R}$, we will denote by $D(\mathcal{R})$ the graded ring in which $\mathcal{R}$ has degree $0$, $f$ has degree $1$, $v$ has degree $-1$, and $fv=p$. The symbol $k$ will always denote a perfect field of positive characteristic, and $W(k)$ the $p$-typical Witt vectors. Letters $X$, $Y$,$Z$ will denote smooth varieties over $k$, while $\mathfrak{X}$,$\mathfrak{Y}$,$\mathfrak{Z}$ will denote smooth formal schemes over $W(k)$. When working with formal schemes, we let $\Omega_{\mathfrak{X}}^{1}$ denote the sheaf of continuous differentials (over $W(k)$), and $\mathcal{T}_{\mathcal{X}}$ denote the continuous $W(k)$-linear derivations; we set $\Omega_{\mathfrak{X}}^{i}=\bigwedge^{i}\Omega_{\mathfrak{X}}^{1}$ and $\mathcal{T}_{\mathfrak{X}}^{i}=\bigwedge^{i}\mathcal{T}_{\mathfrak{X}}$. We denote by $X^{(i)}$ the $i$th Frobenius twist of $X$; i.e., the scheme $X\times_{\text{Spec}(k)}\text{Spec}(k)$, where $k$ map to $k$ via $F^{i}$. Since $k$ is perfect, the natural map $\sigma:X^{(i)}\to X$ is an isomorphism. On the other hand, the relative Frobenius $X\to X^{(i)}$ is a bijection on topological spaces, which allows us to identify $\mathcal{O}_{X^{(i)}}\tilde{=}\mathcal{O}_{X}^{p^{i}}$; we shall tacitly use this below. Now we introduce some conventions on differential operators. If $\mathfrak{X}$ is a smooth formal scheme over $W(k)$, then for each $i\geq0$ we have Berthelot's ring of differential operators of level $i$, $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(i)}$, introduced in \cite{key-1} This is a $p$-adically complete, locally noetherian sheaf of rings on $\mathfrak{X}$. In general, this sheaf is somewhat complicated to define, but when $\mathfrak{X}=\text{Specf}(\mathcal{A})$ is affine and admits local coordinates\footnote{i.e., $\Gamma(\Omega_{\mathfrak{X}}^{1})$ is a free module over $\mathcal{A}$} one has the following description of its global sections: let $D_{\mathcal{A}}^{(\infty)}$ denote the subring of $\text{End}_{W(k)}(\mathcal{A})$ consisting of the the finite order, continuous differential operators on $\mathcal{A}$. Define $D_{\mathcal{A}}^{(i)}\subset D_{\mathcal{A}}^{(\infty)}$ to be the subring generated by differential operators of level $\leq p^{i}$. Then we have \[ \Gamma(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(i)})=\widehat{D_{\mathcal{A}}^{(i)}} \] where $\widehat{?}$ stands for $p$-adic completion. For each $i\geq0$ there is a natural, injective map $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(i)}\to\widehat{\mathcal{D}}_{\mathfrak{X}}^{(i+1)}$; when $\mathfrak{X}=\text{Specf}(\mathcal{A})$ is as above it is given by the $p$-adic completion of the tautological inclusion $D_{\mathcal{A}}^{(i)}\subset D_{\mathcal{A}}^{(i+1)}$. Similarly, we have the sheaves of algebras $\mathcal{D}_{X}^{(i)}$ when $X$ is smooth over $k$. In the case $i=0$, this is simply the usual sheaf of pd-differential operators on $X$ (c.f. \cite{key-10}). This sheaf can be rather rapidly defined (as in \cite{key-3} chapter 1, though there they are called crystalline differential operators) as the enveloping algebroid of the tangent sheaf $\mathcal{T}_{X}$. Finally let us mention that we will be often working with derived categories of graded modules in this work. In that context, the symbol $[i]$ denotes a shift in homological degree, while $(i)$ denotes a shift in the grading degree. \section{Preliminaries} \subsection{Gauges over $W(k)$} In this section we set some basic notation and terminology; all of which is essentially taken from the paper \cite{key-5}. Let $k$ be a perfect field of characteristic $p>0$; and let $W(k)$ be the $p$-typical Witt vectors. Let $S$ be a noetherian $W(k)$-algebra. We recall from \cite{key-5} (also \cite{key-20}) that a gauge over $S$ is a graded module ${\displaystyle M=\bigoplus_{i=\infty}^{\infty}M^{i}}$ over the graded ring $D(S)$ where, (as always) we suppose $\text{deg}(f)=1$, $\text{deg}(v)=-1$, and $fv=p$. A morphism of gauges is a morphism in the category of graded modules. If $M$ is a gauge, we denote the resulting multiplication maps by $f:M^{i}\to M^{i+1}$ and $v:M^{i}\to M^{i-1}$ for all $i$. As explained in \cite{key-5}, lemma 1.1.1, such a module is finitely generated over $R$ iff each $M^{i}$ is finite over $S$ and the maps $f:M^{r}\to M^{r+1}$ and $v:M^{-r}\to M^{-r-1}$ are isomorphisms for $r>>0$. It follows that in this case the map $v:M^{r}\to M^{r-1}$ is $p\cdot$ for $r>>0$, and $f:M^{-r}\to M^{-r+1}$ is $p\cdot$ for $r>>0$. In the terminology of \cite{key-5}, such a gauge is \emph{concentrated in a finite interval}. \begin{defn} \label{def:endpoints} Let $M$ be a gauge. 1) Set ${\displaystyle M^{\infty}:=M/(f-1)M\tilde{\to}\lim_{r\to\infty}M^{r}}$ and ${\displaystyle M^{-\infty}:=M/(v-1)M\tilde{\to}\lim_{r\to-\infty}M^{r}}$. 2) For each $i$, denote by $f_{\infty}:M^{i}\to M^{\infty}$ and $v_{-\infty}:M^{i}\to M^{-\infty}$ the induced maps. 3) Define $F^{i}(M^{\infty}):=\text{image}(M^{i}\to M^{\infty})$ and $C^{i}(M^{-\infty}):=\text{image}(M^{i}\to M^{-\infty})$. In particular, $F^{i}$ is an increasing filtration on $M^{\infty}$ and $C^{i}$ is a decreasing filtration on $M^{-\infty}$. Clearly any morphism of gauges $M\to N$ induces morphisms of filtered modules $(M^{\infty},F^{\cdot})\to(N^{\infty},F^{\cdot})$ and $(M^{-\infty},C^{\cdot})\to(N^{-\infty},C^{\cdot})$. \end{defn} If $M$ is finitely generated we see that $M^{r}\tilde{=}M^{\infty}$ and $M^{-r}\tilde{=}M^{-\infty}$ for all $r>>0$. Many gauges arising in examples posses an additional piece of structure- a Frobenius semi-linear isomorphism from $M^{\infty}$ to $M^{-\infty}$. So let us now suppose that $S$ is equipped with an endomorphism $F$ which extends the Frobenius on $W(k)$. \begin{defn} \label{def:F-gauge} (\cite{key-5}, section 1.4) An $F$-gauge is a gauge $M$ equipped with an isomorphism $\varphi:F^{*}M^{\infty}\tilde{\to}M^{-\infty}$. A morphism of $F$-gauges is required to respect the isomorphism $\varphi$. More precisely, given a morphism $G:M\to N$, it induces $G^{\infty}:M^{\infty}\to N^{\infty}$ and $G^{-\infty}:M^{-\infty}\to N^{-\infty}$, and we demand $\varphi\circ F^{*}G^{\infty}=G^{\infty}\circ\varphi$. This makes the category of $F$-gauges into an additive category, which is abelian if $F^{*}$ is an exact functor. \end{defn} Now suppose in addition that $F:S\to S$ is an isomorphism. Then: \begin{rem} \label{rem:basic-equiv}There is an equivalence of categories from $F$-gauges to $F^{-1}$-gauges; namely, send $M$ to the gauge $N$ where $N^{i}=M^{-i}$, $f:N^{i}\to N^{i+1}$ is defined to be $v:M^{-i}\to M^{-i-1}$ , $v:N^{i}\to N^{i-1}$ is defined to be $f:M^{-i}\to M^{-i+1}$. Then $M^{\infty}=N^{-\infty}$ , $M^{-\infty}=N^{-\infty}$, and the isomorphism $\varphi:F^{*}M^{\infty}\tilde{\to}M^{-\infty}$ yields an isomorphism $\psi^{-1}:F^{*}N^{-\infty}\tilde{\to}N^{\infty}$; which is equivalent to giving an isomorphism $\psi:(F^{-1})^{*}N^{\infty}\tilde{\to}N^{-\infty}$. \end{rem} Finally, we want to quickly review an important construction of gauges. We suppose here that $S=W(k)$; equipped with its Frobenius automorphism $F$. We use the same letter $F$ to denote the induced automorphism of the field $B=W(k)[p^{-1}]$. We will explain how gauges arrive from lattices of $B$-vector spaces: \begin{example} \label{exa:BasicGaugeConstruction}Let $D$ be a finite dimensional $B$-vector space, and let $M$ and $N$ be two lattices (i.e., finite free $W(k)$-modules which span $D$) in $D$. To this situation we may attach a gauge over $W(k)$ as follows: for all $i\in\mathbb{Z}$ define \[ M^{i}=\{m\in M|p^{i}m\in N\} \] We let $f:M^{i}\to M^{i+1}$ be the inclusion, and $v:M^{i}\to M^{i-1}$ be the multiplication by $p$. For $i>>0$ we have $p^{i}M\subset N$ and so $M^{i}=M$ for all such $i$. For $i<<0$ we have $p^{-i}N\subset M$ and so $M^{i}=p^{-i}N\tilde{=}N$ for such $i$. In particular we obtain $M^{-\infty}\tilde{=}N$ and $M^{\infty}\tilde{=}M$. This is evidently a finite-type gauge over $W(k)$. Now suppose that there is an $F$-semi-linear automorphism $\Phi:D\to D$ so that $M=\Phi(N)$. Then the previous construction gives an $F^{-1}$ gauge via the isomorphism $\Phi:N=M^{-\infty}\to M^{\infty}=M$. \end{example} \begin{rem} \label{rem:=00005BFJ=00005D-standard}In \cite{key-5}, section 2.2, there is associated an $F$-gauge to a finite dimensional $B$ vector space $D$, equipped with lattice $M\subset D$ and a semi-linear automorphism $\Phi:D\to D$. We recall that their construction is \end{rem} \[ M^{i}=\{m\in M|\Phi(m)\in p^{i}M\}=\{m\in M|m\in p^{i}\Phi^{-1}(M)\} \] for all $i\in\mathbb{Z}$. In this instance $f:M^{i}\to M^{i+1}$ is the multiplication by $p$, and $v:M^{i}\to M^{i-1}$ is the inclusion. If we set $N=\Phi^{-1}(M)$ then this is exactly the $F$-gauge which corresponds to the $F^{-1}$ gauge constructed in \exaref{BasicGaugeConstruction} above, via the equivalence of categories of \remref{basic-equiv}. In \cite{key-5} this construction is referred to as the standard construction of gauges. We will generalize this below in \subsecref{Standard}. \subsection{Cohomological Completion of Graded Modules} In this section we give some generalities on sheaves of graded modules. Throughout this section, we let $X$ be a noetherian topological space and $\tilde{\mathcal{R}}=\bigoplus_{i\in\mathbb{Z}}\tilde{\mathcal{R}}^{i}$ a $\mathbb{Z}$-graded sheaf of rings on $X$. The noetherian hypothesis ensures that, for each open subset $U\subset X$, the functor $\mathcal{F}\to\mathcal{F}(U)$ respects direct sums; although perhaps not strictly necessary, it simplifies the discussion of graded sheaves (and it always applies in this paper). Denote $\tilde{\mathcal{R}}^{0}=\mathcal{R}$, a sheaf of rings on $X$. Let $\mathcal{G}(\tilde{\mathcal{R}})$ denote the category of graded sheaves of modules over $\tilde{\mathcal{R}}$. This is a Grothendieck abelian category; the direct sum is given by the usual direct sum of sheaves. To construct the product of sheaves $\{\mathcal{M}_{i}\}_{i\in I}$, one takes the sheafification of the pre-sheaf of local sections of the form $(m_{i})_{i\in I}$ for which there is a bound on the degree; i.e. $-N\leq\text{deg}(m_{i})\le N$ for a fixed $N\in\mathbb{N}$ and all $i\in I$. Since $X$ is a noetherian space, this pre-sheaf is actually already a sheaf. It follows formally that $\mathcal{G}(\tilde{\mathcal{R}})$ has enough injectives; this can also be proved in the traditional way by constructing enough injective in the category of modules over a graded ring and then noting that the sheaf ${\displaystyle \prod_{x\in X}\mathcal{I}_{x}}$ is injective if $\mathcal{I}_{x}$ is an injective object in the category of graded $\tilde{\mathcal{R}}_{x}$-modules. We note that an injective in $\mathcal{G}(\tilde{\mathcal{R}})$ might not be an injective $\mathcal{\tilde{R}}$-module. However, from the previous remark it follows that any injective in $\mathcal{G}(\tilde{\mathcal{R}})$ is a summand of a sheaf of the form $\prod_{x\in X}\mathcal{I}_{x}$; as such sheaves are clearly flasque it follows that any injective in $\mathcal{G}(\tilde{\mathcal{R}})$ is flasque. For each $i\in\mathbb{Z}$ we have the exact functor $\mathcal{M}\to\mathcal{M}^{i}$ which takes $\mathcal{G}(\tilde{\mathcal{R}})\to\mathcal{R}-\text{mod}$; the direct sum of all of these functors is isomorphic to the identity (on the underlying sheaves of $\mathcal{R}$-modules). Note that the functor $\mathcal{M}\to\mathcal{M}^{0}$ admits the left adjoint $\mathcal{N}\to\tilde{\mathcal{R}}\otimes_{\mathcal{R}}\mathcal{N}$. Let $D(\mathcal{G}(\tilde{\mathcal{R}}))$ denote the (unbounded) derived category of $\mathcal{G}(\tilde{\mathcal{R}})$. Then the exact functor $\mathcal{M}\to\mathcal{M}^{i}$ derives to a functor $\mathcal{M}^{\cdot}\to\mathcal{M}^{\cdot,i}$, and we have $\mathcal{M}^{\cdot}={\displaystyle \bigoplus_{i}\mathcal{M}^{\cdot,i}}$ for any complex in $D(\mathcal{G}(\tilde{\mathcal{R}}))$. \begin{lem} Let $\varphi:X\to Y$ be a continuous map, and let $\tilde{\mathcal{R}}_{X}$ and $\tilde{\mathcal{R}}_{Y}$ be graded sheaves of algebras on $X$ and $Y$, respectively. Suppose there is a morphism of graded rings $\varphi^{-1}(\tilde{\mathcal{R}}_{Y})\to\tilde{\mathcal{R}}_{X}$. Then we can form the derived functor $R\varphi_{*}:D(\mathcal{G}(\tilde{\mathcal{R}}_{X}))\to D(\mathcal{G}(\tilde{\mathcal{R}}_{Y}))$, as well as $R\varphi_{*}:D(\tilde{\mathcal{R}}_{X}-\text{mod})\to D(\tilde{\mathcal{R}}_{Y}-\text{mod})$. 1) Let $\mathcal{F}_{X}$ denote the forgetful functor from $\mathcal{G}(\tilde{\mathcal{R}}_{X})$ to $\tilde{\mathcal{R}}_{X}-\text{mod}$ (and similarly for $\mathcal{F}_{Y}$). Then for any $\mathcal{M}^{\cdot}\in D^{+}(\mathcal{G}(\tilde{\mathcal{R}}_{X}))$, we have $\mathcal{F}_{Y}R\varphi_{*}(\mathcal{M}^{\cdot})\tilde{\to}R\varphi_{*}(\mathcal{F}_{X}\mathcal{M}^{\cdot})$; where on the right hand side $R\varphi_{*}$ denotes the pushforward $D^{+}(\tilde{\mathcal{R}}_{X}-\text{mod})\to D^{+}(\tilde{\mathcal{R}}_{Y}-\text{mod})$. If $X$ and $Y$ have finite dimension, then this isomorphism holds for all $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}_{X}))$. 2) Again assuming $X$ and $Y$ have finite dimension; for each $i\in\mathbb{Z}$ we have $R\varphi_{*}(\mathcal{M}^{\cdot,i})\tilde{=}R\varphi_{*}(\mathcal{M}^{\cdot})^{i}$ in $D(\mathcal{R}_{Y}-\text{mod})$. 3) For every $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}_{X}))$ and $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}_{Y}))$ we have \[ R\varphi_{*}R\underline{\mathcal{H}om}_{\varphi^{-1}(\tilde{\mathcal{R}}_{Y})}(\varphi^{-1}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}_{Y}}(\mathcal{N}^{\cdot},R\varphi_{*}\mathcal{M}^{\cdot}) \] \end{lem} \begin{proof} 1) The statement about $D^{+}(\mathcal{G}(\tilde{\mathcal{R}}_{X}))$ follows immediately from the fact that injectives are flasque. For the unbounded derived category, the assumption implies $\varphi_{*}$ has finite homological dimension; and by what we have just proved the forgetful functor takes acyclic objects to acyclic objects. Therefore we can apply the composition of derived functors (as in \cite{key-9}, corollary 14.3.5), which implies that, since $\varphi_{*}$, $\mathcal{F}_{X}$, and $\mathcal{F}_{Y}$ have finite homological dimension in this case, there is an isomorphism $R\varphi_{*}\circ\mathcal{F}_{X}\tilde{=}R(\varphi_{*}\circ\mathcal{F}_{X})\tilde{\to}R(\mathcal{F}_{Y}\circ\varphi_{*})\tilde{=}\mathcal{F}_{Y}\circ R\varphi_{*}$. 2) As above this follows from \cite{key-9}, corollary 14.3.5, using $\varphi_{*}\circ\mathcal{M}^{i}\tilde{=}(\varphi_{*}\mathcal{M})^{i}$. 3) This is essentially identical to the analogous fact in the ungraded case. \end{proof} Now we briefly discuss the internal Hom and tensor on these categories. If $\mathcal{M}$ and $\mathcal{N}$ are objects of $\mathcal{G}(\tilde{\mathcal{R}})$, we have the sheaf of $\mathbb{Z}$-modules $\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\mathcal{M},\mathcal{N})$ as well as the sheaf of graded $\mathbb{Z}$-modules ${\displaystyle \underline{\mathcal{H}om}(\mathcal{M},\mathcal{N})=\bigoplus_{i\in\mathbb{Z}}\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\mathcal{M},\mathcal{N}(i))}$; if $\mathcal{M}$ is locally finitely presented this agrees with $\mathcal{H}om$ on the underlying $\tilde{\mathcal{R}}$-modules. Also, if $\mathcal{M}\in\mathcal{G}(\tilde{\mathcal{R}})$ and $\mathcal{N}\in\mathcal{G}(\tilde{\mathcal{R}}^{opp})$, we have the tensor product $\mathcal{N}\otimes_{\tilde{\mathcal{R}}}\mathcal{M}$ which is graded in the natural way. Suppose now that $\tilde{\mathcal{S}}$ is another sheaf of graded algebras on $X$, \begin{lem} \label{lem:basic-hom-tensor}1) Let $\mathcal{N}$ be a graded $(\mathcal{\tilde{\mathcal{R}}},\mathcal{\tilde{\mathcal{S}}})$ bimodule, $\mathcal{M}\in\mathcal{G}(\tilde{\mathcal{S}})$, and $\mathcal{P}\in\mathcal{G}(\tilde{\mathcal{R}})$. Then there is an isomorphism \[ \underline{\mathcal{H}om}_{\mathcal{\tilde{R}}}(\mathcal{N}\otimes_{\mathcal{\tilde{S}}}\mathcal{M},\mathcal{P})\tilde{\to}\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\mathcal{M},\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{N},\mathcal{P})) \] Now, if we consider $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{S}}))$ and $\mathcal{P}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$, we have a map \[ R\underline{\mathcal{H}om}_{\mathcal{\tilde{R}}}(\mathcal{N}\otimes_{\mathcal{\tilde{S}}}^{L}\mathcal{M}^{\cdot},\mathcal{P}^{\cdot})\to R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{N},\mathcal{P}^{\cdot})) \] and if, further, $\mathcal{N}$ is flat over $\tilde{\mathcal{S}}^{opp}$, then this map is an isomorphism. 2) Now suppose $\tilde{\mathcal{S}}\subset\tilde{\mathcal{R}}$ is a central inclusion of graded rings (in particular $\tilde{\mathcal{S}}$ is commutative). Then for any $\mathcal{M}\in\mathcal{G}(\tilde{\mathcal{R}})$, $\mathcal{N}\in\mathcal{G}(\tilde{\mathcal{R}}^{opp})$, and $\mathcal{P}\in\mathcal{G}(\tilde{\mathcal{R}})$ there are isomorphisms \[ \underline{\mathcal{H}om}_{\mathcal{\tilde{S}}}(\mathcal{N}\otimes_{\mathcal{\tilde{R}}}\mathcal{M},\mathcal{P})\tilde{\to}\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{M},\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\mathcal{N},\mathcal{P})) \] the analogous result holds at the level of complexes: if $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$, $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}^{opp}))$, and $\mathcal{P}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ there are isomorphisms \[ R\underline{\mathcal{H}om}_{\mathcal{\tilde{S}}}(\mathcal{N}^{\cdot}\otimes_{\mathcal{\tilde{R}}}^{L}\mathcal{M}^{\cdot},\mathcal{P}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\mathcal{N}^{\cdot},\mathcal{P}^{\cdot})) \] \end{lem} This is proved in a nearly identical way to the ungraded case (c.f. \cite{key-9}, theorem 14.4.8). Throughout this work, we will make extensive use of various sheaves of rings over $W(k)$ and derived categories of sheaves of modules over them. One of our main techniques will be to work with complexes of sheaves which are\emph{ }complete in a suitable sense, and then to apply Nakayama's lemma to deduce properties of those complexes from their $\text{mod}$ $p$ analogues. The technical set-up for this is the theory of cohomologically complete complexes (also called \emph{derived complete complexes }in many places) which has been treated in the literature in many places, e.g., \cite{key-41}, \cite{key-42}, \cite{key-43}, Tag 091N, and \cite{key-82}, section 3.4. We will use the reference \cite{key-8}, chapter 1.5, which deals with non-commutative sheaves of algebras in a very general setting (namely, they work with sheaves of rings over $\mathbb{Z}[h]$, which are $h$-torsion-free). However, we actually have to extend the theory slightly to get exactly what we need, because our interest is in complexes of \emph{graded} modules, and the useful notion of completeness in this setting is to demand, essentially, that each graded piece of a module (or complex) is complete. We will set this up in a way that we can derive the results in a similar way to \cite{key-8} (or even derive them from \cite{key-8} sometimes). From now on, we impose the assumption that $\tilde{\mathcal{R}}$ is a $W(k)$-algebra (where $W(k)$ sits in degree $0$) which is $p$-torsion-free. Note that we have the sheaf of algebras $\tilde{\mathcal{R}}[p^{-1}]$, which we regard as an object of $\mathcal{G}(\tilde{\mathcal{R}})$ via $\tilde{\mathcal{R}}[p^{-1}]=\bigoplus_{i\in\mathbb{Z}}\tilde{\mathcal{R}}^{i}[p^{-1}]$. There is the category $\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}])$ of graded sheaves of modules over $\tilde{\mathcal{R}}[p^{-1}]$, and there is the functor $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))\to D(\mathcal{G}(\tilde{\mathcal{R}}))$; which is easily seen to be fully faithful, with essential image consisting of those complexes in $D(\mathcal{G}(\tilde{\mathcal{R}}))$ for which $p$ acts invertibly on each cohomology sheaf (compare \cite{key-8}, lemma 1.5.2); we shall therefore simply regard $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$ as being a full subcategory of $D(\mathcal{G}(\tilde{\mathcal{R}}))$. Then, following \cite{key-8}, definition 1.5.5, we make the \begin{defn} \label{def:CC}1) An object $\mathcal{M}^{\cdot}\in D(\mathcal{R}-\text{mod})$ is said to be cohomologically complete if $R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{\cdot})=R\mathcal{H}om_{W(k)}(W(k)[p^{-1}],\mathcal{M}^{\cdot})=0$. 2) An object $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\tilde{R}}))$ is said to be cohomologically complete if \linebreak{} $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})=0$. \end{defn} We shall see below that two notions are not quite consistent with one another, however, we shall only use definition $2)$ when working with graded objects, so this will hopefully cause no confusion. Following \cite{key-8}, proposition 1.5.6), we have: \begin{prop} \label{prop:Basic-CC-facts}1) The cohomologically complete objects in $D(\mathcal{G}(\tilde{\mathcal{R}}))$ form a thick triangulated subcategory, denoted $D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$. An object $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\tilde{R}}))$ is in $D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$ iff $R\underline{\mathcal{H}om}(\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})=0$ for all $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$. 2) If $\tilde{\mathcal{S}}$ is any graded sheaf of $p$-torsion-free $W(k)$-algebras equipped with a graded algebra map $\tilde{\mathcal{S}}\to\tilde{\mathcal{R}}$, and $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$, then $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$ iff $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\tilde{\mathcal{S}}))$ 3) For every $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ there is a distinguished triangle \[ R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})\to\mathcal{M}^{\cdot}\to R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}[-1],\mathcal{M}^{\cdot}) \] and we have $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}[-1],\mathcal{M}^{\cdot})\in D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$ while $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})\in D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$. In particular, the category $D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$ is naturally equivalent to the quotient of $D(\mathcal{G}(\tilde{\mathcal{R}}))$ by $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$. 4) Recall that for each object $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ we have, for $i\in\mathbb{Z}$, the $i$th graded piece $\mathcal{M}^{\cdot,i}\in D(\mathcal{R}-\text{mod})$. Then $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ is in $D_{cc}(\mathcal{G}(\mathcal{R}))$ iff each $\mathcal{M}^{\cdot,i}\in D_{cc}(\mathcal{R}-\text{mod})$. \end{prop} \begin{proof} 1) For any $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$ we have \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{=}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}]\otimes_{\tilde{\mathcal{R}}}^{L}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{N}^{\cdot},R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})) \] here, we have used the fact $\tilde{\mathcal{R}}[p^{-1}]$ is an $(\tilde{\mathcal{R}},\tilde{\mathcal{R}})$-bimodule, along with \lemref{basic-hom-tensor}, 1). Thus if $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}]\mathcal{M}^{\cdot})=0$ then $R\underline{\mathcal{H}om}(\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})=0$ as claimed. Therefore $D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$ is the (right) orthogonal subcategory to the thick subcategory $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$; it follows that is a thick triangulated subcategory. 2) We have \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\tilde{\mathcal{S}}[p^{-1}],\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{S}}[p^{-1}]\otimes_{\tilde{\mathcal{S}}}^{L}\tilde{\mathcal{R}},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot}) \] from which the result follows. 3) This triangle follows by applying $R\underline{\mathcal{H}om}$ to the short exact sequence \[ \tilde{\mathcal{R}}\to\tilde{\mathcal{R}}[p^{-1}]\to\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}} \] and noting that $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}},)$ is the identity functor. The complex $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})$ is contained in $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$ via the action of $\tilde{\mathcal{R}}[p^{-1}]$ on itself. On the other hand, as above there is a canonical isomorphism \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}],R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}[-1],\mathcal{M}^{\cdot}))\tilde{\leftarrow}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}]\otimes_{\tilde{\mathcal{R}}}^{L}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}),\mathcal{M}^{\cdot})[1] \] and the term on the right is zero since $\tilde{\mathcal{R}}[p^{-1}]\otimes_{\tilde{\mathcal{R}}}^{L}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}})=0$; therefore \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}[-1],\mathcal{M}^{\cdot})\in D_{cc}(\mathcal{G}(\mathcal{R})) \] This shows that the inclusion $D_{cc}(\mathcal{G}(\mathcal{R}))\to D(\mathcal{G}(\mathcal{R}))$ admits a right adjoint, and the statement about the quotient category follows immediately. 4) For each $\mathcal{M}\in\mathcal{G}(\tilde{\mathcal{R}})$ there is an isomorphism of functors \[ \mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{0})\tilde{=}\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}) \] given by restricting a morphism on the right hand side to degree $0$; this follows from the fact that an local section of $\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M})$ is simply a system $(m_{i})$ of local sections of $\mathcal{M}^{0}$ satisfying $pm_{i}=m_{i-1}$; which is exactly a local section of $\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{0})$. Now, $\mathcal{M}\to\mathcal{M}^{0}$ admits a left adjoint (namely $\mathcal{N}\to\tilde{\mathcal{R}}\otimes_{\mathcal{R}}\mathcal{N}$), and $\mathcal{N}\to\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{N})$ admits a left adjoint (namely $\mathcal{M}\to\mathcal{R}[p^{-1}]\otimes_{\mathcal{R}}\mathcal{M}$). So by \cite{key-9}, proposition 14.4.7, the derived functor of $\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{0})$ is given by the functor $R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{\cdot,0})$ for any $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{R}))$. Therefore there is an isomorphism of functors \[ R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{\cdot,0})\tilde{\to}R\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}) \] Therefore \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})=\bigoplus_{i}R\mathcal{H}om_{\mathcal{G}(\tilde{\mathcal{R}})}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot}(i))\tilde{=}\bigoplus_{i}R\mathcal{H}om_{\mathcal{R}}(\mathcal{R}[p^{-1}],\mathcal{M}^{\cdot,-i}) \] and the result follows. \end{proof} We will refer to the functor $\mathcal{M}^{\cdot}\to R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}]/\tilde{\mathcal{R}}[-1],\mathcal{M}^{\cdot})$ as the \emph{graded derived completion} of $\mathcal{M}^{\cdot}$, or, usually, simply the completion of $\mathcal{M}^{\cdot}$ if no confusion seems likely; we will denote it by $\widehat{\mathcal{M}}^{\cdot}$. A typical example of a cohomologically complete complex in $D(\mathcal{R}-\text{mod})$ is the following: suppose $\mathcal{M}^{\cdot}=\mathcal{M}$ is concentrated in degree $0$. Then if $\mathcal{M}$ is $p$-torsion free, then $\mathcal{M}$ is $p$-adically complete iff $\mathcal{M}^{\cdot}$ is cohomologically complete (c.f. \cite{key-8}, lemma 1.5.4). By part $4)$ of the proposition, if $\mathcal{M}=\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ is concentrated in a single degree, then if each $\mathcal{M}^{i}$ is $p$-torsion free and $p$-adically complete, then $\mathcal{M}^{\cdot}$ is cohomologically complete (in the graded sense). Therefore the two notions are not in general compatible; an infinite direct sum of $p$-adically complete modules is generally not complete. Now we develop this notion a bit more: \begin{lem} \label{lem:reduction-of-completion}Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$. Then the natural map $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\to\mathcal{\widehat{M}}^{\cdot}\otimes_{W(k)}^{L}k$ is an isomorphism. \end{lem} \begin{proof} The cone of the map $\mathcal{M}^{\cdot}\to\mathcal{\widehat{M}}^{\cdot}$ is contained in $D(\mathcal{G}(\tilde{\mathcal{R}}[p^{-1}]))$, and therefore vanishes upon applying $\otimes_{W(k)}^{L}k$. \end{proof} Now we can transfer the Nakayama lemma into the graded setting: \begin{cor} \label{cor:Nakayama}Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\tilde{R}}))$, and let $a\in\mathbb{Z}$. If $\mathcal{H}^{i}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)=0$ for all $i<a$, then $\mathcal{H}^{i}(\mathcal{M}^{\cdot})=0$ for all $i<a$. In particular $\mathcal{M}^{\cdot}=0$ iff $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k=0$. Therefore, if $\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\tilde{R}}))$ and $\eta:\mathcal{M}^{\cdot}\to\mathcal{N}^{\cdot}$ is a morphism such that $\eta\otimes_{W(k)}^{L}k:\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\to\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k$ is an isomorphism, then $\eta$ is an isomorphism. \end{cor} \begin{proof} By part 4) of the previous proposition this follows immediately from the analogous fact for cohomologically complete sheaves over $\mathcal{R}$; which is \cite{key-8}, proposition 1.5.8. \end{proof} For later use, we record a few more useful properties of cohomologically complete sheaves, following \cite{key-8}, propositions 1.5.10 and 1.5.12. \begin{prop} \label{prop:Push-and-complete}1) Suppose $\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$, and let $\tilde{\mathcal{S}}$ be any central graded sub-algebra of $\tilde{\mathcal{R}}$ which contains $W(k)$. Then $R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\in D_{cc}(\mathcal{G}(\tilde{\mathcal{S}}))$. 2) Suppose $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ and $\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$. Then the map $\mathcal{M}^{\cdot}\to\widehat{\mathcal{M}}^{\cdot}$ induces an isomorphism \[ R\underline{\mathcal{H}om}(\widehat{\mathcal{M}}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}) \] 3) Suppose $\varphi:X\to Y$ is a continuous map, and suppose $\tilde{\mathcal{R}}$ is a graded sheaf of algebras on $Y$ (satisfying the running assumptions of the section). Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\varphi^{-1}(\tilde{\mathcal{R}})))$. Then $R\varphi_{*}(\mathcal{M}^{\cdot})\in D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$. Therefore, if $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\varphi^{-1}(\tilde{\mathcal{R}})))$ is any complex, then we have \[ \widehat{R\varphi_{*}(\mathcal{M}^{\cdot})}\tilde{\to}R\varphi_{*}(\widehat{\mathcal{M}^{\cdot}}) \] \end{prop} \begin{proof} 1) As $\tilde{\mathcal{S}}$ is central we have \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\tilde{\mathcal{S}}[p^{-1}],R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}))\tilde{\leftarrow}R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\mathcal{M}^{\cdot}\otimes_{\tilde{\mathcal{S}}}^{L}\tilde{\mathcal{S}}[p^{-1}],\mathcal{N}^{\cdot}) \] \[ \tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\tilde{\mathcal{S}}}(\tilde{\mathcal{S}}[p^{-1}],\mathcal{N}^{\cdot})) \] where the second isomorphism follows from the flatness of $\tilde{\mathcal{S}}[p^{-1}]$ over $\tilde{\mathcal{S}}$, and first isomorphism follows directly from \[ \tilde{\mathcal{S}}[p^{-1}]\tilde{=}\text{lim}(\tilde{\mathcal{S}}\xrightarrow{p}\tilde{\mathcal{S}}\xrightarrow{p}\tilde{\mathcal{S}}\cdots)\tilde{=}{\displaystyle \text{hocolim}(\tilde{\mathcal{S}}\xrightarrow{p}\tilde{\mathcal{S}}\xrightarrow{p}\tilde{\mathcal{S}}\cdots)} \] so the result follows from part $2)$ of \propref{Basic-CC-facts}. 2) This follows since $\text{cone}(\mathcal{M}^{\cdot}\to\widehat{\mathcal{M}}^{\cdot})$ is contained in the orthogonal to $D_{cc}(\mathcal{G}(\tilde{\mathcal{R}}))$, by definition. 3) For the first claim, we use the adjunction \[ R\varphi_{*}R\underline{\mathcal{H}om}_{\varphi^{-1}(\tilde{\mathcal{R}})}(\varphi^{-1}(\tilde{\mathcal{R}})[p^{-1}],\mathcal{M}^{\cdot})\tilde{=}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\tilde{\mathcal{R}}[p^{-1}],R\varphi_{*}(\mathcal{M}^{\cdot})) \] along with part $2)$ of \propref{Basic-CC-facts}. For the second, we use the distinguished triangle \[ R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})\to\mathcal{M}^{\cdot}\to\widehat{\mathcal{M}^{\cdot}} \] Since $p$ acts invertibly on $R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot})$, it will also act invertibly on\linebreak{} $R\varphi_{*}(R\underline{\mathcal{H}om}(\tilde{\mathcal{R}}[p^{-1}],\mathcal{M}^{\cdot}))$, and the result follows from the fact that $R\varphi_{*}(\widehat{\mathcal{M}^{\cdot}})$ is already cohomologically complete. \end{proof} In using this theory, it is also useful to note the following straightforward \begin{lem} \label{lem:Hom-tensor-and-reduce}Let $\tilde{\mathcal{R}}$ be as above. Then, for any $\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}}))$ we have \[ R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}}(\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\otimes_{W(k)}^{L}k\tilde{\to}R\underline{\mathcal{H}om}_{\tilde{\mathcal{R}}/p}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k,\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k) \] If we have $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\tilde{\mathcal{R}})^{\text{opp}})$, then we have \[ (\mathcal{N}^{\cdot}\otimes_{\tilde{\mathcal{R}}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{\to}(\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{\tilde{\mathcal{R}}/p}^{L}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k) \] \end{lem} To close out this section, we will give an explicit description of the (ungraded) cohomological completion functor in a special case. Let $\mathcal{R}$ be a $p$-torsion-free sheaf of rings on $X$ as above; suppose $\mathcal{R}$ is left noetherian. Let us suppose that, in addition, the $p$-adic completion $\widehat{\mathcal{R}}$ is $p$-torsion-free, left noetherian, and that there exists a base of open subsets $\mathcal{B}$ on $X$ such that, for any $U\in\mathcal{B}$ and any coherent sheaf $\mathcal{M}$ of $\mathcal{R}_{0}=\mathcal{R}/p=\widehat{\mathcal{R}}/p$ modules on $U$, we have $H^{i}(U,\mathcal{M})=0$ for all $i>0$ (these are assumptions $1.2.2$ and $1.2.3$ of \cite{key-8}; they are always satisfied in this paper). Then we have \begin{prop} \label{prop:Completion-for-noeth}Let $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{R}-\text{mod})$. Then there is an isomorphism \[ \widehat{\mathcal{M}^{\cdot}}\tilde{=}\widehat{\mathcal{R}}\otimes_{\mathcal{R}}^{L}\mathcal{M}^{\cdot} \] where $\widehat{\mathcal{M}^{\cdot}}$ denotes the derived completion as usual. \end{prop} \begin{proof} Let $\mathcal{M}$ be a coherent $\mathcal{R}$-module, and let $\widehat{\mathcal{M}}$ denote its $p$-adic completion. By \cite{key-8}, lemma 1.1.6 and the assumption on $\mathcal{B}$, we have ${\displaystyle \widehat{\mathcal{M}}(U)=\lim_{\leftarrow}\mathcal{M}(U)/p^{n}}$ for any $U\in\mathcal{B}$. So, by the noetherian hypothesis, we see that the natural map $\widehat{\mathcal{R}}(U)\otimes_{\mathcal{R}(U)}\mathcal{M}(U)\to\widehat{\mathcal{M}}(U)$ is an isomorphism for all $U\in\mathcal{B}$ . It follows that the map $\widehat{\mathcal{R}}\otimes_{\mathcal{R}}\mathcal{M}\to\widehat{\mathcal{M}}$ is an isomorphism of sheaves; and therefore (as $p$-adic completion is exact on $\mathcal{R}(U)-\text{mod}$) that $\widehat{\mathcal{R}}$ is flat over $\mathcal{R}$. Now consider an arbitrary $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{R}-\text{mod})$. The above implies \[ \mathcal{H}^{i}(\widehat{\mathcal{R}}\otimes_{\mathcal{R}}^{L}\mathcal{M}^{\cdot})\tilde{=}\widehat{\mathcal{R}}\otimes_{\mathcal{R}}\mathcal{H}^{i}(\mathcal{M}^{\cdot})\tilde{\to}\widehat{\mathcal{H}^{i}(\mathcal{M}^{\cdot})} \] Therefore $\widehat{\mathcal{R}}\otimes_{\mathcal{R}}^{L}\mathcal{M}^{\cdot}\in D_{coh}^{b}(\widehat{\mathcal{R}}-\text{mod})$, which is contained in $D_{cc}(\mathcal{R}-\text{mod})$ by \cite{key-8}, theorem 1.6.1. Let $\mathcal{C}^{\cdot}$ be the cone of the map $\mathcal{M}^{\cdot}\to\widehat{\mathcal{R}}\otimes_{\mathcal{R}}^{L}\mathcal{M}^{\cdot}$ . Then we have a long exact sequence \[ \mathcal{H}^{i-1}(\mathcal{C}^{\cdot})\to\mathcal{H}^{i}(\mathcal{M}^{\cdot})\to\widehat{\mathcal{H}^{i}(\mathcal{M}^{\cdot})}\to\mathcal{H}^{i}(\mathcal{C}^{\cdot}) \] and since both the kernel and cokernel of $\mathcal{H}^{i}(\mathcal{M}^{\cdot})\to\widehat{\mathcal{H}^{i}(\mathcal{M}^{\cdot})}$ are in $\mathcal{R}[p^{-1}]-\text{mod}$, we conclude that $\mathcal{C}^{\cdot}\in D(\mathcal{R}[p^{-1}]-\text{mod})$. Since $\widehat{\mathcal{R}}\otimes_{\mathcal{R}}^{L}\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{R}-\text{mod})$, the result follows from the fact that $D_{cc}(\mathcal{R}-\text{mod})$ is the quotient of $D(\mathcal{R}-\text{mod})$ by $D(\mathcal{R}[p^{-1}]-\text{mod})$. \end{proof} \section{\label{sec:The-Algebra}The Algebra $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$} To define the algebra $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ and prove \thmref{D01}, we are going to apply the basic gauge construction (\exaref{BasicGaugeConstruction}) to Berthelot's differential operators. Let $\mathfrak{X}$ be a smooth formal scheme over $W(k)$, and $X$ its special fibre. If $\mathfrak{X}$ is affine, then we denote $\mathfrak{X}=\text{Specf}(\mathcal{A})$, and $X=\text{Spec}(A)$. \begin{defn} \label{def:D^(0,1)-in-the-lifted-case} We set \[ \mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i}:=\{\Phi\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(1)}|p^{i}\Phi\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}\} \] We let $f:\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i}\to\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i+1}$ denote the inclusion, and $v:\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i}\to\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i-1}$ denote the multiplication by $p$. If $\Phi_{1}\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i}$ and $\Phi_{2}\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),j}$, then $\Phi_{1}\cdot\Phi_{2}\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i+j}$, and in this way we give \[ \mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}=\bigoplus_{i\in\mathbb{Z}}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i} \] the structure of a sheaf of graded algebras over $D(W(k))$. Now suppose $\mathfrak{X}=\text{Specf}(\mathcal{A})$. Then we have ring theoretic analogue of the above: define $\widehat{D}_{\mathcal{A}}^{(0,1),i}:=\{\Phi\in\widehat{D}_{\mathcal{A}}^{(1)}|p^{i}\Phi\in\widehat{D}_{\mathcal{A}}^{(0)}\}$, and as above we obtain the graded ring \[ \widehat{D}_{\mathcal{A}}^{(0,1)}=\bigoplus_{i}\widehat{D}_{\mathcal{A}}^{(0,1),i} \] over $D(W(k))$. In this case, we also have the finite-order analogue: define $D_{\mathcal{A}}^{(0,1),i}:=\{\Phi\in D_{\mathcal{A}}^{(1)}|p^{i}\Phi\in D_{\mathcal{A}}^{(0)}\}$, and as above we obtain the graded ring \[ D_{\mathcal{A}}^{(0,1)}=\bigoplus_{i}D_{\mathcal{A}}^{(0,1),i} \] over $D(W(k))$. \end{defn} It is easy to see that $\widehat{D}_{\mathcal{A}}^{(0,1),i}:=\Gamma(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i})$ when $\mathfrak{X}=\text{Specf}(\mathcal{A})$. With the help of local coordinates, this algebra is not too difficult to study. We now suppose $\mathfrak{X}=\text{Specf}(\mathcal{A})$ where $\mathcal{A}$ possesses local coordinates; i.e., there is a collection $\{x_{i}\}_{i=1}^{n}\in\mathcal{A}$ and derivations $\{\partial_{i}\}_{i=1}^{n}$ such that $\partial_{i}(x_{j})=\delta_{ij}$ and such that $\{\partial_{i}\}_{i=1}^{n}$ form a free basis for the $\mathcal{A}$-module of $W(k)$-linear derivations. We let $\partial_{i}^{[p]}:=\partial_{i}^{p}/p!$, this is a differential operator of order $p$ on $\mathcal{A}$. \begin{lem} \label{lem:Basic-structure-of-D_A^(i)} For $i\geq0$ we have that $\widehat{D}_{\mathcal{A}}^{(0,1),i}$ is the left $\widehat{D}_{\mathcal{A}}^{(0)}$-module\footnote{In fact, it is also the right $\widehat{D}_{\mathcal{A}}^{(0)}$-module generated by the same elements, as an identical proof shows } generated by $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where ${\displaystyle \sum_{t=1}^{n}j_{t}\leq i}$. For $i\leq0$ we have that $\widehat{D}_{\mathcal{A}}^{(0,1),i}=p^{-i}\cdot\widehat{D}_{\mathcal{A}}^{(0)}$. \end{lem} \begin{proof} Let $i>0$. Clearly the module described is contained in $\widehat{D}_{\mathcal{A}}^{(0,1),i}$. For the converse, we begin with the analogous finite-order version of the statement. Namely, let $\Phi\in D_{\mathcal{A}}^{(1)}$ be such that $p^{i}\Phi\in D_{\mathcal{A}}^{(0)}$. We can write \[ \Phi=\sum_{I,J}a_{I,J}\partial_{1}^{i_{1}}(\partial_{1}^{[p]})^{j_{1}}\cdots\partial_{n}^{i_{n}}(\partial_{n}^{[p]})^{j_{n}}=\sum_{I,J}a_{I,J}\frac{\partial_{1}^{i_{1}+pj_{1}}\cdots\partial_{n}^{i_{n}+pj_{n}}}{(p!)^{|J|}} \] where $|J|=j_{1}+\dots+j_{n}$, and the sum is finite. After collecting like terms together, we may suppose that $0\leq i_{j}<p$. In that case, the $a_{I,J}\in\mathcal{A}$ are uniquely determined by $\Phi$, and $\Phi\in D_{\mathcal{A}}^{(0)}$ iff $p^{|J|}|a_{I,J}$ for all $I,J$. So, if $p^{i}\Phi\in D_{\mathcal{A}}^{(0)}$, we have ${\displaystyle a_{I,J}\frac{p^{i}}{p^{|J|}}\in\mathcal{A}}$. Thus whenever $|J|>i$ we have $a_{I,J}\in p^{|J|-i}\mathcal{A}$. On the other hand \[ p^{|J|-i}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}=u\cdot\partial_{1}^{pj'_{1}}\cdots\partial_{n}^{pj'_{n}}\cdot(\partial_{1}^{[p]})^{j''_{1}}\cdots\partial_{n}^{i_{n}}(\partial_{n}^{[p]})^{j''_{n}} \] where $u$ is a unit in $\mathbb{Z}_{p}$, ${\displaystyle \sum j'_{i}=|J|-i}$, and ${\displaystyle \sum j''_{i}=i}$ (this follows from the relation $p!\partial_{j}^{[p]}=\partial_{j}^{p}$). Therefore if $|J|>i$ we have \[ a_{I,J}\frac{\partial_{1}^{i_{1}+pj_{1}}\cdots\partial_{n}^{i_{n}+pj_{n}}}{(p!)^{|J|}}\in D_{\mathcal{A}}^{(0)}\cdot(\partial_{1}^{[p]})^{j''_{1}}\cdots\partial_{n}^{i_{n}}(\partial_{n}^{[p]})^{j''_{n}} \] It follows that $\Phi$ is contained in the $D_{\mathcal{A}}^{(0)}$-submodule generated by $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where $j_{1}+\dots+j_{n}\leq i$. So this submodule is exactly $\{\Phi\in D_{\mathcal{A}}^{(1)}|p^{i}\Phi\in D_{\mathcal{A}}^{(0)}\}$. Now let $\Phi\in\widehat{D}_{\mathcal{A}}^{(0,1),i}$. Then we can write \[ p^{i}\Phi=\sum_{j=0}^{\infty}p^{j}\Phi_{j} \] where $\Phi_{j}\in D_{\mathcal{A}}^{(0)}$. Therefore, if $j\le i$ we have, by the previous paragraph, that $p^{-i}(p^{j}\Phi_{j})$ is contained in the $D_{\mathcal{A}}^{(0)}$ submodule generated by $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where $j_{1}+\dots+j_{n}\leq i$. So the result follows from ${\displaystyle \Phi=\sum_{j=0}^{i}p^{j-i}\Phi_{j}+\sum_{j=i+1}^{\infty}p^{j-i}\Phi_{j}}$ as the second term in this sum is contained in $\widehat{\mathcal{D}}_{\mathcal{A}}^{(0)}$. This proves the lemma for $i\geq0$; while for $i\leq0$ it follows immediately from the definition. \end{proof} From this it follows that ${\displaystyle \widehat{D}_{\mathcal{A}}^{(0,1),\infty}:=\lim_{\rightarrow}\widehat{D}_{\mathcal{A}}^{(0,1),i}}$ is the sub-algebra of $\text{End}_{W(k)}(\mathcal{A})$ generated by $\widehat{D}_{\mathcal{A}}^{(0)}$ and $\{\partial_{1}^{[p]},\dots,\partial_{n}^{[p]}\}$. We have \begin{lem} \label{lem:Basic-Structure-of-D^(1)} The algebra $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}$ is a (left and right) noetherian ring, whose $p$-adic completion is isomorphic to $\widehat{D}_{\mathcal{A}}^{(1)}$. Further, we have $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}[p^{-1}]\tilde{=}\widehat{D}_{\mathcal{A}}^{(0)}[p^{-1}]$. \end{lem} \begin{proof} First, put a filtration on $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}$ by setting $F^{j}(\widehat{D}_{\mathcal{A}}^{(0,1),\infty})$ to be the $\widehat{D}_{\mathcal{A}}^{(0)}$-submodule generated by $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where $j_{1}+\dots+j_{n}\leq i$. Then $\text{gr}(\widehat{D}_{\mathcal{A}}^{(0,1),\infty})$ is a quotient of a polynomial ring $\widehat{D}_{\mathcal{A}}^{(0)}[T_{1},\dots T_{n}]$ where $T_{i}$ is sent to the class of $\partial_{i}^{[p]}$ in $\text{gr}_{1}(\widehat{D}_{\mathcal{A}}^{(0,1),\infty})$. To see this, we need to show that the image of $\partial_{i}^{[p]}$ in $\text{gr}(\widehat{D}_{\mathcal{A}}^{(0,1),\infty})$ commutes with $\widehat{D}_{\mathcal{A}}^{(0)}=\text{gr}^{0}(\widehat{D}_{\mathcal{A}}^{(0,1),\infty})$; this follows from the relation \[ [\partial_{i}^{[p]},a]=\sum_{j=1}^{p}\partial_{i}^{[j]}(a)\partial_{i}^{[p-j]}\in\widehat{D}_{\mathcal{A}}^{(0)} \] for any $a\in\mathcal{A}$. So the fact that $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}$ is a (left and right) noetherian ring follows from the Hilbert basis theorem and the fact that $\widehat{D}_{\mathcal{A}}^{(0)}$ is left and right noetherian. Now we compute the $p$-adic completion of $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}$. Inside $\text{End}_{W(k)}(\mathcal{A})$, we have \[ D_{\mathcal{A}}^{(1)}\subset\widehat{D}_{\mathcal{A}}^{(0,1),\infty}\subset\widehat{D}_{\mathcal{A}}^{(1)} \] and so, for all $n>0$ we have \[ D_{\mathcal{A}}^{(1)}/p^{n}\to\widehat{D}_{\mathcal{A}}^{(0,1),\infty}/p^{n}\to\widehat{D}_{\mathcal{A}}^{(1)}/p^{n} \] and the composition is the identity. Thus $D_{\mathcal{A}}^{(1)}/p^{n}\to\widehat{D}_{\mathcal{A}}^{(0,1),\infty}/p^{n}$ is injective. On the other hand, suppose $\Phi\in\widehat{D}_{\mathcal{A}}^{(0,1),\infty}$. By definition we can write \[ \Phi=\sum_{I}\varphi_{i}\cdot(\partial_{1}^{[p]})^{i_{1}}\cdots(\partial_{n}^{[p]})^{i_{n}} \] where $I=(i_{1},\dots,i_{n})$ is a multi-index, $\varphi_{i}\in\widehat{D}_{\mathcal{A}}^{(0)}$, and the sum is finite. Choose elements $\psi_{i}\in D_{A}^{(0)}$ such that $\psi_{i}-\varphi_{i}\in p^{n}\cdot\widehat{D}_{\mathcal{A}}^{(0)}$ (this is possible since $\widehat{D}_{\mathcal{A}}^{(0)}$ is the $p$-adic completion of $D_{A}^{(0)}$). Then if we set \[ \Phi'=\sum_{I}\psi_{i}\cdot(\partial_{1}^{[p]})^{i_{1}}\cdots(\partial_{n}^{[p]})^{i_{n}}\in D_{\mathcal{A}}^{(1)} \] we see that the class of $\Phi'$ in $D_{\mathcal{A}}^{(1)}/p^{n}$ maps to the class of $\Phi\in\widehat{D}_{\mathcal{A}}^{(0,1),\infty}/p^{n}$. Thus $D_{\mathcal{A}}^{(1)}/p^{n}\to\widehat{D}_{\mathcal{A}}^{(0,1),\infty}/p^{n}$ is onto and therefore an isomorphism, and the completion result follows by taking the inverse limit. Finally, since each $\partial_{i}^{[p]}=\partial_{i}^{p}/p!$ is contained in $\widehat{D}_{\mathcal{A}}^{(0)}[p^{-1}]$, we must have $\widehat{D}_{\mathcal{A}}^{(0)}\subset D_{\mathcal{A}}^{(0,1),\infty}\subset\widehat{D}_{\mathcal{A}}^{(0)}[p^{-1}]$, so that $\widehat{D}_{\mathcal{A}}^{(0,1),\infty}[p^{-1}]\tilde{=}\widehat{D}_{\mathcal{A}}^{(0)}[p^{-1}]$. \end{proof} \begin{cor} $\widehat{D}_{\mathcal{A}}^{(0,1)}$ is a (left and right) noetherian ring, which is finitely generated as an algebra over $\widehat{D}_{\mathcal{A}}^{(0)}[f,v]$. Therefore the sheaf $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ is a coherent, locally noetherian sheaf of rings which is stalk-wise noetherian. \end{cor} This follows immediately from the above. Set $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),-\infty}:=\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/(v-1)\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$, while $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}:=\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}/(f-1)$ has $p$-adic completion equal to $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. \subsection{Generators and Relations, Local Coordinates} In addition to the description above as via endomorphisms of $\mathcal{O}_{\mathfrak{X}}$, it is also useful to have a more concrete (local) description of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ and, especially, $\mathcal{D}_{\mathfrak{X}}^{(0,1)}/p$ . Suppose $\mathfrak{X}=\text{Specf}(\mathcal{A})$ possesses local coordinates as above. We'll start by describing ${\displaystyle \widehat{D}_{\mathcal{A}}^{(0,1),+}:=\bigoplus_{i=0}^{\infty}\widehat{D}_{\mathcal{A}}^{(0,1),i}}$. \begin{defn} Let $M_{\mathcal{A}}$ be the free graded $\mathcal{A}$-module on generators $\{\xi_{i}\}_{i=1}^{n}$ (in degree $0$), and $f$ and $\{\xi_{i}^{[p]}\}_{i=1}^{n}$ (in degree $1$). Let $\mathcal{B}^{(0,1),+}$ be the quotient of the tensor algebra $T_{\mathcal{A}}(M_{\mathcal{A}})$ by the relations $[f,m]$ (for all $m\in M_{\mathcal{A}}$), $[\xi_{i},a]-\partial_{i}(a)$ (for all $i$, and for any $a\in A$), $[\xi_{i},\xi_{j}]$, $[\xi_{i}^{[p]},\xi_{j}^{[p]}]$, $[\xi_{i}^{[p]},\xi_{j}]$ (for all $i,j$), ${\displaystyle [\xi_{i}^{[p]},a]-f\cdot\sum_{r=0}^{p-1}\frac{\partial_{i}^{p-r}}{(p-r)!}(a)\cdot\frac{\xi_{i}^{r}}{r!}}$ (for all $i$, and for any $a\in A$), $f\xi_{i}^{p}-p!\xi_{i}^{[p]}$ for all $i$, and $\xi_{i}^{p}\xi_{j}^{[p]}-\xi_{j}^{p}\xi_{i}^{[p]}$ for all $i$ and $j$. The algebra $\mathcal{B}^{(0,1),+}$ inherits a grading from $T_{\mathcal{A}}(M_{\mathcal{A}})$. Let $\mathcal{C}^{(0,1),+}$ be the graded ring obtained by $p$-adically completing each component of $\mathcal{B}^{(0,1),+}$. \end{defn} Then we have \begin{lem} \label{lem:Reduction-is-correct}There is an isomorphism of graded algebras $\mathcal{C}^{(0,1),+}\tilde{\to}\widehat{D}_{\mathcal{A}}^{(0,1),+}$. \end{lem} \begin{proof} There is an evident map $T_{\mathcal{A}}(M_{\mathcal{A}})\to\widehat{D}_{\mathcal{A}}^{(0,1),+}$ which is the identity on $\mathcal{A}$ and which sends $\xi_{i}\to\partial_{i}$, $f\to f$ and $\xi_{i}^{[p]}\to\partial_{i}^{[p]}$. Clearly this induces a graded map $\mathcal{B}^{(0,1),+}\to\tilde{\to}\widehat{D}_{\mathcal{A}}^{(0,1),+}$. Since each graded component of $\widehat{D}_{\mathcal{A}}^{(0,1),+}$ is $p$-adically complete, we obtain a map $\mathcal{C}^{(0,1),+}\to\tilde{\to}\widehat{D}_{\mathcal{A}}^{(0,1),+}$. Let us show that is is an isomorphism. We begin with the surjectivity. In degree $0$, we have that $\mathcal{B}^{(0,1),0}$ is generated by $\mathcal{A}$ and $\{\xi_{i}\}_{i=1}^{n}$ and satisfies $[\xi_{i},a]=\partial_{i}(a)$ for all $i$ and all $a\in\mathcal{A}$. Thus the obvious map $\mathcal{B}^{(0,1),0}\to\mathcal{D}_{\mathcal{A}}^{(0)}$ is an isomorphism, and therefore so is the completion $\mathcal{C}^{(0,1),0}\to\widehat{\mathcal{D}}_{\mathcal{A}}^{0}=\widehat{\mathcal{D}}_{\mathcal{A}}^{(0,1),0}$. Further, we saw above (in \lemref{Basic-structure-of-D_A^(i)}) that each $\widehat{\mathcal{D}}_{\mathcal{A}}^{(0,1),i}$ (for $i\geq0$) is generated, as a module over $\widehat{\mathcal{D}}_{\mathcal{A}}^{0}$, by terms of the form $\{f^{i_{0}}\partial_{1}^{[p]i_{1}}\cdots\partial_{n}^{[p]i_{n}}\}$ where $i_{0}+i_{1}\dots i_{n}=i$. By definition, $\mathcal{C}^{(0,1),i}$ is exactly the $\mathcal{C}^{(0,1),0}$-module generated by terms of the form $\{f^{i_{0}}\xi_{1}^{[p]i_{1}}\cdots\xi_{n}^{[p]i_{n}}\}$. Thus we see that the map surjects onto the piece of degree $i$ for all $i$; hence the map is surjective. To show the injectivity, consider the graded ring ${\displaystyle \mathcal{A}[f]=\bigoplus_{i=0}^{\infty}\mathcal{A}}$. The algebra $\widehat{D}_{\mathcal{A}}^{(0,1),+}$ acts on $\mathcal{A}[f]$ as follows: if $\Phi\in\widehat{D}_{\mathcal{A}}^{(0,1),j}$ then $\Phi\cdot(af^{i})=\Phi(a)f^{i+j}$, where $\Phi(a)$ is the usual action of $\Phi$ on $\mathcal{A}$, coming from the fact that $\Phi\in\widehat{D}_{\mathcal{A}}^{(1)}$. In addition, $\mathcal{C}^{(0,1),+}$ acts on $\mathcal{A}[f]$ via $\xi_{i}(af^{j})=\xi_{i}(a)f^{j}$ and $\xi_{i}^{[p]}(af^{j})=\partial_{i}^{[p]}(a)f^{j+1}$. This action agrees with the composed map \[ \mathcal{C}^{(0,1),+}\to\widehat{D}_{\mathcal{A}}^{(0,1),+}\to\text{End}_{W(k)}(\mathcal{A}[f]) \] where the latter map comes from the action of $\widehat{D}_{\mathcal{A}}^{(0,1),+}$ on $\mathcal{A}[f]$. We will therefore be done if we can show that this composition is injective. For this, we proceed by induction on the degree $i$. When $i=0$ it follows immediately from the fact that $\mathcal{C}^{(0,1),0}\tilde{=}\widehat{D}_{\mathcal{A}}^{(0)}$. Let $\Phi\in\mathcal{C}^{(0,1),i}$. If $\Phi$ acts as zero on $\mathcal{A}[f]$, we will show that $\Phi\in f\cdot\mathcal{C}^{(0,1),i-1}$; the induction assumption (and the fact that $f$ acts injectively on $\mathcal{A}[f]$) then implies that $\Phi=0$. Write \[ \Phi=\sum_{J}\Phi_{J}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}}-f^{i}\Psi_{0}-\sum_{s=1}^{i-1}f^{i-s}\sum_{J}\Psi_{sJ}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}} \] where, in the first sum, each $J$ satisfies ${\displaystyle i=|J|=\sum_{i=1}^{n}j_{i}}$, and in the second sum we have $|J|=i-s$, and $\Phi_{J},\Psi_{0},\Psi_{sJ}\in\mathcal{C}^{(0,1),0}\tilde{=}\widehat{D}_{\mathcal{A}}^{(0)}$. We shall show that every term in the first sum is contained in $f\cdot\mathcal{C}^{(0,1),i-1}$. Expanding in terms of monomials in the $\{\xi_{i}\}$, denoted $\{\xi^{I}\}$, we obtain an equation \[ \Phi=\sum_{I,J}a_{J,I}\xi^{I}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}}-f^{i}\sum_{I}b_{0,I}\xi^{I}-\sum_{s=1}^{i-1}f^{i-s}\sum_{I,J}b_{s,I,J}\xi^{I}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}} \] where $a_{J,I}\to0$, $b_{0,I}\to0$, and $b_{s,I,J}\to0$ (in the $p$-adic topology on $\mathcal{A}$) as $|I|\to\infty$. For any multi-index $J=(j_{1},\dots,j_{n})$ let $pJ=(pj_{1},\dots,pj_{n})$. The relations $\xi_{i}^{p}\xi_{j}^{[p]}=\xi_{j}^{p}\xi_{i}^{[p]}$ (for all $i,j$ ) in $\mathcal{C}^{(0,1),+}$ ensure that $\xi^{I}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}}=\xi^{I'}(\xi_{1}^{[p]})^{j'_{1}}\cdots(\xi_{n}^{[p]})^{j'_{n}}$ whenever $I+pJ=I'+pJ'$ and $|J|=|J'|$. Since, in the sum ${\displaystyle \sum_{I,J}a_{J,I}\xi^{I}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}}}$, we have $|J|=i$ for all $J$, we may collect terms together and assume that each multi-index $I+pJ$ is represented only once. Now, the fact that $\Phi$ acts as zero on $\mathcal{A}[f]$ implies that the differential operators ${\displaystyle \sum_{I,J}a_{J,I}\partial^{I}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}}$ and ${\displaystyle \sum_{I}b_{0,I}\partial^{I}+\sum_{s=1}^{i-1}\sum_{I,J}b_{s,I,J}\partial^{I}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}}$ act as the same endomorphism on $\mathcal{A}$. Therefore, for each $a_{J,I}$ which is nonzero, we have \[ a_{J,I}\partial^{I}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}=\sum_{I'=I+pJ}b_{0,I'}\partial^{I'}+\sum_{s=1}^{i-1}\sum_{I'+pJ'=I+pJ}b_{s,I',J'}\partial^{I'}(\partial_{1}^{[p]})^{j'_{1}}\cdots(\partial_{n}^{[p]})^{j'_{n}} \] Now, after inverting $p$, and using $\partial_{i}^{[p]}=\partial_{i}^{p}/p!$ inside $\widehat{\mathcal{D}}_{\mathcal{A}}^{(1)}[p^{-1}]$, we obtain the equation \[ \frac{a_{J,I}}{(p!)^{i}}=b_{0,I'}+\sum_{s=1}^{i-1}\sum_{I',J'}\frac{b_{s,I',J'}}{(p!)^{s}} \] which implies $a_{J,I}\in p\cdot\mathcal{A}$. But we have the relation $f\xi_{i}^{p}-p!\xi_{i}^{[p]}$ in $\mathcal{C}^{(0,1),+}$; i.e., $p\xi_{i}^{[p]}\in f\cdot\mathcal{C}^{(0,1),+}$. Therefore $a_{J,I}\in p\cdot\mathcal{A}$ implies $a_{J,I}\xi^{I}(\xi_{1}^{[p]})^{j_{1}}\cdots(\xi_{n}^{[p]})^{j_{n}}\in f\cdot\mathcal{C}^{(0,1),i-1}$. Since this holds for all $I,J$, we see that in fact $\Phi\in f\cdot\mathcal{C}^{(0,1),i-1}$ as desired. \end{proof} \begin{rem} Given the isomorphism of the theorem, from now on, we shall denote $\xi_{i}$ by $\partial_{i}$ and $\xi_{i}^{[p]}$ by $\partial_{i}^{[p]}$ inside $\mathcal{C}^{(0,1),+}$. \end{rem} Next, we have \begin{lem} \label{lem:linear-independance-over-D_0-bar} Suppose that $\{\Phi_{sJ}\}$ are elements of $\widehat{D}_{\mathcal{A}}^{(0)}$, and suppose that, for some $i\geq1$, we have \[ \sum_{s=0}^{i-1}\sum_{|J|=s}f^{i-s}\Phi_{sJ}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\in p\cdot\widehat{D}_{\mathcal{A}}^{(0,1),i} \] in $\widehat{D}_{\mathcal{A}}^{(0,1),+}$. Then each $\Phi_{sJ}$ is contained in the right ideal generated by $\{\partial_{1}^{p},\dots,\partial_{n}^{p}\}$ and $p$. \end{lem} \begin{proof} As in the previous proof we may expand the $\Phi_{0}$ and $\Phi_{s,J}$ in terms of the $\{\partial^{I}\}$ to obtain \begin{equation} \Phi=\sum_{s=0}^{i-1}f^{i-s}\sum_{I,J,|J|=s}b_{s,I,J}\partial^{I}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\label{eq:first-form-for-phi} \end{equation} where $b_{s,I,J}\to0$ as $|I|\to\infty$. Om the other hand, since $\Phi\in p\cdot\widehat{D}_{\mathcal{A}}^{(0,1),i}$, and $\widehat{D}_{\mathcal{A}}^{(0,1),i}$ is generated over $\widehat{D}_{\mathcal{A}}^{(0)}$ by $\{f^{i-s}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{0\leq s\leq i,|J|=s}$, we also obtain \begin{equation} \Phi=\sum_{s=0}^{i}f^{i-s}\sum_{I,J,|J|=s}a_{s,I,J}\partial^{I}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\label{eq:second-form-for-phi} \end{equation} where each $a_{0I}$ and $a_{s,I,J}$ are contained in $p\cdot\widehat{D}_{\mathcal{A}}^{(0)}$, and $a_{0I}\to0$, $a_{s,I,J}\to0$ as $|I|\to\infty$. For a multi-index $K$, let $\tilde{K}$ denote the multi-index $(\tilde{k}_{1},\dots,\tilde{k}_{n})$ such that $\tilde{k}_{i}\leq k_{i}$ for all $i$ and such that $\tilde{k}{}_{i}$ is the greatest multiple of $p$ less than or equal $k_{i}$. Write $\tilde{K}=p\tilde{J}$, and $K=\tilde{I}+p\tilde{J}$. Then if $K=I'+pJ'$ for some $I'\neq\tilde{I}$ and $J'\neq\tilde{J}$, we must have $j_{m}<\tilde{j}_{m}$ for some $m$; which implies that $\partial^{I'}$ is contained in the right ideal generated by $\partial_{m}^{p}$. Since $f\cdot\partial_{i}^{p}=p!\partial_{i}^{[p]}$, we obtain \[ f^{i-s}b_{s,I',J'}\partial^{I'}(\partial_{1}^{[p]})^{j'_{1}}\cdots(\partial_{n}^{[p]})^{j'_{n}}=f^{i-s-1}b_{s-1,I'',J''}\partial^{I''}(\partial_{1}^{[p]})^{j''_{1}}\cdots(\partial_{n}^{[p]})^{j''_{n}} \] where $I''+pJ''=I'+pJ'=K$, with $j''_{i}=j_{i}'+1$, and $b_{s-1,I'',J''}\in p\cdot\mathcal{A}$. Therefore each such term is in the right ideal generated by $\{\partial_{i}^{p}\}$ and is contained in $p\cdot\widehat{D}_{\mathcal{A}}^{(0,1),i}$, and so we may subtract each of these terms from $\Phi$ without affecting the statement. Thus we may assume that each nonzero $b_{s,I,J}$ in \eqref{first-form-for-phi} is of the form $\tilde{I}+p\tilde{J}$ as above, and so there is only one nonzero $b_{s,I,J}$ for each multi-index $K$. Now, comparing the actions of each of the expressions \eqref{first-form-for-phi} and \eqref{second-form-for-phi} on $\mathcal{A}[f]$, we obtain, for each multi-index $K$, the equality \[ b_{s\tilde{,I},\tilde{J}}=\sum_{I+pJ=K}\sum_{s}a_{s,I,J} \] and since each $a_{s,I,J}\in p\cdot\mathcal{A}$, we see $b_{s\tilde{,I},\tilde{J}}\in p\cdot\mathcal{A}$. Since this is true for all $b_{s\tilde{,I},\tilde{J}}$ the result follows. \end{proof} Using these results, we can give a description of $\widehat{D}_{\mathcal{A}}^{(0,1),+}/p:=D_{A}^{(0,1),+}.$ Let $I$ be the two-sided ideal of $D_{A}^{(0)}:=\widehat{D}_{\mathcal{A}}^{(0)}/p$ generated by $\mathcal{Z}(D_{A}^{(0)})^{+}$, the positive degree elements of the center\footnote{The center of $D_{A}^{(0)}$ is a graded algebra via the isomorphism $\mathcal{Z}(D_{A}^{(0)})\tilde{=}A^{(1)}[\partial_{1}^{p},\dots,\partial_{n}^{p}]$, the degree of each $\partial_{i}^{p}$ is $1$}, and let $\overline{D_{A}^{(0)}}=D_{A}^{(0)}/I$. \begin{thm} \label{thm:Local-Coords-for-D+} Let ${\displaystyle D_{A}^{(0,1),+}=\bigoplus_{i=0}^{\infty}D_{A}^{(0,1),i}}$ be the decomposition according to grading. Then each $D_{A}^{(0,1),i}$ is a module over $D_{A}^{(0)}=D_{A}^{(0,1),0}$, and \[ D_{A}^{(0,1),i}=f\cdot D_{A}^{(0,1),i-1}\oplus\sum_{|J|=i}D_{A}^{(0)}\cdot(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}} \] as $D_{A}^{(0)}$-modules. Further, $f\cdot D_{A}^{(0,1),i-1}$ is free over $\overline{D_{A}^{(0)}}$, and the module\linebreak{} ${\displaystyle \sum_{|J|=i}D_{A}^{(0)}\cdot(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}}$ is isomorphic, as a $D_{A}^{(0)}$-module, to $I^{i}$, via the map which sends $(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}$ to $\partial_{1}^{p}{}^{j_{1}}\cdots\partial_{n}^{p}{}^{j_{n}}$. In particular, on each $D_{A}^{(0,1),i}$ we have $\text{ker}(f)=\text{im}(v)$ and $\text{im}(f)=\text{ker}(v)$. \end{thm} \begin{proof} Let $i\geq1$. By definition $D_{A}^{(0,1),i}$ is generated, over $D_{A}^{(0)}$ terms of the form \linebreak{} $\{f^{i-s}\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{|J|=s}$; and so it is also generated by $f\cdot D_{A}^{(0,1),i-1}$ and $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{|J|=i}$ . Suppose we have an equality of the form \[ \sum_{J=i}\bar{\Phi}_{J}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}=\sum_{s=0}^{i-1}f^{i-s}\sum_{J}\bar{\Psi}_{sJ}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}} \] in $D_{A}^{(0,1),i}$ (here, $\bar{\Phi}_{J},\bar{\Psi}_{sJ}$ are in $D_{A}^{(0)}$). Choosing lifts to $\Phi_{J},\Psi_{J}\in\widehat{D}_{\mathcal{A}}^{(0)}$ yields \[ \sum_{J=i}\Phi_{J}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}-\sum_{s=0}^{i-1}f^{i-s}\sum_{J}\Psi_{sJ}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\in p\cdot D_{\mathcal{A}}^{(0,1),i}\subset f\cdot D_{\mathcal{A}}^{(0,1),i-1} \] (the last inclusion follows from $(p!)\partial_{i}^{[p]}=f\partial_{i}^{p}$); and so (the proof of) \lemref{Reduction-is-correct} now forces $\Phi_{J}\in p\cdot\widehat{D}_{\mathcal{A}}^{(0,1),i}$ for all $J$ so $\bar{\Phi}_{J}=0$ as desired. The isomorphism of ${\displaystyle \sum_{|J|=i}D_{A}^{(0)}\cdot(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}}$ with $I^{i}$ is given by the reduction of the morphism $p^{i}\cdot$ on $D_{\mathcal{A}}^{(0,1),i}$, and \lemref{linear-independance-over-D_0-bar} yields that $f\cdot D_{A}^{(0,1),i-1}$ is free over $\overline{D}_{A}^{(0)}$; a basis is given by $\{f^{i-|J|}(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{0\leq|J|\leq i}$. The last statement follows directly from this description. \end{proof} We now use this to describe the entire graded algebra $D_{A}^{(0,1)}:=D_{\mathcal{A}}^{(0,1)}/p$. \begin{cor} \label{cor:Local-coords-over-A=00005Bf,v=00005D} The algebra $D_{A}^{(0,1)}$ is a free graded module over $D(A)$, with a basis given by the set $\{\partial^{I}(\partial^{[p]})^{J}\}$, where $I=(i_{1},\dots,i_{n})$ is a multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J$ is any multi-index with entries $\geq0$. \end{cor} \begin{proof} By the previous corollary, any element of $D_{A}^{(0,1),+}$ can be written as a finite sum \[ \sum_{I,J}a_{I,J}\partial^{I}(\partial^{[p]})^{J} \] where $a_{I,J}\in A[f]$ and $I$ and $J$ are arbitrary multi-indices. As any element in $D_{A}^{(0,1),-}$ is a sum of the form \[ \sum_{i=1}^{m}v^{i}\sum_{J}b_{i,J}(\partial)^{J} \] we see that in fact any element of $D_{A}^{(0,1)}$ can be written as a finite sum \[ \sum_{I,J}a_{I,J}\partial^{I}(\partial^{[p]})^{J} \] where $a_{I,J}\in A[f,v]$ and $I$ and $J$ are arbitrary multi-indices. Iteratively using the relations $(p-1)!\partial_{i}^{p}=v\partial_{i}^{[p]}$ we see that we may suppose that each entry of $I$ is contained in $\{0,\dots,p-1\}$; this shows that these elements span. To see the linear independence, suppose we have \begin{equation} \sum_{I,J}a_{I,J}\partial^{I}(\partial^{[p]})^{J}=0\label{eq:lin-dep} \end{equation} where now each entry of $I$ is contained in $\{0,\dots,p-1\}$. Write \[ a_{I,J}=\sum_{s\geq0}f^{s}a_{I,J,s}+\sum_{t<0}v^{t}a_{I,J,t} \] We have \[ a_{I,J}\partial^{I}(\partial^{[p]})^{J}=\sum_{s\geq0}f^{s}a_{I,J,s}\partial^{I}(\partial^{[p]})^{J}+\sum_{J',t}a_{I,J,t}\partial^{I+pJ'}(\partial^{[p]})^{J''}+\sum_{t<0}v^{-t-|J|}a_{I,J,t}\partial^{I+pJ} \] where, in the middle sum, $t$ satisfies $0<-t\leq|J|$; for each such $t$ we pick $J'$ such that $J'+J''=J$ and $|J'|=-t$. Now, the previous corollary gives an isomorphism \[ D_{A}^{(0,1),i}\tilde{=}D_{A}^{(0)}/I^{i}\oplus I^{i} \] where $I=C^{1}(D_{A}^{(0)})$, for all $i\geq0$; this in fact holds for all $i\in\mathbb{Z}$ if we interpret $I^{i}=D_{A}^{(0)}$ for $i<0$. This implies that the elements $\{f^{s}\partial^{I}(\partial^{[p]})^{J},\partial^{I+pJ'}(\partial^{[p]})^{J''},v^{-t-|J|}\partial^{I+pJ}\}$ where $I,J$ are multi-indices with each entry of $I$ is contained in $\{0,\dots,p-1\}$, are linearly independent over $A$ (one may look at each degree separately and use the above description). Thus \eqref{lin-dep} implies $a_{I,J,s}=0=a_{I,J,t}$ for all $I,J,s,t$; hence each $a_{I,J}=0$ as desired. \end{proof} Finally, let us apply this result to describe the finite order operators $D_{\mathcal{A}}^{(0,1)}$. Namely, we have \begin{cor} \label{cor:Each-D^(i)-is-free}The algebra $D_{\mathcal{A}}^{(0,1)}$ is free over $D(\mathcal{A})$, with a basis given by the set $\{\partial^{I}(\partial^{[p]})^{J}\}$, where $I=(i_{1},\dots,i_{n})$ is a multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J$ is any multi-index with entries $\geq0$. \end{cor} \begin{proof} By the previous result, the images of these elements in $D_{A}^{(0,1)}=D_{\mathcal{A}}^{(0,1)}/p$ form a basis over $D(\mathcal{A})$. Since $D_{\mathcal{A}}^{(0,1)}$ is $p$-torsion-free, and $D(\mathcal{A})$ is $p$-adically separated, it follows directly that these elements are linearly independent over $D(\mathcal{A})$. The fact that they span follows (as in the previous proof) from \lemref{Basic-structure-of-D_A^(i)}. \end{proof} \subsection{$\mathcal{D}^{(0,1)}$-modules over $k$} Now let $X$ be an arbitrary smooth variety over $k$; in this subsection we make no assumption that there is a lift of $X$; however, if $U\subset X$ is an affine, there is a always a lift of $U$ to as smooth formal scheme $\mathfrak{U}$. In this section we will construct a sheaf of algebras $\mathcal{D}_{X}^{(0,1)}$ such that, on each open affine $U$ which possesses local coordinates, we have $\mathcal{D}_{X}^{(0,1)}(U)=\widehat{\mathcal{D}}_{\mathfrak{U}}^{(0,1)}(\mathfrak{U})/p$. There is a natural action of $\mathcal{D}_{X}^{(0)}$ on $\mathcal{O}_{X}$; inducing a map $\mathcal{D}_{X}^{(0)}\to\mathcal{E}nd_{k}(\mathcal{O}_{X})$, and we let $\overline{\mathcal{D}_{X}^{(0)}}\subset\mathcal{E}nd_{k}(\mathcal{O}_{X})$ denote the image of $\mathcal{D}_{X}^{(0)}$ under this map. It is a quotient algebra of $\mathcal{D}_{X}^{(0)}$, and a quick local calculation gives \begin{lem} \label{lem:Basic-description-of-D-bar} Let $U\subset X$ be an open subset, which possesses local coordinates $\{x_{1},\dots,x_{n}\}$, and let $\{\partial_{1},\dots,\partial_{n}\}$ denote derivations satisfying $\partial_{i}(x_{j})=\delta_{ij}$. Then the kernel of the map $\mathcal{D}_{X}^{(0)}(U)\to\mathcal{E}nd_{k}(\mathcal{O}_{X}(U))$ is the two sided ideal $\mathcal{I}$ generated by $\{\partial_{1}^{p},\dots,\partial_{n}^{p}\}$. The image consists of differential operators of the form ${\displaystyle \sum a_{I}\partial^{I}}$ where the sum ranges over multi-indices $I=(i_{1},\dots,i_{n})$ for which $0\leq i_{j}<p$ (for all $j$), the $a_{I}\in\mathcal{O}_{X}(U)$, and $\partial^{I}=\partial_{1}^{i_{1}}\cdots\partial_{n}^{i_{n}}$. \end{lem} In particular, if $U=\text{Spec}(A)$ then we have $\overline{\mathcal{D}_{X}^{(0)}}(U)=\overline{D_{A}^{(0)}}$ as defined in the previous section. Now let $\mathcal{D}iff_{X}^{\leq n}$ denote the sheaf of differential operators of order $\leq n$ on $X$. This is a sub-sheaf of $\mathcal{E}nd_{k}(\mathcal{O}_{X})$. \begin{defn} \label{def:L}1) Let $\tilde{\mathcal{D}iff}_{X}^{\leq p}$ denote the sub-sheaf of $\mathcal{D}iff_{X}^{\leq p}$ defined by the following condition: a local section $\delta$ of $\mathcal{D}iff_{X}^{\leq p}$ is contained in $\tilde{\mathcal{D}iff}_{X}^{\leq p}$ if, for any local section $\Phi\in\overline{\mathcal{D}_{X}^{(0)}}$, we have $[\delta,\Phi]\in\overline{\mathcal{D}_{X}^{(0)}}$ (Here, the bracket is the natural Lie bracket on $\mathcal{E}nd_{k}(\mathcal{O}_{X})$ coming from the algebra structure). 2) We define the sub-sheaf $\mathfrak{l}_{X}\subset\mathcal{D}iff_{X}$ to be $\tilde{\mathcal{D}iff}_{X}^{\leq p}+\overline{\mathcal{D}_{X}^{(0)}}$. \end{defn} The sections in $\mathfrak{l}_{X}$ can easily be identified in local coordinates. Suppose $U=\text{Spec}(A)$ possess local coordinates $\{x_{1},\dots,x_{n}\}$, and coordinate derivations $\{\partial_{1},\dots,\partial_{n}\}$. \begin{prop} \label{lem:O^p-action} Let $U\subset X$ be an open subset as above. Then we have \[ \mathfrak{l}_{X}(U)=\bigoplus_{i=1}^{n}\mathcal{O}_{U}^{p}\cdot\partial_{i}^{[p]}\oplus\overline{\mathcal{D}_{X}^{(0)}}(U) \] In particular, $\mathfrak{l}_{X}$ is a sheaf of $\mathcal{O}_{X}^{p}$-modules (via the left action of $\mathcal{O}_{X}^{p}$ on $\mathcal{E}nd_{k}(\mathcal{O}_{X})$). \end{prop} \begin{proof} First, let's show that the displayed sum is contained in $\mathfrak{l}_{X}(U)$. By definition $\overline{\mathcal{D}_{X}^{(0)}}(U)\subset\mathfrak{l}_{X}(U)$. Let $\Phi\in\overline{\mathcal{D}_{X}^{(0)}}(U)$, and write ${\displaystyle \Phi=\sum_{I}a_{I}\partial^{I}}$ as in \lemref{Basic-description-of-D-bar}. Then, for any $g\in O_{X}(U)$, we have \[ [g^{p}\partial_{i}^{[p]},\sum_{I}a_{I}\partial^{I}]=\sum_{I}[g^{p}\partial_{i}^{[p]},a_{I}\partial^{I}]=\sum_{I}[g^{p}\partial_{i}^{[p]},a_{I}]\partial^{I}+\sum_{I}a_{I}[g^{p}\partial_{i}^{[p]},\partial^{I}] \] Now, ${\displaystyle [g^{p}\partial_{i}^{[p]},a_{I}]\partial^{I}=g^{p}[\partial_{i}^{[p]},a_{I}]\partial^{I}=g^{p}\sum_{r=0}^{p-1}\partial_{i}^{[p-r]}(a_{I})\partial_{i}^{[r]}\cdot\partial^{I}\in\overline{\mathcal{D}_{X}^{(0)}}(U)}$. Further, $a_{I}[g^{p}\partial_{i}^{[p]},\partial^{I}]=a_{I}g^{p}[\partial_{i}^{[p]},\partial^{I}]+a_{I}[g^{p},\partial^{I}]\partial_{i}^{[p]}=0$. Thus we see that each $g^{p}\partial_{i}^{[p]}\in\mathfrak{l}_{X}(U)$, and the right hand side is contained in the left. For the converse, let $\Phi\in\mathcal{D}iff_{X}^{\leq p}(U)$. It may be uniquely written as \[ \Phi=\sum_{i=1}^{n}a_{i}\partial_{i}^{[p]}+\sum_{I}a_{I}\partial^{I} \] where $a_{i}$ and $a_{I}$ are in $\mathcal{O}_{X}(U)$, and the second sum ranges over multi-indices $I=(i_{1},\dots,i_{n})$ with each $i_{j}<p$ and so that $i_{1}+\dots+i_{n}\leq p$. For any coordinate derivation $\partial_{j}$, we have \[ [\Phi,\partial_{j}]=-(\sum_{i=1}^{n}\partial_{j}(a_{i})\partial_{i}^{[p]}+\sum_{I}\partial_{j}(a_{I})\partial^{I}) \] For this to be contained in $\overline{\mathcal{D}_{X}^{(0)}}(U)$, we must have $\partial_{j}(a_{i})=0$ for all $i$. Therefore, if $[\Phi,\partial_{j}]\in\overline{\mathcal{D}_{X}^{(0)}}(U)$ for all $j$, we must have $\partial_{j}(a_{i})=0$ for all $j$ (and all $i$), which means that each $a_{i}\in\mathcal{O}_{X}(U)^{p}$. Therefore, if $\Phi\in\tilde{\mathcal{D}iff}_{X}^{\leq p}(U)$, then $\Phi$ must be contained in ${\displaystyle \bigoplus_{i=1}^{n}\mathcal{O}_{U}^{p}\cdot\partial_{i}^{[p]}\oplus\overline{\mathcal{D}_{X}^{(0)}}(U)}$, and the result follows. \end{proof} \begin{cor} $\mathfrak{l}_{X}$ is a sheaf of Lie subalgebras of $\mathcal{E}nd_{k}(\mathcal{O}_{X})$. \end{cor} \begin{proof} As the question is local, it suffices to prove that $\mathfrak{l}_{X}(U)$ is closed under the bracket for a neighborhood $U$ which possesses local coordinates. We use the description of the previous lemma. So we must show that all brackets of the form \[ [g^{p}\partial_{i}^{[p]},h^{p}\partial_{j}^{[p]}] \] and \[ [g^{p}\partial_{i}^{[p]},\sum_{I}a_{I}\partial^{I}] \] are contained in $\mathfrak{l}_{X}(U)$. Here the notation is as above; so $g,h\in\mathcal{O}_{X}(U)$, and $I=(i_{1},\dots,i_{n})$ is a multi-index with each $i_{j}<p$. In fact, we already showed that ${\displaystyle [g^{p}\partial_{i}^{[p]},\sum_{I}a_{I}\partial^{I}]\in\overline{\mathcal{D}_{X}^{(0)}}(U)}$ in the course of the proof of the previous lemma. So we are left to analyze the first bracket. Now, \[ [g^{p}\partial_{i}^{[p]},h^{p}\partial_{j}^{[p]}]=h^{p}[g^{p}\partial_{i}^{[p]},\partial_{j}^{[p]}]+[g^{p}\partial_{i}^{[p]},h^{p}]\partial_{j}^{[p]} \] \[ =h^{p}[g^{p},\partial_{j}^{[p]}]\partial_{i}^{[p]}+g^{p}[\partial_{i}^{[p]},h^{p}]\partial_{j}^{[p]} \] and we have \[ [\partial_{i}^{[p]},h^{p}]=\sum_{r=0}^{p-1}\partial_{i}^{[p-r]}(h^{p})\partial_{i}^{[r]}=\partial_{i}^{[p]}(h^{p}) \] and similarly, $[g^{p},\partial_{j}^{[p]}]=-\partial_{j}^{[p]}(g^{p})$. It is a well-known fact that $\partial_{i}^{[p]}(h^{p})=(\partial_{i}(h))^{p}$ (for the sake of completeness, we include a proof directly below). It follows immediately that $[g^{p}\partial_{i}^{[p]},h^{p}\partial_{j}^{[p]}]\in\mathfrak{l}_{X}(U)$, and the corollary follows. To prove that $\partial_{i}^{[p]}(h^{p})=(\partial_{i}(h))^{p}$, recall the following formula for Hasse-Schmidt derivations acting on powers: \[ \partial_{i}^{[j]}(h^{m})=\sum_{i_{1}+\dots+i_{m}=j}\partial_{i}^{[i_{1}]}(h)\cdots\partial_{i}^{[i_{m}]}(h) \] which is easily checked by induction. Put $m=j=p$ in the formula. The set \[ \{(i_{1},\dots,i_{p})\in\mathbb{Z}_{\geq0}^{p}|i_{1}+\dots+i_{p}=p\} \] is acted upon by the symmetric group $S_{p}$, and, after grouping like terms together, we see that each term $\partial_{i}^{[i_{1}]}(h)\cdots\partial_{i}^{[i_{m}]}(h)$ in the sum is repeated $N$ times, where $N$ is the size of the $S_{p}$ orbit of $(i_{1},\dots,i_{p})$. There is a unique orbit of size $1$, namely $i_{1}=i_{2}=\cdots=i_{p}=1$; and for every other orbit, the size is a number of the form ${\displaystyle \frac{p!}{c_{1}!\cdots c_{r}!}}$ for some numbers $c_{i}<p$ such that $\sum c_{i}=p$ . Any such is divisible by $p$, and so all these terms are zero in the sum since we are in characteristic $p$. Thus we obtain \[ \partial_{i}^{[p]}(h^{p})=\partial_{i}^{[1]}(h)\cdots\partial_{i}^{[1]}(h)=(\partial_{i}(h))^{p} \] as claimed. \end{proof} Now we will build the ring $\mathcal{D}_{X}^{(0,1)}$ out of $\mathcal{D}_{X}^{(0)}$ and $\mathfrak{l}_{X}$, in a manner quite analogous to the way in which $\mathcal{D}_{X}^{(0)}$ is built out of $\mathcal{O}_{X}$ and $\mathcal{T}_{X}$ as an enveloping algebra of a Lie algebroid; in the classical case, this construction is given in \cite{key-44} (for schemes) and \cite{key-46} (for rings). Our construction is similar in spirit to these works (c.f. also \cite{key-45}). \begin{defn} \label{def:D-=00005Cplus-L}Let $f:\mathcal{D}_{X}^{(0)}\to\mathfrak{l}_{X}$ denote the map $\mathcal{D}_{X}^{(0)}\to\overline{\mathcal{D}_{X}^{(0)}}\subset\mathfrak{l}_{X}$. Define the sheaf \[ \mathfrak{L}_{X}:=\mathcal{D}_{X}^{(0)}\oplus\bigoplus_{i=1}^{\infty}\mathfrak{l}_{X}=\bigoplus_{i=0}^{\infty}\mathfrak{L}_{X}^{i} \] and make it into a graded $k[f]$-module by letting $f:\mathfrak{l}_{X}\to\mathfrak{l}_{X}$ be the identity in degrees $\geq1$; thus any homogenous element in degree $i\geq1$ can be uniquely written $f^{i-1}\Psi$ for some $\Psi\in\mathfrak{l}_{X}$. For local sections $\Phi\in\mathcal{D}_{X}^{(0)}$ and $f^{i-1}\Psi\in\mathfrak{L}_{X}^{i}$, define $[\Phi,f^{i-1}\Psi]:=f^{i-1}[f\circ\Phi,\Psi]$ where on the right we have the bracket in $\mathfrak{l}_{X}$. We then make $\mathfrak{L}_{X}$ into a sheaf of graded Lie algebras by setting $[f^{i-1}\Psi_{1},f^{j-1}\Psi_{2}]=f^{i+j-1}[\Psi_{1},\Psi_{2}]$ where $\{\Psi_{1},\Psi_{2}\}$ are local sections of $\mathfrak{l}_{X}$. The Jacobi identity can be verified by a direct computation. \end{defn} Next we introduce the action of $v$: \begin{lem} \label{lem:Construction-of-v-1}There is a unique endomorphism $v$ of $\mathfrak{L}_{X}$ satisfying $v(\mathcal{D}_{X}^{(0)})=0$ and, upon restriction to an open affine $U$ which possesses local coordinates, $v(\partial_{i}^{[p]})=(p-1)!\partial_{i}^{p}$ for coordinate derivations $\{\partial_{i}\}_{i=1}^{n}$. This endomorphism vanishes on $f(\mathcal{D}_{X}^{(0)})$, and on ${\displaystyle \bigoplus_{i=2}^{\infty}\mathfrak{L}_{X}^{i}}$. \end{lem} \begin{proof} Since $v(\mathcal{D}_{X}^{(0)})=0$ it suffices to define $v$ on $\mathfrak{l}_{X}$. For any $\Phi\in\mathfrak{l}_{X}$, the action of $\Phi$ preserves $\mathcal{O}_{X}^{p}$, and the restriction of $\Phi$ to $\mathcal{O}_{X}^{p}\tilde{=}\mathcal{O}_{X^{(1)}}$ is a derivation on $\mathcal{O}_{X}^{p}$ (this follows immediately from \lemref{O^p-action} and the fact that $\partial_{i}^{[p]}(g^{p})=(\partial_{i}(g))^{p}$). Further this derivation is trivial iff $\Phi\in f(\mathcal{D}_{X}^{(0)})\subset\mathfrak{l}_{X}$. On the other hand, since $k$ is perfect there is a natural isomorphism between the sheaf of derivations on $\mathcal{O}_{X^{(1)}}$ and the sheaf of derivations on $\mathcal{O}_{X}$, given as follows: if $\partial'$ is a (local) derivation on $\mathcal{O}_{X^{(1)}}$, then we can define a derivation of $\mathcal{O}_{X}$ by $\partial(g)=(\partial'(g^{p}))^{1/p}$; this is possible precisely by the identification $\mathcal{O}_{X}^{p}\tilde{=}\mathcal{O}_{X^{(1)}}$. This association is easily checked to be an isomorphism using local coordinates; let's name it $\tau:\text{Der}(\mathcal{O}_{X^{(1)}})\to\text{Der}(\mathcal{O}_{X})$. Further, there is a map $\sigma:\text{Der}(\mathcal{O}_{X})\to\mathcal{Z}(\mathcal{D}_{X}^{(0)})$ defined by $\partial\to\partial^{p}-\partial^{[p]}$, where $\partial^{[p]}$ is the $p$th iterate of the derivation (c.f. \cite{key-3}, chapter 1). In particular this map takes $\partial_{i}\to\partial_{i}^{p}$ if $\partial_{i}$ is a coordinate derivation as above. Now we define $v(\Phi)=(p-1)!\cdot\sigma\circ\tau(\Phi|_{\mathcal{O}_{X^{(1)}}})$; by the above discussion this satisfies all the properties of the lemma. \end{proof} Now we proceed to the definition of $\mathcal{D}_{X}^{(0,1),+}$. By the functoriality of the enveloping algebra construction, we can now form the pre-sheaf of enveloping algebras $\mathcal{U}(\mathfrak{L}_{X})$; this is a pre-sheaf of graded algebras with the grading inherited from $\mathfrak{L}_{X}$. Inside this pre-sheaf is the pre-sheaf $\mathcal{U}^{+}(\mathfrak{L}_{X})$, which is the pre-sheaf of non-unital algebras generated by $\mathfrak{L}_{X}\subset\mathcal{U}(\mathfrak{L}_{X})$. For any local section $\Phi\in\mathcal{D}_{X}^{(0)}$, let $\Phi'$ denote its image in $\mathcal{U}^{+}(\mathfrak{L}_{X})$, by regarding $\Phi\in\mathcal{D}_{X}^{(0)}\subset\mathfrak{L}_{X}$; similarly, for a local sections $\Psi\in\mathfrak{L}_{X}$, let $\Psi'\in\mathfrak{L}_{X}\subset\mathcal{U}^{+}(\mathfrak{L}_{X})$ denote its image. \begin{defn} Let $\mathcal{J}$ be the pre-sheaf of homogenous two-sided ideals in $\mathcal{U}^{+}(\mathfrak{L}_{X})$ generated by the following sections: for any local sections $\Phi_{1},\Phi_{2}\in\mathcal{D}_{X}^{(0)}$: $(\Phi_{1}\cdot\Phi_{2})'-\Phi_{1}'\cdot\Phi_{2}'$ , $f\cdot\Phi_{1}'-f(\Phi_{1})'$, $\Phi'_{1}\cdot f(\Phi'_{2})-f(\Phi_{1}')\cdot\Phi'_{2}$, $\Phi'_{1}\cdot f(\Phi'_{2})-f\cdot(\Phi_{1}'\cdot\Phi'_{2})$. Further, if $\Psi_{1},\Psi_{2}\in\mathfrak{L}_{X}$ are any local sections, we add the elements $\Psi_{1}'\Psi_{2}'-\Psi_{2}'\Psi_{1}'-[\Psi_{1},\Psi_{2}]'$, as well as $g'\cdot\Psi_{1}=(g\cdot\Psi_{1})'$ for any local section $g\in\mathcal{O}_{X}^{p}$ (the action of $\mathcal{O}_{X}^{p}$ on $\mathfrak{l}_{X}$ is that of \lemref{O^p-action}). Finally, we add $\Phi_{1}'\cdot\Psi'_{1}-\Phi_{2}'\cdot\Psi'_{2}$ where $\Phi_{i}$ are local sections of $\mathcal{Z}(D_{X}^{(0)})$ such that $\Phi_{1}\cdot v(\Psi_{1})=\Phi_{2}\cdot v(\Psi_{2})$. Define $\mathcal{D}_{X}^{(0,1),+}$ to be the sheafification of the presheaf $\mathcal{U}^{+}(\mathfrak{L}_{X})/\mathcal{J}$. It is a graded sheaf of algebras on $X$. \end{defn} Of course, such a definition is only really useful if we can write the algebra out explicitly in the presence of coordinates. Fortunately, this is the case; in fact, if $U=\text{Spec}(A)$ we can compare it with the presentation of $D_{A}^{(0,1),+}=D_{\mathcal{A}}^{(0,1),+}/p$ discussed in the previous section: \begin{thm} \label{thm:D-is-quasi-coherent} Let $\tilde{D}_{A}^{(0,1),+}$ be the quasi-coherent sheaf on $U$ obtained by localizing $D_{A}^{(0,1),+}$. This a sheaf of algebras on $U$. There is an isomorphism (of graded sheaves of algebras) $\mathcal{D}_{X}^{(0,1),+}|_{U}\tilde{=}\tilde{D}_{A}^{(0,1),+}$. In particular, $\mathcal{D}_{X}^{(0,1),+}$ is a quasi-coherent sheaf of algebras on $X$, and we have $\mathcal{D}_{X}^{(0,1),0}\tilde{=}\mathcal{D}_{X}^{(0)}$. \end{thm} \begin{proof} We have the algebra $\mathcal{U}^{+}(\mathfrak{L}_{X})(U)/\mathcal{J}(U)$. It admits a map to $D_{A}^{(0,1),+}$ as follows: by \lemref{O^p-action}, the lie algebra $\mathfrak{L}_{X}(U)$ is equal to \[ \mathcal{D}_{X}^{(0)}(U)\oplus\bigoplus_{i=1}^{\infty}(f^{i}(\overline{\mathcal{D}_{X}^{(0)}}(U))\oplus\bigoplus_{i=1}^{n}f^{i-1}\mathcal{O}_{X}^{p}(U)\cdot\partial_{i}^{[p]}) \] \[ =D_{A}^{(0)}\oplus\bigoplus_{i=1}^{\infty}(f^{i}(\overline{D_{A}^{(0)}})\oplus\bigoplus_{i=1}^{n}f^{i-1}A^{p}\cdot\partial_{i}^{[p]}) \] We map this to $D_{A}^{(0,1),+}$ via the identification of $D_{A}^{(0)}$ with $D_{A}^{(0,1),0}$, and by sending $f^{i}(\overline{D_{A}^{(0)}})$ to $f^{i}\cdot D_{A}^{(0)}\tilde{=}\overline{D_{A}^{(0)}}$ and $f^{i-1}g^{p}\partial_{i}^{[p]}$ to $f^{i-1}g^{p}\partial_{i}^{[p]}\in D_{A}^{(0,1),i}$. By sending $f$ to $f$ we get a map of algebras $\mathcal{U}^{+}(\mathfrak{L}_{X})(U)/\mathcal{J}(U)\to D_{A}^{(0,1),+}$ (one checks the relations directly). Conversely, we get a map $D_{A}^{(0,1),+}\to\mathcal{U}^{+}(\mathfrak{L}_{X})(U)/\mathcal{J}(U)$ by sending $A\to A\subset\mathcal{D}_{X}^{(0)}(U)$, $\partial_{i}\to\partial_{i}\in\mathcal{D}_{X}^{(0)}(U)$, $\partial_{i}^{[p]}\to\partial_{i}^{[p]}\in\mathfrak{l}_{X}(U)$ and $f\to f$. Again checking the relations, this is a morphism of algebras, and the compositions in both directions are the identity on generators. Therefore the presheaf $U\to\mathcal{U}^{+}(\mathfrak{L}_{X})(U)/\mathcal{J}(U)$, when restricted to open affines which admit local coordinates, agrees with the assignment $U\to D_{A}^{(0,1),+}$. But the latter, by the description of \thmref{Local-Coords-for-D+}, clearly agrees with the quasi-coherent sheaf $\tilde{D}_{A}^{(0,1),+}$ on $\text{Spec}(A)$, and the result follows. \end{proof} Finally, we need to define the entire algebra $\mathcal{D}_{X}^{(0,1)}$. This entails extending the operator $v$ to an endomorphism of all of $\mathcal{D}_{X}^{(0,1),+}$. \begin{lem} \label{lem:Construction-of-v} There is a unique $\mathcal{D}_{X}^{(0)}$-linear endomorphism $v$ of $\mathcal{D}_{X}^{(0,1),+}$ satisfying $v(\mathcal{D}_{X}^{(0,1),i})\subset D_{X}^{(0.1),i-1}$ for all $i\geq1$ (and $v(\mathcal{D}_{X}^{(0,1),0})=0$), $v(\Phi_{1}\cdot\Phi_{2})=\Phi_{1}v(\Phi_{2})$ for all $\Phi_{1},\Phi_{2}\in\bigoplus_{i=1}^{\infty}\mathcal{D}_{X}^{(0,1),i}$, $v(f\cdot\Phi)=0$ for all $\Phi$, and such that the restriction of $v$ to $\mathfrak{L}_{X}$ agrees with the map $v$ constructed in \lemref{Construction-of-v-1}. \end{lem} \begin{proof} Define $v$ on $\mathcal{D}_{X}^{(0)}\oplus\mathfrak{l}_{X}$ to be the map constructed in \lemref{Construction-of-v-1}. The claim is that there is a unique extension of this map to all of $\mathcal{D}_{X}^{(0,1),+}$ satisfying the conditions of the lemma. By the uniqueness, it is enough to check this locally. Let $U=\text{Spec}(A)$ posses local coordinates. By \thmref{Local-Coords-for-D+}, if we set $v$ to be zero on $D_{A}^{(0,1),0}$ and $f\cdot(D_{A}^{(0,1),+})$, and we define \[ v((\partial_{j}^{[p]})^{i_{j}}\cdots(\partial_{n}^{[p]})^{i_{n}})=\partial_{j}^{p}\cdot(\partial_{j}^{[p]})^{i_{j}-1}\cdots(\partial_{n}^{[p]})^{i_{n}} \] where $j$ is the first index such that $i_{j}\geq1$, then we have a well-defined $D_{A}^{(0)}$-linear map satisfying all the properties of the lemma, and which agrees with the $v$ defined above on $\mathfrak{L}_{X}(U)$. On the other hand, $D_{A}^{(0,1),+}$ is generated as a $D_{A}^{(0)}$-module by $D_{A}^{(0)}$, $f\cdot(D_{A}^{(0,1),+})$, and elements which are products of $\mathfrak{L}_{X}(U)$ (again by \thmref{Local-Coords-for-D+}). So any map which satisfies the above list of properties and equals $v$ on $\mathfrak{L}_{X}(U)$ is equal to the one we have written down; so the uniqueness follows as well. \end{proof} Now we arrive at \begin{defn} \label{def:D(0,1)}The sheaf of algebras $\mathcal{D}_{X}^{(0,1)}$ is defined as the $\mathbb{Z}$-graded sheaf of $k[v,f]$-algebras, which as a graded sheaf is given by \[ \bigoplus_{i=-\infty}^{-1}\mathcal{D}_{X}^{(0)}\oplus\mathcal{D}_{X}^{(0,1),+} \] and where we extend the action of $f$ (to an operator of degree $1$) from $\mathcal{D}_{X}^{(0,1),+}$ to $\mathcal{D}_{X}^{(0,1)}$ by setting $f=0$ on ${\displaystyle \bigoplus_{i=-\infty}^{-1}\mathcal{D}_{X}^{(0)}}$, and we extend the action of $v$ (to an operator of degree $-1$) on $\bigoplus_{i=1}^{\infty}\mathcal{D}_{X}^{(0,1),i}$ by letting $v:\mathcal{D}_{X}^{(0,1),i}\to\mathcal{D}_{X}^{(0,1),i-1}$ be the identity whenever $i\leq0$. The product on this algebra extends the product on $\mathcal{D}_{X}^{(0,1),+}$ as follows: on the negative half ${\displaystyle \bigoplus_{i=-\infty}^{0}\mathcal{D}_{X}^{(0)}}=\mathcal{D}_{X}^{(0)}\otimes_{k}k[v]$, we use the obvious graded product. For $i\leq0$, if $\Phi\in\mathcal{D}_{X}^{(0,1),i}\tilde{=}D_{X}^{(0)}$ and $\Psi\in\mathcal{D}_{X}^{(0,1),+}$, we set \[ \Phi\cdot\Psi=\Phi_{0}v^{i}(\Psi) \] where $\Phi_{0}$ is the element $\Phi\in D_{X}^{(0)}$, now regarded as an element of degree $0$. \end{defn} From this definition and \thmref{D-is-quasi-coherent}, we see that this is a quasicoherent sheaf of algebras, and we have an isomorphism \[ \widehat{D}_{\mathcal{A}}^{(0,1)}/p\tilde{=}\mathcal{D}_{X}^{(0,1)}(U) \] for any $U=\text{Spec}(A)$ which possesses local coordinates. It follows that $\mathcal{D}_{X}^{(0,1)}$ is a coherent, locally noetherian sheaf of rings which is stalk-wise noetherian. One sees directly the isomorphism $\mathcal{D}_{X}^{(0)}\tilde{=}\mathcal{D}^{(0,1)}/(v-1)$, and we may now define $\mathcal{D}_{X}^{(1)}:=\mathcal{D}^{(0,1)}/(f-1)$. We will see below that this agrees with Berthelot's definition; this is clear if $X$ is liftable but not quite obvious in general. \section{Gauges Over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$} We now have several locally noetherian graded rings and so we can consider categories of modules over them; in particular we have the category of graded $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-modules, (which we refer to a gauges over $\mathfrak{X}$) $\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ and the category of coherent graded modules $\mathcal{G}_{coh}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$. We have the analogous categories in positive characteristic as well as $\mathcal{G}_{qcoh}(\mathcal{D}_{X}^{(0,1)})$, the graded quasicoherent $\mathcal{D}_{X}^{(0,1)}$-modules; as $\mathcal{D}_{X}^{(0,1)}$ is itself a quasi-coherent sheaf of algebras, this is simply the category of sheaves in $\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ which are quasi-coherent over $\mathcal{O}_{X}[f,v]$. In this chapter we develop the basic properties of these categories of gauges; we begin by collecting a few of their most basic properties. For any object in $\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ (or $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$) set $\mathcal{M}^{\infty}:=\mathcal{M}/(f-1)$ and $\mathcal{M}^{-\infty}:=\mathcal{M}/(v-1)$; these are exact functors to the categories of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}$ and $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}(=\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),-\infty})$-modules, respectively; there are obvious maps $f_{\infty}:\mathcal{M}^{i}\to\mathcal{M}^{\infty}$ and $v_{-\infty}:\mathcal{M}^{i}\to\mathcal{M}^{-\infty}$ for each $i$. We use the same notation to denote the analogous constructions for $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$. We have: \begin{lem} \label{lem:Basic-v}Let $\mathcal{M}\in\mathcal{G}_{coh}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$. Then each $\mathcal{M}^{i}$ is coherent as a $\mathcal{\widehat{D}}_{X}^{(0)}$-module. Further, for all $i<<0$, the map $v:\mathcal{M}^{i}\to\mathcal{M}^{i-1}$ is an isomorphism. The same holds for $\mathcal{M}\in\mathcal{G}_{coh}(\mathcal{D}_{X}^{(0,1)})$. \end{lem} \begin{proof} By definition we have, at least locally, an exact sequence \[ \bigoplus_{i=1}^{s}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}(r_{i})\to\bigoplus_{i=1}^{m}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}(l_{i})\to\mathcal{M}\to0 \] Now the result follows as the lemma is true for $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$ by construction. As the same holds for $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$, we may prove $2)$ in an identical manner. \end{proof} This allows us to give: \begin{defn} \label{def:Index!}Let $\mathcal{M}\in\mathcal{G}_{coh}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$. Then the index of $\mathcal{M}$ in $\mathbb{Z}\cup\{\infty\}$ is the largest integer $i$ for which $v:\mathcal{M}^{j}\to\mathcal{M}^{j-1}$ is an isomorphism for all $j\leq i$. The index is $\infty$ if $v$ is an isomorphism for all $i$ (this can indeed happen; c.f. \exaref{Exponential!} below). We can make the same definition for $\mathcal{M}\in\mathcal{G}_{coh}(\mathcal{D}_{X}^{(0,1)})$. \end{defn} We will now use show how cohomological completeness gives a convenient criterion for a complex to be in $D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$. \begin{prop} \label{prop:coh-to-coh}We have $D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))\subset D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$. Further, for $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, we have $\mathcal{M}^{\cdot}\in D_{coh}^{?}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ iff $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{coh}^{?}(\mathcal{G}(\mathcal{\widehat{D}}_{X}^{(0,1)}))$, where $?=+$ or $?=b$. \end{prop} \begin{proof} Recall that if $\mathcal{F}$ is a sheaf of $W(k)$-modules which is $p$-torsion free and $p$-adically complete; or if it is annihilated by $p^{N}$ for some fixed $N\in\mathbb{N}$, then $\mathcal{F}$ (considered as a complex concentrated in one degree) is contained in $D_{cc}(W(k))$. It follows that a coherent $\mathcal{\widehat{D}}_{X}^{(0)}$-module is cohomologically complete; therefore so is an element of $\mathcal{G}_{coh}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ by \propref{Basic-CC-facts}, part $4)$. Since $D_{cc}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ is closed under extensions, the first statement follows directly (c.f. \cite{key-8}, theorem 1.6.1). For the second statement, the forward direction is obvious. For the converse, we note that by \cite{key-8} theorem 1.6.4, since (for either $?=+$ or $?=b$) each \linebreak{} $(\mathcal{M}^{\cdot})^{i}\otimes_{W(k)}^{L}k\in D_{coh}^{+}(\mathcal{D}_{X}^{(0)}-\text{mod})$, we must have $(\mathcal{M}^{\cdot})^{i}\in D_{coh}^{+}(\mathcal{D}_{\mathfrak{X}}^{(0)}-\text{mod})$. In particular $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})$ is $p$-adically complete for each $i$ and $j$. Further, we have the short exact sequences for the functor $\otimes_{W(k)}^{L}k$ \[ 0\to\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p\to\mathcal{H}^{j}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\to\mathcal{T}or_{1}^{W(k)}(\mathcal{H}^{j+1}(\mathcal{M}^{\cdot}),k))\to0 \] which implies also that $\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p$ is coherent over $\mathcal{D}_{X}^{(0,1)}$ for all $j$ (this follows from the fact that $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})/p$ is coherent, and hence quasi-coherent, for all $i$; which implies $\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p$ is a quasicoherent sub-sheaf of the coherent $\mathcal{D}_{X}^{(0,1)}$-module $\mathcal{H}^{j}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$, and hence coherent). Now, for a fixed $j$, we can consider, for any $i$ \[ v:\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})\to\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i-1}) \] and \[ \mathcal{D}_{\mathfrak{X}}^{(0,1),1}\otimes_{\mathcal{D}_{\mathfrak{X}}^{(0)}}\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})\to\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i+1}) \] Since $\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p$ is coherent over $\mathcal{D}_{X}^{(0,1)}$, we have that the reduction mod $p$ of $v$ is surjective for $i<<0$ and the the reduction mod $p$ of the second map is surjective for $i>>0$. By the usual complete Nakayama lemma, we see that $v$ is surjective for $i<<0$ and the second map is surjective for $i>>0$; therefore $\mathcal{H}^{j}(\mathcal{M}^{\cdot})$ is locally finitely generated over $\mathcal{D}_{\mathfrak{X}}^{(0,1)}$ (since each $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})$ is coherent over $\mathcal{D}_{\mathfrak{X}}^{(0)}$). Now, let $U\subset X$ be an open affine and let $D(g)\subset U$ be a principle open inside $U$; let $\tilde{g}$ be a lift of the function $g$ to $\Gamma(\mathcal{O}_{U})$. As each $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})$ is coherent, we have that $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})(D(g))$ is isomorphic to the completion of the localization of $\mathcal{H}^{j}((\mathcal{M}^{\cdot})^{i})(U)$ at $\tilde{g}$. It follows that $\mathcal{H}^{j}(\mathcal{M}^{\cdot})(D(g))$ is given by localizing $\mathcal{H}^{j}(\mathcal{M}^{\cdot})(U)$ at $\tilde{g}$ and then completing each component. If $\mathcal{F}$ is a graded free module over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$, it has the same description; and so the kernel of any map $\mathcal{F}\to\mathcal{H}^{j}(\mathcal{M}^{\cdot})|_{U}$ also has this description (as the functor of localizing and completing is exact on coherent $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules); hence it is locally finitely generated and so $\mathcal{H}^{j}(\mathcal{M}^{\cdot})$ is itself coherent. Finally, we note that $\mathcal{H}^{j}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)=0$ implies $\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p=0$ by the above short exact sequence. So, if $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{coh}^{+}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, we see that $\mathcal{H}^{j}(\mathcal{M}^{\cdot})/p=0$ for all $j<<0$; which implies $\mathcal{H}^{j}(\mathcal{M}^{\cdot})=0$ for $j<<0$ since each $\mathcal{H}^{j}(\mathcal{M}^{\cdot})^{i}$ is $p$-adically complete; i.e., we have $\mathcal{M}^{\cdot}\in D_{\text{Coh}}^{+}(\mathcal{D}_{\mathfrak{X}}^{(0,1)})$; the same argument applies for bounded complexes. \end{proof} This proposition will be our main tool for showing that elements of $D_{cc}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ are actually in $D_{coh}^{b}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$. \subsection{\label{subsec:Standard}Standard Gauges, Mazur's Theorem} In this subsection we discuss the analogue of (the abstract version of) Mazur's theorem in the context of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-gauges. Since the notion of gauge was invented in order to isolate the structures used in the proof of Mazur's theorem, it comes as no surprise that there is a very general version of the theorem available in this context. Before proving it, we discuss some generalities, starting with \begin{defn} \label{def:Standard!}Let $\mathcal{M}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$. We say $\mathcal{M}$ is standard if $\mathcal{M}^{-\infty}$ and $\mathcal{M}^{\infty}$ are $p$-torsion-free, each $f_{\infty}:\mathcal{M}^{i}\to\mathcal{M}^{\infty}$ is injective; and, finally, there is a $j_{0}\in\mathbb{Z}$ so that \[ f_{\infty}(\mathcal{M}^{i+j_{0}})=\{m\in\mathcal{M}^{\infty}|p^{i}m\in f_{\infty}(\mathcal{M}^{j_{0}})\} \] for all $i\in\mathbb{Z}$. \end{defn} The $j_{0}$ appearing in this definition is not unique; indeed, from the definition if $i<0$ we have $f_{\infty}(\mathcal{M}^{i+j_{0}})=p^{-i}\cdot f_{\infty}(\mathcal{M}^{j_{0}})$ which implies that we can replace $j_{0}$ with any $j<j_{0}$. In particular the\emph{ }index\emph{ }of a standard gauge (as in \defref{Index!}) is the maximal $j_{0}$ for which the description in the definition is true (and it takes the value $\infty$ if this description is true for all integers). Note that if $\mathcal{M}$ is standard, so is the shift $\mathcal{M}(j)$, and the index of $\mathcal{M}(j)$ is equal to $\text{index}(\mathcal{M})+j$. As in the case where $\mathfrak{X}$ is a point (which is discussed above in \exaref{BasicGaugeConstruction}), standard gauges are (up to a shift of index) exactly the ones that can be constructed from lattices: \begin{example} \label{exa:Basic-Construction-over-X} Let $\mathcal{N}'$ be a $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}[p^{-1}]$-module, and let $\mathcal{N}$ be a lattice; i.e., a $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-submodule such that $\mathcal{N}[p^{-1}]=\mathcal{N}'$. Recalling the isomorphism $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}[p^{-1}]\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}[p^{-1}]$ (c.f. \lemref{Basic-Structure-of-D^(1)}), we also suppose given a $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}$-lattice of $\mathcal{N}'$ called $\mathcal{M}^{\infty}$. Then we may produce a standard gauge $\mathcal{M}$ via \[ \mathcal{M}^{i}=\{m\in\mathcal{M}^{\infty}|p^{i}m\in\mathcal{N}\} \] If $\mathcal{M}^{\infty}$ is coherent over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}$ and $\mathcal{N}$ is coherent over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$, then $\mathcal{M}$ is a coherent gauge. \end{example} Let us give some general properties of standard gauges: \begin{lem} \label{lem:Standard-is-rigid}Suppose $\mathcal{M}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ is standard; and let $\mathcal{M}_{0}=\mathcal{M}/p$ be its reduction mod $p$. Then $\mathcal{M}_{0}$ has $\text{ker}(f)=\text{im}(v)$ and $\text{ker}(v)=\text{im}(f)$; further, if $\overline{m}_{i}\in\mathcal{M}_{0}^{i}$, then $f\overline{m}_{i}=0=v\overline{m}_{i}$ iff $\overline{m}_{i}=0$. \end{lem} \begin{proof} Since $fv=0$ on $\mathcal{M}_{0}$, we always have $\text{im}(f)\subset\text{ker}(v)$ and $\text{im}(v)\subset\text{ker}(f)$; so we consider the other inclusions. Let $m_{i}\in\mathcal{M}^{i}$, and denote its image in $\mathcal{M}_{0}^{i}$ by $\overline{m}_{i}$. Suppose $v\overline{m}_{i}=0$. Then $vm_{i}=pm_{i-1}$ for some $m_{i-1}\in\mathcal{M}^{i-1}$, so that $f_{\infty}(vm_{i})=pf_{\infty}(m_{i})=pf_{\infty}(m_{i-1})$. Since $\mathcal{M}^{\infty}$ is $p$-torsion-free this yields $f_{\infty}(m_{i})=f_{\infty}(m_{i-1})$ so that $fm_{i-1}=m_{i}$ by the injectivity of $f_{\infty}$. Thus $\overline{m}_{i}\in\text{im}(f)$ and we see $\text{ker}(v)\subset\text{im}(f)$ as required. Now suppose $f\overline{m}_{i}=0$. Then $fm_{i}=pm_{i+1}$ for some $m_{i+1}\in\mathcal{M}^{i+1}$ so that $f_{\infty}(m_{i})=pf_{\infty}(m_{i+1})=f_{\infty}(vm_{i+1})$, and the injectivity of $f_{\infty}$ implies $m_{i}=vm_{i+1}$ so that $\overline{m}_{i}\in\text{im}(v)$ as required. To obtain the last property; since $\mathcal{M}$ is standard, after shifting the grading if necessary, we may identify $f_{\infty}(\mathcal{M}^{i})$ with $\{m\in\mathcal{M}^{\infty}|p^{i}m\in f_{\infty}(\mathcal{M}^{0})\}$. If $m_{i}\in\mathcal{M}^{i}$ and $f\overline{m}_{i}=0=v\overline{m}_{i}$ then $fm_{i}=pm_{i+1}$ and $vm_{i}=pm_{i-1}$; therefore $f_{\infty}(m_{i})=pf_{\infty}(m_{i+1})$ and $pf_{\infty}(m_{i})=pf_{\infty}(m_{i-1})$ so that $p^{2}f_{\infty}(m_{i+1})=pf_{\infty}(m_{i-1})$ which implies $pf_{\infty}(m_{i+1})=f_{\infty}(m_{i-1})$. But $p^{i-1}f_{\infty}(m_{i-1})\in f_{\infty}(\mathcal{M}^{0})$, so that $p^{i}f_{\infty}(m_{i+1})\in f_{\infty}(\mathcal{M}^{0})$ which forces $f_{\infty}(m_{i+1})\in f_{\infty}(\mathcal{M}^{i})$ so that $m_{i+1}=fm'_{i}$ for some $m'_{i}\in\mathcal{M}^{i}$. So $fm_{i}=pm_{i+1}=f(pm_{i}')$ which implies $m_{i}=pm'_{i}$ and so $\overline{m}_{i}=0$. \end{proof} This motivates the \begin{defn} (\cite{key-5}, definition 2.2.2) A gauge $\mathcal{M}_{0}$ over $\mathcal{D}_{X}^{(0,1)}$ is called quasi-rigid if it satisfies $\text{ker}(f)=\text{im}(v)$ and $\text{ker}(v)=\text{im}(f)$, it is called rigid if it is quasi-rigid and, in addition, $\text{ker}(f)\cap$$\text{ker}(v)=0$. \end{defn} By the above lemma, a gauge is rigid if it is of the form $\mathcal{M}/p$ for some standard gauge $\mathcal{M}$. As explained in \cite{key-5}, rigidity is a very nice condition; and in particular we have the following generalization of \cite{key-5}, lemma 2.2.5: \begin{lem} \label{lem:Basic-Facts-on-Rigid}Let $\mathcal{M}_{0}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$. Then $\mathcal{M}_{0}$ is rigid iff $\mathcal{M}_{0}/f$ is $v$-torsion free and $\mathcal{M}_{0}/v$ is $f$-torsion-free. Further, $\mathcal{M}_{0}$ is quasi-rigid iff $\mathcal{M}_{0}\otimes_{k[f,v]}^{L}k[f]\tilde{=}\mathcal{M}_{0}/v$ and $\mathcal{M}_{0}\otimes_{k[f,v]}^{L}k[v]\tilde{=}\mathcal{M}_{0}/f$. \end{lem} \begin{proof} Suppose $\mathcal{M}_{0}$ is rigid. To show $\mathcal{M}_{0}/f$ is $v$-torsion free we have to show that if $m$ is a local section of $\mathcal{M}_{0}$ with $vm=fm'$, then $m\in\text{im}(f)$. Since $\text{im}(f)=\text{ker}(v)$ we have $v(vm)=0$, and since also $f(vm)=0$ we must (by the second condition of rigidity) have $vm=0$. Therefore $m\in\text{ker}(v)=\text{im}(f)$ as desired. The proof that $\mathcal{M}_{0}/v$ is $f$-torsion-free is essentially identical. Now suppose $\mathcal{M}_{0}$ satisfies $\mathcal{M}_{0}/f$ is $v$-torsion free and $\mathcal{M}_{0}/v$ is $f$-torsion-free. Suppose $m\in\text{ker}(f)$. Then the image of $m$ in $\mathcal{M}_{0}/v$ is $f$-torsion, hence $0$; and so $m\in\text{im}(v)$; therefore $\text{ker}(f)=\text{im}(v)$ and similarly $\text{ker}(v)=\text{im}(f)$. If $fm=0=vm$, then $m\in\text{ker}(f)\cap\text{ker}(v)=\text{im}(v)\cap\text{ker}(v)=\text{ker}(f)\cap\text{im}(f)$. Since $m\in\text{im}(v)$ the image of $m$ in $\mathcal{M}_{0}/v$ is zero; also, $m=fm'$, so since $\mathcal{M}_{0}/v$ is $f$-torsion free we see $m'\in\text{im}(v)$. So $m=fm'=fv(m'')=0$ as desired. Now we consider the quasi-rigidity condition: we can write the following free resolution of $k[f]$ over $D(k)$: \[ \cdots\rightarrow D(k)(-1)\xrightarrow{v}D(k)\xrightarrow{f}D(k)(-1)\xrightarrow{v}D(k) \] so that $\mathcal{M}_{0}\otimes_{D(k)}^{L}k[f]$ has no higher cohomology groups iff $\text{ker}(v)=\text{im}(f)$ and $\text{ker}(f)=\text{im}(v)$; the same holds for $\mathcal{M}_{0}\otimes_{D(k)}^{L}k[v]\tilde{=}\mathcal{M}_{0}/f$. \end{proof} Now we turn to conditions for checking that a gauge is standard. \begin{prop} \label{prop:Baby-Mazur}Let $\mathcal{M}\in\mathcal{G}_{\text{coh}}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$, and suppose that $\mathcal{M}^{-\infty}$ and $\mathcal{M}^{\infty}$ are $p$-torsion-free. Set $\mathcal{M}_{0}=\mathcal{M}/p$, and suppose $\mathcal{M}_{0}/v$ is $f$-torsion-free. Then $\mathcal{M}$ is standard; in particular $\mathcal{M}$ is $p$-torsion free. \end{prop} \begin{proof} Let $\mathcal{N}_{i}=\text{ker}(f_{\infty}:\mathcal{M}^{i}\to\mathcal{M}^{\infty})$ and let $\mathcal{N}=\bigoplus_{i}\mathcal{N}_{i}$. Clearly $\mathcal{N}$ is preserved under $W(k)[f]$; further, since $f_{\infty}(vm)=pf_{\infty}(m)$ we see that $\mathcal{N}$ is preserved under $W(k)[f,v]$. For $m$ a local section of $\mathcal{M}^{i}$ let $\overline{m}$ denote its image in $\mathcal{M}_{0}$. If $m\in\mathcal{N}$ then certainly $f_{\infty}(\overline{m})=0$ in $\mathcal{M}_{0}^{\infty}$. Since $\mathcal{M}_{0}/v$ is $f$-torsion-free, the map $f_{\infty}:\mathcal{M}_{0}^{i}/v\mathcal{M}_{0}^{i+1}\to\mathcal{M}_{0}^{\infty}$ is injective; so $\overline{m}\in\text{im}(v)$. Thus there is some $m'$ so that $m-vm'\in p\cdot\mathcal{M}^{i}$; since $p=fv$ we see that $m\in\text{im}(v)$ as well; i.e., we can assume $m=vm'$. Since $0=f_{\infty}(vm')=pf_{\infty}(m')$ and $\mathcal{M}^{\infty}$ is $p$-torsion-free, we see that $m'\in\mathcal{N}$ as well. So in fact $\mathcal{N}=v\cdot\mathcal{N}$. Now, as $\mathcal{M}$ is coherent, we may choose some $j_{0}$ for which $v_{-\infty}:\mathcal{M}^{j}\to\mathcal{M}^{-\infty}$ is an isomorphism for all $j\le j_{0}$. Then, for each such $j$, $\mathcal{M}^{j}$ is $p$-torsion-free (since $\mathcal{M}^{-\infty}$ is). Further, since $fv=p$, we have that $f$ and $v$ are isomorphisms after inverting $p$, which shows $f_{\infty}:\mathcal{M}^{j}[p^{-1}]\tilde{\to}\mathcal{M}^{\infty}[p^{-1}]$. Since $\mathcal{M}^{j}$ and $\mathcal{M}^{\infty}$ are $p$-torsion-free, we see that $f_{\infty}$ is injective on $\mathcal{M}^{j}$. Thus $\mathcal{N}$ is concentrated in degrees above $j_{0}$, and we see that every element of $\mathcal{N}$ is killed by a power of $v$. Since $\mathcal{M}$ is coherent, it is locally noetherian, so that every local section of $\mathcal{M}$ killed by a power of $v$ is actually killed by $v^{N}$ for some fixed $N\in\mathbb{N}$. Therefore, we have $v^{N}\cdot\mathcal{N}=0$. Since also $\mathcal{N}=v\cdot\mathcal{N}$ we obtain $\mathcal{N}=0$. Thus each $f_{\infty}:\mathcal{M}^{i}\to\mathcal{M}^{\infty}$ is injective. It follows that each $\mathcal{M}^{i}$ is $p$-torsion-free, and since $fv=p$ we see that $\mathcal{M}$ is $f$ and $v$-torsion-free as well. Choose $j_{0}$ so that $v:\mathcal{M}^{j}\to\mathcal{M}^{j-1}$ is an isomorphism for all $j\leq j_{0}$. To finish the proof, we have to show that, for all $i\in\mathbb{Z}$, $f_{\infty}(\mathcal{M}^{i+j_{0}})=\{m\in\mathcal{M}^{\infty}|p^{i}m\in f_{\infty}(\mathcal{M}^{j_{0}})\}$. If $i\leq0$, then $v^{-i}:\mathcal{M}^{j_{0}}\to\mathcal{M}^{i+j_{0}}$ is an isomorphism, and $f_{\infty}(\mathcal{M}^{i+j_{0}})=p^{-i}f_{\infty}(\mathcal{M}^{j_{0}})$ as required. If $i>0$, then for $m\in\mathcal{M}^{i+j_{0}}$ we have $f_{\infty}(v^{i}m)=p^{i}f_{\infty}(m)\in f_{\infty}(\mathcal{M}^{j_{0}})$ so that $f_{\infty}(\mathcal{M}^{i+j_{0}})\subseteq\{m\in\mathcal{M}^{\infty}|p^{i}m\in f_{\infty}(\mathcal{M}^{j_{0}})\}$. For the reverse inclusion, let $m\in\mathcal{M}^{\infty}$ be such that $p^{i}m=f_{\infty}(m_{j_{0}})$ for some $m_{j_{0}}\in\mathcal{M}^{j_{0}}$. By definition $\mathcal{M}^{\infty}$ is the union of its sub-sheaves $f_{\infty}(\mathcal{M}^{n})$, so suppose $m=f_{\infty}(m_{l})$ for some $m_{l}\in\mathcal{M}^{l}$, with $l>i+j_{0}$. Since $f_{\infty}(v^{i}m_{l})=p^{i}f_{\infty}(m_{l})=p^{i}m=f_{\infty}(m_{j_{0}})$, we see that \[ f^{l-(i+j_{0})}(m_{j_{0}})=v^{i}m_{l} \] Consider the image of this equation in $\mathcal{M}_{0}$. It shows that that $f^{l-(i+j_{0})}(\overline{m}_{j_{0}})\in v\cdot\mathcal{M}_{0}$. Since $f$ is injective on $\mathcal{M}_{0}/v$, the assumption that $l-(i+j_{0})>0$ implies $\overline{m}_{j_{0}}\in v\cdot\mathcal{M}_{0}$. As above, since $fv=p$ this implies $m_{j_{0}}\in v\cdot\mathcal{M}$; writing $m_{j_{0}}=vm_{j_{0}+1}$ we now have the equation $f^{l-(i+j_{0})}(vm_{j_{0}+1})=v^{i}m_{l}$. Since $v$ acts injectively on $\mathcal{M}$ we see that $f^{l-(i+j_{0})}(m_{j_{0}+1})=v^{i-1}m_{l}$. Applying $f_{\infty}$, we see that $p^{i-1}m\in f_{\infty}(\mathcal{M}^{j_{0}+1})$. If $i=1$, this immediately proves $f_{\infty}(\mathcal{M}^{1+j_{0}})=\{m\in\mathcal{M}^{\infty}|pm\in f_{\infty}(\mathcal{M}^{j_{0}})\}$. For $i>1$, then by induction on $i$ we can suppose $pm\in f_{\infty}(\mathcal{M}^{j_{0}+i-1})$. But then $f_{\infty}(vm_{l})=pf_{\infty}(m_{l})=pm=f_{\infty}(m_{j_{0}+i-1})$ for some $m_{j_{0}+i-1}\in\mathcal{M}^{j_{0}+i-i}$. This implies $f^{l-(j_{0}+i)}(m_{j_{0}+i-1})=vm_{l}$ so if $l>j_{0}+i$ then, arguing exactly as in the previous paragraph, we have $m_{j_{0}+i-1}=vm_{j_{0}+i}$ for some $m_{j_{0}+i}\in\mathcal{M}^{j_{0}+i}$ and so $f^{l-(j_{0}+i)}(m_{j_{0}+i})=m_{l}$ which implies $m=f_{\infty}(m_{l})\in f_{\infty}(\mathcal{M}^{j_{0}+i})$ as required. \end{proof} This result implies a convenient criterium for ensuring that gauges are standard; this is the first analogue of Mazur's theorem in this context: \begin{thm} \label{thm:Mazur!}Let $\mathcal{M}^{\cdot}\in D_{\text{coh}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. Suppose that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{-\infty}$ and $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{\infty}$ are $p$-torsion-free for all $n$, and suppose that $\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free for all $n$. Then $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is standard for all $n$. In particular, $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is $p$-torsion-free, and $\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p$ is rigid for all $n$. We have $\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p\tilde{=}\mathcal{H}^{n}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$ , $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/v\tilde{=}\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$, and $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/f\tilde{=}\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[v])$ for all $n$. Further, $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/f$ is $v$-torsion-free and $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/v$ is $f$-torion-free for all $n$. \end{thm} \begin{proof} Suppose that $b\in\mathbb{Z}$ is the largest integer so that $\mathcal{H}^{b}(\mathcal{M}^{\cdot})\neq0$. Then $b$ is the largest integer for which $\mathcal{H}^{b}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k))\neq0$, and \[ \mathcal{H}^{b}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k))\tilde{=}\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p \] Thus we have a distinguished triangle \[ \tau_{\leq b-1}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\to\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\to(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)[-b] \] to which we may apply the functor $\otimes_{k[f,v]}^{L}k[f]$. This yields \begin{equation} \tau_{\leq b-1}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]\to(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]\to(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)[-b]\otimes_{D(k)}^{L}k[f]\label{eq:triangle!} \end{equation} Since $\otimes_{D(k)}k[f]$ is right exact, the complex on the left is still concentrated in degrees $\leq b-1$, and the middle and right complex are concentrated in degrees $\leq b$. Further \[ \mathcal{H}^{b}((\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)[-b]\otimes_{D(k)}^{L}k[f])\tilde{=}\mathcal{H}^{0}((\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)\otimes_{D(k)}^{L}k[f])\tilde{=}(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)/v \] Therefore $(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)/v\tilde{=}\mathcal{H}^{b}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free by assumption. Thus we may apply the previous proposition to $\mathcal{H}^{b}(\mathcal{M}^{\cdot})$ and conclude that it is standard. Now, to finish the proof that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is standard for all $n$, we proceed by induction on the cohomological length of $\mathcal{M}^{\cdot}$. If the length is $1$ we are done. If not, we have the distinguished triangle \[ \tau_{\leq b-1}(\mathcal{M}^{\cdot})\to\mathcal{M}^{\cdot}\to\mathcal{H}^{b}(\mathcal{M}^{\cdot})[-b] \] which yields the triangle \[ \tau_{\leq b-1}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\to\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\to(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)[-b] \] where we have used that $\mathcal{H}^{b}(\mathcal{M}^{\cdot})$ is $p$-torsion-free to identify $(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)\tilde{=}\mathcal{H}^{b}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k$. As noted above, we have $\mathcal{H}^{b}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k))\tilde{=}\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p$, so this triangle implies the isomorphism \[ \tau_{\leq b-1}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{=}\tau_{\leq b-1}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k) \] Further, since $\mathcal{H}^{b}(\mathcal{M}^{\cdot})$ is standard we have that $\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p$ is rigid; therefore by \lemref{Basic-Facts-on-Rigid} we have $(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)\otimes_{D(k)}^{L}k[f]\tilde{=}(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)/v$ is concentrated in a single degree. Thus, the distinguished triangle \eqref{triangle!} becomes \[ (\tau_{\leq b-1}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]\to(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]\to(\mathcal{H}^{b}(\mathcal{M}^{\cdot})/p)/v[-b] \] and so we have the isomorphism \[ (\tau_{\leq b-1}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]\tilde{=}\tau_{\leq b-1}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]) \] Thus the complex $\tau_{\leq b-1}(\mathcal{M}^{\cdot})$ satisfies the assumption that $(\tau_{\leq b-1}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f]$ has cohomology sheaves which are $f$-torsion-free, and so the complex $\tau_{\leq b-1}(\mathcal{M}^{\cdot})$ satisfies all of the assumptions of the theorem, but has a lesser cohomological length than $\mathcal{M}^{\cdot}$. So we conclude by induction that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is standard for all $n$. For the final part, since standard modules are torsion-free, we see \[ \mathcal{H}^{n}(\mathcal{M}^{\cdot})/p\tilde{=}\mathcal{H}^{n}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k) \] for all $n$, and since each $\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p$ is rigid, the complex $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k$ has cohomology sheaves which are all acyclic for $\otimes_{D(k)}k[f]$ and for $\otimes_{D(k)}k[f]$, by (\lemref{Basic-Facts-on-Rigid}); and the last sentence follows. \end{proof} \begin{rem} As we shall see below, the condition that each cohomology sheaf of $((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free is quite natural; it says that the spectral sequence associated to the Hodge filtration on $(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)^{\infty}$ degenerates at $E^{1}$; this can be checked using Hodge theory in many geometric situations. On the other hand, one conclusion of the theorem is that each cohomology sheaf of $(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[v]$ is $v$-torsion-free; this corresponds to degeneration of the conjugate spectral sequence on $(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)^{-\infty}$. Over a point, one checks in an elementary way (using the finite dimensionality of the vector spaces involved) that these two degenerations are equivalent; this is true irrespective of weather the lift $\mathcal{M}^{\cdot}$ has $p$-torsion-free cohomology groups. This allows one to make various stronger statements in this case (c.f., e.g., \cite{key-10}, proof of theorem 8.26). I don't know if this is true over a higher dimensional base. \end{rem} In most cases of interest, the assumption that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{\infty}$ is $p$-torsion-free is actually redundant, more precisely, it is implied by the assumption that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{-\infty}$ is $p$-torsion-free when one has a Frobenius action; c.f. \thmref{F-Mazur} below. \subsection{Filtrations, Rees algebras, and filtered Frobenius descent} In this section, we consider how the various gradings and filtrations appearing in this paper (in positive characteristic) relate to the more usual Hodge and conjugate filtrations in $\mathcal{D}$-module theory. We start with the basic definitions; as usual $X$ is smooth over $k$. \begin{defn} \label{def:Hodge-and-Con} The decreasing filtration ${\displaystyle \text{image}(\mathcal{D}^{(0,1),i}\to\mathcal{D}_{X}^{(0)})}:=C^{i}(\mathcal{D}_{X}^{(0)})$ is called the conjugate filtration. The increasing filtration ${\displaystyle \text{image}(\mathcal{D}^{(0,1),i}\to\mathcal{D}_{X}^{(1)})}:=F^{i}(\mathcal{D}_{X}^{(1)})$ is called the Hodge filtration. Similarly, for any $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ we may define ${\displaystyle \text{image}(\mathcal{M}^{i}\xrightarrow{v_{\infty}}\mathcal{M}^{-\infty})}:=C^{i}(\mathcal{M}^{-\infty})$ and ${\displaystyle \text{image}(\mathcal{M}^{i}\xrightarrow{f_{\infty}}\mathcal{M}^{\infty})}:=F^{i}(\mathcal{M}^{\infty})$, the conjugate and Hodge filtrations, respectively. \end{defn} \begin{rem} \label{rem:Description-of-conjugate}1) From the explicit description of $v$ given in (the proof of) \lemref{Construction-of-v}, we see that $C^{i}(\mathcal{D}_{X}^{(0)})=\mathcal{I}^{i}\mathcal{D}_{X}^{(0)}$ where $\mathcal{I}$ is the two-sided ideal of $\mathcal{D}_{X}^{(0)}$ generated by $\mathcal{Z}(\mathcal{D}_{X}^{(0)})^{+}$, the positive degree elements of the center\footnote{The center is a graded sheaf of algebras via the isomorphism $\mathcal{Z}(\mathcal{D}_{X}^{(0)})\tilde{=}\mathcal{O}_{T^{*}X^{(1)}}$}. In local coordinates, $\mathcal{I}$ is the just ideal generated by $\{\partial_{1}^{p},\dots,\partial_{n}^{p}\}$, which matches the explicit description of the action of $v$ given above. This is the definition of the conjugate filtration on $\mathcal{D}_{X}^{(0)}$ given, in {[}OV{]} section 3.4, extended to a $\mathbb{Z}$-filtration by setting $C_{i}(\mathcal{D}_{X}^{(0)})=\mathcal{D}_{X}^{(0)}$ for all $i\leq0$. 2) On the other hand, from \thmref{Local-Coords-for-D+}, we see that $F^{i}(\mathcal{D}_{X}^{(1)})$ a locally free, finite $\overline{\mathcal{D}_{X}^{(0)}}$-module; in local coordinates it has a basis $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{0\leq|J|\leq i}$. 3) If $\mathcal{M}$ is a coherent gauge over $X$, then by \lemref{Basic-v} the Hodge filtration of $\mathcal{M}^{\infty}$ is exhaustive and $F^{i}(\mathcal{M}^{\infty})=0$ for $i<<0$, and the conjugate filtration satisfies $C^{i}(\mathcal{M}^{-\infty})=\mathcal{M}^{-\infty}$ for all $i<<0$. \end{rem} \begin{defn} \label{def:Rees-and-Rees-bar}Let $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ denote the Rees algebra of $\mathcal{D}_{X}^{(1)}$ with respect to the Hodge filtration; and let $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$ denote the Rees algebra of $\mathcal{D}_{X}^{(0)}$ with respect to the conjugate filtration. We will denote the Rees parameters (i.e., the element $1\in F^{1}(\mathcal{D}_{X}^{(1)})$, respectively $1\in C^{-1}(\mathcal{D}_{X}^{(0)})$) by $f$ and $v$, respectively. We also let $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ denote the Rees algebra of $\mathcal{D}_{X}^{(0)}$ with respect to the symbol filtration; here the Rees parameter will also be denoted $f$. \end{defn} \begin{lem} We have $\mathcal{D}_{X}^{(0,1)}/v\tilde{=}\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\mathcal{D}_{X}^{(0,1)}/f\tilde{=}\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$ as graded rings. \end{lem} \begin{proof} By \corref{Local-coords-over-A=00005Bf,v=00005D}, we have that $f$ acts injectively on $\mathcal{D}_{X}^{(0,1)}/v$. Since $fv=0$ the map $f_{\infty}:\mathcal{D}_{X}^{(0,1),i}\to\mathcal{D}_{X}^{(1)}$ factors through a map $f_{\infty}:\mathcal{D}_{X}^{(0,1),i}/v\to\mathcal{D}_{X}^{(1)}$, which has image equal to $F^{i}(\mathcal{D}_{X}^{(1)})$ (by definition). The kernel is $0$ since $f$ acts injectively; so $\mathcal{D}_{X}^{(0,1),i}/v\tilde{\to}F^{i}(\mathcal{D}_{X}^{(1)})$ as required. The isomorphism $\mathcal{D}_{X}^{(0,1)}/f\tilde{=}\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$ is proved identically. \end{proof} Therefore we have the natural functors \[ \mathcal{M}^{\cdot}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}k[f]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot} \] from $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ to $D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and \[ \mathcal{M}^{\cdot}\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}k[v]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot} \] from $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ to $D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$. We are going to give some basic results on the derived categories of modules over these rings. As a motivation, we recall general result of Schapira-Schneiders (\cite{key-47}, theorem 4.20; c.f, also example 4.22) \begin{thm} Let $(\mathcal{A},F)$ be a $\mathbb{Z}$-filtered sheaf of rings on a topological space; let $\mathcal{R}(\mathcal{A})$ denote the associated Rees algebra. Let $D((\mathcal{A},F)-\text{mod})$ denote the filtered derived category of modules over $(\mathcal{A},F)$. Then there is an equivalence of categories \[ \mathcal{R}:D((\mathcal{A},F)-\text{mod})\tilde{\to}D(\mathcal{G}(\mathcal{R}(\mathcal{A}))) \] which preserves the subcategories of bounded, bounded below, and bounded above complexes. To a filtered module $\mathcal{M}$ (considered as a complex in degree $0$) this functor attaches the usual Rees module $\mathcal{R}(\mathcal{M})$. \end{thm} Recall that a filtered complex $\mathcal{M}^{\cdot}$ over $(\mathcal{A},F)$ is said to be strict if each morphism $d:(\mathcal{M}^{i},F)\to(\mathcal{M}^{i+1},F)$ satisfies $d(m)\in F_{j}(\mathcal{M}^{i+1})$ iff $m\in F_{j}(\mathcal{M}^{i})$ (for all local sections $m$). Then $\mathcal{M}^{\cdot}$ is quasi-isomorphic to a strict complex iff each cohomology sheaf $\mathcal{H}^{i}(\mathcal{R}(\mathcal{M}^{\cdot}))$ is torsion-free with respect to the Rees parameter. If $\mathcal{M}^{\cdot}$ is a bounded complex, for which the filtration is bounded below (i.e. there is some $j\in\mathbb{Z}$ so that $F_{j}(\mathcal{M}^{i})=0$ for all $i$), then this condition is equivalent to the degeneration at $E_{1}$ of the spectral sequence associated to the filtration. Now we return the discussion to $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$. We begin with the latter; recall that Ogus and Vologodsky in \cite{key-11} have considered the filtered derived category associated to the conjugate filtration on $\mathcal{D}_{X}^{(0)}$; by the above theorem\footnote{The careful reader will note that in their work they require filtrations to be separated; however, this leads to a canonically isomorphic filtered derived category, as explained in \cite{key-59}, proposition 3.1.22 } this category is equivalent to $\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$. After we construct our pushforward on $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, we will show that it is compatible with the one constructed on \cite{key-11}, for now, we will just prove the following basic structure theorem for $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$; following \cite{key-3}, theorem 2.2.3: \begin{prop} We have $\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{O}_{T^{*}X^{(1)}}[v]$; this is a graded ring where $\mathcal{O}_{T^{*}X^{(1)}}$ is graded as usual and $v$ is placed in degree $-1$. The algebra $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is Azumaya over $\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$, of index $p^{\text{dim}(X)}$. In particular, $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})(U)$ has finite homological dimension for each open affine $U$. \end{prop} \begin{proof} The filtered embedding $\mathcal{O}_{T^{*}X^{(1)}}\to\mathcal{D}_{X}^{(0)}$ induces the map $\mathcal{O}_{T^{*}X^{(1)}}[v]\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, by the very definition of the conjugate filtration it is a map of graded rings. To show that this map is an isomorphism onto the center, note that by \corref{Local-coords-over-A=00005Bf,v=00005D}, after choosing etale local coordinates we have that a basis for $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ over $\mathcal{O}_{X}[v]$ is given by $\{\partial^{I}(\partial^{[p]})^{J}\}$ where each entry of $I$is contained in $\{0,\dots,p-1\}$; and the formula for the bracket by $\partial_{i}^{[p]}$ (c.f. \thmref{Local-Coords-for-D+}) shows that $(\partial^{[p]})^{J}$ is now central. Thus the center is given by $\mathcal{O}_{X^{(1)}}[v,\partial_{1}^{[p]},\dots,\partial_{n}^{[p]}]$ which is clearly the (isomorphic) image of the map. The above local coordinates also show that $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is locally free over $\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$, of rank $p^{2\text{dim}(X)}$. Now we can follow the strategy of \cite{key-3}, to show that $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is Azumaya: we consider the commutative subalgebra $\mathcal{A}_{X,v}:=\mathcal{O}_{X}\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{O}_{T^{*}X^{(1)}}[v]$ inside $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$; it acts by right multiplication on $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is a locally free module over it of rank $p^{\text{dim}(X)}$. We have the action map \[ A:\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{T^{*}X^{(1)}}[v]}\mathcal{A}_{X,v}\to\mathcal{E}nd_{\mathcal{A}_{X,v}}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})) \] which is a morphism of algebras, both of which are locally free modules of rank $p^{2\text{dim}(X)}$ over $\mathcal{A}_{X,v}$. Since the left hand side is the pullback of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, considered as a sheaf of algebras over $T^{*}X^{(1)}\times\mathbb{A}^{1}$, to the flat cover $X\times_{X^{(1)}}T^{*}X^{(1)}\times\mathbb{A}^{1}$, we see that $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is Azumaya if $A$ is an isomorphism. To prove that $A$ an isomorphism it suffices to prove it after inverting $v$ and after setting $v=0$. Upon inverting $v$, we have $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})=\mathcal{D}_{X}^{(0)}[v,v^{-1}]$, so the map $A$ simply becomes the analogous map for $\mathcal{D}_{X}^{(0)}$ tensored with $k[v,v^{-1}]$; this is shown to be an isomorphism by \cite{key-3}, proposition 2.2.2. Upon setting $v=0$, we obtain \[ A_{0}:\text{gr}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{T^{*}X^{(1)}}}\mathcal{A}_{X}\to\mathcal{E}nd_{\mathcal{A}_{X}}(\text{gr}(\mathcal{D}_{X}^{(0)})) \] where $\text{gr}(\mathcal{D}_{X}^{(0)})$ is the associated graded of $\mathcal{D}_{X}^{(0)}$ with respect to the conjugate filtration; this is a (split) Azumaya algebra which is easily seen to be isomorphic to $\overline{\mathcal{D}}_{X}^{(0)}\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{O}_{T^{*}X^{(1)}}$ (c.f. \cite{key-11}; the discussion below lemma 3.18). Thus the map $A_{0}$ is again an isomorphism; indeed, we have \[ \text{gr}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{T^{*}X^{(1)}}}\mathcal{A}_{X}\tilde{\to}\overline{\mathcal{D}}_{X}^{(0)}\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{O}_{T^{*}X^{(1)}}\otimes_{\mathcal{O}_{T^{*}X^{(1)}}}\mathcal{A}_{X} \] \[ \tilde{\to}\mathcal{E}nd_{\mathcal{O}_{X^{(1)}}}(\mathcal{O}_{X})\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{A}_{X} \] so that each reduction of $\text{gr}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{T^{*}X^{(1)}}}\mathcal{A}_{X}$ to closed point in $X\times_{X^{(1)}}T^{*}X^{(1)}$ is a matrix algebra of rank $p^{\text{dim}(X)}$, and hence a central simple ring, and the result follows immediately. \end{proof} Next we turn to the category of modules over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, in this case, we can describe them in terms of the familiar filtered $\mathcal{D}_{X}^{(0)}$-modules (in terms of the symbol filtration). The key to doing so is a version of Berthelot's Frobenius descent for filtered $\mathcal{D}_{X}^{(1)}$-modules; while we will consider the more general Frobenius descent over $\mathfrak{X}$ in the next subsection, we will give the basic construction on $X$ for now. To proceed, recall that we have the embedding $\overline{\mathcal{D}_{X}^{(0)}}\subset\mathcal{D}_{X}^{(1)}$ which is simply the image of map $f_{\infty}:\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(1)}$. Let $\mathcal{J}\subset\overline{\mathcal{D}_{X}^{(0)}}$ denote the annihilator of $1\in\mathcal{O}_{X}$ under the action of $\overline{\mathcal{D}_{X}^{(0)}}$ on $\mathcal{O}_{X}$; we have the left ideal $\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$. \begin{prop} \label{prop:Basic-F^*-over-k}There is an isomorphism of $\mathcal{O}_{X}$-modules $\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}\tilde{\to}F^{*}\mathcal{D}_{X}^{(0)}$, thus endowing $F^{*}\mathcal{D}_{X}^{(0)}$ with the structure of a left $\mathcal{D}_{X}^{(1)}$-module; and hence the structure of a $(\mathcal{D}_{X}^{(1)},\mathcal{D}_{X}^{(0)})$-bimodule. Let $F^{i}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J})$ be the filtration induced from the Hodge filtration on $\mathcal{D}_{X}^{(1)}$, and let $F^{i}(\mathcal{D}_{X}^{(0)})$ be the symbol filtration. Then we have \[ F^{i}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J})\cdot F^{j}(\mathcal{D}_{X}^{(0)})=F^{i+j}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}) \] for all $i,j\geq0$. The induced morphism $\mathcal{D}_{X}^{(1)}\to\mathcal{E}nd_{\mathcal{D}_{X}^{(0),\text{opp}}}(F^{*}\mathcal{D}_{X}^{(0)})$ is an isomorphism of filtered algebras. \end{prop} \begin{proof} We put a right $\mathcal{D}_{X}^{(0)}$-module structure on $\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$ as follows: let $\Phi\in\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$ be a section, over some open affine subset $U$ which possesses local coordinates. Let $\partial$ be a derivation over $U$. We may choose a differential operator $\delta$ of order $p$ on $U$ such that $\delta(f^{p})=(\partial f)^{p}$ for all $f\in\mathcal{O}_{X}(U)$; for instance, if $\partial=\sum a_{i}\partial_{i}$ then we may choose $\delta=\sum a_{i}^{p}\partial_{i}^{[p]}$. If $\delta'$ is another such differential operator, then $\delta-\delta'$ is a section of $\overline{\mathcal{D}_{X}^{(0)}}(U)$ which annihilates $\mathcal{O}_{X}(U)^{p}$. In particular, $\delta-\delta'\in\mathcal{J}$, and so $\Phi\delta=\Phi\delta'$ inside $\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$. So ,if we set $\Phi\star f=\Phi\cdot f^{p}$ and $\Phi\star\partial=\Phi\cdot\delta$ we obtain a (semilinear) right action of $\mathcal{D}_{X}^{(0)}$ on $\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$. Since $\mathcal{O}_{X}$ acts on $\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$ on the left, the map \[ (f,\Psi)\to f\star\Psi \] induces a morphism $F^{*}\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$. To show it is an isomorphism, let us consider filtrations: by \thmref{Local-Coords-for-D+} we have that $F^{i}(\mathcal{D}_{X}^{(1)})(U)$ is the free $\overline{\mathcal{D}_{X}^{(0)}}(U)$ module on $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{|J|\leq i}$. Since $\overline{\mathcal{D}_{X}^{(0)}}/\mathcal{J}\tilde{\to}\mathcal{O}_{X}$, we see that $F^{i}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J})$ is the free $\mathcal{O}_{X}(U)$-module on $\{(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}_{|J|\leq i}$. On the other hand, $F^{i}(\mathcal{D}_{X}^{(0)})(U)$ is the free $\mathcal{O}_{X}(U)$-module on $\{\partial_{1}^{j_{1}}\cdots\partial_{n}^{j_{n}}\}_{|J|\leq i}$. Since $1\star\partial_{i}=\partial_{i}^{[p]}$ we deduce $F^{*}F^{i}(\mathcal{D}_{X}^{(0)})=F^{i}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J})$ which implies that the map $F^{*}\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}$ is an isomorphism. The same calculation gives \[ F^{i}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J})\cdot F_{j}(\mathcal{D}_{X}^{(0)})=F^{i+j}(\mathcal{D}_{X}^{(1)}/\mathcal{D}_{X}^{(1)}\cdot\mathcal{J}) \] Therefore, the map \[ \mathcal{D}_{X}^{(1)}\to\mathcal{E}nd_{\mathcal{D}_{X}^{(0),\text{op}}}(F^{*}\mathcal{D}_{X}^{(0)}) \] is a morphism of filtered algebras, where the filtration on the right hand side is defined by \[ F^{i}(\mathcal{E}nd_{\mathcal{D}_{X}^{(0)}}(F^{*}\mathcal{D}_{X}^{(0)}))=\{\varphi\in\mathcal{E}nd_{\mathcal{D}_{X}^{(0)}}(F^{*}\mathcal{D}_{X}^{(0)})|\varphi(F^{j}(F^{*}\mathcal{D}_{X}^{(0)})\subset F^{i+j}(F^{*}\mathcal{D}_{X}^{(0)})\phantom{i}\text{for all}\phantom{i}j\} \] Upon passing to the associated graded, we obtain the morphism \[ \text{gr}(\mathcal{D}_{X}^{(1)})\to\text{gr}\mathcal{E}nd_{\mathcal{D}_{X}^{(0),\text{op}}}(F^{*}\mathcal{D}_{X}^{(0)})\tilde{=}\mathcal{E}nd_{\text{gr}(\mathcal{D}_{X}^{(0)})}(\text{gr}(F^{*}\mathcal{D}_{X}^{(0)})) \] (the last isomorphism follows from the fact that $F^{*}\mathcal{D}_{X}^{(0)}$ is a locally free filtered module over $\mathcal{D}_{X}^{(0)}$). Working in local coordinates, we obtain the morphism \[ \mathcal{\overline{D}}_{X}^{(0)}[\partial_{1}^{[p]},\dots,\partial_{n}^{[p]}]\to\mathcal{E}nd_{\text{Sym}_{\mathcal{O}_{X}}(\mathcal{T}_{X})}(F^{*}(\text{Sym}_{\mathcal{O}_{X}}(\mathcal{T}_{X}))) \] where $\partial_{i}^{[p]}$ is sent to $\partial_{i}\in\mathcal{T}_{X}$. By Cartier descent, there is an isomorphism $\mathcal{\overline{D}}_{X}^{(0)}\tilde{=}\mathcal{E}nd_{\mathcal{O}_{X}}(F^{*}\mathcal{O}_{X})$ (here, the action of $\mathcal{O}_{X}$ on $F^{*}\mathcal{O}_{X}$ is on the right-hand factor in the tensor product; in other words, it is the action of $\mathcal{O}_{X}$ on itself through the Frobenius); and so we see that this map is an isomorphism. Thus the map $\mathcal{D}_{X}^{(1)}\to\mathcal{E}nd_{\mathcal{D}_{X}^{(0),\text{op}}}(F^{*}\mathcal{D}_{X}^{(0)})$ is an isomorphism as claimed. \end{proof} This yields a functor $\mathcal{M}\to F^{*}\mathcal{M}:=F^{*}\mathcal{D}_{X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}\mathcal{M}$ (the Frobenius pullback) from $\mathcal{D}_{X}^{(0)}-\text{mod}$ to $\mathcal{D}_{X}^{(1)}-\text{mod}$; from the last part of the above proposition and standard Morita theory one sees that it is an equivalence of categories. Further: \begin{thm} \label{thm:Filtered-Frobenius} The Frobenius pullback $F^{*}$ can be upgraded to an equivalence from $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))$ to $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. Therefore, the functor $F^{*}$ can also be upgraded to an equivalence of categories from filtered $\mathcal{D}_{X}^{(0)}$-modules to filtered $\mathcal{D}_{X}^{(1)}$-modules. In particular, $\mathcal{R}(\mathcal{D}_{X}^{(1)})(U)$ has finite homological dimension for each open affine $U$. \end{thm} \begin{proof} In \propref{Basic-F^*-over-k}, we showed that $F^{*}\mathcal{D}_{X}^{(0)}$ is filtered, in a way that strictly respects the filtered action of both $\mathcal{D}_{X}^{(1)}$ and $\mathcal{D}_{X}^{(0)}$. So, consider the Rees module $\mathcal{R}(F^{*}\mathcal{D}_{X}^{(0)})$. This is a graded $(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\mathcal{R}(\mathcal{D}_{X}^{(0)}))$-bimodule; and the isomorphism $F^{*}F_{i}(\mathcal{D}_{X}^{(0)}\tilde{=}F_{i}(F^{*}\mathcal{D}_{X}^{(0)})$ proved in loc.cit. shows that $\mathcal{R}(F^{*}\mathcal{D}_{X}^{(0)})\tilde{=}F^{*}\mathcal{R}_{X}^{(0)}$ as a right $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-module. Thus the result will follow if we can show that the action map \begin{equation} \mathcal{R}(\mathcal{D}_{X}^{(1)})\to\mathcal{E}nd_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}(\mathcal{R}(F^{*}\mathcal{D}_{X}^{(0)}))\tilde{=}\underline{\mathcal{E}nd}{}_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}(\mathcal{R}(F^{*}\mathcal{D}_{X}^{(0)}))\label{eq:first-map} \end{equation} is an isomorphism (the latter isomorphism follows from the fact that $\mathcal{R}(F^{*}\mathcal{D}_{X}^{(0)})$ is coherent over $\mathcal{R}(\mathcal{D}_{X}^{(0)})$). Both sides are therefore positively graded algebras over the ring $k[f]$; taking reduction mod $f$ we obtain the map $\text{gr}(\mathcal{D}_{X}^{(1)})\to\mathcal{E}nd_{\text{gr}(\mathcal{D}_{X}^{(0)})}(\text{gr}(F^{*}\mathcal{D}_{X}^{(0)}))$ which we already showed to be an isomorphism. Thus by the graded Nakayama lemma \eqref{first-map} is surjective. As both sides are $f$-torsion free it follows that it is an isomorphism. Thus the first result of $1)$ is proved, the second follows by identifying filtered modules with graded modules over the Rees ring which are torsion-free with respect to $f$. \end{proof} \begin{rem} \label{rem:The-inverse-to-F^*}The inverse to the functor $F^{*}$ can be described as follows: via the embedding $\overline{\mathcal{D}_{X}^{(0)}}\subset\mathcal{R}(\mathcal{D}_{X}^{(1)})$, any module $\mathcal{M}$ over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ possesses a connection which has $p$-curvature $0$. Apply this to $\mathcal{R}(\mathcal{D}_{X}^{(1)})/\mathcal{J}\cdot\mathcal{R}(\mathcal{D}_{X}^{(1)})$, where as above $\mathcal{J}\subset\overline{\mathcal{D}_{X}^{(0)}}$ denotes the annihilator of $1\in\mathcal{O}_{X}$ under the action of $\overline{\mathcal{D}_{X}^{(0)}}$ on $\mathcal{O}_{X}$. We obtain from the above argument the isomorphism \[ (\mathcal{R}(\mathcal{D}_{X}^{(1)})/\mathcal{J}\cdot\mathcal{R}(\mathcal{D}_{X}^{(1)}))^{\nabla}\tilde{=}\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)}) \] Thus for any the sheaf $\mathcal{M}^{\nabla}:=\text{ker}(\nabla:\mathcal{M}\to\mathcal{M})$ inherits the structure of a module over $\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)})$. As $k$ is perfect we have an isomorphism of schemes $\sigma:X^{(1)}\to X$, and so composing with this we can obtain from $\mathcal{M}^{\nabla}$ an $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-module; this is the inverse functor to $F^{*}$. \end{rem} To close out this subsection, we'd like to discuss an important tool for studying $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$; namely, reducing statements to their analogues in $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. For any $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$, we have a short exact sequence \[ 0\to\text{ker}(f)\to\mathcal{M}\to\mathcal{M}/\text{ker}(f)\to0 \] the module on the left is annihilated by $f$; i.e., it is a module over $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$ while the module on the right is annihilated by $v$; i.e., it is a module over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$. This allows us to deduce many basic structural properties of $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ from properties of $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$ and $\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))$. We now give the key technical input; to state it, we will abuse notation slightly, so that if $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ (or in $D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$) we will use the same symbol $\mathcal{M}^{\cdot}$ to denote its image in $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ \begin{prop} \label{prop:Sandwich!}1) Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$. Suppose $\mathcal{N}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1),\text{opp}})$ is quasi-rigid. Then \[ \mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\mathcal{N}/v\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M}^{\cdot} \] Similarly, if $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$ we have \[ \mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\mathcal{N}/f\otimes_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}^{L}\mathcal{M}^{\cdot} \] The analogous statement holds for $\mathcal{N}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ quasi-rigid and $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})^{\text{opp}}))$, resp. $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})^{\text{opp}}))$. 2) As above, let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and suppose $\mathcal{N}$ is quasi-rigid. Then \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}^{\cdot},\mathcal{N})\tilde{=}R\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{M}^{\cdot},\text{ker}(v:\mathcal{N}\to\mathcal{N})) \] Similarly, if $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$ then \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M},\mathcal{N})\tilde{=}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{M},\text{ker}(f:\mathcal{N}\to\mathcal{N})) \] \end{prop} \begin{proof} 1) Choose a flat resolution $\mathcal{F}^{\cdot}\to\mathcal{N}$ (in the category of right $\mathcal{D}_{X}^{(0,1)}$-gauges); concretely, the terms of $\mathcal{F}^{\cdot}$ are direct sums of sheaves of the form $j_{!}(\mathcal{D}_{X}^{(0,1)}(i)|_{U})$ (where $U\subset X$ is open and $j_{!}$ denotes extension by zero). Then $\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}$ is represented by the complex \[ \mathcal{F}^{\cdot}\otimes_{\mathcal{D}_{X}^{(0,1)}}\mathcal{M}^{\cdot}\tilde{=}(\mathcal{F}/v)^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}\mathcal{M}^{\cdot} \] where the isomorphism follows from the fact that each term of $\mathcal{M}^{\cdot}$ is annihilated by $v$. On the other hand, $(\mathcal{F}/v)^{\cdot}$ is a complex, whose terms are direct sums of sheaves of the form $j_{!}(\mathcal{R}(\mathcal{D}_{X}^{(1)}(i)|_{U})$, which computes $\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})$. However, we have $\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})\tilde{=}\mathcal{N}/v$ by the assumption on $\mathcal{N}$ (c.f. \lemref{Basic-Facts-on-Rigid}) Therefore $(\mathcal{F}/v)^{\cdot}$ is a flat resolution (in the category of graded right $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules) of $\mathcal{N}$, and so \[ (\mathcal{F}/v)^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}\mathcal{M}^{\cdot}\tilde{=}\mathcal{N}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M}^{\cdot} \] as claimed. The case $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$ is essentially identical. 2) Choose an injective resolution $\mathcal{N}\to\mathcal{I}^{\cdot}$. Then we have that $R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}^{\cdot},\mathcal{N})$ is represented by \[ \underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}^{\cdot},\mathcal{I}^{\cdot})=\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}^{\cdot},\mathcal{I}^{\cdot,v=0})=\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{M}^{\cdot},\mathcal{I}^{\cdot,v=0}) \] where $\mathcal{I}^{j,v=0}=\{m\in\mathcal{I}^{j}|vm=0\}$. From the isomorphism $\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M},\mathcal{I}^{\cdot})=\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{M},\mathcal{I}^{\cdot,v=0})$ we see that the the functor $\mathcal{I}\to\mathcal{I}^{v=0}$ takes injectives in $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ to injectives in $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. On the other hand, we have $\mathcal{I}^{j,v=0}=\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\mathcal{I}^{j,v=0})$. Thus the functor $\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{R}(\mathcal{D}_{X}^{(1)}),)$ takes injectives in $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ to injectives in $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and so \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}^{\cdot},\mathcal{N})\tilde{=}R\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\mathcal{N})) \] On the other hand, using the resolution \[ \cdots\to\mathcal{D}_{X}^{(0,1)}(-1)\xrightarrow{v}\mathcal{D}_{X}^{(0,1)}\xrightarrow{f}\mathcal{D}_{X}^{(0,1)}(-1)\xrightarrow{v}\mathcal{D}_{X}^{(0,1)}\to\mathcal{R}(\mathcal{D}_{X}^{(1)}) \] one deduces \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\mathcal{N})\tilde{=}\text{ker}(v:\mathcal{N}\to\mathcal{N}) \] and the first statement in $2)$ follows; the second statement is proved in an identical fashion. \end{proof} Here is a typical application: \begin{prop} \label{prop:Quasi-rigid=00003Dfinite-homological}A quasicoherent gauge $\mathcal{N}\in\mathcal{G}_{qcoh}(\mathcal{D}_{X}^{(0,1)})$ is quasi-rigid iff, for each open affine $U\subset X$, $\mathcal{N}(U)$ has finite projective dimension over $\mathcal{D}_{X}^{(0,1)}(U)$. \end{prop} \begin{proof} Let $\mathcal{N}$ be quasi-rigid. Then for any quasicoherent $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1),\text{opp}})$, we have the short exact sequence \[ 0\to\text{ker}(f)\to\mathcal{M}\to\mathcal{M}/\text{ker}(f)\to0 \] which yields the distinguished triangle \[ 0\to\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\text{ker}(f)\to\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}\to\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}/\text{ker}(f)\to0 \] Applying the previous result; we see that the outer two tensor products are isomorphic to tensor products over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$, respectively. As these algebras have finite homological dimension (the dimension is $2\text{dim}(X)+1$, in fact) over any open affine, we see that $\mathcal{N}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}$ is a bounded complex; since this is true for all $\mathcal{M}$ we obtain the forward implication. For the reverse, note that by \lemref{Basic-Facts-on-Rigid}, the functor $\mathcal{M}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}\mathcal{M}\tilde{\to}k[f]\otimes_{D(k)}\mathcal{M}$ has infinite homological dimension when $\mathcal{M}$ is not quasi-rigid. \end{proof} \subsection{\label{subsec:Frobenius-Descent,--Gauges}Frobenius Descent, $F^{-1}$-Gauges} In this section we recall Berthelot's theory of Frobenius descent for $\mathcal{D}$-modules and give the definition of an $F^{-1}$-gauge over a higher dimensional base. We begin by briefly recalling Berthelot's theory of the Frobenius action in mixed characteristic. This is developed using the theory of (mixed) divided powers in \cite{key-2}; for the reader's convenience we will recall a simple description in the case of interest to us (this point of view is emphasized in \cite{key-48}). First suppose that $\mathfrak{X}$ admits an endomorphism $F$ which lifts the Frobenius on $X$, and whose restriction to $W(k)$ agrees with the Witt-vector Frobenius on $W(k)$. This is equivalent to giving a morphism $\mathfrak{X}\to\mathfrak{X}^{(1)}$ whose composition with the natural map $\mathfrak{X}^{(1)}\to\mathfrak{X}$ agrees with $F$ (here, $\mathfrak{X}^{(1)}$ denotes the first Frobenius twist of $\mathfrak{X}$ over $W(k)$); we will also denote the induced morphism $\mathfrak{X}\to\mathfrak{X}^{(1)}$ by $F$. On the underlying topological spaces (namely $X$ and $X^{(1)}$), this map is a bijection, and we shall consistently consider $\mathcal{O}_{\mathfrak{X}^{(1)}}$ as a sheaf of rings on $X$, equipped with an injective map of sheaves of algebras $F^{\#}:\mathcal{O}_{\mathfrak{X}^{(1)}}\to\mathcal{O}_{\mathfrak{X}}$ which makes $\mathcal{O}_{\mathfrak{X}}$ into a finite $\mathcal{O}_{\mathfrak{X}^{(1)}}$-module. Now consider the sheaf $\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}^{(1)}},\mathcal{O}_{\mathfrak{X}})$. For any $i\geq0$, this is a $(\mathcal{D}_{\mathfrak{X}}^{(i+1)},\mathcal{D}_{\mathfrak{X}^{(1)}}^{(i)})$ bi-module (via the actions of these rings of differential operators on $\mathcal{O}_{\mathfrak{X}}$ and $\mathcal{O}_{\mathfrak{X}^{(1)}}$, respectively). Then we have the \begin{thm} \label{thm:Berthelot-Frob}(Berthelot) The $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)},\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)})$ bi-sub-module of $\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}^{(1)}},\mathcal{O}_{\mathfrak{X}})$ locally generated by $F^{\#}$ is isomorphic to $\mathcal{O}_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}^{(1)}}}\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}$, via the map \[ (f,\Phi)\to f\circ F^{\#}\circ\Phi\in\mathcal{H}om_{W(k)}(\mathcal{O}_{\mathfrak{X}^{(1)}},\mathcal{O}_{\mathfrak{X}}) \] for local sections $f\in\mathcal{O}_{\mathfrak{X}}$ and $\Phi\in\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}$. This gives the sheaf $\mathcal{O}_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}^{(1)}}}\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}=F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}$ the structure of a $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)},\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)})$ bimodule. The associated functor, denoted $F^{*}$, \[ \mathcal{M}\to F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}\otimes_{\mathcal{D}_{\mathfrak{X}^{(1)}}^{(i)}}\mathcal{M}\tilde{=}F^{*}\mathcal{M} \] is an equivalence of categories from $\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)}-\text{mod}\to\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}-\text{mod}$ ; which induces an equivalence $\text{Coh}(\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)})\to\text{Coh}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)})$. In particular, the map $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(i+1)}\to\mathcal{E}nd_{\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i),\text{op}}}(F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}^{(1)}}^{(i)})$ is an isomorphism of sheaves of algebras. As $W(k)$ is perfect, we have an isomorphism $\mathfrak{X}^{(1)}\tilde{\to}\mathfrak{X}$; and we may therefore regard $F^{*}$ as being an equivalence of categories from $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}-\text{mod}$ to $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}-\text{mod}$ \end{thm} This is proved in \cite{key-2}, section 2.3. In fact, in the case where $\mathfrak{X}=\text{Specf}(\mathcal{A})$ is affine and admits etale local coordinates, and the map $F$ acts on coordinates $\{t_{i}\}_{i=1}^{n}$ via $F(t_{i})=t_{i}^{p}$, then the first assertion can be proved quite directly. The second is the theory of \cite{key-2}. Note that this description implies that the reduction mod $p$ of the bimodule $F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}$ agrees with the bimodule $F^{*}\mathcal{D}_{X}^{(0)}$ constructed in \propref{Basic-F^*-over-k}. \begin{rem} \label{rem:Compare-With-Berthelot}1) Let $\mathcal{D}_{X,\mathbf{Ber}}^{(1)}$ denote Berthelot's ring of divided power differential operators of level $1$ on $X$. Then the Frobenius descent theory of the previous theorem gives an isomorphism \[ \mathcal{D}_{X,\text{Ber}}^{(1)}\to\mathcal{E}nd_{\mathcal{D}_{X}^{(0),\text{op}}}(F^{*}\mathcal{D}_{X}^{(0)}) \] It follows that $\mathcal{D}_{X,\mathbf{Ber}}^{(1)}\tilde{=}\mathcal{D}_{X}^{(1)}$ even if $X$ is not liftable to $W(k)$. 2) The Frobenius descent over $X$ implies the Frobenius descent over $\mathfrak{X}$, once one constructs the $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(1)},\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)})$ bimodule structure on $F^{*}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$. Indeed, this structure yields a morphism \[ \widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}\to\mathcal{E}nd_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0),\text{op}}}(F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}) \] as both sides are $p$-adically complete and $p$-torsion-free, to check that this map is an isomorphism one simply has to reduce mod $p$. \end{rem} Now let us return to a general $\mathfrak{X}$. It is an fundamental fact that Frobenius descent doesn't really depend on the existence of the lift $F$: \begin{thm} (Berthelot) Suppose $F_{1},F_{2}$ are two lifts of Frobenius on $\mathfrak{X}$. Then there is an isomorphism of bimodules $\sigma_{1,2}:F_{1}^{*}\mathcal{D}_{\mathfrak{X}}^{(i)}\tilde{\to}F_{2}^{*}\mathcal{D}_{\mathfrak{X}}^{(i)}$. If $F_{3}$ is a third lift, we have $\sigma_{2,3}\circ\sigma_{12}=\sigma_{1,3}$. \end{thm} This is \cite{key-2}, theorem 2.2.5; c.f. also \cite{key-21}, corollary 13.3.8. As lifts of Frobenius always exist locally, this implies that there is a globally defined bimodule $F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$, which induces an equivalence $F^{*}:\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}-\text{mod}\to\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}-\text{mod}$; we use the same letter to denote the derived equivalence $F^{*}:D(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}-\text{mod})\to D(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}-\text{mod})$. The equivalence of categories $F^{*}$ has many remarkable properties, in particular its compatibility with the push-forward, pullback, and duality functors for $\mathcal{D}$-modules; we will recall these properties in the relevant sections below. It will also be useful to recall some basic facts about the right-handed version of the equivalence. Recall that we have equivalences of categories $\mathcal{M}\to\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}$ from $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}-\text{mod}$ to $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i),\text{op}}-\text{mod}$ for any $i$ (c.f, \cite{key-1}, or \propref{Left-Right-Swap} below). This implies that there is a functor $\mathcal{M}\to F^{!}\mathcal{M}:=\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}F^{*}(\omega_{\mathfrak{X}}^{-1}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M})$ which is an equivalence from $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i),\text{op}}-\text{mod}$ to $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1),\text{op}}-\text{mod}$. By basic Grothendieck duality theory (c.f. \cite{key-2}, 2.4.1), there is an isomorphism \[ F^{!}\mathcal{M}\tilde{=}F^{-1}\mathcal{H}om_{\mathcal{O}_{\mathfrak{X}}}(F_{*}\mathcal{O}_{\mathfrak{X}},\mathcal{M}) \] of sheaves of $\mathcal{O}_{\mathfrak{X}}$-modules (this justifies the notation). If we put $\mathcal{M}=\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$ this isomorphism exhibits the left $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$-module structure on $F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$. \begin{prop} \label{prop:F^*F^!}1) The equivalence of categories $F^{!}:\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i),\text{op}}-\text{mod}\to\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1),\text{op}}-\text{mod}$ is given by $\mathcal{M}\to\mathcal{M}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}}F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$. 2) There are isomorphisms of $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+!)},\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)})$ bimodules $F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}}F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}=F^{*}F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}\tilde{\to}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}$ and $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}\tilde{\leftarrow}F^{!}F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}=F^{*}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}}F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$. In particular, for a $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}$-module $\mathcal{M}$, we have $\mathcal{M}=F^{*}\mathcal{N}$ iff $F^{!}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i+1)}}\mathcal{M}\tilde{=}\mathcal{N}$. \end{prop} This is proved in \cite{key-2}, 2.5.1 (c.f. also \cite{key-21}, lemma 13.5.1). Further, by applying the Rees functor it directly implies the analogue for the filtered Frobenius descent of \thmref{Filtered-Frobenius}: \begin{cor} \label{cor:Filtered-right-Frob}There is an equivalence of categories $F^{!}:\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})^{\text{op}})\to\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})^{\text{op}})$; which yields a $(\mathcal{R}(\mathcal{D}_{X}^{(0)}),\mathcal{R}(\mathcal{D}_{X}^{(1)}))$ bimodule $F^{!}\mathcal{R}(\mathcal{D}_{X}^{(0)})$. We have isomorphisms of $(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\mathcal{R}(\mathcal{D}_{X}^{(1)}))$ bimodules \[ F^{!}F^{*}\mathcal{R}(\mathcal{D}_{X}^{(0)})\tilde{\to}\mathcal{R}(\mathcal{D}_{X}^{(1)})\leftarrow F^{*}F^{!}\mathcal{R}(\mathcal{D}_{X}^{(0)}) \] \end{cor} Now we proceed to the \begin{defn} \label{def:Gauge-Defn!}An $F^{-1}$-gauge over $\mathfrak{X}$ is an object of $\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ equipped with an isomorphism $F^{*}\mathcal{M}^{-\infty}\tilde{\to}\widehat{\mathcal{M}^{\infty}}$ (here $\widehat{?}$ denotes $p$-adic completion). A coherent $F^{-1}$-gauge is an $F^{-1}$-gauge whose underlying $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module is coherent. We define the category of $F^{-1}$-gauges, $\mathcal{G}_{F^{-1}}(\mathcal{D}_{\mathfrak{X}}^{(0,1)})$ by demanding that morphisms between $F^{-1}$-gauges respect the $F^{-1}$-structure (as in \defref{F-gauge}), and similarly for the category of coherent $F^{-1}$-gauges, $\mathcal{G}_{F^{-1},coh}(\mathcal{D}_{\mathfrak{X}}^{(0,1)})$. Similarly, An $F^{-1}$-Gauge over $X$ is an object of $\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ equipped with an isomorphism $F^{*}\mathcal{M}^{-\infty}\tilde{\to}\widehat{\mathcal{M}^{\infty}}$, and for the category $\mathcal{G}_{F^{-1}}(\mathcal{D}_{X}^{(0,1)})$ we demand that morphisms between $F^{-1}$-gauges respect the $F^{-1}$-structure. We have the obvious subcategories of quasi-coherent and coherent gauges. \end{defn} In the world of coherent gauges, we have seen in \propref{Completion-for-noeth} that completion is an exact functor. Therefore, the category of coherent $F^{-1}$-gauges over $\mathfrak{X}$ is abelian; the same does not seem to be true for the category of all gauges over $\mathfrak{X}$. On the other hand, the category of all $F^{-1}$-gauges over $X$ is abelian, as are the categories of coherent and quasicoherent $F^{-1}$-gauges. Now let us turn to the derived world: \begin{defn} \label{def:F-gauge-for-complexes}A complex $\mathcal{M}^{\cdot}$ in $D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ is said to admit the structure of an $F^{-1}$-gauge if there is an isomorphism $F^{*}(\mathcal{M}^{\cdot})^{-\infty}\tilde{\to}\widehat{(\mathcal{M}^{\cdot})^{\infty}}$ where $\widehat{}$ denotes the cohomological completion. Similarly, we say that a complex $\mathcal{M}^{\cdot}$ in $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ admits the structure of an $F^{-1}$-gauge if there is an isomorphism $F^{*}(\mathcal{M}^{\cdot})^{-\infty}\tilde{\to}(\mathcal{M}^{\cdot})^{\infty}$. We will use a subscript $F^{-1}$ to denote the relevant categories; e.g. $D_{F^{-1}}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$. \end{defn} These are not triangulated categories in general, though there is an obvious functor $D^{b}(\mathcal{G}_{F^{-1},coh}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))\to D_{coh,F^{-1}}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ (and similarly for $X$). To give the correct triangulated analogue of \defref{Gauge-Defn!} one must use higher homotopy theory; namely, the glueing of $\infty$-categories along a pair of functors. I intend to peruse this in a later project. For the purposes of this paper, \defref{F-gauge-for-complexes} will suffice. \begin{rem} \label{rem:Cut-off-for-F-gauges}Suppose $\mathcal{M}^{\cdot}\in D_{coh,F^{-1}}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$, then by \propref{Completion-for-noeth} $\widehat{\mathcal{H}^{i}(\mathcal{M}^{\cdot})^{\infty}}\tilde{=}\mathcal{H}^{i}(\widehat{\mathcal{M}^{\cdot,\infty}})$. Therefore $\mathcal{H}^{i}(\mathcal{M}^{\cdot})$ admits the structure of an $F^{-1}$-gauge for each $i$. Further, as both $F^{*}$ and the completion functor are exact, we have that $\tau_{\leq i}(\mathcal{M}^{\cdot})$ and $\tau_{\geq i}(\mathcal{M}^{\cdot})$ are contained in $\mathcal{M}^{\cdot}\in D_{coh,F^{-1}}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$, where $\tau_{\leq i},\tau_{\geq i}$ are the cut-off functors. \end{rem} Given this, we can give the more refined version of Mazur's theorem for $F^{-1}$-gauges: \begin{thm} \label{thm:F-Mazur}Let $\mathcal{M}^{\cdot}\in D_{\text{coh},F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. Suppose that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{-\infty}$ is $p$-torsion-free for all $n$, and suppose that $\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free for all $n$. Then $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is standard for all $n$. In particular, $\mathcal{H}^{n}(\mathcal{M}^{\cdot})$ is $p$-torsion-free, and $\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p$ is rigid for all $n$. We have $\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p\tilde{=}\mathcal{H}^{n}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$ , $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/v\tilde{=}\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$, and $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/f\tilde{=}\mathcal{H}^{n}((\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[v])$ for all $n$. Further, $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/f$ is $v$-torsion-free and $(\mathcal{H}^{n}(\mathcal{M}^{\cdot})/p)/v$ is $f$-torion-free for all $n$. \end{thm} \begin{proof} This follows from \thmref{Mazur!} if we can show that $\mathcal{H}^{n}(\mathcal{M}^{\cdot})^{\infty}\tilde{=}\mathcal{H}^{n}(\mathcal{M}^{\cdot,\infty})$ is also $p$-torsion-free for all $n$. Since $\mathcal{M}^{\cdot}\in D_{coh,F^{-1}}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ , we have that the cohomological completion of the complex $\mathcal{\widehat{M}}^{\cdot,\infty}$ is isomorphic to $F^{*}(\mathcal{M}^{\cdot,-\infty})$; and this complex has $p$-torsion-free cohomologies by the assumption. Since $\mathcal{M}^{\cdot,\infty}$ is a bounded complex with coherent cohomology sheaves, by \propref{Completion-for-noeth} we have that $\mathcal{H}^{n}(\mathcal{\widehat{M}}^{\cdot,\infty})\tilde{=}\widehat{\mathcal{H}^{n}(\mathcal{M}^{\cdot,\infty})}$, where the completion on the right denotes the usual $p$-adic completion. But the module $\mathcal{H}^{n}(\mathcal{M}^{\cdot,\infty})$, being coherent, is $p$-torsion-free iff its $p$-adic completion is. Thus each $\mathcal{H}^{n}(\mathcal{M}^{\cdot,\infty})$ is $p$-torsion-free as desired. \end{proof} In the case where $\mathfrak{X}=\text{Specf}(W(k))$ is a point, and $\mathcal{M}^{\cdot}$ is the gauge coming from cohomology of some smooth proper $\mathfrak{X}$ (this exists by \thmref{=00005BFJ=00005D}, and we'll construct it, in the language of this paper, in \secref{Push-Forward} below), this is exactly the content of \thmref{(Mazur)}; indeed, the first assumption is that $\mathbb{H}_{dR}^{i}(\mathfrak{X})$ is $p$-torsion-free for all $i$, and the second assumption is the degeneration of the Hodge to de Rham spectral sequence. \subsection{Examples of Gauges} We close out this chapter by giving a few important examples of gauges, beyond $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ itself. \begin{example} Let $\mathfrak{X}$ be a smooth formal scheme. Then $D(\mathcal{O}_{\mathfrak{X}})\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ by the very definition of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ ; indeed, we have $D(\mathcal{O}_{\mathfrak{X}}){}^{i}=\{g\in\mathcal{O}_{\mathfrak{X}}|p^{i}g\in\mathcal{O}_{\mathfrak{X}}\}$ so that the natural action of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}[p^{-1}]$ on $\mathcal{O}_{\mathfrak{X}}[p^{-1}]$ induces the action of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ on $\mathcal{O}_{\mathfrak{X}}[f,v]$. This is an $F^{-1}$-gauge via the isomorphism $F^{*}\mathcal{O}_{\mathfrak{X}}\tilde{\to}\mathcal{O}_{\mathfrak{X}}$. \end{example} To generalize this, suppose $\mathfrak{D}\subset\mathfrak{X}$ is a locally normal crossings divisor. Let $\mathfrak{U}$ be the compliment of $\mathfrak{D}$. Denote the inclusion map by $j$. We are going to define a coherent $F^{-1}$-gauge ${\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{U}})}$, whose cohomology is the gauge version of the log de Rham cohomology of $\mathfrak{X}$ with respect to $\mathfrak{D}$. To proceed, let $\mathfrak{V}\subset\mathfrak{X}$ be an affine open, on which there are local coordinates $\{x_{1},\dots,x_{n}\}$ in which the divisor $\mathfrak{D}$ is given by $\{x_{1}\cdots x_{j}=0\}$. Then (starting with the action of finite-order differential operators), we may consider the $D_{\mathfrak{V}}^{(0)}$-submodule of $\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]$ generated by $x_{1}^{-1}\cdots x_{j}^{-1}$; it is easily seen to be independent of the choice of coordinates; hence we obtain a well-defined $D_{\mathfrak{V}}^{(0)}$-module denoted ${\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{U}})}^{\text{fin}}$; and we define the $\mathcal{\widehat{D}}_{\mathfrak{V}}^{(0)}$ module, denoted $(j_{\star}\mathcal{O}_{\mathfrak{U}})^{-\infty}|_{\mathfrak{V}}$ to be the $p$-adic completion of ${\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{U}})}^{\text{fin}}$. By glueing we obtain a coherent $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}$-module $(j_{\star}\mathcal{O}_{\mathfrak{U}})^{-\infty}$. We have \begin{lem} \label{lem:Injectivity-of-completion}For any $\mathfrak{V}$ as above, the natural map $\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ (where $\widehat{}$ denotes $p$-adic completion) is injective. \end{lem} We'll give a proof of this rather technical result in \secref{Appendix:-an-Inectivity}. From this we deduce \begin{lem} \label{lem:Hodge-filt-on-log}Let $F$ be a lift of Frobenius satisfying $F(x_{i})=x_{i}^{p}$ for all $1\leq i\leq n$. Then the natural map $F^{*}\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ is injective, and its image is the $\widehat{\mathcal{D}}_{\mathfrak{V}}^{(1)}$-submodule generated by $x_{1}^{-1}\cdots x_{j}^{-1}$. \end{lem} \begin{proof} For each $r>0$ we have an isomorphism $F^{*}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]/p^{r})\tilde{\to}\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]/p^{r}$; upon taking the inverse limit we obtain $F^{*}\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}\tilde{\to}\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$. Since $F^{*}$ is an exact, conservative functor on $\mathcal{O}_{\mathfrak{V}}-\text{mod}$, the previous lemma implies that $F^{*}\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ is injective. Since the image of $({\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ is the $\widehat{\mathcal{D}}_{\mathfrak{V}}^{(0)}$-submodule generated by $x_{1}^{-1}\cdots x_{j}^{-1}$, the image of $F^{*}(j_{\star}{\displaystyle \mathcal{O}_{\mathfrak{U}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ is the $\widehat{\mathcal{D}}_{\mathfrak{V}}^{(1)}$-submodule generated by $F(x_{1}^{-1}\cdots x_{j}^{-1})=x_{1}^{-p}\cdots x_{j}^{-p}$. But since $\partial_{i}^{[p]}x_{i}^{-1}=-x_{i}^{-p-1}$ we see that $\widehat{\mathcal{D}}_{\mathfrak{V}}^{(1)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}=\widehat{\mathcal{D}}_{\mathfrak{V}}^{(1)}\cdot x_{1}^{-p}\cdots x_{j}^{-p}$ as claimed. \end{proof} We can now construct the full gauge ${\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}[f,v]}$ as follows: denote by $\widehat{\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{\infty}}$ the $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$-submodule of $\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ locally generated by $x_{1}^{-1}\cdots x_{j}^{-1}$; as above this is independent of the choice of coordinates for the divisor $\mathfrak{D}$. Then we have \begin{example} \label{exa:Integral-j} Define $({\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{U}})})^{i}:=\{m\in\widehat{\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{\infty}}|p^{i}m\in\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}\}$. By the above discussion this is an object in $\text{Coh}_{F^{-1}}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ via the isomorphism \linebreak{} $F^{*}\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}\tilde{\to}\widehat{\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{\infty}}$. Let ${\displaystyle j_{\star}D(\mathcal{O}_{U}})$ denote the reduction mod $p$. We claim that the $l$th term of the Hodge filtration on $({\displaystyle j_{\star}D(\mathcal{O}_{U})})^{\infty}$ is given by $F^{*}(F^{l}(\mathcal{D}_{X}^{(0)})\cdot(x_{1}^{-1}\cdots x_{j}^{-1}))$, where $F^{l}\mathcal{D}_{X}^{(0)}$ is the $l$th term of the symbol filtration. To see this, we work again in local coordinates over $\mathfrak{V}$. One computes that $(\partial_{i}^{[p]})^{l}(x_{i}^{-p})=u\cdot l!x_{i}^{-p(l+1)}$ where $u$ is a unit in $\mathbb{Z}_{p}$. Therefore the module \linebreak{} $D_{\mathfrak{V}}^{(1)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}=D_{\mathfrak{V}}^{(1)}\cdot x_{1}^{-p}\cdots x_{j}^{-p}$ is spanned over $\mathcal{O}_{\mathfrak{V}}$ by terms of the form $I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}$; the $p$-adic completion of this module is ${\displaystyle \widehat{\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{\infty}}}$. For a multi-index $I$, set $\tilde{I}=(pi_{1}+p-1,\dots,pi_{j}+p-1)$. Then \linebreak{} $I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}\in({\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{U}})})^{r}$ iff $p^{r}\cdot I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}\in\mathcal{\widehat{D}}_{\mathfrak{V}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}$ . Furthermore, it is not difficult to see that $p^{r}\cdot I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}\in\mathcal{\widehat{D}}_{\mathfrak{V}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}$ iff $p^{r}\cdot I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}\in\mathcal{D}_{\mathfrak{V}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}$; in turn, this holds iff $r\geq\text{val}(\tilde{I}!)-\text{val}(I!)$ (since $\mathcal{D}_{\mathfrak{V}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}$ is spanned by terms of the form $I!x_{1}^{-(i_{1}+1)}\cdots x_{j}^{-(i_{j}+1)}$); here $\text{val}$ denotes the usual $p$-adic valuation; so that $\text{val}(p)=1$. On the other hand one has \[ \text{val}((pi+p-1)!)-\text{val}(i!)=i \] for all $i\geq0$. So ${\displaystyle \text{val}(\tilde{I}!)-\text{val}(I!)=\sum_{t=1}^{j}i_{t}}$ which implies $I!\cdot x_{1}^{-p(i_{1}+1)}\cdots x_{j}^{-p(i_{j}+1)}\in({\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{U}})})^{r}$ iff ${\displaystyle r\geq\sum_{t=1}^{j}i_{t}}$. On the other hand, $(F^{l}(\mathcal{D}_{\mathfrak{X}}^{(0)})\cdot(x_{1}^{-1}\cdots x_{j}^{-1})$ is spanned over $\mathcal{O}_{\mathfrak{V}}$ by terms of the form $I!\cdot x_{1}^{-(i_{1}+1)}\cdots x_{j}^{-(i_{j}+1)}$ where ${\displaystyle \sum_{t=1}^{j}i_{t}\leq l}$. Thus the module $F^{*}(F^{l}(\mathcal{D}_{X}^{(0)})\cdot(x_{1}^{-1}\cdots x_{j}^{-1}))$ is exactly the image in $({\displaystyle j_{\star}\mathcal{O}_{U}[f,v]})^{\infty}$ of $({\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}[f,v]})^{r}$, which is the claim. \end{example} Finally, we end with an example of a standard, coherent gauge which definitely does not admit an $F^{-1}$-action: \begin{example} \label{exa:Exponential!} Let $\mathfrak{X}=\widehat{\mathbb{A}_{W(k)}^{1}}$. Consider the $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-module $e^{x}$; i.e., the sheaf $\mathcal{O}_{\mathfrak{X}}$ equipped with the action determined by \[ \sum_{i=0}^{\infty}a_{i}\partial^{i}\cdot1=\sum_{i=0}^{\infty}a_{i} \] (here $a_{i}\to0$ as $i\to\infty$). Then $\mathcal{O}_{\mathfrak{X}}[p^{-1}]$ is a coherent $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}$-module since $\partial^{[p]}\cdot1=(p!)^{-1}$; it has a $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-lattice given by $\mathcal{O}_{\mathfrak{X}}$. Thus by \exaref{Basic-Construction-over-X} we may define \[ (e^{x})^{i}:=\{m\in\mathcal{O}_{\mathfrak{X}}[p^{-1}]|p^{i}m\in\mathcal{O}_{\mathfrak{X}}\} \] and we obtain a gauge, also denoted $e^{x}$, such that $(e^{x})^{i}\tilde{=}\mathcal{O}_{\mathfrak{X}}$ for all $i$, and such that $v$ is an isomorphism for all $i$, while $f$ is given by multiplication by $p$. We have $(e^{x})^{-\infty}\tilde{=}\mathcal{O}_{\mathfrak{X}}$ while $(e^{x})^{\infty}=\mathcal{O}_{\mathfrak{X}}[p^{-1}]$. \end{example} This example indicates that the ``exponential Hodge theory'' appearing, e.g., in Sabbah's work \cite{key-22}, could also be a part of this story; this should be interesting to pursue in future work. \section{\label{sec:Operations:PullBack}Operations on Gauges: Pull-back} Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism of smooth formal schemes over $W(k)$. Let us begin by setting our conventions on the pullback of $\mathcal{O}$-modules: \begin{defn} \label{def:Correct-Pullback}1) If $\mathcal{M}\in\mathcal{O}_{\mathfrak{Y}}-\text{mod}$, we set $\varphi^{*}\mathcal{M}:=\mathcal{O}_{\mathfrak{X}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}})}\varphi^{-1}(\mathcal{M}^{\cdot})$, the $p$-adic completion of the naive tensor product. If $\mathcal{M}^{\cdot}\in D(\mathcal{O}_{\mathfrak{Y}})$, then we define $L\varphi^{*}\mathcal{M}^{\cdot}:=\mathcal{O}_{\mathfrak{X}}\widehat{\otimes}_{\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$; the cohomological completion of the usual derived tensor product. 2) Consider $D(\mathcal{O}_{\mathfrak{X}})$ and $D(\mathcal{O}_{\mathfrak{Y}})$ as graded sheaves of rings as usual. If $\mathcal{M}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{\mathfrak{Y}})))$, then we define $L\varphi^{*}\mathcal{M}^{\cdot}:=D(\mathcal{O}_{\mathfrak{X}})\widehat{\otimes}_{\varphi^{-1}(D(\mathcal{O}_{\mathfrak{Y}}))}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$, the graded cohomological completion of the usual derived tensor product. \end{defn} \begin{rem} The reader will note several inconsistencies in these notations. First of all, we do not, in general, have $\mathcal{H}^{0}(L\varphi^{*}\mathcal{M})=\varphi^{*}\mathcal{M}$. Furthermore, the functor $L\varphi^{*}$ does not commute with the forgetful functor from graded $\mathcal{O}[f,v]$-modules to $\mathcal{O}$-modules. However, we will only use the underived $\varphi^{*}$ in a few very special cases (c.f. the lemma directly below), when in fact the equality $\mathcal{H}^{0}(L\varphi^{*}\mathcal{M})=\varphi^{*}\mathcal{M}$ does hold. Further, we will only apply the graded functor when working with a graded module; and this will almost always be the case. Hopefully this notational scheme does not cause any undue confusion. \end{rem} Now we should check that this operation behaves well on the basic objects of interest in our paper: \begin{lem} \label{lem:phi-pullback-of-D^i}For each $i\in\mathbb{Z}$ we have \[ L\varphi^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i})\tilde{=}\varphi^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i})\tilde{=}\lim_{n}(\mathcal{O}_{\mathfrak{X}_{n}}\otimes_{\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}_{n}})}\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}/p^{n})) \] In particular, we have \[ \mathcal{H}^{0}(L\varphi^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}))\tilde{=}\varphi^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}) \] under the conventions of \defref{Correct-Pullback}. The same holds if we replace $\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}$ by $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)}$ for any $j\geq0$. \end{lem} \begin{proof} As this question is local, we can assume $\mathfrak{X}=\text{Specf}(\mathcal{B})$ and $\mathfrak{Y}=\text{Specf}(\mathcal{A})$ where $\mathcal{A}$ possess local coordinates $\{t_{1},\dots,t_{n}\}$. By definition we have that $\widehat{D}_{\mathcal{A}}^{(0,1),i}$ is the $p$-adic completion of $D_{\mathcal{A}}^{(0,1),i}$. By \corref{Each-D^(i)-is-free} we have that $D_{\mathcal{A}}^{(0,1),i}$ is free over $\mathcal{A}$. In particular, it is $p$-torsion free and $p$-adically separated; and so by \cite{key-8}, lemma 1.5.4 its cohomological completion is equal to $\widehat{D}_{\mathcal{A}}^{(0,1),i}$. Therefore we have the short exact sequence \[ D_{\mathcal{A}}^{(0,1),i}\to\widehat{D}_{\mathcal{A}}^{(0,1),i}\to K \] where $p$ acts invertibly on $K$. Now we apply the functor $\mathcal{B}\otimes_{\mathcal{A}}^{L}$. By \cite{key-8}, theorem 1.6.6, we have that $\widehat{D}_{\mathcal{A}}^{(0,1),i}$ is flat over $\mathcal{A}$. Thus we see that $\mathcal{B}\widehat{\otimes}_{\mathcal{A}}^{L}\mathcal{\widehat{D}}_{\mathcal{A}}^{(0,1),i}$, the cohomological completion of $\mathcal{B}\otimes_{\mathcal{A}}^{L}\widehat{D}_{\mathcal{A}}^{(0,1),i}$, is isomorphic to the cohomological completion of $\mathcal{B}\otimes_{\mathcal{A}}^{L}\mathcal{D}_{\mathcal{A}}^{(0,1),i}$, which is just the usual $p$-adic completion since this is a free $\mathcal{B}$-module, and the statement follows. An identical argument works for $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)}$. \end{proof} Now let $j\geq0$. We recall that, for each $j\geq0$, Berthelot has constructed a pullback functor $\varphi^{!,(j)}$ from $\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(j)}-\text{mod}$ to $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(j)}-\text{mod}$. In fact, in \cite{key-2}, section 3.2, he has shown that $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(j)}:=\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)})$ carries the structure of a left $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(j)}$-module. By definition $\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)}$ carries the structure of a right $\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)})$-module. This, in turn allows one to define the functor $\varphi^{*,(j)}$ via \[ L\varphi^{*,(j)}(\mathcal{M})=\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(j)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}})}^{L}\varphi^{-1}(\mathcal{M})\tilde{=}L\varphi^{*}(\mathcal{M}) \] (where the last isomorphism is as sheaves of $\mathcal{O}_{\mathfrak{X}}$-modules). One sets $\varphi^{!,(j)}:=L\varphi^{*,(j)}[d_{X/Y}]$ (where $d_{X/Y}=\text{dim}(X)-\text{dim}(Y)$). In fact, this is not quite Berthelot's definition, as he does not use the cohomological completion; rather, he first defines the functor in the case of a morphism $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ (the reductions mod $p^{n}$ of $\mathfrak{X}$ and $\mathfrak{Y}$, respectively), and then applies the $\text{R}\lim$ functor. However, the two notions agree on bounded complexes of coherent $\widehat{\mathcal{D}}_{\mathfrak{Y}}$-modules; the version introduced here seems better suited to very general complexes. In order to upgrade this to gauges, we must upgrade the bimodule $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)}$ to a bimodule $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}$: \begin{defn} \label{def:Transfer-Bimod} We set \[ \mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}:=\bigoplus_{i\in\mathbb{Z}}\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i} \] The sheaf ${\displaystyle \bigoplus_{i\in\mathbb{Z}}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1),i}}$ is a graded sheaf of $D(W(k))$-modules; induced from the $D(W(k))$ action on $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}$. Note that $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1),-\infty}=\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)}$. \end{defn} Let us analyze this sheaf: \begin{prop} \label{prop:Basic-properties-of-the-transfer-module}1) For each $i\in\mathbb{Z}$ , the natural map $\iota:\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}\to\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)}$ (induced from the inclusion $\eta:\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}\to\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)}$) is injective. 2) The image $\iota(\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i})$ is equal to the sheaf whose local sections are given by $\{\Psi\in\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(1)}|p^{i}\Psi\in\iota(\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)})\}$. In particular, $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}$ is a standard gauge. 3) The sheaf ${\displaystyle \mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}}$ carries the structure of a graded $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)},\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))$-bimodule as follows: we have the inclusions $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}\subset\mathcal{\widehat{D}}_{\mathfrak{X}}^{(1)}$, so if $\Phi\in\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),i}$ and $\Psi\in\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1),j}$ are local sections, then $p^{i+j}(\Phi\cdot\Psi)=(p^{i}\Phi)\cdot(p^{j}\Psi)\in\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)}$. Similarly, $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}$ becomes a right $\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})$-module via $\varphi^{-1}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)}\subset\varphi^{-1}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(1)}$. \end{prop} \begin{proof} 1) As the statement is local, we can suppose $\mathfrak{Y}=\text{Specf}(\mathcal{A})$ and $\mathfrak{X}=\text{Specf}(\mathcal{B})$ where $\mathcal{A}$ and $\mathcal{B}$ admit local coordinates; let the reductions mod $p$ be $Y=\text{Spec}(A)$ and $X=\text{Spec}(B)$. By \corref{Local-coords-over-A=00005Bf,v=00005D} we know that $D_{A}^{(0,1)}$ is a free graded $A[f,v]$-module, therefore $\varphi^{*}D_{A}^{(0,1)}=B\otimes_{A}D_{A}^{(0,1)}$ is a free graded $B[f,v]$-module; and we have that the kernel of $f_{\infty}:\varphi^{*}D_{A}^{(0,1),i}\to\varphi^{*}D_{A}^{(0,1),\infty}=\varphi^{*}D_{A}^{(1)}$ is exactly the image of $v:\varphi^{*}D_{A}^{(0,1),i+1}\to\varphi^{*}D_{A}^{(0,1),i}$. Now consider $m\in\text{ker}(\iota:\varphi^{*}\widehat{D}_{\mathcal{A}}^{(0,1),i}\to\varphi^{*}\widehat{D}_{\mathcal{A}}^{(1)})$. The reduction mod $p$ of $\iota$ agrees with $f_{\infty}:\varphi^{*}D_{A}^{(0,1),i}\to\varphi^{*}D_{A}^{(1)}$. Let $\overline{m}$ denote the image of $m$ in $\varphi^{*}D_{A}^{(0,1),i}$. Then $\overline{m}\in\text{ker}(\varphi^{*}D_{A}^{(0,1),i}\to\varphi^{*}D_{A}^{(1)})=v\cdot\varphi^{*}D_{A}^{(0,1),i+1}$. So, since $fv=p,$ we have $m\in v\cdot\varphi^{*}\widehat{D}_{\mathcal{A}}^{(0,1),i+1}$; write $m=vm'$. By definition, the composition $\widehat{D}_{\mathcal{A}}^{(0,1),i+1}\xrightarrow{v}\widehat{D}_{\mathcal{A}}^{(0,1),i}\xrightarrow{\eta}\widehat{D}_{\mathcal{A}}^{(1)}$ is equal to $p\cdot\eta:\widehat{D}_{\mathcal{A}}^{(0,1),i+1}\to\widehat{D}_{\mathcal{A}}^{(1)}$; thus also $\iota\circ v=p\cdot\iota$ and so $\iota(m)=\iota(vm')=p\iota(m')=0$; therefore $m'\in\text{ker}(\iota)$ as $\varphi^{*}\widehat{D}_{\mathcal{A}}^{(1)}$ is $p$-torsion-free\footnote{Indeed, it is the inverse limit of the $W_{n}(k)$-flat modules $(\varphi^{*}\widehat{D}_{\mathcal{A}}^{(1)})/p^{n}=(\mathcal{B}/p^{n})\otimes_{\mathcal{A}/p^{n}}(\widehat{D}_{\mathcal{A}}^{(1)}/p^{n})$}. Iterating the argument, we see that $m\in v^{N}\varphi^{*}\widehat{D}_{\mathcal{A}}^{(0,1),i+N}$ for all $N>0$; reducing mod $p$, this forces $\overline{m}=0$ since (again by \corref{Local-coords-over-A=00005Bf,v=00005D}) $\varphi^{*}D_{A}^{(0,1)}$ is $v$-adically seperated. Thus $m=pm_{1}$; and then $\iota(m_{1})=0$ since $\varphi^{*}\widehat{D}_{\mathcal{A}}^{(1)}$ is $p$-torsion-free; continuing in this way we obtain $m\in\bigcap_{n}p^{n}\cdot\varphi^{*}\widehat{D}_{\mathcal{A}}^{(0,1),i}=0$. 2) For each $i\geq0$ we have a short exact sequence \[ 0\to\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}\to\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i+1}\to\mathcal{F}_{i}\to0 \] where $\mathcal{F}_{i}$ is a sheaf which is annihilated by $p$. By the injectivity just proved (and the equality $L\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}=\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1),i}$) we obtain the short exact sequence \[ 0\to\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}\to\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i+1}\to\mathcal{H}^{0}(L\varphi^{*}\mathcal{F}_{i})\to0 \] and, since $\mathcal{F}_{i}$ is annihilated by $p$, we have $\mathcal{H}^{0}(L\varphi^{*}\mathcal{F}_{i})=\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\mathcal{F}_{i})$. So we obtain $p\cdot\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i+1}\subset\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}$, and since $\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),0}=\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)}$, we see inductively that $\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1),i}\subset\{\Psi\in\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(1)}|p^{i}\Psi\in\iota(\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)})\}$ for all $i$. For the converse direction, we work locally and assume $\mathfrak{X}=\text{Specf}(\mathcal{B})$ and $\mathfrak{Y}=\text{Specf}(\mathcal{A})$ where $\mathcal{A}$ possess etale local coordinates $\{t_{1},\dots,t_{n}\}$. Then we have that $\Gamma(\varphi^{*}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(1)})=\mathcal{B}\widehat{\otimes}_{\mathcal{A}}\widehat{D}_{\mathcal{A}}^{(1)}\tilde{=}\mathcal{B}\widehat{\otimes}_{\mathcal{A}}D_{\mathcal{A}}^{(1)}$. As in the proof of \lemref{Basic-structure-of-D_A^(i)}, we will consider the finite-order analogue first. From (the proof of) that lemma, it follows that, any element of $\mathcal{B}\otimes_{\mathcal{A}}D_{\mathcal{A}}^{(1)}$ admits a unique expression of the form \[ \Psi=\sum_{I,J}b_{I,J}\frac{\partial_{1}^{i_{1}+pj_{1}}\cdots\partial_{n}^{i_{n}+pj_{n}}}{(p!)^{|J|}} \] for which $0\leq i_{j}<p$, all $b_{I,J}\in\mathcal{B}$, and the sum is finite. We have that $p^{i}\Psi\in\mathcal{B}\otimes_{\mathcal{A}}D_{\mathcal{A}}^{(0)}$ iff ${\displaystyle \frac{p^{i}}{p^{|J|}}b_{I,J}}\in\mathcal{B}$. So, if $|J|>i$ we can conclude (again, as in the proof of \lemref{Basic-structure-of-D_A^(i)}) that \[ b_{I,J}\frac{\partial_{1}^{i_{1}+pj_{1}}\cdots\partial_{n}^{i_{n}+pj_{n}}}{(p!)^{|J|}}=\tilde{b}_{I,J}\cdot\partial_{1}^{i_{1}+pj'_{1}}\cdots\partial_{n}^{i_{n}+pj'_{n}}\cdot(\partial_{1}^{[p]})^{j''_{1}}\cdots\partial_{n}^{i_{n}}(\partial_{n}^{[p]})^{j''_{n}} \] where $\tilde{b}_{I,J}\in\mathcal{B}$, and $j''_{1}+\dots+j_{n}''=i$. In particular $\Psi$ is contained in the $\mathcal{B}$-submodule spanned by $\{\partial_{1}^{i_{1}}\cdots\partial_{n}^{i_{n}}\cdot(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where $j_{1}+\dots+j_{n}\le i$, which is exactly the image of $\mathcal{B}\otimes_{\mathcal{A}}D_{\mathcal{A}}^{(0,1),i}$ in $\mathcal{B}\otimes_{\mathcal{A}}D_{\mathcal{A}}^{(1)}$. Now, if $\Psi\in\mathcal{B}\widehat{\otimes}_{\mathcal{A}}\widehat{D}_{\mathcal{A}}^{(1)}$ is such that $p^{i}\Psi\in\mathcal{B}\widehat{\otimes}_{\mathcal{A}}\widehat{D}_{\mathcal{A}}^{(0)}$, then we can write ${\displaystyle p^{i}\Psi=\sum_{j=0}^{\infty}p^{j}\Psi_{j}}$ where $\Psi_{j}\in\mathcal{B}\otimes_{\mathcal{A}}D_{\mathcal{A}}^{(0)}$. Therefore \[ \Psi=\sum_{j=0}^{i}p^{j-i}\Psi_{j}+\sum_{j=i+1}^{\infty}p^{j-i}\Psi_{j} \] where, by the previous paragraph, the first sum is contained in the $\mathcal{B}$-submodule spanned by $\{\partial_{1}^{i_{1}}\cdots\partial_{n}^{i_{n}}\cdot(\partial_{1}^{[p]})^{j_{1}}\cdots(\partial_{n}^{[p]})^{j_{n}}\}$ where $j_{1}+\dots+j_{n}\le i$, and the second sum is contained in $\mathcal{B}\widehat{\otimes}_{\mathcal{A}}\widehat{D}_{\mathcal{A}}^{(0)}$. Thus $\Psi$ is in the image of $\mathcal{B}\widehat{\otimes}_{\mathcal{A}}\widehat{D}_{\mathcal{A}}^{(0,1),i}$ as required. It follows directly from the definition that $\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}$ is standard. Part $3)$ of the proposition follows immediately. \end{proof} \begin{rem} \label{rem:Direct-defn-of-transfer-bimodule}Combining the previous proposition with \lemref{phi-pullback-of-D^i}, we also obtain the description \[ \mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\tilde{=}L\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})=D(\mathcal{O}_{\mathfrak{X}})\widehat{\otimes}_{\varphi^{-1}(D(\mathcal{O}_{\mathfrak{Y}}))}^{L}\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}) \] in the category $D_{cc}(\mathcal{G}(D(\mathcal{O}_{\mathfrak{X}}))$. \end{rem} This leads to the \begin{defn} \label{def:Pullback!}Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))$. Then we define \[ L\varphi^{*}(\mathcal{M}^{\cdot}):=\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})\in\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})) \] where, as usual $\widehat{?}$ denotes graded derived completion. The induced left action of $\mathcal{D}_{\mathfrak{X}}^{(0,1)}$ given by the above definition; set $\varphi^{!}:=L\varphi^{*}[d_{X/Y}]$. \end{defn} In order to study this definition, we shall use the corresponding mod $p$ theory; as usual this can be defined by reduction mod $p$ when the schemes $X$ and $Y$ are liftable, but it actually exists for all $\varphi:X\to Y$. This is contained in the the following \begin{prop} \label{prop:pull-back-in-pos-char}Let $\varphi:X\to Y$ be a morphism of smooth varieties over $k$. 1) There is a map of sheaves $\alpha:\mathfrak{l}_{X}\to\varphi^{*}\mathcal{D}_{Y}^{(0,1),1}$ (where $\mathfrak{l}_{X}$ is defined in \defref{L}). 2) Let $\beta:\mathcal{T}_{X}\to\varphi^{*}\mathcal{D}_{Y}^{(0,1),0}=\varphi^{*}\mathcal{D}_{Y}^{(0)}$ denote the natural map. There is a left action of $\mathcal{D}_{X}^{(0,1)}$ on $\varphi^{*}\mathcal{D}_{Y}^{(0,1)}$ satisfying $\partial\cdot(1\otimes1)=\beta(\partial)$ for all $\partial\in\mathcal{T}_{X}$ and $\delta\cdot(1\otimes1)=\alpha(\delta)$ for all $\delta\in\mathfrak{l}_{X}$. This action commutes with the right action of $\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$ on $\varphi^{*}\mathcal{D}_{Y}^{(0,1)}$. \end{prop} \begin{proof} 1) Let $\Phi$ be a local section of $\mathfrak{l}_{X}$. Composing the map $\varphi^{\#}:\varphi^{-1}(\mathcal{O}_{Y})\to\mathcal{O}_{X}$ with $\Phi$ gives a differential operator from $\varphi^{-1}(\mathcal{O}_{Y})$ to $\mathcal{O}_{X}$; call this operator $\Phi'$. We claim $\Phi'\in\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}\mathfrak{l}_{Y}$ (here, we are using the fact that the sheaf $\mathfrak{l}_{Y}$ is a subsheaf of $\mathcal{D}iff_{Y}$ and that $\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\mathcal{D}iff_{Y})\tilde{=}\mathcal{D}iff(\varphi^{-1}(\mathcal{O}_{Y}),\mathcal{O}_{X})$). Let $U\subset X$ and $V\subset Y$ be open subsets which possess local coordinates, such that $\varphi(U)\subset V$. As in \lemref{O^p-action} write \[ \Phi=\sum_{i=1}^{n}a_{i}^{p}\partial_{i}^{[p]}+\sum_{I}a_{I}\partial^{I} \] where $a_{i},a_{I}\in\mathcal{O}_{X}(U)$. The map $(\sum_{I}a_{I}\partial^{I})\circ\varphi^{\#}:\varphi^{-1}(\mathcal{O}_{V})\to\mathcal{O}_{U}$ is a differential operator which satisfies $((\sum_{I}a_{I}\partial^{I})\circ\varphi^{\#})(g^{p}\cdot h)=\varphi^{\#}(g^{p})\cdot((\sum_{I}a_{I}\partial^{I})\circ\varphi^{\#})(h)$ for all $g,h\in\mathcal{O}_{V}$. From this we conclude \[ (\sum_{I}a_{I}\partial^{I})\circ\varphi^{\#}=\sum b_{J}\partial^{J} \] where $b_{J}\in\mathcal{O}_{X}(U)$ and now $\partial^{J}=\partial_{1}^{j_{1}}\cdots\partial_{r}^{j_{r}}$ are coordinate derivations on $V$ (to prove this, write the differential operator $(\sum_{I}a_{I}\partial^{I})\circ\varphi^{\#}$ in terms of $\partial_{1}^{[j_{1}]}\cdots\partial_{r}^{[j_{r}]}$ and then use the linearity over $\varphi^{\#}(g^{p})$ to deduce that there are no terms with any $j_{i}\geq p$). Similarly, the map ${\displaystyle \sum_{i=1}^{n}a_{i}^{p}\partial_{i}^{[p]}}\circ\varphi^{\#}:\varphi^{-1}(\mathcal{O}_{V})\to\mathcal{O}_{U}$ is a differential operator of order $\leq p$, whose action on any $p$th power in $\varphi^{-1}(\mathcal{O}_{V})$ is a $p$th power in $\mathcal{O}_{U}$. From this one easily sees \[ (\sum_{i=1}^{n}a_{i}^{p}\partial_{i}^{[p]})\circ\varphi^{\#}=\sum_{j=1}^{r}b_{j}^{p}\partial_{j}^{[p]}+\sum_{J}b_{J}\partial^{J} \] for some $b_{j},b_{J}\in\mathcal{O}_{U}$. So we conclude $\Phi'\in\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}\mathfrak{l}_{Y}$ as desired. Further, since $\mathfrak{l}_{Y}\subset\mathcal{D}_{Y}^{(0,1),1}$ we obtain $\varphi^{-1}(\mathfrak{l}_{Y})\subset\varphi^{-1}(\mathcal{D}_{Y}^{(0,1),1})$ and therefore a map $\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\mathfrak{l}_{Y})\to\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\mathcal{D}_{Y}^{(0,1),1})$; we can now define $\alpha$ as the composition. 2) It suffices to check this locally. Restrict to an open affine $U\subset X$ which posses etale local coordinates, and we may suppose $\varphi(U)\subset V$, where $V$ also possesses etale local coordinates. Writing $U=\text{Spec}(A)$ and $V=\text{Spec}(B)$, we let $\mathcal{A}$ and $\mathcal{B}$ be flat lifts of $A$ and $B$ to $W(k)$, as in the proof of \lemref{linear-independance-over-D_0-bar} above. Let $\varphi^{\#}:\mathcal{B}\to\mathcal{A}$ be a lift of $\varphi^{\#}:B\to A$ (these always exist for affine neighborhoods which posses local coordinates, by the infinitesimal lifting property). Then the construction of \defref{Transfer-Bimod} provides an action of $D_{\mathcal{B}}^{(0,1)}$ on $\varphi^{*}(D_{\mathcal{A}}^{(0,1)})$ which commutes with the obvious right action of $D_{\mathcal{A}}^{(0,1)}$. The reduction mod $p$ of this action, when restricted to $\mathcal{T}_{X}\subset\mathcal{D}_{X}^{(0)}$ and $\mathfrak{l}_{X}\subset\mathcal{D}_{X}^{(0,1),1}$ clearly agrees with the map described above. Thus the map extends (uniquely) to an action, as claimed. \end{proof} Thus we have \begin{defn} Let $\mathcal{D}_{X\to Y}^{(0,1)}:=\varphi^{*}\mathcal{D}_{Y}^{(0,1)}$, equipped with the structure of a graded $(\mathcal{D}_{X}^{(0,1)},\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)}))$-bimoddule as above. Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. Then we define $L\varphi^{*}(\mathcal{M}^{\cdot}):=\mathcal{D}_{X\to Y}^{(0,1)}\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$ with the induced left action of $\mathcal{D}_{X}^{(0,1)}$ given by the above. Set $\varphi^{!}=L\varphi^{*}[d_{X/Y}]$. The functor $L\varphi^{*}$ takes $D_{\text{qcoh}}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ to $D_{\text{qcoh}}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. \end{defn} \begin{rem} In fact, as an object in $D(\mathcal{G}(D(\mathcal{O}_{X})))$, we have that $L\varphi^{*}(\mathcal{M}^{\cdot})$ agrees with the usual pullback of $\mathcal{O}$-modules. This follows directly from the isomorphism $\varphi^{*}\mathcal{D}_{Y}^{(0,1)}\tilde{=}\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$, and the fact that $\mathcal{D}_{Y}^{(0,1)}$ is flat over $\mathcal{O}_{Y}$. The analogous fact is also true for $\varphi:\mathfrak{X}\to\mathfrak{Y}$; making use of \remref{Direct-defn-of-transfer-bimodule}. It follows that $L\varphi^{*}$ has finite homological dimension. \end{rem} Now we record some basic properties of these functors: \begin{lem} \label{lem:composition-of-pullbacks}If $\psi:\mathfrak{Y}\to\mathfrak{Z}$, there is an isomorphism of functors \linebreak{} $L\varphi^{*}\circ L\psi^{*}\tilde{=}L(\psi\circ\varphi)^{*}$. The same result holds for $\varphi:X\to Y$ and $\psi:Y\to Z$. \end{lem} \begin{proof} (compare \cite{key-49}, proposition 1.5.11) We have, by \remref{Direct-defn-of-transfer-bimodule}, \[ \mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{D}_{\mathfrak{Y\to\mathfrak{Z}}}^{(0,1)}) \] \[ =(D(\mathcal{O}_{\mathfrak{X}})\widehat{\otimes}_{\varphi^{-1}(D(\mathcal{O}_{\mathfrak{Y}}))}^{L}\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{O}_{\mathfrak{Y}}[f,v]\widehat{\otimes}_{\psi^{-1}(D(\mathcal{O}_{\mathfrak{Z}}))}^{L}\psi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(0,1)})) \] \[ \tilde{=}D(\mathcal{O}_{\mathfrak{X}})\widehat{\otimes}_{\varphi^{-1}(D(\mathcal{O}_{\mathfrak{Y}}))}^{L}(\varphi^{-1}D(\mathcal{O}_{\mathfrak{Y}})\widehat{\otimes}_{(\psi\circ\varphi)^{-1}(D(\mathcal{O}_{\mathfrak{Z}}))}^{L}(\psi\circ\varphi)^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(0,1)})) \] \[ \tilde{=}D(\mathcal{O}_{\mathfrak{X}})\widehat{\otimes}_{(\psi\circ\varphi)^{-1}(D(\mathcal{O}_{\mathfrak{Z}}))}^{L}(\psi\circ\varphi)^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(0,1)})=\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Z}}^{(0,1)} \] as $(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)},(\psi\circ\varphi)^{-1}\mathcal{\widehat{D}}_{\mathfrak{Z}}^{(0,1)})$-bimodules. This yields \[ L(\psi\circ\varphi)^{*}(\mathcal{M}^{\cdot})=\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Z}}^{(0,1)}\widehat{\otimes}_{(\psi\circ\varphi)^{-1}(\mathcal{D}_{\mathfrak{Z}}^{(0,1)})}^{L}(\psi\circ\varphi)^{-1}(\mathcal{M}^{\cdot}) \] \[ \tilde{=}(\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{D}_{\mathfrak{Y\to\mathfrak{Z}}}^{(0,1)}))\widehat{\otimes}_{(\psi\circ\varphi)^{-1}(\mathcal{D}_{\mathfrak{Z}}^{(0,1)})}^{L}(\psi\circ\varphi)^{-1}(\mathcal{M}^{\cdot}) \] \[ \tilde{=}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}((\mathcal{D}_{\mathfrak{Y\to\mathfrak{Z}}}^{(0,1)})\widehat{\otimes}_{\psi^{-1}(\mathcal{D}_{\mathfrak{Z}}^{(0,1)})}^{L}\psi^{-1}(\mathcal{M}^{\cdot}))=L\varphi^{*}(L\psi^{*}\mathcal{M}^{\cdot}) \] An identical argument works for $\varphi:X\to Y$ and $\psi:Y\to Z$. \end{proof} Next, we have \begin{prop} \label{prop:Basic-base-change-for-pullback}1) Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$. Then $L\varphi^{*}(\mathcal{M}^{\cdot})^{-\infty}\tilde{\to}L\varphi^{*,(0)}(\mathcal{M}^{\cdot,-\infty})$ and $\widehat{L\varphi^{*}(\mathcal{M}^{\cdot})^{\infty}}\tilde{\to}L\varphi^{*,(1)}(\widehat{\mathcal{M}^{\cdot,\infty}})$. The analogous result holds for $\varphi:X\to Y$. 2) Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$. Then $L\varphi^{*}(\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{\to}L\varphi^{*,(0)}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)$. \end{prop} \begin{proof} 1) By construction we have \[ (\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{D(W(k))}^{L}W(k)[f,v]/(f-1)\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(1)} \] and \[ (\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)})\widehat{\otimes}_{D(W(k))}^{L}W(k)[f,v]/(v-1)\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)} \] from which the result follows directly. Similarly, for part $2)$ one uses \[ (\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\otimes_{W(k)}^{L}k\tilde{=}\mathcal{\widehat{D}}_{X\to Y}^{(0,1)} \] \end{proof} Specializing to the case of positive characteristic, it is also useful to have comparisons with the pullbacks of $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules. First, we need to give the relevant definitions: \begin{defn} Suppose $\varphi:X\to Y$. We let $\mathcal{R}_{X\to Y}^{(1)}:=\varphi^{*}\mathcal{D}_{Y}^{(0,1)}/(v)$ and $\mathcal{\overline{R}}_{X\to Y}^{(0)}:=\varphi^{*}\mathcal{D}_{Y}^{(0,1)}/(f)$; considered as a graded $(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$ bimodule (respectively a graded $(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}),\varphi^{-1}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))$ bimodule). Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(1)})))$. Then we define $L\varphi^{*,(1)}(\mathcal{M}^{\cdot}):=\mathcal{R}_{X\to Y}^{(1)}\otimes_{\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$ with the induced left action of $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ given by the bimodule structure.. Set $\varphi^{\dagger,(1)}=L\varphi^{*,(1)}[d_{X/Y}]$. We make the analogous definition for $\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$-modules; and denote the corresponding functors $L\varphi^{*,(0)}$ and $\varphi^{\dagger,(1)}$. \end{defn} We note that the functor $L\varphi^{*,(1)}$ takes $D_{qc}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$ to $D_{qc}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$; and similarly for $\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$. Then we have the \begin{prop} \label{prop:pullback-and-R}Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. There is an isomorphism of functors \[ \mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\varphi^{\dagger}\mathcal{M}^{\cdot}\tilde{=}\varphi^{\dagger,(1)}(\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] and similarly for $\varphi_{0}^{\dagger}$. \end{prop} \begin{proof} We have \[ \varphi^{\dagger,(1)}(\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\mathcal{M}^{\cdot})=\mathcal{R}_{X\to Y}^{(1)}\otimes_{\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))}^{L}\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\mathcal{M}^{\cdot})[d_{X/Y}] \] \[ \tilde{=}\mathcal{R}_{X\to Y}^{(1)}\otimes_{\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))}^{L}\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[d_{X/Y}] \] \[ \tilde{=}\mathcal{R}_{X\to Y}^{(1)}\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[d_{X/Y}] \] Now, by definition, the module $\mathcal{D}_{X\to Y}^{(0,1)}$, admits, locally on $X$ and $Y$, a lift $\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}$ which we have constructed above in \defref{Transfer-Bimod}. This lift is a standard gauge, and so $\mathcal{D}_{X\to Y}^{(0,1)}$ is quasi-rigid. So, using the resolution (c.f. \lemref{Basic-Facts-on-Rigid}) \[ \cdots\to\mathcal{D}_{X}^{(0,1)}(-1)\xrightarrow{v}\mathcal{D}_{X}^{(0,1)}\xrightarrow{f}\mathcal{D}_{X}^{(0,1)}(-1)\xrightarrow{v}\mathcal{D}_{X}^{(0,1)}\to\mathcal{R}(\mathcal{D}_{X}^{(1)}) \] for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ over $\mathcal{D}_{X}^{(0,1)}$, this tell us that \begin{equation} \mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{D}_{X\to Y}^{(0,1)}\tilde{=}\mathcal{D}_{X\to Y}^{(0,1)}/v=\mathcal{R}_{X\to Y}^{(1)}\label{eq:transfer-iso-1} \end{equation} i.e., this complex is concentrated in degree $0$ and is equal to $\mathcal{R}_{X\to Y}^{(1)}$ there. Thus \[ \mathcal{R}_{X\to Y}^{(1)}\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[d_{X/Y}] \] \[ \tilde{=}\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{D}_{X\to Y}^{(0,1)}\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[d_{X/Y}]=\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\varphi^{\dagger}\mathcal{M}^{\cdot} \] as desired. The case of $\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$-modules is essentially identical. \end{proof} Finally, we also have \begin{prop} \label{prop:Smooth-pullback-preserves-coh}If $\varphi$ is smooth, then $L\varphi^{*}$ takes $D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$ to $D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$. The same holds for a smooth morphism $\varphi:X\to Y$. \end{prop} \begin{proof} By part $2)$ of \propref{Basic-base-change-for-pullback}, as well as \propref{coh-to-coh}, the first statement reduces to the second. We may assume that $X=\text{Spec}(B)$ and $Y=\text{Spec}(A)$ both possess local coordinates. After further localizing if necessary we can suppose that there are local coordinates $\{\partial_{1},\dots,\partial_{n}\}$ on $B$ such that the $A$-linear derivations of $B$ are $\{\partial_{1},\dots,\partial_{d}\}$. In this case, if we let $J\subset\mathcal{D}_{B}^{(0,1)}$ be the ideal generated by $\{\partial_{1},\dots,\partial_{d},\partial_{1}^{[p]},\dots,\partial_{d}^{[p]}\}$, then we have \[ \mathcal{D}_{B}^{(0,1)}/J\tilde{=}B\otimes_{A}\mathcal{D}_{A}^{(0,1)} \] which shows that $B\otimes_{A}\mathcal{D}_{A}^{(0,1)}=\varphi^{*}\mathcal{D}_{A}^{(0,1)}$ is a coherent $\mathcal{D}_{B}^{(0,1)}$-module, which is flat as a module over $\mathcal{D}_{A}^{(0,1),\text{opp}}$. This shows that $\varphi^{*}$ is exact; and the coherence of the pullback for an arbitrary coherent $\mathcal{D}_{A}^{(0,1)}$-module $\mathcal{M}$ follows by taking a finite presentation for $\mathcal{M}$. \end{proof} \section{\label{sec:Operations:Swap-Tensor}Operations on Gauges: Left-Right Interchange and tensor Product} The first goal of this subsection is to prove \begin{prop} \label{prop:Left-Right-Swap} Let $\mathcal{M}\in\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$. Then $\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}$ carries the structure of a right graded $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module. This functor defines an equivalence of categories, which preserves coherent modules. The derived functor preserves the subcategories of derived complete complexes. The analogous result holds for $X$ (i.e., in positive characteristic); there, the functor preserves the category of quasi-coherent sheaves as well. \end{prop} In order to prove this, we first recall that $\omega_{\mathfrak{X}}$ naturally carries the structure of a right $\mathcal{D}_{\mathfrak{X}}^{(i)}$-module for all $i\geq0$; indeed, $\omega_{\mathfrak{X}}[p^{-1}]$ carries a right $\mathcal{D}_{\mathfrak{X}}^{(i)}[p^{-1}]=\mathcal{D}_{\mathfrak{X}}^{(0)}[p^{-1}]$ structure via the Lie derivative (c.f., e.g. \cite{key-4}, page 8). In local coordinates, this action is simply given by \[ (gdx_{1}\wedge\cdots\wedge dx_{n})\partial=-\partial(g)dx_{1}\wedge\cdots\wedge dx_{n} \] for any derivation $\partial$. It follows that $\mathcal{D}_{\mathfrak{X}}^{(i)}$ preserves $\omega_{\mathfrak{X}}$ (for all $i$). As $\omega_{\mathfrak{X}}$ is $p$-adically complete, we see that it also inherits a right $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(i)}$-module structure. \begin{lem} Let $D(\omega_{\mathfrak{X}})=\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}D(\mathcal{O}_{\mathfrak{X}})$. Then $D(\omega_{\mathfrak{X}})$ has a natural right graded $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module structure. Similarly, $D(\omega_{X})$ admits a right graded $\mathcal{D}_{X}^{(0,1)}$-module structure, for any smooth $X$ over $k$. \end{lem} \begin{proof} We note that $(\omega_{\mathfrak{X}}[f,v])^{i}=\{m\in\omega_{\mathfrak{X}}[p^{-1}]|p^{i}m\in\omega_{\mathfrak{X}}\}$. Thus the first result follows by using the right $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(1)}$-module structure on $\omega_{\mathfrak{X}}$. To prove the second result, we choose on open affine $\text{Spec}(A)\subset X$ which possesses etale local coordinates. In coordinates, the required action is given by \[ (gdx_{1}\wedge\cdots\wedge dx_{n})\partial=-\partial(g)dx_{1}\wedge\cdots\wedge dx_{n} \] and \[ (gdx_{1}\wedge\cdots\wedge dx_{n})\partial^{[p]}=-f\cdot\partial^{[p]}(g)dx_{1}\wedge\cdots\wedge dx_{n} \] for any $g\in D(\mathcal{O}_{\mathfrak{X}})$. If we choose a lift $\mathcal{A}$ of $A$, then, after lifting the coordinates, we see that this action is the reduction mod $p$ of the action just defined; in particular it is actually independent of the choice of coordinates and therefore glues to define an action on all of $X$. \end{proof} Now we recall a very general construction from \cite{key-4}, section 1.4b \begin{lem} Let $\mathcal{L}$ be any line bundle on $\mathfrak{X}$. Placing $\mathcal{L}$ and $\mathcal{L}^{-1}$ in degree $0$, the sheaf $\mathcal{\widehat{D}}_{\mathfrak{X},\mathcal{L}}^{(0,1)}:=\mathcal{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{L}^{-1}$ carries the structure of a graded algebra on $\mathfrak{X}$, via the multiplication \[ (s_{1}\otimes\Phi_{1}\otimes t_{1})\cdot(s_{2}\otimes\Phi_{2}\otimes t_{2})=s_{1}\otimes\Phi_{1}<t_{1},s_{1}>\Phi_{2}\otimes t_{2} \] There is a functor $\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})\to\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X},\mathcal{L}}^{(0,1)})$ given by $\mathcal{M}\to\mathcal{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}$; the action of $\mathcal{\widehat{D}}_{\mathfrak{X},\mathcal{L}}^{(0,1)}$ on $\mathcal{L}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}$ is defined by \[ (s\otimes\Phi\otimes t)\cdot(s_{1}\otimes m)=s\otimes\Phi_{1}<t,s_{1}>m \] This functor is an equivalence of categories, whose inverse is given by $\mathcal{N}\to\mathcal{L}^{-1}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{N}$. \end{lem} So, \propref{Left-Right-Swap} follows directly from \begin{lem} There is an isomorphism of algebras $\mathcal{\widehat{D}}_{\mathfrak{X},\omega_{\mathfrak{X}}}^{(0,1)}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\text{op}}$. The same is true over $X$. \end{lem} \begin{proof} We have the isomorphism $\mathcal{\widehat{D}}_{\mathfrak{X},\omega_{\mathfrak{X}}}^{(0,1)}\tilde{=}D(\omega_{\mathfrak{X}})\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}D(\mathcal{\omega}_{\mathfrak{X}}^{-1})$. This yields a left action of $\mathcal{\widehat{D}}_{\mathfrak{X},\omega_{\mathfrak{X}}}^{(0,1)}$ on $\omega_{\mathfrak{X}}[f,v]$, given by \[ (s\otimes\Phi\otimes t)\cdot s_{1}=s\otimes\Phi\cdot<t,s_{1}> \] where $<,>$ refers to the pairing $D(\mathcal{\omega}_{\mathfrak{X}})\otimes_{D(\mathcal{O}_{\mathfrak{X}})}D(\mathcal{\omega}_{\mathfrak{X}}^{-1})\to D(\mathcal{O}_{\mathfrak{X}})$. Computing in local coordinates, one sees that the image of $\mathcal{\widehat{D}}_{\mathfrak{X},\omega_{\mathfrak{X}}}^{(0,1)}$ in $\mathcal{E}nd_{W(k)}(D(\omega_{\mathfrak{X}}))$ is the same as the image of $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\text{op}}$ in $\mathcal{E}nd_{W(k)}(D(\omega_{\mathfrak{X}}))$ via the right action defined above. This yields the isomorphism over $\mathfrak{X}$. To deal with $X$, one first obtains the isomorphism locally (via a local lifting of the variety), and then shows that the resulting isomorphism is independent of the choice of coordinates (as in the proof of the previous lemma). \end{proof} Next, we define tensor products of (left) $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-modules. The first step is to define the external product of sheaves: \begin{defn} 1) Let $\mathfrak{X}$ and $\mathfrak{Y}$ be smooth formal schemes, and let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{\mathfrak{X}})))$, $\mathcal{N}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{\mathfrak{Y}})))$. Then we define \[ \mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot}:=Lp_{1}^{*}(\mathcal{M}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}})}^{L}Lp_{2}^{*}(\mathcal{N}^{\cdot})\in D_{cc}(\mathcal{G}(D(\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}))) \] where $p_{i}$ ($i\in\{1,2\}$) are the projections and $Lp_{1}^{*},Lp_{2}^{*}$ are defined as in \defref{Correct-Pullback}. 2) Let $X$ and $Y$ be smooth schemes over $k$. Then for $\mathcal{M}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{X})))$, $\mathcal{N}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{Y})))$. Then we define \[ \mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot}:=Lp_{1}^{*}(\mathcal{M}^{\cdot})\otimes_{D(\mathcal{O}_{X\times Y})}^{L}Lp_{2}^{*}(\mathcal{N}^{\cdot})\in D(\mathcal{G}(D(\mathcal{O}_{X\times Y}))) \] where for $\mathcal{M}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{X})))$ we have $Lp_{1}^{*}\mathcal{M}^{\cdot}=D(\mathcal{O}_{X\times Y})\otimes_{p_{1}^{-1}(D(\mathcal{O}_{X}))}^{L}\mathcal{M}^{\cdot}\in D(\mathcal{G}(D(\mathcal{O}_{X\times Y})))$ ( and similarly for $p_{2}$). \end{defn} The relationship with $\mathcal{D}$-modules is the following: \begin{lem} 1) There is an isomorphism \[ \widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(0,1)} \] of sheaves of algebras on $\mathfrak{X}\times\mathfrak{Y}$. 2) There is an isomorphism \[ \mathcal{D}_{X}^{(0,1)}\boxtimes\mathcal{D}_{Y}^{(0,1)}\tilde{=}\mathcal{D}_{X\times Y}^{(0,1)} \] of sheaves of algebras on $X\times Y$. \end{lem} \begin{proof} First suppose $\mathfrak{X}=\text{Specf}(\mathcal{A})$ and $\mathfrak{Y}=\text{Specf}(\mathcal{B})$. Then there is a morphism $\mathcal{D}_{\mathcal{A}}^{(\infty)}\otimes_{W(k)}\mathcal{D}_{\mathcal{B}}^{(\infty)}\to\mathcal{D}_{\mathcal{A}\widehat{\otimes}_{W(k)}\mathcal{B}}^{(\infty)}$ defined as follows: for sections $a\in\mathcal{A}$ and $b\in\mathcal{B}$, we set \[ (\Phi_{1}\otimes\Phi_{2})(a\otimes b)=\Phi_{1}(a)\otimes\Phi_{2}(b) \] and we extend to $\mathcal{A}\widehat{\otimes}_{W(k)}\mathcal{B}$ by linearity and continuity. For a fixed integer $j\geq0$, this yields a map $\mathcal{D}_{\mathcal{A}}^{(j)}\otimes_{W(k)}\mathcal{D}_{\mathcal{B}}^{(j)}\to\mathcal{D}_{\mathcal{A}\widehat{\otimes}_{W(k)}\mathcal{B}}^{(j)}$; these maps are compatible with localization at any element of $\mathcal{A}$ or $\mathcal{B}$. After $p$-adically completing we get a map $\widehat{\mathcal{D}}_{\mathcal{A}}^{(j)}\widehat{\otimes}_{W(k)}\widehat{\mathcal{D}}_{\mathcal{B}}^{(j)}\to\widehat{\mathcal{D}}_{\mathcal{A}\widehat{\otimes}_{W(k)}\mathcal{B}}^{(j)}$, and these maps sheafifiy to a map \linebreak{} $p_{1}^{-1}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(j)})\widehat{\otimes}_{W(k)}p_{2}^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(j)})\to\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(j)}$. Note that since $\mathcal{D}_{\mathcal{A}}^{(j)}\otimes_{W(k)}\mathcal{D}_{\mathcal{B}}^{(j)}$ is $p$-torsion-free (as is $\mathcal{D}_{\mathcal{A}\widehat{\otimes}_{W(k)}\mathcal{B}}^{(j)}$), the usual $p$-adic completion of these sheaves agrees with the cohomological completion. It follows that $p_{1}^{-1}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(j)})\widehat{\otimes}_{W(k)}p_{2}^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(j)})\tilde{=}p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(j)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(j)}))$. 1) We claim that the map \[ p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(j)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(j)}))\to\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(j)} \] is an isomorphism; indeed, both sides are $p$-adically complete and $p$-torsion-free, so it suffices to check this after reduction mod $p$, where it becomes an easy computation in local coordinates. Thus we obtain isomorphisms \[ \{\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)})|p^{i}\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)})\} \] \[ \tilde{\to}\{\Phi\in\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(1)}|p^{i}\Phi\in\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(0)}\}=\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(0,1),i} \] for each $i\in\mathbb{Z}$. On the other hand, we claim that there is an isomorphism \[ (\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})^{i}\tilde{\to}\{\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)})|p^{i}\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)})\} \] Combined with the above, this proves $(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})^{i}\tilde{\to}\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{Y}}^{(0,1),i}$ as required. To see it, note that we have the map \[ f_{\infty}:(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})^{i}\to(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})^{\infty} \] The completion of the right hand side is $p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)})$; so we obtain a map \[ (\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})^{i}\to\{\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(1)})|p^{i}\Phi\in p_{1}^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{X}\times\mathfrak{Y}}}p_{2}^{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)})\} \] and to see that it is an isomorphism, one may check it after reduction mod $p$; then it follows from the result of part $2)$ proved directly below. 2) As above we have the map $p_{1}^{-1}\mathcal{D}_{X}^{(\infty)}\otimes_{k}p_{2}^{-1}\mathcal{D}_{Y}^{(\infty)}\to\mathcal{D}_{X\times Y}^{(\infty)}$. Restricting to $\mathcal{T}_{X}$ and $\mathfrak{l}_{X}$ (a defined in \defref{L} above)we get maps $p_{1}^{\#}:p_{1}^{-1}(\mathcal{T}_{X})\to\mathcal{T}_{X\times Y}$ and $p_{1}^{\#}:p_{1}^{-1}(\mathfrak{l}_{X})\to\mathfrak{l}_{X\times Y}$; and similarly for $p_{2}$. Thus we get a map \[ A:(\mathcal{T}_{X}\boxtimes1)\oplus(1\boxtimes\mathcal{T}_{Y})\oplus(\mathfrak{l}_{X}\boxtimes1)\oplus(1\boxtimes\mathfrak{l}_{Y})\to\mathcal{D}_{X\times Y}^{(0,1)} \] defined by \[ A(\partial_{1}\boxtimes1+1\boxtimes\partial_{2}+\delta_{1}\boxtimes1+1\boxtimes\delta_{2})=p_{1}^{\#}(\partial_{1})+p_{2}^{\#}(\partial_{2})+p_{1}^{\#}(\delta_{1})+p_{2}^{\#}(\delta_{2}) \] On the other hand, the sheaf $(\mathcal{T}_{X}\boxtimes1)\oplus(1\boxtimes\mathcal{T}_{Y})\oplus(\mathfrak{l}_{X}\boxtimes1)\oplus(1\boxtimes\mathfrak{l}_{Y})$ generates $\mathcal{D}_{X}^{(0,1)}\boxtimes\mathcal{D}_{Y}^{(0,1)}$ as a sheaf of algebras over $\mathcal{O}_{X\times Y}[f,v]$. Thus to show that $A$ extends (necessarily uniquely) to an isomorphism of algebras, we can so do locally. So, let $\{x_{1},\dots,x_{n}\}$ and $\{y_{1},\dots,y_{m}\}$ be local coordinates on $X$ and $Y$, respectively, with associated derivations $\{\partial_{x_{1}},\dots,\partial_{x_{n}}\}$ and $\{\partial_{y_{1}},\dots,\partial_{y_{m}}\}$. Then by \corref{Local-coords-over-A=00005Bf,v=00005D} an $D(\mathcal{O}_{X})$-basis for $\mathcal{D}_{X}^{(0,1)}$ is given by the set $\{\partial_{x}^{I}(\partial_{x}^{[p]})^{J}\}$ for multi-indices $I,J$ such that each entry of $I$ is contained in $\{0,1,\dots,p-1\}$; the analogous statement holds over $Y$. Therefore the set $\{\partial_{x}^{I_{1}}(\partial_{x}^{[p]})^{J_{1}}\otimes\partial_{y}^{I_{2}}(\partial_{y}^{[p]})^{J_{2}}\}$ is an $\mathcal{O}_{X\times Y}[f,v]$-basis for $\mathcal{D}_{X}^{(0,1)}\boxtimes\mathcal{D}_{Y}^{(0,1)}$; but also $\{\partial_{x}^{I_{1}}\partial_{y}^{I_{2}}(\partial_{x}^{[p]})^{J_{1}}(\partial_{y}^{[p]})^{J_{2}}\}$ is certainly an $D(\mathcal{O}_{X\times Y})$-basis for $\mathcal{D}_{X\times Y}^{(0,1)}$ and so the result follows immediately. \end{proof} Now we can define the tensor product: \begin{defn} Let $\Delta:\mathfrak{X}\to\mathfrak{X}\times\mathfrak{X}$ denote the diagonal morphism. 1) Then for $\mathcal{M}^{\cdot},\mathcal{N}^{\cdot}\in D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ we define $\mathcal{M}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}^{\cdot}:=L\Delta^{*}(\mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot})\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$, where $\mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot}$ is regarded as an element of $D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{X}}^{(0,1)}))$ via the isomorphism $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\boxtimes\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{X}}^{(0,1)}$. 2) For $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{op}}))$ and $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$, we define $\mathcal{M}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}^{\cdot}:=\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}((\omega_{\mathfrak{X}}^{-1}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}^{\cdot})\in D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{op}}))$ One has the analogous constructions for a smooth $X$ over $k$. \end{defn} From the construction, one sees directly that, as an $D(\mathcal{O}_{\mathfrak{X}})$-module, the module $\mathcal{M}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}^{\cdot}$ agrees with the $D(\mathcal{O}_{\mathfrak{X}})$-module denoted in the same way. The issue that this construction resolves is how to put a $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-module structure on this object. To proceed further, it is useful to note some explicit formulas in coordinates: \begin{rem} \label{rem:Two-actions-agree}Suppose we have local coordinates $\{x_{i}\}_{i=1}^{n}$ and $\{\partial_{i}\}_{i=1}^{n}$ on $\mathfrak{X}$. Then for modules $\mathcal{M},\mathcal{N}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ we can put an action of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ on $\mathcal{M}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{N}$ via the following formulas: \[ \partial_{i}(m\otimes n)=\partial_{i}m\otimes n+m\otimes\partial_{i}n \] and \[ \partial_{i}^{([p]}(m\otimes n)=f\sum_{j=1}^{p-1}\partial^{[j]}(m)\otimes\partial^{[p-j]}(m)+\partial^{[p]}(m)\otimes n+m\otimes\partial^{[p]}(n) \] Taking a flat resolution of $\mathcal{N}$, this gives $\mathcal{M}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}$ the structure of an element of $D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$, which means that $\mathcal{M}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{N}$ belongs to $D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. This object is isomorphic to the tensor product defined above. Indeed, in local coordinates the action of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ on $\Delta^{*}(\widehat{\mathcal{D}}_{\mathfrak{X}\times\mathfrak{X}}^{(0,1)})$ is given as follows: let $\{\partial_{i},\partial'_{i}\}_{i=1}^{n}$ be local coordinate derivations on $\mathfrak{X}\times\mathfrak{X}$. Then the action is given by $\partial_{i}\cdot1=\partial_{i}+\partial_{i}'$ and $\partial_{i}^{[p]}\cdot1=f\sum_{j=1}^{p-1}\partial_{i}^{[j]}\cdot(\partial'_{i})^{[p-j]}+\partial_{i}^{[p]}+(\partial'_{i})^{[p]}$, which agrees with the above formula. \end{rem} This allows us to prove the following useful \begin{lem} \label{lem:Juggle}(Compare \cite{key-50}, lemma 2.2.5) Let $\mathcal{M}^{\cdot},\mathcal{P}^{\cdot}$ be elements of $D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{opp}}))$. Then there is an isomorphism \[ \mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}(\mathcal{M}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{P}^{\cdot})\tilde{\to}(\mathcal{N}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{M}^{\cdot})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{P}^{\cdot} \] \end{lem} \begin{proof} Let $\mathcal{M},\mathcal{P}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ and $\mathcal{N}\in\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{opp}})$. We have a map of $D(\mathcal{O}_{\mathfrak{X}})$-modules \[ \mathcal{N}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}(\mathcal{M}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{P})\to(\mathcal{N}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{M})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}\mathcal{P} \] simply because $D(\mathcal{O}_{\mathfrak{X}})$ is a sub-algebra of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. Using the local description of the $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-module action on $\mathcal{N}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{M}$ given by \remref{Two-actions-agree}, one sees that this map factors through $\mathcal{N}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{P})$ and we obtain a morphism \[ \mathcal{N}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{P})\to(\mathcal{N}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}\mathcal{M})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}\mathcal{P} \] Since $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ is flat over $D(\mathcal{O}_{\mathfrak{X}})$, we can compute the associated derived functors using K-flat resolutions over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ of $\mathcal{N}$, and $\mathcal{P}$, respectively. Doing so gives a map in the derived category \[ \mathcal{N}^{\cdot}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}(\mathcal{M}^{\cdot}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{P}^{\cdot})\to(\mathcal{N}^{\cdot}\otimes_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{M}^{\cdot})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{P}^{\cdot} \] and passing to the derived completions gives the map in the statement of the lemma; to show it is an isomorphism we may reduce mod $p$ and, taking K-flat resolutions, assume that each term of both $\mathcal{N}^{\cdot}$ and $\mathcal{P}^{\cdot}$ is stalk-wise free over $\mathcal{D}_{X}^{(0,1)}$; thus the statement comes down to the claim that \[ \mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{M}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)})\tilde{\to}(\mathcal{D}_{X}^{(0,1)}\otimes_{D(\mathcal{O}_{X})}\mathcal{M})\otimes_{\mathcal{D}_{X}^{(0,1)}}\mathcal{D}_{X}^{(0,1)} \] which is immediate. \end{proof} Finally, we note the following compatibility of tensor product and pull-back, which follows directly from unpacking the definitions. \begin{lem} \label{lem:Tensor-and-pull}Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism. Then there is a canonical isomorphism $L\varphi^{*}(\mathcal{M}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{N}^{\cdot})\tilde{\to}L\varphi^{*}(\mathcal{M}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}(\mathcal{N}^{\cdot})$. The analogous statement holds for a morphism of smooth $k$-schemes $\varphi:X\to Y$. \end{lem} \section{\label{sec:Push-Forward}Operations on Gauges: Push-Forward} As above let $\varphi:\mathfrak{X}\to\mathfrak{Y}$. Now that we have both the pull-back and the left-right swap, we can define the push-forward. We start by noting that $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}$ carries a natural right module structure over itself (by right multiplication). Therefore, by \propref{Left-Right-Swap} there is a natural left $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}$ gauge structure on $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1}$. By \defref{Pullback!} there is a natural left $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module structure on $\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1})=L\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1})$. \begin{defn} \label{def:Push!}1) Define the $(\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}),\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)})$ bimodule $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}:=\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1})\otimes\omega_{\mathfrak{X}}$; here, the right $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module structure comes from the left $\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}$-module structure on $\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1})$; the left $\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})$-structure comes from the left multiplication of $\varphi^{-1}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})$ on $\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}\otimes\omega_{\mathfrak{Y}}^{-1})$. 2) Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$. Then we define ${\displaystyle \int_{\varphi}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})}\in D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))$. 3) If we instead have $\varphi:X\to Y$ over $k$; then for $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ we define ${\displaystyle \int_{\varphi}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})}\in D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ where $\mathcal{D}_{Y\leftarrow X}^{(0,1)}$ is defined analogously to $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}$. 4) if $\mathfrak{Y}=\text{Specf}(W(k))$, then we denote ${\displaystyle \mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot}):=\int_{\varphi}\mathcal{M}^{\cdot}}$ for any $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$. Similarly, there are push-forwards in the category of right $\mathcal{\widehat{D}}^{(0,1)}$-modules defined by ${\displaystyle \int_{\varphi}\mathcal{M}_{r}^{\cdot}:=R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})}$ for $\mathcal{M}_{r}^{\cdot}\in D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))^{\text{op}}$; clearly the left-right interchange intertwines the two pushforwards. Similar remarks apply to a morphism $\varphi:X\to Y$ over $k$. \end{defn} We begin by recording some basic compatibilities; for these note that we have the transfer bimodule $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}:=\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}/(v-1)$ in the category of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules, and $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}:=(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}/(f-1))^{\widehat{}}$ (here the $()^{\widehat{}}$ denotes $p$-adic completion, which is the same as cohomological completion in this case by \propref{Basic-properties-of-the-transfer-module}). One may therefore define ${\displaystyle \int_{\varphi,0}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot})}$ for $\mathcal{M}^{\cdot}\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$ and ${\displaystyle \int_{\varphi,1}\mathcal{M}^{\cdot}:=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot})}$ for $\mathcal{M}^{\cdot}\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. As in the case of the pullback, this is not quite Berthelot's definition of these functors; because he uses the more traditional $\text{R}\lim$. However, they do agree in important cases, such as when $\varphi$ is proper and $\mathcal{M}^{\cdot}$ is coherent. We have \begin{prop} \label{prop:push-and-complete-for-D} Let $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$. 1) ${\displaystyle (\int_{\varphi}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k\tilde{=}\int_{\varphi}(\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k)}$ in the category $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. 2) $(\int_{\varphi}\mathcal{M}^{\cdot})^{-\infty}\tilde{=}(\int_{\varphi,0}\mathcal{M}^{\cdot,-\infty})$ where the pushforward on the right is defined as $R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0)}}^{L}\mathcal{M}^{\cdot,-\infty})$. 3) If $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, then $\widehat{((\int_{\varphi}\mathcal{M}^{\cdot})^{\infty})}\tilde{=}\int_{\varphi,1}\widehat{(\mathcal{M}^{\cdot,\infty})}$ where both uses of $\widehat{}$ denote derived completion. \end{prop} \begin{proof} 1) We have \[ \int_{\varphi}\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k=R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k \] \[ \tilde{=}R\varphi_{*}((\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k) \] (since $k$ is a perfect complex over $W(k)$, this is a special case of the projection formula where we consider $X$ and $Y$ as ringed spaces with the locally constant sheaf of rings $W(k)$; c.f. {[}Stacks{]}, tag 0B54). We have the isomorphism \[ k\otimes_{W(k)}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\tilde{=}\mathcal{D}_{Y\leftarrow X}^{(0,1)} \] since $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}$ is a $p$-torsion-free sheaf; and so \[ R\varphi_{*}((\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\otimes_{W(k)}^{L}k)\tilde{=}R\varphi_{*}(k\otimes_{W(k)}^{L}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] \[ R\varphi_{*}(k\otimes_{W(k)}^{L}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))\tilde{=}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] where we used that for any complex $\mathcal{N}^{\cdot}$ we have $k\otimes_{W(k)}^{L}\mathcal{N}^{\cdot}\tilde{=}k\otimes_{W(k)}^{L}\widehat{\mathcal{N}^{\cdot}}$ (c.f. \lemref{reduction-of-completion}). Now, we have \[ R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{=}\int_{\varphi}\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}(\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] But since $\mathcal{D}_{X}^{(0,1)}=\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}/p$ we have $\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k$ and the result follows. 2) For any complex we have $\mathcal{M}^{\cdot,-\infty}=\mathcal{M}^{\cdot}\otimes_{D(W(k))}^{L}(D(W(k))/(v-1))$. Thus the proof is an easier variant of that of $1)$, replacing $\otimes_{W(k)}^{L}k$ with $\otimes_{D(W(k))}^{L}D(W(k))/(v-1)$. 3) We have \[ \mathcal{K}^{\cdot}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)} \] where $\mathcal{K}^{\cdot}$ is a complex of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}[p^{-1}]$-modules; indeed, $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}$ is a $p$-torsion-free sheaf whose completion is exactly $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}$ (c.f. \propref{Basic-properties-of-the-transfer-module} and \lemref{Basic-Structure-of-D^(1)}).Thus there is a distinguished triangle \[ \mathcal{K}^{\cdot}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\to\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty} \] and the term on the left is a complex of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\infty}[p^{-1}]$-modules. Thus the derived completion of $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$ is isomorphic to the derived completion of $\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$. Further, we have \[ \mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}}^{L}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}) \] And, since $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}))$, we have (by \propref{Completion-for-noeth}) that $\widehat{\mathcal{M}^{\cdot,\infty}}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}$ as modules over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}$. Therefore we obtain \[ \mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(1)}}^{L}\widehat{(\mathcal{M}^{\cdot,\infty})} \] and so, taking $R\varphi_{*}$ yields \[ R\varphi_{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1),\infty}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1),\infty}}^{L}\mathcal{M}^{\cdot,\infty})\tilde{\to}\int_{\varphi,1}\widehat{(\mathcal{M}^{\cdot,\infty})} \] But the term on the left is isomorphic to the derived completion of ${\displaystyle \int_{\varphi}\mathcal{M}^{\cdot,\infty}}$ by \propref{Push-and-complete}. \end{proof} Now we will discuss the relationship between the $\mathcal{D}_{X}^{(0,1)}$ pushforward and the push-forwards over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. As usual we'll work with the functors $\text{\ensuremath{\mathcal{M}}}^{\cdot}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}k[f]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot}$ and $\text{\ensuremath{\mathcal{M}}}^{\cdot}\to\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}k[v]\otimes_{D(k)}^{L}\mathcal{M}^{\cdot}$ which take $D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ to $D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and $D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$, respectively (as in \propref{Quasi-rigid=00003Dfinite-homological}). Both of the algebras $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ and $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ possess transfer bimodules associated to any morphism $\varphi:X\to Y$, and hence are equipped with a push-pull formalism. In the case of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ this is well known (c.f., e.g \cite{key-22}, chapter $1$), while in the case of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ this theory is developed in \cite{key-11}, in the language of filtered derived categories. We shall proceed using the push-pull formalism for $\mathcal{D}_{X}^{(0,1)}$-modules that we have already developed, and discuss the relations with the other theories in section \subsecref{Hodge-and-Conjugate} below. \begin{defn} Let $\varphi:X\to Y$ be a morphism. We define a $(\varphi^{-1}\mathcal{R}(\mathcal{D}_{Y}^{(1)}),\mathcal{R}(\mathcal{D}_{X}^{(1)}))$ bimodule $\mathcal{R}_{Y\leftarrow X}^{(1)}:=\mathcal{D}_{Y\leftarrow X}^{(0,1)}/v$. Define a $(\varphi^{-1}\mathcal{\overline{R}}(\mathcal{D}_{Y}^{(1)}),\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$ bimodule $\mathcal{R}_{Y\leftarrow X}^{(1)}:=\mathcal{D}_{Y\leftarrow X}^{(0,1)}/f$. Define ${\displaystyle \int_{\varphi,1}}\mathcal{M}^{\cdot}=R\varphi_{*}(\mathcal{R}_{Y\leftarrow X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M}^{\cdot})$ on the category $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$, and analogously ${\displaystyle \int_{\varphi,0}}$ for $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$-modules. As above, there is also a push-forward for right modules defined by ${\displaystyle \int_{\varphi,1}}\mathcal{M}_{r}^{\cdot}=R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to Y}^{(1)})$, and analogously for right $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$-modules. We have the basic compatibility: \end{defn} \begin{prop} If $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, then we have \[ {\displaystyle \mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\int_{\varphi}\mathcal{M}^{\cdot}\tilde{=}\int_{\varphi,1}(\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})} \] The analogous result holds for $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$. \end{prop} \begin{proof} We have \[ \mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}\int_{\varphi}\mathcal{M}^{\cdot}=\mathcal{R}(\mathcal{D}_{Y}^{(1)})\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] \[ \tilde{=}R\varphi_{*}(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] (we will prove this last isomorphism in the lemma directly below). We have the isomorphism \[ \varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}\mathcal{D}_{Y\leftarrow X}^{(0,1)}\tilde{=}\mathcal{R}_{Y\leftarrow X}^{(1)} \] which is proved in the same way as \eqref{transfer-iso-1} above. Therefore \[ R\varphi_{*}(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}))\tilde{=}R\varphi_{*}(\mathcal{R}_{Y\leftarrow X}^{(1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] \[ \tilde{=}R\varphi_{*}(\mathcal{R}_{Y\leftarrow X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})=\int_{\varphi,1}(\mathcal{R}(\mathcal{D}_{X}^{(1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] as claimed. The proof for the case of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ is essentially identical. \end{proof} In the previous proof we used the \begin{lem} \label{lem:baby-projection-1}Let $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, and $\mathcal{N}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$. Then there is an isomorphism \[ \mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{=}R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] \end{lem} \begin{proof} (c.f. the proof of \cite{key-17}, proposition 5.3). First, we construct a canonical map \[ \mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\to R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] as follows: one may replace $\mathcal{N}^{\cdot}$ with a complex of $K$-flat graded $\mathcal{D}_{Y}^{(0,1)}$-modules, $\mathcal{F}^{\cdot}$. Choosing a quasi-isomorphism $\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}\mathcal{I}^{\cdot}$, a $K$-injective complex of graded $\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$-modules, one obtains the quasi-isomorphism \[ \mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{F}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}\varphi_{*}\mathcal{I}^{\cdot} \] Then there is the obvious isomorphism \[ \mathcal{F}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}\varphi_{*}\mathcal{I}^{\cdot}\tilde{\to}\varphi_{*}(\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot}) \] and a canonical map \[ \varphi_{*}(\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot})\to R\varphi_{*}((\varphi^{-1}(\mathcal{F}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}\mathcal{I}^{\cdot})) \] \[ \tilde{\to}R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] Thus we obtain the canonical map \[ \mathcal{N}^{\cdot}\otimes_{\mathcal{D}_{Y}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})\to R\varphi_{*}(\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] this map exists for all $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{op}})$ and $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. To check whether it is an isomorphism, we may work locally on $Y$ and suppose that $Y$ is affine from now on. To prove this, we proceed in a similar manner to the proof of the projection formula for quasi-coherent sheaves, in the general version of \cite{key-17}, proposition 5.3. Fix $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. For any $\mathcal{N}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$, we claim that $\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$ is quasi-isomorphic to a complex in $D_{qcoh}(D(\mathcal{O}_{X}))$. To see this, we observe that any quasicoherent $\mathcal{D}_{X}^{(0,1)}$-module $\mathcal{M}$ is a quotient of the $\mathcal{D}_{X}^{(0,1)}$-module $\mathcal{D}_{X}^{(0,1)}\otimes_{D(\mathcal{O}_{X})}\mathcal{M}$ (where the $\mathcal{D}_{X}^{(0,1)}$-module is via the action on the left hand factor on the tensor product). It follows that any bounded-above complex in $D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ is quasi-isomorphic to a complex, whose terms are of the form $\mathcal{D}_{X}^{(0,1)}\otimes_{D(\mathcal{O}_{X})}\mathcal{M}$ for quasi-coherent $\mathcal{M}$. Therefore any complex in $D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ is a homotopy colimit of such complexes. Therefore $\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}$ is quasi-isomorphic to a complex of quasicoherent $D(\mathcal{O}_{X})$-modules. In addition, since $Y$ is affine, $\mathcal{N}^{\cdot}$ is quasi-isomorphic to a $K$-projective complex of $\mathcal{D}_{Y}^{(0,1)}$-modules; in particular, a complex whose terms are projective $\mathcal{D}_{Y}^{(0,1)}$-modules. It follows that $\varphi^{-1}(\mathcal{N}^{\cdot})\otimes_{\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})}^{L}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot})$ is quasi-isomorphic to a complex in $D_{qcoh}(D(\mathcal{O}_{X}))$ as claimed. Now, since $R\varphi_{*}$ commutes with arbitrary direct sums on $D_{qcoh}(D(\mathcal{O}_{X}))$ (by \cite{key-17}, lemma 1.4), we see that both sides of arrow commute with arbitrary direct sums (over objects in $D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{opp}})$); so the set of objects on which the arrow is an isomorphism is closed under arbitrary direct sums. Since $Y$ is affine, the category $D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)})^{\text{op}})$ is generated by the compact objects $\{\mathcal{D}_{Y}^{(0,1)}[i]\}_{i\in\mathbb{Z}}$; therefore (as in the proof of \cite{key-17}, lemma 5.3), it actually suffices to show that the arrow is an isomorphism on $\mathcal{D}_{Y}^{(0,1)}$ itself, but this is obvious. \end{proof} This type of projection formula is so useful that we will record here a minor variant: \begin{lem} \label{lem:proj-over-D}Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism. Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1),\text{opp}}))$, such that $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)})^{\text{opp}})$. Then we have \[ (\int_{\varphi}\mathcal{N}^{\cdot})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{\to}R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}\mathcal{M}^{\cdot}) \] The analogous statement holds for $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1),\text{opp}}))$; as well as for the Rees algebras $\mathcal{R}(\mathcal{D}^{(1)})$ and $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$. \end{lem} \begin{proof} We have that ${\displaystyle \int_{\varphi}\mathcal{N}^{\cdot}=R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})}$. As in the proof of \lemref{baby-projection-1}, there is a morphism \begin{equation} R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\to R\varphi_{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot}))\label{eq:adunction} \end{equation} Indeed, one constructs the map \[ R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\to R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\otimes_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})) \] exactly as above; and then passes to the cohomological completion. Since $L\varphi^{*}\mathcal{M}^{\cdot}=\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})$ by definition, the result will follow if \eqref{adunction} is an isomorphism. To prove that, apply $\otimes_{W(k)}^{L}k$ and quote the previous result. The proof in the case of the Rees algebras is completely analogous. \end{proof} Here is an important application of these ideas: \begin{lem} \label{lem:Composition-of-pushforwards}Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ and $\psi:\mathfrak{Y}\to\mathfrak{Z}$ be morphisms. There is a canonical map \[ \int_{\psi}\circ\int_{\varphi}\mathcal{M}^{\cdot}\to\int_{\psi\circ\varphi}\mathcal{M}^{\cdot} \] for any $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$, which is an isomorphism if $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. If $\varphi:X\to Y$ and $\psi:Y\to Z$ are morphisms, we have the analogous statements in $D(\mathcal{G}(\mathcal{D}_{Z}^{(0,1)}))$. \end{lem} \begin{proof} As in \lemref{composition-of-pullbacks}, we have an isomorphism \[ \varphi^{-1}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)})\widehat{\otimes}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\tilde{=}\mathcal{\widehat{D}}_{\mathfrak{Z}\leftarrow\mathfrak{X}}^{(0,1)} \] as $((\psi\circ\varphi)^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(0,1)}),\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ bimodules. Then we have \[ \int_{\psi}\circ\int_{\varphi}\mathcal{M}^{\cdot}=R\psi_{*}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}R\varphi_{*}(\mathcal{D}_{\mathfrak{Y\leftarrow\mathfrak{X}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})) \] \[ \to R\psi_{*}R\varphi_{*}(\varphi^{-1}(\mathcal{D}_{\mathfrak{Z\leftarrow\mathfrak{Y}}}^{(0,1)})\widehat{\otimes}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}^{L}\mathcal{D}_{\mathfrak{Y\leftarrow\mathfrak{X}}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}) \] \[ \tilde{\to}R(\psi\circ\varphi)_{*}(\mathcal{\widehat{D}}_{\mathfrak{Z}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})=\int_{\psi\circ\varphi}\mathcal{M}^{\cdot} \] where the first arrow is constructed as in \lemref{proj-over-D} and the second isomorphism is given above. Applying the functor $\otimes_{W(k)}^{L}k$ and using \propref{Push-and-complete}, part $1)$, we reduce to proving the analogous statement for $\varphi:X\to Y$ and $\psi:Y\to Z$; where it follows exactly as in \lemref{baby-projection-1}. \end{proof} We shall also need results relating the pushforwards when $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ is already annihilated by $f$ (or $v$): \begin{prop} \label{prop:Sandwich-push}Suppose $\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$ satisfies $v\mathcal{M}=0$. Then ${\displaystyle \int_{\varphi}\mathcal{M}}$ is contained in the image of the functor $D(\mathcal{R}(\mathcal{D}_{X}^{(1)})-\text{mod})\to D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. In fact, there is an isomorphism of graded sheaves of $\mathcal{O}_{X}[f,v]$-modules \[ R\varphi_{*}(\mathcal{R}_{Y\leftarrow X}^{(1)}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{M})\tilde{=}R\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}) \] In other words, the pushforward of $\mathcal{M}$, regarded as a module over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, agrees with its pushforward as a $\mathcal{D}_{X}^{(0,1)}$-module. The analogous result hold when $f\mathcal{M}=0$. \end{prop} \begin{proof} This is an immediate consequence of \propref{Sandwich!} \end{proof} As a consequence of these results, we obtain: \begin{thm} \label{thm:phi-push-is-bounded}Let $\varphi:X\to Y$ be a morphism. Then, for each of the algebras $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, and $\mathcal{D}_{X}^{(0,1)}$, the pushforward along $\varphi$ takes $D_{qcoh}^{b}$ to $D_{qcoh}^{b}$. If $\varphi$ is proper, then the pushforward along $\varphi$ takes $D_{coh}^{b}$ to $D_{coh}^{b}$. \end{thm} \begin{proof} Let us start with the statement, that the pushforward takes $D_{qcoh}$ to $D_{qcoh}$ in all of these cases. For this, we can argue as in the proof of \lemref{baby-projection-1}: namely, one may assume $Y$ is affine, and then if $\mathcal{M}^{\cdot}\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, we may replace $\mathcal{M}^{\cdot}$ by a homotopy colimit of $\mathcal{D}_{X}^{(0,1)}$-modules of the form $\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{O}_{X}[f,v]}\mathcal{M}$, for quasi-coherent $\mathcal{M}$. Therefore $\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{M}^{\cdot}$ is quasi-isomorphic to a complex of quasicoherent $\mathcal{O}_{X}[f,v]$-modules, which implies that the cohomology sheaves of its pushforward are quasi-coherent $\mathcal{O}_{Y}[f,v]$-modules; and therefore quasi-coherent $\mathcal{D}_{Y}^{(0,1)}$-modules. The same argument works for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. To prove the boundedness, we can factor $\varphi$ as a closed immersion (the graph $X\to X\times Y$) followed by the projection $X\times Y\to Y$, and, applying \lemref{Composition-of-pushforwards}, we see that it suffices to consider separately the case of a closed immersion and the case of a smooth morphism. For a closed immersion $\iota:X\to Y$ we have that the bimodule $\mathcal{D}_{X\to Y}^{(0)}$ is locally free over $\mathcal{D}_{X}^{(0,1)}$ (this elementary fact will be checked below in \lemref{transfer-is-locally-free}) and so the tensor product $\otimes_{\mathcal{D}_{X}^{(0,1)}}\mathcal{D}_{X\to Y}^{(0)}$ takes quasicoherent sheaves to quasicoherent sheaves. Now, if $X\to Y$ is smooth, we have by (the proof of) \propref{Smooth-pullback-preserves-coh}, that $\mathcal{D}_{Y\leftarrow X}^{(0,1)}$ is a coherent $\mathcal{D}_{X}^{(0,1),\text{opp}}$-module. Further, since it is locally the reduction mod $p$ of a standard module, it is rigid, so that by \propref{Quasi-rigid=00003Dfinite-homological} it is locally of finite homological dimension; and the result follows directly. Thus we see that ${\displaystyle \int_{\varphi}}$ is bounded on $D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$, the same holds for the pushforward on $D_{qcoh}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$ and $D_{qcoh}(\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})))$ by the previous proposition. Now suppose $\varphi$ is proper. Let us say that a right $\mathcal{D}_{X}^{(0,1)}$-module is induced if it is of the form $\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}$ for some coherent $\mathcal{F}$ over $D(\mathcal{O}_{X})$. In this case we have \[ \int_{\varphi}\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}=R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{D}_{X\to Y}^{(0,1)}) \] \[ \tilde{=}R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}^{L}\mathcal{D}_{X}^{(0,1)}\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\varphi^{*}(\mathcal{D}_{Y}^{(0,1)}))\tilde{\to}R\varphi_{*}(\mathcal{F}\otimes_{D(\mathcal{O}_{X})}^{L}\varphi^{*}(\mathcal{D}_{Y}^{(0,1)})) \] \[ \tilde{\to}R\varphi_{*}(\mathcal{F})\otimes_{D(O_{Y})}^{L}\mathcal{D}_{Y}^{(0,1)} \] Thus the result is true for any induced module. If $\mathcal{M}$ is an arbitrary coherent right $\mathcal{D}_{X}^{(0,1)}$-module, then, as a quasicoherent sheaf over $\mathcal{O}_{X}[f,v]$, it is the union of its $\mathcal{O}_{X}[f,v]$ coherent sub-sheaves. Selecting such a subsheaf which generates $\mathcal{M}$ as a $\mathcal{D}_{X}^{(0,1)}$-module, we obtain a short exact sequence \[ 0\to\mathcal{K}\to\mathcal{F}\otimes_{D(\mathcal{O}_{X})}\mathcal{D}_{X}^{(0,1)}\to\mathcal{M}\to0 \] where $\mathcal{K}$ is also coherent. Since the functor ${\displaystyle \int_{\varphi}}$ is concentrated in homological degrees $\leq d_{X/Y}$ for all coherent $\mathcal{D}_{X}^{(0,1)}$-modules, we can now deduce the coherence of ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}\mathcal{M})}$ by descending induction on $i$. This proves the result for $\mathcal{D}_{X}^{(0,1)}$-modules, and we can deduce the result for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules by again invoking \propref{Sandwich-push}. \end{proof} From this and the formalism of cohomological completion (specifically, \propref{coh-to-coh}), we deduce \begin{cor} \label{cor:proper-push-over-W(k)}Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be proper. Then ${\displaystyle \int_{\varphi}}$ takes $D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ to $D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. If $\mathcal{M}^{\cdot}\in D_{coh,F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ then ${\displaystyle \int_{\varphi}\mathcal{M}}\in D_{coh,F^{-1}}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. \end{cor} \begin{proof} The first part follows immediately from the proceeding theorem by applying $\otimes_{W(k)}^{L}k$. The second part follows from \propref{push-and-complete-for-D} (part $3)$), as well as Berthelot's theorem that ${\displaystyle \int_{\varphi,0}F^{*}\tilde{\to}F^{*}\int_{\varphi,1}}$ (c.f. \cite{key-2}, section 3.4, and also \thmref{Hodge-Filtered-Push} below). \end{proof} \subsection{\label{subsec:Hodge-and-Conjugate}Push-forwards for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. } In this section we take a close look at the theory over $k$. In particular, we study the pushforwards of modules over $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, and compare them with more traditional filtered pushforwards found in the literature. For $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ modules themselves, we will construct the analogue of the relative de Rham resolution. This will allow us to exhibit an adjunction between ${\displaystyle \int_{\varphi}}$ and $\varphi^{\dagger}$ when $\varphi$ is smooth. We begin with $\mathcal{R}(\mathcal{D}_{X}^{(1)})$, where we can reduce everything to the more familiar situation of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules using the fact that $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ is Morita equivalent to $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ (c.f. \thmref{Filtered-Frobenius}). Let $\varphi:X\to Y$. Recall that Laumon constructed in \cite{key-19} the push-forward in the filtered derived category of $\mathcal{D}_{X}^{(0)}$-modules (with respect to the symbol filtration); essentially, his work upgrades the bimodule $\mathcal{D}_{Y\leftarrow X}^{(0)}$ to a filtered $(\varphi^{-1}(\mathcal{D}_{Y}^{(0)}),\mathcal{D}_{X}^{(0)})$-bimodule via \[ F_{i}(\mathcal{D}_{Y\leftarrow X}^{(0)}):=\varphi^{-1}(F_{i}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\omega_{X} \] (c.f. \cite{key-19}, formula 5.1.3); then one may define ${\displaystyle \int_{\varphi}}$ via the usual formula, but using the tensor product and push-forward in the filtered derived categories. On the other hand, we can apply the Rees construction to the above filtered bimodule to obtain $\mathcal{R}(\mathcal{D}_{Y\leftarrow X}^{(0)})$, a graded $(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})),\mathcal{R}(\mathcal{D}_{X}^{(0)}))$ bimodule, which (again by the usual formula) yields a push-forward functor ${\displaystyle \int_{\varphi}}:D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))\to D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$, and we have the following evident compatibility: \begin{lem} Let $\mathcal{M}^{\cdot}\in D((\mathcal{D}_{X}^{(0)},F)-\text{mod})$. Then we have \[ \mathcal{R}(\int_{\varphi}\mathcal{M}^{\cdot})\tilde{\to}\int_{\varphi}\mathcal{R}(\mathcal{M}^{\cdot}) \] In particular, the Hodge-to-deRham spectral sequence for ${\displaystyle \int_{\varphi}\mathcal{M}^{\cdot}}$ degenerates at $E_{1}$ iff each of the sheaves ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}\mathcal{R}(\mathcal{M}^{\cdot}))}$ is torsion-free over the Rees parameter $f$. \end{lem} Next, we relate this to the pull-back and push-forward for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ modules; starting with the analogous statement for pull-back: \begin{lem} \label{lem:Hodge-Filtered-Pull}Let $\varphi:X\to Y$ and suppose $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. Then $L\varphi^{*}\circ F_{Y}^{*}\mathcal{M}^{\cdot}\tilde{=}F_{X}^{*}\circ L\varphi^{*}\mathcal{M}^{\cdot}$. Here, the pullback on the left is in the category of $\mathcal{R}(\mathcal{D}^{(1)})$-modules, while the pullback on the right is in the category of $\mathcal{R}(\mathcal{D}^{(0)})$-modules. \end{lem} \begin{proof} Since $\varphi\circ F_{X}=F_{Y}\circ\varphi$ we have \[ L\varphi^{*}\circ F_{Y}^{*}\mathcal{M}^{\cdot}\tilde{\to}F_{X}^{*}\circ L\varphi^{*}\mathcal{M}^{\cdot} \] as (graded) $\mathcal{O}_{X}$-modules; we need to check that this map preserves the $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-module structures on both sides.This question is local, so we may suppose $X=\text{Spec}(B)$ and $Y=\text{Spec}(A)$ both posses local coordinates. Further, by taking a K-flat resolution of $\mathcal{M}^{\cdot}$ we may suppose that $\mathcal{M}^{\cdot}=\mathcal{M}$ is concentrated in a single degree. Now, as an $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-module, $F_{Y}^{*}\mathcal{M}$ possesses the structure of a connection with $p$-curvature $0$, and so the induced connection on $\varphi^{*}F_{Y}^{*}\mathcal{M}$ also has $p$-curvature $0$; and the kernel of this connection is equal to $(\varphi^{(1)})^{*}\mathcal{M}^{(1)}\subset\varphi^{*}F_{Y}^{*}\mathcal{M}$ (here $\mathcal{M}^{(1)}$ denotes $\sigma^{*}\mathcal{M}$ where $\sigma:X^{(1)}\to X$ is the natural isomorphism of schemes). Note that $\mathcal{M}^{(1)}$ possesses the action of $\mathcal{R}(\mathcal{D}_{Y^{(1)}}^{(0)})$ (c.f. \remref{The-inverse-to-F^*}). Let $\{\partial_{i}\}_{i=1}^{n}$ be coordinate derivations on $X$. Then the action of $\partial_{i}^{[p]}$ on $\varphi^{*}F_{Y}^{*}\mathcal{M}$ is given (by \propref{pull-back-in-pos-char}) by first restricting $\partial_{i}^{[p]}$ to a differential operator $\varphi^{-1}(\mathcal{O}_{Y})\to\mathcal{O}_{X}$, writing the resulting operator as \[ \sum_{j=1}^{r}b_{j}^{p}\partial_{j}^{[p]}+\sum_{J}b_{J}\partial^{J} \] (where $\{\partial_{j}\}_{j=1}^{r}$ are coordinate derivations on $Y$, and $b_{j},b_{J}\in B$) and then letting $\partial_{i}^{[p]}$ act as \[ \sum_{j=1}^{r}b_{j}^{p}\partial_{j}^{[p]}+\sum_{J}b_{J}\partial^{J} \] therefore, the action of $\partial_{i}^{[p]}$ preserves $(\varphi^{(1)})^{*}\mathcal{M}^{(1)}$ and it acts there as ${\displaystyle \partial_{i}^{[p]}(1\otimes m)=\sum_{j=1}^{r}b_{j}^{p}\cdot\partial_{j}^{[p]}(m)}$. But the action of $\{\partial_{j}^{[p]}\}$ on $\mathcal{M}^{(1)}$ defines the action of $\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)})$ on $\mathcal{M}^{(1)}$, and this formula simply defines the pullback from $\mathcal{R}(\mathcal{D}_{Y^{(1)}}^{(0)})$ to $\mathcal{R}(\mathcal{D}_{X^{(1)}}^{(0)})$-modules; in other words, $(\varphi^{(1)})^{*}\mathcal{M}^{(1)}=((\varphi)^{*}\mathcal{M})^{(1)}$ where $\varphi^{*}\mathcal{M}$ is the usual pullback of $\mathcal{R}(\mathcal{D}^{(0)})$-modules. Thus we see that $\varphi^{*}F_{Y}^{*}\mathcal{M}=F_{X}^{*}((\varphi)^{*}\mathcal{M})$ as $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules, as desired. \end{proof} Now we discuss push-forward: \begin{thm} \label{thm:Hodge-Filtered-Push}Let $\mathcal{M}^{\cdot}$ be a complex of graded $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules, and via \thmref{Filtered-Frobenius} write $\mathcal{M}^{\cdot}\tilde{=}F_{X}^{*}\mathcal{N}^{\cdot}$, where $\mathcal{N}^{\cdot}$ is a complex of graded $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules. There is an isomorphism \[ \int_{\varphi,1}\mathcal{M}^{\cdot}\tilde{\to}F_{X}^{*}\int_{\varphi}\mathcal{N}^{\cdot} \] where ${\displaystyle \int_{\varphi}\mathcal{N}}^{\cdot}$ is the pushforward of $\mathcal{N}^{\cdot}$ over $\mathcal{R}(\mathcal{D}_{X}^{(0)})$. \end{thm} \begin{proof} (following \cite{key-2}, theoreme 3.4.4) By left-right interchange it is equivalent to prove the right-handed version \[ \int_{\varphi,1}F_{X}^{!}\mathcal{N}^{\cdot}\tilde{\to}F_{Y}^{!}\int_{\varphi}\mathcal{N}^{\cdot} \] for any $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})^{\text{opp}}))$. We have \[ \int_{\varphi,1}F_{X}^{!}(\mathcal{N}^{\cdot})=R\varphi_{*}(F_{X}^{!}(\mathcal{N}^{\cdot})\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to Y}^{(1)})=R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}F_{X}^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to Y}^{(1)}) \] Now, recall \[ \mathcal{R}_{X\to Y}^{(1)}=\varphi^{*}\mathcal{R}(\mathcal{D}_{X}^{(1)})\tilde{=}\varphi^{*}F_{Y}^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(1)})\tilde{=}F_{X}^{*}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(1)}) \] where the second isomorphism is \corref{Filtered-right-Frob}, and the third is by the lemma above; note that this isomorphism preserves the natural right $\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$ -module structures on both sides. It follows (c.f. \propref{F^*F^!}, part $2)$, and \corref{Filtered-right-Frob}) that \[ F_{X}^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to Y}^{(1)}\tilde{=}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)}) \] (as $(\mathcal{R}(\mathcal{D}_{X}^{(1)}),\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(1)}))$ bimodules). Therefore \[ R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}F^{!}(\mathcal{R}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}^{L}\mathcal{R}_{X\to Y}^{(1)})\tilde{=}R\varphi_{*}(\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}^{L}\varphi^{*}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)})) \] \[ \tilde{=}\int_{\varphi}\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}^{L}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)})) \] where the last line is \lemref{proj-over-D}. However, \[ \int_{\varphi}\mathcal{N}^{\cdot}\otimes_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}^{L}F_{Y}^{!}\mathcal{R}(\mathcal{D}_{Y}^{(0)}))=F_{Y}^{!}\int_{\varphi}\mathcal{N}^{\cdot} \] whence the result. \end{proof} To exploit this result, we recall that the formalism of de Rham cohomology applies to $\mathcal{R}(\mathcal{D}_{X}^{(0)})$: \begin{prop} Let $\varphi:X\to Y$ be smooth of relative dimension $d$. Then the induced connection $\nabla:\mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{1}(1)$ is a morphism of filtered right $\mathcal{D}_{X}^{(0)}$-modules (with respect to the symbol filtration; the symbol $(1)$ denotes a shift in the filtration). The associated de Rham complex \[ \mathcal{D}_{X}^{(0)}\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{1}(1)\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{2}(2)\to\dots\to\mathcal{D}_{X}^{(0)}\otimes\Omega_{X/Y}^{d}(d) \] is exact except at the right most term, where the cokernel is $\mathcal{D}_{Y\leftarrow X}^{(0)}(d)$ (as a filtered module). After applying the left-right swap and a shift in the filtration, we obtain the Spencer complex \[ \mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}^{d}(-d)\to\mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}^{d-1}(-d+1)\to\dots\to\mathcal{D}_{X}^{(0)}\otimes\mathcal{T}_{X/Y}(-1)\to\mathcal{D}_{X}^{(0)} \] of left filtered $\mathcal{D}_{X}^{(0)}$-modules, which is exact except at the right most term, and the cokernel is $\mathcal{D}_{X\to Y}^{(0)}$ (as a filtered module). Applying the Rees functor yields a complex \[ \mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}^{d}(-d)\to\mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}^{d-1}(-d+1)\to\dots\to\mathcal{R}(\mathcal{D}_{X}^{(0)})\otimes\mathcal{T}_{X/Y}(-1)\to\mathcal{R}(\mathcal{D}_{X}^{(0)}) \] which is exact except at the right most term, and the cokernel is $\mathcal{R}(\mathcal{D}_{X\to Y}^{(0)})$. \end{prop} The proof of this is identical to that of the corresponding result in characteristic zero (\cite{key-4}, proposition 4.2); one notes that the associated graded is a Koszul resolution. Applying this resolution in the definition of the filtered push-forward, one deduces \begin{cor} Let $\varphi:X\to Y$ be smooth of relative dimension $d$. Let $\mathcal{M}$ be a filtered $\mathcal{D}_{X}^{(0)}$-module (with respect to the symbol filtration). Then there is an isomorphism \[ \int_{\varphi}\mathcal{M}[-d]\tilde{=}R\varphi_{*}(\mathcal{M}(-d)\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{1}(1-d)\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{2}(2)\xrightarrow{\nabla}\dots\xrightarrow{\nabla}\mathcal{M}\otimes\Omega_{X/Y}^{d}) \] in the filtered derived category of $\mathcal{O}_{Y}$-modules. \end{cor} In fact, with a little more work, one can show that, for any $i$, the $\mathcal{D}_{Y}^{(0)}$-module structure on the sheaf ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}\mathcal{M})}$ is given by the Gauss-Manin connection (c.f., e.g. \cite{key-51}, proposition 1.4). Thus the push-forward for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules is exactly the ``Frobenius pullback of Gauss-Manin.'' As another corollary, we have \begin{cor} \label{cor:sm-adunction-for-filtered-D}Let $\varphi:X\to Y$ be smooth of relative dimension $d$. 1) There is an isomorphism $R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(0})}(\mathcal{R}(\mathcal{D}_{X\to Y}^{(0)}),\mathcal{R}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{R}(\mathcal{D}_{Y\leftarrow X}^{(0)})(d)[-d]$ as $(\varphi^{-1}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})),\mathcal{R}(\mathcal{D}_{X}^{(0)}))$ bimodules. 2) There is an isomorphism of functors \[ R\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{R}(\mathcal{D}_{X}^{(0)})}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{Y}^{(0)})}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d)) \] for any $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$ and any $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. The analogous isomorphism holds for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules. \end{cor} \begin{proof} Part $1)$ follows directly from the previous proposition; compare \cite{key-4}, propositions 4.2 and 4.19. Then $2)$ follows from $1)$, as in \cite{key-4}, Theorem 4.40 (we'll give the argument below in a slightly different context in \corref{smooth-adjunction}) Finally, the statement for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules follows from the Frobenius descent (\lemref{Hodge-Filtered-Pull} and \thmref{Hodge-Filtered-Push}). \end{proof} Next we are going to give the analogue of these results for the push-forward of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules; and compare with the constructions of \cite{key-11}, section 3.4. We start with the analogues of the de Rham resolution and the adjunction for smooth morphisms. Although $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ does possess a canonical flat connection, the resulting (relative) de Rham complex is not a resolution of a transfer bimodule. Instead, we consider the action of the center \[ \mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{O}_{T^{*}X^{(1)}}[v] \] The action map $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{T}_{X^{(1)}}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ yields (by dualizing) a map \[ \Theta:\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\Omega_{X^{(1)}}^{1}(1) \] which makes $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ into a Higgs sheaf over $X^{(1)}$. In particular we have $\Theta\circ\Theta=0$ and so we can form the complex $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\Omega_{X^{(1)}}^{i}(i)$ with the differential induced from $\Theta$. In addition, we can form the analogue of the Spencer complex, whose terms are $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X^{(1)}}}\mathcal{T}_{X^{(1)}}^{i}(-i)$. Now let $\varphi:X\to Y$ be a smooth morphism of relative dimension $d$. Let $X_{Y}^{(1)}\to Y$ be the base change of this morphism over the absolute Frobenius on $Y$. Then we can perform the above constructions for $\Omega_{X_{Y}^{(1)}/Y}^{i}$ instead of $\Omega_{X^{(1)}}^{i}$. We have \begin{lem} \label{lem:Koszul-Res-For-R-bar} The complex $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X_{Y}^{(1)}/Y}^{i}(-i)$ is exact except at the right-most term. The image of the map \[ \overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X_{Y}^{(1)}/Y}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}) \] is the central ideal $\mathcal{J}$ generated by $\mathcal{T}_{X_{Y}^{(1)}/Y}\subset\mathcal{Z}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$. The cokernel of the map, $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$, carries the structure of a right $\mathcal{D}_{X/Y}^{(0)}$-module, of $p$-curvature zero; this action commutes with the natural left $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-module structure on $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$. The cokernel of the associated map $(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J})\otimes_{\mathcal{O}_{X/Y}}\mathcal{T}_{X/Y}\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$ is isomorphic to $\mathcal{\overline{R}}_{X\to Y}^{(0)}$. The complex $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{i}(i)$ is exact except at the right-most term. The cokernel of the map \[ \overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{d-1}(d-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\Omega_{X_{Y}^{(1)}/Y}^{d}(d) \] denoted $\mathcal{K}_{X/Y}$, carries the structure of a left $\mathcal{D}_{X/Y}^{(0)}$-module, of $p$-curvature zero; this action commutes with the natural right $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-module structure on $\mathcal{K}_{X/Y}$. The kernel of the associated connection on $\mathcal{K}_{X/Y}$ is isomorphic to $\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)$. \end{lem} \begin{proof} Choose local coordinates on $X$ for which $\text{Der}(\mathcal{O}_{X})$ is the free module on $\{\partial_{1},\dots\partial_{n}\}$ and $\text{Der}_{\mathcal{O}_{Y}}(\mathcal{O}_{X})=\{\partial_{n-d+1},\dots\partial_{n}\}$. Then the complex under consideration is simply the Koszul complex for the elements $\{\partial_{n-d+1}^{p},\dots,\partial_{n}^{p}\}$, which proves the exactness statements. Furthermore, as the elements $\{\partial_{n-d+1}^{p},\dots,\partial_{n}^{p}\}$ are central in $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, we see that $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$ has the structure of a left and right $\mathcal{D}_{X}^{(0)}$-module (we are here using the fact that $\mathcal{D}_{X}^{(0)}$ is the degree $0$ part of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$). Now, we have \[ \mathcal{\overline{R}}_{X\to Y}^{(0)}=\text{coker}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X_{Y}^{(1)}}}\mathcal{T}_{X/Y}(-1)\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})) \] \[ =\text{coker}((\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J})\otimes_{\mathcal{O}_{X/Y}}\mathcal{T}_{X/Y}\to\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}) \] since $X\to Y$ is smooth. The second statement follows similarly. \end{proof} Now we can give the analogue of \corref{sm-adunction-for-filtered-D}. It reads: \begin{cor} \label{cor:sm-adjunction-for-R-bar}Let $\varphi:X\to Y$ be smooth of relative dimension $d$. 1) There is an isomorphism $R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)[-d]$ as $(\varphi^{-1}(\mathcal{\overline{R}}(\mathcal{D}_{Y}^{(0)})),\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$ bimodules. 2) There is an isomorphism of functors \[ R\varphi_{*}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{\dagger,(0)}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{N}^{\cdot},\int_{\varphi,0}\mathcal{M}^{\cdot}(d)) \] for any $\mathcal{N}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{Y}^{(0)})))$ and any $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. \end{cor} \begin{proof} As in the proof of \corref{smooth-adjunction} below, $2)$ follows formally from $1)$. To see $1)$, we note that for any $\mathcal{N}\in\mathcal{G}(\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))$ the complex $R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J},\mathcal{N})$ can be considered a complex of left $\mathcal{D}_{X/Y}^{(0)}$-modules with $p$-curvature $0$ (as $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J}$ is a right $\mathcal{D}_{X/Y}^{(0)}$-module of $p$-curvature zero, and this action commutes with the left $\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})$-action). As Cartier descent for $\mathcal{D}_{X/Y}^{(0)}$-modules of $p$-curvature $0$ is an exact functor, applying the previous lemma we obtain \[ R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})/\mathcal{J},\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))^{\nabla} \] \[ (\mathcal{K}_{X/Y}[-d])^{\nabla}=\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)[-d] \] as desired. \end{proof} Now we'll give the relation of our pushforward to the constructions of \cite{key-11}, section 3.4. We recall that to any morphism $\varphi:X\to Y$ we may attach the diagram \[ T^{*}X\xleftarrow{d\varphi}X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y \] and we use the same letters to denote the products of these morphisms with $\mathbb{A}^{1}$; We have the following analogue of \cite{key-52}, proposition 3.7 (c.f. also \cite{key-11}, theorem 3.11) \begin{lem} \label{lem:Bez-Brav}There is an equivalence of graded Azumaya algebras $(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\sim(\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$. \end{lem} \begin{proof} Consider the (graded) Azumaya algebra $\mathcal{A}:=(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]}(\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})^{\text{opp}}$ on $(X\times_{Y}T^{*}Y)^{(1)}\times\mathbb{A}^{1}$. It is enough to find a (graded) splitting module for $\mathcal{A}$; i.e., a graded $\mathcal{A}$-module which is locally free of rank $p^{\text{dim}(X)+\text{dim}(Y)}$ over $\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]$. The graded $(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\varphi^{-1}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})))$ bimodule $\overline{\mathcal{R}}_{X\to Y}:=\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$ inherits the structure of an $\mathcal{A}$-module; we claim it is locally free over $\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}[v]$ of the correct rank. This can be checked after inverting $v$ and setting $v=0$; upon inverting $v$ it becomes (via the isomorphisms $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})[v^{-1}]\tilde{=}\mathcal{D}_{X}^{(0)}[v,v^{-1}]$ and $\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})[v^{-1}]\tilde{=}\mathcal{D}_{Y}^{(0)}[v,v^{-1}]$) a direct consequence of \cite{key-52}, proposition 3.7. After setting $v=0$ we have $\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v=\text{gr}(\mathcal{D}_{Y}^{(0)})=\overline{\mathcal{D}}_{Y}^{(0)}\otimes_{\mathcal{O}_{Y^{(1)}}}\mathcal{O}_{T^{*}Y^{(1)}}$; and similarly for $X$. Therefore \[ \varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v\tilde{=}\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)}\otimes_{\mathcal{O}_{Y^{(1)}}}\mathcal{O}_{T^{*}Y^{(1)}}) \] \[ \tilde{=}\mathcal{O}_{X}\otimes_{\varphi^{-1}(\mathcal{O}_{Y})}\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)})\otimes_{\varphi^{-1}(\mathcal{O}_{Y^{(1)}})}\varphi^{-1}(\mathcal{O}_{T^{*}Y^{(1)}}) \] But $\varphi^{-1}(\overline{\mathcal{D}}_{Y}^{(0)})$ is locally free of rank $p^{\text{dim}(Y)}$ over $\varphi^{-1}(\mathcal{O}_{Y})$, and $\mathcal{O}_{X}$ is locally free of rank $p^{\text{dim}(X)}$ over $\mathcal{O}_{X^{(1)}}$; so $\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})/v$ is locally free of rank $p^{\text{dim}(X)+\text{dim}(Y)}$ over $\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}$ as claimed. \end{proof} Next, we have the following straightforward: \begin{lem} Let $\varphi:X\to Y$ be smooth. Then $d\varphi$ is a closed immersion, and we may regard the algebra $(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ as a (graded) central quotient of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. The obvious functor $(d\varphi)_{*}:D(\mathcal{G}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ admits a right adjoint $(d\varphi)^{!}$ defined by $\mathcal{M}^{\cdot}\to R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot})$. \end{lem} Therefore we obtain \begin{cor} \label{cor:Filtered-Bez-Brav}Let $C:D(\mathcal{G}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to D(\mathcal{G}((\pi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})))$ denote the equivalence of categories resulting from \lemref{Bez-Brav}. Then, when $\varphi:X\to Y$ is smooth of relative dimension $d$, there is an isomorphism of functors \[ \int_{\varphi,0}\tilde{\to}R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}[-d]:D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))\to D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))) \] Therefore, the functor ${\displaystyle \int_{\varphi,0}}[d]$ agrees, under the application of the Rees functor, with the pushforward of conjugate-filtered derived categories constructed in \cite{key-11}, section 3.4. \end{cor} \begin{proof} (in the spirit of \cite{key-11}, proposition 3.12) We have, for any $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$, \[ C\circ(d\varphi)^{!}(\mathcal{M}^{\cdot})=C\circ R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot}) \] \[ \tilde{=}\underline{\mathcal{H}om}_{(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}((d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot})) \] Since $\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})$ is locally projective over $(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$, this is canonically isomorphic to \[ R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(d\varphi^{(1)})^{*}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}),\mathcal{M}^{\cdot})) \] \[ =R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot})) \] so that \[ R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}(\mathcal{M}^{\cdot})\tilde{\to}R\pi_{*}^{(1)}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot})) \] \[ \tilde{=}R\varphi_{*}R\underline{\mathcal{H}om}_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\varphi^{*}\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}),\mathcal{M}^{\cdot})) \] But the right-hand functor is canonically isomorphic to ${\displaystyle \int_{\varphi,0}}$ by smooth adjunction (\corref{sm-adjunction-for-R-bar}). \end{proof} From this description it follows directly (c.f. \cite{key-11}, lemma 3.18) that (up to a renumbering) the spectral sequence associated to the filtration on ${\displaystyle \int_{\varphi}\mathcal{O}_{X}}$ agrees with the usual conjugate spectral sequence; i.e., the ``second spectral sequence'' for $R\varphi_{dR,*}(\mathcal{O}_{X})$ as discussed in \cite{key-12}. \subsection{Adjunction for a smooth morphism, base change, and the projection formula} In this section, we prove adjunction for a smooth morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ and the projection formula for an arbitrary morphism; as consequences we obtain the smooth base change and the and the Kunneth formula, in fairly general contexts. To start off, let us recall: \begin{prop} For a smooth morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ there is an isomorphism of sheaves $\mathcal{R}\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[-d_{X/Y}]$. \end{prop} This is proved identically to the analogous fact for $\mathcal{D}_{X}^{(0)}$ and $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules, as discussed above in \corref{sm-adunction-for-filtered-D}. \begin{prop} For a smooth morphism $\varphi:\mathfrak{X}\to\mathfrak{Y}$ of relative dimension $d$, there is an isomorphism $\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}(d)[-d]$ \end{prop} \begin{proof} We have \[ \mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)} \] \[ \tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1) \] \[ \tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1),\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}\otimes_{D(W(k))}^{L}(W(k)[f,v]/(v-1)) \] \[ \tilde{=}R\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}) \] and \[ \mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{D}_{X}^{(0,1)}\tilde{=}\mathcal{R}\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\otimes_{W(k)}^{L}k \] \[ \tilde{=}R\mathcal{H}om_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}) \] By the same token, we have \[ R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})\tilde{=}R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{R}_{X\to Y}^{(1)},\mathcal{R}(\mathcal{D}_{X}^{(1)})) \] and the analogous statement for $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. Applying the smooth adjunction for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules (\corref{sm-adunction-for-filtered-D}, part $2)$) to the case where $\mathcal{N}^{\cdot}=\mathcal{R}(\mathcal{D}_{Y}^{(1)})$ and $\mathcal{M}^{\cdot}=\mathcal{R}(\mathcal{D}_{X}^{(1)})$, we have an isomorphism \[ R\underline{\mathcal{H}om}{}_{\mathcal{R}_{X}}(\mathcal{R}_{X\to Y}^{(1)},\mathcal{R}(\mathcal{D}_{X}^{(1)}))\tilde{=}\mathcal{R}_{Y\leftarrow X}^{(1)}(d)[-d] \] and by \corref{sm-adjunction-for-R-bar} we have \[ R\underline{\mathcal{H}om}{}_{\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0})}(\mathcal{\overline{R}}(\mathcal{D}_{X\to Y}^{(0)}),\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\tilde{=}\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)[-d] \] Furthermore, using the relative de Rham resolution for $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules (or, equivalently, the previous proposition) we have $\mathcal{R}\mathcal{H}om_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)})\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[-d_{X/Y}]$. On the other hand, we have the short exact sequence \[ \mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})\to\mathcal{D}_{X}^{(0,1)}\to\mathcal{R}(\mathcal{D}_{X}^{(1)})(-1) \] which by \propref{Sandwich!} yields the distinguished triangle \[ R\underline{\mathcal{H}om}{}_{\bar{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{\overline{R}}_{X\to Y},\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)}))\to R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}) \] \[ \to R\underline{\mathcal{H}om}{}_{\mathcal{R}(\mathcal{D}_{X}^{(1)})}(\mathcal{R}_{X\to Y},\mathcal{R}(\mathcal{D}_{X}^{(1)}))(-1) \] which implies that $R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})$ is concentrated in a single homological degree (namely $d$). So, since $R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$ is cohomologically complete, we see that $\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ is $p$-torsion-free and concentrated in degree $0$. We also see, by \propref{coh-to-coh}, that this module is coherent over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ (each of $\mathcal{R}_{Y\leftarrow X}^{(1)}$ and $\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}$ are coherent, since $X\to Y$ is smooth). Further, since $R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)})\otimes_{\mathcal{D}_{X}^{(0,1)}}^{L}\mathcal{\overline{R}}(\mathcal{D}_{X}^{(0)})$ are concentrated in degree $d$ as well, we see that $\text{im}(f)=\text{ker}(v)$ and $\text{im}(v)=\text{ker}(f)$ on $\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}))$ (by \lemref{Basic-Facts-on-Rigid}). Furthermore, the distinguished triangle above now yields the short exact sequence \[ \mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)\to\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\mathcal{D}_{X}^{(0,1)}}(\mathcal{D}_{X\to Y}^{(0,1)},\mathcal{D}_{X}^{(0,1)}))\to\mathcal{R}_{Y\leftarrow X}^{(1)}(d-1) \] and since $\mathcal{R}_{Y\leftarrow X}^{(1)}$ is $f$-torsion-free, we see that $\text{im}(v)=\text{ker}(f)=\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)$ and so $\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ satisfies the conditions of \propref{Baby-Mazur}. So we may conclude that the module $\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ is standard. Furthermore, we see that the grading on $\mathcal{\overline{R}}_{Y\leftarrow X}^{(0)}(d)$ is zero in degrees $<-d$ and is nontrivial in degree $-d$ and above. Therefore, the index (as defined directly below \defref{Standard!}) is $d$. Since we identified $\mathcal{H}^{d}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})^{-\infty})$ with $\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}$ we see that \[ \mathcal{H}^{d_{X/Y}}(R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})^{i-d})=\{m\in\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}[p^{-1}]|p^{i}m\in\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0)}\} \] which is exactly the definition of $\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}(d)$. \end{proof} From this one deduces \begin{cor} \label{cor:smooth-adjunction}Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be smooth of relative dimension $d$; let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$. Then there is an isomorphism of functors \[ R\varphi_{*}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d)) \] In particular, if $\varphi$ is also proper, then since both $\varphi^{\dagger}$ and ${\displaystyle \int_{\varphi}}$ preserve $D_{coh}^{b}$, we obtain that these functors form an adjoint pair on $D_{coh}^{b}$. Further, the analogous isomorphism for $\varphi:X\to Y$ holds, and in this situation the functors are adjoint on $D_{qcoh}^{b}$ in this setting (even if $\varphi$ is not proper). \end{cor} \begin{proof} (following \cite{key-4}, Theorem 4.40). We have \[ R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}\widehat{\otimes}_{\varphi^{-1}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}\varphi^{-1}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot})[d] \] \[ \tilde{\to}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot}))[d] \] To prove the last isomorphism, one may reduce mod $p$, and then apply \lemref{basic-hom-tensor} (part $1$), noting that $\mathcal{D}_{X\to Y}^{(0,1)}$ is faithfully flat over $\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$. Further, we have \[ R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot})\tilde{\leftarrow}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}\tilde{=}\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)[-d] \] where the first isomorphism again follows by reduction mod $p$ and then applying the fact that $\mathcal{D}_{X\to Y}^{(0,1)}$ is (locally) isomorphic to a bounded complex of projective $\mathcal{D}_{X}^{(0,1)}$-modules (by \propref{Quasi-rigid=00003Dfinite-homological}) and the second isomorphism is the previous proposition. Applying this to the previous isomorphism we obtain \[ R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)},\mathcal{M}^{\cdot}))[d]\tilde{=}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)) \] Then applying $R\varphi_{*}$ we obtain \[ R\varphi_{*}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\varphi^{\dagger}\mathcal{N}^{\cdot},\mathcal{M}^{\cdot}) \] \[ \tilde{=}R\varphi_{*}R\underline{\mathcal{H}om}{}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\varphi^{-1}\mathcal{N}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)) \] \[ \tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}(d)))\tilde{\to}R\underline{\mathcal{H}om}{}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\mathcal{N}^{\cdot},\int_{\varphi}\mathcal{M}^{\cdot}(d)) \] where the final isomorphism, is the adjunction between $\varphi^{-1}$ and $R\varphi_{*}$. One applies analogous reasoning for $\varphi:X\to Y$. \end{proof} Now we prove the projection formula, and then give the the smooth base change and Kunneth formulas in this context. We start with \begin{thm} \label{thm:Projection-Formula}(Projection Formula) Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a morphism. Let $\mathcal{M}^{\cdot}\in D_{cc}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{cc}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$, be such that $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. Then we have \[ \int_{\varphi}(L\varphi^{*}(\mathcal{N}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{M}^{\cdot})\tilde{\to}\mathcal{N}^{\cdot}\otimes_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\int_{\varphi}\mathcal{M}^{\cdot} \] \end{thm} The proof works essentially the same way as the complex analytic one (c.f. \cite{key-50}, theorem 2.3.19). In particular, we use \lemref{proj-over-D}, as well as the tensor product juggling lemma \lemref{Juggle} \begin{proof} By the left-right interchange it suffices to prove \[ \int_{\varphi}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}\mathcal{N}^{\cdot})\tilde{=}\int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{N}^{\cdot} \] where $\mathcal{M}_{r}^{\cdot}=\omega_{\mathfrak{X}}\otimes_{\mathcal{O}_{\mathfrak{X}}}\mathcal{M}^{\cdot}$. We have \[ \int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{N}^{\cdot}\tilde{=}\int_{\varphi}(\mathcal{M}_{r}^{\cdot})\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}}^{L}(\mathcal{N}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}) \] \[ \tilde{=}R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{Y}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)}))\tilde{=}R\varphi_{*}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot})\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\varphi^{*}(\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0,1)})) \] \[ \tilde{=}R\varphi_{*}((\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}))\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\tilde{=}R\varphi_{*}((\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}(\mathcal{N}^{\cdot}))\widehat{\otimes}_{\mathcal{\widehat{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{\widehat{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}) \] \[ =\int_{\varphi}(\mathcal{M}_{r}^{\cdot}\widehat{\otimes}_{D(\mathcal{O}_{\mathfrak{X}})}^{L}L\varphi^{*}\mathcal{N}^{\cdot}) \] as claimed; note that the second isomorphism is \lemref{proj-over-D} which uses the assumption on $\mathcal{M}^{\cdot}$ and $\mathcal{N}^{\cdot}$. \end{proof} Now we turn to the smooth base change. Consider the fibre square of smooth formal schemes $$ \begin{CD} \mathfrak{X}_{\mathfrak{Z}} @>\tilde{\psi} >> \mathfrak{X} \\ @VV\tilde{\varphi}V @VV{\varphi}V \\ \mathfrak{Z} @>\psi >> \mathfrak{Y} \end{CD} $$where the bottom row $\psi:\mathfrak{Z}\to\mathfrak{Y}$ is smooth of relative dimension $d$. We have also the analogous square for smooth varieties over $k$. \begin{thm} \label{thm:Smooth-base-change}Suppose that $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. There is an isomorphism \[ \psi^{\dagger}\int_{\varphi}\mathcal{M}^{\cdot}\tilde{\to}\int_{\tilde{\varphi}}\tilde{\psi}{}^{\dagger}\mathcal{M}^{\cdot} \] inside $D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Z}}^{(0,1)}))$. The analogous statement holds for smooth varieties over $k$. \end{thm} \begin{proof} By the adjunction for ${\displaystyle (\tilde{\psi}^{\dagger},\int_{\tilde{\psi}}(d))}$ there is a morphism of functors \[ \int_{\varphi}\to\int_{\varphi}\circ\int_{\tilde{\psi}}(\tilde{\psi})^{\dagger}(d)\tilde{=}\int_{\psi}\circ\int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger}(d) \] where the last isomorphism follows from the composition of push-forwards (\lemref{Composition-of-pushforwards}). Now, applying the adjunction for ${\displaystyle (\psi^{\dagger},\int_{\psi}(d))}$, we obtain a morphism \[ \psi^{\dagger}\int_{\varphi}\to\int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger} \] After applying $\otimes_{W(k)}^{L}k$ we obtain the analogous map over $k$. So it suffices to show that the map is an isomorphism for varieties over $k$. Furthermore, working locally on $Z$, we reduce to the case where the map $\psi:Z\to Y$ factors as an etale morphism $Z\to Z'$ followed by a projection $Z'\tilde{=}Y\times\mathbb{A}^{d}\to Y$. In the case of an etale morphism, the functor ${\displaystyle \int_{\varphi}}$ agrees with $R\varphi_{*}$, so the result follows from the usual flat base change for quasicoherent sheaves. In the case of the projection, we have \[ \int_{\tilde{\varphi}}(\tilde{\psi})^{\dagger}\mathcal{M}^{\cdot}\tilde{=}\int_{\text{id}\times\varphi}D(\mathcal{O}_{\mathbb{A}_{k}^{d}})\boxtimes\mathcal{M}^{\cdot}[d]\tilde{=}D(\mathcal{O}_{\mathbb{A}_{k}^{d}})\boxtimes\int_{\varphi}\mathcal{M}^{\cdot}[d]\tilde{=}\psi^{\dagger}\mathcal{M}^{\cdot} \] where the second isomorphism follows directly from the definition of the pushforward; this implies the result in this case. \end{proof} From this we deduce the Kunneth formula: \begin{cor} Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}))$, so that $\mathcal{M}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\otimes_{W(k)}^{L}k\in D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. Then there is an isomorphism \[ \mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot}\boxtimes\mathcal{N}^{\cdot})\tilde{=}\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{M}^{\cdot})\widehat{\otimes}_{W(k)[f,v]}^{L}\mathbb{H}_{\mathcal{G}}^{\cdot}(\mathcal{N}^{\cdot}) \] (where $\mathbb{H}_{\mathcal{G}}^{\cdot}$ is defined in \defref{Push!})The analogous statement holds for complexes in $D_{qcoh}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ and $D_{qcoh}(\mathcal{G}(\mathcal{D}_{Y}^{(0,1)}))$. \end{cor} This is a formal consequence of the projection formula and the smooth base change (compare, e.g. \cite{key-53}, corollary 2.3.30). \section{Operations on Gauges: Duality} In this section we study the duality functor on $D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ (and on $D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$. Although neither $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ nor $\mathcal{D}_{X}^{(0,1)}$ have finite homological dimension, we shall show (using \propref{Sandwich!}) that there is a well-behaved duality functor $\mathbb{D}$ which takes bounded complexes of coherent modules to bounded complexes of coherent modules. Further, under suitable conditions this functor commutes with push-forward, in the following sense: \begin{thm} Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be either a smooth proper morphism or a projective morphism. Then there is an isomorphism of functors \[ \int_{\varphi}\mathbb{D}_{\mathfrak{X}}\tilde{\to}\mathbb{D}_{\mathfrak{Y}}\int_{\varphi} \] The analogous statement holds for either a smooth proper or a projective morphism $\varphi:X\to Y$. In particular; when $\varphi$ is smooth proper the functors $(\int_{\varphi},\varphi^{\dagger})$ form an adjoint pair on $D_{coh}^{b}$. \end{thm} The proof, which will essentially occupy this section of the paper, is somewhat unsatisfactory. The key point is to construct a trace morphism \[ \text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}})[d_{X}]\to D(\mathcal{O}_{\mathfrak{Y}})[d_{Y}] \] When $\varphi$ is smooth proper this is done by first constructing the map in $\mathcal{D}^{(0)}$ and $\mathcal{D}^{(1)}$ modules (using the Hodge to de Rham spectral sequence), and then deducing its existence for $\mathcal{D}^{(0,1)}$-modules. When $\varphi$ is a closed immersion the construction of the trace follows from a direct consideration of the structure of ${\displaystyle \int_{\varphi}}$ (the transfer bimodule is easy to describe in this case). For a projective $\varphi$ one defines the trace by breaking up the map into an immersion followed by a projection. Presumably there is a way to construct the trace for all proper morphisms at once, but I have been unable to find it. To kick things off, we need to define the duality functor and show that it has finite homological dimension. \begin{defn} Let $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. We define $\mathbb{D}_{\mathfrak{X}}(\mathcal{M}^{\cdot}):=\omega_{\mathfrak{X}}^{-1}\otimes_{\mathcal{O}_{\mathfrak{X}}}R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})[d_{X}]\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ (where we have used the natural right $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$-module structure on $R\underline{\mathcal{H}om}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})$). The same formula defines $\mathbb{D}_{X}$ for a smooth variety $X$ over $k$; and in the analogous way we define the duality functors for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. \end{defn} This is really a duality on the category of coherent modules: \begin{prop} Suppose $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$ then $\mathbb{D}_{\mathfrak{X}}(\mathcal{M}^{\cdot})\in D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. Further, the natural transformation $\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{X}}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}$ is an isomorphism. The same result holds for a smooth variety $X$ over $k$. \end{prop} \begin{proof} By reduction mod $p$ it suffices to prove the result for $X$. Using \propref{Sandwich!}, and the fact that $\text{ker}(f:\mathcal{D}_{X}^{(0,1)}\to\mathcal{D}_{X}^{(0,1)})\tilde{=}\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})(1)$ and $\text{ker}(v:\mathcal{D}_{X}^{(0,1)}\to\mathcal{D}_{X}^{(0,1)})\tilde{=}\mathcal{R}(\mathcal{D}_{X}^{(1)})(-1)$ one reduces to proving the analogous result for $\mathcal{R}(\mathcal{D}_{X}^{(1)})$ and $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. But these algebras have finite homological dimension, as noted above, and the results follow at once. \end{proof} \subsection{Duality for a smooth proper morphism} Now we turn to defining the trace morphism, and proving the duality, for a smooth proper map $\mathfrak{X}\to\mathfrak{Y}$ of relative dimension $d$. In this case the usual Grothendieck duality theory gives us a canonical morphism \[ \text{tr}:R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}})\to\mathcal{O}_{\mathfrak{Y}} \] Now consider $\mathcal{O}_{\mathfrak{X}}$ as a module over $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$. As the pushforward ${\displaystyle \int_{\varphi,0}\mathcal{O}_{\mathfrak{X}}}$ can be computed by the relative de Rham complex, looking at the Hodge-to-de Rham spectral sequence in degree $2d$ yields an isomorphism of $\mathcal{O}_{\mathfrak{Y}}$-modules \[ \mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}})\tilde{=}R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}}) \] so; composing with the trace morphism above, we obtain a map \[ \text{tr}:\mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}})\to\mathcal{O}_{\mathfrak{Y}} \] of $\mathcal{\widehat{D}}_{\mathfrak{Y}}^{(0)}$-modules. Now consider $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$, the reduction mod $p^{n}$ of $\varphi$ for each $n\geq0$. Repeating the argument, we can construct \[ \text{tr}:\mathcal{H}^{d}(\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}_{n}})\to\mathcal{O}_{\mathfrak{Y}_{n}} \] and, in fact, the inverse limit of these maps is the trace constructed above. In this setting, the de Rham complex $\Omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}^{\cdot}$ has the structure of a complex of coherent sheaves over the scheme $W_{n}(\mathcal{O}_{X^{(n)}})$ (here we are identifying the underlying topological spaces of $\mathfrak{X}_{n}$ and $W_{n}(\mathcal{O}_{X^{(n)}})$). Thus we may also consider the second spectral sequence for the pushforward of this complex, and we obtain an isomorphism \[ R^{d}\varphi_{*}(\text{coker}(d:\Omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}^{d-1}\to\omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}))\tilde{\to}R^{d}\varphi_{*}(\omega_{\mathfrak{X}/\mathfrak{Y}}) \] or, equivalently, \[ R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0)}}\mathcal{O}_{\mathfrak{X}_{n}})\tilde{\to}R^{d}\varphi_{*}(\omega_{\mathfrak{X}_{n}/\mathfrak{Y}_{n}}) \] Now we consider the the pushforward of $\mathcal{O}_{\mathfrak{X}_{n}}$, in the category of $\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}$-modules. By the commutativity of Frobenius descent with push-forward (\cite{key-2}, theoreme 3.4.4), we have \[ \int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}}\tilde{=}\int_{\varphi,1}F^{*}\mathcal{O}_{\mathfrak{X}_{n}}\tilde{\to}F^{*}\int_{\varphi,0}\mathcal{O}_{\mathfrak{X}_{n}} \] Therefore we obtain a trace map \[ \text{tr}:\mathcal{H}^{d}(\int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}})\to\mathcal{O}_{\mathfrak{Y}_{n}} \] in the category of $\mathcal{D}_{\mathfrak{Y}_{n}}^{(1)}$-modules; and, using the second spectral sequence for the pushforward as above, we have \[ R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}}\mathcal{O}_{\mathfrak{X}_{n}})\tilde{\to}\mathcal{H}^{d}(\int_{\varphi,1}\mathcal{O}_{\mathfrak{X}_{n}}) \] Using these maps, we construct a trace for $\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}$-modules: \begin{lem} There is a canonical morphism \[ \text{tr}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))\to D(\mathcal{O}_{\mathfrak{Y}_{n}}) \] which has the property that the map $\text{tr}^{\infty}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))^{\infty}\to D(\mathcal{O}_{\mathfrak{Y}_{n}}){}^{\infty}$ agrees with the trace map for $\mathcal{D}_{\mathfrak{X}_{n}}^{(1)}$-modules constructed above; and the map $\text{tr}^{-\infty}:R^{d}\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes{}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}D(\mathcal{O}_{\mathfrak{X}_{n}}))^{-\infty}\to D(\mathcal{O}_{\mathfrak{Y}_{n}})^{-\infty}$ agrees with the trace map for $\mathcal{D}_{\mathfrak{X}_{n}}^{(0)}$-modules constructed above. We have the analogous statement for a proper morphism $\varphi:X\to Y$, as well as in the categories of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules and $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules. This map yields a trace map in the derived category: \[ \text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}_{n}})[d]\to D(\mathcal{O}_{\mathfrak{Y}_{n}}) \] Upon taking the inverse limit over $n$, we obtain a map \[ \text{tr}:\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}})[d]\to D(\mathcal{O}_{\mathfrak{Y}}) \] \end{lem} \begin{proof} We begin with the case $n=1$; i.e., $\mathfrak{X}_{n}=X$ and $\mathfrak{Y}_{n}=Y$. We claim that the $\varphi^{-1}(\mathcal{D}_{Y}^{(0,1)})$-gauge $\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X})$ satisfies the property that $v$ is an isomorphism in degrees $0$ and below and $f$ is an isomorphism in degrees $1$ and above. This can be checked in local coordinates, where we have the isomorphism \[ \mathcal{D}_{Y\leftarrow X}^{(0,1)}=\mathcal{J}\backslash\mathcal{D}_{X}^{(0,1)} \] where $\mathcal{J}$ is the right ideal generated by $\{\partial_{n-d+1},\dots,\partial_{n},\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]}\}$. In degrees below $0$, the elements $\{\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]}\}$ act trivially; so that \[ (\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}=\mathcal{O}_{X}/(\partial_{n-d+1},\dots,\partial_{n}) \] for all $i\leq0$. On the other hand we have \[ (\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}=\mathcal{O}_{X}/(\partial_{n-d+1},\dots,\partial_{n},\partial_{n-d+1}^{[p]},\dots,\partial_{n}^{[p]}) \] for $i>0$; and the claim about $f$ and $v$ follows immediately. As the functor $R^{d}\varphi_{*}$ commutes with direct sums, we see that the gauge \[ R^{d}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X})) \] has the same property: $v$ is an isomorphism in degrees $0$ and below and $f$ is an isomorphism in degrees $1$ and above. Thus we may define \[ \text{tr}:R^{d}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\to\mathcal{O}_{Y} \] for any $i$ as follows: if $i\leq0$ we have $v_{-\infty}:R^{d}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\tilde{=}R^{d}\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(0)}\otimes_{\mathcal{D}_{X}^{(0)}}\mathcal{O}_{X})$ and so we define the trace as the composition $\text{tr}\circ v_{-\infty}$, where here $\text{tr}$ denotes the trace for $\mathcal{D}_{X}^{(0)}$-modules constructed above. If $i>0$ we have $f_{\infty}:R^{d}(\mathcal{D}_{Y\leftarrow X}^{(0,1)}\otimes{}_{\mathcal{D}_{X}^{(0,1)}}D(\mathcal{O}_{X}))^{i}\tilde{=}R^{d}\varphi_{*}(\mathcal{D}_{Y\leftarrow X}^{(1)}\otimes_{\mathcal{D}_{X}^{(1)}}\mathcal{O}_{X})$ and so we define the trace as the composition $\text{tr}\circ f_{\infty}$, where here $\text{tr}$ denotes the trace for $\mathcal{D}_{X}^{(1)}$-modules constructed above. In a similar way, we construct the trace map in the categories of $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$-modules and $\mathcal{R}(\mathcal{D}_{X}^{(1)})$-modules. Now we consider $\mathfrak{X}_{n}$ for $n>1$. Since the functor $\varphi_{*}$ has homological dimension $d$ (on the category of quasicoherent sheaves), we have that $(R^{d}\varphi_{*}\mathcal{F})\otimes_{W(k)}k\tilde{=}(R^{d}\varphi_{*}\mathcal{F}\otimes_{W(k)}^{L}k)$ for any $\mathcal{F}\in\text{Qcoh}(\mathfrak{X}_{n})$. So, by Nakayama's lemma and the result of the previous paragraph, we see that $f$ is onto in degrees $1$ and above while $v$ is onto in degrees $0$ and below; by the coherence of the sheaves involved we see that these maps are isomorphisms for $i<<0$ and $i>>0$. Since the target of the trace map, $D(\mathcal{O}_{\mathfrak{Y}_{n}})$, has the property that $v$ is an isomorphism in degrees $0$ and below and $f$ is an isomorphism in degrees $1$ and above, we may define the trace map in the exact same way as above. \end{proof} \begin{rem} \label{rem:trace-and-compose}If $\varphi:\mathfrak{X}\to\mathfrak{Y}$ and $\psi:\mathfrak{Y}\to\mathfrak{Z}$, then the trace map for the composition satisfies ${\displaystyle \text{tr}_{\psi\circ\varphi}=\text{tr}_{\psi}\circ\int_{\psi}\text{tr}_{\varphi}}$. This follows from the analogous result for the trace map in coherent sheaf theory. \end{rem} Now, following the usual method of algebraic $\mathcal{D}$-module theory (c.f. \cite{key-49}, theorem 2.7.2), we have \begin{prop} There is a canonical morphism \[ \int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot} \] for any $\mathcal{M}^{\cdot}\in D_{cc}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. The same holds for $\mathcal{M}^{\cdot}\in D(\mathcal{G}(\mathcal{D}_{X}^{(0,1)}))$ when we have a proper map $\varphi:X\to Y$. Further, these maps are compatible under application of $\otimes_{W(k)}^{L}k$. \end{prop} \begin{proof} We have \[ \int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}=R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)})\widehat{\otimes}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{X}] \] \[ =R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{X}] \] while \[ \mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}=R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}}^{L}\omega_{\mathfrak{Y}}^{-1}[d_{Y}] \] To construct a canonical map between these complexes, we begin by considering ${\displaystyle \int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}}$. By $\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}=L\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}$, we may apply \thmref{Projection-Formula} to obtain \[ \int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}=\int_{\varphi}L\varphi^{*}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}\tilde{\to}(\int_{\varphi}D(\mathcal{O}_{\mathfrak{X}}))\widehat{\otimes}_{\mathcal{O}_{\mathfrak{Y}}[f,v]}^{L}\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)} \] so applying the trace map yields a canonical morphism \[ \int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d]\to\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)} \] and since $d=d_{X}-d_{Y}$ we have \[ \int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d_{X}]\to\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}[d_{Y}] \] Then we have \[ R\varphi_{*}(R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})[d_{X}] \] \[ \to R\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)})}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)})[d_{X}] \] \[ \to R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),R\varphi_{*}(\widehat{\mathcal{D}}_{\mathfrak{Y}\leftarrow\mathfrak{X}}^{(0,1)}\otimes_{\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}}^{L}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}))[d_{X}] \] \[ =R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\int_{\varphi}\widehat{\mathcal{D}}_{\mathfrak{X}\to\mathfrak{Y}}^{(0,1)}[d_{X}])\to R\underline{\mathcal{H}om}_{\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0,1)}[d_{Y}]) \] where the last map is the trace. Combining with the above description yields the canonical map \[ \int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot} \] as desired; the case of a proper map $\varphi:X\to Y$ is identical. \end{proof} Now we turn to \begin{thm} \label{thm:Duality-for-smooth-proper}The canonical map $\int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot}$ is an isomorphism for $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}))$. The same is true for a proper map $\varphi:X\to Y$. \end{thm} The proof of this result will make use of several auxiliary results. First, we recall a basic computation for pushforwards of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$-modules; as in the previous section we have the diagram \[ T^{*}X\xleftarrow{d\varphi}X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y \] and the result reads \begin{lem} Let $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. There is an isomorphism \[ (\int_{\varphi}\mathcal{M}^{\cdot})\otimes_{k[f]}^{L}k\tilde{=}R\pi_{*}((d\varphi)^{!}(\mathcal{M}^{\cdot}\otimes_{k[f]}^{L}k)) \] inside $D_{coh}^{b}(T^{*}Y)$; in this formula $d\varphi^{!}$ is the extraordinary inverse image in coherent sheaf theory. \end{lem} This is a result of Laumon, (c.f. \cite{key-19}, construction 5.6.1). For a proof in the Rees algebra language, see \cite{key-70}, corollary 3.9. Next, we need the following Grothendieck duality statement for Azumaya algebras: \begin{lem} \label{lem:GD-for-Az}Let $X$ and $Y$ be smooth varieties over $k$ and let $\pi:X\to Y$ be a smooth proper morphism, of relative dimension $d$. Suppose that $\mathcal{A}_{Y}$ be an Azumaya algebra on $Y$. Set $\mathcal{A}_{X}=\pi^{*}\mathcal{A}_{Y}$, an Azumaya algebra on $X$. Then there is a trace map $R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}$ which induces, for any $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$ a functorial isomorphism \[ R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{\to}R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}) \] inside $D_{coh}^{b}(\mathcal{O}_{Y}-\text{mod})$. \end{lem} \begin{proof} Via the projection formula we have \[ R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})=R\pi_{*}(\pi^{*}\mathcal{A}_{Y}\otimes_{\mathcal{O}_{X}}\omega_{X})\tilde{\to}\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}^{L}R\pi_{*}(\omega_{X}) \] so the usual trace $\text{tr}:R^{d}\pi_{*}(\omega_{X})\to\omega_{Y}$ induces a trace $\text{tr}:R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}$. Since $\pi$ has homological dimension $d$, we have $R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to R^{d}\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})$ so that there is a map \[ R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y} \] Thus we obtain \[ R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\to R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},R\pi_{*}(\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]) \] \[ \to R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}) \] for any $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$. To prove that this map is an isomorphism, we can can work in the etale (or flat) topology on $Y$ and so assume that $\mathcal{A}_{Y}$ is split; i.e., $\mathcal{A}_{Y}=\mathcal{E}nd(\mathcal{E}_{Y})$ for some vector bundle $\mathcal{E}_{Y}$. This implies $\mathcal{A}_{X}=\mathcal{E}nd(\mathcal{E}_{X})$ where $\mathcal{E}_{X}=\pi^{*}\mathcal{E}_{Y}$. Then for any $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{A}_{X}-\text{mod})$ we have $\mathcal{M}^{\cdot}=\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{N}^{\cdot}$ for a complex $\mathcal{N}^{\cdot}\in D_{coh}^{b}(\mathcal{O}_{X}-\text{mod})$. Therefore \[ R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{M}^{\cdot},\mathcal{A}_{X}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{A}_{X}}(\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{N}^{\cdot},\mathcal{E}_{X}\otimes_{\mathcal{O}_{X}}(\mathcal{E}_{X}^{*}\otimes_{\mathcal{O}_{X}}\omega_{X}))[d] \] \[ \tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{N}^{\cdot},\mathcal{E}_{X}^{*}\otimes_{\mathcal{O}_{X}}\omega_{X})[d]\tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{M}^{\cdot},\omega_{X}[d]) \] \[ \tilde{\to}R\mathcal{H}om_{\mathcal{O}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\omega_{Y})\tilde{=}R\mathcal{H}om_{\mathcal{A}_{Y}}(\mathcal{E}_{Y}\otimes_{\mathcal{O}_{Y}}R\pi_{*}\mathcal{M}^{\cdot},\mathcal{E}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}) \] \[ \tilde{\to}R\mathcal{H}om_{\mathcal{A}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\mathcal{A}_{Y}\otimes_{\mathcal{O}_{Y}}\omega_{Y}) \] where the isomorphism $R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X}}(\mathcal{M}^{\cdot},\omega_{X}[d])\tilde{\to}R\mathcal{H}om_{\mathcal{O}_{Y}}(R\pi_{*}\mathcal{M}^{\cdot},\omega_{Y})$ is Grothendieck duality for coherent sheaves. \end{proof} Now we can proceed to the \begin{proof} (of \thmref{Duality-for-smooth-proper}) By applying $\otimes_{W(k)}^{L}k$ we reduce to the characteristic $p$ situation of a smooth proper morphism $\varphi:X\to Y$. By induction on the cohomological length, we may suppose that $\mathcal{M}^{\cdot}$ is concentrated in a single degree; i.e., $\mathcal{M}^{\cdot}=\mathcal{M}\in\mathcal{G}(\mathcal{D}_{X}^{(0,1)})$. Then $\mathcal{M}$ admits a short exact sequence \[ \mathcal{M}_{0}\to\mathcal{M}\to\mathcal{M}_{1} \] where $\mathcal{M}_{0}\in\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))$ and $\mathcal{M}_{1}\in\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. By \propref{Sandwich!} and \propref{Sandwich-push}, we see that is suffices to prove the analogous statements in $D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ and $D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)})))$. By Frobenius descent (\thmref{Hodge-Filtered-Push}), one sees that it suffices to prove the result for $D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$ and $D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. These two cases require similar, but slightly different techniques; we begin with the case of $\mathcal{R}(\mathcal{D}_{X}^{(0)})$. In this case, since the grading on $\mathcal{R}(\mathcal{D}_{X}^{(0)})$ is concentrated in degrees $\geq0$, the graded Nakayama lemma applies and so it suffices to prove that the map is an isomorphism after applying $\otimes_{k[f]}^{L}k$; i.e., we have to prove \[ R\pi_{*}(d\varphi)^{!}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}X})[d] \] \[ \tilde{\to}R\mathcal{H}om_{\mathcal{O}_{T^{*}Y}}(R\pi_{*}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}Y}) \] Since $d\varphi$ is a closed immersion of smooth schemes, we have \[ (d\varphi)^{!}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}X})[d] \] \[ \tilde{=}R\mathcal{H}om_{\mathcal{O}_{X\times_{Y}T^{*}Y}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),(d\varphi)^{!}\omega_{T^{*}X})[d] \] Furthermore, $(d\varphi)^{!}\omega_{T^{*}X}=\omega_{X\times_{Y}T^{*}Y}\tilde{=}\pi^{!}(\omega_{T^{*}Y})$. Therefore \[ R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{T^{*}X}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),(d\varphi)^{!}\omega_{T^{*}X})[d] \] \[ \tilde{=}R\pi_{*}R\mathcal{H}om_{\mathcal{O}_{X\times_{Y}T^{*}Y}}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\pi^{!}\omega_{T^{*}Y}[d]) \] \[ \tilde{\to}R\mathcal{H}om_{\mathcal{O}_{T^{*}Y}}(R\pi_{*}((d\varphi)^{!}(\mathcal{M}\otimes_{k[f]}^{L}k),\omega_{T^{*}Y}) \] where the last isomorphism is induced by the trace for $X\times_{Y}T^{*}Y\xrightarrow{\pi}T^{*}Y$, i.e., it is given by Grothendieck duality for $\pi$; this proves the result for $D_{coh}^{b}(\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(0)})))$. In order to handle $D_{coh}^{b}(\mathcal{G}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})))$, we apply a similar technique, but working directly\footnote{As the grading on $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$ in unbounded, the graded Nakayama lemma does not apply} with the Azumaya algebra $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. Since the morphism $\varphi$ is smooth, we can make use of \corref{Filtered-Bez-Brav} and work with the functor $R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}$. We therefore have to prove \[ R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}[d] \] \[ \tilde{\to}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})}(R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1} \] We proceed as above. We have an isomorphism \[ C\circ(d\varphi^{(1)})^{!}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})}(\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)}))\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1} \] \[ \tilde{=}R\mathcal{H}om_{(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1})) \] Applying the definition of $(d\varphi^{(1)})^{!}$ and $C$, one deduces \[ C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1})\tilde{=}(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}}\omega_{(X\times_{Y}T^{*}Y)^{(1)}} \] Therefore \[ R\mathcal{H}om_{\pi^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},C\circ(d\varphi^{(1)})^{!}(\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{X}}\omega_{X}^{-1}))[d] \] \[ \tilde{\to}R\mathcal{H}om_{\pi^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)}))}(C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},(\pi^{(1)})^{*}(\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1})\otimes_{\mathcal{O}_{(X\times_{Y}T^{*}Y)^{(1)}}}\omega_{(X\times_{Y}T^{*}Y)^{(1)}}) \] \[ \tilde{\to}R\mathcal{H}om_{\overline{\mathcal{R}}(\mathcal{D}_{Y}^{(0)})}(R\pi_{*}^{(1)}\circ C\circ(d\varphi^{(1)})^{!}\mathcal{M}^{\cdot},\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})\otimes_{\mathcal{O}_{Y}}\omega_{Y}^{-1}) \] where the last isomorphism follows from \lemref{GD-for-Az}; this proves the result for $\overline{\mathcal{R}}(\mathcal{D}_{X}^{(0)})$. \end{proof} This implies, by an identical argument to theorem 2.7.3 of \cite{key-49}: \begin{cor} \label{cor:Smooth-proper-adunction}There is a functorial isomorphism \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot}) \] for all $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$. \end{cor} \subsection{Duality for a Projective morphism} Now we turn to constructing the trace map in the case where $\varphi:\mathfrak{X}\to\mathfrak{Y}$ is a closed embedding, of relative dimension $d$. In this case the pushforward is fairly easy to describe: \begin{lem} \label{lem:transfer-is-locally-free}Let $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ be the reduction to $W_{n}(k)$ of the closed embedding $\varphi$. Then the transfer bimodule $\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}$ is locally free over $\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}$ and is coherent over $\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}}$. Thus the functor $\int_{\varphi}^{0}:\mathcal{G}_{coh}(\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)})\to\mathcal{G}_{coh}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})$ is exact. \end{lem} \begin{proof} Working locally, we can assume that $\mathfrak{X}_{n}=\text{Spec}(B_{n})$, $\mathfrak{Y}_{n}=\text{Spec}(A_{n})$, and $A_{n}$ admits local coordinates $\{x_{1},\dots,x_{n}\}$ for which $B_{n}=A_{n}/(x_{1},\dots,x_{m})$. Then \[ \Gamma(\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)})=\mathcal{D}_{A_{n}}^{(0,1)}/I\cdot\mathcal{D}_{A_{n}}^{(0,1)} \] is coherent over $\mathcal{D}_{A_{n}}^{(0,1),\text{opp}}$. Now, \corref{Local-coords-over-A=00005Bf,v=00005D} implies that $\mathcal{D}_{A_{n}}^{(0,1)}$ is free over $D(A_{n})$ (c.f. also the proof of \corref{Each-D^(i)-is-free}), with basis given by the set $\{\partial^{I}(\partial^{[p]})^{J}\}$, where $I=(i_{1},\dots,i_{n})$ is a multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J$ is any multi-index with entries $\geq0$. So $\mathcal{D}_{A_{n}}^{(0,1)}/I\cdot\mathcal{D}_{A_{n}}^{(0,1)}$ is free over $\mathcal{D}_{B_{n}}^{(0,1)}$with basis given by $\{\partial^{I}(\partial^{[p]})^{J}\}$, where $I=(i_{1},\dots,i_{m})$ is a multi-index with $0\leq i_{j}\leq p-1$ for all $j$ and $J=(j_{1},\dots,j_{m})$ is any multi-index with entries $\geq0$. \end{proof} Now we can proceed to analyze this functor, and the pullback $\varphi^{\dagger}$, in exactly the same way as is done in the usual algebraic $\mathcal{D}$-module theory. In this case, the existence of the trace map is essentially deduced from the duality. To start off, we have \begin{prop} Let $\varphi:\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ be as above. Define $\varphi^{\sharp}(\mathcal{M}^{\cdot}):=R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{M}^{\cdot}))$. Then there is an isomorphism of functors $\varphi^{\dagger}\tilde{=}\varphi^{\sharp}$ on $D(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$. \end{prop} \begin{proof} This is very similar to \cite{key-49}, propositions 1.5.14 and 1.5.16. One first shows \[ R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}})}(\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\tilde{=}\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}[-d] \] by using the Koszul complex to write a locally free resolution for $\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}$ over $\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1),\text{opp}})$; note that by the left-right interchange this implies \[ R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\tilde{=}\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}[-d] \] Then, we have \[ \varphi^{\dagger}(\mathcal{M}^{\cdot})=\mathcal{D}_{\mathfrak{X}_{n}\to\mathfrak{Y}_{n}}^{(0,1)}\otimes_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot})[-d] \] \[ \tilde{=}R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))\otimes_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}^{L}\varphi^{-1}(\mathcal{M}^{\cdot}) \] \[ \tilde{\to}R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(\mathcal{M}^{\cdot})) \] where the last isomorphism uses the fact that $\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}$ admits, locally, a finite free resolution over $\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})$. \end{proof} In turn, this implies \begin{cor} We have a functorial isomorphism \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot}) \] for all $\mathcal{M}^{\cdot}\in D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$. \end{cor} \begin{proof} (Just as in \cite{key-49}, proposition 1.5.25). By the previous proposition, it suffices to prove the result for $\varphi^{\sharp}$ instead of $\varphi^{\dagger}$. To proceed, note that we have the local cohomology functor $\mathcal{N}^{\cdot}\to R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})$ which takes $\mathcal{N}^{\cdot}\in D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$ to $D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$. We have \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})=R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),\mathcal{N}^{\cdot}) \] \[ \tilde{=}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})) \] \[ \tilde{=}\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\varphi^{-1}(\varphi_{*}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot})),\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot}))) \] \[ \tilde{=}\varphi_{*}(R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)}\otimes_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}^{L}\mathcal{M}^{\cdot}),\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot}))) \] \[ \tilde{=}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},R\underline{\mathcal{H}om}_{\varphi^{-1}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)})}(\mathcal{D}_{\mathfrak{Y}_{n}\leftarrow\mathfrak{X}_{n}}^{(0,1)},\varphi^{-1}(R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})))) \] \[ \tilde{=}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}_{n}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\sharp}\mathcal{N}^{\cdot}) \] where, in both the second isomorphism and the last, we have used the existence of an exact triangle \[ R\Gamma_{\mathfrak{X}_{n}}(\mathcal{N}^{\cdot})\to\mathcal{N}^{\cdot}\to\mathcal{K}^{\cdot} \] where $\mathcal{K}^{\cdot}\in D_{qcoh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}_{n}}^{(0,1)}))$ is isomorphic to $Rj_{*}(\mathcal{N}^{\cdot}|_{\mathfrak{Y}_{n}\backslash\mathfrak{X}_{n}})$; here $j:\mathfrak{Y}_{n}\backslash\mathfrak{X}_{n}\to\mathfrak{Y}_{n}$ is the inclusion. In particular, we have that $R\underline{\mathcal{H}om}(\mathcal{C}^{\cdot},\mathcal{K}^{\cdot})=0$ for any $\mathcal{C}^{\cdot}$ supported along $\mathfrak{X}_{n}$. \end{proof} \begin{cor} There is a canonical map \[ \text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}_{n}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}_{n}}[d_{Y}] \] After taking inverse limit, we obtain a trace map ${\displaystyle \text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}}[d_{Y}]}$. If $\psi:\mathfrak{Y}\to\mathfrak{Z}$ is a smooth morphism, We also have a trace map \[ \text{tr}:\int_{\varphi}\mathcal{O}_{\mathfrak{X}}[d_{X}]\to\mathcal{O}_{\mathfrak{Y}}[d_{Y}] \] given by taking the inverse limit of the above maps; the same compatibility holds for this trace as well. \end{cor} \begin{proof} The previous corollary gives an adjunction ${\displaystyle \int_{\varphi}\varphi^{\dagger}\to\text{Id}}$. Since $\varphi^{\dagger}(\mathcal{O}_{\mathfrak{Y}_{n}})=\mathcal{O}_{\mathfrak{X}_{n}}[d_{X}-d_{Y}]$ we obtain the trace map via this adjunction. \end{proof} Now, by factoring an arbitrary projective morphism as a closed immersion followed by a smooth projective map, we obtain by composing the trace maps a trace map for an arbitrary projective morphism. Arguing as in the classical case (c.f. \cite{key-54}, section 2.10), we see that this map is independent of the choice of the factorization. Therefore we obtain \begin{thm} Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a projective morphism. Then we have a functorial morphism \[ \int_{\varphi}\mathbb{D}_{\mathfrak{X}}\mathcal{M}^{\cdot}\to\mathbb{D}_{\mathfrak{Y}}\int_{\varphi}\mathcal{M}^{\cdot} \] which is an isomorphism for $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$. Further, we have a functorial isomorphism \[ R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{Y}}^{(0,1)}}(\int_{\varphi}\mathcal{M}^{\cdot},\mathcal{N}^{\cdot})\tilde{\to}\varphi_{*}R\underline{\mathcal{H}om}_{\mathcal{D}_{\mathfrak{X}}^{(0,1)}}(\mathcal{M}^{\cdot},\varphi^{\dagger}\mathcal{N}^{\cdot}) \] for all $\mathcal{M}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{X}}^{(0,1)}))$ and $\mathcal{N}^{\cdot}\in D_{coh}^{b}(\mathcal{G}(\mathcal{D}_{\mathfrak{Y}}^{(0,1)}))$. \end{thm} \section{Applications} In this section we put things together and give the statement and proof of our generalization of Mazur's theorem for a mixed Hodge module. We begin with a brief review of the pushforward operation in the world of mixed Hodge modules. Let $X_{\mathbb{C}}$ be a smooth complex variety, and suppose that $(\mathcal{M}_{\mathbb{C}},F^{\cdot},\mathcal{K}_{\mathbb{Q}},W_{\cdot})$ is a mixed Hodge module on $X_{\mathbb{C}}$. We won't attempt to recall a complete definition here, instead referring the reader to\cite{key-15},\cite{key-16}, and the excellent survey \cite{key-56}. We will only recall that $\mathcal{M}_{\mathbb{C}}$ is a coherent $\mathcal{D}$-module which comes equipped with a good filtration $F^{\cdot}$, a weight filtration $W_{\cdot}$, and $\mathcal{K}_{\mathbb{Q}}$ is a perverse sheaf defined over $\mathbb{Q}$ which corresponds to $\mathcal{M}_{\mathbb{C}}$ under the Riemann-Hilbert correspondence. In this paper, our attention is on the filtration $F^{\cdot}$ and we will mostly suppress the other aspects of the theory. For the sake of notational convenience, we will denote simply by $\mathcal{O}_{X_{\mathbb{C}}}$ the mixed Hodge module whose underlying filtered $\mathcal{D}$-module is $\mathcal{O}_{X_{\mathbb{C}}}$ with its trivial filtration: $F^{i}(\mathcal{O}_{X_{\mathbb{C}}})=\mathcal{O}_{X_{\mathbb{C}}}$ for all $i\geq0$, while $F^{i}(\mathcal{O}_{X_{\mathbb{C}}})=0$ for $i<0$. Now let $\varphi:X_{\mathbb{C}}\to Y_{\mathbb{C}}$ be a morphism of smooth complex varieties. By Nagata's compatification theorem, combined with Hironaka's resolution of singularities, we can find an open immersion $j:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ into a smooth variety, whose compliment is a normal crossings divisor, and a proper morphism $\overline{\varphi}:\overline{X}_{\mathbb{C}}\to Y_{\mathbb{C}}$, with $\varphi=\overline{\varphi}\circ j$. Then, the following is one of the main results of \cite{key-16} (c.f. theorem 4.3 and theorem 2.14) \begin{thm} Let $\varphi,\overline{\varphi},j$ be morphisms as above. 1) There is a mixed Hodge module $(j_{\star}(\mathcal{M}_{\mathbb{C}}),F^{\cdot}j_{*}\mathcal{K}_{\mathbb{Q}},W_{\cdot})$, whose underlying $\mathcal{D}$-module agrees with the usual pushforward of $\mathcal{D}$-modules under $j$. This defines an exact functor $j_{\star}:\text{MHM}(X_{\mathbb{C}})\to\text{MHM}(\overline{X}_{\mathbb{C}})$. 2) There is an object of $D^{b}(\text{MHM}(Y_{\mathbb{C}}))$, $R\overline{\varphi}_{\star}(j_{\star}(\mathcal{M}_{\mathbb{C}}),F^{\cdot}j_{*}\mathcal{K}_{\mathbb{Q}},W_{\cdot})$, whose underlying complex of filtered $\mathcal{D}$-modules agrees with ${\displaystyle \int_{\overline{\varphi}}(j_{\star}\mathcal{M}_{\mathbb{C}})}$. This object of $D^{b}(\text{MHM}(Y_{\mathbb{C}}))$ is, up to isomorphism, independent of the choice of factorization $\varphi=\overline{\varphi}\circ j$. Furthermore, the filtration on this complex is strict. \end{thm} The reason for stating the theorem this way is that, if $\varphi$ is not proper, the filtered pushforward ${\displaystyle \int_{\varphi}}$ of filtered $\mathcal{D}$-modules does not agree with the pushforward of mixed Hodge modules. The issue appears already if $Y_{\mathbb{C}}$ is a point and $\mathcal{M}_{\mathbb{C}}=\mathcal{O}_{X_{\mathbb{C}}}$. In that case, the pushforward $R\varphi_{\star}$ returns\footnote{up to a homological shift, and a re-indexing of the Hodge filtration} Deligne's Hodge cohomology of $X_{\mathbb{C}}$, while ${\displaystyle {\displaystyle \int_{\varphi}}}$ returns the de Rham cohomology of $X_{\mathbb{C}}$ equipped with the naive Hodge-to-de Rham filtration; these disagree, e.g., if $X_{\mathbb{C}}$ is affine. The construction of the extension $j_{\star}(\mathcal{M}_{\mathbb{C}})$ is, in general, quite deep, and relies on the detailed study of the degenerations of Hodge structures given in \cite{key-60} and \cite{key-61}. However, when $\mathcal{M}_{\mathbb{C}}=\mathcal{O}_{X_{\mathbb{C}}}$ is the trivial mixed Hodge module, one can be quite explicit: \begin{lem} \label{lem:Hodge-filt-on-j_push}Let $j:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ be an open immersion of smooth varieties, whose compliment is a normal crossings divisor $D_{\mathbb{C}}$. Let $x\in X_{\mathbb{C}}$ be a point, about which $D_{\mathbb{C}}$ is given by the equation $\{x_{1}\cdots x_{j}=0\}$. Then as filtered $\mathcal{D}$-modules we have $j_{\star}\mathcal{O}_{X_{\mathbb{C}}}=(j_{*}(\mathcal{O}_{X_{\mathbb{C}}}),F^{\cdot})$ where $F^{l}(j_{*}(\mathcal{O}_{X_{\mathbb{C}}})):=F^{l}(\mathcal{D}_{X_{\mathbb{C}}})\cdot(x_{1}\cdots x_{j})^{-1}$. In particular, $F^{l}(j_{*}(\mathcal{O}_{X_{\mathbb{C}}}))$ is spanned over $\mathcal{O}_{X_{\mathbb{C}}}$ by terms of the form $x_{1}^{-(i_{1}+1)}\cdots x_{j}^{-(i_{j}+1)}$ where ${\displaystyle \sum_{t=1}^{j}i_{t}\leq l}$. \end{lem} For a proof, see \cite{key-6}, section 8. This implies that the Hodge cohomology of $X_{\mathbb{C}}$, as an object in the filtered derived category of vector spaces, can be computed as ${\displaystyle \int_{\overline{\varphi}}j_{\star}\mathcal{O}_{X_{\mathbb{C}}}(d)[d]}$ where $\overline{\varphi}:\overline{X}_{\mathbb{C}}\to\{*\}$. Of course, this can be checked directly by comparing the log de Rham complex with the de Rham complex of a the filtered $\mathcal{D}$-module $j_{\star}\mathcal{O}_{X_{\mathbb{C}}}$. Combining \thmref{Mazur!}with \corref{proper-push-over-W(k)} gives: \begin{prop} 1) Let $\varphi:\mathfrak{X}\to\mathfrak{Y}$ be a projective morphism, and let $\mathfrak{D}\subset\mathfrak{X}$ be a (possibly empty) normal crossings divisor. Let ${\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}})}$ be the gauge of \exaref{Integral-j} on $\mathfrak{X}$. Suppose that each $\mathcal{H}^{i}({\displaystyle \int_{\varphi}(j_{\star}\mathcal{O}_{\mathfrak{X}})^{-\infty}})$ is a $p$-torsion-free $\widehat{\mathcal{D}}_{\mathfrak{Y}}^{(0)}$-module, and that each $\mathcal{H}^{i}({\displaystyle (\int_{\varphi}{\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}}})}\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])$ is $f$-torsion-free. Then each $\mathcal{H}^{i}{\displaystyle (\int_{\varphi}{\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}}}}))$ is a standard gauge on $\mathfrak{Y}$. 2) Let ${\displaystyle j_{!}D(\mathcal{O}_{\mathfrak{X}}):=\mathbb{D}_{\mathfrak{X}}j_{\star}D(\mathcal{O}_{\mathfrak{X}})}$. The same conclusion holds for $j_{!}D(\mathcal{O}_{\mathfrak{X}})$. \end{prop} When $\mathfrak{Y}$ is a point this recovers the log-version of Mazur's theorem, as discussed in Ogus' paper \cite{key-18}. Now let $R$ be a finite type algebra over $\mathbb{Z}$ so that there exists smooth (over $R$) models $X_{R},Y_{R}$ for $X_{\mathbb{C}}$ and $Y_{\mathbb{C}}$, respectively, and a projective morphism $\varphi:X_{R}\to Y_{R}$ whose base change to $\mathbb{C}$ is the original morphism. We may suppose the divisor $D_{\mathbb{C}}$ is defined over $R$ as well. Let $\mathcal{D}_{X_{R}}^{(0)}$ be the level zero differential operators over $X_{R}$, equipped with the symbol filtration; let the associated Rees algebra be $\mathcal{R}(\mathcal{D}_{X_{R}}^{(0)})$ (as usual we will use $f$ for the Rees parameter). Since $\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$ is a coherent $\mathcal{R}(\mathcal{D}_{X_{\mathbb{C}}})$-module, we can by generic flatness choose a flat model for $\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$; in fact, we can describe it explicitly as follows: if $D_{R}$ is given, in local coordinates, by $\{x_{1}\cdots x_{j}=0\}$, then we may consider \[ \mathcal{D}_{X_{R}}^{(0)}\cdot x_{1}^{-1}\cdots x_{j}^{-1}\subset j_{*}\mathcal{O}_{U_{R}} \] with the filtration inherited from the symbol filtration on $\mathcal{D}_{X_{R}}^{(0)}$. The Rees module of this filtered $\mathcal{D}_{X_{R}}^{(0)}$-module is a flat $R$-model for $\text{Rees}(j_{*}\mathcal{O}_{U_{\mathbb{C}}})$. Let us call this sheaf ${\displaystyle j_{\star}\mathcal{O}_{U_{R}}[f]}$; we will denote the associated filtered $\mathcal{D}_{X_{R}}^{(0)}$-module by ${\displaystyle j_{\star}\mathcal{O}_{U_{R}}}$. Then, localizing $R$ if necessary, we have that \[ \int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f] \] is an $f$-torsion-free complex inside $D_{coh}^{b}(\mathcal{D}_{Y_{R}}^{(0)}-\text{mod})$ (since it becomes $f$-torsion-free after base change to $\mathbb{C}$, as remarked above). By generic flatness, we may also suppose (again, localizing $R$ if necessary), that each cohomology sheaf ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}})}$ is flat over $R$. Let $k$ be a perfect field of characteristic $p>0$, for which there is a morphism $R\to W(k)$ (so that $R/p\to k$)\footnote{If we extend $R$ so that it is smooth over $\mathbb{Z}$, then any map $R/p\to k$ lifts to $R\to W(k)$}. Then, combining this discussion with the previous proposition, we obtain \begin{cor} \label{cor:Mazur-for-Hodge-1}Let $\mathfrak{X}$ be the formal completion of $X_{R}\times_{R}W(k)$, and similarly for $\mathfrak{Y}$. Then each gauge $\mathcal{H}^{i}({\displaystyle (\int_{\varphi}{\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}})}}))$ is a standard, coherent, $F^{-1}$-gauge on $\mathfrak{Y}$. There is an isomorphism \[ \mathcal{H}^{i}(({\displaystyle (\int_{\varphi}{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})\otimes_{W(k)}^{L}k)\otimes_{D(k)}^{L}k[f])\tilde{\to}F^{*}\mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f]\otimes_{R}^{L}k) \] in $\mathcal{G}(\mathcal{R}(\mathcal{D}_{X}^{(1)}))$. In particular, the Hodge filtration on ${\displaystyle \mathcal{H}^{i}({\displaystyle (\int_{\varphi}{\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}}})}))^{\infty}/p}$ is the Frobenius pullback of the Hodge filtration on ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}\otimes_{R}^{L}k)}$. The same holds if we replace ${\displaystyle j_{\star}D(\mathcal{O}_{\mathfrak{X}}})$ by ${\displaystyle j_{!}D(\mathcal{O}_{\mathfrak{X}}})$. The same statement holds for the pushforward of ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j_{\star}\mathcal{O}_{U_{R}})}$ under another proper morphism $\psi:Y\to Z$. \end{cor} \begin{proof} The displayed isomorphism follows immediately from \thmref{Hodge-Filtered-Push}. Since ${\displaystyle \int_{\varphi}j_{\star}\mathcal{O}_{U_{R}}[f]}$ has $f$-torsion free cohomology sheaves, which are also flat over $R$, we deduce that ${\displaystyle {\displaystyle ((\int_{\varphi}{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})\otimes_{W(k)}^{L}k})\otimes_{D(k)}^{L}k[f]$ has $f$-torsion free cohomology sheaves. Comparing the description of the Hodge filtration on ${\displaystyle ({\displaystyle j_{\star}\mathcal{O}_{\mathfrak{X}}[f,v]}})^{\infty}/p$ with the result of \lemref{Hodge-filt-on-j_push}, the result now follows from \thmref{F-Mazur}. \end{proof} Let us give some first applications of these results. Suppose that $X_{\mathbb{C}}$ is an arbitrary (possibly singular) quasi-projective variety. Let $V_{\mathbb{C}}$ be a smooth quasi-projective variety such that there is a closed embedding $X_{\mathbb{C}}\to V_{\mathbb{C}}$, and let $\overline{V}_{\mathbb{C}}$ be a projective compatification of $V_{\mathbb{C}}$ (i.e., $\overline{V}_{\mathbb{C}}\backslash V_{\mathbb{C}}$ is a normal crossings divisor). Let $U_{\mathbb{C}}\subset X_{\mathbb{C}}$ be an affine open \emph{smooth} subset. Let $\varphi:\tilde{X}_{\mathbb{C}}\to X_{\mathbb{C}}$ denote a resolution of singularities so that $\varphi$ is an isomorphism over $U_{\mathbb{C}}$ and $\varphi^{-1}(X_{\mathbb{C}}\backslash U_{\mathbb{C}})$ is a normal crossings divisor $\tilde{D}_{\mathbb{C}}\subset\tilde{X}_{\mathbb{C}}$. The decomposition theorem for Hodge modules implies that the complex ${\displaystyle \int_{\varphi}\mathcal{O}_{\tilde{X}_{\mathbb{C}}}}\in D^{b}(\text{MHM}_{X})$ is quasi-isomorphic to the direct sum of its cohomology sheaves, and that each such sheaf is a direct sum of simple, pure Hodge modules. Therefore, if $j:U_{\mathbb{C}}\to X_{\mathbb{C}}$ (resp. $j':U_{\mathbb{C}}\to\tilde{X}_{\mathbb{C}}$) denotes the inclusion, then the image of the natural map \[ \mathcal{H}^{0}({\displaystyle \int_{\varphi}\mathcal{O}_{\tilde{X}_{\mathbb{C}}}})\to\mathcal{H}^{0}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{\mathbb{C}}})\tilde{\to}\mathcal{H}^{0}(j_{\star}\mathcal{O}_{U_{\mathbb{C}}}) \] is the Hodge module $\text{IC}_{X}$; indeed, ${\displaystyle \mathcal{H}^{0}({\displaystyle \int_{\varphi}\mathcal{O}_{\tilde{X}_{\mathbb{C}}}})}=\text{IC}_{X}\oplus\mathcal{M}$ where $\mathcal{M}$ is a pure Hodge module supported on $X_{\mathbb{C}}\backslash U_{\mathbb{C}}$; its image in $\mathcal{H}^{0}({\displaystyle j_{\star}\mathcal{O}_{U_{\mathbb{C}}})}$ is therefore isomorphic to $\text{IC}_{X}$ (as a Hodge module, and so in particular as a filtered $\mathcal{D}$-module). Now let $\overline{X}_{\mathbb{C}}$ denote the closure of $X_{\mathbb{C}}$ in $\overline{V}_{\mathbb{C}}$, and let $\varphi:\tilde{\overline{X}}_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ be a resolution of singularities, whose restriction to $X_{\mathbb{C}}\subset\overline{X}_{\mathbb{C}}$ is isomorphic to $\varphi:\tilde{X}_{\mathbb{C}}\to X_{\mathbb{C}}$, and so that the inverse image of $\overline{X}_{\mathbb{C}}\backslash X_{\mathbb{C}}$ is a normal crossings divisor (we can modify $\varphi$ if necessary to ensure that this happens). Let $i:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ and $i':\tilde{X}_{\mathbb{C}}\to\tilde{\overline{X}}_{\mathbb{C}}$ denote the inclusions. Since Hodge modules on $X_{\mathbb{C}}$ are, by definition, Hodge modules on $V_{\mathbb{C}}$ which are supported on $X_{\mathbb{C}}$, the fact that $\overline{V}_{\mathbb{C}}\backslash V_{\mathbb{C}}$ is a divisor implies that $i_{*}$ is an exact functor on the category of mixed Hodge modules. Therefore the image of the natural map \[ \mathcal{H}^{0}(\int_{\varphi}i'_{\star}\mathcal{O}_{\tilde{X}_{\mathbb{C}}})\tilde{=}i_{\star}\mathcal{H}^{0}({\displaystyle \int_{\varphi}\mathcal{O}_{\tilde{X}_{\mathbb{C}}}})\to i_{\star}\mathcal{H}^{0}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{\mathbb{C}}})\tilde{=}\mathcal{H}^{0}(i\circ j)_{\star}\mathcal{O}_{U_{\mathbb{C}}} \] is isomorphic to $i_{\star}(\text{IC}_{X})$ (again, as a Hodge module, and so in particular as a filtered $\mathcal{D}$-module). As above, we now select a finite type $\mathbb{Z}$-algebra $R$ so that everything in sight is defined and flat over $R$, and let $R\to W(k)$ for some perfect $k$ of characteristic $p>0$. Let $\tilde{\mathfrak{\overline{X}}}\to\mathfrak{\overline{X}}\subset\overline{\mathcal{V}}$ be the formal completion of $\tilde{\overline{X}}_{R}\times_{R}W(k)\to\overline{X}_{R}\times_{R}W(k)\subset\overline{V}_{R}\times_{R}W(k)$. Abusing notation slightly we'll also denote by $\varphi$ the composed map $\tilde{\mathfrak{\overline{X}}}\to\widehat{\overline{\mathcal{V}}}$. \begin{cor} \label{cor:Mazur-for-IC}1) The image of the map \[ \mathcal{H}^{0}(\int_{\varphi}i'_{\star}D(\mathcal{O}_{\tilde{\mathfrak{X}}}))\to\mathcal{H}^{0}(\int_{\varphi}(i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}})) \] defines a coherent, standard $F^{-1}$-gauge on $\widehat{\mathbb{P}^{n}}$, denoted $\text{IC}_{\mathfrak{X}}$. The $\widehat{\mathcal{D}}_{\overline{\mathcal{V}}}^{(0)}$-module $\text{IC}_{\mathfrak{X}}^{-\infty}$ is isomorphic to the $p$-adic completion of $\text{IC}_{X_{R}}\otimes_{R}W(k)$, where $\text{IC}_{X_{R}}$ is an $R$-model for $\text{IC}_{X_{\mathbb{C}}}$. The Hodge filtration on the $\mathcal{D}_{\overline{V}_{k}}^{(1)}$-module $\widehat{\text{IC}_{\mathfrak{X}}^{\infty}}/p\tilde{=}F^{*}\text{IC}_{\mathfrak{X}}^{-\infty}/p$ is equal to the Frobenius pullback of the Hodge filtration on $\text{IC}_{\mathfrak{X}}^{-\infty}/p\tilde{=}\text{IC}_{X_{R}}\otimes_{R}k$ coming from the Hodge filtration on $\text{IC}_{X_{R}}$. 2) The intersection cohomology groups $\text{IH}^{i}(X_{R})\otimes_{R}W(k):=\mathbb{H}_{dR}^{i}(\text{IC}_{X_{R}})\otimes_{R}W(k)$ satisfy the conclusions of Mazur's theorem; as in \thmref{Mazur-for-IC-Intro} \end{cor} \begin{proof} Since the displayed map is a map of coherent gauges, the image, $i_{\star}\text{IC}_{\mathfrak{X}}$, is a coherent gauge. Since both ${\displaystyle i'_{\star}D(\mathcal{O}_{\tilde{\mathfrak{X}}})}$ and $(i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}}))$ are $F^{-1}$gauges, and the natural map $i'_{\star}D({\displaystyle \mathcal{O}_{\tilde{\mathfrak{X}}})\to(i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}})}$ is $F^{-1}$-equivariant, the same is true of the displayed map, and so $i_{\star}\text{IC}_{\mathfrak{X}}$ is an $F^{-1}$-gauge. By \propref{push-and-complete-for-D} (and the exactness of the functor $\mathcal{M}\to\mathcal{M}^{-\infty}$) we have that the image of \[ \mathcal{H}^{0}(\int_{\varphi}i'_{\star}D(\mathcal{O}_{\tilde{\mathfrak{X}}}))^{-\infty}\to\mathcal{H}^{0}(\int_{\varphi}(i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}}))^{-\infty} \] is equal to the image of \[ \mathcal{H}^{0}(\int_{\varphi}(i_{\star}\mathcal{O}_{\tilde{\mathfrak{X}}})^{-\infty})\to\mathcal{H}^{0}\int_{\varphi}((i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}}))^{-\infty} \] in the category of $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-modules. On the other hand, we have $R$-flat filtered $\mathcal{D}_{X_{R}}^{(0)}$-modules ${\displaystyle \mathcal{H}^{0}(\int_{\varphi}i_{\star}\mathcal{O}_{\tilde{X}_{R}})}$ and ${\displaystyle \mathcal{H}^{0}(\int_{\varphi}(i'\circ j')_{*}\mathcal{O}_{U_{R}})}$ such that the $p$-adic completion of $\mathcal{H}^{0}(\int_{\varphi}i_{\star}\mathcal{O}_{\tilde{X}_{R}}){\displaystyle \otimes_{R}W(k)}$ is ${\displaystyle \mathcal{H}^{0}(\int_{\varphi}(i_{\star}\mathcal{O}_{\tilde{\mathfrak{X}}})^{-\infty})}$, and the $p$-adic completion of ${\displaystyle \mathcal{H}^{0}(\int_{\varphi}(i'\circ j')_{*}\mathcal{O}_{U_{R}})}$ is ${\displaystyle \mathcal{H}^{0}\int_{\varphi}((i'\circ j')_{*}D(\mathcal{O}_{\mathfrak{U}}))^{-\infty}}$, and, after localizing $R$ if necessary, we may further suppose that the kernel of the map \begin{equation} {\displaystyle \mathcal{H}^{0}(\int_{\varphi}i_{\star}\mathcal{O}_{\tilde{X}_{R}})}\to\mathcal{H}^{0}(\int_{\varphi}(i'\circ j')_{*}\mathcal{O}_{U_{R}})\label{eq:natural-map-over-R} \end{equation} is a summand (in the category of filtered $\mathcal{D}_{X_{R}}^{(0)}$-modules) of ${\displaystyle {\displaystyle \mathcal{H}^{0}(\int_{\varphi}\mathcal{O}_{\tilde{X}_{R}})}}$ (as this is true over $\mathbb{C}$). Thus the image is flat over $R$, and so its $p$-adic completion is $p$-torsion-free; therefore $\text{IC}_{\mathfrak{X}}^{-\infty}$ is $p$-torsion-free, as is $\text{IC}_{\mathfrak{X}}^{\infty}$ (since $\text{IC}_{\mathfrak{X}}^{-\infty}$ is an $F^{-1}$-gauge; c.f. the proof of \thmref{F-Mazur}) Further, the map \eqref{natural-map-over-R} is strict with respect to the Hodge filtration, and so the same is true after taking reduction mod $p$ and applying $F^{*}$. It follows that $\text{IC}_{\mathfrak{X}}^{\infty}/p/v$ is $f$-torsion-free. Thus by \propref{Baby-Mazur}, we see that $\text{IC}_{\mathfrak{X}}$ is a standard gauge; the statement about the Hodge filtration follows from \corref{Mazur-for-Hodge-1}. This proves part $1)$, and part $2)$ follows from taking the pushforward to a point. \end{proof} \begin{rem} The construction above involved a few auxiliary choices- namely, the ring $R$ and the resolution $\tilde{X}_{R}$. However, any two resolutions of singularities can be dominated by a third. Therefore, after possibly localizing $R$, any two definitions of $\text{IC}_{X_{R}}$ agree. Further, we if we have an inclusion of rings $R\to R'$ then $\text{IC}_{X_{R}}\otimes_{R}R'=\text{IC}_{X_{R'}}$. Therefore we have $\mathbb{H}_{dR}^{i}(\text{IC}_{X_{R}})\otimes_{R}R'\tilde{\to}\mathbb{H}_{dR}^{i}(\text{IC}_{X_{R'}})$ when both are flat. Since any two finite-type $\mathbb{Z}$-algebras can be embedded into a third, we also obtain a comparison for any two such algebras. \end{rem} Now suppose $X_{\mathbb{C}}$ is a smooth (quasiprojective) scheme, and let $i:Y_{\mathbb{C}}\to X_{\mathbb{C}}$ be a closed immersion; here, $Y_{\mathbb{C}}$ can be singular; let $j:X_{\mathbb{C}}\backslash Y_{\mathbb{C}}\to X_{\mathbb{C}}$ be the open immersion. Now, let $\tilde{j}:X_{\mathbb{C}}\to\overline{X}_{\mathbb{C}}$ be a smooth proper compactification of $X_{\mathbb{C}}$, so that $\overline{X}_{\mathbb{C}}\backslash X_{\mathbb{C}}$ is a normal crossings divisor. Choose flat $R$ models for everything in sight. Then we have \begin{cor} \label{cor:Mazur-for-Ordinary}For each $i$, the Hodge cohomology group $H^{i}(Y_{\mathbb{C}})$ admits a flat model $H^{i}(Y_{R})$ (as a filtered vector space). Let $k$ is a perfect field such that $R\to W(k)$. Then there is a standard $F^{-1}$-gauge $H^{i}(Y_{R})_{W(k)}^{\cdot}$ such that $H^{i}(Y_{R})_{W(k)}^{-\infty}\tilde{=}H^{i}(Y_{R})\otimes_{R}W(k)$, and such that the Hodge filtration on $H^{i}(Y_{R})_{W(k)}^{\infty}/p$ agrees with the Frobenius pullback of the Hodge filtration on $H^{i}(Y_{R})\otimes_{R}k$. In particular, there is a Frobenius-linear isomorphism of $H^{i}(Y_{R})_{W(k)}[p^{-1}]$ for which the Hodge filtration on $H^{i}(Y_{R})_{W(k)}$ satisfies the conclusions of Mazur's theorem. The same holds for the compactly supported Hodge cohomology $H_{c}^{i}(Y_{\mathbb{C}})$. \end{cor} \begin{proof} As the usual Hodge cohomology and the compactly supported Hodge cohomology are interchanged under applying the filtered duality functor, it suffices to deal with the case of the compactly supported cohomology. Let us recall how to define this in the language of mixed Hodge modules. We have the morphism \[ Rj_{!}(\mathcal{O}_{X_{\mathbb{C}}\backslash Y_{\mathbb{C}}})\to\mathcal{O}_{X_{\mathbb{C}}} \] in the category of mixed Hodge modules (where $\mathcal{O}$ has its usual structure as the trivial mixed Hodge module). The cone of this map is, by definition, the complex of mixed Hodge modules representing the unit object on $Y_{\mathbb{C}}$; we denote it by $\mathbb{I}_{Y_{\mathbb{C}}}$. Then we have \[ H_{c}^{i}(Y_{\mathbb{C}})=\int_{\varphi}^{d+i}R\tilde{j}_{!}\mathbb{I}_{Y_{\mathbb{C}}}=\int_{\varphi}^{d+i}R\tilde{j}_{!}(\text{cone}(Rj_{!}(\mathcal{O}_{X_{\mathbb{C}}\backslash Y_{\mathbb{C}}})\to\mathcal{O}_{X_{\mathbb{C}}})) \] \[ \tilde{=}\int_{\varphi}^{d+i}\text{cone}(R(\tilde{j}\circ j)_{!}(\mathcal{O}_{X_{\mathbb{C}}\backslash Y_{\mathbb{C}}})\to R\tilde{j}_{!}\mathcal{O}_{X_{\mathbb{C}}}) \] Now, after spreading out both $R(\tilde{j}\circ j)_{!}(\mathcal{O}_{X_{\mathbb{C}}\backslash Y_{\mathbb{C}}})$ and $R\tilde{j}_{!}\mathcal{O}_{X_{\mathbb{C}}})$ over $R$, we can apply \corref{Mazur-for-Hodge-1}. \end{proof} \begin{rem} The previous two corollaries also hold for quasiprojective varieties defined over $\overline{\mathbb{Q}}$. Although the theory of mixed Hodge modules only exists over $\mathbb{C}$, its algebraic consequences, such as the strictness of the pushforward of modules of the form $j_{\star}(\mathcal{O}_{X})$, hold over any field of characteristic $0$. So the above results go through in this case as well. \end{rem} Finally, we wish to give some relations of the theory of this paper to the Hodge structure of the local cohomology sheaves $\mathcal{H}_{Y_{\mathbb{C}}}^{i}(\mathcal{O}_{X_{\mathbb{C}}})$, as developed in {[}MP1{]}, {[}MP2{]}. Here, $X_{\mathbb{C}}$ is a smooth affine variety and $Y_{\mathbb{C}}\subset X_{\mathbb{C}}$ is a subscheme defined by $(Q_{1},\dots,Q_{r})$. In this case, the nontrivial sheaf is \[ \mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}})\tilde{=}\mathcal{O}_{X_{\mathbb{C}}}[Q_{1}^{-1}\cdots Q_{r}^{-1}]/\sum_{i=1}^{r}\mathcal{O}_{X_{\mathbb{C}}}\cdot Q_{1}^{-1}\cdots\widehat{(Q_{i}^{-1})}\cdots Q_{r}^{-1} \] where $\widehat{?}$ stands for ``omitted.'' As above, these sheaves admits a Hodge structure via \[ \mathcal{H}_{Y_{\mathbb{C}}}^{i}(\mathcal{O}_{X_{\mathbb{C}}})\tilde{=}\mathcal{H}^{i}(\int_{\varphi}\int_{j'}\mathcal{O}_{U_{\mathbb{C}}}) \] where $\varphi:\tilde{X}_{\mathbb{C}}\to X_{\mathbb{C}}$ is a resolution of singularities such that $\varphi^{-1}(Y_{\mathbb{C}})$ is a normal crossings divisor; and $j':U_{\mathbb{C}}\to\tilde{X}_{\mathbb{C}}$ is the inclusion. The resulting Hodge filtration is independent of the choice of the resolution. Taking $R$-models for everything in sight at above, we obtain a filtered $\mathcal{D}_{X_{R}}^{(0)}$-module ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})}$ which (localizing $R$ if necessary) is a flat $R$-model for $\mathcal{H}_{Y_{\mathbb{C}}}^{i}(\mathcal{O}_{X_{\mathbb{C}}})$. Now let $R\to W(k)$, and let $\mathfrak{X}$, $\tilde{\mathfrak{X}}$, etc. be the formal completions of the base-change to $W(k)$ as usual. Then we have a gauge \[ \mathcal{M}_{Y}:=\mathcal{H}^{i}(\int_{\varphi}j'_{\star}D(\mathcal{O}_{\mathfrak{U}})) \] which satisfies $\mathcal{M}_{Y}^{-\infty}={\displaystyle \mathcal{H}^{i}(\int_{\varphi}(j'_{\star}\mathcal{O}_{\mathfrak{U}_{W(k)}})^{-\infty})}$. \begin{lem} \label{lem:injectivity-for-local-coh}Let $\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}:=\mathcal{H}^{i}(Rj_{*}\mathcal{O}_{\mathfrak{U}})$. (This is simply the $p$-adic completion of the $i$th algebraic local cohomology of $\mathfrak{X}$ along $\mathfrak{Y}$). Then the natural map \[ \mathcal{M}_{Y}^{-\infty}\to\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})} \] is injective. If $F$ is a lift of Frobenius, the natural map $F^{*}\mathcal{M}_{Y}^{-\infty}\to\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}$ is also injective. \end{lem} \begin{proof} We have the Hodge filtration on ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})}$, which is is a filtration by coherent $\mathcal{O}_{X_{R}}$-modules; base changing to $W(k)$ yields a Hodge filtration on ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{W(k)}})}$. The map in question is the $p$-adic completion of the natural map \[ \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{W(k)}})\to\mathcal{H}_{Y_{W(k)}}^{i}(\mathcal{O}_{X_{W(k)}}) \] and the right hand module also has a Hodge filtration, which is simply the restriction of the Hodge filtration on $\mathcal{H}_{Y_{B}}^{r}(\mathcal{O}_{X_{B}})$ where $B=\text{Frac}(W(k))$. So the proof proceeds in an essentially identical manner to \lemref{Injectivity-of-completion}. \end{proof} Now, fix an integer $m\geq0$. Let us explain how to use this gauge to obtain an arithmetic description of the Hodge filtration, up to level $m$. Since $m$ is fixed, we may, after localizing $R$ as needed, suppose that the image of the map ${\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})})\to\mathcal{H}_{Y_{R}}^{i}(\mathcal{O}_{X_{R}})$ is equal to $F^{m}(\mathcal{H}_{Y_{F}}^{i}(\mathcal{O}_{X_{F}}))\cap\mathcal{H}_{Y_{R}}^{i}(\mathcal{O}_{X_{R}})$. In particular the map \[ F^{m}({\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})})\otimes_{R}k\to\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}}) \] is injective; under the isomorphism $F^{*}\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}})\tilde{\to}\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}})$, we also obtain an injection $F^{*}(F^{m}({\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})}))\otimes_{R}k\to\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}})$. Then \begin{prop} \label{prop:Hodge-for-local-coh!}Let the be notation as above. We have that the image of $\{g\in F^{*}\mathcal{M}_{Y}^{-\infty}|p^{j}g\in\mathcal{M}_{Y}^{-\infty}\}$ in $\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}})$ is exactly $F^{*}(F^{j}({\displaystyle \mathcal{H}^{i}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})}))\otimes_{R}k)$. For each $0\leq j\leq m$, this is also the image of $\{g\in\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}|p^{j}g\in\mathcal{M}_{Y}^{-\infty}\}$. \end{prop} \begin{proof} By construction $\mathcal{M}_{Y}$ is a standard, coherent, $F^{-1}$-gauge of index $0$ (this can be easily seen as the Hodge filtration is concentrated in degrees $\geq0$). Therefore, we have $\widehat{\mathcal{M}_{Y}^{\infty}}\tilde{=}F^{*}\mathcal{M}_{Y}$, and by the previous lemma $F^{*}\mathcal{M}_{Y}\to F^{*}\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}\tilde{\to}\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}$ is injective. Since $\mathcal{M}_{Y}$ is standard of index $0$, we have \[ \mathcal{M}_{Y}^{j}=\{m\in\mathcal{M}_{Y}^{\infty}|p^{j}m\in f_{\infty}(\mathcal{M}_{Y}^{0})\} \] Note that if $j\leq0$, this means $\mathcal{M}_{Y}^{j}\tilde{=}\mathcal{M}_{Y}^{-\infty}$, and the map \[ \eta_{i}:\mathcal{M}_{Y}^{j}\xrightarrow{f_{\infty}}\mathcal{M}_{Y}^{\infty}\xrightarrow{\widehat{?}}F^{*}\mathcal{M}_{Y}\to\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})} \] is simply $p^{-j}$ times the injection $\mathcal{M}_{Y}^{-\infty}\to\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}$, and is therefore injective by \lemref{injectivity-for-local-coh} . If $j>0$, then $p^{j}\cdot\eta_{j}=\eta_{0}\circ v^{j}$ is injective for the same reason, and so $\eta_{j}$ is injective since everything in sight is $p$-torsion -free. Thus the entire gauge embeds into $\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}$ and the first result follows. For the second result, consider the standard gauge $\mathcal{N}_{Y}$ defined by $\mathcal{N}_{Y}^{j}:=\{m\in\widehat{\mathcal{H}_{\mathfrak{Y}}^{i}(\mathcal{O}_{\mathfrak{X}})}|p^{j}m\in\mathcal{M}_{Y}^{0}\}$ (the actions of $f$ and $v$ are inclusion and multiplication by $p$ as usual). We have the natural injection for each $j$ $\mathcal{M}_{Y}^{j}\to\mathcal{N}_{Y}^{j}$, which yields a morphism of gauges $\mathcal{M}_{Y}\to\mathcal{N}_{Y}$. Let us show that for $j\leq m$ the map $\psi:\mathcal{M}_{Y}^{j}\to\mathcal{N}_{Y}^{j}$ is an isomorphism. For $j\leq0$ this is clear by definition, so suppose it is true for some $j-1\leq m-1$. For any $j$ let $\mathcal{M}_{Y,0}^{j}:=\mathcal{M}_{Y}^{j}/p$ and similarly define $\mathcal{N}_{Y,0}^{j}$. We first claim that the isomorphism $\psi:\mathcal{M}_{Y,0}^{j-1}\to\mathcal{N}_{Y,0}^{j-1}$ induces an isomorphism $\text{ker}(f:\mathcal{M}_{Y,0}^{j-1}\to\mathcal{M}_{Y,0}^{j})\tilde{\to}\text{ker}(f:\mathcal{N}_{Y,0}^{j-1}\to\mathcal{N}_{Y,0}^{j})$. Indeed, we have that $\mathcal{M}_{Y,0}^{j-1}/\text{ker}(f)\tilde{=}F^{j-1}(\mathcal{M}_{Y,0}^{\infty})$ and $\mathcal{N}_{Y,0}^{j-1}/\text{ker}(f)\tilde{=}F^{j-1}(\mathcal{N}_{Y,0}^{\infty})$ (as $\mathcal{M}_{Y}$ and $\mathcal{N}_{Y}$ are standard gauges). Further, the composed morphism \[ F^{j-1}(\mathcal{M}_{Y,0}^{\infty})\to F^{j-1}(\mathcal{N}_{Y,0}^{\infty})\to\mathcal{H}_{Y_{k}}^{i}(\mathcal{O}_{X_{k}}) \] is injective (since $j-1\leq m$); therefore $F^{j-1}(\mathcal{M}_{Y,0}^{\infty})\to F^{j-1}(\mathcal{N}_{Y,0}^{\infty})$ is injective, and it is clearly surjective since $\mathcal{M}_{Y,0}^{j-1}\to\mathcal{N}_{Y,0}^{j-1}$ is surjective. Therefore it is an isomorphism; and hence so is $\text{ker}(f:\mathcal{M}_{Y,0}^{j-1}\to\mathcal{M}_{Y,0}^{j})\to\text{ker}(f:\mathcal{N}_{Y,0}^{j-1}\to\mathcal{N}_{Y,0}^{j})$ as claimed. Now suppose $m\in\text{ker}(\psi:\mathcal{M}_{Y,0}^{j}\to\mathcal{N}_{Y,0}^{j})$. Then $vm\in\text{ker}(\psi:\mathcal{M}_{Y,0}^{j-1}\to\mathcal{N}_{Y,0}^{j-1})=0$, so that $m\in\text{ker}(v)=\text{im}(f)$. If $m=fm'$, then we see $\psi m'\in\text{ker}(f)$; but by the above paragraph this implies $m'\in\text{ker}(f)$; therefore $m=0$ and $\psi:\mathcal{M}_{Y,0}^{j}\to\mathcal{N}_{Y,0}^{j}$ is injective. Thus the cokernel of $\psi:\mathcal{M}_{Y}^{j}\to\mathcal{N}_{Y}^{j}$ is $p$-torsion-free. On the other hand, we clearly have $p^{j}\cdot\mathcal{N}_{Y}^{j}\subset\mathcal{M}_{Y}^{j}$; so that the cokernel of $\psi$ is annihilated by $p^{j}$; therefore the cokernel is $0$ and we see that $\psi:\mathcal{M}_{Y}^{j}\to\mathcal{N}_{Y}^{j}$ is an isomorphism as claimed. \end{proof} Note that this gives a description of the reduction mod $p$ of the Hodge filtration (up to $F^{m}$) which makes no reference to a resolution of singularities. It does depend on an $R$-model for the $\mathcal{D}$-module $\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}})$, though any two such models agree after localizing $R$ at an element. Now let us further suppose that $Y_{\mathbb{C}}\subset X_{\mathbb{C}}$ is a complete intersection of codimension $r$. By {[}MP2{]}, proposition 7.14, (c.f. also section $9$ of loc. cit.) we have \[ F^{m}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))\subset O^{m}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=\text{span}_{\mathcal{O}_{X_{\mathbb{C}}}}\{Q_{1}^{-a_{1}}\cdots Q_{r}^{-a_{r}}|\sum a_{i}\leq m+r\} \] In {[}MP2{]} the condition $F^{m}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O^{m}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$ is discussed at length; and the point of view developed there shows that the largest $m$ for which there is equality is a subtle measure of the singularities of $Y_{\mathbb{C}}$. In fact, equality for any $m$ already implies serious restrictions on the singularities; indeed, $F^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$ is equivalent to $Y_{\mathbb{C}}$ having du Bois singularities (this is the first case of theorem F of loc. cit.). Now, using the methods of this paper, let us show \begin{cor} \label{cor:Canonical-Singularities} Suppose $F^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$, i.e., $Q_{1}^{-1}\cdots Q_{r}^{-1}\in F^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$. Then the log-canonical threshold of $Y_{\mathbb{C}}$ is $r$. \end{cor} Combined with the above, this gives a new proof of the famous fact that du Bois singularities are canonical, in the l.c.i. case at least (c.f. \cite{key-57}, \cite{key-58}). It is also a (very) special case of {[}MP2{]}, conjecture 9.11; of course, it also follows from theorem C of {[}MP2{]}, using the results of \cite{key-57}. To prove this result, we will recall a few facts from positive characteristic algebraic geometry, following {[}BMS{]}. We return to a perfect field $k$ of positive characteristic and $X$ smooth over $k$. Let $\mathcal{I}\subset\mathcal{O}_{X}$ be an ideal sheaf. For each $m>0$ we let $\mathcal{I}^{[1/p^{m}]}$ be the minimal ideal sheaf such that $\mathcal{I}\subset(F^{m})^{*}(\mathcal{I}^{[1/p^{m}]})$ (here we are using the isomorphism $(F^{m})^{*}\mathcal{O}_{X}\tilde{\to}\mathcal{O}_{X}$; for any ideal sheaf $\mathcal{J}$ we have $(F^{m})^{*}\mathcal{J}=\mathcal{J}^{[p^{m}]}$, the ideal locally generated by $p^{m}$th powers of elements of $\mathcal{J}$). Then, for each $i>0$ one has inclusions \[ (\mathcal{I}^{i})^{[1/p^{m}]}\subset(\mathcal{I}^{i'})^{[1/p^{m'}]} \] whenever $i/p^{m}\leq i'/p^{m'}$ and $m\leq m'$ (this is {[}BMS{]}, lemma 2.8). These constructions are connected to $\mathcal{D}$-module theory as follows: for any ideal sheaf $\mathcal{I}$, we have $\mathcal{D}_{X}^{(m)}\cdot\mathcal{I}=(F^{m+1})^{*}(\mathcal{I}^{[1/p^{m+1}]})$ (c.f. \cite{key-64}, remark 2.6, and \cite{key-62}, lemma 3.1). Now, fix a number $c\in\mathbb{R}^{+}$. If $x\to\lceil x\rceil$ denotes the ceiling function, then the previous discussion implies inclusions \[ (\mathcal{I}^{\lceil cp^{m}\rceil})^{[1/p^{m}]}\subset(\mathcal{I}^{\lceil cp^{m+1}\rceil})^{[1/p^{m+1}]} \] for all $m$. Thus we have a chain of ideals, which necessarily stabilizes, and so we can define \[ \tau(\mathcal{I}^{c})=(\mathcal{I}^{\lceil cp^{m}\rceil})^{[1/p^{m}]} \] for all $m>>0$. These ideals are called generalized test ideals. There is a deep connection to the theory of multiplier ideals in complex algebraic geometry, which is due to Hara and Yoshida ({[}HY{]}, theorems 3.4 and 6.8). Suppose we have a complex variety $X_{\mathbb{C}}$, and flat $R$-model $X_{R}$, and an ideal sheaf $\mathcal{I}_{R}$ which is also flat over $R$. Fix a rational number $c$; we may then choose a flat model $\mathcal{J}(\mathcal{I}_{R}^{c})$ for the multiplier ideal $\mathcal{J}(\mathcal{I}_{\mathbb{C}}^{c})$. Then for all perfect fields $k$ of sufficiently large positive characteristic, we have \[ \mathcal{J}(\mathcal{I}_{R}^{c})\otimes_{R}k=\tau(\mathcal{I}_{k}^{c}) \] Finally, we note that since $\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}})$ is a coherent $\mathcal{D}_{X_{\mathbb{C}}}$-module, there exists some $l>0$ such that $\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}})=\mathcal{D}_{X_{\mathbb{C}}}\cdot(Q_{1}\cdot Q_{r})^{-l}$ . Therefore we may obtain an $R$-model by taking the sheaf \[ \mathcal{D}_{X_{R}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l}\subset\mathcal{H}_{Y_{R}}^{r}(\mathcal{O}_{X_{R}}) \] After base change to $F=\text{Frac}(R)$ this agrees with ${\displaystyle \mathcal{H}^{r}(\int_{\varphi}j'_{\star}\mathcal{O}_{U_{R}})}$; therefore the two models agree after possibly localizing $R$. In particular, $\mathcal{M}_{Y}^{-\infty}$ is the $p$-adic completion of $\mathcal{D}_{X_{W(k)}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l}$. Now let us turn to the \begin{proof} (of \corref{Canonical-Singularities}) Let $\mathcal{I}_{\mathbb{C}}=(Q_{1},\dots,Q_{r})$ and let us fix a rational number $0<c<r$. Suppose that the ideal $\mathcal{J}(\mathcal{I}_{\mathbb{C}}^{c})\subsetneq\mathcal{O}_{X_{\mathbb{C}}}$. We spread everything out over $R$, and reduce to $k$ of large positive characteristic. Then the above implies $\tau(\mathcal{I}_{k}^{c})\subsetneq\mathcal{O}_{X_{k}}$. Now, recall that we have fixed an $R$-model $\mathcal{D}_{X_{R}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l}$ of $\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}})$. Then the description of the Hodge filtration in \propref{Hodge-for-local-coh!} implies that $F^{*}(F_{0}(\mathcal{D}_{X_{R}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l}))\otimes_{R}k)$ is the image of $\mathcal{D}_{X_{R}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l}\otimes_{R}k$ in $\mathcal{H}_{Y_{k}}^{r}(\mathcal{O}_{X_{k}})$; in other words, the $\mathcal{D}_{X_{k}}^{(0)}$-submodule generated by $(Q_{1}\cdots Q_{r})^{-l}$. Thus the assumption $F_{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O_{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$ is equivalent to the statement \[ (Q_{1}\cdots Q_{r})^{-p}\in\mathcal{D}_{X_{k}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l} \] inside $\mathcal{H}_{Y_{k}}^{r}(\mathcal{O}_{X_{k}})$. Since $\mathcal{H}_{Y_{k}}^{r}(\mathcal{O}_{X_{k}})$ is the quotient of $\mathcal{O}_{X_{k}}[(Q_{1}\cdots Q_{r})^{-1}]$ by the submodule generated by $\{Q_{1}^{-1}\cdots\widehat{Q_{i}^{-1}}\cdots Q_{r}^{-1}\}_{i=1}^{r}$, which is contained in the $\mathcal{D}_{X_{k}}^{(0)}$-submodule generated by $(Q_{1}\cdots Q_{r})^{-l}$, we see that the assumption actually implies \[ (Q_{1}\cdots Q_{r})^{-p}\in\mathcal{D}_{X_{k}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{-l} \] inside $\mathcal{O}_{X_{k}}[(Q_{1}\cdots Q_{r})^{-1}]$. To use this, note that the map $(Q_{1}\cdots Q_{r})^{p}\cdot$ is a $\mathcal{D}_{X_{k}}^{(0)}$-linear isomorphism on $\mathcal{O}_{X_{k}}[(Q_{1}\cdots Q_{r})^{-1}]$. Thus we see \[ \mathcal{O}_{X_{k}}=\mathcal{D}_{X_{k}}^{(0)}\cdot(Q_{1}\cdots Q_{r})^{p-l}\subset F^{*}(\mathcal{I}^{r(p-l)})^{[1/p]} \] so that $(\mathcal{I}^{r(p-l)})^{[1/p]}=\mathcal{O}_{X_{k}}$ which implies $\tau(\mathcal{I}^{r(1-l/p)})=\mathcal{O}_{X_{k}}$. Taking $p$ large enough so that $r(1-l/p)>c$, we deduce $\tau(\mathcal{I}_{k}^{c})=\mathcal{O}_{X_{k}}$ (the test ideals form a decreasing filtration, by {[}BMS{]}, proposition 2.11); contradiction. Therefore in fact $\mathcal{J}(\mathcal{I}_{\mathbb{C}}^{c})=\mathcal{O}_{X_{\mathbb{C}}}$ for all $c\in(0,r)$ which is the statement. \end{proof} As a corollary of this argument, we have: \begin{cor} Suppose $r=1$ in the previous corollary (so that $\mathcal{I}=(Q)$). Then, under the assumption that $F^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O^{0}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$, we have that, for all $p>>0$, $\tau(Q^{(1-l/p)})=\mathcal{O}_{X_{k}}$. \end{cor} This says that, after reducing mod $p$, for $p>>0$ the $F$-pure threshold of $Q$ is $\geq1-l/p$. Recall that $l$ is any integer for which $Q^{-l}$ generates the $\mathcal{D}_{X_{\mathbb{C}}}$-module $j_{*}(\mathcal{O}_{U_{\mathbb{C}}})$; thus we may take $l$ to be the least natural number such that $b_{Q}(-l-t)\neq0$ for all $t\in\mathbb{N}$ (here $b_{Q}$ is the $b$-function for $Q$). In this language, this result was recently reproved (and generalized) in \cite{key-65}, by completely different techniques. To finish off this section, we'll spell out how the description of the Hodge filtration in \propref{Hodge-for-local-coh!} relates to the condition $F_{i}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))=O_{i}(\mathcal{H}_{Y_{\mathbb{C}}}^{r}(\mathcal{O}_{X_{\mathbb{C}}}))$ when $Y_{\mathbb{C}}$ is a hypersurface inside $X_{\mathbb{C}}$; $Y_{\mathbb{C}}=Z(Q)$. In this case, we get an intriguing description in terms of the behavior of $\mathcal{H}_{Y}^{1}(\mathcal{O}_{X})$ in mixed characteristic: \begin{cor} We have $F_{i}(\mathcal{H}_{Y}(\mathcal{O}_{X}))=O_{i}(\mathcal{H}_{Y}^{1}(\mathcal{O}_{X}))$ iff $p^{i}Q^{-(i+1)p}\in\mathcal{D}_{X_{W_{i+1}(k)}}^{(0)}\cdot Q^{-l}$ inside $\mathcal{H}_{Y_{W_{i+1}(k)}}^{1}(\mathcal{O}_{X_{W_{i+1}(k)}})$ for $p>>0$. \end{cor} \begin{proof} Applying the condition of \propref{Hodge-for-local-coh!}, we see that $F_{i}(\mathcal{H}_{Y}(\mathcal{O}_{X}))=O_{i}(\mathcal{H}_{Y}^{1}(\mathcal{O}_{X}))$ iff, for all $p>>0$, there exists some $g\in\widehat{\mathcal{H}_{\mathfrak{Y}}^{1}(\mathcal{O}_{\mathfrak{X}})}$, with $p^{i}g\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}\cdot Q^{-l}$ whose image, in $\mathcal{H}_{Y_{k}}^{1}(\mathcal{O}_{X_{k}})$ is $Q^{-(i+1)p}$. This holds iff there is some $g_{1}\in\widehat{\mathcal{H}_{\mathfrak{Y}}^{1}(\mathcal{O}_{\mathfrak{X}})}$ so that \[ Q^{-(i+1)p}=g+pg_{1} \] inside $\widehat{\mathcal{H}_{\mathfrak{Y}}^{1}(\mathcal{O}_{\mathfrak{X}})}$. But this is equivalent to \[ p^{i}Q^{-(i+1)p}=p^{i}g+p^{i+1}g_{1}=\Phi\cdot Q^{-l}+p^{i+1}g_{1} \] for some $\Phi\in\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$. This, in turn, is a restatement of the corollary. \end{proof} \section{\label{sec:Appendix:-an-Inectivity}Appendix: an Injectivity Result} In this appendix we give a proof of the following technical result used in \lemref{Hodge-filt-on-log}: \begin{lem} The natural map $\text{(\ensuremath{{\displaystyle j_{\star}\mathcal{O}_{\mathfrak{U}}}}})^{-\infty}|_{\mathfrak{V}}\to\widehat{(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}$ (where $\widehat{}$ denotes $p$-adic completion) is injective. \end{lem} Recall that $j_{\star}(\mathcal{O}_{\mathfrak{U}})$ was defined as the $\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0)}$-module locally generated by $x_{1}^{-1}\cdots x_{j}^{-1}$, where $x_{1}\cdots x_{j}$ is a local equation for the divisor $\mathfrak{D}\subset\mathfrak{X}$. \begin{proof} Let $\mathfrak{V}$ be an open affine. On $\mathfrak{V}$, the map in question is the $p$-adic completion of the inclusion ${\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}\to\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]$, where ${\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}$ is the $D_{\mathfrak{V}}^{(0)}$-submodule of $j_{*}(\mathcal{O}_{\mathfrak{V}})$ generated by $x_{1}^{-1}\cdots x_{j}^{-1}$. This is a map of $p$-torsion-free sheaves, let $\mathcal{C}$ denote its cokernel. Then the kernel of the completion is given by \[ \lim_{\leftarrow}\mathcal{C}[p^{n}] \] where $\mathcal{C}[p^{n}]=\{m\in\mathcal{C}|p^{n}m=0\}$, and the maps in the inverse system are multiplication by $p$. Now, both ${\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}$ and $\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]$ are filtered by the Hodge filtration; on ${\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}$ it is given by $F^{l}(\mathcal{D}_{\mathfrak{V}}^{(0)})\cdot(x_{1}^{-1}\cdots x_{r}^{-1})$, which is precisely the span over $\mathcal{O}_{\mathfrak{V}}$ of terms of the form $I!\cdot x_{1}^{-i_{1}-1}\cdots x_{j}^{-i_{j}-1}=\partial^{I}x_{1}^{-1}\cdots x_{j}^{-1}$ for $|I|\leq l$, here we have denoted $I!=i_{1}!\cdots i_{j}!$. The Hodge filtration $F^{l}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])$ is defined to be the span over $\mathcal{O}_{\mathfrak{V}}$ of terms of the form $x_{1}^{-i_{1}-1}\cdots x_{j}^{-i_{j}-1}$ for $|I|\leq l$. From this description it follows that, in both cases, all of the terms $F^{i}$ and $F^{i}/F^{i-1}$ are $p$-torsion-free; and the morphism is strict with respect the the filtrations; i.e., \[ F^{i}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])\cap{\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}=F^{i}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}}) \] Now we consider the inclusion $\mathcal{R}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}})\to\mathcal{R}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])$ (where $\mathcal{R}$ stands for the Rees functor with respect to the Hodge filtrations on both sides). The strictness of the map implies \[ \text{coker}(\mathcal{R}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}})\to\mathcal{R}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]))=\mathcal{R}(\mathcal{C}) \] We now shall show that the $p$-adic completion\footnote{In this appendix only, we use the completion of the \emph{entire} Rees module, NOT the graded completion of the rest of the paper; similarly, the product is the product in the category of all modules, not the category of graded modules} of this map is injective. The natural map \[ \mathcal{R}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}})=\bigoplus_{i=0}^{\infty}F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}}\to\prod_{i=0}^{\infty}F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}} \] is injective, and the cokernel is easily seen to be $p$-torsion-free; therefore we obtain an injection \[ \widehat{\mathcal{R}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}})}\to\widehat{\prod_{i=0}^{\infty}F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}}} \] and the analogous statement holds for $\mathcal{R}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])$. Further, one has an isomorphism \[ \widehat{\prod_{i=0}^{\infty}F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}}}\tilde{=}\prod_{i=0}^{\infty}(\widehat{F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}})}=\prod_{i=0}^{\infty}F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}} \] where the last equality is because $F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}}$ is a coherent $\mathcal{O}_{\mathfrak{V}}$-module and therefore $p$-adically complete; similarly \[ \widehat{\prod_{i=0}^{\infty}F^{i}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]}\tilde{=}\prod_{i=0}^{\infty}(\widehat{F^{i}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])}=\prod_{i=0}^{\infty}F^{i}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}] \] So, since each $F^{i}({\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})^{\text{fin}}}\to F^{i}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}]$ is injective, we obtain an injection \[ \widehat{\mathcal{R}({\displaystyle {\displaystyle (j_{\star}\mathcal{O}_{\mathfrak{V}})}}^{\text{fin}})}\to\widehat{\mathcal{R}(\mathcal{O}_{\mathfrak{V}}[x_{1}^{-1}\cdots x_{j}^{-1}])} \] This means that ${\displaystyle \lim_{\leftarrow}\mathcal{R}(\mathcal{C})[p^{n}]}=0$. Let $f$ denote the parameter in the Rees ring. Then, for each $n$ we have a short exact sequence \[ \mathcal{R}(\mathcal{C})[p^{n}]\xrightarrow{f-1}\mathcal{R}(\mathcal{C})[p^{n}]\to C[p^{n}] \] Since ${\displaystyle \lim_{\leftarrow}\mathcal{R}(\mathcal{C})[p^{n}]}=0$, to prove ${\displaystyle \lim_{\leftarrow}\mathcal{C}[p^{n}]=0}$ we must show that $f-1$ acts injectively on ${\displaystyle \text{R}^{1}\lim_{\leftarrow}\mathcal{R}(\mathcal{C})[p^{n}]}$. Recall that this module is the cokernel of \[ \eta:\prod_{n=1}^{\infty}\mathcal{R}(C)[p^{n}]\to\prod_{n=1}^{\infty}\mathcal{R}(C)[p^{n}] \] where $\eta(c_{1},c_{2},c_{3},\dots)=(c_{1}-pc_{2},c_{2}-pc_{3},\dots)$. Now, since each $\mathcal{R}(C)[p^{n}]$ is graded, we may define a homogenous element of degree $i$ in ${\displaystyle \prod_{n=1}^{\infty}\mathcal{R}(C)[p^{n}]}$ to be an element $(c_{1},c_{2},\dots)$ such that each $c_{j}$ has degree $i$. Any element of $d\in{\displaystyle \prod_{n=1}^{\infty}\mathcal{R}(C)[p^{n}]}$ has a unique representation of the form ${\displaystyle \sum_{i=0}^{\infty}d_{i}}$ where $d_{i}$ is homogenous of degree $i$ (this follows by looking at the decomposition by grading of each component). Since the map $\eta$ preserves the set of homogenous elements of degree $i$, we have ${\displaystyle \eta(\sum_{i=0}^{\infty}d_{i})=\sum_{i=0}^{\infty}\eta(d_{i})}$. Suppose that $(f-1)d=\eta(d')$. Write ${\displaystyle d=\sum_{i=j}^{\infty}d_{i}}$ where $d_{j}\neq0$. Then \[ (f-1){\displaystyle \sum_{i=j}^{\infty}d_{j}}=-d_{j}+\sum_{i=j+1}^{\infty}(fd_{i-1}-d_{i})=\sum_{i=0}^{\infty}\eta(d_{i}') \] So we obtain $d_{j}=-\eta(d_{j}')$, and $d_{i}=fd_{i-1}+\eta(d_{i}')$ for all $i>j$, which immediately gives $d_{i}\in\text{image}(\eta)$ for all $i$; so $d\in\text{image }(\eta)$ and $f-1$ acts injectively on $\text{coker}(\eta)$ as required. \end{proof} The University of Illinois at Urbana-Champaign, [email protected] \end{document}

\begin{document} \title{Forman-Ricci curvature and Persistent homology of unweighted complex networks} \author{Indrava Roy} \email{Correspondence to: [email protected]} \affiliation{The Institute of Mathematical Sciences (IMSc), Homi Bhabha National Institute (HBNI), Chennai 600113 India} \author{Sudharsan Vijayaraghavan} \affiliation{Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore 641004 India} \author{Sarath Jyotsna Ramaia} \affiliation{Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore 641004 India} \author{Areejit Samal} \email{Correspondence to: [email protected]} \affiliation{The Institute of Mathematical Sciences (IMSc), Homi Bhabha National Institute (HBNI), Chennai 600113 India} \affiliation{Max Planck Institute for Mathematics in the Sciences, Leipzig 04103 Germany} \begin{abstract} We present the application of topological data analysis (TDA) to study unweighted complex networks via their persistent homology. By endowing appropriate weights that capture the inherent topological characteristics of such a network, we convert an unweighted network into a weighted one. Standard TDA tools are then used to compute their persistent homology. To this end, we use two main quantifiers: a local measure based on Forman's discretized version of Ricci curvature, and a global measure based on edge betweenness centrality. We have employed these methods to study various model and real-world networks. Our results show that persistent homology can be used to distinguish between model and real networks with different topological properties. \end{abstract} \maketitle \section{Introduction} Recent advances in topological data analysis (TDA) \cite{Zomorodian2005,Edelsbrunner2008,Carlsson2009} have made it a powerful tool in data science. TDA has lead to important applications in different areas of science. For example, in astrophysics, TDA has be used for analysis of the Cosmic Microwave Background (CMB) radiation data \cite{Pranav2016}; in imaging, TDA has been used for feature detection in 3D gray-scale images \cite{Gunther2011}; in biology, TDA has been used for detection of breast cancer type with high survival rates \cite{Nicolau2011} and understanding cell fate from single-cell RNA sequencing data \cite{Rizvi2017}. The main tool in TDA is that of \textit{persistent homology} \cite{Zomorodian2005,Edelsbrunner2008,Carlsson2009}, which has the power to detect the topology of the underlying data. The field of algebraic topology \cite{Munkres2018} provides the basic mathematical tool required for TDA, namely that of homology. The conceptual roots of persistent homology, however, are in \textit{differential} topology, in particular Morse theory \cite{Edelsbrunner2008}. Network science \cite{Watts1998,Barabasi1999,Albert2002,Newman2010,Barabasi2016}, on the other hand, investigates the topological and dynamical properties of various complex networks, that encode interactions between various agents in the natural as well as artificial setting. The ability to understand and predict the nature of these interactions is a key challenge. Historically, graph theory \cite{Bollobas1998,Newman2010,Barabasi2016} has provided the main tools and techniques for studying such networks, via their graph representation. Although graph theory has provided significant insights into such problems, recent studies have shown that such techniques do not adequately capture higher-order interactions and correlations arising in networks \cite{De2007,Horak2009,Petri2013,Petri2014,Bianconi2015, Wu2015,Sizemore2016,Courtney2017,Ritchie2017,Courtney2018,Kartun-Giles2019,Iacopini2019,Kannan2019}. These higher-order phenomena can be encoded in \textit{hypergraph} \cite{Klamt2009,Zlatic2009} and \textit{simplicial complex} \cite{De2007,Horak2009,Lee2012,Petri2013,Petri2014,Sizemore2016,Iacopini2019} representations of networks. The tools of TDA are applicable to any simplicial complex and can be used to determine the important topological characteristics of networks. In this work, we employ TDA to study the persistent homology of unweighted and undirected simple graphs arising from model and real-world networks. Previous research in this direction have investigated the persistent homology of weighted and undirected networks \cite{Petri2013,Petri2014}. The filtration scheme required to compute persistent homology in weighted networks was then provided by the edge weights \cite{Petri2013,Petri2014}. However, this technique is not immediately applicable to unweighted graphs due to the absence of edge weights. At present, due to insufficient information, the interaction networks underlying many real-world complex systems are available only as unweighted and undirected graphs. Examples of such unweighted and undirected real networks include the Yeast protein interaction network \cite{Jeong2001}, the US Power Grid network \cite{Leskovec2007} and the Euro road network \cite{Subelj2011}. In order to reveal the higher-order topological features of such real-world networks, it is important to develop methods to study persistent homology in unweighted and undirected networks. A simple way to devise such a method would be to transform the given unweighted graph into an edge-weighted graph by assigning certain weights to all edges, and then, using the induced filtration to compute persistent homology. However, \textit{a priori} it is not evident which edge weighting scheme would capture the topological characteristics of different types of unweighted networks. Previously, Horak \textit{et al.} \cite{Horak2009} used a dimension-based weighting scheme for unweighted networks where the weights are simply the dimension of the simplices. In particular, Horak \textit{et al.} assign all edges with the weight $+1$ to study persistent homology in unweighted networks. However, we have recently shown that the dimension-based filtration scheme of Horak \textit{et al.}, though computationally fast, may not be able to conclusively distinguish between various model networks \cite{Kannan2019}. In recent work \cite{Kannan2019}, we gave another weighting method based on a \textit{discrete Morse function} as introduced by Robin Forman \cite{Forman1998,Forman2002}, which assigns weights to each simplex in the clique complex corresponding to the unweighted graph according to a global acyclicity constraint. This method \cite{Kannan2019} simplifies the topological structure of the underlying simplicial complex, that leads to a computationally efficient way to compute homology and persistent homology. Moreover, we also showed that the persistent homology computed using this method was able to distinguish various unweighted model networks having different topological characteristics, the difference being quantified by the averaged bottleneck distance between the corresponding persistence diagrams \cite{Kannan2019}. A natural question then is whether other choices of weights can also be used to distinguish such unweighted networks via persistent homology. In the present contribution, we shall use both local and global network quantifiers for obtaining edge weighting schemes to compute persistent homology, namely that of \textit{discrete Ricci curvature} \cite{Forman2003, Sreejith2016,Sreejith2017,Samal2018}, also introduced by R. Forman \cite{Forman2003}, which plays the role of local curvature in a discrete setting, and \textit{edge betweenness centrality} \cite{Freeman1977,Girvan2002, Newman2010} which is an edge-based measure analogous to the classical betweenness centrality for vertices of a graph. We shall show that the simpler methods introduced here to study persistent homology based on Forman-Ricci curvature or edge betweenness centrality are also able to distinguish unweighted model networks like our recent method \cite{Kannan2019} based on discrete Morse functions. However, note that the advantages in topological simplification and computational efficiency that result from using a discrete Morse function are lost with the simpler method presented here. Nevertheless, if the sole goal is to compute persistent homology in unweighted networks, the weighting schemes presented here are likely to be much simpler to use in practice. In this context, we have also applied our methods to study the persistent homology of some real-world networks. Note that our recent method based on discrete Morse functions \cite{Kannan2019} and the simpler methods presented here based on Forman-Ricci curvature or edge betweenness centrality are much better at distinguishing between different types of model networks in comparison to dimension-based method of Horak \textit{et al.} \cite{Horak2009}. The remainder of the paper is organized as follows. In the Theory section, we present an overview of the concepts needed to study persistent homology in unweighted networks based on Forman-Ricci curvature and edge betweenness centrality. In the Datasets section, we describe the model and real networks analyzed here. In the Results section, we describe our new methods to study persistent homology in unweighted networks, and its application to both model and real-world networks. In the last section, we conclude with a brief summary and future outlook. \begin{figure*} \caption{Schematic figure illustrating our method to study persistent homology in an unweighted and undirected network using Forman-Ricci curvature. (a) An example of an unweighted graph $G$. (b) Transformation of the unweighted graph into an edge-weighted graph using Forman-Ricci curvature. (c) Assignment of normalized filtration weights to edges in the weighted graph shown in (b) based on Forman-Ricci curvature. (d) Weighted clique simplicial complex $K$ corresponding to the unweighted graph $G$. (e) Assignment of normalized filtration weights to vertices ($0$-simplices) and $2$-simplices in the weighted clique complex shown in (d) based on edge weights. (f) Filtration of the weighted clique complex $K$ based on the ascending sequence of weights assigned to simplices. Barcodes depict that there is a $0$-hole (or connected component) that persists across the 6 stages of the filtration while another $0$-hole is born at the last stage on addition of the isolated vertex $v_9$. Moreover, a $1$-hole is born at stage 4 on addition of the edge $[v_3,v_4]$.} \label{schemfig} \end{figure*} \section{Theory} \subsection{Clique complex of a graph} Let $G(V,E)$ be a finite simple graph with $V$ being the set of vertices and $E$ being the set of edges. Each edge in the graph $G$ is an unordered pair of distinct vertices. We remark that a simple graph does not contain self-loops or multi-edges \cite{Bollobas1998}. An induced subgraph $K$ of $G$ that is complete is called a \textit{clique}. We can view $G$ as a finite clique simplicial complex $K$ where a $p$-dimensional simplex (or $p$-simplex) is determined by a set of $p+1$ vertices that form a clique \cite{Zomorodian2005,Edelsbrunner2008}. Specifically, a $p$-simplex is a polytope which is the convex hull of its $p+1$ vertices. Note that a \textit{simplex} can be thought of as a generalization of points, lines, triangles, tetrahedron, and so on in higher dimensions. In the clique complex $K$, $0$-simplices correspond to vertices in $G$, $1$-simplices to edges in $G$, $2$-simplices to triangles in $G$, and so on. Given a $p$-simplex $\alpha$ in $K$, a \textit{face} $\gamma$ of $\alpha$ is determined by a subset of the vertex set of $\alpha$ of cardinality less than or equal to $p+1$. Dually, a \textit{co-face} $\beta$ of $\alpha$ is a simplex that contains $\alpha$ as a face. The dimension of a clique simplicial complex $K$ is the maximum dimension of its constituent simplices. An orientation of a $p$-simplex is given by an ordering of its constituent vertices \cite{Munkres2018}. Moreover, two orientations of a simplex are equivalent if they differ by an even permutation of its vertices. \subsection{Persistent homology of a simplicial complex} A simplicial complex is a collection $K$ of simplices which satisfies following two properties \cite{Munkres2018}. Firstly, any face $\gamma$ of a simplex $\alpha$ in $K$ is also included in $K$. Secondly, if two simplices $\alpha$ and $\beta$ in $K$ have a non-empty intersection $\gamma$, then $\gamma$ is a common face of $\alpha$ and $\beta$. A subcomplex $K'$ of a simplicial complex $K$ is a collection of simplices in $K$ such that $K'$ is also a simplicial complex. A \textit{filtration} on a simplicial complex $K$ is given by a nested sequence of subcomplexes $K_i$, $i=0,1,\ldots,n$, such that: \begin{equation*} \emptyset=K_0\subseteq K_1\subseteq \ldots \subseteq K_n=K. \end{equation*} For a simplicial complex $K$ with a given filtration, one can define its persistent homology groups as follows. First we fix a base field $\mathbb{F}$ \cite{Munkres2018}. The set of all oriented $p$-simplices in $K$ generate a free group $C_p$ over $\mathbb{F}$, called $p^{\text{th}}$-chain group \cite{Munkres2018}. An element in $C_p$ is called a $p$-chain, and is given by a finite formal sum: \begin{equation*} C_p =\sum_{i=1}^{N} c_i\alpha_i \end{equation*} where the coefficients $c_i$ are in $\mathbb{F}$, and $\alpha_i$ are oriented $p$-simplices in $K$ \cite{Munkres2018}. Component-wise addition endows $C_p$ with the structure of a group, whose identity element is given by the unique $p$-chain with all coefficients $c_i$ equal to zero. If a $p$-simplex $\alpha$ is given an opposite orientation, then it is represented as $-\alpha$ in $C_p$, and gives the inverse of $\alpha$ in $C_p$. To define the persistent homology groups, we use the so-called boundary operator $\partial_p$, which is a map $\partial_p: C_p\rightarrow C_{p-1}$. For an oriented $p$-simplex $\alpha = [x_0,x_1,\ldots,x_p]$ (i.e., the ordered vertex set $\{x_0, x_1,\ldots,x_p\}$ of $\alpha$), we define the boundary operator $\partial_p$ as: \begin{equation*} \partial_p(\alpha)=\sum_{i=0}^{p}(-1)^i[x_0,\ldots,\hat{x}_i, \ldots,x_p] \end{equation*} where $[x_0,\ldots,\hat{x}_i,\ldots,x_p]$ denotes the $(p-1)$-face of $\alpha$ obtained by removing the vertex $x_i$ \cite{Munkres2018}. Since the right hand side of the above equation is a linear combination of $(p-1)$-simplices, it belongs to $C_{p-1}$. One can then extend the definition of $\partial_p$ to all elements of $C_p$ by linearity. The boundary operators satisfy the fundamental property: \begin{equation*} \partial_p\circ \partial_{p+1}=0. \end{equation*} The kernel of the boundary operator is called the group of $p$-cycles and denoted by $Z_p$ \cite{Munkres2018}. It is given by the set of elements in $C_p$ that is mapped to $0$ in $C_{p-1}$ by the boundary operator $\partial_p$: \begin{equation*} Z_p =\operatorname{Ker}(\partial_p)=\{c\in C_p|\partial_p(c)=0\}. \end{equation*} A $p$-boundary is a $p$-cycle which lies in the image of the boundary operator $\partial_{p+1}$. The set of $p$-boundaries is denoted by $B_p$ and is a subgroup of $Z_p$ \cite{Munkres2018}. \begin{equation*} B_p =\operatorname{Image}(\partial_{p+1})=\{c\in C_p| \exists b \in C_{p+1}, \partial_{p+1}(b)=c\}. \end{equation*} Thus, the $p$-homology group is defined as \cite{Munkres2018}: \begin{equation*} H_p(K)=\frac{Z_p(K)}{B_p(K)}. \end{equation*} Note that $H_p$ is a vector space over the field $\mathbb{F}$. The $p$-Betti number $\beta_p$ is given by the dimension of the homology group $H_p$. Informally, $\beta_p$ represents the number of holes in the $p$-homology group. Now, every subcomplex $K_i$ in the filtration of the simplicial complex $K$ has an index $i$ associated with it. Also, for each $K_i$ there exists its corresponding $p$-chain, $p$-boundary operators, and thus, $p$-boundaries and $p$-cycles. We shall denote the $p$-cycles of $K_i$ as $Z^i_p$ and the $p$-boundaries of $K_i$ as $B^i_p$. The $j$-persistent $p$-homology of $K_i$ is defined as \cite{Zomorodian2005,Edelsbrunner2008}: \begin{equation} H^{i,j}_p = \frac{Z^i_p}{(B^{i+j}_p \cap Z^i_p)} \end{equation} and the corresponding $j$-persistent $p$-Betti number as: \begin{equation} \beta^{i,j}_p = \operatorname{dim}(H^{i,j}_p). \end{equation} A $p$-homology class $\alpha$ is \textit{born} at $K_i$ if it is not in the image of the map induced on $p$-homology by the inclusion $K_{i-1} \subseteq K_i$ . Furthermore, if $\alpha$ is born at $K_i$, we say that it \textit{dies} entering $K_{i+j}$, if it becomes the boundary of a $(p+1)$-chain in $K_{i+j}$. The persistent homology group $H^{i,j}_p$ thus encodes information of $p$-homology classes that are born at the filtration index $i$ and survive until the index $i+j$. Each $p$-hole across the filtration can be characterized by its birth and death. By studying persistent homology, the persistence of such holes can be quantified, thus revealing the importance of the corresponding topological features across the filtration. \subsubsection{Filtration of a weighted simplicial complex} Let $K$ be a simplicial complex, endowed with a set of numerical numbers called \textit{weights} associated with its simplices, i.e. to each of its constituent simplices $\alpha$ is assigned a number $w(\alpha)$. To study the persistent homology of such simplicial complexes, we can consider the following filtration on $K$ \cite{Zomorodian2005,Edelsbrunner2008}. Given a real number $r$, we define the subcomplex $K(r)$ as: \begin{equation} K(r)= \bigcup_{\{\alpha: w(\alpha)\leq r\}} \bigcup_{\beta\leq \alpha} \beta \end{equation} In simple terms, $K(r)$ is the smallest simplicial subcomplex of $K$ containing simplices which have weight less than or equal to $r$. Note that all faces $\beta$ of an simplex $\alpha$ of weight less than $r$ are admitted to this subcomplex, irrespective of the weight of $\beta$. In the particular case of a finite simplicial complex $K$ with simplices $\alpha_i, i=1,2,\ldots,n$, we can arrange the corresponding weights $w(\alpha_i)$ in ascending order, say $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Then the associated filtration of $K$ is given by: \begin{equation} \label{filtration} \emptyset\subseteq K(\lambda_1)\subseteq K(\lambda_2)\subseteq \ldots \subseteq K(\lambda_n)=K \end{equation} Throughout this work, we shall consider the particular case of a finite weighted simplicial complex arising from a finite \textit{edge-weighted} simple graph, i.e. a graph with weights assigned to all its edges. The clique complex of such an edge-weighted graph already has weights on its $1$-simplices corresponding to its edges, and we can extend this weighting scheme to any $0$-simplex $\beta$ or $2$-simplex $\sigma$ by defining their weights $w(\beta)$ and $w(\sigma)$ by the following \textit{min/max} formulae: \begin{eqnarray} w(\beta)= \min\{ w(\alpha) : \alpha \text{ is a 1-dimensional co-face of } \beta\}\ \text{and} \nonumber \\ w(\sigma)= \max\{ w(\alpha) : \alpha \text{ is a 1-dimensional face of } \sigma\}. \label{simplexweight} \end{eqnarray} Weights of higher-dimensional simplices can be defined in a similar way. Note that with these weights, we enforce the following conditions. Any vertex of an edge $e$ that is included in a subcomplex containing $e$ has weight less than or equal to $e$. Moreover, if a collection of edges forms a higher-dimensional simplex $\gamma$, then $\gamma$ is included in a subcomplex that includes the edge with the maximum weight. With respect to the filtration induced by this weighting scheme arranged in increasing order $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$, one can now compute the persistent homology groups of $K$ as defined above. The \textit{persistence} of a class in $p$-homology that is born at the $i^{\text{th}}$-stage of the filtration and dies at the $j^{\text{th}}$-stage is then defined to be $\lambda_j- \lambda_i$, where $\lambda_i$ is the weight associated to the $i^{\text{th}}$-subcomplex $K(\lambda_i)$ as above. \subsubsection{Barcode diagrams} The $p^{\text{th}}$-barcode diagram for a given filtration of a finite simplicial complex $K$ gives a graphical summary of the birth and death of $p$-holes across the filtration \cite{Ghrist2008}. In this work, the x-axis of the $p^{\text{th}}$-barcode diagram corresponds to the filtration weights of $p$-simplices in $K$; the filtration weights have been normalized to lie in the range 0 to 1. A horizontal line in the $p^{\text{th}}$-barcode diagram of $K$ is referred to as a barcode. A barcode that begins at a x-axis value of $w_i$ and ends at a x-axis value of $w_j$ represents a $p$-hole in $K$ whose birth and death weights are $w_i$ and $w_j$, respectively. The number of barcodes between $w_i$ and $w_j$ in the diagram is precisely the $p$-Betti number $\beta^{i,j-i}$, i.e., the dimension of the persistent homology group $H^{i,j-i}_p$. \subsubsection{Persistence diagrams and bottleneck distance between them} Given two multisets $X$ and $Y$ in $\mathbb{R}^2$, the $\infty$-Wasserstein distance or bottleneck distance between them is defined as: \begin{equation*} \label{botdist} W_\infty(X,Y) = \inf_{\eta:X \rightarrow Y} \text{sup}_{x \in X} || x - \eta(x) ||_{\infty}. \end{equation*} In the above equation, the supremum is taken over all bijections $\eta:X\rightarrow Y$ (with the convention that a point with multiplicity $k \in \mathbb N$ is considered as $k$ individual points) and for $(a,b) \in \mathbb{R}^2$, $||(a,b)||_\infty:= \max\{|a|, |b|\}$ \cite{DiFabio2015}. Given a filtration of a weighted simplicial complex $K$ with weights $w_i, i= 1, 2, \cdots, n$, the $p^{\text{th}}$-persistence diagram $D^pK$, is defined as follows. Consider the multiset of points $W^pK:=\{(w_i, w_j): w_i<w_j, i, j = 1, 2, \cdots, n\}$ with each point $(w_i, w_j)$ endowed with the multiplicity $\mu_p(w_i, w_j)$ given by \cite{DiFabio2015}: \begin{equation*} \mu_p(w_i, w_j) := \lim_{\epsilon \rightarrow 0^+} (\beta_{w_i+\epsilon}^{w_j-\epsilon} - \beta_{w_i+\epsilon}^{w_j+\epsilon} + \beta_{w_i-\epsilon}^{w_j+\epsilon} - \beta_{w_i-\epsilon}^{w_j-\epsilon}) \end{equation*} where $\beta_x^y$ is the dimension of the image of the induced map in $p$-homology from $K(x)$ to $K(y)$ for $x, y \in \mathbb R$ with $x<y$. Denote by $\Delta$ the diagonal in $\mathbb R^2$ considered as a multiset with infinite multiplicity given to each of its points. The persistence diagram $D^pK$ is the subset of $W^p_K \cup \Delta$ consisting of points $(u, v)$ with $\mu_p(u, v)>0$. In this work, we consider the total persistence diagram \cite{Cohen-Steiner2007} given by the union of all $D^pK$ for $0 \leq p \leq \operatorname{dim}{K}$. Thereafter, we consider the bottleneck distance \cite{Cohen-Steiner2007} between total persistence diagrams considered as multisets in $\mathbb{R}^2$. \subsection{Forman-Ricci curvature} In previous work \cite{Sreejith2016,Sreejith2017,Samal2018}, some of us have ported a discretization of the classical notion of Ricci curvature due to Robin Forman \cite{Forman2003} to graphs or networks. Briefly, in Riemannian geometry, curvature measures the amount of deviation of a smooth Riemannian manifold from being Euclidean. The Ricci curvature tensor quantifies the dispersion of geodesic lines in the neighbourhood of a given tangential direction as well as volume growth of metric balls. Forman \cite{Forman2003} has proposed a discretization of the classical Ricci curvature based on the \textit{Bochner-Weitzenb\"{o}ck formula} which measures the difference between the \textit{Laplace-Beltrami operator} and the \textit{connection Laplacian} \cite{Jost2017}. Forman's discretized version of the Ricci curvature is applicable to a large class of topological objects, namely, \textit{weighted $CW$-complexes} which includes graphs and simplicial complexes \cite{Forman2003,Sreejith2016}. Starting from a graph or network, one may construct a two-dimensional polyhedral complex by inserting a solid triangle into any connected triple of vertices or cycle of length 3, a solid quadrangle into a cycle of length 4, a solid pentagon into a cycle of length 5, and so on. The mathematical definition of Forman-Ricci curvature \cite{Forman2003} for general weighted $CW$-complexes is also applicable to such a two-dimensional polyhedral complex constructed from a graph, and is given by: \begin{equation*} {\rm F} (e) = w_e \left[ \sum_{e < f} \frac{w_e}{w_f}+\sum_{v < e} \frac{w_v}{w_e} \right. - \left. \sum_{\hat{e} \parallel e} \left| \sum_{\hat{e},e < f} \frac{\sqrt{w_e \cdot w_{\hat{e}}}}{w_f} - \sum_{v < e, v < \hat{e}} \frac{w_v}{\sqrt{w_e \cdot w_{\hat{e}}}} \right| \right] \; ; \end{equation*} where $w_e$ denotes the weight of edge $e$, $w_v$ denotes the weight of vertex $v$, $w_f$ denotes the weight of face $f$, $\sigma < \tau$ means that $\sigma$ is a face of $\tau$, and $||$ signifies \textit{parallelism}, i.e. the two cells have a common \textit{parent} (higher dimensional co-face) or a common \textit{child} (lower dimensional face), but not both a common parent and common child \cite{Samal2018}. For the particular case of restricted two-dimensional complexes containing only triangular faces $t$ while ignoring faces consisting of more than 3 vertices, the above equation simplifies to \cite{Samal2018}: \begin{equation} \label{AugmentedFormanRicciEdge} {\rm F} (e) = w_e \left[ \sum_{e < t} \frac{w_e}{w_t} + (\frac{w_{v_1}}{w_e} + \frac{w_{v_2}}{w_e}) \right. - \left. \sum_{e_{v_1},e_{v_2}\ \sim\ e,\ e_{v_1},e_{v_2}\ \nless\ t} \left( \frac{w_{v_1}}{\sqrt{w_e \cdot w_{e_{v_1}}}} + \frac{w_{v_2}}{\sqrt{w_e \cdot w_{e_{v_2}}}} \right) \right] \; ; \end{equation} where $w_e$ is the weight of the edge $e$ under consideration, $w_{v_1}$ and $w_{v_2}$ denote the weights associated with the vertices $v_1$ and $v_2$, respectively, which anchor the edge $e$ under consideration. In the above equation, $e_{v_1} \sim e$ and $e_{v_2} \sim e$ denote the set of edges incident on vertices $v_1$ and $v_2$, respectively, after \textit{excluding} the edge $e$ under consideration which connects the two vertices $v_1$ and $v_2$. While computing the Forman-Ricci curvature of an edge in an unweighted graph $G$, we substitute in the above equation $w_t = w_e = w_v = 1, \; \forall\ t \in T(G), e \in E(G), v \in V(G)$, where $T(G)$, $E(G)$ and $V(G)$ represent the set of triangular faces, edges and vertices, respectively. Note that the above definition (Eq. \ref{AugmentedFormanRicciEdge}) of the Forman-Ricci curvature of an edge or $1$-simplex in the restricted two-dimensional complex constructed from a graph was referred to as \textit{augmented} Forman-Ricci curvature in earlier contributions \cite{Samal2018,Saucan2019}. For brevity, we here refer to the quantity defined in Eq. \ref{AugmentedFormanRicciEdge} as Forman-Ricci curvature of an edge. From a geometric point of view, the Forman-Ricci curvature quantifies the information spread at the ends of edges in a network. Higher information spread at the ends of an edge implies more negative value for its Forman-Ricci curvature. In this work, we employ Forman-Ricci curvature of an edge (given by Eq. \ref{AugmentedFormanRicciEdge}) to transform an unweighted graph into a weighted graph which captures the local curvature properties (Figure \ref{schemfig}). \subsection{Edge Betweenness Centrality} Edge betweenness centrality \cite{Freeman1977,Girvan2002,Newman2010} quantifies the importance of edges for global information flow in networks. For any edge $e$, this measure is computed based on the number of shortest paths between different pairs of vertices in the network that contain the considered edge $e$. Formally, in a graph $G(V,E)$, the edge betweenness centrality of an edge $e \in E$ is given by: \begin{equation} \label{EBC} {\rm EBC} (e) = \sum_{v_i} \sum_{v_j, v_j \neq v_i} \frac{\sigma_{v_i v_j}(e)}{\sigma_{v_i v_j}} \end{equation} where $\sigma_{v_i v_j}$ gives the number of shortest paths between vertices $v_i$ and $v_j$ in the network and $\sigma_{v_i v_j}(e)$ gives the number of shortest paths between vertices $v_i$ and $v_j$ in the network that contain the considered edge $e$. Note that an edge with a high edge betweenness centrality is critical for maintaining information flow in the network. \section{Datasets} The proposed method for studying persistent homology in unweighted networks has been investigated in different network models, namely, Erd\"{o}s-Renyi (ER) model \cite{Erdos1961}, Watts-Strogatz (WS) model \cite{Watts1998}, Barab\'{a}si-Albert (BA) model \cite{Barabasi1999}, and the Hyperbolic Graph Generator (HGG) \cite{Krioukov2010}. We give brief descriptions of each below. \begin{itemize} \setlength\itemsep{0em} \item \textit{ER model}: The ER model has two parameter $n$ and $p$, where $n$ is the number of vertices and $p$ is the probability for the existence of an edge between distinct pairs of vertices. ER graph is obtained by starting with a set of vertices and connecting a distinct pair of vertices by an edge with probability $p$. The presence of an edge between any two pairs of vertices is independent of the other edges. \item \textit{WS model}: The WS model can be characterized by three parameters: $n$, the number of vertices; $k$, the number of neighbours the vertex has before rewiring; and $p$, the rewiring probability. The construction of the WS graph begins with a graph with $n$ vertices where each vertex has $k$ nearest neighbours. Thereafter, the endpoints of each edge is chosen randomly based on the rewiring probability and it is rewired to another randomly chosen vertex with uniform probability. \item \textit{BA model}: The BA model generates a scale-free graph with $n$ vertices by satisfying the so-called preferential attachment condition. Under preferential attachment scheme, at each iteration step, the graph expansion takes place by addition of a new vertex with $m$ edges to existing vertices in such a way that existing vertices with higher degree have more probability to gain additional edges to the new vertex than the vertices with lower degree. In the BA model, the probability of connecting the new vertex to an existing vertex is directly proportional to the degree of that vertex at that time. The BA model generates graphs with power-law degree distribution \cite{Barabasi1999}. \item \textit{Hyperbolic random graphs}: The input parameters of HGG are the number of vertices $n$, the target average degree $k$, the target exponent $\gamma$ of the power-law degree distribution and temperature $T$. For the construction of a hyperbolic random graph, HGG scatters $n$ vertices on a hyperbolic space and the existence of an edge between the vertices is based on a probability value, which is given by a function of the hyperbolic distance between the vertices. The vertex degree distribution in the hyperbolic random graphs produced by HGG follow a power-law. The HGG generates a hyperbolic random graph for $\gamma=[0,\infty)$. \item \textit{Spherical random graphs}: Similar to the hyperbolic random graph, a spherical random graph can be constructed using HGG by scattering $n$ vertices on a sphere of radius $R$, and the probability for existence of an edge between two vertices is a function of the spherical distance between the vertices. The HGG model produces a spherical random graph for $\gamma=\infty$. \end{itemize} In addition to model networks, the proposed method has also been studied in the following seven real-world networks. \begin{itemize} \setlength\itemsep{0em} \item \textit{Yeast protein interaction} network \cite{Jeong2001} with 1870 vertices representing proteins and 2277 edges signifying protein-protein interactions. \item \textit{Human protein interaction} network \cite{Rual2005} with 3133 vertices representing proteins and 6726 edges signifying protein-protein interactions. \item \textit{US Power Grid} network \cite{Leskovec2007} with 4941 vertices representing generators, transformers and substations in the Western states of USA and the 6594 edges signifying power links between them. \item \textit{Euro road} network \cite{Subelj2011} with 1174 vertices corresponding to cities in Europe and the 1417 edges signifying roads linking the cities. \item \textit{Email} network \cite{Guimera2003} with 1133 vertices representing users in the University of Rovira i Virgili and 5451 edges signifying the existence of at least one Email communication between pairs of users. \item \textit{Route views} network \cite{Leskovec2007} with 6474 vertices corresponding to autonomous systems and 13895 edges signifying communication between the autonomous systems or vertices. \item \textit{Hamsterster friendship} network \cite{Kunegis2013} with 1858 vertices representing the users and 12534 edges signifying friendships between the users. \end{itemize} We remark that self-loops have been omitted while constructing the clique complexes from the undirected graphs corresponding to real networks. Note that the dataset of model and real-world networks analyzed here using the proposed methods based on Forman-Ricci curvature or edge betweenness centrality were also studied using our previous method \cite{Kannan2019} based on discrete Morse theory. \begin{figure*} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model networks with average degree 4. (a) ER model with $n=1000$ and $p=0.004$. (b) WS model with $n=1000$, $k=4$ and $p=0.5$. (c) BA model with $n=1000$ and $m=2$. (d) Spherical random graphs produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=\infty$. (e) Hyperbolic random graphs produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=2$.} \label{fig2} \end{figure*} \begin{figure*} \caption{$H_2$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model and real networks. (a) Spherical random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=\infty$. (b) Hyperbolic random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=2$. (c) Email communication. (d) Route views. (e) Hamsterster friendship. (f) Human protein interaction.} \label{fig3} \end{figure*} \begin{figure*} \caption{$H_3$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model and real networks. (a) Spherical random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=\infty$. (b) Hyperbolic random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=2$. (c) US Power Grid. (d) Email communication. (e) Route views. (f) Yeast protein interaction. (g) Hamsterster friendship. (h) Human protein interaction.} \label{fig4} \end{figure*} \begin{figure*} \caption{Bottleneck distance between persistence diagrams obtained using our new method based on Forman-Ricci curvature in model networks with average degree 4. For each of the five model networks, 10 random samples are generated by fixing the number of vertices $n$ and other parameters of the model. We report the distance (rounded to two decimal places) between two different models as the average of the distance between each of the possible pairs of the 10 sample networks corresponding to the two models along with the standard error.} \label{fig5} \end{figure*} \begin{figure*} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in real networks. (a) US Power Grid. (b) Email communication. (c) Route views. (d) Yeast protein interaction. (e) Hamsterster friendship. } \label{fig6} \end{figure*} \section{Results and Discussion} \subsection{Persistent homology of unweighted networks using Forman-Ricci curvature} We here present a new method based on Forman-Ricci curvature \cite{Sreejith2016,Sreejith2017,Samal2018} to study persistent homology in unweighted and undirected networks. Essentially, our method relies on transforming an unweighted and undirected graph $G$ into an edge-weighted network followed by construction of a weighted clique simplicial complex $K$. We begin by transforming a given unweighted and undirected network into an edge-weighted network by assigning weights to edges based on their Forman-Ricci curvature (Figure \ref{schemfig}). The Forman-Ricci curvature of each edge in an unweighted network can be computed using Equation \ref{AugmentedFormanRicciEdge} as described in the Theory section. Thereafter, we assign weights to edges in the network by normalization of the associated Forman-Ricci curvatures using the following formula: \begin{equation} \label{FormanWeight} w(e) = \frac { \rm{F}(e) - ( \rm{F}_{\text{min}} - \epsilon)} { ( \rm{F}_{\text{max}} + \epsilon ) - ( \rm{F}_{\text{min}} - \epsilon ) } \end{equation} where $w(e)$ is the weight of edge $e$, $\rm{F}(e)$ is the Forman-Ricci curvature of edge $e$, $\rm{F}_{\text{min}}$ and $\rm{F}_{\text{max}}$ are the minimum value and maximum value, respectively, of the Forman-Ricci curvature across all edges in the network, and $\epsilon$ is a positive number which is taken here to be 1. In sum, the above formula gives the weights of edges in the weighted network corresponding to the given unweighted and undirected network. In schematic Figure \ref{schemfig}, we show this transformation of an unweighted and undirected graph into an edge-weighted network using an example. Next, we construct a weighted clique simplicial complex starting from the edge-weighted network as follows (Figure \ref{schemfig}). The $1$-simplices or edges in the clique complex corresponding to the edge-weighted network already have normalized weights based on their Forman-Ricci curvature. Based on the weights of $1$-simplices or edges, we assign weights to $0$-simplices or vertices such that the weight of a vertex in the clique complex is equal to the minimum of the weights of edges incident on the vertex (Equation \ref{simplexweight}). In other words, the weight of the $0$-simplex is the minimum of the weights of its $1$-dimensional co-faces in the clique complex. Similarly, we assign weights to $2$-simplices such that the weight of a $2$-simplex in the clique complex is equal to the maximum of the weights of its $1$-dimensional faces or edges (Equation \ref{simplexweight}). In the same manner, we can assign weights to higher-dimensional simplices (see Theory section). For example, the weight of a $3$-simplex in the clique complex is equal to the maximum of the weights of its $2$-dimensional faces. In schematic Figure \ref{schemfig}, we show this construction of a weighted clique simplicial complex starting from an edge-weighted network using an example. To construct a weighted clique simplicial complex corresponding to an unweighted and undirected network, our scheme hinges on assignment of weights to $0$-simplices (vertices) based on weights of their $1$-dimensional co-faces (i.e., edges attached to vertices). In many real-world networks, there are isolated vertices which are not attached to any edges in the graph. In our scheme, isolated vertices ($0$-simplices) are assigned weights equal to the maximum of the weights given to any simplex in the clique complex. In the example shown in schematic Figure \ref{schemfig}, we assign weight to the isolated vertex $v_9$ in the weighted clique complex as described above. After constructing the weighted clique complex $K$ corresponding to a given unweighted and undirected graph $G$, we investigate the persistent homology of the simplicial complex via the associated filtration described in Equation \ref{filtration} in the Theory section. In order to construct this filtration of the weighted clique complex $K$, the assigned weights $w(\alpha_i)$ to simplices $\alpha_i$ in $K$ are arranged in an increasing order, say $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$, and thereafter, the sequence of subcomplexes, $K(\lambda_1) \subseteq K(\lambda_2) \subseteq \cdots \subseteq K(\lambda_n)$ is used to compute the persistent homology groups of $K$ as described in the Theory section. In schematic Figure \ref{schemfig}, we show this filtration of the weighted clique complex corresponding to an unweighted and undirected network using an example. In previous work \cite{Sreejith2016,Sreejith2017,Samal2018}, it was shown that edges critical for the robustness of a complex network have highly negative Forman-Ricci curvature. From Equation \ref{FormanWeight}, it follows that the assigned weights to edges or $1$-simplices in the weighted clique complex $K$ constructed by our scheme is likely to be inversely proportional to their importance from robustness perspective. Noteworthy, critical edges for the integrity of the network are likely to be added in the initial stages of the filtration of the weighted clique complex $K$, and thus, our method for studying the persistent homology revolves around the central idea that the more important features of the network are included in the initial stages of filtration. We emphasize that the proposed method summarized in Figure \ref{schemfig} to study persistent homology in unweighted networks basically relies on transforming an unweighted network into an edge-weighted graph which is then used to construct a weighted clique complex. In principle, an edge-weighted graph can be obtained from an unweighted graph by assigning weights to edges based on any edge-centric measure. In our method summarized in Figure \ref{schemfig}, we have chosen the edge-centric measure, Forman-Ricci curvature, for this transformation. Another possible and attractive choice of an edge-centric measure for this transformation is the edge betweenness centrality \cite{Freeman1977,Girvan2002,Newman2010}. In this work, we have also explored this alternate choice of edge betweenness centrality to construct edge-weighted networks and study the persistent homology of unweighted networks. The edge betweenness centrality of an edge in an unweighted network can be computed using Equation \ref{EBC} as described in the Theory section. Thereafter, we can assign weights to edges in the network by normalization of the associated edge betweenness centralities using the following formula: \begin{equation} \label{EBCWeight} w(e) = \frac { ( \rm{EBC}_{\text{max}} + \epsilon) - \rm{EBC}(e)} { ( \rm{EBC}_{\text{max}} + \epsilon ) - ( \rm{EBC}_{\text{min}} - \epsilon ) } \end{equation} where $w(e)$ is the weight of edge $e$, $\rm{EBC}(e)$ is the edge betweenness centrality of edge $e$, $\rm{EBC}_{\text{min}}$ and $\rm{EBC}_{\text{max}}$ are the minimum value and maximum value, respectively, of the edge betweenness centrality across all edges in the network, and $\epsilon$ is a positive number which is taken here to be 1. Since the edges with high edge betweenness centrality are highly critical for information flow in the network, the above equation assigns lower weights to such critical edges to ensure their addition during initial stages of the filtration. In the main text of this paper, we report results from the investigation of persistent homology in unweighted networks using our method based on Forman-Ricci curvature. In the supplementary information (SI) of this paper, we report results from the investigation of persistent homology in unweighted networks using our method based on alternate choice of edge betweenness centrality. In the sequel, we will show that the qualitative and quantitative results obtained in unweighted model and real networks using our method based on Forman-Ricci curvature is very similar to our method based on edge betweenness centrality. However, the calculation of edge betweenness centrality requires computing all shortest paths between every distinct pair of vertices in the network, and thus, it is much more computationally expensive than Forman-Ricci curvature. Therefore, our method based on Forman-Ricci curvature is a better choice from a computational perspective to study persistent homology in unweighted and undirected networks. \subsection{Implementation in model and real networks} In this work, we have investigated five model networks and seven real-world networks using our methods based on Forman-Ricci curvature and edge betweenness centrality described in the preceding section to study persistent homology in unweighted and undirected networks. For a given unweighted and undirected network $G$, either model or real-world, we first construct the corresponding edge-weighted network based on Forman-Ricci curvature (Equation \ref{FormanWeight}) or edge betweenness centrality (Equation \ref{EBCWeight}); thereafter, we construct a weighted clique simplicial complex $K$ and then study the corresponding filtration based on the edge weights as described in the preceding section. We remark that our investigation of the persistent homology in model and real networks is limited to the $3$-dimensional clique simplicial complex $K$ corresponding to $G$. That is, we only include $p$-simplices which have $0 \leq p \leq 3$ while constructing the weighted clique simplicial complex $K$ starting from an unweighted graph $G$. For these computations of persistent homology in model and real networks, we use GUDHI \cite{Maria2014} which is a C++ based library for Topological Data Analysis (http://gudhi.gforge.inria.fr/). In the following, we present our results from application of our method based on Forman-Ricci curvature to study persistent homology in model and real networks (Figure \ref{schemfig}). We have studied here five different model networks, namely, ER, WS, BA, spherical random graphs and hyperbolic random graphs (see Datasets section). The $0$-holes or $H_0$ barcode diagram gives the number of connected components in the network at every stage of the filtration. We find that the ER and WS networks possess a large number of components throughout the filtration in comparison to BA networks where there is typically a single component which persists across the entire filtration (Figure \ref{fig2}, SI Figures S1 and S2). This suggests that the simplices which are critical for the overall connectivity of the network are introduced during the initial stages of the filtration of BA networks in contrast to ER and WS networks. The $H_1$ barcode diagram indicates the presence of $1$-holes in the network. We find that the $1$-holes appear earlier during filtration in ER and WS networks in comparison to BA networks where $1$-holes appear towards the end of filtration (Figure \ref{fig2}, SI Figures S1 and S2). Thus, the $H_0$ and $H_1$ barcode diagrams are able to qualitatively distinguish scale-free BA networks from random ER networks and small-world WS networks (Figure \ref{fig2}, SI Figures S1 and S2). Lastly, $H_2$ and $H_3$ barcode diagrams do not provide any insight into the structure of ER, WS and BA networks due to the lack of $2$-holes and $3$-holes. The ER, WS and BA networks can be distinguished from both spherical and hyperbolic random graphs based on significantly larger number of $0$-holes or connected components in the later (Figure \ref{fig2}, SI Figures S1 and S2). Though the number of components in the spherical and hyperbolic random graphs are similar, the patterns of filtration sequence is a distinguishing factor. The $0$-holes in the spherical random graph are more distributed across the filtration sequence, while there are very few $0$-holes in the hyperbolic random graph for most part of the filtration with the introduction of a large number of $0$-holes just before the end of the filtration. This can be understood by the presence of many isolated vertices in the hyperbolic random graphs which are assigned maximum weight in the corresponding weighted clique complex, and thus, appear at the end of filtration (Figure \ref{fig2}, SI Figures S1 and S2). Moreover, $1$-holes and $2$-holes appear during intermediate stages of filtration in both spherical and hyperbolic random graphs, however, these holes do not persist till the end of filtration (Figures \ref{fig2}-\ref{fig3}, SI Figures S1-S3). Noteworthy, there are many more $3$-holes in hyperbolic random graphs in comparison to spherical graphs (Figure \ref{fig4}, SI Figure S3). Overall, these features enable qualitative distinction between hyperbolic and spherical model graphs with very different underlying geometry. The barcode diagrams for the five model networks obtained from our method based on Forman-Ricci curvature can be used to make a qualitative distinction between different types of networks (Figure \ref{fig2}-\ref{fig4}, SI Figures S1-S3). For a quantitative distinction between the topological features of the five model networks, we use the bottleneck distance between total persistence diagrams obtained from our method based on Forman-Ricci curvature as described in the Theory section. From Figure \ref{fig5}, it is seen that the bottleneck distance between BA and ER or WS networks of similar average degree is higher than the distance between ER and WS networks. Furthermore, the bottleneck distance is high between spherical and hyperbolic random graphs of similar average degree. Overall, the bottleneck distances between the total persistence diagrams for the five model networks provide a quantitative validation of the applicability of our method based on Forman-Ricci curvature to reveal distinct topological features in unweighted and undirected networks. We have also studied here seven well-known real-world networks (see Datasets section). From the $H_0$ barcode of the US Power Grid network, it is clear that there is one connected component which persists across the entire filtration despite many transient components appearing and disappearing during intermediate stages of filtration (Figure \ref{fig6}). The $H_0$ barcode diagrams of the E-mail and Route views networks are similar in the sense that the number of connected components across the entire filtration is low (Figure \ref{fig6}). The $H_0$ barcode diagrams of the two biological networks, namely Human protein interaction and Yeast protein interaction show a sudden increase in the number of connected components at the final stage of filtration (Figure \ref{fig6}, SI Figure S4). The $H_0$ barcode diagram of the Euro road network displays a distributed pattern with components spanning across varied intervals of filtration (SI Figure S4). The $H_1$ barcode diagrams of the seven real networks are similar in the respect that there is typically an increase in the number of $1$-holes around the middle to later stages of filtration (Figure \ref{fig6}, SI Figure S4). We also display the $H_2$ and $H_3$ barcode diagrams for real networks in Figures \ref{fig3} and \ref{fig4}, respectively. Note that the Euro road network is devoid of $2$-holes and $3$-holes, while the Yeast protein interaction network and US Power Grid network are devoid or $2$-holes (Figures \ref{fig3} and \ref{fig4}). In sum, the barcode diagrams obtained using our method based on Forman-Ricci curvature can be used to reveal differences between model and real networks. In the above paragraphs, we reported our results from application of our method based on Forman-Ricci curvature to study persistent homology in unweighted networks, both model and real-world. As described in the preceding section, we can also apply an alternate method based on edge betweenness centrality to study persistent homology in unweighted networks, both model and real-world. In SI Figures S5-S9, we present the $H_0$, $H_1$, $H_2$ and $H_3$ barcode diagrams obtained using our alternate method based on edge betweenness centrality in five model and seven real-world networks. Based on SI Figures S5-S9, it is evident that we are also able to make qualitative distinction between varied model networks using the barcode diagrams obtained from our method based on edge betweenness centrality, and these results are similar to results described above from our method based on Forman-Ricci curvature. Moreover, in SI Figure S10, we display the bottleneck distances between persistence diagrams corresponding to the five model networks obtained using the alternate method based on edge betweenness centrality, and these results are also similar to those shown in Figure \ref{fig5} which are obtained using our method based on Forman-Ricci curvature. \subsection{Comparison with method based on discrete Morse theory} Classical Morse theory on smooth manifolds has been a rich theory to detect the topology of the underlying space \cite{Morse1934}. However, it requires a smooth structure to probe the topology via real-valued smooth functions. Robin Forman introduced \textit{discrete Morse theory}, the discrete counterpart of classical Morse theory \cite{Forman1998,Forman2002}, which is applicable to a large class of topological objects called $CW$-complexes, even those which lack smoothness. Similar to classical Morse theory, this discretized version also captures the topology of the underlying object. A fundamental notion in discrete Morse theory is that of \textit{critical cells}, which are the discrete analogues of equilibrium points of a Morse function, i.e. points on which its gradient vanishes. The number of such critical cells is intricately related to the Betti numbers and the Euler characteristic of the topological space via the Morse inequalities \cite{Forman1998,Forman2002}. In our previous work \cite{Kannan2019}, we have used discrete Morse theory of Robin Forman to set weights on the clique complex of an unweighted graph. We also gave an algorithm to produce a near-optimal discrete Morse function, in the sense that the number of so-called critical simplices of the function is close to the theoretical minimum given by Betti numbers. The advantages of this approach lies in the ability for \textit{preprocessing} of the simplicial complex, which leads to significant simplification that in turn, leads to computational efficiency for homology calculations \cite{Mischaikow2013,Harker2014}. In principle, one can use this method independent of TDA, for example to study combinatorial topology aspects of such complexes \cite{Shareshian2001}. As an application to TDA, we \cite{Kannan2019} used this method to compute the persistent homology of the model and real-world networks which are also analyzed in this contribution, and showed that it was able to distinguish between various networks with different topological features. Our present method focusses on the computation of persistent homology only, setting aside the computational advantages of using a near-optimal discrete Morse function. We have shown that even though we lose the simplified topological structure provided by discrete Morse theory, we can still apply other weighting schemes using local and global measures like Forman-Ricci curvature and edge betweenness centrality, to compute persistent homology of various networks. Indeed, the new method also distinguishes the various model networks as well as the previous method using discrete Morse theory. Therefore, using these new weighting schemes can be seen as a tradeoff between computational efficiency and applicability of TDA to unweighted graphs. \section{Conclusions} This work is meant to provide techniques to apply TDA, for the investigation of topological properties of unweighted networks. We employ two methods to convert an unweighted network into a weighted network. The first one is a discretized version of Ricci curvature which takes into account only local information around an edge in the network, while the second one is a global quantifier which measures the importance of an edge based on the number of shortest paths between any two distinct vertices of the network passing through that edge. Once we have a weighted graph, standard techniques allow us to obtain a filtration of the associated clique complexes, and thereby compute their persistent homology (Figure \ref{schemfig}). We have applied this method to study five different kinds of model networks. We also show that this method can distinguish different model networks via the averaged bottleneck distance between the corresponding persistence diagrams (Figure \ref{fig5}). We also apply the same techniques to study some well-known real-world networks to obtain insights into their underlying topology. In future work, we plan to apply these techniques to networks arising out of physical systems, with the goal of trying to find correlations between dynamics of such systems and their underlying topology via the application of TDA. \subsection*{Funding} This work was supported by Max Planck Society, Germany, through the award of a Max Planck Partner Group in Mathematical Biology (to A.S.) and Science and Engineering Research Board (SERB), Department of Science and Technology (DST) India through the award of a MATRICS grant [MTR/2017/000835] (to I.R.). \subsection*{Declaration of Competing Interest} None. \subsection*{Acknowledgments} S.V. and S.J.R. thank the Institute of Mathematical Sciences (IMSc), Chennai, India for their local hospitality and R. Nadarajan for encouragement. \subsection*{Author contributions} I.R. and A.S. designed the study and developed the method. S.V. and S.J.R. performed the simulations. I.R. and A.S. analyzed results. I.R., S.V., S.J.R. and A.S. wrote the manuscript. 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{167--174}\BibitemShut {NoStop} \bibitem [{\citenamefont {Morse}(1934)}]{Morse1934} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Morse}},\ }\href@noop {} {\emph {\bibinfo {title} {The calculus of variations in the large}}},\ Vol.~\bibinfo {volume} {18}\ (\bibinfo {publisher} {American Mathematical Society},\ \bibinfo {year} {1934})\BibitemShut {NoStop} \bibitem [{\citenamefont {Mischaikow}\ and\ \citenamefont {Nanda}(2013)}]{Mischaikow2013} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Mischaikow}}\ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Nanda}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Discrete \& Computational Geometry}\ }\textbf {\bibinfo {volume} {50}},\ \bibinfo {pages} {330} (\bibinfo {year} {2013})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Harker}\ \emph {et~al.}(2014)\citenamefont {Harker}, \citenamefont {Mischaikow}, \citenamefont {Mrozek},\ and\ \citenamefont {Nanda}}]{Harker2014} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Harker}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Mischaikow}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Mrozek}}, \ and\ \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Nanda}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Foundations of Computational Mathematics}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo {pages} {151} (\bibinfo {year} {2014})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Shareshian}(2001)}]{Shareshian2001} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Shareshian}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Topology}\ }\textbf {\bibinfo {volume} {40}},\ \bibinfo {pages} {681} (\bibinfo {year} {2001})}\BibitemShut {NoStop} \end{thebibliography} \begin{center} \section*{\large \bf SUPPLEMENTARY INFORMATION (SI)} \end{center} \renewcommand{S.\arabic{equation}}{S.\arabic{equation}} \renewcommand{S\arabic{figure}}{S\arabic{figure}} \setcounter{equation}{0} \setcounter{figure}{0} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model networks with average degree 6.} \end{figure} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model networks with average degree 8.} \end{figure} \begin{figure} \caption{$H_2$ and $H_3$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in model networks with average degree 6 and 8.} \end{figure} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on Forman-Ricci curvature in real networks. (a) Euro road. (b) Human protein interaction. } \end{figure} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on edge betweenness centrality in model networks with average degree 4.} \end{figure} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on edge betweenness centrality in real networks. (a) US Power Grid. (b) Email communication. (c) Route views. (d) Yeast protein interaction. (e) Hamsterster friendship.} \end{figure} \begin{figure} \caption{$H_0$ and $H_1$ barcode diagrams obtained using our new method based on edge betweenness centrality in real networks. (a) Euro road. (b) Human protein interaction. } \end{figure} \begin{figure*} \caption{$H_2$ barcode diagrams obtained using our new method based on edge betweenness centrality in model and real networks. (a) Spherical random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=\infty$. (b) Hyperbolic random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=2$. (c) Email communication. (d) Route views. (e) Hamsterster friendship. (f) Human protein interaction.} \end{figure*} \begin{figure*} \caption{$H_3$ barcode diagrams obtained using our new method based on edge betweenness centrality in model and real networks. (a) Spherical random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=\infty$. (b) Hyperbolic random graph produced from HGG model with $n=1000$, $T=0$, $k=4$ and $\gamma=2$. (c) US Power Grid. (d) Email communication. (e) Route views. (f) Yeast protein interaction. (g) Hamsterster friendship. (h) Human protein interaction.} \end{figure*} \begin{figure*} \caption{Bottleneck distance between persistence diagrams obtained using our new method based on edge betweenness centrality in model networks with average degree 4. For each of the five model networks, 10 random samples are generated by fixing the number of vertices $n$ and other parameters of the model. We report the distance (rounded to two decimal places) between two different models as the average of the distance between each of the possible pairs of the 10 sample networks corresponding to the two models along with the standard error.} \end{figure*} \end{document}

\begin{document} \title[Nondivergence form degenerate parabolic equations]{Nondivergence form degenerate linear parabolic equations on the upper half space} \author[H. Dong]{Hongjie Dong} \address[H. Dong]{Division of Applied Mathematics, Brown University, 182 George Street, Providence RI 02912, USA} \email{hongjie\[email protected]} \author[T. Phan]{Tuoc Phan} \address[T. Phan]{Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Drive, Knoxville, TN 37996-1320, USA} \email{[email protected]} \author[H. V. Tran]{Hung Vinh Tran} \address[H. V. Tran]{Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall 480 Lincoln Drive Madison, WI 53706, USA} \email{[email protected]} \thanks{ H. Dong is partially supported by the NSF under agreement DMS-2055244 and the Simons Fellows Award 007638. H. Tran is supported in part by NSF CAREER grant DMS-1843320 and a Vilas Faculty Early-Career Investigator Award. } \subjclass[2020]{35K65, 35K67, 35K20, 35D30} \keywords{Degenerate linear parabolic equations; degenerate viscous Hamilton-Jacobi equations; nondivergence form; boundary regularity estimates; existence and uniqueness; weighted Sobolev spaces} \begin{abstract} We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times \mathbb{R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $\mu(x_d)$ and bounded positive definite matrices, where $\mu(x_d)$ behaves like $x_d^\alpha$ for some given $\alpha \in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in \cite{DPT21}. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations. \end{abstract} \dedicatory{Dedicated to Professor Mikhail Safonov on the occasion of his $70^{\text{th}}$ birthday} \maketitle \section{Introduction and main results} \subsection{Settings} Let $T\in (-\infty,\infty]$ and $\Omega_T=(-\infty,T)\times \mathbb{R}^d_+$. We study the following degenerate parabolic equation in nondivergence form \begin{equation}\label{eq:main} \begin{cases} \sL u=\mu(x_d) f \quad &\text{ in } \Omega_T,\\ u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+, \end{cases} \end{equation} where $u: \Omega_T \rightarrow \mathbb{R}$ is an unknown solution, $f: \Omega_T \rightarrow \mathbb{R}$ is a given measurable forcing term, and \begin{equation} \label{L-def} \sL u = a_0(z) u_t+\lambda c_0(z)u-\mu(x_d) a_{ij}(z)D_i D_j u. \end{equation} Here in \eqref{L-def}, $\lambda\ge 0$ is a constant, $z=(t,x) \in \Omega_T$ with $x = (x', x_d) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$, $D_i$ denotes the partial derivative with respect to $x_i$, and $a_0, c_0: \Omega_T \rightarrow \mathbb{R}$ and $\mu: \mathbb{R}_+ \rightarrow \mathbb{R}$ are measurable and satisfy \begin{equation} \label{con:mu} a_0(z), \ c_0(z), \ \frac{\mu(x_d)}{x_d^\alpha} \in[\nu,\nu^{-1}], \quad \forall \ x_d \in \mathbb{R}_+, \quad \forall \ z \in \Omega_T, \end{equation} for some given $\alpha\in (0,2)$ and $\nu \in (0,1)$. Moreover, $(a_{ij}): \Omega_T \rightarrow \mathbb{R}^{d\times d}$ are measurable and satisfy the uniform ellipticity and boundedness conditions \begin{equation} \label{con:ellipticity} \nu|\xi|^2 \leq a_{ij}(z) \xi_i \xi_j, \quad |a_{ij}(z)| \leq \nu^{-1}, \quad \forall \ z \in \Omega_T, \end{equation} for all $\xi = (\xi_1, \xi_2, \ldots, \xi_d) \in \mathbb{R}^d$. We observe that due to \eqref{con:mu} and \eqref{con:ellipticity}, the diffusion coefficients in the PDE in \eqref{eq:main} are degenerate when $x_d \rightarrow 0^+$, and singular when $x_d \rightarrow \infty$. We also note that the PDE in \eqref{eq:main} can be written in the form \[ [a_0(z) u_t + \lambda c_0(z) u]/\mu(x_d) - a_{ij}(z) D_iD_j u = f \quad \text{in} \quad \Omega_T, \] in which the singularity and degeneracy appear in the coefficients of the terms involving $u_t$ and $u$. In the special case when $a_0 = c_0 =1, \mu(x_d) = x_d^\alpha$, and $(a_{ij})$ is an identity matrix, the equation \eqref{eq:main} is reduced to \begin{equation} \label{simplest-eqn} \left\{ \begin{array}{cccl} u_t + \lambda u - x_d^\alpha \Delta u & = & x_d^\alpha f & \quad \text{in} \quad \Omega_T, \\ u & =& 0 & \quad \text{on} \quad (-\infty, T) \times \partial \mathbb{R}^d_+, \end{array} \right. \end{equation} in which the results obtained in this paper are still new. The theme of this paper is to study the existence, uniqueness, and regularity estimates for solutions to \eqref{eq:main}. To demonstrate our results, let us state the following theorem which gives prototypical estimates of our results in a special weighted Lebesgue space $L_p(\Omega, x_d^\gamma\, dz)$ with the power weight $x_d^\gamma$ and norm \[ \|f\|_{L_p(\Omega, x_d^\gamma dz)} = \left( \int_{\Omega_T} |f(t,x)|^p x_d^\gamma\, dx dt \right)^{1/p}. \] {For any measurable function $f$ and $s \in \mathbb{R}$, we define the multiplicative operator $(\mathbf{M}^s f)(\cdot)=x_d^s f(\cdot)$.} \begin{theorem} \label{thm:demo} Let $\alpha \in (0,2), \lambda >0$, $p\in (1,\infty)$, and $\gamma \in \big(p(\alpha-1)_+-1,2p-1\big)$. Then, for every $f \in L_p(\Omega, x_d^\gamma\, dz)$, there exists a unique strong solution $u$ to \eqref{simplest-eqn}, which satisfies \begin{align} \label{show-est-1} \|\mathbf{M}^{-\alpha}u_t\|_{L_p}+\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p} +\|D^2 u\|_{L_p}\le N\|f\|_{L_p}; \end{align} and for $\gamma\in (\alpha p/2-1,2p-1)$, \begin{equation}\label{show-est-2} \lambda^{1/2}\|\mathbf{M}^{-\alpha/2}Du\|_{L_p} \le N\|f\|_{L_p}, \end{equation} where $\|\cdot\|_{L_p} = \|\cdot \|_{L_p(\Omega, x_d^\gamma dz)}$ and $N=N(d,\nu,\alpha, \gamma, p)>0$. \end{theorem} \noindent See Corollary \ref{cor1} and Theorem \ref{thm:xd} for more general results. We note that the ranges of $\gamma$ in \eqref{show-est-1}--\eqref{show-est-2} are optimal as pointed out in Remarks \ref{remark-1-range}--\ref{remark-2-range} below. In fact, in this paper, a much more general result in weighted mixed-norm spaces is established in Theorem \ref{main-thrm}. As an application, we obtain a regularity result for solutions to degenerate viscous Hamilton-Jacobi equations in Theorem \ref{example-thrm}. To the best of our knowledge, our main results (Theorems \ref{thm:demo}, \ref{main-thrm}, \ref{thm:xd}, Corollary \ref{cor1}, and Theorem \ref{example-thrm}) appear for the first time in the literature. \subsection{Relevant literature} The literature on regularity theory for solutions to degenerate elliptic and parabolic equations is extremely rich, and we only describe results related to \eqref{eq:main}. The divergence form of \eqref{eq:main} was studied by us in \cite{DPT21} with motivation from the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations of the form \begin{equation}\label{eq:HJ-intro} u_t+\lambda u-\mu(x_d) \Delta u=H(z,Du) \qquad \text{ in } \Omega_T. \end{equation} Here, $H:\Omega_T \times \mathbb{R}^d \to \mathbb{R}$ is a given Hamiltonian. Under some appropriate conditions on $H$, we obtain a regularity {and solvability} result for \eqref{eq:HJ-intro} in Theorem \ref{example-thrm}. Another class of divergence form equations, which is closely related to that in \cite{DPT21}, was analyzed recently in \cite{JinXiong2} when $\alpha<1$. When $\alpha=2$ and $ d=1$, a specific version of \eqref{eq:HJ-intro} gives the well-known Black-Scholes-Merton PDE that appears in mathematical finance. The analysis for \eqref{eq:main} when $\alpha\geq 2$ is completely open. A similar equation to \eqref{eq:main}, \eqref{simplest-eqn}, and \eqref{eq:HJ-intro} \[ u_t+\lambda u-\beta D_du - x_d \Delta u = f \qquad \text{ in } \Omega_T \] with an additional structural condition $\beta>0$, an important prototype equation in the study of porous media equations and parabolic Heston equation, was studied extensively in the literature (see \cite{DaHa, FePo, Koch, JinXiong1, JinXiong2} and the references therein). We stress that we do not require this structural condition in the analysis of \eqref{eq:main} and \eqref{eq:HJ-intro}, and thus, our analysis is rather different from those in \cite{DaHa, FePo, Koch}. We note that similar results on the wellposedness and regularity estimates in weighted Sobolev spaces for a different class of equations with singular-degenerate coefficients were established in a series of papers \cite{DP-20, DP-21, DP-JFA, DP-AMS}. There, the weights of singular/degenerate coefficients of $u_t$ and $D^2u$ appear in a balanced way, which plays a crucial role in the analysis and functional space settings. If this balance is lost, then Harnack's inequalities were proved in \cite{Chi-Se-1, Chi-Se-2} to be false in certain cases. However, with an explicit weight $x_d^\alpha$ as in our setting, it is not known if some version of Harnack's inequalities and H\"{o}lder estimates of the Krylov-Safonov type as in \cite{K-S} still hold for in \eqref{eq:main}. Of course, \eqref{eq:main} does not have this balance structure, and our analysis is quite different from those in \cite{DP-20, DP-21, DP-JFA, DP-AMS}. Finally, we emphasize again that the literature on equations with singular-degenerate coefficients is vast. Below, let us give some references on other closely related results. The H\"{o}lder regularity for solutions to elliptic equations with singular and degenerate coefficients, which are in the $A_2$-Muckenhoupt {class}, were proved in the classical papers \cite{Fabes, FKS}. See also the books \cite{Fichera, OR}, the papers \cite{KimLee-Yun, Sire-1, Sire-2}, and the references therein for other results on the wellposedness, H\"{o}lder, and Schauder regularity estimates for various classes of degenerate equations. Note also that the Sobolev regularity theory version of the results in \cite{Fabes, FKS} was developed and proved in \cite{Men-Phan}. In addition, we would like to point out that equations with degenerate coefficients also appear naturally in geometric analysis \cite{Lin, WWYZ}, in which H\"{o}lder and Schauder estimates for solutions were proved. \subsection{Main ideas and approaches} The main ideas of this paper are along the lines with those in \cite{DPT21}. However, at the technical level, the proofs of our main results are quite different from those in \cite{DPT21}. More precisely, instead of the $L_2$-estimates as in \cite{DPT21}, the starting point in this paper is the weighted $L_p$-result in Lemma \ref{l-p-sol-lem} which is based on the weighted $L_p$ for divergence form equations established in \cite{DPT21}, an idea introduced by Krylov \cite{Kr99}, together with a suitable scaling. Moreover, while the proofs in \cite{DPT21} use the Lebesgue measure as an underlying measure, in this paper we make use of more general underlying measure $\mu_1 (dz) = x_d^{\gamma_1}$ with an appropriate parameter $\gamma_1$. In particular, this allows us to obtain an optimal range of exponents for power weights in Corollary \ref{cor1}. See Remarks \ref{remark-1-range} - \ref{remark-2-range}. Several new H\"{o}lder estimates for higher order derivatives of solutions to a class of degenerate homogeneous equations are proved in Subsections \ref{subsec:boundary}--\eqref{subsec:int}. The results and techniques developed in these subsections {might be} of independent interest. \subsection*{Organization of the paper} The paper is organized as follows. In Section \ref{sec:2}, we introduce various function spaces, assumptions, and then state our main results. The filtration of partitions, a quasi-metric, the weighted mixed-norm Fefferman-Stein theorem and Hardy-Littlewood theorem are recalled in Section \ref{Feffer}. Then, in Section \ref{sec:3}, we consider \eqref{eq:main} in the case when the coefficients in \eqref{eq:main} only depend on the $x_d$ variable. A special version of Theorem \ref{main-thrm}, Theorem \ref{thm:xd}, will be stated and proved in this section. The proofs of Theorem \ref{main-thrm} and Corollary \ref{cor1} are given in Section \ref{sec:4}. Finally, we study the degenerate viscous Hamilton-Jacobi equation \eqref{eq:HJ-intro} in Section \ref{sec:5}. \section{Function spaces, parabolic cylinders, mean oscillations, and main results}\label{sec:2} \subsection{Function spaces} Fix $p,q \in [1, \infty)$, $-\infty\le S<T\le +\infty$, and a domain $\cD \subset \mathbb{R}^d_+$. Denote by $L_p((S,T)\times \cD)$ the usual Lebesgue space consisting of measurable functions $u$ on $(S,T)\times \cD$ such that \[ \|u\|_{L_p( (S,T)\times \cD)}= \left( \int_{(S,T)\times \cD} |u(t,x)|^p\, dxdt \right)^{1/p} <\infty. \] For a given weight $\omega$ on $(S,T)\times \cD$, let $L_{p}((S,T)\times \cD,\omega)$ be the weighted Lebesgue space on $(S,T)\times \cD$ equipped with the norm \begin{equation*} \|u\|_{L_{p}((S,T)\times \cD, \omega)}=\left(\int_{(S,T)\times \cD} |u(t,x)|^p \omega (t,x)\, dx dt\right)^{1/p} < \infty. \end{equation*} For the weights $\omega_0=\omega_0(t)$, $\omega_1=\omega_1(x)$, and a measure $\sigma$ on $\cD$, set $\omega(t,x)=\omega_0(t)\omega_1(x)$ and define $L_{q,p}((S,T)\times \cD,\omega d\sigma)$ to be the weighted and mixed-norm Lebesgue space on $(S,T)\times \cD$ equipped with the norm \[ \|u\|_{L_{q,p}((S,T)\times \cD, \omega d\sigma)}=\left(\int_S^T \left(\int_{\cD} |u(t,x)|^p \omega_1(x)\, \sigma(dx)\right)^{q/p} \omega_0(t)\,dt \right)^{1/q} < \infty. \] \subsubsection{Function spaces for nondivergence form equations} Consider $\alpha>0$. We define the solution spaces as follows. Firstly, define \[ W_{p}^{1,2}((S,T)\times \cD, \omega) =\left\{u \,:\, \mathbf{M}^{-\alpha} u, \mathbf{M}^{-\alpha} u_t, D^2u \in L_p((S,T) \times \cD,\omega)\right\}, \] where, for $u\in W_{p}^{1,2}((S,T)\times \cD, \omega)$, \begin{multline*} \|u\|_{W^{1,2}_p((S,T)\times \cD,\omega)}\\ =\| \mathbf{M}^{-\alpha} u\|_{L_p((S,T)\times \cD,\omega)}+\| \mathbf{M}^{-\alpha} u_t\|_{L_p((S,T)\times \cD,\omega)}+\|D^2u\|_{L_p((S,T)\times \cD,\omega)}. \end{multline*} Let $\sW^{1,2}_p((S,T)\times \cD,\omega)$ be the closure in $W^{1,2}_p((S,T)\times \cD,\omega)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty. The space $\sW^{1,2}_p((S,T)\times \cD,\omega)$ is equipped with the same norm $\|\cdot\|_{\sW^{1,2}_p((S,T)\times \cD,\omega)}=\|\cdot\|_{W^{1,2}_p((S,T)\times \cD,\omega)}$. When there is no time dependence, we write these two spaces as $W^2_p(\cD,\omega)$ and $\sW^2_p(\cD,\omega)$, respectively. Next, denote by \[ \begin{split} & W_{q,p}^{1,2}((S,T)\times \cD, \omega\, d\sigma)\\ & \qquad =\left\{u \,:\, \mathbf{M}^{-\alpha} u, \mathbf{M}^{-\alpha} u_t,D^2u \in L_{q,p}((S,T) \times \cD,\omega\, d\sigma)\right\}, \end{split} \] which is equipped with the norm \begin{multline*} \|u\|_{W^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)} =\| \mathbf{M}^{-\alpha} u\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}\\ +\| \mathbf{M}^{-\alpha} u_t\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}+\|D^2u\|_{L_{q,p}((S,T)\times \cD,\omega\, d\sigma)}. \end{multline*} Let $\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)$ be the closure in $W^{1,2}_{q,p}((S,T)\times \cD,\omega d\sigma)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty. The space $\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)$ is equipped with the same norm $\|\cdot\|_{\sW^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)}=\|\cdot\|_{W^{1,2}_{q,p}((S,T)\times \cD,\omega\, d\sigma)}$. \subsubsection{Function spaces for divergence form equations} We also need function spaces for divergence form equations in this paper, which are taken from \cite{DPT21}. Set $$ W^1_p((S,T)\times \cD, \omega)=\left\{u\,:\, \mathbf{M}^{-\alpha/2} u, Du\in L_p((S,T)\times \cD, \omega)\right\}, $$ which is equipped with the norm $$ \|u\|_{W^1_p((S,T)\times \cD,\omega)}=\| \mathbf{M}^{-\alpha/2} u\|_{L_p((S,T)\times \cD,\omega)}+\|Du\|_{L_p((S,T)\times \cD,\omega)}. $$ We denote by $\sW^1_p((S,T)\times \cD,\omega)$ the closure in $W^1_p((S,T)\times \cD,\omega)$ of all compactly supported functions in $C^\infty((S,T)\times \overline{\cD})$ vanishing near $\overline{\cD} \cap \{x_d=0\}$ if $\overline{\cD} \cap \{x_d=0\}$ is not empty. The space $\sW^1_p((S,T)\times \cD,\omega)$ is equipped with the same norm $\|\cdot\|_{\sW^1_p((S,T)\times \cD,\omega)}=\|\cdot\|_{W^1_p((S,T)\times \cD,\omega)}$. Set \[ \begin{split} & \mathbb{H}_{p}^{-1}( (S,T)\times \cD, \omega) \\ & =\big\{u\,:\, u = \mu(x_d) D_iF_i +f_1+f_2, \ \ \text{where}\ \mathbf{M}^{1-\alpha} f_1,\mathbf{M}^{-\alpha/2}f_2\in L_{p}( (S,T)\times \cD, \omega)\\ & \qquad\text{and } F= (F_1,\ldots,F_d) \in L_{p}((S,T)\times \cD, \omega)^{d}\big\}, \end{split} \] equipped with the norm \begin{align*} &\|u\|_{\mathbb{H}_{p}^{-1}((S,T)\times \cD, \omega)} \\ &=\inf\big\{\|F\|_{L_{p}((S,T)\times \cD, \omega)} +\|| \mathbf{M}^{1-\alpha} f_1|+|\mathbf{M}^{-\alpha/2}f_2|\|_{L_{p}((S,T)\times \cD, \omega)}\,:\\ &\qquad u= \mu(x_d) D_iF_i +f_1+f_2\big\}. \end{align*} Define \[ \cH_{p}^1((S,T)\times \cD, \omega) =\big\{u \,:\, u \in \sW^1_p((S,T) \times \cD,\omega)), u_t\in \mathbb{H}_{p}^{-1}( (S,T)\times \cD, \omega)\big\}, \] where, for $u\in \cH_{p}^1((S,T)\times \cD, \omega)$, \begin{align*} \|u\|_{\cH_{p}^1((S,T)\times \cD, \omega)} &= \|\mathbf{M}^{-\alpha/2} u\|_{L_{p}((S,T)\times \cD, \omega)} + \|Du\|_{L_{p}((S,T)\times \cD, \omega)} \\ & \qquad +\|u_t\|_{\mathbb{H}_{p}^{-1}((S,T)\times \cD, \omega)}. \end{align*} \subsection{Parabolic cylinders} We use the same setup as that in \cite{DPT21}. For $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$ and $\rho>0$, denote by $B_\rho(x_0)$ the usual ball with center $x_0$ radius $\rho$ in $\mathbb{R}^d$, $B_\rho'(x_0')$ the ball center $x_0'$ radius $\rho$ in $\mathbb{R}^{d-1}$, and \[ B_\rho^+(x_0) = B_\rho(x_0) \cap \mathbb{R}^d_+. \] We note that \eqref{eq:main} is invariant under the scaling \begin{equation} \label{scaling} (t,x) \mapsto (s^{2-\alpha} t, sx), \qquad s > 0. \end{equation} For $x_d \sim x_{0d} \gg 1$, $a_{ij} = \delta_{ij}$, and $\lambda =f=0$, then \eqref{eq:main} behaves like a heat equation \[ u_t -x_{0d}^{\alpha} \Delta u = 0, \] which can be reduced to the heat equation with unit heat constant under the scaling \[ (t,x) \mapsto (s^{2-\alpha} t, s^{1-\alpha/2} x_{0d}^{-\alpha/2}x), \quad s>0. \] It is thus natural to use the following parabolic cylinders in $\Omega_T$ in this paper. For $z_0 = (t_0, x_0) \in (-\infty, T) \times \mathbb{R}^d_+$ with $x_0= (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$ and $\rho>0$, set \begin{equation} \label{def:Q} \begin{split} & Q_{\rho}(z_0) = (t_0 - \rho^{2-\alpha}, t_0) \times B_{r(\rho, x_{0d})} (x_0), \quad \\ &Q_{\rho}^+(z_0) = Q_\rho(z_0) \cap \{x_d>0\}, \end{split} \end{equation} where \begin{equation} \label{def:r} r(\rho,x_{0d}) = \max\{\rho, x_{0d}\}^{\alpha/2} \rho^{1-\alpha/2}. \end{equation} Of course, $Q_{\rho}(z_0) = Q_{\rho}^+(z_0) \subset (-\infty, T) \times \mathbb{R}^d_+$ for $\rho \in (0,x_{0d})$. Finally, for $z' = (t, x') \in \mathbb{R} \times \mathbb{R}^{d-1}$, we write \[ Q_{\rho}'(z') = (t-\rho^{2-\alpha}, t_0) \times B_{\rho}'(x'). \] \subsection{Mean oscillations and main results} \label{main-result-sect} Throughout the paper, for a locally integrable function $f$, a locally finite measure $\omega$, and a domain $Q\subset \mathbb{R}^{d+1}$, we write \begin{equation} \label{everage-def} (f)_{Q} = \fint_{Q} f(s,y)\, dyds, \qquad (f)_{Q,\omega} = \frac{1}{\omega(Q)}\int_{Q} f(s,y) \,\omega(dyds). \end{equation} Also, for a number $\gamma_1 \in (-1, \infty)$ to be determined, we define \[ \mu_1(dz) = x_d^{\gamma_1}\, dxdt. \] We impose the following assumption on the partial mean oscillations of the coefficients $(a_{ij})$, $a_0$, and $c_0$. \begin{assumption}[$\rho_0,\gamma_1, \delta$] \label{assumption:osc} For every $\rho \in (0, \rho_0)$ and $z_0= (z_0', z_{0d}) \in \overline{\Omega}_T$, there exist $[a_{ij}]_{\rho, z'}, [a_{0 }]_{\rho, z'}, [c_{0 }]_{\rho, z'}: ((x_{d} -r(\rho, x_d))_+, x_d + r(\rho, x_d)) \rightarrow \mathbb{R}$ such that \eqref{con:mu}--\eqref{con:ellipticity} hold on $((x_{d} -r(\rho, x_d))_+, x_d + r(\rho, x_d))$ with $[a_{ij}]_{\rho, z'}$, $[a_{0 }]_{\rho, z'}$, $[c_{0 }]_{\rho, z'}$ in place of $(a_{ij})$, $a_0$, $c_0$, respectively, and \begin{align*} a_\rho^{\#}(z_0):= & \max_{1 \leq i, j \leq d}\fint_{Q_\rho^+(z_0)} | a_{ij}(z) -[a_{ij}]_{\rho,z'}(x_d)|\, \mu_1(dz) \\ & \qquad + \fint_{Q_\rho^+(z)} | a_{0}(z) -[a_{0}]_{\rho,z'}(x_d)|\, \mu_1(dz) \\ & \qquad + \fint_{Q_\rho^+(z)} | c_{0}(z) -[c_{0}]_{\rho,z'}(x_d)|\, \mu_1(dz) < \delta. \end{align*} \end{assumption} \noindent We note that the un-weighted partial mean oscillation was introduced in \cite{Kim-Krylov} to study a class of elliptic equations with uniformly elliptic and bounded coefficients (i.e., $\gamma_1=\alpha=0$). Note also that by dividing the equation \eqref{eq:main} by $a_{dd}$ and adjusting $\nu$, we can assume without loss of generality throughout the paper that \begin{equation} \label{add-assumption} a_{dd} \equiv 1. \end{equation} The theorem below is the first main result of our paper, in which the definition of the $A_p$ Muckenhoupt class of weights can be found in Definition \ref{Def-Muck-wei} below. \begin{theorem} \label{main-thrm} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist $\delta = \delta(d, \nu, p, q, K, \alpha, \gamma_1)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, K, \alpha, \gamma_1)>0$ sufficiently large such that the following assertion holds. Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} are satisfied, $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ with \[ [\omega_0]_{A_q(\mathbb{R})} \leq K \quad \text{and} \quad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K, \quad \text{where} \,\, \mu_1(dz) = x_d^{\gamma_1} dxdt. \] Suppose also that Assumption \ref{assumption:osc} $(\rho_0, \gamma_1,\delta)$ holds for some $\rho_0>0$. Then, for any function $f \in L_{q, p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ and $\lambda \geq \lambda_0 \rho_0^{-(2-\alpha)}$, there exists a strong solution $u{\in \sW^{1,2}_{q, p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)}$ to the equation \eqref{eq:main}, which satisfies \begin{equation} \label{main-est-1} \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}}, \end{equation} where $\omega(t, x) = \omega_0(t) \omega_1(x)$ for $(t,x) \in \Omega_T$, $L_{q,p} = L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega \, d\mu_1)$, and $N = N(d, \nu, p, q, \alpha, \gamma_1)>0$. Moreover, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, then it also holds that \begin{equation} \label{main-est-2} \begin{split} & \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \| D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} \\ & \leq N \|f\|_{L_{q,p}}. \end{split} \end{equation} \end{theorem} The following is an important corollary of Theorem \ref{main-thrm} in which $\omega_1$ is a power weight of the $x_d$ variable and $\beta_0$ and $\gamma_1$ are specifically chosen. \begin{corollary} \label{cor1} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma \in (p(\alpha-1)_+ -1, 2p-1)$. Then, there exist $\delta = \delta(d, \nu, p, q, \alpha, \gamma)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, \alpha, \gamma)>0$ sufficiently large such that the following assertion holds. Suppose that \eqref{con:mu}, \eqref{con:ellipticity} hold and suppose also that Assumption \ref{assumption:osc} $(\rho_0, 1-(\alpha-1)_+, \delta)$ holds for some $\rho_0>0$. Then, for any $f \in L_{q, p}(\Omega_T, x_d^{\gamma} dz)$ and $\lambda \geq \lambda_0 \rho_0^{-(2-\alpha)}$, there exists a strong solution $u \in \sW^{1,2}_{q, p}(\Omega_T, x_d^\gamma\, dz)$ to the equation \eqref{eq:main}, which satisfies \begin{equation} \label{cor-est-1} \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}}, \end{equation} where $L_{q,p} = L_{q,p}(\Omega_T, x_d^\gamma dz)$ and $N = N(d, \nu, p, q, \alpha, \gamma)>0$. Additionally, if Assumption \ref{assumption:osc} $(\rho_0, 1-\alpha/2, \delta)$ also holds and $\gamma \in (\alpha p/2 -1, 2p-1)$, then we have \begin{equation} \label{cor-est-2} \begin{split} &\|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} \\ & \leq N \|f\|_{L_{q,p}}. \end{split} \end{equation} \end{corollary} \begin{remark} By viewing solutions to elliptic equations as stationary solutions to parabolic equations, from Theorem \ref{main-thrm} and Corollary \ref{cor1}, we derive the corresponding results for elliptic equations. Furthermore, it follows from Corollary \ref{cor1} and the weighted Morrey embedding (see, for instance, \cite[Theorem 5.3]{RO}), we obtain the $C^{1,\alpha}$ regularity of solutions to the corresponding elliptic equations when $p>d+\gamma$. \end{remark} In the remarks below, we give examples showing that the ranges of $\gamma$ in \eqref{show-est-1}--\eqref{show-est-2} as well as \eqref{cor-est-1}--\eqref{cor-est-2} are optimal. We note that the range of $\gamma$ for the estimate of $Du$ in \eqref{show-est-2}, \eqref{main-est-2}, and \eqref{cor-est-2} is smaller than that for $u, u_t, D^2u$ in \eqref{show-est-1}, \eqref{main-est-1}, and \eqref{cor-est-1}. See Remark \ref{remark-3-range} below to see the necessity of such different ranges. \begin{remark} \label{remark-1-range} When $\alpha \in (0,1)$, the range $(p(\alpha-1)_+ -1, 2p-1)$ for the power $\gamma$ in \eqref{show-est-1} becomes $(-1,2p-1)$, which agrees with the range in \cite{KN} for equations with uniformly elliptic and bounded coefficients. See also \cite{DP-JFA} and \cite{MNS} in which a similar range of the power $\gamma$ is also used in for a class of equations of extensional type. When $\alpha \in [1,2)$, the lower bound $p(\alpha-1)_+ -1$ for $\gamma$ in \eqref{show-est-1} is optimal. To see this, consider an explicit example when $d=1$, $\lambda>0$, $T < \infty$, and \[ u(t,x)=\left(x + c x^{3-\alpha}\right) \xi(x) e^{\lambda t} \qquad \text{ for } (t,x) \in \Omega_T. \] Here, $\xi \in C^\infty([0,\infty),[0,\infty))$ is a cutoff function such that $\xi=1$ on $[0,1]$, $\xi=0$ on $[3,\infty)$, $\|\xi'\|_{L^\infty(\mathbb{R})} \leq 1$, and \[ c=\frac{2 \lambda}{(3-\alpha)(2-\alpha)}. \] Set \[ f(t,x)=x^{-\alpha}(u_t +\lambda u) - u_{xx}. \] Then, $u$ solves \[ u_t + \lambda u - x^\alpha u_{xx} = x^\alpha f \qquad \text{ in } \Omega_T. \] We see that $\mathbf{M}^{-\alpha}u_t, \mathbf{M}^{-\alpha}u \in L_p(\Omega_T, x^\gamma)$ for $\gamma>p(\alpha-1)-1$. On the other hand, \[ \begin{split} & \int_{\Omega_T} |x^{-\alpha}u|^p x^{p(\alpha-1)-1}\,dz =\int_{\Omega_T} |x^{-1}u|^p x^{-1}\,dz \\ & \geq \int_{0}^1 \int_{-\infty}^T x^{-1} e^{p\lambda t}\,dt dx = N\int_{0}^1 x^{-1} \,dx=\infty. \end{split} \] Thus, $\mathbf{M}^{-\alpha}u_t, \mathbf{M}^{-\alpha}u \notin L_p(\Omega_T, x^{p(\alpha-1)-1})$. We next note that $f(t,x)=0$ for $(t,x) \in (-\infty,T] \times [3,\infty)$, and \[ f(t,x)=2 c\lambda x^{3-2\alpha} e^{\lambda t} \qquad \text{ for } (t,x) \in (-\infty,T] \times [0,1]. \] From this and \[ \int_{0}^1 \int_0^T |x^{3-2\alpha}|^p x^{p(\alpha-1)-1} e^{p \lambda t} dt dx = N \int_0^1 x^{p(2-\alpha)-1} dx < \infty, \] it follows that $f\in L_p(\Omega_T, x^{p(\alpha-1)-1})$. \end{remark} \begin{remark} \label{remark-3-range} When $\alpha \in (0,2)$, the lower bound $\alpha p/2 -1$ for $\gamma$ in \eqref{show-est-2} is optimal. Indeed, consider the same example as that in Remark \ref{remark-1-range} above. It is clear that $\mathbf{M}^{-\alpha/2}u_x \in L_p(\Omega_T, x^\gamma)$ for $\gamma> \alpha p/2-1$. On the other hand, $\mathbf{M}^{-\alpha/2}u_x \notin L_p(\Omega_T, x^{\alpha p/2-1})$ as \[ \int_{\Omega_T} |x^{-\alpha/2}u_x|^p x^{\alpha p/2-1}\,dz =\int_{\Omega_T} |u_x|^p x^{-1}\,dz \geq \int_{0}^1 \int_{-\infty}^T x^{-1} e^{p\lambda t}\,dt dx =\infty. \] Besides, $f(t,x)=0$ for $(t,x) \in (-\infty,T] \times [3,\infty)$, and \[ f(t,x)=2 c\lambda x^{3-2\alpha} e^{\lambda t} \qquad \text{ for } (t,x) \in (-\infty,T] \times [0,1]. \] Hence, $f \in L_p(\Omega_T, x^{\alpha p/2-1})$ as \[ \int_{0}^1 \int_0^T |x^{3-2\alpha}|^p x^{\alpha p/2-1} e^{p \lambda t} dt dx = N \int_0^1 x^{p(3-3\alpha/2)-1} dx < \infty. \] \end{remark} \begin{remark} \label{remark-2-range} We also have that the upper bound $\gamma < 2p-1$ in \eqref{show-est-1}--\eqref{show-est-2} is optimal. Indeed, for $\gamma=2p-1$, the trace of $W_{p}^{2}(\cD,x_d^{2p-1})$ is not well defined. For simplicity, let $d=1$, $\cD=[0,1/2]$, and consider \[ \phi(x) = \log (|\log x|). \] Then, \[ \phi_{xx} = \frac{1}{x^2} \left( |\log x|^{-1} - |\log x|^{-2} \right). \] It is clear that $\phi \in W_{p}^{2}([0,1/2],x^{2p-1})$, and $\phi$ is not finite at $0$. \end{remark} \section{A filtration of partitions and a quasi-metric} \label{Feffer} We recall the construction of a filtration of partitions $\{\mathbb{C}_n\}_{n \in \mathbb{Z}}$ (i.e., dyadic decompositions) of $\mathbb{R}\times \mathbb{R}^d_+$ in \cite{DPT21}, which satisfies the following three basic properties (see \cite{Krylov}): \begin{enumerate}[(i)] \item The elements of partitions are ``large'' for big negative $n$'s and ``small'' for big positive $n$'s: for any $f\in L_{1,\text{loc}}$, $$ \inf_{C\in \mathbb{C}_n}|C|\to \infty\quad\text{as}\,\,n\to -\infty,\quad \lim_{n\to \infty}(f)_{C_n(z)}=f(z)\quad\text{a.e.}, $$ where $C_n(z)\in \mathbb{C}_n$ is such that $z\in C_n(z)$. \item The partitions are nested: for each $n\in \mathbb{Z}$, and $C \in \mathbb{C}_n$, there exists a unique $C' \in \mathbb{C}_{n-1}$ such that $C \subset C'$. \item The following regularity property holds: For $n,C, C'$ as in (ii), we have $$ |C'|\le N_0|C|, $$ where $N_0>0$ is independent of $n$, $C$, and $C'$. \end{enumerate} For $s\in \mathbb{R}$, denote by $\lfloor s \rfloor$ the integer part of $s$. For a fixed $\alpha\in (0,2)$ and $n\in \mathbb{Z}$, let $k_0=\lfloor -n/(2-\alpha) \rfloor$. The partition $\mathbb{C}_n$ contains boundary cubes in the form $$ ((j-1)2^{-n},j2^{-n}]\times (i_12^{k_0},(i_1+1)2^{k_0}] \times\cdots\times (i_{d-1}2^{k_0},(i_{d-1}+1)2^{k_0}]\times (0, 2^{k_0}], $$ where $j,i_1,\ldots,i_{d-1}\in \mathbb{Z}$, and interior cubes in the form $$ ((j-1)2^{-n},j2^{-n}]\times (i_12^{k_2},(i_1+1)2^{k_2}] \times\cdots \times (i_d2^{k_2}, (i_d+1)2^{k_2}], $$ where $j,i_1,\ldots,i_{d}\in \mathbb{Z}$ and \begin{equation} \label{eq:part1} i_d2^{k_2}\in [2^{k_1},2^{k_1+1})\, \text{for some integer}\, k_1\ge k_0, \quad k_2=\lfloor (-n+k_1\alpha)/2 \rfloor-1. \end{equation} It is clear that $k_2$ increases with respect to $k_1$ and decreases with respect to $n$. As $k_1\ge k_0>-n/(2-\alpha)-1$, we have $(-n+k_1\alpha)/2-1\le k_1$, which implies $k_2\le k_1$ and $(i_d+1)2^{k_2}\le 2^{k_1+1}$. According to \eqref{eq:part1}, we also have $$ (2^{k_2}/2^{k_1})^2\sim 2^{-n}/(2^{k_1})^{2-\alpha}, $$ which allows us to apply the interior estimates after a scaling. The quasi-metric $\varrho: \Omega_\infty\times \Omega_\infty\to [0,\infty)$ is defined as $$ \varrho((t,x),(s,y))=|t-s|^{1/(2-\alpha)} +\min\big\{|x-y|,|x-y|^{2/(2-\alpha)}\min\{x_d,y_d\}^{-\alpha/(2-\alpha)}\big\}. $$ There exists a constant $K_1=K_1(d,\alpha)>0$ such that $$ \varrho((t,x),(s,y))\le K_1\big(\varrho((t,x),(\hat t,\hat x))+ \varrho((\hat t,\hat x),(s,y))\big) $$ for any $(t,x),(s,y),(\hat t,\hat x)\in \Omega_\infty$, and $ \varrho((t,x),(s,y))=0$ if and only if $(t,x)=(s,y)$. Besides, the cylinder $Q_\rho^+(z_0)$ defined in \eqref{def:Q} is comparable to $$ \{(t,x)\in \Omega: t<t_0,\, \varrho((t,x),(t_0,x_0))<\rho \}. $$ Of course, $(\Omega_T, \varrho)$ equipped with the Lebesgue measure is a space of homogeneous type and we have the above dyadic decomposition. The dyadic maximal function and sharp function of a locally integrable function $f$ and a given weight $\omega$ in $\Omega_\infty$ are defined as \begin{align*} \cM_{\text{dy},\omega} f(z)&=\sup_{n<\infty}\frac{1}{\omega(C_n(z))}\int_{C_n(z)\in \mathbb{C}_n}|f(s,y)| \omega(s,y)\,dyds,\\ f_{\text{dy},\omega}^{\#}(z)&=\sup_{n<\infty}\frac{1}{\omega(C_n(z))}\int_{C_n(z)\in \mathbb{C}_n}|f(s,y)-(f)_{C_n(z),\omega}|\omega(s,y)\,dyds. \end{align*} Observe that the average notation in \eqref{everage-def} is used in the above definition. Similarly, the maximal function and sharp function over cylinders are given by \begin{align*} \cM_\omega f(z)&=\sup_{z\in Q^+_\rho(z_0), z_0\in \overline{\Omega_\infty}} \frac{1}{\omega(Q_\rho^+(z_0))} \int_{Q_\rho^+(z_0)}|f(s,y)|\omega(s,y)\,dyds,\\ f^{\#}_\omega(z)&=\sup_{z\in Q^+_\rho(z_0),z_0\in \overline{\Omega_\infty}}\frac{1}{\omega(Q_\rho^+(z_0))}\int_{Q_\rho^+(z_0)}|f(s,y)-(f)_{Q^+_\rho(z_0)}|\omega(s,y)\,dyds. \end{align*} We have, for any $z\in \Omega_\infty$, $$ \cM_{\text{dy},\omega} f(z)\le N\cM_{\omega} f(z) \qquad \text{ and } \qquad f_{\text{dy},\omega}^{\#}(z)\le Nf^{\#}_\omega(z), $$ where $N=N(d,\alpha)>0$. We also recall the following definition of the $A_p$ Muckenhoupt class of weights. \begin{definition} \label{Def-Muck-wei} For each $p \in (1, \infty)$ and for a nonnegative Borel measure $\sigma$ on $\mathbb{R}^d$, a locally integrable function $\omega : \mathbb{R}^d \rightarrow \mathbb{R}_+$ is said to be in the $A_p( \mathbb{R}^d, \sigma)$ Muckenhoupt class of weights if and only if $[\omega]_{A_p(\mathbb{R}^d, \sigma)} < \infty$, where \begin{equation} \label{Ap.def} \begin{split} & [\omega]_{A_p(\mathbb{R}^d, \sigma)} \\ & = \sup_{\rho >0,x =(x', x_d)\in \mathbb{R}^d } \bigg[\fint_{B_{\rho} (x)} \omega(y)\, \sigma(dy) \bigg]\bigg[\fint_{B_{\rho}(x)} \omega(y)^{\frac{1}{1-p}}\, \sigma(dy) \bigg]^{p-1}. \end{split} \end{equation} Similarly, the class of weights $A_p(\mathbb{R}^d_+, \sigma)$ can be defined in the same way in which the ball $B_{\rho} (x)$ in \eqref{Ap.def} is replaced with $B_\rho^+(x)$ for $x\in \overline{\mathbb{R}^d_+}$. For weights with respect to the time variable, the definition is similar with the balls replaced with intervals $(t_0 -\rho^{2-\alpha}, t_0 + \rho^{2-\alpha})$ and $\sigma(dy)$ replaced with $dt$. If $\sigma$ is a Lebesgue measure, we simply write $A_p(\mathbb{R}^d_+) = A_p(\mathbb{R}^d_+, dx)$ and $A_p(\mathbb{R}^d) = A_p(\mathbb{R}^d, dx)$. Note that if $\omega \in A_p(\mathbb{R})$, then $\tilde{\omega} \in A_p(\mathbb{R}^d)$ with $[\omega]_{A_p(\mathbb{R})} = [\tilde{\omega}]_{A_p(\mathbb{R}^d)}$, where $\tilde{\omega}(x) = \omega(x_d)$ for $x = (x', x_d) \in \mathbb{R}^d$. Sometimes, if the context is clear, we neglect the spatial domain and only write $\omega \in A_p$. \end{definition} The following version of the weighted mixed-norm Fefferman-Stein theorem and Hardy-Littlewood maximal function theorem can be found in \cite{Dong-Kim-18}. \begin{theorem} \label{FS-thm} Let $p, q \in (1,\infty)$, $\gamma_1 \in (-1, \infty)$, $K\geq 1$, and $\mu_1(dz) = x_d^{\gamma_1}\, dxdt$. Suppose that $\omega_0\in A_q(\mathbb{R})$ and $\omega_1 \in A_p(\mathbb{R}^{d}_{+},\mu_1)$ satisfy $$ [\omega_0]_{A_q}, \,\, [\omega_{1}]_{A_p(\mathbb{R}_+^d, \mu_1)}\le K.$$ Then, for any $f \in L_{q, p}(\Omega_T, \omega\, d\mu_1)$, we have \begin{align*} & \|f\|_{L_{q, p}(\Omega_T, \omega\, d\mu_1)} {\leq N \| f^{\#}_{\text{dy},\mu_1}\|_{L_{q,p}(\Omega_T, \omega\, d \mu_1)}} \leq N \| f^{\#}_{\mu_1}\|_{L_{q,p}(\Omega_T, \omega\, d \mu_1)}, \\ & \|\mathcal{M}_{\mu_1}(f)\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \leq N \|f\|_{L_{q, p}(\Omega_T, \omega\, d \mu_1)}, \end{align*} where $N = N(d, q, p, \gamma_1, K)>0$ and $\omega(t,x) = \omega_0(t)\omega_1(x)$ for $(t,x) \in \Omega_T$. \end{theorem} \section{Equations with coefficients depending only on the \texorpdfstring{$x_d$}{} variable} \label{sec:3} In this section, we consider \eqref{eq:main} when the coefficients in \eqref{eq:main} only depend on the $x_d$ variable. Let us denote \begin{equation} \label{L0-def} \sL_0 u = \bar{a}_0(x_d) u_t+\lambda \bar{c}_0(x_d) u-\mu(x_d) \bar{a}_{ij}(x_d)D_iD_j u. \end{equation} where $\mu, \bar{a}_0, \bar{c}_0, \bar{a}_{ij}: \mathbb{R}_+ \rightarrow \mathbb{R}$ are given measurable functions and they satisfy \eqref{con:mu}-\eqref{con:ellipticity}. We consider \begin{equation}\label{eq:xd} \left\{ \begin{array}{cccl} \sL_0 u & = & \mu(x_d) f \quad &\text{ in } \Omega_T,\\ u & = & 0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+. \end{array} \right. \end{equation} The main result of this section is the following theorem, which is a special case of Corollary \ref{cor1}. \begin{theorem}\label{thm:xd} Assume that $\bar{a}_0, \bar{c}_0, (\bar{a}_{ij})$ satisfy \eqref{con:mu}--\eqref{con:ellipticity} and assume further that $f \in {L_p(\Omega_T,x_d^\gamma\, dz)}$ for some given $p>1$ and \[ \gamma \in \big(p(\alpha-1)_+-1,2p-1\big). \] Then, \eqref{eq:xd} admits a strong unique solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$. Moreover, \begin{align} \notag & \|\mathbf{M}^{-\alpha}u_t\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} + \|D^2 u\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} \\ \label{eq:xd-main} & \quad \qquad +\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} \le N\|f\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)}; \end{align} and if $\gamma\in (\alpha p/2-1,2p-1)$, we also have \begin{equation} \label{eq3.09} \lambda^{1/2}\|\mathbf{M}^{-\alpha/2}Du\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)} \le N\|f\|_{L_p(\Omega_T,x_d^{\gamma}\, dz)}, \end{equation} where $N=N(d,\nu,\alpha, \gamma, p)>0$. \end{theorem} The proof of Theorem \ref{thm:xd} requires various preliminary results and estimates. Our starting point is Lemma \ref{l-p-sol-lem} below which gives Theorem \ref{thm:xd} when $\gamma$ is large. See Subsection \ref{subsec:L2} below. Then, in Subsections \ref{subsec:boundary} and \ref{subsec:int}, we derive pointwise estimates for solutions to the corresponding homogeneous equations. Afterwards, we derive the oscillation estimates for solutions in Subsection \ref{subsec:osc-est}. The proof of Theorem \ref{thm:xd} will be given in the last subsection, Subsection \ref{proof-xd}. Before starting, let us point out several observations as well as recall several needed definitions. Note that by dividing the PDE in \eqref{eq:xd} by $\bar{a}_0$ and then absorbing $\bar{a}_{dd}$ into $\mu(x_d)$, without loss of generality, we may assume that \begin{equation} \label{add-cond} \bar{a}_{dd}=1 \qquad \text{ and } \qquad \bar{a}_0 =1. \end{equation} Observe that \eqref{eq:xd} can be rewritten into a divergence form equation \begin{equation} \label{eq:xd-div} \bar{a}_0 u_t+\lambda \bar{c}_0(x_d)u-\mu(x_d) D_i(\tilde a_{ij}(x_d) D_{j} u)=\mu(x_d)f \quad \text{ in } \Omega_T, \end{equation} where \begin{equation} \label{eq:change} \tilde a_{ij}=\left\{ \begin{array}{ll} \bar{a}_{ij}+ \bar{a}_{ji} & \hbox{for $i\neq d$ and $j=d$;} \\ 0 & \hbox{for $i=d$ and $j\neq d$;} \\ \bar{a}_{ij} & \hbox{otherwise.} \end{array} \right. \end{equation} We note that even though $(\tilde a_{ij})$ is not symmetric, it still satisfies the ellipticity condition \eqref{con:ellipticity} and also $\tilde a_{dd} =1$ when \eqref{add-cond} holds. Due to the divergence form as in \eqref{eq:xd-div}, we need the definition of its weak solutions. In fact, sometimes in this section, we consider the following class of equations in divergence form which are slightly more general than \eqref{eq:xd-div} \begin{equation} \label{eq:dx-loc} u_t + \lambda \bar{c}_0(x_d) u - \mu(x_d)D_i (\tilde{a}_{ij}(x_d)D_j u - F_i) = \mu(x_d) f \quad \text{in} \quad (S, T) \times \mathcal{D} \end{equation} with the boundary condition \begin{equation*} u = 0 \quad \text{on} \quad (S, T) \times (\overline{\mathcal{D}} \cap \{x_d =0\}) \end{equation*} for some open set $\mathcal{D} \subset \mathbb{R}^d_+$ and $-\infty \leq S < T \leq \infty$. \begin{definition} For a given weight $\omega$ defined on $(S, T) \times \mathcal{D} $ and for given $F= (F_1, F_2, \ldots, F_2) \in L_{p, \text{loc}}((S, T) \times \mathcal{D} )^d$ and $f \in L_{p, \text{loc}}((S, T) \times \mathcal{D} )$, we say that a function $u \in \cH_p^1((S, T) \times \mathcal{D} , \omega)$ is a weak solution of \eqref{eq:dx-loc} if \begin{equation} \label{def-local-weak-sol} \begin{split} & \int_{(S, T) \times \mathcal{D} }\mu(x_d)^{-1}(-u \partial_t \varphi + \lambda \overline{c}_0 u \varphi)dz + \int_{(S, T) \times \mathcal{D} } (\tilde{a}_{ij} D_ju - F_i) D_i \varphi dz \\ & = \int_{(S, T) \times \mathcal{D} } f(z) \varphi(z) dz, \quad \forall \ \varphi \in C_0^\infty((S, T) \times \mathcal{D} ). \end{split} \end{equation} \end{definition} \subsection{\texorpdfstring{$L_p$}{} strong solutions when the powers of weights are large} \label{subsec:L2} The following lemma is the main result of this subsection, which gives Theorem \ref{thm:xd} when $\gamma \in (p-1, 2p-1)$. \begin{lemma} \label{l-p-sol-lem} Let $\nu \in (0,1)$, $\lambda>0$, $\alpha \in (0,2)$, $p \in (1, \infty)$, and $\gamma \in (p-1, 2p-1)$. Assume that $\bar{a}_0, \bar{c}_0, (\bar{a}_{ij})$, and $\mu$ satisfy the ellipticity and boundedness conditions \eqref{con:mu}--\eqref{con:ellipticity}. Then, for any $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, there exists a unique strong solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ to \eqref{eq:xd}. Moreover, for every solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ of \eqref{eq:xd} with $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, it holds that \begin{align} \notag & \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}dz)} \\ \label{est-0405-1} & \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}dz)} \leq N \|f\|_{L_p(\Omega_T, x_d^{\gamma}dz)}, \end{align} where $N = N(d, \alpha, \nu, \gamma, p)>0$. \end{lemma} \begin{proof} The key idea is to apply \cite[Theorem 2.4]{DPT21} to the divergence form equation \eqref{eq:xd-div}, and then use an idea introduced by Krylov in \cite[Lemma 2.2]{Kr99} with a suitable scaling. To this end, we assume that \eqref{add-cond} holds, and let us denote $\gamma' = \gamma -p \in (-1, p-1)$ and we observe that \[ x_d^{1-\alpha}\mu(x_d) |f(z)| \sim x_d |f(z)| \in L_p(\Omega_T, x_d^{\gamma'}dz). \] As $\gamma' \in (-1, p-1)$, we have $x_d^{\gamma'} \in A_p$. Moreover, the equation \eqref{eq:xd} can be written in divergence form as \eqref{eq:xd-div}. Therefore, we apply \cite[Theorem 2.4]{DPT21} to \eqref{eq:xd-div} with $f_1=\mu(x_d) f$ and $f_2= 0$ to yield the existence of a unique weak solution $u \in \mathscr{H}^1_p(\Omega_T, x_d^{\gamma'}dz)$ of \eqref{eq:xd-div} satisfying \begin{align} \notag & \|Du\|_{L_{p}(\Omega_T,x_d^{\gamma'} dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}u\|_{L_{p}(\Omega_T,x_d^{\gamma'}dz)} \\ \label{Du-0226} & \le N\|{x_d^{1-\alpha}f_1}\|_{L_{p}(\Omega_T,x_d^{\gamma'}dz)} = N \|f\|_{L_p(\Omega_T, x_d^\gamma dz)}, \end{align} with $N = N(d, \nu, \gamma, p)>0$. We note here that because the coefficients $\bar{c}_0, \bar{a}_{ij}$ only depend on $x_d$, \cite[Theorem 2.4]{DPT21} holds for any $\lambda>0$ by a scaling argument. From \eqref{Du-0226}, the zero boundary condition, and the weighted Hardy inequality (see \cite[Lemma 3.1]{DP-AMS} for example), we infer that \begin{align} \notag \|u\|_{L_p(\Omega_T, x_d^{\gamma -2p}dz)} & = \|\mathbf{M}^{-1}u\|_{L_p(\Omega_T, x_d^{\gamma'}dz)} \leq N \|Du\|_{L_{p}(\Omega_T,x_d^{\gamma'})} \\ \label{u-op-wei-0226} & \leq \|f\|_{L_p(\Omega_T, x_d^\gamma)}. \end{align} It remains to prove that \eqref{est-0405-1} holds as it also implies that $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma)$. We apply the idea introduced by Krylov in \cite[Lemma 2.2]{Kr99} and combine it with a scaling argument to remove the degeneracy of the coefficients. See also \cite[Theorem 3.5]{DK15} and \cite[Lemma 4.6]{DP-JFA}. To this end, let us fix a standard non-negative cut-off function $\zeta \in C_0^\infty((1,2))$. For each $r >0$, let $\zeta_r(s) =\zeta(rs)$ for $s \in \mathbb{R}_+$. Note that with a suitable assumption on the integrability of a given function $v: \Omega_T \rightarrow \mathbb{R}$ and for $\beta \in \mathbb{R}$, by using the substitution $r^{\alpha}t \mapsto s$ for the integration with respect to the time variable, and then using the Fubini theorem, we have \begin{equation} \label{weight-kry} \begin{split} & \int_0^\infty \left(\int_{\Omega_{r^{-\alpha}T}}|\zeta_r(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_1\int_{\Omega_T} |v(z)|^p x_d^{\beta+\alpha}\, dz, \\ & \int_0^\infty \left(\int_{\Omega_{r^{-\alpha} T}}|\zeta_r'(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_2\int_{\Omega_T} |v_r(z)|^p x_d^{\beta + \alpha -p}\, dz, \\ & \int_0^\infty \left(\int_{\Omega_{r^{-\alpha} T}}|\zeta_r''(x_d) v_r(z)|^p\, dz\right) r^{-\beta-1}\, dr =N_3\int_{\Omega_T} |v_r(z)|^p x_d^{\beta + \alpha-2p}\, dz, \end{split} \end{equation} where $v_r(z) = v(r^{\alpha}t, x)$ for $z = (t, x) \in \Omega_{r^{-\alpha}T}$, \[ N_1 = \int_0^\infty |\zeta (s)|^p s^{-\beta-\alpha-1} ds, \quad N_2 = \int_0^\infty |\zeta'(s)|^p s^{p-\beta-\alpha-1} ds, \] and \[ N_3 = \int_0^\infty |\zeta''(s)|^p s^{2p-\beta-\alpha -1} ds. \] Next, for $r>0$, we denote $u_r(z) = u(r^{\alpha}t,x)$, \[ \hat{a}_{ij} (x_d)= r^{\alpha}\mu(x_d) \bar{a}_{ij}(x_d), \quad \bar{\lambda} = \lambda r^{\alpha}, \quad \text{and} \quad f_r (z) = r^{\alpha} \mu(x_d) f(r^{\alpha}t, x). \] Note that $u_r$ solves the equation \[ \partial_t u_r + \bar{\lambda} \bar{c}_0 u_r - \hat{a}_{ij}(x_d) D_i D_j u_r = f_r \quad \text{in} \quad \Omega_{r^{-\alpha}T}. \] Let $w(z) =\zeta_r(x_d) u_r(z)$, which satisfies \begin{equation} \label{w-eqn-0405-1} w_t + \bar{\lambda} \bar{c}_0(x_d) w - \hat{a}_{ij}(x_d) D_i D_j w = \hat{g} \quad \text{in} \quad \Omega_{r^{-\alpha}T} \end{equation} with the boundary condition $w(z', 0) =0$ for $z' \in (-\infty, r^{-\alpha}T)\times \mathbb{R}^{d-1}$, where \[ \begin{split} \hat{g}(z) & = \zeta_r f_r(z) - \hat{a}_{dd} \zeta''_r u_r - \sum_{i \neq d} \big(\hat{a}_{id} + \hat{a}_{di}\big)\zeta '_rD_i u_r. \end{split} \] We note that $\text{supp}(w) \subset (-\infty, r^{-\alpha}T) \times \mathbb{R}^{d-1} \times(1/r, 2/r)$, and on this set the coefficient matrix $(\hat{a}_{ij})$ is uniformly elliptic and bounded as $r^{\alpha}\mu (x_d) \sim 1$ due to \eqref{con:mu}. We now prove \eqref{est-0405-1} with the extra assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$. Under this assumption and as $\zeta_r$ is compactly supported in $(0, \infty)$, we see that $w \in W^{1,2}_p(\Omega_{r^{\alpha}T})$, the usual parabolic Sobolev space. Then by applying the $W^{1, 2}_{p}$-estimate for the uniformly elliptic and bounded coefficient equation \eqref{w-eqn-0405-1} (see, for instance, \cite{D12}), we obtain \[ \bar{\lambda} \|w\| + \bar{\lambda}^{1/2}\, \|Dw\| + \|D^2 w\| + \|w_t\| \leq N \|\hat{g}\|, \] where $\|\cdot \| = \| \cdot \|_{L_p(\Omega_{r^{-\alpha}T})}$ and $N = N(d, \nu, p)>0$. From this, the definition of $\hat{g}$, and a simple manipulation, we obtain \[ \begin{split} & \lambda r^{\alpha} \|\zeta_r u_r \| + \sqrt{\lambda} r^{\alpha/2} \|\zeta_r Du_r\| + \|\zeta_r D^2u_r\| +\|\zeta_r \partial_t u_r \| \\ & \leq N\Big[ \|\zeta_r f_r\| + \sqrt{\lambda} r^{\alpha/2} \|\zeta_r' u_r\| + \|\zeta''_r u_r\| + \|\zeta_r' Du_r\| \Big]. \end{split} \] Now, we raise this last estimate to the power $p$, multiply both sides by $r^{-(\gamma-\alpha)-1}$, integrate the result with respect to $r$ on $(0,\infty)$, and then apply \eqref{weight-kry} to obtain \[ \begin{split} & \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} +\sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \\ & \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}\\ & \leq N \Big[ \|f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} +\sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}u\|_{L_p(\Omega_T, x_d^{\gamma -p}\, dz)} + \|u\|_{L_p(\Omega_T, x_d^{\gamma -2p}\, dz)} \\ & \qquad + \|Du\|_{L_p(\Omega_T, x_d^{\gamma -p}\, dz)}\Big]. \end{split} \] From the last estimate, \eqref{Du-0226}, \eqref{u-op-wei-0226}, and the fact that $\gamma' = \gamma-p$, we infer that \[ \begin{split} & \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \sqrt{\lambda} \|\mathbf{M}^{-\alpha/2}Du \|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} \\ & \qquad + \| D^2u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} + \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} \leq N \|f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)}. \end{split} \] This proves \eqref{est-0405-1} under the additional assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}\,dz)$. It remains to remove the extra assumption that $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$. By mollifying the equation \eqref{eq:xd} in $t$ and $x'$ and applying \cite[Theorem 2.4]{DPT21} to the equations of $u^{(\varepsilon)}_t$ and $D_{x'}u^{(\varepsilon)}$, we obtain $$ \mathbf{M}^{-\alpha}u^{(\varepsilon)}, \mathbf{M}^{-\alpha} u_t^{(\varepsilon)}, DD_{x'}u^{(\varepsilon)} \in L_p(\Omega_T, x_d^{\gamma'} dz). $$ This and the PDE in \eqref{eq:xd} for $u^{(\varepsilon)}$ imply that \[ D_{dd} u^{(\varepsilon)} \in L_p(\Omega_T, x_d^{\gamma'} dz). \] Therefore $u^{(\varepsilon)} \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma'}dz)$ is a strong solution of \eqref{eq:xd} with $f^{(\varepsilon)}$ in place of $f$. From this, we apply the a priori estimate \eqref{est-0405-1} that we just proved for $u^{(\varepsilon)}$ and pass to the limit as $\varepsilon \rightarrow 0^+$ to obtain the estimate \eqref{est-0405-1} for $u$. The proof of the lemma is completed. \end{proof} \subsection{Boundary H\"older estimates for homogeneous equations} \label{subsec:boundary} Recall the operator $\sL_0$ defined in \eqref{L0-def}. In this subsection, we consider the homogeneous equation \begin{equation} \label{eq:hom} \begin{cases} \sL_0u=0 \quad &\text{ in } Q_1^+,\\ u=0 \quad &\text{ on } Q_1\cap \{x_d=0\}. \end{cases} \end{equation} As above, {without loss of generality we assume \ref{add-cond} so that} \eqref{eq:hom} can be written in divergence form as \begin{equation} \label{eq:hom-div} \left\{ \begin{array}{cccl} u_t+\lambda \bar{c}_0(x_d)u-\mu(x_d) D_i(\tilde a_{ij}(x_d) D_{j} u) & = &0 & \quad \text{ in } Q_1^+,\\ u & = & 0 & \quad \text{on} \quad Q_1\cap \{x_d=0\}. \end{array} \right. \end{equation} A function $u \in \cH_{p}^1(Q_1^+)$ with $p \in (1,\infty)$ is said to be a weak solution of \eqref{eq:hom} if it is a weak solution of \eqref{eq:hom-div} in the sense defined in \eqref{def-local-weak-sol} and $u=0$ on $Q_1\cap \{x_d=0\}$ in the sense of trace. For each $\beta \in (0,1)$, the $\beta$-H\"older semi-norm in the spatial variable of a function $u$ on an open set $Q\subset \mathbb{R}^{d+1}$ is given by \[ \llbracket u\rrbracket_{C^{0, \beta}(Q)} = \sup\left\{ \frac{|u(t,x) - u(t,y)|}{|x-y|^{\beta}}: x \not =y, \ (t,x), (t,y) \in Q \right\}. \] For $k, l \in \mathbb{N} \cup \{0\}$, we denote \[ \|u\|_{C^{k, l}(Q)} = \sum_{i=0}^k \sum_{|j| \leq l}\|\partial_t^i D_{x}^j u\|_{L_\infty(Q)} . \] We also use the following H\"{o}lder norm of $u$ on $Q$ \[ \|u\|_{C^{k, \beta}(Q)} = \|u\|_{C^{k,0}(Q)} + \sum_{i=0}^{k} \llbracket \partial_t^i u\rrbracket_{C^{0, \beta}(Q)}. \] We begin with the following Caccioppoli type estimate. \begin{lemma} \label{caccio} Suppose that $u\in \cH^1_2(Q_1^+)$ is a weak solution of \eqref{eq:hom}. Then, for any integers $k,j\ge 0$ and $l=0,1$, \begin{equation} \label{eq:hom-b1} \int_{Q_{1/2}^+}|\partial_t^k D_{x'}^j D_{d}^l u|^2 \,dz \le N \int_{Q_{1}^+} u^2 \,dz \end{equation} where $N = N(d,\nu,\alpha,k,j,l)>0$. \end{lemma} \begin{proof} Again, we can assume \eqref{add-cond} holds. The estimate \eqref{eq:hom-b1} follows from \cite[(4.12)]{DPT21} applied to \eqref{eq:hom-div}. \end{proof} \begin{lemma} \label{lem:boundary} Let $p_0 \in (1, \infty)$ and suppose that $u\in \cH^1_{p_0}(Q_1^+)$ is a weak solution of \eqref{eq:hom}. Then, \begin{equation} \label{eq:hom-b2} \begin{split} & \|u\|_{C^{1,1}(Q_{1/2}^+)}+\|D_{x'}u\|_{C^{1,1}(Q_{1/2}^+)}+\|D_d u\|_{C^{1,\delta_0}(Q_{1/2}^+)} \\ & + \sqrt{\lambda}\|\mathbf{M}^{-\alpha/2}u\|_{C^{1,1-\alpha/2}(Q_{1/2}^+)} \le N \|Du\|_{L_{p_0}(Q_1^+)}, \end{split} \end{equation} where $N=N(d,\nu,\alpha, p_0)>0$ and $\delta_0=\min\{2-\alpha,1\}$. \end{lemma} \begin{proof} As explained, we can assume that \eqref{add-cond} holds. We apply \cite[Lemma 5.5]{DPT21} to \eqref{eq:hom-div} by noting that $U:=\tilde a_{dj}D_j u=D_du$ in view of \eqref{add-cond} and \eqref{eq:change}. \end{proof} \begin{lemma} \label{prop:boundary} Let $p_0 \in (1, \infty)$, $\beta_0\in {(-\infty}, \min\{1,\alpha\}]$, and $\alpha_0 > -1$ be fixed constants. There exists a number $\beta_1=\beta_1(\alpha,\beta_0) \in (0,1]$ such that for every weak solution $u\in \cH^1_{p_0}(Q_1^+)$ to \eqref{eq:hom}, it holds that \begin{align} \label{eq:hom-b3} \|\mathbf{M}^{-\beta_0} u\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{-\beta_0} u\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},\\ \label{eq:hom-b4} \|\mathbf{M}^{-\beta_0} u_t\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{-\beta_0} u_t\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)},\\ \label{eq:hom-b5} \|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u\|_{C^{1,\beta_1}(Q_{1/2}^+)}&\le N\|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}, \end{align} and \begin{equation} \label{eq:hom-b-Du} \|\mathbf{M}^{\beta_0}Du\|_{C^{1,\beta_1}(Q_{1/2}^+)} \le N\|\mathbf{M}^{\beta_0}Du\|_{L_{p_0}(Q_{3/4}^+,x_d^{\alpha_0}dz)}, \end{equation} where $N=N(d,\nu,\alpha,\alpha_0,\beta_0, p_0)>0$. \end{lemma} \begin{proof} Again, we assume \eqref{add-cond}. Note that once the lemma with $\alpha_0 \geq 0$ is proved, the case when $\alpha_0 \in (-1, 0)$ will follow immediately. Hence, we only need to prove the lemma with the assumption that $\alpha_0 \geq 0$. We first assume $p_0 =2$. Since $\beta_0\le \min\{1,\alpha\}$, by \eqref{eq:hom-b1} and the boundary Poincar\'e inequality, the right-hand sides of \eqref{eq:hom-b3}, \eqref{eq:hom-b4}, and \eqref{eq:hom-b5} are all finite. We consider two cases. \noindent {\em Case 1: $\beta_0=0$.} When $\alpha_0=0$, \eqref{eq:hom-b3} and \eqref{eq:hom-b-Du} follow from \eqref{eq:hom-b2} and \eqref{eq:hom-b1}. For general $\alpha_0\ge 0$, by \eqref{eq:hom-b3} with $\beta_0=0$ and $\alpha_0=0$ and H\"older's inequality, we have \begin{align*} \|u\|_{L_\infty(Q_{1/2}^+)}& \le N \|u\|_{L_2(Q_{2/3}^+)} \leq N\|u\|^{2\alpha_0/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{-1/2}\,dz)} \|u\|^{1/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)}\\ &\le N\|u\|^{2\alpha_0/(1+2\alpha_0)}_{L_\infty(Q_{2/3}^+)} \|u\|^{1/(1+2\alpha_0)}_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)} \\ & \leq \frac{1}{2} \|u\|_{L_\infty(Q_{2/3}^+)} + N\|u\|_{L_2(Q_{2/3}^+,x_d^{\alpha_0}\,dz)}, \end{align*} where $N = N(d, \nu, \alpha, \alpha_0)>0$. From this and the standard iteration argument (see \cite[p. 75]{HanLin} for example), we obtain \begin{equation} \label{est.alpha-zero} \|u\|_{L_\infty(Q_{1/2}^+)} \leq N \|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}. \end{equation} The above, together with Lemma \ref{caccio}, yields \begin{equation} \label{eq:hom-b6} \int_{Q_{1/2}^+}|\partial_t^k D_{x'}^j D_{d}^l u|^2 \,dz \le N(d,\nu,\alpha, \alpha_0,k,j,l) \int_{Q_{3/4}^+} u^2 x_d^{\alpha_0}\,dz \end{equation} for any integers $k,j\ge 0$ and $l=0,1$. Using this last estimate, \eqref{eq:hom-b2}, and by suitably adjusting the sizes of the cylinders, we obtain \eqref{eq:hom-b3} with $\beta_1 = 1$. Similar to \eqref{est.alpha-zero}, we have \[ \|Du\|_{L_\infty(Q_{1/2}^+)} \leq N \|Du\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}. \] From this, \eqref{eq:hom-b2}, and by shrinking the cylinders, we obtain \[ \|Du\|_{C^{1,\delta_0}(Q_{1/2}^+)} \leq N \|Du\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}dz)}, \quad \text{where} \,\, \delta_0=\min\{2-\alpha,1\}, \] which is \eqref{eq:hom-b-Du} when $\beta_0 =0$. Since $u_t$ and $D_{x'} u$ satisfy the same equation with the same boundary condition, similarly we also obtain \eqref{eq:hom-b4} as well as \begin{equation} \label{eq:hom-b7} \|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|DD_{x'}u\|_{L_2(Q_{2/3}^+)}, \end{equation} by Lemma \ref{lem:boundary}. This together with \eqref{eq:hom-b6} implies \eqref{eq:hom-b5} with $$\beta_1=\min\{\delta_0,\alpha \} = \min\{\alpha, 2-\alpha,1\}.$$ \noindent {\em Case 2: $\beta_0\neq 0$.} We first prove \eqref{eq:hom-b5}. By \eqref{eq:hom-b7} and by using the iteration argument as in \eqref{est.alpha-zero}, we have \[ \|DD_{x'} u\|_{L_\infty(Q_{1/2}^+)} \leq N\|DD_{x'} u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)}, \] where $N = N(d, \nu, \alpha, \alpha_0)>0$. Then, it follows from \eqref{eq:hom-b7} that \begin{equation} \label{eq:hom-b7-bis} \|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|DD_{x'}u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)}. \end{equation} Therefore, if $\beta_0=\alpha$, \eqref{eq:hom-b5} with $\beta_1=\delta_0$ follows from \eqref{eq:hom-b7-bis}. If $\beta_0<\alpha$, it follows from \eqref{eq:hom-b7-bis} that \[ \|DD_{x'}u\|_{C^{1,\delta_0}(Q_{1/2}^+)}\le N\|\mathbf{M}^{{\alpha-\beta_0}}DD_{x'}u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\, dz)} \] where $N = N(d, \nu, \alpha, \beta_0, \alpha_0)>0$. Then we also have \eqref{eq:hom-b5} with \[ \beta_1=\min\{\delta_0,\alpha-\beta_0\} = \min\{2-\alpha,1,\alpha-\beta_0\}. \] Similarly, \eqref{eq:hom-b-Du} can be deduced from \eqref{eq:hom-b-Du} when $\beta_0 =0$ by taking $\beta_1 = \min\{\delta_0, \beta_0\}$. Hence, both \eqref{eq:hom-b5} and \eqref{eq:hom-b-Du} hold with \[ \beta_1=\min\{\delta_0,\alpha-\beta_0,{\beta_0}\} = \min\{2-\alpha,1,\alpha-\beta_0, \beta_0\}. \] Next we show \eqref{eq:hom-b3}. Since $\beta_0\le 1$, using the zero boundary condition, \eqref{eq:hom-b2}, and \eqref{eq:hom-b6}, we get \begin{equation} \label{eq:hom-b8} \|\mathbf{M}^{-\beta_0} u\|_{L_\infty(Q_{1/2}^+)} \le N \|D_d u\|_{L_\infty(Q_{1/2}^+)}\le N\|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}. \end{equation} Since $u_t$ and $D_{x'} u$ satisfy the same equation and the same boundary condition, we have \begin{align} \label{eq:hom-b9} &\|\mathbf{M}^{-\beta_0} u_t\|_{L_\infty(Q_{1/2}^+)} +\|\mathbf{M}^{-\beta_0}D_{x'} u\|_{L_\infty(Q_{1/2}^+)}\notag\\ &\le \ N \|D_d u_t\|_{L_\infty(Q_{1/2}^+)}+N \|D_dD_{x'} u\|_{L_\infty(Q_{1/2}^+)}\notag\\ &\le \ N \|D u\|_{L_2(Q_{2/3}^+,x_d^{\alpha_0})}\le N\|u\|_{L_2(Q_{3/4}^+,x_d^{\alpha_0}\,dz)}, \end{align} where we used \eqref{eq:hom-b2}. To estimate the H\"older semi-norm of $\mathbf{M}^{-\beta_0} u$ in $x_d$, we write $$ x_d^{-\beta_0} u(t,x)=x_d^{1-\beta_0}\int_0^1 (D_d u)(t,x',sx_d)\,ds $$ and use \eqref{eq:hom-b2} and \eqref{eq:hom-b6}. Then we see that \[ \llbracket \mathbf{M}^{-\beta_0} u \rrbracket_{C^{0, \beta_1}(Q_{1/2}^+)} + \llbracket \mathbf{M}^{-\beta_0} \partial_t u \rrbracket_{C^{0, \beta_1}(Q_{1/2}^+)} \leq N \|\mathbf{M}^{-\beta_0} u\|_{L_2(Q_{3/4}^+, x_d^{\alpha_0}\,dz)} \] where $\beta_1 = \min\{\delta_0, 1-\beta_0\}$. Combining this with \eqref{eq:hom-b8} and \eqref{eq:hom-b9}, we reach \eqref{eq:hom-b3}. Note that $u_t$ satisfies the same equation and the same boundary condition, we deduce \eqref{eq:hom-b4} from \eqref{eq:hom-b3}. The proof of the lemma when $p_0 =2$ is completed. Next, we observe that when $p_0 >2$, the estimates \eqref{eq:hom-b3}--\eqref{eq:hom-b-Du} can be derived directly from the case $p_0 =2$ that we just proved using H\"{o}lder's inequality. On the other hand, when $p_0 \in (1, 2)$, it follows from Lemma \ref{lem:boundary} that $u \in \cH_2^1(Q_{3/4}^+)$. Then, by shrinking the cylinders, we apply the assertion when $p_0=2$ that we just proved and an iteration argument as in the proof of \eqref{est.alpha-zero} to obtain the estimates \eqref{eq:hom-b3}--\eqref{eq:hom-b-Du}. \end{proof} \begin{remark} The number $\beta_1$ defined in Lemma \ref{prop:boundary} can be found explicitly. However, we do not need to use this in the paper. \end{remark} \subsection{Interior H\"older estimates for homogeneous equations} \label{subsec:int} Fix a point $z_0 = (t_0, x_0) \in \Omega_T$, where $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$. For $0<\rho < x_{0d}$ and $\beta \in (0,1)$, we define the weighted $\beta$-H\"{o}lder semi-norm of a function $u$ on $Q_\rho(z_0)$ by \[ \begin{split} \llbracket u\rrbracket_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))} & = \sup \Big\{ \frac{|u(s,x) - u(t, y)|}{\big(x_{0d}^{-\alpha/2}|x-y| + |t-s|^{1/2}\big)^{\beta}}: (s,x) \not=(t,y) \\ & \qquad \qquad \qquad \text{and } (s,x), (t,y) \in Q_\rho(z_0) \Big\}. \end{split} \] As usual, we denote the corresponding weighted norm by \[ \|u\|_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))} = \|u\|_{L_\infty(Q_\rho(z_0))} + \llbracket u\rrbracket_{C^{\beta/2, \beta}_{\alpha}(Q_\rho(z_0))}. \] The following result is the interior H\"older estimates of solutions to the homogeneous equation \eqref{eq:hom-div}. \begin{lemma}\label{prop:int} Let $z_0 = (t_0, x_0) \in \Omega_T$, where $x_0 = (x_0', x_{0d}) \in \mathbb{R}^{d-1} \times \mathbb{R}_+$, and $\rho \in (0, x_{0d}/4)$. Let $u \in \sW_{p_0}^{1,2}(Q_{2\rho}(z_0))$ be a strong solution of \begin{equation*} \sL_0 u=0 \quad \text{ in } Q_{2\rho}(z_0) \end{equation*} with some $p_0 \in (1, \infty)$. Then for any $\beta \in \mathbb{R}$, \begin{align*} & \|\mathbf{M}^{\beta} u\|_{L_\infty(Q_{\rho}(z_0))} + \rho ^{(1-\alpha/2)/2} \llbracket \mathbf{M}^{\beta} u\rrbracket_{C^{1/4, 1/2}_\alpha(Q_{\rho}(z_0))} \\ &\leq N \left(\fint_{ Q_{2\rho}( z_0)} |x_{d}^{\beta}u|^{p_0} \mu_0(dz)\right)^{1/p_0}, \end{align*} and \begin{align*} & \|\mathbf{M}^{\beta}Du\|_{L_\infty(Q_{\rho}(z_0))} + \rho^{(1-\alpha/2)/2} \llbracket {\mathbf{M}^{\beta} Du}\rrbracket_{C^{1/4, 1/2}_\alpha(Q_{\rho}(z_0))} \\ &\leq N \left(\fint_{ Q_{2\rho}( z_0)} |x_d^{\beta}Du|^{p_0} \mu_0(dz) \right)^{1/p_0}, \end{align*} where $\mu_0(dz) = x_d^{\alpha_0}dtdx$ with some $\alpha_0 > -1$, and $N = N(\nu, d,\alpha, \alpha_0)>0$. \end{lemma} \begin{proof} As in the proof of Lemma \ref{prop:boundary}, we may assume that $p_0 =2$. Without loss of generality, we assume that $x_{0d}=1$. Note that when $\beta = 0$, the assertions follow directly from \cite[Proposition 4.6]{DPT21}. In general, the assertions follow from the case when $\beta =0$ and the fact that \[ \left(\fint_{ Q_{2\rho}( z_0)} |\mathbf{M}^{\beta}f(z)|^{p_0} \mu_0(dz) \right)^{1/p_0} \approx \left(\fint_{ Q_{2\rho}( z_0)} |f(z)|^{p_0} dz \right)^{1/p_0}. \] The lemma is proved. \end{proof} \subsection{Mean oscillation estimates} \label{subsec:osc-est} In this subsection, we apply Lemmas \ref{prop:boundary} and \ref{prop:int} to derive the mean oscillation estimates of $$ U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u) \qquad \text{and} \qquad Du $$ respectively with the underlying measure \begin{equation} \label{mu-1-def} \mu_1(dz) = x_d^{\gamma_1}\,dx dt \qquad \text{and} \qquad \bar{\mu}_1(dz) = x_d^{\bar{\gamma}_1} dxdt, \end{equation} where $u$ is a strong solution of \eqref{eq:xd}, \[ \gamma_1 \in (p_0(\beta_0-\alpha +1)-1, p_0(\beta_0-\alpha+2)-1) \quad \text{and} \quad \bar{\gamma}_1 =\gamma_1 + p_0 (\alpha /2-\beta_0) \] with some $p_0 \in (1, \infty)$ and $\beta_0 \in (\alpha-1, \min\{1, \alpha\}]$. The main result of the subsection is Lemma \ref{oscil-lemma-2} below. Let us point out that both $\mu_1$ and $\bar{\mu}_1$ depend on the choice of $\beta_0$, and \begin{equation} \label{mu1=bar-mu-1} \mu_1= \bar{\mu}_1 \quad \text{when} \quad \beta_0 = \alpha/2. \end{equation} To get the weighted estimate of $U$ in $L_p(\Omega_T, x_d^\gamma\, dz)$ with the optimal range for $\gamma$ as in Theorem \ref{thm:xd}, we will use $\beta_0 = \min\{1, \alpha\}$. On the other hand, to derive the estimate for $Du$, we will use $\beta_0 = \alpha/2$ and \eqref{mu1=bar-mu-1}. For the reader's convenience, let us also recall that for a cylinder $Q \subset \mathbb{R}^{d+1}$, a locally finite measure $\omega$, and an $\omega$-integrable function $g$ on $Q$, we denote the average of $g$ on $Q$ with respect to the measure $\omega$ by \[ (g)_{Q, \omega} = \frac{1}{\omega(Q)}\int_{Q} g(z)\, \omega(dz) \] and the average of $g$ on $Q$ with respect to the Lebesgue measure by \[ (g)_{Q} =\frac{1}{|Q|} \int_{Q} g(z)\, dz. \] We begin with the following lemma on the mean oscillation estimates of solutions to the homogeneous equations. \begin{lemma} \label{oscil-lemma-1} Let $\nu \in (0,1)$, $\alpha \in (0,2)$, $p_0 \in (1, \infty)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. There exists $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$ such that if $u \in \sW^{1,2}_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'}\, dz)$ is a strong solution of \[ \left\{ \begin{array}{cccl} \sL_0 u & =& 0 & \quad \text{in} \quad Q_{14\rho}^+(z_0) \\ u & = & 0 & \quad \text{on} \quad Q_{14\rho}(z_0) \cap \{x_d =0\} \end{array} \right. \] for some $\lambda>0, \rho>0$, $z_0 =(z_0', x_{d0}) \in \overline{\Omega}_T$, and for $\gamma_1' = \gamma_1 -p_0(\beta_0-\alpha)$, then \begin{equation} \label{osc-h} (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} \end{equation} and \begin{equation} \label{Du-osc-h} (|Du - (Du)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1} \leq N \kappa^{\theta} (|Du|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0} \end{equation} for every $\kappa \in (0,1)$, where $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def}, \[ U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u), \] and $\theta = \min\{\beta_1(\alpha, \beta_0), (2-\alpha)/4, 2-\alpha, 1\} \in (0,1)$ in which $\beta_1$ is defined in lemma \ref{prop:boundary}. \end{lemma} \begin{proof} By using the scaling \eqref{scaling}, we assume that $\rho =1$. We consider two cases: the boundary case and the interior one. \noindent {\em Boundary case.} Consider $x_{0d} <4$. Let $\bar{z} =(t_0, x_0', 0)$ and note that from the definition of cylinders in \eqref{def:Q}, we have \[ Q_{1}^+(z_0) \subset Q_{5}^+(\bar{z}_0) \subset Q_{10}^+(\bar{z}_0) \subset Q_{14}^+(z_0). \] Then, we apply the mean value theorem and the estimates \eqref{eq:hom-b3}-\eqref{eq:hom-b5} in Lemma \ref{prop:boundary} with $\gamma_1$ in place of $\alpha_0$, $\beta_0$ in place of $\beta$ in the estimates of $u$, $u_t$, and $DD_{x'}u$. We infer that \[ \begin{split} & (|U - (U)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\ & \leq N \kappa^{2-\alpha} \|\partial_t U\|_{L^\infty(Q_1(z_0))} + N \kappa^{\beta_1} \llbracket U \rrbracket_{C^{0, \beta_1}(Q_{1}^+(z_0))} \\ & \leq N \kappa^{\theta}\big[ \|\partial_t U\|_{L^\infty(Q_{5}^+(\bar{z}))} + \llbracket U \rrbracket_{C^{0, \beta_1}(Q_{5}^+(\bar{z}))} \big] \\ & \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{10}^+(\bar{z}), \mu_1}^{1/p_0} \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14}^+(z_0), \mu_1}^{1/p_0}, \end{split} \] where we used the doubling property of $\mu_1$ in the last step. This implies the estimate \eqref{osc-h} as $\kappa \in (0,1)$. To estimate the oscillation of $Du$ as asserted in \eqref{Du-osc-h}, we note that $$\bar{\gamma}_1 = \gamma_1 - p_0(\beta_0 -\alpha/2) > p_0 (1-\alpha/2)-1 >-1. $$ Therefore, \eqref{Du-osc-h} can be proved in a similar way as that of \eqref{osc-h} using the estimate \eqref{eq:hom-b-Du} in Lemma \ref{prop:boundary} with $\beta =0$ and $\alpha_0 = \bar{\gamma}_1 >-1$. \ \\ \noindent {\em Interior case.} Consider $x_{0d} >4\rho=4$. By using Lemma \ref{prop:int} with $\beta = -\beta_0$ and the doubling property of $\mu_1$, we see that \[ \begin{split} & (|\mathbf{M}^{-\beta_0} u - (\mathbf{M}^{-\beta_0} u)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\ & \leq N \kappa^{1/2-\alpha/4}\llbracket \mathbf{M}^{-\beta_0} u \rrbracket_{C_\alpha^{1/4, 1/2}(Q_{1}^+(z_0))} \\ & \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{2}^+(z_0)}|\mathbf{M}^{-\beta_0} u|^{p_0}\mu_1(dz) \right)^{1/p_0} \\ & \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{-\beta_0} u|^{p_0} \mu_1(dz) \right)^{1/p_0}. \end{split} \] Similarly, by using the finite difference quotient, we can apply Lemma \ref{prop:int} to $u_t$ and obtain \[ \begin{split} & (|\mathbf{M}^{-\beta_0} u_t - (\mathbf{M}^{-\beta_0} u_t)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\ & \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{-\beta_0} u_t|^{p_0} \mu_1(dz) \right)^{1/p_0}. \end{split} \] In the same way, by applying Lemma \ref{prop:int} to $D_{x'}u$ with $\gamma=\alpha-\beta_0$ and $\alpha_0 = \gamma_1$, we infer that \[ \begin{split} & (|\mathbf{M}^{\alpha -\beta_0} DD_{x'} u - (\mathbf{M}^{\alpha -\beta_0} DD_{x'} u)_{Q_{\kappa}^+(z_0), \mu_1}|)_{Q_{\kappa}^+(z_0), \mu_1} \\ & \leq N \kappa^{1/2-\alpha/4} \left(\fint_{Q_{14}^+(z_0)}|\mathbf{M}^{\alpha -\beta_0} DD_{x'} u|^{p_0} \mu_1(dz) \right)^{1/p_0}. \end{split} \] The oscillation estimate of $Du$ can be proved in a similar way. Therefore, we obtain \eqref{osc-h}. The proof of the lemma is completed. \end{proof} Now, we recall that for a given number $a \in \mathbb{R}$, $a_+ = \max\{a, 0\}$. We derive the oscillation estimates of solutions to the non-homogeneous equation \eqref{eq:xd}, which is the main result of the subsection. \begin{lemma} \label{oscil-lemma-2} Let $\nu \in (0,1)$, $\alpha \in (0,2)$, $p_0 \in (1, \infty)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. There exists $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$ such that the following assertions hold. Suppose that $u \in \sW^{1,2}_{p_0, \textup{loc}}(\Omega_T, x_d^{\gamma_1'}\, dz)$ is a strong solution of \eqref{eq:xd} with $f \in L_{p_0, \textup{loc}}(\Omega_T, x_d^{\gamma_1'}\, dz)$ and $\gamma_1' = \gamma_1 - p_0(\beta_0-\alpha)$. Then, for every $z_0 \in \overline{\Omega}_T$, $\rho \in (0, \infty)$, $\kappa \in (0,1)$, we have \[ \begin{split} & (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\ &\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \end{split} \] and \[ \begin{split} & \lambda^{1/2}(|Du - (Du)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0),\bar{\mu}_1} \\ & \leq N \kappa^{\theta} \lambda^{1/2} (|Du|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha/2} f|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0}, \end{split} \] where $\theta{\in (0,1)}$ is defined in Lemma \ref{oscil-lemma-1}, \[ U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u), \] and $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def}. \end{lemma} \begin{proof} As $\gamma_1 \in (p_0(\beta_0-\alpha +1)-1, p_0(\beta_0-\alpha+2)-1)$, we see that \[ \gamma_1' = \gamma_1 - p_0(\beta_0-\alpha) \in (p_0 -1, 2p_0 -1). \] Therefore, by Lemma \ref{l-p-sol-lem}, there is a strong solution $v \in \sW^{1,2}_{p_0}(\Omega_T, x_d^{\gamma_1'}\, dz)$ to \begin{equation} \label{v-sol-1} \left\{ \begin{array}{cccl} \sL_0 v & =& f \mathbf{1}_{Q_{14\rho}^+(z_0)} & \quad \text{in} \quad \Omega_T,\\ v & = & 0 & \quad \text{on} \quad \{x_d =0\} \end{array} \right. \end{equation} satisfying \begin{equation} \label{v-sol-est-1} \begin{split} \|\mathbf{M}^{-\alpha} v_t\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} & + \|D^2 v\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Dv\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)}\\ & + \lambda \|\mathbf{M}^{-\alpha} v\|_{L_{p_0}(\Omega_T, x_d^{\gamma_1'} dz)} \leq N \|f\|_{L_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'} dz)}. \end{split} \end{equation} Let us denote \[ V = (\mathbf{M}^{-\beta_0} v_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} v, \lambda \mathbf{M}^{-\beta_0}v). \] Then, it follows from \eqref{v-sol-est-1} and the definitions of $\mu_1$ and $\gamma_1'$ that \begin{equation} \label{V-osc-est-1} (|V|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \leq N (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}. \end{equation} Note also that due to \eqref{v-sol-est-1} and the definition of $\bar{\gamma}_1$, \[ \begin{split} \lambda^{1/2} \left( \int_{Q_{14\rho}^+(z_0)} | Dv|^{p_0} x_d^{\bar{\gamma}_1}dz \right)^{1/p_0} & = \lambda^{1/2} \left( \int_{Q_{14\rho}^+(z_0)} |\mathbf{M}^{-\alpha/2}Dv|^{p_0} x_d^{\gamma_1'} dz \right)^{1/p_0} \\ & \leq N \left( \int_{Q_{14\rho}^+(z_0)} |\mathbf{M}^{ \alpha/2} f|^{p_0} x_d^{\bar{\gamma}_1}dz \right)^{1/p_0}. \end{split} \] Then, \begin{equation} \label{os-Dv-18} \lambda^{1/2} (|Dv|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0} \leq N (|\mathbf{M}^{\alpha/2}f|^{p_0})_{Q_{14\rho}^+(z_0), \bar{\mu}_1}^{1/p_0}. \end{equation} Now, let $w = u- v$. From \eqref{v-sol-1}, we see that $w \in \sW^{1,2}_{p_0}(Q_{14\rho}^+(z_0), x_d^{\gamma_1'}\, dz)$ is a strong solution of \[ \left\{ \begin{array}{cccl} \sL_0 w & =& 0 & \quad \text{in} \quad Q_{14\rho}^+(z_0),\\ w & = & 0 & \quad \text{on} \quad Q_{14\rho}(z_0) \cap \{x_d =0\}. \end{array} \right. \] Then, by applying Lemma \ref{oscil-lemma-1} to $w$, we see that \begin{equation} \label{W-osc-est-1} (|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \leq N \kappa^{\theta} (|W|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} \end{equation} and \begin{equation} \label{os-Dw-18} (|Dw - (Dw)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1}|)_{Q_{\kappa \rho}^+(z_0), \bar{\mu}_1} \leq N \kappa^{\theta} (|Dw|^{p_0})_{Q_{14 \rho}^+(z_0), \bar{\mu}_1}^{1/p_0}, \end{equation} where \[ \begin{split} & W= (\mathbf{M}^{-\beta_0} w_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} w, \lambda \mathbf{M}^{-\beta_0}w). \end{split} \] Now, note that from \eqref{def:Q} and \eqref{def:r} we have \begin{align} \notag \frac{\mu_1(Q_{14 \rho}^+(z_0))}{\mu_1(Q_{\kappa \rho}^+(z_0))} & = N(d) \kappa^{\alpha-2} \Big(\frac{r(14\rho, x_{0d})}{r(\kappa \rho, x_{0d})}\Big)^{d + (\gamma_1)_+} \\ \label{Q-compared} & \leq N (d)\kappa^{-(d+ (\gamma_1)_+ + 2-\alpha)}. \end{align} Then, it follows from the triangle inequality, H\"{o}lder's inequality, \eqref{W-osc-est-1}, and \eqref{Q-compared} that \[ \begin{split} & (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\ & \leq (|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} + (|V - (V)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\ & \leq (|W - (W)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1} \\ & \qquad \quad+ N(d) \kappa^{-(d+ (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})^{1/p_0}_{Q_{14\rho}^+(z_0), \mu_1}\\ & \leq N \kappa^{\theta} (|W|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N(d) \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})^{1/p_0}_{Q_{14\rho}^+(z_0), \mu_1}. \end{split} \] As $W = U -V$ and $\kappa \in (0,1)$, we apply the triangle inequality again to see that \[ \begin{split} & (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\ &\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0) , \mu_1}^{1/p_0} \\ & \qquad + N\big( \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} + \kappa^{\theta}\big)(|V|^{p_0})_{Q_{14\rho}^+(z_0) , \mu_1}^{1/p_0} \\ & \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|V|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}. \end{split} \] From this and \eqref{V-osc-est-1}, it follows that \[ \begin{split} & (|U - (U)_{Q_{\kappa \rho}^+(z_0) , \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\ & \leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0) , \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_++2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0) , \mu_1}^{1/p_0}, \end{split} \] where $N = N(d,\nu, \alpha, \gamma_1, p_0)>0$. This proves the assertion on the oscillation of $U$. The oscillation estimate of $Du$ can be proved similarly using \eqref{os-Dv-18} and \eqref{os-Dw-18}. The proof of the lemma is completed. \end{proof} We now conclude this subsection by pointing out the following important remark, which can be proved in the same way as Lemma \ref{oscil-lemma-2} with minor modifications. \begin{remark} \label{all-oscilla-est} Under the assumptions as in Lemma \ref{oscil-lemma-2}, and if $\beta_0 \in {(\alpha-1}, \alpha/2]$, it holds that \[ \begin{split} & (|U' - (U')_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\ &\leq N \kappa^{\theta} (|U'|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \end{split} \] where \[ U' = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda^{1/2}\mathbf{M}^{\alpha/2-\beta_0} Du, \lambda \mathbf{M}^{-\beta_0}u). \] \end{remark} \subsection{Proof of Theorem \ref{thm:xd}} \label{proof-xd} We are now ready to give the proof of Theorem \ref{thm:xd}. \begin{proof}[Proof of Theorem \ref{thm:xd}] We begin with the proof of the a priori estimates \eqref{eq:xd-main}--\eqref{eq3.09} assuming that $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$ is a strong solution to the equation \eqref{eq:xd} with \begin{equation} \label{gamma-alla-range} \gamma \in (p (\alpha-1)_{+} -1, 2p-1), \quad \text{where} \,\, (\alpha -1)_+ = \max\{\alpha-1, 0\}. \end{equation} In our initial step, we prove \eqref{eq:xd-main}--\eqref{eq3.09} with an extra assumption that $u$ is compactly supported. We first prove \eqref{eq:xd-main}. Let $\beta_0 = \min\{1, \alpha\}$, and we will apply Lemma \ref{oscil-lemma-2} with this $\beta_0$. Let $p_0 \in (1, p)$ and $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha+2)-1)$. We choose $p_0$ to be sufficiently close to $1$ and $\gamma_1$ to be sufficiently close to $p_0(\beta_0-\alpha+2)-1$ so that \begin{equation} \label{nature-choice-2} \gamma - [\gamma_1 +p(\alpha-\beta_0)] < (1+\gamma_1)(p/p_0 -1). \end{equation} We note that this is possible because $\alpha-\beta_0 = (\alpha-1)_+$ and \[ \gamma - [ \gamma_1 + p(\alpha-\beta_0)] < p[2 - (\alpha-1)_+] -1 -\gamma_1, \] and also from our choices of $p_0$ and $\gamma_1$, \[ \begin{split} (1+\gamma_1)(p/p_0 -1) & \sim p (1+\gamma_1) -1 -\gamma_1 \sim p[2 - (\alpha-1)_+] -1 -\gamma_1. \end{split} \] Now, let us denote \begin{equation} \label{gamma-1-pri} \gamma_1' : = \gamma_1 + p(\alpha-\beta_0) = \gamma_1 + p (\alpha-1)_+. \end{equation} Due to \eqref{gamma-alla-range} and the definition of $\gamma_1'$, it follows that \begin{equation} \label{nature-choice-1} \gamma - \gamma_1' = \gamma - p(\alpha-1)_+ - \gamma_1 > -1 - \gamma_1. \end{equation} From \eqref{nature-choice-1} and \eqref{nature-choice-2}, it holds that \begin{equation} \label{gamma-0414} \gamma' : = \gamma - \gamma_1' \in (-1-\gamma_1, (1+\gamma_1)(p/p_0 -1)). \end{equation} Now, since $u$ has compactly support in $\Omega_T$, we have $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma_1'} dz)$. Therefore, it follows from Lemma \ref{oscil-lemma-2} that \[ U^{\#}_{\mu_1} \leq N \Big[ \kappa^{\theta} \cM_{\mu_1}(|U|^{p_0}) ^{1/p_0} + \kappa^{-(d+ (\gamma_1)_+ + 2-\alpha)/2} \cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})^{1/p_0} \Big], \] where $\mu_1(dz) = x_d^{\gamma_1}dxdt$ and \[ U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u). \] Next, due to \eqref{gamma-0414}, we see that $x_d^{\gamma'} \in A_{p/p_0}(\mu_1)$. It then follows from the weighted Fefferman-Stein theorem and Hardy-Littlewood theorem (i.e., Theorem \ref{FS-thm}) that \begin{align} \notag & \|U\|_{L_p(\Omega_T, x_d^{\gamma'}\, d\mu_1)} \leq N \|U^{\#}_{\mu_1}\|_{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)} \\ \notag & \leq N \Big[\kappa^{\theta}\|\cM_{\mu_1}(|U|^{p_0})^{1/p_0}\|_{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1 )} \\ \notag & \qquad + \kappa^{-(d + (\gamma_1)_++2-\alpha)/2} \|\cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})^{1/p_0}\|_{L_p(\Omega_T, x_d^{\gamma'}\, d\mu_1)} \Big] \\ \label{est:0414-1} & \leq N \Big[ \kappa^{\theta} \|U\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} + \kappa^{-(d+ (\gamma_1)_++2-\alpha)/2} \| \mathbf{M}^{\alpha-\beta_0} f\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} \Big]. \end{align} From the definition of $U$, the choices of $\gamma'$ in \eqref{gamma-0414} and $\gamma_1'$ in \eqref{gamma-1-pri}, we have \[ \begin{split} \|U\|_{{L_p(\Omega_T, x_d^{\gamma'}d\mu_1)}} & = \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|DD_{x'}u\|_{L_p(\Omega_T, x_d^\gamma dz)} \\ & \quad + \lambda \|\mathbf{M}^{-\alpha} u \|_{L_p(\Omega_T, x_d^\gamma\, dz)} <\infty. \end{split} \] Then, by choosing $\kappa \in (0,1)$ sufficiently small so that $N \kappa^{\theta} < 1/2$, we obtain from \eqref{est:0414-1} that \[ \begin{split} & \|\mathbf{M}^{-\alpha} u_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|DD_{x'}u\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \lambda \|\mathbf{M}^{-\alpha} u \|_{L_p(\Omega_T, x_d^\gamma\, dz)} \\ & \leq N \| f\|_{{L_p(\Omega_T, x_d^{\gamma'}\,d\mu_1)}} = N \| f\|_{{L_p(\Omega_T, x_d^{\gamma}\,dz)}}. \end{split} \] Also, from the PDE in \eqref{eq:xd}, we see that \[ |D_{dd} u| \leq N[|DD_{x'}u| + (|u_t| + \lambda |u|)x_d^{-\alpha} + |f|], \] and therefore \[ \begin{split} & \|\mathbf{M}^{-\alpha}u_t\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}+\lambda\|\mathbf{M}^{-\alpha}u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)} +\|D^2 u\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}\\ &\le N\|f\|_{L_p(\Omega_T, x_d^{\gamma}\,dz)}, \end{split} \] which is \eqref{eq:xd-main}. Next, we prove the estimate \eqref{eq3.09} also with the extra assumption that $u$ has compact support. We observe that if $\gamma \in (p -1, 2p -1)$, \eqref{eq3.09} follows from \eqref{est-0405-1}. Therefore, it remains to consider the case that $\gamma \in (\alpha p/2-1, p -1]$ or equivalently \begin{equation} \label{gamma-range-2} \gamma - \alpha p/2 \in (-1, p(1-\alpha/2) -1]. \end{equation} The main idea is to apply Lemma \ref{oscil-lemma-2} with this $\beta_0 = \alpha/2$. Let $p_0, \gamma_1$ be as before but with the new choice of $\beta_0$. As noted in \eqref{mu1=bar-mu-1}, we have \[ \bar{\gamma}_1 = \gamma_1 - p_0(\beta_0 -\alpha/2) = \gamma_1 \qquad \text{and} \qquad \bar{\mu}_1 = \mu_1. \] Because of \eqref{gamma-range-2}, we can perform the same calculation as the one that yields \eqref{gamma-0414} to obtain \[ \bar{\gamma}' := \gamma - (\bar{\gamma}_1 + p\alpha/2 ) \in (-1 - \bar{\gamma}_1, (1+\bar{\gamma}_1)(p/p_0 -1)) \] and therefore $x_d^{\bar{\gamma}'} \in A_{p/p_0}(\bar{\mu}_1)$. By using Lemma \ref{oscil-lemma-2} , we have \begin{equation} \label{Du-sharp} \begin{split} \lambda^{1/2} (Du)^{\#}_{\bar{\mu}_1} & \leq N \Big[ \kappa^{\theta} \lambda^{1/2}\cM_{\bar{\mu}_1}(|Du|^{p_0}) ^{1/p_0} \\ & \quad + \kappa^{-(d+ \bar{\gamma}_1 + 2-\alpha)/2} \cM_{\bar{\mu}_1}(|\mathbf{M}^{\alpha/2} f|^{p_0})^{1/p_0} \Big], \end{split} \end{equation} where $\bar{\mu}_1(dz) = x_d^{\bar{\gamma}_1}dxdt$. We apply Theorem \ref{FS-thm} to \eqref{Du-sharp}, and then choose $\kappa>0$ sufficiently small as in the proof of \eqref{eq:xd-main} to obtain \[ \lambda^{1/2}\|Du\|_{L_p(\Omega_T, x_d^{\bar{\gamma}'} d\bar{\mu}_1)} \leq N \|\mathbf{M}^{\alpha/2} f\|_{L_p(\Omega_T, x_d^{\bar{\gamma}'}\, d\bar{\mu}_1)}. \] This implies \[ \lambda^{1/2} \|\mathbf{M}^{-\alpha/2}Du\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \| f\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \] as $\gamma - p\alpha/2= \bar{\gamma}' + \bar{\gamma}_1$. The estimate \eqref{eq3.09} is proved. Now, we prove \eqref{eq:xd-main}--\eqref{eq3.09} without the assumption that $u$ is compactly supported. As $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma dz)$, there is a sequence $\{u_n\}$ in $C_0^\infty(\Omega_T)$ such that \begin{equation} \label{u-approx-compact} \lim_{n\rightarrow \infty} \|u_n -u\|_{\sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)} =0. \end{equation} Let $f_n = f + \sL_0 (u_n - u)/\mu(x_d)$ and observe that $u_n$ is a strong solution of \[ \sL_0 u_n = \mu(x_d) f_n \quad \text{in} \quad \Omega_T \quad \text{and} \quad u_n =0 \quad \text{on} \quad \{x_d =0\}. \] Then, applying the estimates \eqref{eq:xd-main}--\eqref{eq3.09} to $u_n$, we obtain \begin{equation} \label{un-supported} \|u_n\|_{\sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)} \leq N\|f_n\|_{L_p(\Omega_T, x_d^\gamma\, dz)}. \end{equation} Note that \[ \begin{split} & \|f_n\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \leq \|f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + N\lambda \|\mathbf{M}^{-\alpha} (u-u_n)\|_{L_p(\Omega_T, x_d^\gamma\, dz)}\\ & \qquad + N \Big [ \|D^2(u-u_n)\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha}(u-u_n)_t\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \| \Big] \\ & \rightarrow \|f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \quad \text{as} \quad n \rightarrow \infty. \end{split} \] Therefore, by taking $n\rightarrow \infty$ in \eqref{un-supported} and using \eqref{u-approx-compact}, we obtain the estimates \eqref{eq:xd-main}--\eqref{eq3.09} for $u$. Hence, the proof of \eqref{eq:xd-main}--\eqref{eq3.09} is completed. It remains to prove the existence of a strong solution $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ to \eqref{eq:xd} assuming that $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, for $p \in (1, \infty)$ and $\gamma \in (p (\alpha-1)_{+} -1, 2p-1)$. We observe when $\gamma \in (p-1, 2p-1)$, the existence of solution is already proved in Lemma \ref{l-p-sol-lem}. Therefore, it remains to consider the case when $$\gamma \in (p (\alpha-1)_{+} -1, p-1].$$ We consider two cases. \noindent {\em Case} 1. Consider $\gamma \in (p(\alpha-1)_+ -1, p-1)$. As $f \in L_p(\Omega_T, x_d^\gamma\, dz)$, there is a sequence $\{f_k\}_k \subset C_0^\infty(\Omega_T)$ such that \begin{equation} \label{fk-approx} \lim_{k\rightarrow \infty}\|f_k - f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} =0. \end{equation} For each $k \in \mathbb{N}$, because $f_k$ has compact support, we see that \[ x_d^{1-\alpha} \mu(x_d) f_k \sim x_d f_k \in L_p(\Omega_T, x_d^{\gamma}\, dz). \] Then, as in the proof of Lemma \ref{l-p-sol-lem}, we apply \cite[Theorem 2.4]{DPT21} to find a weak solution $u_k \in \cH^1_p(\Omega_T, x_d^{\gamma}\, dz)$ to the divergence form equation \eqref{eq:xd-div} with $f_k$ in place of $f$. Moreover, \begin{equation} \label{uk-Hp-est} \|Du_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} < \infty. \end{equation} We claim that $u_k \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ for each $k \in \mathbb{N}$. Note that if the claim holds, we can apply the a priori estimate that we just proved for the equations of $u_k$ and of $u_k - u_l$ to get \[ \begin{split} & \|u_k\|_{\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \|f_k\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \quad \text{and} \\ & \|u_k - u_l\|_{\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)} \leq N \|f_k - f_l\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} \end{split} \] for any $k, l \in \mathbb{N}$, where $N = N(\nu, \gamma, \alpha, p)>0$ which is independent of $k, l$. The last estimate and \eqref{fk-approx} imply that the sequence $\{u_k\}_k$ is convergent in $\sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$. Let $u \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ be the limit of such sequence, we see that $u$ solves \eqref{eq:xd}. Hence, in this case, it remains to prove the claim that $u_k \in \sW^{1,2}_p(\Omega_T, x_d^{\gamma}\, dz)$ for every $k \in \mathbb{N}$. Also, let us fix $k \in \mathbb{N}$, and let us denote $\Omega_T' = (-\infty, T) \times \mathbb{R}^{d-1}$. Let $0 < r_0 <R_0$ such that \begin{equation} \label{fk-support} \text{supp}(f_k) \subset {\Omega_T'} \times (r_0, R_0). \end{equation} Without loss of generality, we assume that $ r_0 =2$. From \eqref{uk-Hp-est}, it follows directly that \[ \begin{split} & \|Du_k\|_{L_p(\Omega_T' \times (1, \infty), x_d^{\gamma -p}\, dz)} + \|u_k\|_{L_p(\Omega_T' \times (1, \infty), x_d^{\gamma-2p}\, dz)} \\ & \qquad + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (1,\infty), x_d^{\gamma-p}\, dz)} <\infty. \end{split} \] Then, we can follow the proof of Lemma \ref{l-p-sol-lem} to show that \[ \|u_k\|_{\sW^{1,2}_{p}(\Omega_T' \times (1,\infty), x_d^{\gamma}\, dz)} <\infty. \] It now remains to prove that $u_k \in \sW^{1,2}_{p}({\Omega_T'\times (0,1)}, x_d^{\gamma}\, dz)$ and \begin{equation} \label{near-est-uk} \|u_k\|_{\sW^{1,2}_{p}(\Omega_T' \times (0, 1), x_d^{\gamma}\, dz)} < \infty. \end{equation} To this end, because of \eqref{fk-support}, we note that $u_k$ solves the homogeneous equation \begin{equation} \label{uk-ne-zero} \sL_0 u_k =0 \quad \text{in} \quad \Omega_T' \times (0, 2) \end{equation} with the boundary condition $u_k =0$ on $\{x_d =0\}$. Let us denote \[ \begin{split} & C_r = [-1, 0) \times \big\{ x = (x_1, \ldots, x_d) \times \mathbb{R}^{d}_+ : {\max_{1 \leq i \leq d}|x_i|}<r\big\}, \\ & C_r(t,x) = C_r + (t,x), \quad r >0. \end{split} \] Consider $\alpha \in (0, 1)$. By using Lemmas \ref{caccio}, and \ref{prop:boundary} with a scaling argument and translation, we obtain \begin{equation*} \begin{split} & \|\mathbf{M}^{-\alpha} u_k\|_{L_\infty(C_{1}(z_0))} + \|Du_k\|_{L_\infty(C_{1}(z_0))} + \|\mathbf{M}^{-\alpha} \partial_t u_k\|_{L_\infty(C_{1}(z_0))} \\ & \quad + \|DD_{x'} u_k\|_{L_\infty(C_{1}(z_0))} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma dz)} \Big] \end{split} \end{equation*} for every $z_0 = (t_0, x_0', 0) \in \Omega_T' \times\{0\}$. Note that $N$ depends on $k$, but is independent of $z_0$. This and the PDE in \eqref{uk-ne-zero} imply that \[ \begin{split} & \|\mathbf{M}^{-\alpha} u_k\|_{L_\infty(C_{1}(z_0))} + \|Du_k\|_{L_\infty(C_{1}(z_0))} + \|\mathbf{M}^{-\alpha} \partial_t u_k\|_{L_\infty(C_{1}(z_0))} \\ & \quad + \|D^2 u_k\|_{L_\infty(C_{1}(z_0))} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} \Big]. \end{split} \] Then, as $\gamma > -1$, we see that \[ \begin{split} & \|\mathbf{M}^{-\alpha}u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} + \|Du_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha}\partial_t u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} \\ & + \|\ D^2u_k\|_{L_p(C_1(z_0), x_d^{\gamma}\, dz)} \leq N \Big[\|Du_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|_{L_{p}(C_{2}(z_0), x_d^\gamma\, dz)} \Big]. \end{split} \] Then, with $z_0 = (t_0, x_0', 0)$ and with $\mathcal{I} = ({(\mathbb{Z}+T)} \cap (-\infty, T{]}) \times (2\mathbb{Z})^{d-1}$, we have \[ \begin{split} \|u_k\|^p_{\sW^{1,2}_p(\Omega_T' \times (0,1))} & = \sum_{(t_0', x_0') \in \mathcal{I} } \|u_k\|^p_{\sW^{1,2}_p(C_1(z_0))} \\ & \leq N \sum_{(t_0', x_0') \in \mathcal{I} }\Big[\|Du_k\|^p_{L_{p}(C_{2}(z_0))} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(C_{2}(z_0))} \Big] \\ & = N\Big[ \|Du_k\|^p_{L_{p}(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(\Omega_T, x_d^\gamma\, dz)}\Big] <\infty. \end{split} \] Hence, \eqref{near-est-uk} holds. Now, we consider $\alpha \in [1, 2)$. As $\gamma + p (1-\alpha) >-1$, we see that \[ \begin{split} & \int_{C_1(z_0)} |x_d^{-\alpha} u_k(z)|^p x_d^\gamma dz = \int_{C_1(z_0)} |x_d^{-1}u_k(z)|^p x_d^{\gamma + p (1-\alpha)} dz \\ & \leq N \|Du_k\|^p_{L_\infty(C_1(z_0))} \\ & \leq N\Big[ \|Du_k\|^p_{L_{p}(C_2(z_0), x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha/2} u_k\|^p_{L_{p}(C_2(z_0), x_d^\gamma\, dz)}\Big]. \end{split} \] Then, by taking the sum of this inequality for $(t_0, x_0') \in \mathcal{I}$, we also obtain \[ \|\mathbf{M}^{-\alpha} u_k\|_{L_p(\Omega_T' \times (0,1), x_d^\gamma\, dz)} \leq N \Big[ \|Du_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} + \|\mathbf{M}^{-\alpha}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} \Big]. \] Similarly, we also have $\mathbf{M}^{-\alpha} (u_k)_t, Du_k\in L_p(\Omega_T' \times (0,1), x_d^\gamma\,dz)$. By using the different quotient, we also get $DD_{x'} u_k \in L_p(\Omega_T' \times (0,1), x_d^\gamma\,dz)$. From this, and the PDE of $u_k$, we have $D^2u_k \in L_p(\Omega_T' \times (0,1), x_d^\gamma\, dz)$. Therefore, \eqref{near-est-uk} holds. The proof of the claim in this case is completed. \noindent {\em Case} 2. We consider $\gamma =p-1$. Let $\{f_k\}_k$ be as in \eqref{fk-approx} and let $\bar{\gamma} \in (p(\alpha-1)_+ -1, p-1)$. As in {\em Case 1}, we can find a weak solution $u_k \in \cH^1_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)$ to the divergence form equation \eqref{eq:xd-div} with $f_k$ in place of $f$, and \begin{equation} \label{uk-Hp-est-b} \|Du_k\|_{L_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^{\bar{\gamma}}\, dz)} < \infty. \end{equation} We claim that for each $k \in \mathbb{N}$, \begin{equation} \label{uk-Hp-est-b-1} \|Du_k\|_{L_p(\Omega_T, x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T, x_d^\gamma\, dz)} < \infty. \end{equation} Once this claim is proved, we can follow the proof in {\em Case 1} to obtain the existence of a solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma\, dz)$. Therefore, we only need to prove \eqref{uk-Hp-est-b-1}. Let us fix $k \in \mathbb{N}$ and let $0 < r_0 < R_0$ such that \eqref{fk-support} holds. As $\bar{\gamma} < \gamma$, we see that \[ \begin{split} & \|Du_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^\gamma\, dz)} \\ & \leq N\Big[ \|Du_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\bar\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (0, 2R_0), x_d^{\bar\gamma}\, dz)}\Big] <\infty \end{split} \] due to \eqref{uk-Hp-est-b}. Hence, it remains to prove \begin{equation} \label{uk-Hp-est-b-2} \|Du_k\|_{L_p(\Omega_T' \times (2R_0, \infty), x_d^{\gamma}\, dz)} + \|\mathbf{M}^{-\alpha/2}u_k\|_{L_p(\Omega_T' \times (2R_0, \infty), x_d^\gamma\, dz)} < \infty. \end{equation} To prove \eqref{uk-Hp-est-b-2}, we use the localization technique along the $x_d$ variable. See \cite[Proof of Theorem 4.5, Case II]{DP-JFA}. We skip the details. \end{proof} \section{Equations with partially VMO coefficients} \label{sec:4} We study \eqref{eq:main} in this section. Precisely, we consider the equation \begin{equation}\label{eq:main-1} \begin{cases} \sL u=\mu(x_d) f \quad &\text{ in } \Omega_T,\\ u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+, \end{cases} \end{equation} where $\sL$ is defined in \eqref{L-def} in which the coefficients $a_0$, $c_0$, and $a_{ij}$ are measurable functions depending on $z = (z', x_d) \in \Omega_T$. We employ the perturbation method by freezing the coefficients. For $z_0 = (z'_0, x_{0d}) \in \overline{\Omega}_T$, let $[{a}_{ij}]_{Q_{\rho}'(z'_0)}, [a_{0}]_{Q_{\rho}'(z'_0)}$, and $[c_{0}]_{Q_{\rho}'(z'_0)}$ be functions defined in Assumption \ref{assumption:osc} $(\delta, \gamma_1, \rho_0)$, and we denote \begin{equation} \label{a-sharp-def} \begin{split} a^{\#}_{\rho_0}(z_0) & =\sup_{\rho\in(0,\rho_0)}\left[ \max_{i,j=1, 2,\ldots, d}\fint_{Q_{\rho}^+(z_0)}|a_{ij}(z) -[{a}_{ij}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \right. \\ & \qquad + \fint_{Q_{\rho}^+(z_0)}|a_{0}(z) -[{a}_{0}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \\ & \qquad \left. + \fint_{Q_{\rho}^+(z_0)}|c_{0}(z) -[{c}_{0}]_{Q_{\rho}'(z'_0)}(x_d)| \mu_1(dz) \right]. \end{split} \end{equation} For the reader's convenience, recall that $\mu_1, \bar{\mu}_1$ are defined in \eqref{mu-1-def}. We also recall that for a given $u$, we denote \[ U = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda \mathbf{M}^{-\beta_0}u). \] We also denote \[ U' = (\mathbf{M}^{-\beta_0} u_t, \mathbf{M}^{\alpha-\beta_0}DD_{x'} u, \lambda^{1/2} \mathbf{M}^{\alpha/2-\beta_0}Du, \lambda \mathbf{M}^{-\beta_0}u). \] We begin with the following oscillation estimates for solutions to \eqref{eq:main-1} that have small supports in the time-variable. \begin{lemma} \label{osc-est-small} Let $\nu, \rho_0 \in (0,1)$, $p_0 \in (1, \infty)$, $\alpha \in (0,2)$, $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$, $\gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha +2)-1)$, and $\gamma_1' = \gamma_1-p_0(\beta_0-\alpha)\in (p_0-1,2p_0-1)$. Assume that $u \in \sW^{1,2}_{p}(Q_{6\rho}^+(z_0), x_d^{\gamma_1'}dz)$ is a strong solution of \[ \left\{ \begin{array}{cccl} \sL u & = & \mu(x_d) f & \quad \text{in} \quad Q_{6\rho}^+(z_0),\\ u & = & 0 & \quad \text{on} \quad Q_{6\rho}(z_0) \cap \{x_d =0\} \end{array} \right. \] for $f \in L_{p_0}(Q_{6\rho}^+(z_0), x_d^{{\gamma_1'}}dz)$. Assume in addition that $\textup{supp}(u) \subset (t_1 -(\rho_0 \rho_1)^{2-\alpha}, t_1 +(\rho_0 \rho_1)^{2-\alpha})$ for some $t_1 \in \mathbb{R}$ and $\rho_0 >0$. Then, \begin{align} \notag &\big (|U - (U)_{Q_{\kappa\rho}^+(z_0), \mu_1}|\big)_{Q_{\kappa\rho}^+(z_0), \mu_1} \\ \notag & \leq N \Big[\kappa^{\theta} + \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} \big( a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] (|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\ \label{U-osc-gen} & \qquad + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}, \end{align} where $\theta>0$ is defined in Lemma \ref{oscil-lemma-1}, $p\in (p_0,\infty)$, and $N = N(p, p_0, \gamma_1, \alpha, \beta_0, d, \nu)>0$. In addition, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, we also have \begin{align} \notag &\big (|U' - (U')_{Q_{\kappa\rho}^+(z_0), \mu_1}|\big)_{Q_{\kappa\rho}^+(z_0), \mu_1} \\ \notag & \leq N \Big[\kappa^{\theta} + \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} \big( a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] (|U'|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\ \label{Uall-osc-gen} & \qquad + N \kappa^{-(d + (\gamma_1)_+ +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}. \end{align} \end{lemma} \begin{proof} We split the proof into two cases. \noindent {\em Case 1.} We consider $\rho < \rho_0/14$. We denote \[ \sL_{\rho, z_0} u =[a_0]_{Q_{6\rho}'(z_0')}(x_d) u_t + \lambda [c_0]_{Q_{6\rho}'(z_0')}(x_d)u - \mu(x_d) [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d) D_i D_j u \] and \[ \begin{split} \tilde{f}(z) & = f(z) + [a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)] D_i D_ju \\ & \qquad + \big[ \lambda ([c_0]_{Q_{6\rho}'(z_0')} - c_0) u + ([a_0]_{Q_{6\rho}'(z_0')} - a_0) u_t \big]/\mu(x_d). \end{split} \] Then, $u \in \sW^{1,2}_{p}(Q_{6\rho}^+(z_0), x_d^{\gamma_1'}dz)$ is a strong solution of \[ \left\{ \begin{array}{cccl} \sL_{\rho, z_0} u & = &\mu(x_d) \tilde{f} & \quad \text{in} \quad Q_{6\rho}^+(z_0)\\ u & = & 0 & \quad \text{on} \quad Q_{6\rho}^+(z_0) \cap \{x_d =0\}. \end{array} \right. \] We note that due to \eqref{add-assumption}, the term $a_{dd} - \bar{a}_{dd} =0$. Therefore, by using H\"{o}lder's inequality and \eqref{con:ellipticity}, we obtain \[ \begin{split} & \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} \big(a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)\big) D_i D_ju|^{p_0} \mu_1(dz) \right)^{1/p_0} \\ & \leq \left(\fint_{Q_{14\rho}^+(z_0)}|a_{ij} - [a_{ij}]_{Q_{6\rho}'(z_0')}(x_d)|^{pp_0/(p-p_0)} \mu_1(dz) \right)^{\frac{1}{p_0} -\frac{1}{p}} \\ & \qquad \times \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u|^{p} \mu_1(dz)\right)^{1/p} \\ & \leq N a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} \left(\fint_{Q_{14\rho}^+(z_0)}|\mathbf{M}^{\alpha-\beta_0} DD_{x'}u|^{p} \mu_1(dz)\right)^{1/p}. \end{split} \] By a similar calculation using \eqref{con:mu}, we also obtain the estimate for the term $\big[ \lambda ([c_0]_{Q_{6\rho}'(z_0')}(x_d) - c_0) u + ([a_{0}]_{Q_{6\rho}'(z_0')}(x_d) - a_0) u_t \big]/\mu(x_d)$. Thus, \[ \begin{split} (|\mathbf{M}^{\alpha-\beta_0} \tilde{f}|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} & \leq (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \\ & \qquad + N a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} (|U|^{p})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p}. \end{split} \] Then, applying Lemma \ref{oscil-lemma-2}, we obtain \[ \begin{split} & (|U - (U)_{Q_{\kappa \rho}^+(z_0), \mu_1}|)_{Q_{\kappa \rho}^+(z_0), \mu_1}\\ &\leq N \kappa^{\theta} (|U|^{p_0})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p_0} + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} \tilde{f}|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0} \\ & \leq N \big(\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p}} \big) (|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} \\ & \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} (|\mathbf{M}^{\alpha-\beta_0} f|^{p_0})_{Q_{14\rho}^+(z_0), \mu_1}^{1/p_0}. \end{split} \] Therefore, \eqref{U-osc-gen} holds. In a similar way but applying Remark \ref{all-oscilla-est}, we also obtain \eqref{Uall-osc-gen}. \noindent {\em Case 2.} Consider $\rho \geq \rho_0/14$. Denoting $\Gamma = (t_1 -(\rho_0 \rho_1)^{2-\alpha}, t_1 + (\rho_0 \rho_1)^{2-\alpha})$, we apply \eqref{Q-compared} and the triangle inequality to infer that \[ \begin{split} & \fint_{Q_{\kappa\rho}^+(z_0)} |U - (U)_{Q_{\kappa\rho}^+(z_0), \mu_1}| \mu_1(dz)\leq 2 \fint_{Q_{\kappa\rho}^+(z_0)} |U(z)|\mu_1(dz) \\ & \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \left(\fint_{Q_{14\rho}^+(z_0)} |U(z)|^{p_0}\mu_1(dz)\right)^{\frac 1 {p_0}} \left(\fint_{Q_{14\rho}^+(z_0)} \mathbf{1}_{\Gamma}(z) \mu_1(dz)\right)^{1-\frac 1 {p_0}} \\ & \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \rho_1^{(2-\alpha)(1-1/p_0)}\left(\fint_{Q_{14\rho}^+(z_0)} |U(z)|^{p_0}\mu_0(dz)\right)^{1/p_0} \\ & \leq N \kappa^{-(d+2 -\alpha + (\gamma_1)_+)} \rho_1^{(2-\alpha)(1-1/p_0)}(|U|^{p})_{Q_{14 \rho}^+(z_0), \mu_1}^{1/p} . \end{split} \] Therefore, \eqref{U-osc-gen} follows. Similarly, \eqref{Uall-osc-gen} can be proved. \end{proof} Our next lemma gives the a priori estimates of solutions having small supports in $t$. \begin{lemma}[Estimates of solutions having small supports] \label{small-support-sol} Let $T \in (-\infty, \infty]$, $\nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist sufficiently small positive numbers $\delta$ and $\rho_1$, depending on $d, \nu, p, q, K, \alpha{,\beta_0}$, and $\gamma_1$, such that the following assertion holds. Suppose that $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ with \[ [\omega_0]_{A_q(\mathbb{R})} \leq K \qquad \text{and} \qquad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K. \] Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} hold, and \textup{Assumption \ref{assumption:osc}}$(\delta, \gamma_1, \rho_0)$ holds with some $\rho_0>0$. If $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ is a strong solution to \eqref{eq:main} with some $\lambda>0$ and a function $f\in L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$, and $u$ vanishes outside $(t_1 - (\rho_0\rho_1)^{2-\alpha}, t_1+(\rho_0\rho_1)^{2-\alpha})$ for some $t_1 \in \mathbb{R}$, then \begin{equation} \label{est-1-small-supp} \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}}, \end{equation} where $N = N(d,\nu, p, q, \alpha,{ \beta_0,}\gamma_1, K)>0$, $L_{q,p}=L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$, $\omega(t,x) =\omega_0(t)\omega_1(x)$ for $(t,x) \in \Omega_T$, and $\mu_1(dz) = x_d^{\gamma_1}\, dxdt$. Moreover, if $\beta_0 \in [0, \alpha/2]$, then it also holds that \begin{equation} \label{est-2-small-supp} \begin{split} & \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \\ & \leq N \|f\|_{L_{q,p}}. \end{split} \end{equation} \end{lemma} \begin{proof} As $\omega_0 \in A_q((-\infty,T))$ and $\omega_1 \in A_p(\mathbb{R}^d_+, d\mu_1)$, by the reverse H\"older's inequality \cite[Theorem 3.2]{MS1981}, we find $p_1=p_1(d,p,q,\gamma_1,K)\in (1,\min(p,q))$ such that \begin{equation} \label{eq0605_13} \omega_0 \in A_{q/p_1}((-\infty,T)),\quad \omega_1 \in A_{p/p_1}(\mathbb{R}^d_+, \mu_1). \end{equation} Because $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$, we can choose $p_0 \in (1, p_1)$ sufficiently closed to $1$ so that \[ \gamma_1 \in (p_0(\beta_0-\alpha +1) -1, p_0(\beta_0-\alpha +2) -1). \] By \eqref{U-osc-gen} of Lemma \ref{osc-est-small} and H\"{o}lder's inequality, we have \[ \begin{split} U^{\#}_{\mu_1} \leq & N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(a_{\rho_0}^{\#}(z_0)^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \cM_{\mu_1}(|U|^{p_1})^{1/p_1} \\ & \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \cM_{\mu_1}(|\mathbf{M}^{\alpha-\beta_0} f|^{p_1})^{1/p_1} \quad \text{in} \quad \overline{\Omega_T} \end{split} \] for any $\kappa\in (0,1)$, where $N = N(\nu, d, p_0, p_1, \alpha,\beta_0, \gamma_1) >0$ and $a_{\rho_0}^{\#}$ is defined in \eqref{a-sharp-def}. Therefore, it follows from Theorem \ref{FS-thm} and \eqref{eq0605_13} that \[ \begin{split} & \norm{U}_{L_{q,p}(\Omega_T, \omega\, d\mu_1)}\\ & \leq N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \times \\ & \qquad \qquad \times \|\cM_{\mu_1}(|U|^{p_1})^{1/p_1}\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\ & \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \| \cM_{\mu_1} (|\mathbf{M}^{\alpha -\beta_0} f|^{p_1})^{\frac 1 {p_1}}\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\ & \leq N \Big[\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] \|U\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \\ & \qquad + N \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \|\mathbf{M}^{\alpha-\beta_0} f\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)}, \end{split} \] where $N = N(d,\nu, p, q, \alpha,\beta_0, \gamma_1, K)>0$. Now, by choosing $\kappa$ sufficiently small and then $\delta$ and $\rho_1$ sufficiently small depending on $d,\nu, p, q,\alpha, \gamma_1$, and $K$ such that \[ N\Big [\kappa^{\theta} + \kappa^{-(d + \gamma_1 +2-\alpha)/p_0} \big(\delta^{\frac{1}{p_0} - \frac{1}{p_1}} + \rho_1^{(2-\alpha)(1-1/p_0)}\big) \Big] <1/2, \] we obtain \[ \begin{split} & \norm{U}_{L_{q,p}(\Omega_T, \omega\, d\mu_1)} \leq N(d, \nu, p, q, \alpha,\beta_0 \gamma_0, K) \|\mathbf{M}^{\alpha -\beta_0} f\|_{L_{q,p}(\Omega_T, \omega\, d\mu_1)}. \end{split} \] From this and the PDE in \eqref{eq:main}, we obtain \[ \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}}. \] This proves \eqref{est-1-small-supp}. The proof of \eqref{est-2-small-supp} is similar by applying \eqref{Uall-osc-gen} instead of \eqref{U-osc-gen}. \end{proof} Below, we provide the proof of Theorem \ref{main-thrm}. \begin{lemma}[A priori estimates of solutions] \label{apriori-est-lemma} Let $T \in (-\infty, \infty]$, $ \nu \in (0,1)$, $p, q, K \in (1, \infty)$, $\alpha \in (0, 2)$, and $\gamma_1 \in (\beta_0 -\alpha, \beta_0 -\alpha +1]$ for $\beta_0 \in {(\alpha-1}, \min\{1, \alpha\}]$. Then, there exist $\delta = \delta(d, \nu, p, q, K, \alpha, \beta_0,\gamma_1)>0$ sufficiently small and $\lambda_0 = \lambda_0(d, \nu, p, q, K, \alpha,\beta_0, \gamma_1)>0$ sufficiently large such that the following assertions hold. Let $\omega_0 \in A_q(\mathbb{R})$, $\omega_1 \in A_p(\mathbb{R}^d_+, \mu_1)$ satisfy \[ [\omega_0]_{A_q(\mathbb{R})} \leq K \qquad \text{and} \qquad [\omega_1]_{A_p(\mathbb{R}^d_+, \mu_1)} \leq K. \] Suppose that \eqref{con:mu}, \eqref{con:ellipticity}, and \eqref{add-assumption} hold, and suppose that \textup{Assumption \ref{assumption:osc}}$ (\delta, \gamma_1, \rho_0)$ holds with some $\rho_0>0$. If $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\,d\mu_1)$ is a strong solution to \eqref{eq:main} with some $\lambda{\ge \lambda_0\rho_0^{\alpha-2}}$ and $f \in L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)$, then \begin{equation} \label{main-est-1-b} \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \|f\|_{L_{q,p}}, \end{equation} where $\omega(t, x) = \omega_0(t) \omega_1(x)$ for $(t,x) \in \Omega_T$, $L_{q,p} = L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega \,d\mu_1)$, and $N = N(d, \nu, p, q, \alpha,{\beta_0,K,} \gamma_1)>0$. Moreover, if $\beta_0 \in {(\alpha-1}, \alpha/2]$, then it also holds that \begin{equation} \label{main-est-2-b} \begin{split} & \|\mathbf{M}^{-\alpha} u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda^{1/2} \|\mathbf{M}^{-\alpha/2} Du\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \\ & \leq N \|f\|_{L_{q,p}}. \end{split} \end{equation} \end{lemma} \begin{proof} Let $\delta, \rho_1$ be positive numbers in Lemma \ref{small-support-sol}, and let $\lambda_0>$ be a number sufficiently large to be determined, depending on $d, p, q, \alpha,{\beta_0,\nu,} \gamma_1, K$. As the proof of \eqref{main-est-1-b} and of \eqref{main-est-2-b} are similar, we only prove the a priori estimate \eqref{main-est-1-b}. We use a partition of unity argument in the time variable. Let $\delta>0$ and $\rho_1>0$ be as in Lemma \ref{small-support-sol} and let $$ \xi=\xi(t) \in C_0^\infty( -(\rho_0\rho_1)^{2-\alpha}, (\rho_0\rho_1)^{2-\alpha}) $$ be a non-negative cut-off function satisfying \begin{equation} \label{xi-0702} \int_{\mathbb{R}} \xi(s)^q\, ds =1 \qquad \text{and} \qquad \int_{\mathbb{R}}|\xi'(s)|^q\,ds \leq \frac{N}{(\rho_0\rho_1)^{q(2-\alpha)}}. \end{equation} For fixed $s \in (-\infty, \infty)$, let $u^{(s)}(z) = u(z) \xi(t-s)$ for $z = (t, x) \in \Omega_T$. We see that $u^{(s)} \in \sW^{1,2}_p(\Omega_T,x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)$ is a strong solution of \[ \sL u^{(s)}(z) =\mu(x_d) f^{(s)} (z) \quad \text{in} \quad \Omega_T \] with the boundary condition $u^{(s)} =0$ on $\{x_d =0\}$, where \[ f^{(s)}(z) = \xi(t-s) f(z) + \xi'(t-s) u(z)/\mu(x_d). \] As $\text{spt}(u^{(s)}) \subset (s -(\rho_0\rho_1)^{2-\alpha}, s+ (\rho_0\rho_1)^{2-\alpha}) \times \mathbb{R}^{d}_{+}$, we apply Lemma \ref{small-support-sol} to get \[ \begin{split} \|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}} + \|D^2u^{(s)}\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}} \leq N \|f^{(s)}\|_{L_{q,p}}. \end{split} \] Then, by integrating the $q$-th power of this estimate with respect to $s$, we get \begin{align}\notag & \int_{\mathbb{R}}\Big(|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}}^q + \|D^2u^{(s)}\|_{L_{q,p}}^q + \lambda^q \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}}^q\Big)\, ds\\ \label{par-int-0515} & \leq N\int_{\mathbb{R}} \|f^{(s)}\|_{L_{q,p}}^q\, ds. \end{align} Now, by the Fubini theorem and \eqref{xi-0702}, it follows that \[ \begin{split} & \int_{\mathbb{R}}\|\mathbf{M}^{-\alpha} \partial_tu^{(s)}\|_{L_{q,p}}^q\, ds\\ & = \int_{\mathbb{R}} \left(\int_{-\infty}^T \|\mathbf{M}^{-\alpha}u_t(t,\cdot)\|_{L_p(\mathbb{R}^d_+, x_d^{p(\alpha-\beta_0)} \omega_1\, d\mu_1)}^q \omega_0(t) \xi^q(t-s)\, dt \right)\, ds \\ &= \int_{-\infty}^T \left( \int_{\mathbb{R}}\xi^q(t-s)\, ds \right) \|\mathbf{M}^{-\alpha}u_t(t,\cdot)\|_{L_p(\mathbb{R}^d_+,x_d^{p(\alpha-\beta_0)} \omega_1\, d\mu_1)}^q \omega_0(t)\, dt \\ & = \|\mathbf{M}^{-\alpha}u_t \|_{L_{q,p}(\mathbb{R}^d_+,x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)}^q, \end{split} \] and similarly \[ \begin{split} & \int_{\mathbb{R}} \| D^2u^{(s)}\|_{L_{q,p}}^q\, ds = \|\mathbf{M}^{\alpha-\beta_0} D^2u\|_{L_{q,p}}^q, \\ & \int_{\mathbb{R}} \|\mathbf{M}^{-\alpha} u^{(s)}\|_{L_{q,p}}^q\, ds = \|\mathbf{M}^{-\beta_0} u\|_{L_{q,p}}^q . \end{split} \] Moreover, \[ \int_{\mathbb{R}} \|f^{(s)}\|_{L_{q,p}}^q\, ds \leq \|f\|_{L_{q,p}}^q + \frac{N}{(\rho_0\rho_1)^{q(2-\alpha)}} \|\mathbf{M}^{-\alpha}u\|_{L_{q,p}}^q, \] where \eqref{xi-0702} is used and $N = N(q)>0$. As $\rho_1$ depends on $d, \nu, p, q, K, \alpha{,\beta_0,\gamma_1}$, by combining the estimates we just derived, we infer from \eqref{par-int-0515} that \[ \begin{split} & \|\mathbf{M}^{-\alpha} \partial_tu \|_{L_{q,p}} + \| D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N\Big(\|f\|_{L_{q,p}} + \rho_0 ^{\alpha-2} \|\mathbf{M}^{-\alpha}u\|_{L_{q,p}} \Big) \end{split} \] with $N=N(d, \nu, \alpha, p, q, \gamma_1) >0$. Now we choose $\lambda_0 = 2N$. Then, with $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$, we have \[ \begin{split} & \|\mathbf{M}^{-\alpha} \partial_tu \|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_{q,p}} \leq N \| f\|_{L_{q,p}} . \end{split} \] This estimate yields \eqref{main-est-1-b}. \end{proof} Now, we have all ingredients to complete the proof of Theorem \ref{main-thrm}. \begin{proof}[Proof of Theorem \ref{main-thrm}] The a priori estimates \eqref{main-est-1} and \eqref{main-est-2} follow from Lemma \ref{apriori-est-lemma}. Hence, it remains to prove the existence of solutions. We employ the the technique introduced in \cite[Section 8]{Dong-Kim-18}. See also \cite[Proof of Theorem 2.3]{DP-JFA}. The proof is split into two steps, and we only outline the key ideas in each step. \noindent {\em Step 1.} We consider the case $p =q$, $\omega_0 \equiv 1$, and $\omega_1 \equiv 1$. We employ the method of continuity. Consider the operator \[ \sL_\tau = (1-\tau)\big(\partial_t + \lambda - \mu(x_d) \Delta\big) + \tau \sL, \qquad \tau \in [0, 1]. \] It is a simple calculation to check that the assumptions in Theorem \ref{main-thrm} are satisfied uniformly with respect to $\tau \in [0,1]$. Then, using the solvability in Theorem \ref{thm:xd} and the a priori estimates obtained in Lemma \ref{apriori-est-lemma}, we get the existence of a solution $u \in \sW^{1,2}_p(\Omega_T, x_d^{p(\alpha-\beta_0)}\, d\mu_1)$ to \eqref{eq:main} when $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$, where $\lambda_0$ is the constant in Lemma \ref{apriori-est-lemma}. \noindent {\em Step 2.} We combine {\em Step 1} and Lemma \ref{apriori-est-lemma} to prove the existence of {a strong} solution $u$ satisfying \eqref{main-est-1}. Let $p_1 > \max\{p,q\}$ be sufficiently large and let $\varepsilon_1, \varepsilon_2 \in (0,1)$ be sufficiently small depending on $K, p, q$, and $\gamma_1$ such that \begin{equation} \label{epsilon12-def} 1-\frac{p}{p_1} = \frac{1}{1+\varepsilon_1} \qquad \text{and} \qquad 1 - \frac{q}{p_1} = \frac{1}{1+\varepsilon_2}, \end{equation} and both $\omega_1^{1+\varepsilon_1}$ and $\omega_0^{1+\varepsilon_2}$ are locally integrable and satisfy the doubling property. Specifically, there is $N_0>0$ such that \begin{equation} \label{omega-0} \int_{\Gamma_{2r}(t_0)} \omega_0^{1+\varepsilon_2}(s)\, ds \leq N_0 \int_{\Gamma_{r}(t_0)} \omega_0^{1+\varepsilon_2}(s)\, ds \end{equation} for any $r>0$ and $t_0 \in \mathbb{R}$, where $\Gamma_{r}(t_0) = (t_0 -r^{2-\alpha}, \min\{t_0 + r^{2-\alpha}, T\})$. Similarly \begin{equation} \label{omega-1-0308} \int_{B_{2r}^+(x_0)} \omega_1^{1+\varepsilon_1}(x)\, d\mu_1 \leq N_0\int_{B_{r}^+(x_0)} \omega_1^{1+\varepsilon_1}(x)\, d\mu_1 \end{equation} for any $r >0$ and $x_0 \in \overline{\mathbb{R}^d_+}$. Next, let $\{f_k\}$ be a sequence in $C_0^\infty(\Omega_T)$ such that \begin{equation} \label{f-k-converge-0227} \lim_{k\rightarrow \infty} \|f_k - f\|_{L_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)}\omega\, d\mu_1)} =0. \end{equation} By {\em Step 1}, for each $k \in \mathbb{N}$, we can find a solution $u_k \in \sW^{1,2}_{p_1}(\Omega_T,x_d^{p_1(\alpha-\beta_0)}\, d\mu_1)$ of \eqref{eq:main} with $f_k$ in place of $f$, where $\lambda \geq \lambda_0 \rho_0^{\alpha-2}$ for $\lambda_0 = \lambda_0(d, \nu, p_1, p_1,\alpha,\beta_0, \gamma_1, K)>0$. Observe that if the sequence $\{u_k\}$ is in $\sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega \, d\mu_1)$, then by applying the a priori estimates in Lemma \ref{apriori-est-lemma}, \eqref{f-k-converge-0227}, and the linearity of the equation \eqref{eq:main}, we conclude that $\{u_k\}$ is Cauchy in $\sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$. Let $u \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$ be the limit of the sequence $\{u_k\}$. Then, by letting $k \rightarrow \infty$ in the equation for $u_k$, we see that $u$ solves \eqref{eq:main}. It remains to prove that for each fixed $k \in \mathbb{N}$, $u_k \in \sW^{1,2}_{q,p}(\Omega_T, x_d^{p(\alpha-\beta_0)} \omega\, d\mu_1)$. To this end, let us denote \[ D_{R} = (-R^{2-\alpha}, \min\{R^{2-\alpha}, T\}) \times B_R^+.\] Then, let $R_0>0$ be sufficiently large such that \begin{equation} \label{fk-spt} \operatorname{supp}(f_k) \subset D_{R_0}. \end{equation} We note that $R_0$ depends on $k$. It follows from \eqref{epsilon12-def}, \eqref{omega-0}, \eqref{omega-1-0308}, and H\"{o}lder's inequality that \[ \begin{split} & \|u_k\|_{\sW^{1,2}_{q,p}(D_{2R_0},x_d^{p(\alpha-\beta_0)} \omega d\mu_1)} \\ & \leq N(d, p, q, p_1, \alpha, \gamma_1, R_0) \|u_k\|_{\sW^{1,2}_{p_1}(D_{2R_0}, x_d^{p_1(\alpha-\beta_0)} d\mu_1)} <\infty. \end{split} \] Hence, we only need to prove \[ \|u_k\|_{\sW^{1,2}_{q,p}(\Omega_T\setminus D_{R_0},x_d^{p(\alpha-\beta_0)} \omega d\mu_1)} <\infty. \] This is done by the localization technique employing \eqref{epsilon12-def}, \eqref{omega-0}, \eqref{omega-1-0308}, \eqref{fk-spt}, and H\"{o}lder's inequality, using the fast decay property of solutions when the right-hand side is compactly supported. We skip the details as the calculation is very similar to that of \cite[Section 8]{Dong-Kim-18}, and also of \cite[Step II - Proof of Theorem 2.3]{DP-JFA}. The proof of Theorem \ref{main-thrm} is completed. \end{proof} Next, we prove Corollary \ref{cor1}. \begin{proof}[Proof of Corollary \ref{cor1}] It is sufficient to show that we can make the choices for $\gamma_1, \beta_0$, and $\omega_1$ to apply Theorem \ref{main-thrm} to obtain \eqref{cor-est-1} and \eqref{cor-est-2}. Indeed, the choices are similar to those in the proof of Theorem \ref{thm:xd}. To obtain \eqref{cor-est-1}, we take $\beta_0 = \min\{1, \alpha\}$, and with this choice of $\beta_0$, we have \[ \alpha - \beta_0 = (\alpha-1)_+ \quad \text{and} \quad (\beta_0 -\alpha, \beta_0-\alpha +1] = (-(\alpha -1)_+, 1- (\alpha-1)_+]. \] Then, let $\gamma_1 = 1- (\alpha-1)_+$ and $\gamma' = \gamma - [\gamma_1 + p(\alpha-1)_+]$. From the choice of $\gamma_1$ and the condition on $\gamma$, we see that \begin{equation} \label{cond-gamma-1} -1-\gamma_1 < \gamma' < (1+\gamma_1) (p-1). \end{equation} Now, let $ \omega_1(x) = x_d^{\gamma'}$ for $x \in \mathbb{R}^d_+$. It follows from \eqref{cond-gamma-1} that $\omega_1 \in A_p(\mu_1)$. As Assumption $(\rho_0, \gamma_1, \delta)$ holds, we can apply \eqref{main-est-1} to obtain \eqref{cor-est-1}. Next, we prove \eqref{cor-est-2}. In this case, we choose $\beta_0 = \alpha/2$, $\gamma_1 = 1-\alpha/2$, and \begin{equation} \label{cond-gamma-2} \gamma' =\gamma - [\gamma_1 + p\alpha/2]. \end{equation} We use the fact that $\gamma \in (p\alpha/2-1, 2p-1)$ and \eqref{cond-gamma-2} to get \eqref{cond-gamma-1}. As Assumption $(\rho_0, 1-\alpha/2, \delta)$ holds, by taking $\omega_1(x) = x_d^{\gamma'}$, we obtain \eqref{cor-est-2} from \eqref{main-est-2}. The proof is complete. \end{proof} \section{Degenerate viscous Hamilton-Jacobi equations}\label{sec:5} To demonstrate an application of the results in our paper, we consider the following degenerate viscous Hamilton-Jacobi equation \begin{equation}\label{eq:nonlinear} \begin{cases} u_t+\lambda u-\mu(x_d) \Delta u=H(z,Du) \quad &\text{ in } \Omega_T,\\ u=0 \quad &\text{ on } (-\infty, T) \times \partial \mathbb{R}^d_+, \end{cases} \end{equation} where $\mu$ satisfies \eqref{con:mu} and $H:\Omega_T \times \mathbb{R}^d \to \mathbb{R}$ is a given Hamiltonian. We assume that there exist $\beta, \ell >0$, and $h:\Omega_T \to {\overline{\mathbb{R}_+}}$ such that, for all $(z,P) \in \Omega_T \times \mathbb{R}^d$, \begin{equation} \label{G-cond} |H(z,P)| \leq{ \nu^{-1} (\min\{x_d^\beta,1\} |P|^{\ell}+x_d^\alpha h(z))}. \end{equation} The following is the main result in this section. \begin{theorem} \label{example-thrm} Let $p \in (1, \infty)$, $\alpha \in (0,2)$, and $\gamma \in (p(\alpha-1)_+-1, 2p-1)$. Assume that \eqref{G-cond} holds with $\ell =1$, $\beta \geq 1$, and $h \in L_p(\Omega_T, x_d^\gamma\, dz)$. Then, there exists $\lambda_0 = \lambda_0(d, p, \alpha, \beta, \gamma)>0$ sufficiently large such that the following assertion holds. For any $\lambda \geq \lambda_0$, there exists a unique solution $u \in \sW^{1,2}_p(\Omega_T, x_d^\gamma \,dz)$ to \eqref{eq:nonlinear} such that \[ \|\mathbf{M}^{-\alpha} u_t\|_{L_p} + \|D^2 u\|_{L_p} + \lambda \|\mathbf{M}^{-\alpha} u\|_{L_p} \leq N \|h\|_{L_p} \] where $\|\cdot\|_{L_p} = \|\cdot \|_{L_p(\Omega_T, x_d^\gamma\, dz)}$ and $N = N(d, p, \alpha, \beta, \gamma)>0$. \end{theorem} \begin{proof} The proof follows immediately from Theorem \ref{thm:xd} and the interpolation inequality in Lemma \ref{interpolation-inq} (i) below. \end{proof} \begin{remark} Overall, it is meaningful to study \eqref{eq:nonlinear} for general Hamiltonians $H$. It is typically the case that if we consider \eqref{eq:nonlinear} in $(0,T)\times \mathbb{R}^d_+$ with a nice given initial data, then we can obtain Lipschitz a priori estimates on the solutions via the classical Bernstein method or the doubling variables method under some appropriate conditions on $H$. See \cite{CIL, AT, LMT} and the references therein. In particular, $\|Du\|_{L^\infty([0,T]\times \mathbb{R}^d_+)} \leq N$, and hence, the behavior of $H(z,P)$ for $|P|>2N+1$ is unrelated and can be modified according to our purpose. As such, if we assume \eqref{G-cond}, then it is natural to require that $\ell=1$ because of the above. We note however that assuming \eqref{G-cond} with $\ell=1$ and $\beta \geq 1$ in Theorem \ref{example-thrm} is rather restrictive. It is not yet clear to us what happens when $0\leq \beta<1$, and we plan to revisit this point in the future work. \end{remark} To obtain a priori estimates for solutions to \eqref{eq:nonlinear}, we consider the nonlinear term $H$ as a perturbation. We prove the following interpolation inequalities when the nonlinear term satisfies \eqref{G-cond} with $\ell=1$ and $\ell=2$, which might be of independent interests. \begin{lemma} \label{interpolation-inq} Let $p \in (1, \infty), \beta\ge 0, \gamma>-1$, $1 \leq \ell \leq \frac{d}{d-p}$, and $\theta = \frac{1}{2}(1+\frac{d}{p}-\frac{d}{\ell p})$. Assume that $H$ satisfies \eqref{G-cond}. The following interpolation inequalities hold for every $u \in C_0^\infty(\Omega_T)$ and $\tilde f(z) = x_d^{-\alpha} {\min\{x_d^\beta,1\}|Du|^\ell}$, \begin{itemize} \item[(i)] If $\ell=1$ and $\beta \geq 1$, \[ \begin{split} \|\tilde f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} & \leq N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{1/2}\|D^2 u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{1/2} \\ & \qquad +N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}, \end{split} \] where $N = N(d, p, \beta, \gamma) >0$. \item[(ii)] If $\ell=2$, $p \geq \frac{d}{2}$, and $\beta \geq \max\{\frac{\gamma}{p}+\frac{d\alpha}{2p}, \frac{\gamma}{p}+2 +\frac{\alpha}{d} - \frac{d\alpha}{p}\}$, then \[ \begin{split} \|\tilde f\|_{L_p(\Omega_T, x_d^\gamma\, dz)} & \leq N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{2(1-\theta)}\|D^2 u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^{2 \theta} \\ & \qquad +N \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_T,x_d^\gamma\, dz)}^2, \end{split} \] where $N = N(d, p, \beta, \gamma) >0$. \end{itemize} \end{lemma} \begin{proof} For $m\in \mathbb{Z}$, set $ \Omega_m=\{z\in \Omega_T\,:\, 2^{-m-1} < x_d \leq 2^{-m}\}$. By the Gagliado-Nirenberg interpolation inequality, for $m\in \mathbb{Z}$, \[ \|Du\|_{L_{p\ell}(\Omega_m)} \leq N \left (\|u\|_{L_p(\Omega_m)}^{1-\theta} \|D^2u\|_{L_p(\Omega_m)}^\theta + 2^{2m\theta} \|u\|_{L_p(\Omega_m)} \right). \] Hence, for $m\geq 0$, \begin{align*} &\|\mathbf{M}^{\beta-\alpha} |Du|^\ell\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^p =\int_{\Omega_m} x_d^{p(\beta-\alpha)+\gamma}|Du|^{p \ell}\,dz\\ &\leq \, 2^{-m(p(\beta-\alpha)+\gamma)} \int_{\Omega_m} |Du|^{p\ell}\,dz\\ &\leq \, N2^{-m(p(\beta-\alpha)+\gamma)} \left(\int_{\Omega_m} |u|^{p}\,dz\right)^{\ell(1-\theta)} \left(\int_{\Omega_m} |D^2u|^{p}\,dz\right)^{\ell \theta} \\ &\qquad+ N2^{-m(p(\beta-\alpha)+\gamma+d-p\ell-d\ell)} \left(\int_{\Omega_m} |u|^{p}\,dz\right)^\ell\\ &\leq \, N2^{-m(p(\beta-\alpha)+\gamma+ p \ell \alpha(1-\theta)-\ell\gamma )} \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell (1-\theta)}\|D^2 u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell\theta} \\ &\qquad+ N2^{-m(p(\beta-\alpha)+\gamma+d-p \ell-d\ell+p \ell\alpha-\ell \gamma)}\|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell}. \end{align*} By performing similar computations, we get that, for $m< 0$, \begin{align*} &\|\mathbf{M}^{-\alpha} |Du|^\ell \|_{L_p(\Omega_m,x_d^\gamma)}^p\\ &\leq \, N2^{-m(-p\alpha+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma )} \|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell(1-\theta)}\|D^2 u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p \ell \theta} \\ &\qquad + N2^{-m(-p\alpha+\gamma+d-p\ell -d\ell +p\ell\alpha-\ell \gamma)}\|\mathbf{M}^{-\alpha} u\|_{L_p(\Omega_m,x_d^\gamma\, dz)}^{p\ell}. \end{align*} Then, if $\ell=1$ and $\beta \geq 1$, we have \[ \begin{cases} p(\beta-\alpha)+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma =p(\beta - \frac{\alpha}{2}) \geq 0,\\ p(\beta-\alpha)+\gamma+d-p\ell-d\ell+p\ell\alpha-\ell\gamma =p(\beta-1) \geq 0,\\ -p\alpha+\gamma+ p \ell \alpha(1-\theta)-\ell \gamma = -\frac{p\alpha}{2} \leq 0,\\ -p\alpha+\gamma+d-p\ell -d\ell +p\ell \alpha-\ell \gamma=-p \leq 0. \end{cases} \] We thus obtain (i). Similarly, the above four inequalities hold true when $\ell=2$, $p \geq \frac{d}{2}$, and $\beta \geq \max\{\frac{\gamma}{p}+\frac{d\alpha}{2p}, \frac{\gamma}{p}+2 +\frac{\alpha}{d} - \frac{d\alpha}{p}\}$, which yield (ii). \end{proof} \def$'${$'$} \end{document}

\begin{document} \begin{frontmatter} \title{An Induced Natural Selection Heuristic for Finding Optimal Bayesian Experimental Designs} \author[cam,uofm,doherty]{David J. Price\corref{cor1}} \cortext[cor1]{[email protected]} \author[uofa,acems]{Nigel G. Bean} \author[uofa,acems]{Joshua V. Ross} \author[uofa,acems]{Jonathan Tuke} \address[cam]{Disease Dynamics Unit, Department of Veterinary Medicine, University of Cambridge, Madingley Road Cambridge CB3 0ES, United Kingdom} \address[uofm]{Centre for Epidemiology and Biostatistics, Melbourne School of Population and Global Health, The University of Melbourne, VIC 3010, Australia} \address[doherty]{Victorian Infectious Diseases Reference Laboratory Epidemiology Unit at the Peter Doherty Institute for Infection and Immunity, The University of Melbourne and Royal Melbourne Hospital, VIC 3000, Australia} \address[uofa]{School of Mathematical Sciences, University of Adelaide, SA 5005, Australia} \address[acems]{ARC Centre of Excellence for Mathematical \& Statistical Frontiers, School of Mathematical Sciences, University of Adelaide, SA 5005, Australia} \begin{abstract} Bayesian optimal experimental design has immense potential to inform the collection of data so as to subsequently enhance our understanding of a variety of processes. However, a major impediment is the difficulty in evaluating optimal designs for problems with large, or high-dimensional, design spaces. We propose an efficient search heuristic suitable for general optimisation problems, with a particular focus on optimal Bayesian experimental design problems. The heuristic evaluates the objective (utility) function at an initial, randomly generated set of input values. At each generation of the algorithm, input values are ``accepted'' if their corresponding objective (utility) function satisfies some acceptance criteria, and new inputs are sampled about these accepted points. We demonstrate the new algorithm by evaluating the optimal Bayesian experimental designs for the previously considered death, pharmacokinetic and logistic regression models. Comparisons to the current ``gold-standard'' method are given to demonstrate the proposed algorithm as a computationally-efficient alternative for moderately-large design problems (i.e., up to approximately 40-dimensions). \end{abstract} \begin{keyword} Bayesian optimal design \sep Optimisation heuristic \sep Stochastic models \sep Sampling windows \end{keyword} \end{frontmatter} \section{Introduction} \label{section:intro} Optimising the design of an experiment is an important consideration in many areas of science, including, but not limited, to: biology \citep{Faller:2003}, clinical trials \citep{Berry:2004} and epidemiology \citep{Pagendam:2013}. The theory of optimal experimental design is a statistical framework that allows us to determine the optimal experimental protocol to gain the most information about model parameters, given constraints on resources. In evaluating an optimal Bayesian design, there are two main components: the search across the design space, and the evaluation of the utility. There have been many approaches to improving the efficiency of both aspects, summarised by \citet{Ryan:2015}. Recently, \citet{Overstall:2017} proposed the Approximate Coordinate Exchange (ACE) algorithm to address the search aspect of the Bayesian experimental design problem. The method utilises a coordinate exchange algorithm to update one dimension of the design at a time, coupled with a Gaussian process in order to search each dimension efficiently. It has been asserted that the future of optimal Bayesian experimental design lies in the ability to evaluate the optimal designs for large-scale problems (\emph{i.e.}, large or high-dimensional design spaces), in a computationally-efficient manner \citep{Ryan:2015}. In this paper, we address this by proposing a new search algorithm targeted at finding optimal Bayesian experimental designs. The search heuristic we present performs targeted sampling of the design space to find high utility designs, without making any assumptions about the shape of the utility function. An initial population of random designs is generated -- synonymous with multiple algorithm runs from random initial conditions as in other optimisation routines. Our method borrows the idea of targeting regions of high utility, as per the MCMC approach of \citet{Muller:1999}, by sampling new designs at each iteration around the ``best'' designs; chosen according to some acceptance criteria. We describe this algorithm using the notion of ``survival-of-the-fittest", as the ``fittest'' individuals -- according to their objective (utility) function value -- survive at each iteration (generation) based on a user-defined acceptance criteria, to produce offspring for the next generation. Hence, we propose this as a new type of evolutionary algorithm (\emph{e.g.}, \citealp{Goldberg:1989}), and refer to it herein as the Induced Natural Selection Heuristic (INSH). By independently sampling new designs around each accepted design, we aim to avoid the pitfalls associated with some other optimisation routines. For example, INSH is able to sample multiple regions of high utility at a time, thus exploring multiple local optima simultaneously, rather than potentially being stuck at a single local optima. Furthermore, by not combining the retained designs in any way, INSH avoids the potential to move to a region of low utility that is at the ``centre" of multiple local optima -- as may occur in a cross-entropy or genetic algorithm. By taking a sampling approach, as opposed to trying to approximate the function, INSH makes no assumptions about the shape of the utility function -- thus, it is not limited to utility functions that are, for example, smooth. Utilising (embarrassingly) parallel computation tools, the method can efficiently evaluate the utility for a large number of designs in each iteration. The ACE algorithm has allowed the consideration of Bayesian optimal designs for a larger, more-complex class of statistical models and experiments than was possible with previous algorithms. There are a number of drawbacks to ACE, however. By searching in one-dimension at a time, ACE risks missing the globally-optimal design, and instead may find only local optima. An approach to avoid this is to re-run the algorithm from a number of randomly generated initial designs \citep{Overstall:2017}. Similarly, as noted by the authors, by searching in one-dimension at a time, the algorithm will be inefficient in scenarios where there is a large correlation between the design variables -- a problem which adds to the difficulty in choosing a suitable number of iterations for each phase of the algorithm. The algorithm requires a sufficiently-good estimate of the utility when determining whether to accept the candidate design -- spurious estimates may lead to sub-optimal candidate designs being accepted, and thus push the algorithm away from regions of high utility. Alternatively, a large improvement in the computation time arises from the estimation of the utility surface in each dimension in the form of a Gaussian process based on a number of candidate points. This approximation to the utility surface based on noisy evaluations of the utility aims to provide a smooth approximation to the surface. When the surface is not smooth, or has a discontinuity (\emph{e.g.}, as exists in the utility surface for the death model in Figure \ref{deathmodel:fullutilitysurface} at $\boldsymbol{t}\approx(2.75,t_2)$ and $\boldsymbol{t}\approx(t_1,2.75)$), this has the potential to cause problems for the ACE algorithm. In the following, we present the INSH search algorithm in a general framework, and we note that efficient evaluation of the utility is another problem that needs to be addressed. We consider two existing approaches to evaluating the utility: an Approximate Bayesian Computation (ABC) approach used by \citet{Price:2016}, in a scenario where the benefits of this approach are realised; and a nested Monte-Carlo approximation using code from the \verb+acebayes+ package \citep{Rpack:acebayes}, otherwise. We consider the problem of finding the optimal design for the death model, a pharmacokinetic (PK) model tracking the concentration of a drug or treatment in the blood, and a four-factor logistic regression model. In the death and PK examples, a design $d$ consists of $n$ sampling times $(t_1,\dots,t_n)$, subject to some problem-specific constraints. First, we address the question of when to observe the stochastic process in order to gain the most information about the model parameters governing the death model. The Markovian death model has been considered previously in a Bayesian framework by \citet{Cook:2008}, \citet{Drovandi:2013}, and \citet{Price:2016}. We compare the optimal designs for 1-4 observation times in order to demonstrate the efficacy of the method. Second, we consider the question of sampling times for a PK model -- a process where the design space is higher-dimensional -- in order to demonstrate the efficiency of the INSH algorithm for larger design spaces. The optimal designs are compared to those evaluated using the ``gold-standard'' Approximate Coordinate Exchange (ACE) algorithm of \citet{Overstall:2017}. We also consider the idea of sampling windows for this example, which have been considered previously by \citet{Duffull:2003}, \citet{Chenel:2005}, \citet{Graham:2006}, \citet{McGree:2012}, and \citet{Duffull:2012}, for example. Finally, we compare the results of the INSH algorithm to those of the ACE algorithm for a standard four-factor logistic regression model \citep{Overstall:2017} -- a considerably higher-dimensional problem. We consider examples with $n=6,10,24$, and $48$ (independent) replicates in each experiment; corresponding to a design space with up to 192 dimensions (i.e., when $n=48$ replicates). \subsection{Bayesian Optimal Experimental Design} The aim of optimal experimental design is to determine the best experimental protocol in order to maximise some utility of the experiment. To achieve this aim, we specify a utility function $U(\boldsymbol{\theta},\boldsymbol{y},d)$ representing how we `value' the experimental design $d$, chosen from the set of all designs $\mathcal{D}$, where $\boldsymbol{\theta}$ represents the model parameters and $\boldsymbol{y}$ is the data. We are interested in the expected utility of using design $d$, over the unknown model parameters and data. That is, we wish to evaluate, \begin{align} u(d) &= E_{\boldsymbol{\theta},\boldsymbol{y}}[ U(\boldsymbol{\theta},\boldsymbol{y}, d)] \notag \\ &= \int_{\boldsymbol{y}} \int_{\boldsymbol{\theta}} U(\boldsymbol{\theta},\boldsymbol{y}, d) p(\boldsymbol{y} \mid \boldsymbol{\theta},d) p(\boldsymbol{\theta}) d\boldsymbol{\theta} d\boldsymbol{y}, \label{utilitydefn} \end{align} where $p(\boldsymbol{y} \mid \boldsymbol{\theta}, d)$ is the likelihood function of the unobserved data $\boldsymbol{y}$, under design $d$, and $p(\boldsymbol{\theta})$ is the prior distribution of the model parameters. The optimal design $d^*$ maximises the expected utility over the design space $\mathcal{D}$, that is, $d^* = \text{argmax}_{d\in\mathcal{D}} u(d)$. The utility function we use throughout this work is the Kullback-Leibler divergence \citep{Kullback} from the prior distribution to the posterior distribution (which is independent of $\boldsymbol{\theta}$), \begin{equation*} U(\boldsymbol{y}, d) = \int_{\boldsymbol{\theta}} \log \left( \frac{p(\boldsymbol{\theta} \mid \boldsymbol{y},d)}{ p(\boldsymbol{\theta})} \right) p(\boldsymbol{\theta} \mid \boldsymbol{y}, d) d\boldsymbol{\theta}, \end{equation*} which leads to an expected utility: \begin{align} u(d) = \int_{\boldsymbol{y}} \int_{\boldsymbol{\theta}} \log \left( \frac{p(\boldsymbol{\theta} \mid \boldsymbol{y},d)}{ p(\boldsymbol{\theta})} \right) p( \boldsymbol{y} \mid \boldsymbol{\theta},d) p(\boldsymbol{\theta}) d\boldsymbol{\theta} d\boldsymbol{y}. \label{ekld} \end{align} See \citet{Price:2016} for details of the derivation. Alternatively, it is commonplace to consider the Shannon Information Gain (SIG), which can be written as: \begin{align} U(\boldsymbol{\theta}, \boldsymbol{y}, d) =& \log p(\boldsymbol{\theta} \mid \boldsymbol{y}, d) - \log p(\boldsymbol{\theta}) \notag \\ =& \log p(\boldsymbol{y} \mid \boldsymbol{\theta}, d) - \log p(\boldsymbol{y} \mid d), \label{eqn:SIG} \end{align} through the application of Bayes' theorem. Maximisation of the expected SIG is equivalent to maximisation of the expected Kullback-Leibler divergence above. Unfortunately, it is often not possible to obtain an analytic evaluation of the expected utility function $u(d)$ (Equation \eqref{utilitydefn}), and approximate methods are required (see Section \ref{subsection:evaluate_utility}). \subsection{ACE Algorithm} The Approximate Coordinate Exchange algorithm of \citet{Overstall:2017} directly addresses the need for a computationally-efficient algorithm for determining optimal Bayesian experimental designs in high-dimensional design spaces \citep{Ryan:2015}. The reader is directed to \citet{Overstall:2017} for full details of the algorithm. Briefly, the algorithm considers each dimension of the experimental design one-at-a-time (\emph{e.g.}, the first observation time in an observation schedule), and evaluates the utility at a number of new, candidate values in that dimension (\emph{e.g.}, consider the utility at each of $q$ equally-spaced times across the feasible range of observation times, conditional on the other elements of the design). Having obtained these approximate utilities across the feasible range for the particular dimension of the design, a Gaussian process is fit to these candidate values to find an approximate ``optimal'' value as an update to this dimension of the design (accepted with some probability). The algorithm cycles through each design variable (probabilistically) updating them to the best value according to the Gaussian process approximation to the utility. The ACE algorithm is the first algorithm capable of dealing with high-dimensional design problems, in a computationally feasible amount of time. \section{Proposed Method: INSH Algorithm} In the following, we present a new algorithm to find optimal Bayesian experimental designs efficiently. We describe an algorithm that can utilise the current advantages of parallel computing -- which are rapidly improving as parallel-computing becomes more widely-available, more easy to implement, and more powerful. Simultaneously, we embrace an advantageous aspect of the inherently sequential, and thus difficult to parallelise efficiently, MCMC algorithms implemented by \citet{Muller:1999}, \citet{Cook:2008}, and \citet{Drovandi:2013}: namely, we seek to spend less computational effort evaluating designs in low-utility regions. This forms the crux of the efficiency of an MCMC approach, and is achieved by sampling from a function proportional to the utility. The new algorithm we propose instead evaluates the utility of multiple designs simultaneously -- in order to realise the benefits of parallel computing -- and samples new designs at each iteration of the algorithm around designs that satisfy some acceptance criteria. The acceptance criteria for designs at each iteration can be chosen in a number of different ways. In this paper, we demonstrate the acceptance of a fixed number of the ``best'' designs, similar to the proportion of ``elite'' samples in a cross-entropy algorithm \citep{DeBoer:2005}. In contrast to these existing optimisation algorithms, the algorithm presented here considers multiple designs at each iteration, allowing us to explore the design space more efficiently. The INSH algorithm is detailed in Algorithm \ref{INSH_algorithm}. Note that in Step 6, the best design considered in any previous iteration is reintroduced into the set of designs that are to be sampled around, in order to continue to explore this region. \begin{algorithm}[h] \caption{INSH Algorithm}\label{INSH_algorithm} \begin{algorithmic}[1] \State Choose an initial set of designs, $D$ (\emph{e.g.}, a coarse grid of design points across the design space, or randomly sample). \State Specify the number of generations (iterations) of the algorithm $W$, a perturbation function $f(d\mid d')$, and the acceptance criteria. \For{$w=1$ to $W$} \State \parbox[t]{0.95\textwidth}{\strut For each design $d^i\in D$, sample parameters $\boldsymbol{\theta}\sim p(\boldsymbol{\theta})$, and simulate data $\boldsymbol{y}^i$ from the model.\strut } \State \strut Evaluate utility $u(d^i)$, for each design $d^i \in D$. \label{insh_evaluate_utility} \State \parbox[t]{0.95\textwidth}{\strut Set $D'$ to be the designs which satisfy the acceptance criteria, and the current optimal design $d^*$ (even if it occurred in a previous generation).\strut } \State \parbox[t]{0.95\textwidth}{\strut Sample $m$ designs from $f(d\mid d')$, for each $d'\in D'$. Set $D$ to be these newly sampled designs.\strut } \EndFor \Ensure Set of designs $d$, and corresponding approximate utilities $u(d)$ (and hence, the optimal design $d^* = \underset{d\in\mathcal{D}}{\text{argmax}}(u(d))$). \end{algorithmic} \end{algorithm} \subsection{Evaluation of the Utility} \label{subsection:evaluate_utility} An efficient approach to evaluate the utility of a design in Step \ref{insh_evaluate_utility} of Algorithm \ref{INSH_algorithm} is that of the ABCdE algorithm \citep{Price:2016}. \comment{We state the ABCdE algorithm in Algorithm 2 of Online Resource A, and direct the reader to \citet{Price:2016} for a complete description of the algorithm}. In particular, we use Steps 3 to 9 of Algorithm 2, in Online Resource A. Note that this approach is suitable for discrete data, and for low-dimensional design spaces. This is due to the majority of the efficiency coming from having to evaluate a posterior distribution only once for each unique data set. As the number of possible unique data sets increases -- for example, either by observing the process more often (increasing the size of the design space), or having a larger population -- this approach to evaluating the utility becomes less efficient. We use this approach to evaluate the utility for the Markovian death model, in order to demonstrate the INSH algorithm. For cases where the dimension of the data is too large (or continuous), we must consider an alternative approach to evaluating the utility for each design. As noted previously, this is one of the two main challenges when searching for optimal Bayesian designs. A suitable and efficient method for evaluation of the utility for a design is often problem-specific, and a number of different approaches have been considered -- a summary of these approaches can be found in \citet{Ryan:2015}. For the PK and logistic regression examples we consider subsequently, we implement the utility function of \citet{Overstall:2017}, as provided in the \verb+acebayes+ package in R \citep{Rpack:acebayes}. Briefly, the SIG utility in equation \eqref{eqn:SIG}, is estimated by a nested Monte-Carlo approximation of the values $p(\boldsymbol{y} \mid \boldsymbol{\theta}, d)$ and $p(\boldsymbol{y} \mid d)$, within the Monte-Carlo approximation to the expected utility, ${u}(d)$. Borrowing the notation of \citet{Overstall:2017}, define $\boldsymbol{\psi} = (\boldsymbol{\theta}, \boldsymbol{\gamma})$ to be the combination of the parameters of interest, $\boldsymbol{\theta}$, and nuisance parameters, $\boldsymbol{\gamma}$. Then, we use $\tilde{B}$ simulations to approximate the inner Monte-Carlo estimates: \begin{align*} \tilde{p}(\boldsymbol{y} \mid \boldsymbol{\theta}, d) = \frac{1}{\tilde{B}} \sum_{b=1}^{\tilde{B}} p(\boldsymbol{y} \mid \boldsymbol{\theta}, \tilde{\boldsymbol{\gamma}}_b, d ), \quad \text{and } \quad \tilde{p}(\boldsymbol{y} \mid d) = \frac{1}{\tilde{B}} \sum_{b=1}^{\tilde{B}} p(\boldsymbol{y} \mid \tilde{\boldsymbol{\theta}}_b, \tilde{\boldsymbol{\gamma}}_b,d ), \end{align*} where $(\tilde{\boldsymbol{\theta}}_b, \tilde{\boldsymbol{\gamma}}_b)$ are the $\tilde{B}$ parameters sampled from the prior distribution of $\boldsymbol{\psi}$. Similarly, $B$ simulations are used to evaluate the outer Monte-Carlo estimate, \begin{align*} \tilde{u}(d)=\frac{1}{B} \sum_{l=1}^B\left[ \log \tilde{p}(\boldsymbol{y}_l \mid \boldsymbol{\theta}_l, d) - \log \tilde{p}(\boldsymbol{y}_l \mid d) \right], \end{align*} with $\{\boldsymbol{y}_l, \boldsymbol{\theta}_l\}$ parameters, and corresponding simulations, sampled from the prior and simulated from the model, respectively. In the work of \citet{Overstall:2017}, the authors use $\tilde{B}=B=1{,}000$ to evaluate the candidate designs' utilities in the one-dimensional search (Step 1b of the ACE Algorithm in \citealp{Overstall:2017}), and $\tilde{B}=B=20{,}000$ to evaluate the utility when determining whether to accept the candidate design (Steps 1d and 3e of the ACE Algorithm in \citet{Overstall:2017}; note that Step 3 is not implemented for the compartmental model). \subsection{Choice of Acceptance Criteria} There are a number of ways in which we can choose to retain designs; taking inspiration from other optimisation routines. For example, one could retain all designs that are greater than some percentage of the current maximum (\emph{e.g.}, retain all designs that have at least $95\%$ of the information compared to the current ``optimal"), although this approach requires some insight to how ``flat" the utility surface is in order to avoid retaining too many or too few designs at each iteration. While we do not present the results here, testing this approach for the Markovian death model showed promising results. The approach that we implement in this work is similar to the ``elite" samples of a cross-entropy algorithm \citep{DeBoer:2005}. That is, at each generation, the algorithm accepts the best $r$ designs according to their utility. At the next generation of the algorithm, we sample $m$ designs from the perturbation kernel from each of these $r$ designs. In order to balance the trade-off between exploration and exploitation, one can specify a sequence of decreasing and increasing values for $m_w$ and $r_w$, respectively. Specifying the number of designs that are retained and sampled at each iteration ensures full control over the number of designs considered at each generation of the algorithm, allowing specification of the computational effort spent in searching for the optimal design. Thus, one may reasonably evaluate the optimal (or near-optimal) Bayesian design in a computationally-efficient time-frame. For the high-dimensional design spaces considered in the logistic regression example, we choose to modify this acceptance step slightly. Specifically, we choose to retain the best $r_w$ designs from the current \emph{and} previous iterations of the INSH algorithm. This acts as a failsafe, in instances where the newly proposed designs end up in regions of lower utility than the original (retained) design, which is more likely to occur with particular high-dimensional designs. This acceptance criteria allows the algorithm to start again from the previous iteration (for a subset of the designs), rather than end up missing regions of high utility through a poor round of sampling. \subsection{Perturbation Kernel} The \emph{perturbation kernel} is a probability distribution used to sample new designs at each generation of the INSH algorithm, by \emph{perturbing} (\emph{i.e.}, adding some noise to) previous designs. In the death and PK examples we consider in this work, we use a truncated, multivariate-Normal distribution (where the dimension is given by the dimension of the design space, and the truncation is to ensure constraints are satisfied). For the logistic regression example, we demonstrate the flexibility of choice in this aspect, by using a uniform distribution centred on each design point. \comment{There are no explicit guidelines on how to choose the kernel, and the choice is often driven by knowledge and experience of the problem at hand, as with sequential importance sampling methods (e.g., \citealp{Toni:2009}), however one can reasonably sample from any distribution, centred on the current design points, which can suitably explore the design space. The authors propose that without any knowledge of the relationship between the design points, a symmetric perturbation kernel is a sensible starting point.} A standard cross-entropy algorithm uses the accepted samples to define the mean and (co-) variance structure of a (multivariate-) Normal distribution, and all new samples are generated by this distribution. We prefer to avoid this approach, instead allowing the region surrounding each accepted point to be explored individually. Combining all accepted samples into a single distribution from which to sample, may result in new samples not being generated in regions of high utility (for example, when considering multi-modal utility surfaces), and requires re-evaluation of the (co-)variance matrix at each generation. \subsection{Stopping Criteria} A common feature of optimisation tools is a criterion for stopping the algorithm. It would be straight-forward for the user to implement a stopping criteria based on the change in utility of newly sampled designs at each iteration of the algorithm, based on the level of accuracy desired. In the examples considered herein, we choose to demonstrate the algorithm by running it for a fixed number of iterations, and assessing convergence graphically through box-plots of the estimated utility across each generation of the algorithm (similar to the trace plots of \citealp{Overstall:2017}). \section{Examples} \subsection{Markovian Death Model} Consider the Markovian death model as defined by \citet{Cook:2008}. There is a population of $N$ individuals which, independently, move to an infectious class $I$ at constant rate $b_1$ -- for example, due to infection from an environmental source. The Markov chain models the number of infectious individuals at time $t$, $I(t)$ (where the number of susceptible individuals is $S(t)=N-I(t)$). The positive transition rates of the Markov chain are given by $q_{i,i+1}=b_1(N-i)$, for $i=0,\dots,N-1$. The prior distribution we consider is $b_1 \sim \log\text{-}N(-0.005, 0.01)$, chosen such that the mean lifetime of individuals in the population is one, with an approximate variance of 0.01 (as per \citealp{Cook:2008}). The optimal experimental design for the Markovian Death model has previously been considered in a Bayesian framework by \citet{Cook:2008}, \citet{Drovandi:2013}, and \citet{Price:2016}. \citet{Cook:2008} utilised the MCMC approach of \citet{Muller:1999}, and used an exact posterior, hence, the designs of \citet{Cook:2008} provide a gold-standard with which to compare our results. \citet{Drovandi:2013} also utilised the MCMC approach of \citet{Muller:1999}, however, coupled with an approximate posterior distribution evaluated via an ABC approach. We note however, that the MCMC approach struggles to evaluate the optimal design once considering more than four design parameters. This is due to the increasing computational difficulty associated with the evaluation of the mode of the multi-dimensional utility surface (\citealp{Drovandi:2013}). \citet{Price:2016} provided an exhaustive-search across a grid on the design space, where the utility was evaluated using the ABCdE method. The INSH code for the death model is implemented in MATLAB R2015b. \subsection{Pharmacokinetic Model} Consider the PK experiment considered by \citet{Ryan:2014} and \citet{Overstall:2017}. In these PK experiments, individuals are administered a fixed amount of a drug. Blood samples are taken in order to understand the concentration of the drug within the body over time. Let $y_t$ represent the observed concentration of the drug at time $t$. We model the concentration as $ y_t = \mu(t)(1 + \epsilon_{1t}) + \epsilon_{2t}$, where, $$\mu(t) = \frac{400 \theta_2}{\theta_3 (\theta_2 - \theta_1)} \left( e^{-\theta_1 t} - e^{-\theta_2t} \right),$$ is the mean concentration at time $t$, and $\epsilon_{1t} \sim N(0, \sigma^2_{prop})$, $\epsilon_{2t} \sim N(0, \sigma^2_{add})$, $\sigma^2_{prop}=0.01$ and $\sigma^2_{add}=0.1$. That is, $y_t \sim N\left( \mu(t), \sigma_{add}^2 + \sigma_{prop}^2\mu(t)^2 \right).$ The blood samples are taken within the first 24 hours after the drug is administered (that is, $t\in[0,24]$), and it is not practical to take blood samples less than 15 minutes apart (hence, $t_{i+1}-t_i\geq0.25$). We wish to make 15 observations of this system in order to obtain information about the model parameters $\boldsymbol{\theta} = (\theta_1, \theta_2, \theta_3)$, where $\theta_1$ represents the first-order elimination rate constant, $\theta_2$ represents the first-order absorption rate constant, and $\theta_3$ represents the \emph{volume of distribution} -- a theoretical volume that a drug would have to occupy in order to provide the same concentration as is currently present in the blood plasma, assuming the drug is uniformly distributed \citep{Ryan:2014}. As per \citet{Ryan:2014} and \citet{Overstall:2017}, the model parameters $\boldsymbol{\theta}=(\theta_1,\theta_2,\theta_3)$ are assumed \emph{a priori} to be independently, normally distributed on the log-scale, with mean $\log(0.1)$, $\log(1)$, and $\log(20)$ respectively, and variance 0.05. \citet{Duffull:2012}, \citet{McGree:2012}, \citet{Ryan:2014}, and \citet{Ryan:2015pk} have previously evaluated optimal Bayesian experimental designs for pharmacokinetic models, either for only a few sampling times ($<5$), or more sampling times via dimension reduction schemes (\emph{e.g.}, search across the two-parameters of a Beta distribution, where the quantiles are scaled to give the observation times). \citet{Overstall:2017} are currently the only example of a method efficient enough to establish optimal Bayesian designs for a design-problem of this magnitude directly (i.e., without implementing a dimension reduction scheme), in a feasible amount of computation time. Furthermore, we show how the output of the INSH algorithm can be used simply to construct \emph{sampling windows} -- a range of values for each observation, rather than a fixed value for each observation time. Sampling windows allows those implementing an optimally-chosen design some flexibility in choosing the sampling times, such that the resulting design is more practically feasible. By defining sampling windows, we can dictate a set of near-optimal designs -- which are practically feasible -- which can be implemented more easily. This avoids the scenario where an inferior design is chosen preferentially by those that are implementing the design, having been supplied with an impractical optimal design. Sampling windows have been considered previously for similar types of models, for example, in \citet{Duffull:2003}, \citet{Chenel:2005}, \citet{Graham:2006}, \citet{Duffull:2012}, and \citet{McGree:2012}, to name a few. As the output of the INSH algorithm consists of a large number of designs sampled around regions of high utility -- as opposed to a single design, as in ACE -- the construction of sampling windows is a simple extension to the algorithm. The INSH algorithm for the PK example is implemented in R (version 3.3.0). \subsection{Logistic Regression in Four Factors} Finally, we consider the logistic regression model of \citet{Overstall:2017} in order to demonstrate the benefits of INSH for a considerably higher-dimensional design problem. We consider only the case with independent groups (\emph{i.e.}, no random effects). Let $y_s \sim \text{Bernoulli}(\rho_s)$ be the $s^{th}$ response ($s=1,\dots,n$), and, \begin{equation*} \log \left( \frac{\rho_{s}}{1-\rho_{s}} \right) = \beta_0 + \beta_1 x_{1s} + \beta_2 x_{2s} + \beta_3 x_{3s} + \beta_4 x_{4s}, \end{equation*} where $\beta_i$ ($i=0,\dots,4$) are the parameters of interest. The design matrix is $\mathbf{D}=(\boldsymbol{X}_1, \dots, \boldsymbol{X}_4)$, where $\boldsymbol{X}_i$ is a column vector containing the $x_{is}$ values ($s=1,\dots,n$), with $x_{is}\in[-1,1]$. We define the following, independent prior distributions for each of the parameters of interest $\beta_i\sim U[a,b]$, $i=0,\dots,4$, where $a=(-3,4,5,-6,-2.5)$, and $b=(3,10,11,0,3.5)$. We consider the cases where $n=6,10,24$, and $48$. The INSH algorithm for the logistic regression example is implemented in R (version 3.3.0). \subsection{Code to Implement INSH} \comment{ The online repository} \verb+http://www.github.com/DJPrice10/INSH_Code+ \comment{contains code to implement the INSH algorithm in MATLAB (Markovian death model), and R (PK model).} \section{Results} \label{section:results} \subsection{Markovian Death Model} We consider the optimal observation schedule when the number of observations permitted is $n=1,\dots4,6$ or $8$. The designs for $n=1,\dots,4$ observation times are compared to existing results, and $n=6$ and $8$ are reported as it was not computationally feasible using previous methods. First, however, we provide a graphical demonstration of the INSH algorithm by considering two observation times for the death model. We choose to implement the INSH algorithm for $W=10$ iterations. We start with 100 randomly chosen designs across the feasible region, and retain the best $r_w=10$ and then $5$ designs (for 5 iterations each). At each generation, $m=3$ and then $6$ designs (for 5 iterations each), are sampled around each accepted design from the perturbation kernel, in order to sufficiently explore the space around each retained design. That is, we consider $m_w\times r_w=30$ designs at each iteration of the algorithm. The perturbation kernel in this example is a Normal distribution centred on the accepted design, with fixed standard deviation 0.1 for each design point (to allow reasonable exploration around each design point), zero covariances, and then truncated subject to the design constraints, \emph{i.e.}, $t_{i+1}-t_i>0$, $i=1,\dots,n-1$. Figure \ref{deathmodel:stagesofINSH} shows the progression of the INSH algorithm at each of the first six generations. For comparison, Figure \ref{deathmodel:fullutilitysurface} shows the full utility surface for the death model, evaluated using the ABCdE algorithm at all observation times across a grid with spacing 0.1, with $t_i\in[0.1,10]$. We can clearly see the optimal design is on a ridge at approximately (0.9, 2.8). There is also a region of high utility around (0.7, 2.0). Regions of low utility exist for very small $t_1$ (and in particular, $t_2>4$), or where both $t_1$ and $t_2$ are large (\emph{e.g.}, both above 3.5). In Figure \ref{deathmodel:stagesofINSH}, Generation 2 (Figure \ref{INSH_G2}) shows that regions of low utility are discarded early, and high utility regions are retained. Generations 2-6 (Figures \ref{INSH_G2}-\ref{INSH_G6}) demonstrate the convergence of the samples towards the region containing the optimal design. Generation 5 demonstrates the samples converging about the two ``peaks" observed in Figure \ref{deathmodel:fullutilitysurface} -- clearly demonstrating the ability to investigate multiple regions of high utility simultaneously. Figure \ref{PK_insh_final_samples} shows all design points considered throughout the INSH algorithm, with each point shaded by the utility value (darker corresponds to higher utility). The regions of high utility have been sampled more thoroughly. \begin{figure} \caption{Demonstration of the design regions being considered by the INSH algorithm at each of the first six generations, and the convergence to regions of high utility. The shaded region corresponds to the infeasible design region (i.e., where $t_2<t_1$).} \label{INSH_G2} \label{INSH_G3} \label{INSH_G4} \label{INSH_G5} \label{INSH_G6} \label{deathmodel:stagesofINSH} \end{figure} \begin{figure} \caption{(a) Full utility surface for two observations of the death model evaluated on a grid using the ABCdE algorithm. (b) Samples from every generation of the INSH algorithm for two observations of the death model. In each figure, darker regions/points correspond to higher utility values.} \label{deathmodel:fullutilitysurface} \label{PK_insh_final_samples} \end{figure} Online Resource B contains: box-plots illustrating the convergence of the sampled observation times towards the optimal, and the corresponding utilities towards the maximum in Figure S1; the optimal designs determined via INSH compared to the existing methods in Table S1, and; the corresponding INSH algorithm inputs in Table S2. \subsection{Pharmacokinetic Model} \label{section:PKresults} Due to the physical constraints on the frequency at which sampling can be performed (at least 15 minutes apart), we restrict the designs such that $t_{i+1}-t_i \geq 0.25$, $i=1,\dots,14$. We sample designs from a multivariate-Normal perturbation kernel with fixed standard deviation 0.20, zero covariance, and truncated subject to the design constraints. The standard deviation was chosen such that one standard deviation was roughly the minimum distance between any two design points. The first generation of designs were sampled uniformly from the viable design space, $[0,24]$, subject to the constraints. As with the previous example, we specify a decreasing sequence of the number of retained designs $r_w$, and an increasing sequence of the number of sampled designs $m_w$. In order to compare the run time of the ACE algorithm to the INSH algorithm, we implemented the ACE algorithm as detailed in \citet{Overstall:2017}, (\emph{i.e.}, running 20 instances of the ACE algorithm from the \verb+acebayes+ package in (embarrassingly) parallel fashion across four cores). On an iMac running OSX 10.11.4 with 4.0GHz Intel Core i7 processor and 32GB memory, this took 15.53 hours. We did not include the run time of the post-processing utility evaluation of the 20 candidate designs, 20 times each, in order to establish the overall optimal design, for reasons we state shortly. The ACE algorithm for this example in \citet{Overstall:2017} was performed 20 times from random initial conditions, each for a total of 20 iterations. Each iteration searches across each of the 15 dimensions of the design, and considers 20 candidate times to fit the Gaussian process. Thus, a total of $120{,}000$ designs are considered (\emph{i.e.}, utility evaluations) in the ACE algorithm, where $6{,}000$ of these utility evaluations are completed using significantly more Monte Carlo simulations. Specifically, the utility for the 20 candidate times used to train the Gaussian process are evaluated using $\tilde{B}=B=1{,}000$ Monte Carlo simulations, while the utility corresponding to the design with the proposed new observation time is evaluated using $\tilde{B}=B=20{,}000$. The advantage of the INSH algorithm is in its ability to consider a large number of designs in multiple regions simultaneously, and so it is sufficient to use less effort to evaluate the utility of each design, as a noisy estimate of the utility will have less influence on the output of the algorithm. Hence, we used $\tilde{B}=B=5{,}000$ for the evaluation of the utility of each design, which was completed in parallel on four cores (using \verb+foreach+ and \verb+doParallel+ packages in R), on the same machine as stated above. In particular, at each generation of the algorithm, the calculation of the utilities of the designs in the current wave was split across the number of available cores (\emph{i.e.}, 1/4 of the required utility calculations were allocated to each core). We ran the INSH algorithm for $W=60$ iterations, with $1{,}200$ randomly generated initial designs. At each iteration, we retained the ``best" 150, 75, 50, 25, and 10 designs, and proposed two, four, six, 12 and 30 new designs around each accepted design, for 12 iterations of each combination -- maintaining consideration of 300 designs at each iteration, while regularly increasing the exploitation and reducing exploration. These values of $r_w$ and $m_w$ were chosen such that earlier generations of the algorithm retained a reasonable number of designs -- thus, not excluding regions of the design space too quickly -- and as the algorithm progressed, focussed computational effort on high-utility regions of the design space. Given the larger dimension of the design in this example (compared to the death model), we chose to sample a large number of designs around each retained design in later generations of the algorithm, in order to sufficiently explore the design space in proximity to the optimal. This run of the INSH algorithm took 2.23 hours (approximately 7 times faster than the ACE algorithm). Having obtained the designs and utility evaluations from the INSH algorithms, we perform the same post-processing utility evaluation on the 20 best considered designs, with 20 evaluations of the utility of each design with $\tilde{B}=B=20{,}000$, in order to identify the overall optimal. The total number of designs considered by the INSH algorithm with this selection criteria is approximately: $(\text{No.\ initial designs}) + (W-1)\times r_w\times m_w=1200 + (60-1)\times 300 = 18{,}900$ -- that is, the number of initial designs, plus how many were retained at each generation multiplied by the number that were sampled around each retained design. In practice, this number is often slightly higher, as the $r_w^{th}$ ranked design can be a tie, and the optimal design is re-introduced into the set of designs being considered if it occurred in a previous generation (this run of the INSH algorithm actually considered $19{,}428$ designs). Figure \ref{PK_all_INSH_designs} shows box-plots of the 20 utility evaluations for each of the 20 best designs that were considered by the INSH algorithm, compared to the same number of evaluations of the ACE optimal design reported in \verb+optdescomp15sig()+ in the \verb+acebayes+ package (each utility evaluation using $\tilde{B}=B=20{,}000$). We can see from this figure that there are three designs (5, 8, 19), that perform similarly well to the design found using the ACE algorithm. Online Resource C contains: these three designs from INSH in Table S3; summaries of the utilities for the top 20 designs evaluated by INSH in Table S4; box-plots demonstrating the convergence of the INSH algorithm to the optimal region in Figure S3, and; a comparison of the ACE and INSH optimal designs performance with regards to inference in Figure S4 (in particular, the posterior variance and bias in posterior mode). \begin{figure} \caption{Box-plots of the utility for the optimal design found by the ACE algorithm, compared to the top 20 designs considered by the INSH algorithm. The utility of each design is evaluated 20 times, using $\tilde{B}=B=20{,}000$ Monte Carlo simulations.} \label{PK_all_INSH_designs} \end{figure} \subsubsection{Sampling Windows} The population-based approach of INSH means that we retain a large number of designs with high utility. We use these ``best'' designs to construct the sampling windows for each sampling time, similar to the approach of \citet{McGree:2012}. \citet{McGree:2012} used percentiles of the designs evaluated once a stopping-criteria has been reached in their algorithm to form the sampling windows -- we choose a fixed number of ``best'' designs to form the windows. Given the windows, those implementing the design can choose observation times from these windows, ensuring that the physical constraint, $t_{i+1}-t_i\geq0.25$, is satisfied. As an example of this process, we arbitrarily consider the top 20 designs from the output of the INSH algorithm for the PK example, and form sampling windows as the range of values considered at each observation time for these ``best'' designs. Alternatively, one could consider all designs that were within some percentage of the utility corresponding to the maximum, or, use a weighting based on the average utility for each design to approximate a distribution for each sampling time which could subsequently be sampled. In order to construct the sampling window designs, we ``bootstrap" an observation schedule by randomly selecting each of the 15 sampling times (with equal probability), from the 20 candidate observation times, subject to the constraints. A new design is sampled for each of the 20 utility evaluations to demonstrate the range of potential outputs from this approach. Figures \ref{PK_all_plots:sw_range} and \ref{PK_all_plots:sw_density} show the INSH sampling windows for each observation time. Figure \ref{PK_all_plots:opt_comparison} shows the optimal observation schedules evaluated using the ACE and INSH algorithms. Note that the optimal design returned from the INSH method corresponded to the $19^{th}$ highest utility value from the original output of the INSH algorithm (\emph{i.e.}, using $\tilde{B}=B=5{,}000$). It was deemed the optimal design as it corresponded to the highest mean utility, from 20 utility evaluations using $\tilde{B}=B=20{,}000$ (Figure \ref{PK_all_INSH_designs}). Figure \ref{PK_all_plots:util_comparison} shows box-plots of 20 utility evaluations (using $\tilde{B}=B=20{,}000$) for the ACE and INSH optimal designs, and the 20 randomly selected designs from the sampling windows. Note that the average efficiency of the sampling windows designs compared to the INSH optimal design is 99.07\%. \begin{figure} \caption{(a) Comparison of the optimal designs from the ACE and INSH methods. (b) Boxplots of 20 utility evaluations for the ACE and INSH optimal designs, and the INSH sampling windows designs, using $\tilde{B}=B=20000$. (c) Sampling windows for each observation time obtained from the 20 designs corresponding to the highest utilities found during the INSH algorithm, plotted over 50 realisations of the PK model simulated at parameter values randomly drawn from the prior distribution. The error bars show the sampling window range. (d) Density plot of the sampling windows for each observation time. } \label{PK_insh_v_ace} \label{PK_all_plots:opt_comparison} \label{PK_utils} \label{PK_all_plots:util_comparison} \label{PK_sw_lineplots} \label{PK_all_plots:sw_range} \label{PK_sw_util} \label{PK_all_plots:sw_density} \label{PK_all_plots} \end{figure} \subsection{Logistic Regression in Four Factors} We implement the INSH design for $n=6,10, 24$, and $48$, and compare the utility for the best performing design found via INSH to those reported from ACE. INSH is implemented in the same way as for the PK example. We step down the value of $r_w$, and increase the value of $m_w$ as $w$ increases, such that early iterations are geared towards exploration, while later iterations are focussed on exploitation. Table S5 in Online Resource D contains the values of $m_w$ and $r_w$ that are used for the INSH algorithm. In this example, we utilise a uniform perturbation kernel with a fixed-width. In order to further increase exploitation as the algorithm progresses, we step down the width of this proposal distribution in line with the change in $r_w$ and $m_w$. For each example, we retained a reasonable number of designs in early generations of the algorithm in order to exclude less of the design space, and increase computational effort on regions of high utility in later generations, as with the PK example. Given the high-dimensional design space for the $n=48$ example, we sampled more designs around each retained design at each generation (\emph{i.e.}, larger $m_w$), in order to better explore the design space. Furthermore, as we are considering very high-dimensional design spaces, we run the risk of randomly stepping in the wrong direction from a given design -- and one cannot feasibly explore satisfactorily around each design point. To avoid the potential for stepping in a poor direction and not being able to get back to a region of potentially high utility, we slightly alter our acceptance step to consider the best $r_w$ designs out of the current iteration, \emph{and} those that were accepted in the previous iteration. This way, should we move from a region of high utility to a region of low utility through a poorly proposed design, we are able to essentially take one-step back, and propose a new design from the previous design. \comment{ Initially, our approach was to run INSH for the same computation time as ACE in order to establish the utility of the best designs found via each method. However, the INSH algorithm was unsuccessful in converging to designs with the same utility of those found via ACE for scenarios with a larger design space (INSH designs contained approximately 98.5\% of the utility relative to the ACE designs). Instead, we ran INSH for a reasonable set of parameter values to determine near-optimal designs in a reasonable amount of computation time, and then adjusted the input parameters for ACE to run for the same computation time as INSH. In particular, we reduced the number of iterations of the two phases of the algorithm ($N_I,N_{II}$), or the amount of effort used in evaluating the utility in both training the Gaussian process, and choosing to update a coordinate ($B$). We denote these implementations of ACE as ACE$_N$ and ACE$_B$, respectively. Figure \ref{LR_util} shows box-plots of the utility evaluated at designs found via each implementation of INSH and ACE. For each $n$, the designs found via ACE$_N$ appear to perform as well as those found via INSH, however, ACE$_B$ designs perform better than those found via INSH for larger design spaces (i.e., $n>10$). We note that for $n=6$, and $10$, INSH finds similarly-performing designs to ACE for the same computation time. In Online Resource D: Tables S6-S9 show the designs found via the INSH algorithm; Figures S5-S8 show box-plots of the utility of each design considered at each iteration of the INSH algorithm for each $n$; Table S10 shows the mean and 2.5-97.5\%-percentiles of the designs found by each of the ACE, INSH, ACE$_N$ and ACE$_B$ algorithms, and; Table S11 shows the input parameters for ACE$_N$ and ACE$_B$ to achieve the same computation time as INSH for each $n$.} \comment{ We do not believe that the discrepancy in performance of the designs found via the ACE and INSH methods is due solely to the increased dimension of the design space. Rather, we believe that it is a combination of the high-dimensional setting rendering the perturbation step less effective, and that the optimal design in each case resides on the boundary. In considerably high-dimension problems, the perturbation step results in designs considered by INSH routinely moving away from some, or all, of the boundary values that would otherwise result in a more informative design. In other words, the resampling approach of INSH means that it is highly unlikely to stay at a large number of boundary values simultaneously. Conceptually, one can see that designs that reside away from the boundary values can be approached from any direction, whereas boundary values can only be approached from, in a loose sense, ``one direction". In these examples, the designs contain many boundary values -- for n=6, 10, 24 and 48 the optimal designs via ACE contain 16/24, 28/40, 70/96, and 143/192 boundary values (i.e., -1 and 1's), respectively. We acknowledge that the optimal design existing on a boundary is a common feature of multi-factor experiments such as this, and that this example has highlighted a shortcoming of the INSH algorithm. However, it is the authors belief that INSH is still a suitable method for high-dimensional problems, however we are unable to demonstrate this with an existing high-dimensional design problem at this stage. } \comment{ Given that INSH out-performs ACE for small-moderate design spaces -- as demonstrated for both the PK example and LR example (for a fixed computation time) -- we propose that INSH is a suitable, computationally-efficient alternative to the ACE algorithm for up to 40-dimensional design spaces (i.e., corresponding to $n=10$ in this example). Otherwise, for truly high-dimensional design problems (i.e., more than 40-dimensions), the authors propose that ACE is implemented given it has been shown to perform well in these scenarios. } \begin{figure} \caption{(a) Box plots of 20 utility evaluations for the optimal designs found by each of the ACE, INSH (solid), ACE$_N$, and ACE$_B$ (dashed) methods, for $n=6,10,24$ and $48$.} \label{LR_util} \end{figure} \section{Discussion} \label{section:discussion} In this paper, we have considered three common types of statistical model: a Markovian death model, a one-compartment PK model, and a four-factor logistic regression model. Our results for the death model provide a simple demonstration of the efficacy of the proposed INSH algorithm, and gives equivalent answers to previously applied methods. The PK model allowed us to consider a larger design space, and show that the INSH algorithm is able to return designs that marginally outperformed those found using the ``gold-standard" ACE algorithm, in considerably less computation time -- illustrating that for moderate-size design spaces, INSH is more computationally efficient than ACE. We also showed the simple extension to the INSH algorithm that allows the construction of sampling windows. Finally, the logistic regression example provided an example of a truly high-dimensional design space. We were unable to find designs that performed as well as those found via ACE, however, we were able to demonstrate that INSH provides a suitable, computationally-efficient approach for up to approximately 40-dimensions -- which encompasses a large range of experimental design problems. Alternative examples of considerably high-dimension with optimal designs residing away from the boundary should be considered in order to demonstrate the performance of INSH relative to ACE in these instances. We have not provided a proof that the INSH algorithm will converge to the optimal design, however, one can see that in the limit (\emph{i.e.}, $W\rightarrow\infty$, $\alpha_0=0$ and $\alpha_w\rightarrow1$ as $w\rightarrow\infty$, and sufficiently large $m$), the INSH algorithm will identify the optimal design. However, as with many optimisation routines, the aim of this algorithm is to find near-optimal designs in a computationally feasible amount of time. Thus, practical algorithm inputs must be chosen, which may not guarantee convergence to the optimal solution. However, this trade-off is apparent in a number of existing optimisation routines -- for example, simulated-annealing, cross-entropy, and genetic algorithms all have the potential to converge to local, rather than global, optima. The INSH algorithm we have presented here is quite general, and there exist many aspects of the algorithm which can be explored in order to improve the efficiency of the algorithm for different optimisation problems. For example: one could update the perturbation kernel based on the correlation/covariance that exists between design parameters of the same design, or; randomly incorporate a sample in a region of the design space that has either not been considered previously, or was dismissed earlier in the algorithm, in order to increase exploration of the design space and maximise the chance of obtaining the optimal design. Another important consideration will be to provide some general rules regarding the choice of algorithm inputs for a particular utility surface, or magnitude and dimension of design space. For example, the initial samples could be used to approximate some characteristics of the utility surface, and provide some insight into sensible choices of the inputs for the algorithm. While we did not consider it here, increasing the number of utility evaluations which form the approximate expected utility could also be increased as the algorithm progresses, \emph{i.e.}, specify a sequence for $\tilde{B}$ and $B$ in the SIG utility evaluation -- ensuring more effort is spent evaluating a more precise estimate of the utility in regions near to the optimal design. While we have added commentary around our choice of parameters for each example, we note that many of the parameters are problem-specific, and require the user to specify sensible values based on their understanding of the system and design space. As with other stochastic optimisation routines, some trial-and-error may be required in order to choose suitable INSH input values for different problem types. Given the drastic increase in computational-efficiency of the INSH algorithm for small-to-moderately sized design problems, the authors believe that one could be very cautious with some parameter choices (\emph{e.g.}, choose a large number of accepted designs, and number of newly sampled designs), in order to ensure satisfactory exploration of the design space, and obtain designs more efficiently than existing algorithms. \section*{References} \begin{thebibliography}{21} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\defURL {URL }\fi \bibitem[{Berry(2004)}]{Berry:2004} Berry, D.~A., 2004. Bayesian statistics and the efficiency and ethics of clinical trials. 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Journal of The Royal Society Interface 6, 187--202. \end{thebibliography} \setcounter{figure}{0} \renewcommand\thefigure{S\arabic{figure}} \renewcommand\thetable{S\arabic{table}} \section*{Appendix A} \textit{Existing Algorithms} Algorithm \ref{ourabc} describes the ABC algorithm used by ABCdE and the INSH algorithm (for the death model) to evaluate the posterior distribution. \begin{algorithm}[htbp] \caption{ABC Algorithm: Fixed tolerance}\label{ourabc} \begin{algorithmic}[1] \Require Observed data $\boldsymbol{x}$, simulated data $\boldsymbol{y}=(\boldsymbol{y}^1,\dots,\boldsymbol{y}^N)$, corresponding parameter values $\boldsymbol{\theta}^i, i=1,\dots,N$, and tolerance $\epsilon$. \State Evaluate discrepancies $\rho^i = \rho(\boldsymbol{x}, \boldsymbol{y}^i)$, creating particles $\{ \boldsymbol{\theta}^i, \rho^i \}$ for $i=1,\dots,N$. \State Using the posterior sample of parameters $\boldsymbol{\theta}^i$ such that $\rho^i<\epsilon$, evaluate utility. \Ensure Utility for current design, having observed $\boldsymbol{x}$, $U(d,\boldsymbol{x})$. \end{algorithmic} \end{algorithm} Algorithm \ref{ABCdEalgorithm} describes the ABCdE algorithm of \citet{Price:2016}, to evaluate the optimal Bayesian experimental design. \begin{algorithm}[H] \caption{ABCdE Algorithm}\label{ABCdEalgorithm} \begin{algorithmic}[1] \State Choose grid over the parameter space for the discrete estimate of the utility, number of simulations $N_{pre}$, and tolerance $\epsilon$. \State Sample $N_{pre}$ parameters $\boldsymbol{\theta}$ from $p(\boldsymbol{\theta})$. \State For each of the $N_{pre}$ parameters, and under every design $d$ in the design space $\mathcal{D}$, simulate process and store $X_{N_{pre}\times |\mathcal{D}|}(\boldsymbol{\theta}, d)$. \label{abcde_algorithm_line3} \For{$i=1$ to $|\mathcal{D}|$} \State \parbox[t]{0.925\linewidth}{Consider the unique rows of data $Y(\boldsymbol{\theta}, d^i) = \text{ unique}(X(\boldsymbol{\theta}, d^i))$.\\ \emph{Note: We let $K^i$ be the number of such unique data, and $n_{k^i}$ be the number of repetitions of the ${k^i}^{th}$ unique data, for $k^i=1,\dots,K^i$}.\label{abcdealg:uniquedata} } \For{$k^i=1$ to $K^i$} \State \parbox[t]{0.925\linewidth}{Pass `observed data' $\boldsymbol{y}^{k^i}=[Y(\boldsymbol{\theta},d^i)]_{k^i}$, `simulated data' $X(\boldsymbol{\theta},d^i)$, $N_{pre}$ sampled parameters, and tolerance $\epsilon$ to Algorithm \ref{ourabc}, and return contribution $U(\boldsymbol{y}^{k^i},d^i)$ to the expected utility, for ${k^i}^{th}$ unique datum (`observed data') and $i^{th}$ design\label{abcdealg:createposterior}.} \EndFor \State {Store $u(d^i) = \frac{1}{N_{pre}} \sum_{k^i} {n_{k^i}} U(\boldsymbol{y}^{k^i}, d^i)$; the average utility over all parameters and data for design $d^i$. \label{abcde_algorithm_line9}} \EndFor \Ensure The optimal design $d^* = \underset{d\in\mathcal{D}}{\text{argmax}}(u(d))$. \end{algorithmic} \end{algorithm} Algorithm \ref{mullersalgorithm} details the MCMC algorithm for determining Bayesian optimal designs proposed by Muller [1999]. \begin{algorithm}[htbp] \caption{MCMC with stationary distribution $h(\boldsymbol{\theta},\boldsymbol{x},d)$, Muller [1999]}\label{mullersalgorithm} \begin{algorithmic}[1] \Require Number of samples $m$, prior distribution of model parameters $p(\boldsymbol{\theta})$, and proposal density $q(\cdot)$. \State Choose, or simulate an initial design, $d^1$. \State Sample $\boldsymbol{\theta}^1\sim p(\boldsymbol{\theta})$, simulate $\boldsymbol{x}^1\sim p(\boldsymbol{x}\mid \boldsymbol{\theta}^1, d^1)$, and evaluate $u^1=U(\boldsymbol{\theta}^1, \boldsymbol{x}^1, d^1)$.\label{mulleralgorithm:initial} \For{$i=1:m$} \State Generate a candidate design, $\tilde{d}$, from a proposal density $q(\tilde{d} \mid d^i)$. \State Sample $\tilde{\boldsymbol{\theta}}\sim p(\boldsymbol{\theta})$, simulate $\tilde{\boldsymbol{x}}\sim p(\boldsymbol{x}\mid \tilde{\boldsymbol{\theta}}, \tilde{d})$, and evaluate $\tilde{u}=U(\tilde{\boldsymbol{\theta}}, \tilde{\boldsymbol{x}}, \tilde{d})$. \label{mulleralgorithm:simdata} \State Calculate, \begin{align} \alpha &=\min\left\{ 1, \frac{\tilde{u}\ q(d^i\mid \tilde{d})}{u^i\ q(\tilde{d} \mid d^i)} \right\}. \notag \end{align} \State Generate $a\sim U(0,1)$ \If{$a<\alpha$} \State Set $(d^{i+1}, u^{i+1}) = (\tilde{d}, \tilde{u})$ \Else{} \State Set $(d^{i+1}, u^{i+1}) =(d^i, u^i)$ \EndIf \EndFor\\ \Ensure Sample of $m$ designs, $d$. \end{algorithmic} \end{algorithm} \section*{Appendix B} \textit{Markovian Death Model ABC Choices} {We provide the parameter choices for the ABC algorithm used to evaluate the approximate posterior distributions when evaluating the utility for the Markovian death model example. Prior to running the ABC algorithm (Algorithm \ref{ourabc}), we sample $N=50,000$ parameter values from the prior distribution, and simulate data corresponding to each under each design. For each of 1, 2, 3, 4, 6, and 8 observation times, we use a tolerance of 0.25, 0.50, 0.75, 1.00, 1.50, 1.50, respectively.} {We note however, that these choices are problem specific, and suggest that researchers undertake a pilot-study in order to determine sensible parameter choices, as one would do prior to using ABC for inference.} \textit{Markovian Death Model Results} Figure \ref{Death_boxplots} demonstrates the convergence of the INSH algorithm to the optimal observation times, and the maximum utility, for two observation times. \begin{figure} \caption{Boxplots of the two observation times, and the utility corresponding to the considered designs at each generation of the INSH algorithm. The horizontal lines in (a) and (b) correspond to the optimal observation times evaluated using the ABCdE method.} \label{Death_boxplots} \end{figure} \begin{table}[H] \caption{Comparison of the optimal observation times for the death process, from \citet{Cook:2008}, \citet{Drovandi:2013}, \citet{Price:2016}, and the INSH algorithm. $|t|$ is the pre-determined number of observation times, and $i$ is the $i^{th}$ time.} \label{table:deathmodel_oeds} \begin{center} \begin{tabular}{cccccc} \hline & & \multicolumn{4}{c}{Design Method}\\ $|t|$ & $i$ & Cook, \emph{et al.}\ & Drovandi $\&$ Pettitt & ABCdE & INSH\\ \hline 1 & 1 & 1.70 & 1.60 & 1.50 & 1.45 \\ \hline 2 & 1 & 0.90 & 1.15 & 0.80 & 0.95 \\ - & 2 & 2.40 & 3.05 & 2.80 & 2.80 \\ \hline 3 & 1 & 0.70 & 0.75 & 0.40 & 0.60 \\ - & 2 & 1.50 & 1.90 & 1.30 & 1.15 \\ - & 3 & 2.90 & 3.90 & 2.60 & 2.70 \\ \hline 4 & 1 & 0.80 & 0.75 & 0.30 & 0.10 \\ - & 2 & 1.70 & 1.70 & 0.70 & 0.50 \\ - & 3 & 3.10 & 2.75 & 1.30 & 1.20 \\ - & 4 & 5.30 & 4.35 & 2.70 & 2.75 \\ \hline 6 & (1,2) & - & - & - & (0.05,0.15) \\ - & (3,4) & - & - & - & (0.45,1.15) \\ - & (5,6) & - & - & - & (1.75,3.05) \\ \hline 8 & (1,2) & - & - & - & (0.05,0.15) \\ - & (3,4) & - & - & - & (0.25,0.45) \\ - & (5,6) & - & - & - & (0.80,1.40) \\ - & (7,8) & - & - & - & (2.20,2.90) \\ \hline \end{tabular} \end{center} \end{table} Table \ref{table:deathmodel_oeds} contains the optimal experimental designs for different numbers of observations of the Markovian death model, evaluated by \citet{Cook:2008}, \citet{Drovandi:2013}, \citet{Price:2016} (where computationally feasible), and the INSH algorithm. Table \ref{death:insh_pars} contains the input parameters for the INSH algorithm, applied to the death model. \begin{table}[H] \caption{Input parameters for the INSH algorithm, applied to the Markovian death model. Note that $m_w$ and $r_w$ are applied each for $W/2$ iterations.} \label{death:insh_pars} \begin{center} \begin{tabular}{rrrrr} $|t|$ & $W$ & $m_w$ & $r_w$ & No. initial designs \\ \hline 1 & 8 & $(3, 5)$ & $(10,6)$ & 20 \\ 2 & 10 & $(3, 5)$ & $(20,12)$ & 50 \\ 3 & 16 & $(3, 5)$ & $(20,12)$ & 120 \\ 4 & 20 & $(3, 5)$ & $(20,12)$ & 250 \\ 6 & 30 & $(3, 5)$ & $(25,15)$ & 400 \\ 8 & 50 & $(3, 5)$ & $(25,15)$ & 600 \\ \end{tabular} \end{center} \end{table} \section*{Appendix C} \textit{INSH Results for the Pharmacokinetic Model} Figure \ref{PK_50_mean_concentrations} demonstrates the mean concentrations over time of the pharmacokinetic model evaluated for 50 parameter sets sampled from the prior distribution. \begin{figure} \caption{Plot of 50 mean concentrations over time of the pharmacokinetic model simulated using values sampled from the prior distribution.} \label{PK_50_mean_concentrations} \end{figure} Table \ref{PKexample:opttimes} gives the 15 optimal observation times from the top three designs considered by the INSH algorithm. Each chosen optimal design shows the same pattern -- four early observation times ($<1.2$), followed by a cluster of observation times around 4-7, and the remaining observations grouped together towards the final permitted time. \begin{table}[H] \caption{Three best sampling schedules evaluated from the INSH algorithm for the pharmacokinetic model.}\label{PKexample:opttimes} \begin{center} \begin{tabular}{lc} Original rank & Design \\ \hline 19 & $(0.1961, 0.4840, 0.7506, 1.176, 4.069, 4.780, 5.281,$ \\ & $ 6.030, 6.377, 18.22, 18.85, 19.72, 20.33, 21.52, 22.04)$ \\ \\ 2 & $(0.2460, 0.5054, 0.8017, 1.211, 4.035, 4.477, 5.173, 6.101, $ \\ & $ 6.632, 17.82, 18.63, 19.71, 20.32, 21.57, 21.98)$ \\ \\ 3 &$(0.1989, 0.4801, 0.7778, 1.103, 4.465, 4.754, 5.776, 6.270,$ \\ &$ 6.754, 18.50, 18.99, 20.19, 20.87, 21.16, 21.87)$ \\ \end{tabular} \end{center} \end{table} Figure \ref{INSH_pk_convergence} shows box plots of the observation times, and the utility evaluations of the corresponding designs considered at each wave of the INSH algorithm. The figure for $t_9$, for example, depicts the ability of the INSH algorithm to search multiple regions simultaneously. In particular, iterations 17-27 are considering observation times in approximately three clusters -- around times of 5, 12 and 15. \begin{figure} \caption{Figure showing the convergence of the sampled designs towards the region near the optimal design. Each panel represents an individual aspect of the sampled designs, the x-axis is the iteration of the INSH algorithm, and the y-axis is the value of the design aspect. The final panel shows the utilities corresponding to the sampled designs.} \label{INSH_pk_convergence} \end{figure} Table \ref{table_pk_klds} contains the estimated expected utility, median utility, and the $10^{th}$ and $90^{th}$ percentiles, corresponding to each of the top 20 designs considered by INSH, and the optimal returned by the ACE algorithm. \begin{table}[H] \caption{Summary statistics of estimated utilities corresponding to the top 20 designs from the INSH algorithm, and the optimal design returned by the ACE algorithm. The design highlighted in bold is the design considered to be the optimal from the INSH algorithm.} \label{table_pk_klds} \centering \begin{tabular}{r|cccc} \hline & \multicolumn{4}{c}{Utility}\\ Design & Mean & Median & 10\% & 90\% \\ \hline ACE & 4.4987 & 4.5004 & 4.4844 & 4.5102 \\ 1 & 4.4874 & 4.4865 & 4.4715 & 4.5040 \\ 2 & 4.4725 & 4.4710 & 4.4596 & 4.4894 \\ 3 & 4.4707 & 4.4685 & 4.4598 & 4.4864 \\ 4 & 4.4700 & 4.4686 & 4.4576 & 4.4835 \\ 5 & 4.4991 & 4.4995 & 4.4870 & 4.5111 \\ 6 & 4.4739 & 4.4719 & 4.4638 & 4.4900 \\ 7 & 4.4707 & 4.4733 & 4.4558 & 4.4866 \\ 8 & 4.5034 & 4.5015 & 4.4956 & 4.5156 \\ 9 & 4.4633 & 4.4648 & 4.4506 & 4.4758 \\ 10 & 4.4595 & 4.4633 & 4.4444 & 4.4736 \\ 11 & 4.4652 & 4.4633 & 4.4526 & 4.4803 \\ 12 & 4.4733 & 4.4742 & 4.4608 & 4.4868 \\ 13 & 4.4508 & 4.4497 & 4.4349 & 4.4654 \\ 14 & 4.4748 & 4.4754 & 4.4616 & 4.4878 \\ 15 & 4.4702 & 4.4690 & 4.4527 & 4.4941 \\ 16 & 4.4537 & 4.4523 & 4.4426 & 4.4725 \\ 17 & 4.4625 & 4.4633 & 4.4439 & 4.4846 \\ 18 & 4.4853 & 4.4877 & 4.4702 & 4.4991 \\ {\bf 19} & {\bf 4.5052} & {\bf 4.5076} & {\bf 4.4866} & {\bf 4.5204} \\ 20 & 4.4799 & 4.4780 & 4.4676 & 4.4975 \\ \hline \end{tabular} \end{table} Figure \ref{PK_inference} provides a comparison of the inferential performance of the two optimal designs -- corresponding to INSH and ACE -- with regards to bias in a point estimate, and the posterior standard deviation. We simulated 100 experiments from random parameters drawn from the prior distribution, and evaluated an approximate posterior distribution using a Metropolis-Hastings algorithm (retaining 100{,}000 samples from the posterior, following a burn-in of 10{,}000). The bias is estimated as the difference between the MAP (\emph{maximum a posteriori}) estimate and the true parameter value that created the simulated data. Recall, the prior variance was 0.05 for each parameter (prior standard deviation is approximately 0.224). It appears as though the design evaluated by the INSH algorithm performs marginally better with respect to the posterior standard deviation -- that is, the estimated standard deviations are slightly lower for each parameter. The bias in the parameter estimates appears roughly equivalent between the two designs. \begin{figure} \caption{Comparison of the bias in MAP estimate, and posterior standard deviation of each parameter in the pharmacokinetic model.} \label{PK_inference} \end{figure} \section{Appendix D} \textit{INSH Results for the Logistic Regression Example} Table \ref{LR_INSH_pars} contains the choices of parameters $r_w$ and $m_w$ for the INSH algorithm. Overall, for $n=6,10$ and $24$, $W=120, 132$ and $240$ iterations were used, and $r_w$ and $m_w$ were chosen such that a total of 600 designs were evaluated at each iteration. For $n=48$, $W=360$, and a total of 1200 designs were considered at each iteration, and more emphasis was placed on exploration early on -- to account for the larger-dimensional design space. In each example, we initiated the INSH algorithm with 10000 designs -- with probability 0.5, uniformly sampled from the design space, otherwise, on a boundary (i.e., all elements of the design consisted of randomly selected -1's and 1's). \begin{table}[H] \caption{INSH algorithm parameter choices for the Logistic Regression example.} \label{LR_INSH_pars} \begin{center} \begin{tabular}{cccc} $n$ & Parameter & Values & Iterations per value\\ \hline & $r_w$ & (200,100,50 ,25,15,10) & \\ $6$ & $m_w$ & $(3, 6,12,24,40,60)$ & 20 \\ & $\sigma_w$ & (0.20, 0.10, 0.05, 0.025, 0.01, 0.005) & \\ \hline & $r_w$ & (200,100,50 ,25,15,10) & \\ $10$ & $m_w$ & $(3,6,12,24,40, 60)$ & 22 \\ & $\sigma_w$ & (0.20, 0.10, 0.05, 0.025, 0.01, 0.005) & \\ \hline & $r_w$ & (200,100,50 ,25,15,10) & \\ $24$ & $m_w$ & $(3,6,12,24,40, 60)$ & 40 \\ & $\sigma_w$ & (0.20, 0.10, 0.05, 0.025, 0.01, 0.005) & \\ \hline & $r_w$ & (200,100,50 ,25,15,10) & \\ $48$ & $m_w$ & $(6,12,24,48,80,120)$ & 60 \\ & $\sigma_w$ & (0.20, 0.10, 0.05, 0.025 0.010, 0.0025) & \\ \end{tabular} \end{center} \end{table} Tables \ref{LR_OD_n6}, \ref{LR_OD_n10}, \ref{LR_OD_n24} and \ref{LR_OD_n48} show the optimal designs found by the INSH algorithm for $n=6,10,24$, and $48$, respectively. \begin{table}[H] \caption{Optimal design from the INSH algorithm for the Logistic regression model with $n=6$.} \label{LR_OD_n6} \begin{center} \begin{tabular}{c|cccc} $n$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ \\ \hline 1 & -0.80 & 0.98 & 0.99 & 0.98 \\ 2 & 1.00 & -0.47 & 0.97 & -0.99 \\ 3 & 1.00 & -0.63 & 1.00 & 1.00 \\ 4 & -0.89 & 1.00 & 0.56 & -1.00 \\ 5 & 0.81 & -1.00 & -0.99 & -0.85 \\ 6 & -0.97 & 0.45 & -1.00 & 0.99 \\ \end{tabular} \end{center} \end{table} \begin{table}[H] \caption{Optimal design from the INSH algorithm for the Logistic regression model with $n=10$.} \label{LR_OD_n10} \begin{center} \begin{tabular}{c|cccc} $n$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ \\ \hline 1 & -0.80 & 1.00 & 1.00 & 0.98 \\ 2 & 0.72 & -1.00 & -0.98 & 0.99 \\ 3 & -1.00 & 0.53 & -0.86 & -0.99 \\ 4 & 1.00 & -0.85 & 0.39 & -1.00 \\ 5 & 0.97 & -0.32 & 0.99 & -0.92 \\ 6 & 0.91 & -0.46 & 1.00 & 0.99 \\ 7 & -0.72 & 1.00 & 0.90 & -0.97 \\ 8 & -1.00 & 0.51 & -0.93 & 0.99 \\ 9 & 0.92 & -0.99 & -1.00 & -0.87 \\ 10 & -0.97 & 0.64 & -0.99 & -0.96 \\ \end{tabular} \end{center} \end{table} \begin{table}[H] \caption{Optimal design from the INSH algorithm for the Logistic regression model with $n=24$.} \label{LR_OD_n24} \begin{center} \begin{tabular}{c|cccc} $n$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ \\ \hline 1 & -0.33 & 0.50 & 0.89 & 0.89 \\ 2 & -0.25 & 0.78 & 0.89 & -0.96 \\ 3 & -0.95 & 0.94 & 0.93 & 0.97 \\ 4 & 0.68 & -0.98 & -0.92 & -0.81 \\ 5 & -0.93 & 0.99 & 0.86 & -0.88 \\ 6 & -0.77 & 0.98 & 0.82 & 0.87 \\ 7 & 0.82 & -0.96 & -0.83 & -0.99 \\ 8 & -0.64 & 0.11 & -0.66 & 0.98 \\ 9 & -0.48 & 0.86 & 0.96 & 0.92 \\ 10 & -0.76 & 0.40 & -0.95 & -0.94 \\ 11 & 1.00 & -0.74 & 0.97 & 1.00 \\ 12 & -0.99 & 0.46 & -0.97 & 0.96 \\ 13 & 0.96 & -0.44 & 0.97 & -0.97 \\ 14 & 0.78 & -0.93 & -0.89 & 0.96 \\ 15 & -0.34 & 0.74 & 0.99 & -0.98 \\ 16 & 1.00 & -0.95 & -0.92 & -0.99 \\ 17 & -0.22 & -0.14 & -0.86 & -0.96 \\ 18 & -0.96 & 0.85 & -0.59 & -0.81 \\ 19 & -0.70 & 0.12 & -0.98 & -0.93 \\ 20 & -0.74 & 0.30 & -0.98 & 0.96 \\ 21 & 0.99 & -0.50 & 0.83 & -0.60 \\ 22 & 0.53 & -0.99 & -0.91 & 0.99 \\ 23 & -0.21 & -0.26 & -1.00 & 0.85 \\ 24 & 1.00 & -0.55 & 0.97 & 0.96 \\ \end{tabular} \end{center} \end{table} \begin{table}[H] \caption{Optimal design from the INSH algorithm for the Logistic regression model with $n=48$.} \label{LR_OD_n48} \begin{center} \begin{tabular}{c|cccc} $n$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ \\ \hline 1 & -0.98 & 0.99 & 1.00 & -0.60 \\ 2 & -0.70 & 0.67 & 0.73 & 0.99 \\ 3 & -0.97 & 0.95 & 0.38 & -0.95 \\ 4 & -0.95 & 0.92 & -0.18 & 0.94 \\ 5 & 0.76 & -0.50 & 0.94 & -0.88 \\ 6 & 0.95 & -0.95 & -0.60 & 0.98 \\ 7 & -1.00 & 0.38 & -1.00 & -0.88 \\ 8 & 0.42 & -0.84 & -0.94 & 0.89 \\ 9 & -0.91 & 0.75 & -0.85 & -0.98 \\ 10 & 0.51 & -0.95 & -0.99 & 1.00 \\ 11 & 0.52 & -0.13 & 0.97 & -0.89 \\ 12 & 0.46 & -0.71 & -0.87 & 0.92 \\ 13 & -0.70 & 0.94 & 0.84 & 0.92 \\ 14 & -0.95 & 0.59 & -1.00 & -0.98 \\ 15 & -0.36 & 0.90 & 0.99 & -0.91 \\ 16 & 0.99 & -0.65 & 0.90 & 0.93 \\ 17 & 0.81 & -0.50 & 0.65 & 0.74 \\ 18 & -0.52 & 1.00 & 0.92 & 0.91 \\ 19 & 0.29 & -0.79 & -0.68 & 0.72 \\ 20 & 0.98 & -0.96 & -0.93 & -0.93 \\ 21 & 0.96 & -0.74 & 0.93 & 0.81 \\ 22 & 0.93 & -0.74 & -0.68 & -0.90 \\ 23 & 0.63 & -0.69 & -0.96 & -0.94 \\ 24 & 0.99 & -0.89 & 0.92 & 1.00 \\ 25 & -0.01 & -0.21 & -0.94 & 0.65 \\ 26 & 0.98 & -0.85 & -0.55 & -0.91 \\ 27 & -0.85 & 0.98 & 0.98 & -0.98 \\ 28 & -0.64 & 0.64 & -0.40 & 0.98 \\ 29 & 0.94 & -0.78 & 0.85 & -0.43 \\ 30 & 0.82 & 0.01 & 0.97 & -0.94 \\ 31 & -0.98 & 0.42 & -0.97 & -0.91 \\ 32 & 0.38 & -0.89 & -1.00 & -1.00 \\ 33 & 0.99 & -0.61 & 0.58 & 0.97 \\ 34 & -0.96 & 0.31 & -0.95 & 0.99 \\ 35 & 0.32 & -0.73 & -0.99 & -0.06 \\ 36 & -0.39 & -0.31 & -1.00 & 0.75 \\ 37 & 0.97 & -0.15 & 0.78 & -1.00 \\ 38 & -0.99 & 0.31 & -0.96 & 0.96 \\ 39 & -0.51 & 0.94 & 0.92 & -1.00 \\ 40 & 0.74 & -0.98 & -0.95 & -0.25 \\ 41 & -0.81 & 0.46 & -0.62 & 0.99 \\ 42 & -0.87 & 0.99 & 0.16 & -0.50 \\ 43 & 0.87 & -0.95 & -0.68 & -0.93 \\ 44 & 0.37 & -0.90 & -0.92 & -0.75 \\ 45 & 0.82 & -0.43 & 0.93 & -0.95 \\ 46 & 0.90 & -0.27 & 1.00 & 0.94 \\ 47 & 0.91 & -0.25 & 0.98 & -0.71 \\ 48 & -0.93 & 0.94 & 0.88 & 0.93 \\ \end{tabular} \end{center} \end{table} Note that each optimal design contains many values that are close to the boundary values (-1 and 1). The optimal designs reported in \citet{Overstall:2017} can be found in the \verb+acebayes+ package in R, using the command \verb+optdeslrsig(n)+, where $n$ is the number of replicates. Figures \ref{LR_trace_n_6}, \ref{LR_trace_n_24}, and \ref{LR_trace_n_48} show the progression of the INSH algorithm for the Logistic regression example with $n=6,24$, and $48$, respectively. It appears as though the algorithm has converged to a optimal design region in each case. \begin{figure} \caption{Box plots of utility evaluations from the INSH algorithm, for the Logistic regression example with $n=6$.} \label{LR_trace_n_6} \end{figure} \begin{figure} \caption{Box plots of utility evaluations from the INSH algorithm, for the Logistic regression example with $n=10$.} \label{LR_trace_n_10} \end{figure} \begin{figure} \caption{Box plots of utility evaluations from the INSH algorithm, for the Logistic regression example with $n=24$.} \label{LR_trace_n_24} \end{figure} \begin{figure} \caption{Box plots of utility evaluations from the INSH algorithm, for the Logistic regression example with $n=48$.} \label{LR_trace_n_48} \end{figure} \begin{table}[H] \caption{The average utility (and 2.5, 97.5-percentiles) of the optimal design found via the ACE, INSH, ACE$_N$ and ACE$_B$ algorithms. For each design, 20 evaluations of the utility were made with $\tilde{B}=B=20000$} \label{LR_ace_insh_opt_utils} \begin{center} \begin{tabular}{c|cccc} $n$ & ACE & INSH & ACE$_N$ & ACE$_B$ \\ \hline 6 & 1.99 (1.97, 2.00) & 1.99 (1.97, 2.01) & 1.94 (1.92, 1.96) & 1.96 (1.95, 1.97) \\ 10 & 2.67 (2.66, 2.69) & 2.66 (2.65, 2.68) & 2.64 (2.62, 2,66) & 2.68 (2.66, 2.69) \\ 24 & 3.97 (3.95, 3.98) & 3.88 (3.86, 3.90) & 3.88 (3.86, 3.90) & 3.96 (3.94, 3.98) \\ 48 & 5.11 (5.10, 5.12) & 5.01 (4.98, 5.02) & 5.01 (4.98, 5.02) & 5.06 (5.04, 5.09) \end{tabular} \end{center} \end{table} \begin{table}[H] \caption{Inputs used to run ACE for the same computation time as INSH: Number of phases of the ACE algorithm ($N_I, N_{II}$), and effort used to evaluate the utility in change step ($B_1$) and for fitting the Gaussian process ($B_2$). Numbers are presented as $(N_I,N_{II}),(B_1,B_2)$. The default settings specified by the authors are $(20,100)$, $(20000,1000)$.} \label{} \begin{center} \begin{tabular}{c|cc} $n$ & ACE$_N$ & ACE$_B$\\ \hline 6 & $(4,20),(20000,1000)$ & $(20,100),(10000,500)$ \\ 10 & $(4,20),(20000,1000)$ & $(20,100),(8000,400)$ \\ 24 & $(3,15),(20000,1000)$ & $(20,100),(5000,250)$ \\ 48 & $(5,25),(20000,1000)$ & $(20,100),(10000,500)$ \\ \end{tabular} \end{center} \end{table} \end{document}

\begin{document} \title{Rectangle Sweepouts and Coincidences} \begin{abstract} We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry classes of inscribed rectangles, grows linearly with the number of positively oriented extremal chords -- a.k.a. diameters -- in a polygon. \end{abstract} \section{Introduction} A {\it Jordan loop\/} is the image of a circle under a continuous injective map into the plane. Toeplitz conjectured in 1911 that every Jordan loop contains $4$ points which are the vertices of a square. This is sometimes called the {\it Square Peg Problem\/}. For historical details and a long bibliography, we refer the reader to the excellent survey article [{\bf M\/}] by B. Matschke, written in 2014, and also Chapter 5 of I. Pak's online book [{\bf P\/}]. Some interesting work on problems related to the Square Peg Problem has been done very recently. The paper of C. Hugelmeyer [{\bf H\/}] shows that a smooth Jordan loop always has an inscribed rectangle of aspect ratio $\sqrt 3$. The paper [{\bf AA\/}] proves that any cyclic quadrilateral can (up to similarity) be inscribed in any convex smooth curve. The paper [{\bf ACFSST\/}] proves, among other things, that a dense set of points on an embedded loop in space are vertices of a (possibly degenerate) inscribed parallelogram. Say that a rectangle $R$ {\it graces\/} a Jordan loop $\gamma$ if the vertices of $R$ lie in $\gamma$ and if the cyclic ordering on the vertices induced by $R$ coincides with the cyclic ordering induced by $\gamma$. Let $G(\gamma)$ denote the space of labeled gracing rectangles. In [{\bf S1\/}] we prove the following result. \begin{theorem} \label{threepoint} Let $\gamma$ be a Jordan loop. Then $G(\gamma)$ contains a connected set $S$ such that all but at most $4$ vertices of $\gamma$ are vertices of members of $S$. \end{theorem} We have a more precise characterization of the possibilities for $S$ in [{\bf S1\/}]. We proved Theorem \ref{threepoint} by taking a limit of a result for polygons. We now describe this result. Given a polygon $P$, we say that a chord $d$ of $P$ is a {\it diameter\/} if $d$ if the two perpendiculars to $d$ based at $\partial P$ do not locally separate $\partial P$ into two arcs. Each diameter can be positively oriented or negatively oriented, but not both. To explain the condition, we rotate the picture so that $d$ is vertical. The endpoints of $d$ divide $P$ into two arcs $P_1$ and $P_2$. Given the non-separating condition associated to a chord, we can say whether $P_1$ locally lies to the left or right of $P_2$ in a neighborhood of each endpoint of $d$. We call $d$ {\it positively oriented\/} if the left/right answer is the same at both endpoints. That is, either $P_1$ locally lies to the left at both endpoints or $P_1$ locally lies to the right at both endpoints. Figure 1 some examples of positive diameters. \begin{center} \resizebox{!}{2.5in}{\includegraphics{fig1.eps}} \newline {\bf Figure 1:\/} Some positive diameters of polygons. \end{center} With respect to the distance function on $P$, a diameter can be a minimum, a maximum, or neither. We call the third kind {\it saddles\/}. Let $\Delta_+(P)$ denote the number of positively oriented diameters of $P$. Let $\Pi_N$ denote the space of embedded $N$-gons. The set $\Pi_N$ is naturally an open subset of $(\mbox{\boldmath{$R$}}^2)^N$ and as such inherits the structure of a smooth manifold. We call a subset $\Pi_N^* \subset \Pi_N$ {\it fat\/} if $\Pi_N-\Pi_N^*$ is a finite union of positive codimension submanifolds. In particular, a fat set is open and has full measure. \begin{theorem} \label{polygon} There exists a fat subset $\Pi_N^* \subset \Pi_N$ with the following property. For every $N$-gon $P \in \Pi_N^*$ the space $\Gamma(P)$ is a piecewise-smooth $1$-manifold. Each arc component of $\Gamma(P)$ connects two positive diameters of $P$, and every positive diameter arises as the end of $4$ arc components. of $\Gamma(P)$. In particular, there are $2\Delta_+(P)$ arc components of $\Gamma(P)$. \end{theorem} The reason that there are $4$ arc components connecting every pair of positive diameters that is that we are considering cyclically labeled rectangles. Each of the $4$ components is obtained from each other one by cyclically relabeling. Now we describe the results we prove in this paper. Given a rectangle $R$, we let $X(R)$ and $Y(R)$ respectively denote the lengths of the first and second sides of $R$. For any continuous path of rectangles in $\Gamma(P)$ which is either a closed loop or which connects two diameters of $P$, we define the {\it shape curve\/} $Z(\alpha)$. This curve is given by \begin{equation} Z(\alpha,t)=(X(R_t),Y(R_t)). \end{equation} Here $t \to R_t$ is a parametrization of $\alpha$. When $\alpha$ is a closed loop, $Z(\alpha)$ is a closed loop as well. When $\alpha$ is an arc component, $Z(\alpha)$ is an arc, not necessarily embedded, that starts and ends on the coordinate axes. Figure 2 shows two of the possibilities. \begin{center} \resizebox{!}{1.5in}{\includegraphics{fig2.eps}} \newline {\bf Figure 2:\/} Shape curves associated to hyperbolic and null arcs. \end{center} In the first case, one endpoint of $\alpha$ lies on the $X$-axis and the second endpoint lies on the $Y$-axis. As in [{\bf S1\/}] we call such arcs {\it hyperbolic arcs\/}. In the other cases, both ends lie on the same axis. We call such components {\it null arcs\/}. In the arc cases, we augment $Z(\alpha)$ by adjoining the relevant parts of the coordinate axes so as to create a closed loop. We have shaded in the regions bounded by these closed loops. We call this augmented loop the {\it shape loop\/} associated to $\alpha$ and give it the same name. In [{\bf S2\/}] we found a kind of integral formula associated to the shape loop, though we stated it in a different context. This invariant is quite similar to the integral invariant in [{\bf Ta\/}], though we use it in a different context. (In \S \ref{squeeze} we give a sample result from [{\bf S2\/}].) Here we adapt the invariant to the present situation and prove the following theorem. \begin{theorem} \label{sweep} Let $P$ be any piecewise smooth Jordan loop. Let $\alpha$ be a piecewise smooth path in $\Gamma(P)$. If $\alpha$ is a hyperbolic arc then the signed area of the region bounded by $Z(\alpha)$ equals (up to sign) the area of the region bounded by $P$. If $\alpha$ is either a null arc or a closed loop, then the signed area of the region bounded by $Z(\alpha)$ is $0$. \end{theorem} Theorem \ref{sweep} says something about the number of coincidences that appear amongst the inscribed rectangles. We will give an example which explains the connection. Since the shape loop associated to a null component bounds a region of area $0$, the shape curve must have a self-intersection. This self-intersection corresponds to a pair of isometric rectangles inscribed in the polygon. Now we formulate a general result. We call two labeled rectangles {\it really distinct\/} if their unlabeled versions are also distinct. Thus, two relabelings of the same rectangle are not really distinct. We define the multiplicity of the pair $(X,Y)$ as follows. \begin{itemize} \item $\mu(X,Y)=n-1$ if there are $n>1$ really distinct labeled rectangles $R_1,...,R_n$ inscribed in $P$ such $X(R_j)=X$ and $Y(R_j)=Y$ for all $j=1,...,n$. We also allow $n=\infty$, \item $\mu(X,Y)=0$ if there are $0$ or $1$ such rectangles. \end{itemize} We define \begin{equation} \label{coincidence} M(P)=\sum \mu(X,Y), \end{equation} where the sum is taken over all pairs $(X,Y)$. Typically this is a sum with finitely many finite nonzero terms. There is a more natural (but somewhat informal) way to think about $M(P)$. Suppose that we color all the points in $\Gamma(P)$ according to the isometry class of rectangles they represent. Then $M(P)$ is the number of points minus the number of colors. \begin{theorem} \label{main} For each $P \in \Pi_N^*$ we have $M(P) \geq 2(\Delta_+(P)-2)$. \end{theorem} When $P$ is an obtuse triangle we have $M(P)=0$ and $\Delta_+(P)=2$, so the result is sharp in a trivial way. Some version of Theorem \ref{main} is true for an arbitrary polygon, but here we place a mild constraint so as to make the proof easier. Let $P$ be a polygon. We call a diameter $S$ of $P$ {\it tricky\/} if the endpoints of $S$ are vertices of $P$ and if at least one of the edges of $P$ incident to $S$ is perpendicular to $S$. \begin{theorem} \label{main2} If $P$ has no tricky diameters, $M(P) \geq \frac{1}{16}(\Delta_+(P)-2)$. \end{theorem} The rest of the paper is devoted to proving the results above. \section{The Integral Formula} \subsection{The Differential Version} Let $J$ be a piecewise smooth Jordan loop and let $R$ be a labeled rectangle that graces $J$. For each $j=1,2,3,4$ we let $A_j$ denote the signed area of the region $R_j^*$ bounded by the segment $\overline{R_{j}R_{j+1}}$ and the arc of $J$ that connects $R_j$ to $R_{j+1}$ and is between these two points in the counterclockwise order. Figure 3 shows a simple example. The signs are taken so that the signed areas are positive in the convex case, and then in general we define the signs so that the signed areas vary continuously. \begin{center} \resizebox{!}{1.9in}{\includegraphics{fig3.eps}} \newline {\bf Figure 3:\/} The curve $J$, the rectangle $R$ and the regions $R_j^*$ for $j=1,2,3,4$. \end{center} Assuming that $J$ is fixed, we introduce the quantity \begin{equation} A(R)=(A_1+A_3)-(A_2+A_4). \end{equation} We also have the point $(X,Y) \in \mbox{\boldmath{$R$}}^2$, where \begin{equation} X={\rm length\/}(\overline{R_1R_2}), \hskip 15 pt Y={\rm length\/}(\overline{R_2R_3}), \hskip 15 pt \end{equation} Assuming that we have a piecewise smooth path $t \to R_t$ of rectangles gracing $J$, we have the two quantities \begin{equation} A_t=A(R_t), \hskip 30 pt (X_t,Y_t)=(X(R_t),Y(R_t)). \end{equation} If $t$ is a point of differentiability, we may take derivatives of all these quantities. Here is the main formula. \begin{equation} \frac{dA}{dt}=Y \frac{dX}{dt} - X \frac{dY}{dt}. \end{equation} It suffices to prove this result for $t=0$. This formula is rotation invariant, so for the purposes of derivation, we rotate the picture so that the first side of $R_0$ is contained in a horizontal line, as shown in Figures 3 and 4. When we differentiate, we evaluate all derivatives at $t=0$. We write \begin{equation} \frac{dR_j}{dt}=(V_j,W_j). \end{equation} Up to second order, the region $R_1^*(t)$ is obtained by adding a small quadrilateral with base $X_0$ and adjacent sides parallel to $t(V_1,W_1)$ and $t(V_2,W_2)$. Up to second order, the area of this quadrilateral is $$\frac{X(W_1+W_2)}{2}.$$ \begin{center} \resizebox{!}{2in}{\includegraphics{fig4.eps}} \newline {\bf Figure 4:\/} The change in area. \end{center} From this equation, we conclude that \begin{equation} \frac{dA_1}{dt}=-\frac{X(W_1+W_2)}{2}. \end{equation} We get the negative sign because the area of the region increases when $W_1$ and $W_2$ are negative. A similar derivation gives \begin{equation} \frac{dA_3}{dt}=+\frac{X(W_3+W_4)}{2}. \end{equation} Adding these together gives $$ \frac{dA_1}{dt}+\frac{dA_3}{dt}= X \times \bigg[\frac{W_3-W_1}{2}\bigg] + X \times \bigg[\frac{W_4-W_2}{2}\bigg]=$$ \begin{equation} \label{term1} -X \times \bigg[\frac{1}{2}\frac{dY}{dt}\bigg]+ -X \times \bigg[\frac{1}{2}\frac{dY}{dt}\bigg]= -X \frac{dY}{dt}. \end{equation} A similar derivation gives \begin{equation} \frac{dA_2}{dt}=-\frac{X(V_2+V_3)}{2}, \hskip 30 pt \frac{dA_4}{dt}=+\frac{X(V_4+V_1)}{2}. \end{equation} Adding these together gives \begin{equation} \label{term2} \frac{dA_2}{dt}+\frac{dA_4}{dt}= -Y \frac{dX}{dt}. \end{equation} Subtracting Equation \ref{term2} from Equation \ref{term1} gives \begin{equation} \label{diff} \frac{dA}{dt}=-X \frac{dY}{dt}+Y \frac{dX}{dt}, \end{equation} as claimed. \subsection{The Integral Version} Let $\omega=-XdY+YdX$. Here we think of $\omega$ as a $1$-form. Suppose that we have parameterized our curve of rectangles so that the parameter $t$ runs from $0$ to $1$. Integrating Equation \ref{diff} over the piecewise smooth path, we see that \begin{equation} \label{int} A_1-A_0=\int_Z \omega. \end{equation} Here $Z$ is the shape curve associated to the path of rectangles. We can interpret this integral geometrically. Letting $O=(0,0)$, consider the closed loop \begin{equation} Z'=\overline{O, Z_0} \cup Z \cup \overline {Z_1,O}. \end{equation} Since $\omega$ vanishes on vectors of the form $(h,h)$, we see that \begin{equation} A_1-A_0=\int_Z\omega=\int_{Z'} \omega= -\int \int_{\Omega} 2dxdy = -2\ {\rm area\/}(\Omega). \end{equation} Here $\Omega$ is the region bounded by $Z'$. The last line of the equation refers to the signed area of $\Omega$. \newline \noindent {\bf Proof of Theorem \ref{sweep}:\/} Suppose first that $\alpha$ is a piecewise smooth loop rectangles which grace the Jordan curve $J$. Then the curve $Z$ is already a closed loop, and the signed area of the region bounded by $Z$ is the same as the signed area bounded by $Z'$. Since $A_1=A_0$ in this case, we see that $Z$ bounds a region of signed area $0$. If $\alpha$ is a null arc, then $R_0$ and $R_1$ both have the same aspect ratio, either $0$ or $\infty$. In either case, we have $A_0=A_1$. The common value is, up to sign, the area of the region bounded by $J$. In this case, $Z$ starts and stops on one of the coordinate axes, and the region bounded by $Z$ has the same area as the shape loop, $Z \cup \overline{Z_0Z_1}$. So, in this case we also see that the shape loop bounds a region of area $0$. If $\alpha$ is a hyperbolic arc, then $A_0=-A_1$ and both quantities up to sign equal the area of the region bounded by $J$. At the same time $Z'$ is precisely the shape loop in this case. So, we see that twice the area of the region bounded by $J$ equals twice the area of the region bounded by $Z$, up to sign. Cancelling the factor of $2$ gives the desired result. $\spadesuit$ \newline \subsection{Generic Coincidences} \label{generic} In this section we prove Theorem \ref{main}. Suppose that $P$ is an $N$-gon that satisfies the conclusions of Theorem \ref{polygon}. This happens if $P \in \Pi_N^*$, but it might happen more generally. In any case, the space $\Gamma(P)$ of gracing rectangles has $2\Delta(P)$ arc components. There is a $\mbox{\boldmath{$Z$}}/4$ action on $\Gamma(P)$ and this action freely permutes the arc components of $\Gamma(P)$. We let $\delta=\Delta/2$ and we let $\alpha_1,...,\alpha_{\delta}$ denote a complete set of representatives of these arc components modulo the $\mbox{\boldmath{$Z$}}/4$ action. It suffices to show that the sum in Equation \ref{coincidence} is at least $\delta-1$ when we restrict our attention to the components just listed. Consider those arcs on our list which are null arcs. The shape loops associated to each of these arcs bound regions of area $0$ and hence the corresponding loop has a double point. Each double point corresponds to a distinct pair that adds $1$ to the total count for $M(J)$. The remaining rectangle coincidences involve rectangles not associated to these arcs or to their images under the $\mbox{\boldmath{$Z$}}/4$ action. Now consider those arcs on our list which are hyperbolic arcs whose shape loops are not embedded. In exactly the same way as above, each of these arcs contributes $1$ to the count for $M(J)$ and the rectangle pairs involved are distinct from the ones we have already considered. Again, the remaining rectangle coincidences involve rectangles not associated to these arcs or to their images under the $\mbox{\boldmath{$Z$}}/4$ action. \newline \newline {\bf Remark:\/} Before we move on to the last case, we mention that the count above might be an under-approximation, even in case there is just one double point per shape loop considered. Consider the simple situation where there are just $2$ null arcs. It might happen that the rectangle pairs corresponding to these $2$ arcs are congruent to each other. This would give us a $4$ congruent gracing rectangles and would contribute $3$ rather than $2$ to the total count. \newline Finally, consider the $d$ hyperbolic arcs on our list which have embedded shape loops. If $\alpha_1$ and $\alpha_2$ are two such arcs, then $Z(\alpha_1)$ and $Z(\alpha_2)$ are two closed loops which bound the same area. If these loops did not intersect in the positive quadrant, then either the region bounded by $Z(\alpha_1)$ would strictly contain the region bounded by $Z(\alpha_2)$ or the reverse. This contradicts the fact that these two regions have the same area. Hence $Z(\alpha_1)$ and $Z(\alpha_2)$ intersect in the positive quadrant, and the intersection point corresponds to a coincidence involving a rectangle associated to $\alpha_1$ and a rectangle associated to $\alpha_2$. Call this the {\it intersection property\/}. We label so that $\alpha_1,...,\alpha_d$ are the hyperbolic arcs having embedded shape loops. We argue by induction that these $d$ arcs contribute at least $d-1$ to the count for $M(J)$. If $d=1$ then there is nothing to prove. By induction, rectangle coincidences associated to the arcs $\alpha_1,...,\alpha_{d-1}$ contribute $d-2$ to the count for $M(J)$. By the intersection property, $\alpha_d$ intersects each of the other arcs, and $\Gamma(J)$ is a manifold, there is at least one new rectangle involved in our count, namely one that corresponds to a point on $Z(\alpha_d)$ that is also on some of the shape loop. The corresponding rectangle adds $1$ to the count in Equation \ref{coincidence}, one way or another. So, all in all, we add $d-1$ to the count for $M(J)$ by considering the rectangle coincidences associated to $\alpha_1,...,\alpha_d$. This proves what we want. \subsection{A Non-Squeezing Result} \label{squeeze} Here we explain how the invariant above implies one of our main results in [{\bf S2\/}]. Really, it is the same proof. The material in this section plays no role in the rest of the paper. Suppose that $\gamma_1$ and $\gamma_2$ are $2$ piecewise smooth curves which are disjoint. Suppose also that at each end, $\gamma_j$ coincides with a line segment. Finally suppose that these line segments are parallel at each end, so to speak. Figure 5 shows what we mean. \begin{center} \resizebox{!}{1.5in}{\includegraphics{fig5.eps}} \newline {\bf Figure 5:\/} Sliding a square along a track. \end{center} Suppose that we have a piecewise smooth family of rectangles, all having the same aspect ratio, that starts at one end, finishes at the other, and remains inscribed in $\gamma_1 \cup \gamma_2$ the whole time. We imagine $\gamma_1 \cup \gamma_2$ as being a kind of track that the rectangle slides along (changing its size and orientation along the way). Figure $5$ shows an example in which case the rectangle is a square. In Figure 5 we show the starting rectangle $R_0$, the ending rectangle $R_1$, and some $R_t$ for $t \in (0,1)$. This is just a hypothetical example. We can complete the union $\gamma_1 \cup \gamma_2$ to a piecewise smooth Jordan loop by extending the ends of one or both of these curves, if necessary, and then dropping perpendiculars. Let $\Omega$ be the region bounded by this loop. The shape curve associated to our path lies on a line through the origin, and our $1$-form $\omega$ vanishes on such lines. Referring to the invariant above, we therefore have $A(R_0)=A(R_1)$. But, after suitably labeling the rectangles in our family, we have $$A(R_j)={\rm area\/}(\Omega)-{\rm area\/}(R_j).$$ Hence $R_0$ and $R_1$ have the same area. Since they also have the same aspect ratio, they have the same side-lengths. This is to say that the perpendicular distance between the end of $\gamma_1$ and the end of $\gamma_2$ is the same at either end. This is a kind of non-squeezing result. In particular, our result shows that Figure 5 depicts an impossible situation. There is no way to slide a square continuously through the shown ``track'' because the widths are different at the $2$ ends. \section{The General Case} \subsection{Rectangles Inscribed in Lines} \label{conn} The goal of this chapter is to prove Theorem \ref{main2}. We plan to take a limit of the result in Theorem \ref{main}. Let $E=(E_1,E_2,E_3,E_4)$ be a collection of $4$ line segments, not necessarily distinct. We say that a rectangle $R$ {\it graces\/} $E$ if the vertices $R_1,R_2,R_3,R_4$ of $R$ go in cyclic order, either clockwise or counterclockwise, and $R_i \in E_i$ for all $i=1,2,3,4$. We allow $R$ to be degenerate. Let $\Gamma(E) \subset (\mbox{\boldmath{$R$}}^2)^4$ denote the set of rectangles gracing $E$. Note We call a point $p \in \Gamma(E)$ {\it degenerate\/} if every neighborhood of $p$ in $\Gamma_E$ contains points corresponding to infinitely many distinct but isometric rectangles. We call $E$ {\it degenerate\/} if there is some $p \in \Gamma(E)$ which is degenerate. \begin{lemma} Suppose that $E$ is nondegenerate. $\Gamma(E)$ is the intersection of a conic section with a rectangular solid. \end{lemma} {\bf { }{\noindent}Proof: } Let $E=(E_1,E_2,E_3,E_4)$ be a $4$-tuple of lines. We rotate so that none of the segments is vertical, so that we may parameterize the lines containing our segments by their first coordinates. Let $L_j$ be the line extending $E_j$. We identify $\mbox{\boldmath{$R$}}^3$ with triples $(x_1,x_2,x_3)$ where $p_j=(x_j,y_j) \in L_j$. We let $p_4$ be such that $p_1+p_3=p_2+p_4$. In other words, we choose $p_4$ to that $(p_1,p_2,p_3,p_4)$ is a parallelogram. Let $\Gamma(L)$ denote the set of rectangles gracing $L$. We describe the subset $\Gamma'(L) \subset \mbox{\boldmath{$R$}}^3$ corresponding to $\Gamma(L)$. The actual set $\Gamma(L)$ is the image of $\Gamma'(L)$ under a linear map from $\mbox{\boldmath{$R$}}^3$ into $(\mbox{\boldmath{$R$}}^2)^4$. The condition that $p_4 \in L_4$ is a linear condition. Therefore, the set $(x_1,x_2,x_3) \in \mbox{\boldmath{$R$}}^3$ corresponding to parallelograms inscribed in $L$ is a hyperplane $\Pi$. The condition that our parallelogram is a rectangle is $(p_3-p_2) \cdot (p_1-p_2)=0.$ This condition defines a quadric hypersurface $H$ in $\mbox{\boldmath{$R$}}^3$. The intersection $\Gamma'(L)=\Pi \cap H$ corresponds to the inscribed rectangles. $\Pi \cap H$ is either a plane or a conic section. In the former case, $E$ is degenerate. In the latter case, every point $\Pi \cap H$ is either an analytic curve or two crossing lines. Since $\Gamma(L)$ is the image of $\Gamma'(L)$ under a linear map, the set $\Gamma(L)$ is also a conic section. Let $[E]=E_1 \times E_2 \times E_3 \times E_4.$ The $[E]$ is a rectangular solid. We have $\Gamma(E)=\Gamma(L) \cap [E]$. $\spadesuit$ \newline \begin{lemma} When $E$ is non-degenerate, $\Gamma(E)$ has at most $64=2^8$ connected components. \end{lemma} {\bf { }{\noindent}Proof: } We use the notation from the previous lemma. Note $[E]$ is bounded by $8$ hyperplanes and a conic section either lies in a hyperplane or intersects it at most twice. So, each boundary component of $[E]$ cuts $\Gamma(L)$ into at most $2$ components. $\spadesuit$ \newline We call a polygon $P$ {\it degenerate\/} if some $4$-tuple of edges associated to $P$ is degenerate. Otherwise we call $P$ {\it non-degenerate\/}. \begin{lemma} Let $P$ be a non-degenerate polygon. The space $\Gamma(E)$ is a graph having analytic edges and degree at most $32$. \end{lemma} {\bf { }{\noindent}Proof: } Each rectangle $R$ can grace at most $16$ different $4$-tuples of edges of $P$, because each vertex can lie in at most $2$ segments. Hence, each $p \in \Gamma(P)$ lies in the intersection of at most $16$ distinct $\Gamma(E)$. Since $\Gamma(E)$ is the intersection of a conic section with a rectangular solid, $\Gamma(E)$ is a graph with analytic edges and maximum degree $4$. From what we have said above, $\Gamma(P)$ is a graph with analytic edges and maximum degree $64=16 \times 4$. We can cut down by a factor of $2$ as follows. The only time a point of $\Gamma(P)$ lies in more than $8$ spaces $\Gamma(E)$ is when $p$ corresponds to a gracing rectangle whose every vertex is a vertex of $P$. In this case, $p$ is a vertex of $[E]$ for each $4$-tuple $E$ that the rectangle graces. But then $p$ has degree at most $2$ in each $\Gamma(E)$. So, this exceptional case produces vertices of degree at most $32$. $\spadesuit$ \newline \subsection{The Inscribing Sequence} A generic polygon $P$ satisfies the conclusions of Theorem \ref{main}. For such polygons, any $4$-tuple which supports a gracing rectangle is nice. We label the sides of $P$ by $\{1,...,N\}$. Let $\Omega$ denote the set of ordered $4$-element subsets of $\{1,...,N\}$, not necessarily distinct. Consider some embedded arc $\alpha \subset \Gamma(P)$ of inscribed rectangles. $\alpha$ defines a finite sequence $\Sigma$ of elements of $\Omega$. We simply note which edges of $P_n$ contain any given rectangle and then we order the elements of $\Omega$ we get. We call $\Sigma$ the {\it inscribing sequence\/} for $\alpha$. \begin{lemma} \label{inscribing} $\Sigma$ has length at most $\kappa N^4$. \end{lemma} {\bf { }{\noindent}Proof: } If $\Sigma$ had length longer than this, then we could find a single $4$-tuple $E$ of edges such that the subset of $\alpha$ supported by $E$ has at least $82$ components. In other words the sequence would have to return to the $4$-tuple describing $E$ at least $82$ times. The arcs of $\Gamma(E)$ corresponding to these returns are disconnected from each other, because otherwise $\alpha$ would be a loop rather than an arc. This contradiction proves our claim. $\spadesuit$ \newline \subsection{Stable Diameters} For the rest of the chapter, we use the word {\it diameter\/} to mean a positively oriented diameter, in the sense discussed in the introduction. Let $P$ be a polygon and let $S$ be a diameter of $P$. We call $S$ {\it stable\/} if \begin{itemize} \item At least one endpoint of $S$ is a vertex of $P$. \item If $v$ is an endpoint of $S$ and $e$ is an edge of $P$ incident to $P$ at $v$, then $S$ and $e$ are not perpendicular. \end{itemize} \begin{lemma} Suppose that $P$ has no tricky diameters. If $P$ has an unstable diameter, then $P$ is non-degenerate. \end{lemma} {\bf { }{\noindent}Proof: } This is a case-by-case analysis. Suppose first that $P$ has a diameter $S$ whose endpoints are not vertices of $P$. Then the endpoints of $S$ lie in the interior of a pair of parallel edges of $P$. But then $P$ is degenerate. Suppose that $P$ has a diameter $S$ having one endpoint which is a vertex $v$ of $P$. The other endpoint of $S$ lies in the interior of an edge $e'$ of $P$. By definition $e'$ and $S$ are perpendicular. If $S$ is not stable, then one of the edges $e$ of $P$ is perpendicular to $S$ and hence parallel to $e'$. But then we can shift $S$ over a bit and produce a diameter of $P$ whose endpoints lie in the interior of $e$ and $e'$. Again, $P$ is degenerate. The remaining unstable diameters are (in the technical sense) tricky. $\spadesuit$ \newline In view of the preceding result, it suffices to prove Theorem \ref{main2} under the assumption that $P$ is non-degenerate and has all stable diameters. \subsection{Limits of Diameters} Let $P$ be an $N$-gon with stable diameters. We can find a sequence $\{P_n\}$ of generic $N$-gons converging to $P$. Each $P_n$ satisfies the conclusions of Theorem \ref{main}. \begin{lemma} Let $D$ be a diameter of $P$. The polygon $P_n$ has a diameter $D_n$ such that $\{D_n\}$ converges to $D$. \end{lemma} {\bf { }{\noindent}Proof: } Since $P$ only has stable diameters, there are just $2$ cases to consider. Suppose first that $D$ connects two vertices $v$ and $w$ of $P$. The polygon $P_n$ has vertices $v_n$ and $w_n$ which converge respectively to $v$ and $w$ as $n \to \infty$. Let $D_n$ be the chord whose endpoints are $v_n$ and $w_n$. By construction, $D_n$ converges to $D$ and for large $n$ this chord is a diameter. Suppose now that $D$ connects a vertex $v$ to a point in the interior of an edge $e$. Let $v_n$ and $e_n$ be the corresponding vertex and edge of $P_n$. Since $v_n \to v$ and since $e_n \to e$ we see that eventually there is a chord $D_n$ that has $v_n$ as one endpoint and has the other endpoint perpendicular to $e_n$. By construction $D_n \to D$ and eventually $D_n$ is a diameter of $P_n$. $\spadesuit$ \newline \begin{lemma} If $\{D_n\}$ is a sequence of diameters of $P_n$, then $\{D_n\}$ converges on a subsequence to a diameter of $P$. \end{lemma} {\bf { }{\noindent}Proof: } Given the sequence $\{D_n\}$ we can pass to a subsequence so that the endpoints of these diameters converge. The limiting segment $D$, provided that it has nonzero length, must be a diameter of $P$ because the required condition is a closed condition. We just have to see that the length of $\{D_n\}$ does not shrink to $0$. Note that $D_n$ is at least as long as the shortest diameter of $P_n$. Furthermore, there is a positive lower bound to the length of any edge of $P_n$, independent of $n$. So, if the length of $D_n$ converges to $0$, there are two non-adjacent vertices of $D_n$ whose distance converges to $0$. This contradicts the fact that $\{P_n\}$ converges to the embedded polygon $P$. $\spadesuit$ \newline We think of a diameter as a subset of $(\mbox{\boldmath{$R$}}^2)^2$, and in this way we can talk about the distance between two diameters of $P_n$. \begin{lemma} Suppose that $\{D_n\}$ and $\{D_n'\}$ are two sequences of diameters such that the distance from $D_n$ to $D_n'$ converges to $0$ as $n \to \infty$. Then $D_n=D_n'$ for $n$ sufficiently large. \end{lemma} {\bf { }{\noindent}Proof: } Let $v_n$ and $w_n$ be the endpoints of $D_n$ and let $v_n'$ and $w_n'$ be the endpoints of $D_n'$. We label so that $\|v_n-v_n'\|$ and $\|w_n-w_n'\|$ both tend to $0$. In all cases, we can re-order so that $v_n$ is a vertex of $P_n$ and $v_n'$ is not. In other words, $v_n'$ lies in the interior of an edge $e_n'$ of $P_n$. Since $v_n'$ converges to $v_n$, a vertex of $P_n$, the segment $e_n'$ becomes perpendicular to $D_n'$ in the limit. This contradicts the fact that $P$ has only stable diameters. $\spadesuit$ \newline \begin{corollary} \label{stable} For $n$ sufficiently large, there is a bijection between the diameters of $P_n$ and the diameters of $P$ such that each diameter of $P$ is match with a sequence of diameters of $P_n$ which converges to $P$. \end{corollary} {\bf { }{\noindent}Proof: } This is an immediate consequence of the preceding $3$ lemmas. $\spadesuit$ \newline We truncate our sequence of polygons so that the last corollary holds for all $n$. For each $n$, these diameters are paired together by the arc components of the manifold $\Gamma(P_n)$. We pass to a further subsequence so that the same pairs arise for each $n$. This gives us a well defined way to pair the diameters of $D$. We say that two diameters of $D$ are {\it partners\/} if and only if the corresponding diameters of $D_n$ are paired together. \begin{lemma} \label{connect} Each pair of partner diameters in $P$ is connected by a piecewise smooth path in $J(P)$. \end{lemma} {\bf { }{\noindent}Proof: } Let $A$ and $B$ be two partner diameters of $P$. Let $A_n$ and $B_n$ be the corresponding diameters of $P_n$. Let $\alpha_n$ be the arc in $\Gamma(P_n)$ which connects $A_n$ and $B_n$. To understand the convergence of $\{\alpha_n\}$ we work in the Hausdorff topology on the set of compact subsets of $(\mbox{\boldmath{$R$}}^2)^4$. This ambient space contains $\Gamma(J)$ for any Jordan loop. We consistently label the sides of $P_n$ and $P$. Let $\Sigma_n$ be the inscribing sequence of $\alpha_n$. By Lemma \ref{inscribing} there is a uniform upper bound of $\kappa N^4$ on the length of $\Sigma_n$. Therefore, we may pass to a subsequence so that the inscribing sequence associated to $\alpha_n$ is independent of $n$. We write $$\alpha_n=\alpha_{n1},...,\alpha_{nk},$$ where $\alpha_{nj}$ is the arc of rectangles corresponding to the $j$th element of the sequence in $\Omega$. Here $k$ is the length of the inscribing sequence. We pass to a subsequence so that $\{\alpha_{nj}\}$ converges in the Hausdorff topology to a subset $\alpha_j \subset \alpha$. The set $\alpha_j$ is connected and contained in a subset of $\Gamma(E)$, where $E$ is the $4$-tuple of edges corresponding to the $j$th element of $\Omega$. From the discussion in \S \ref{conn}, we see that $\alpha_j$ is a compact, connected algebraic arc. By construction $\alpha_j$ and $\alpha_{j+1}$ share one point common for all $j$. This vertex is the limit of the sequence $\{\alpha_{nj} \cap \alpha_{n,j+1}\}$. The description above reveals $\alpha$ to be a piecewise smooth arc connecting the two diameters $A$ and $B$. $\spadesuit$ \newline \subsection{The End of the Proof} Let $P$ be a polygon. We still assume that $P$ has stable diameters, so that the results from the previous section apply. We know from Lemma \ref{connect} that the diameters of $P$ are paired in some way, and each pair is connected by some piecewise smooth path of gracing rectangles. We can erase any loops that these paths have and thereby assume that all these paths are embedded. Next, we can assume that every $2$ arcs in the collection intersect each other in at most one point. Otherwise, we can do a splicing operation to decrease the number of intersection points. (See Figure 6 below.) The splicing operation may change the way that the diameters are paired up, but this doesn't bother us. Finally, we can make our choice of connectors invariant under the $\mbox{\boldmath{$Z$}}/4$ re-labelling action. As in the proof of Theorem \ref{main} we let $\delta=\Delta_+(P)/2$ and we chose a collection $\alpha_1,...,\alpha_{\delta}$ of connecting arcs which has one representative in each orbit of the $\mbox{\boldmath{$Z$}}/4$ action. Suppose that our collection of paths contains two hyperbolic arcs $\alpha_1$ and $\alpha_2$ that intersect. Each path connects a (degenerate) rectangle of aspect ratio $0$ to a (degenerate) rectangle of aspect ratio $\infty$. By splicing the paths together and then re-dividing them, we produce $2$ new paths $\beta_1$ and $\beta_2$ such that each $\beta_j$ connects two degenerate rectangles of the same aspect ratio. In other words, we can do a cut-and-paste operation at an intersection point to replace the two hyperbolic arcs by null arcs. If necessary, we can erase any loops created in this process. Figure 6 shows this operation. \begin{center} \resizebox{!}{1.2in}{\includegraphics{fig6.eps}} \newline {\bf Figure 6:\/} The splicing operation. \end{center} Suppose first that there are $\delta/2$ arcs in our collection that are hyperbolic arcs. Then this collection is an embedded $1$-manifold contained in $\Gamma(P)$. Just using these arcs, the same argument as in the proof of Theorem \ref{main} shows that $$M(P) \geq \Delta_+(P)-2.$$ That is, we get the same answer as in Theorem \ref{main} except for the factor of $1/2$. Now suppose that there are at least $\delta/2$ null arcs. For the rest of the proof we just deal with these null arcs. Let $\Gamma_1(P)$ denote the union of these null arcs. We know that $\Gamma_1(P)$ is a subset of $\Gamma(P)$ and also a graph with algebraic edges and maximim valence at most $32$. Let $\widehat \Gamma_1$ denote the formal disjoint union of these embedded null arcs. The space $\widehat \Gamma_1$ is a $1$-manifold, just a union of arcs, and the ``forgetful map'' $\phi: \widehat \Gamma_1 \to \Gamma_1$ is at most $16$ to $1$. The same argument as in the proof of Theorem \ref{generic} says that there are $\delta$ distinct points in $\widehat \Gamma_1$, two per arc, corresponding to rectangle coincidences. Let $S$ be the set of these points. The image $\phi(S)$ contains at least $\delta/16$ points. For each of these points, there is a second point corresponding to an isometric rectangle. We know this because the map $\phi$ is injective on each null arc, and each null arc contains $2$ points of $S$. So, we can match our $\delta/16$ points into $\delta/32$ distinct pairs of points, corresponding to pairs of isometric but distinct rectangles in $\Gamma(P)$. This adds a count of $\delta/32$ to $M(P)$. To make the comparison with Theorem \ref{main} cleaner, we work with $(\delta-1)/32$ instead. In the case at hand, we get the same bound as in Theorem \ref{main} except for the factor of $1/32$. Going back to the count of labeled rectangles, we have $$M(P) \geq \frac{1}{16}(\Delta_+(P)-2).$$ This completes the proof of Theorem \ref{main2}. [{\bf AA\/}] A. Akopyan and S Avvakumov, {\it Any cyclic quadrilateral can be inscribed in any closed convex smooth curve.\/} arXiv: 1712.10205v1 (2017) \newline \newline [{\bf ACFSST\/}] J. Aslam, S. Chen, F. Frick, S. Saloff-Coste, L. Setiabrate, H. Thomas, {\it Splitting Loops and necklaces: Variants of the Square Peg Problem\/}, arXiv 1806.02484 (2018) \newline \newline [{\bf CH\/}] D. Hilbert and S. Cohn-Vossen, {\it Geometry and The Imagination\/}, \newline Chelsea Publishing Company (American Math Society), 1990 \newline \newline [{\bf H\/}] C. Hugelmeyer, {\it Every Smooth Jordan Curve has an inscribed rectangle with aspect ratio equal to $\sqrt 3$.\/} arXiv 1803:07417 (2018) \newline \newline [{\bf M\/}] B. Matschke, {\it A survey on the Square Peg Problem\/}, Notices of the A.M.S. {\bf Vol 61.4\/}, April 2014, pp 346-351. \newline \newline [{\bf S1\/}] R. E. Schwartz, {\it A Trichotomy for Rectangles Inscribed in Jordan Loops\/}, preprint, 2018 \newline \newline [{\bf S2\/}] R. E. Schwartz, {\it Four lines and a Rectangle\/}, preprint, 2018 \newline \newline [{\bf Ta\/}], T. Tao, {\it An integration approach to the Toeplitz square peg conjecture\/} \newline Forum of Mathematics, Sigma, 5 (2017) \newline \newline [{\bf W\/}] S. Wolfram, {\it The Mathematica Book\/}, 4th ed. Wolfram Media/Cambridge University Press, Champaign/Cambridge (1999) \end{document}

\begin{document} \begin{abstract} We determine the asymptotic distribution of the $p$-rank of the sandpile groups of random bipartite graphs. We see that this depends on the ratio between the number of vertices on each side, with a threshold when the ratio between the sides is equal to $\frac{1}{p}$. We follow the approach of Wood in \cite{mw} and consider random graphs as a special case of random matrices, and rely on a variant the definition of min-entropy given by Maples in \cite{km} in order to obtain useful results about these random matrices. Our results show that unlike the sandpile groups of {Erd\H{o}s--R\'{e}nyi } random graphs, the distribution of the sandpile groups of random bipartite graphs depends on the properties of the graph, rather than coming from some more general random group model. \end{abstract} \title{Sandpile Groups of Random Bipartite Graphs} \section{Introduction} \label{sec:intro} \subsection{The Main Theorem} In this paper, we study the sandpile group of a random bipartite graph. Recall that the sandpile group $\Gamma(G)$ of a connected graph $G$ is the cokernel of the reduced laplacian matrix $\Delta'$. Let $0<\alpha,q<1$ be constants. We define a random bipartite graph $G=G(n,\alpha,q)$ as follows: Take two sets of vertices $L$ and $R$ with $|L|=n,|R|=\lfloor\alpha n\rfloor$, and for each pair of vertices $v\in L$ and $u\in R$, include the edge between $v$ and $u$ independently with probability $q$. We now state our main result about the $p$-rank of $\Gamma(G)$: \begin{thm} \label{thm:main} Let $G=G(n,\alpha,q)$ be a random bipartite graph, and $p$ a prime. Then as $n\rightarrow \infty$, the expected value of the $p$-rank of the sandpile group $\Gamma(G)$ is: \begin{enumerate} \item $\left(\frac{1}{p}-\alpha\right)n+O(1)$ if $\alpha<\frac{1}{p}$ \item $O(1)$ if $\alpha>\frac{1}{p}$ \item $\sqrt{\frac{\frac{1}{p}(1-\frac{1}{p})n}{2\pi}}$+$O(1)$ if $\alpha=\frac{1}{p}.$ \end{enumerate} \end{thm} \noindent It is worth noting that the limits in the theorem do not depend on the value of $q$. Theorem~\ref{thm:strongMain} will also give us explicit information about the distribution of the $p$-ranks. From numerical computations, it appears that the $O(1)$ constants in the first two cases of the theorem are at most $1$, and the $O(1)$ constant in the third case is around $2$. The proof of Theorem~\ref{thm:main} relies on the assumption that $\alpha<1$. Based on numerical computations of random graphs, we conjecture that Theorem~\ref{thm:main} also holds when $\alpha=1$. This implies that the expected $p$-rank of the sandpile group of a balanced bipartite graphs should be $O(1)$ for all primes $p$. However, the best that can be done with our methods is: \begin{cor} \label{cor:balanced} Let $G=G(n,1,q)$ be a random balanced bipartite graph, $p$ prime. Then as $n\rightarrow \infty$, the expected value of the $p$-rank of the sandpile group $\Gamma(G)$ is $o(n)$. \end{cor} \noindent Which we prove in Section~\ref{sec:details}. \subsection{Connection to {Erd\H{o}s--R\'{e}nyi } Random Graphs} It is interesting to ask what the distribution of the sandpile groups of random graphs looks like. The authors of \cite{pk} noted that the sandpile group of a graph comes with a canonical symmetric perfect bilinear pairing $\langle\cdot,\cdot\rangle_{G}$, and conjectured that for an {Erd\H{o}s--R\'{e}nyi } random graph $G$, the pair $(\Gamma(G),\langle\cdot,\cdot\rangle)$ of the sandpile group and its associated pairing can be predicted by certain heuristics of Cohen-Lenstra type. The Cohen-Lenstra heuristics are an attempt to model what a generic ``random'' group should look like. In \cite{pk}, the authors show that the cokernel of a random symmetric matrix over ${\mathbb Z}_{p}$, distributed according to the Haar measure, follows heuristics of Cohen-Lenstra type, and conjectured that the sandpile groups of {Erd\H{o}s--R\'{e}nyi } random graphs should follow the same heuristics. In \cite{mw}, Melanie Wood proves several results in this direction. In particular, she shows that for an {Erd\H{o}s--R\'{e}nyi } random graph $G$, the $p$-part of $\Gamma(G)$ follows these heuristics for any finite collection of primes $p$. However, Theorem~\ref{thm:main} shows that sufficiently unbalanced random bipartite graphs do not follow any similar type of Cohen-Lenstra heuristics: For example, the Cohen-Lenstra heuristics predict that for any $p$, the expected $p$-rank of $\Gamma(G)$ should stay low as $n$ grows. However, Theorem~\ref{thm:main} implies that for sufficiently unbalanced bipartite graphs, the $p$-rank grows linearly with $n$. Furthermore, the Cohen-Lenstra heuristics predict that the probability that $\Gamma(G)$ is cyclic should converge to a constant between $0$ and $1$, but in Section~\ref{sec:details} we prove that this is not the case for sufficiently unbalanced bipartite graphs. \begin{cor} \label{cor:cyclic} Let $G=G(n,\alpha,q)$ be a random bipartite graph with $\alpha<\frac{1}{2}$. Then as $n\rightarrow\infty$, the probability that $\Gamma(G)$ is cyclic goes to zero exponentially fast. \end{cor} \noindent Because of the $O(1)$ factor in Theorem~\ref{thm:main}, the theorem gives us no information on the probability that $\Gamma(G)$ is cyclic when $\alpha\geq\frac{1}{2}$. Numerical computations suggest that this probability converges to a constant around $0.60$ when $\alpha>\frac{1}{2}$, and to a constant around $0.29$ when $\alpha=\frac{1}{2}$. Here is a brief outline of the paper: In Section~\ref{sec:closeness}, we define when sequences of random variables are ``usually within small distance'', which will give us a useful equivalence relation for random variables. We also give Theorem~\ref{thm:strongMain}, which describes the distribution of the $p$-rank of $S(\Gamma)$, and show that it implies Theorem~\ref{thm:main}. In Section~\ref{sec:entropy}, we introduce our notion of min-entropy, which is a variant on the one used by Maples in \cite{km}. This notion is meant to replace independence; the matrices we will work with are not independent, but they are ``almost independent'', in the sense described by min-entropy, which will suffice for our purposes. In Section~\ref{sec:proof} we introduce a random matrix $M$, whose corank is usually within small distance of the $p$-rank of $\Gamma(G)$. Using the min-entropy properties of $M$, we will show that its corank is also usually within small distance of the distribution given in Theorem~\ref{thm:strongMain}, which will complete our proof. Section~\ref{sec:background} contains some background information, and Section~\ref{sec:details} contains proof of the corollaries of Theorem~\ref{thm:main}. \textbf{Acknowledgments}. The author is grateful to Sam Payne and Nathan Kaplan for suggesting the problem, as well as their many helpful suggestions along the way. Also to Dan Carmon, for suggesting the proof of Claim~\ref{claim:estimation}. This work was partially supported by NSF CAREER DMS-1149054. \section{The Sandpile Group, Binomial Distributions, and Schur Complements} \label{sec:background} \subsection{The Sandpile Group.} In this section, we define the sandpile group. For a more thorough introduction to the subject with some lovely pictures, see \cite{wias}. Let $G$ be a connected graph on $n$ vertices, numbered $1$ through $n$. The \textbf{laplacian matrix} of $G$ is the $n\times n$ matrix $\Delta=D-A$, where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal degree matrix of $G$. In other words, $\Delta_{ii}={\operatorname{deg}}(v_{i})$ and for $i\neq j$, $\Delta_{ij}=-1$ if $G$ has an edge between vertices $i$ and $j$, ans to $0$ otherwise. Note that $\Delta$ is a symmetric matrix whose rows and columns sum to zero, so it is singular. In fact, ${\operatorname{corank}}(\Delta)$ is equal to the number of connected components of $G$, where we define the corank of an $n\times m$ matrix $A$ as $\min(n,m)-{\operatorname{rank}}(A)$. Choose a vertex $i$. The \textbf{reduced laplacian matrix} $\Delta'$ is the $(n-1)\times(n-1)$ matrix obtained by removing row $i$ and its corresponding column from $\Delta$. The \textbf{sandpile group} $\Gamma(G)$ is the cokernel of $\Delta'$, that is, $\Gamma(G)={\mathbb Z}^{n-1}/\Delta'({\mathbb Z}^{n-1})$. It is shown in \cite{wias} that the sandpile group of a graph is independent of the choice of the vertex $i$. Moreover, the Matrix Tree Theorem shows that for a connected graph $G$, the determinant $\det(\Delta')$ is equal to the number of spanning trees of $G$. In particular, $\Delta'$ has full rank, so ${\mathbb Z}^{n-1}/\Delta'({\mathbb Z}^{n-1})$ is a finite group of order $\det(\Delta')$ and rank at most $n-1$. If $G$ is disconnected, we define its sandpile group to be the direct sum of the sandpile groups of its connected components. It is easy to see that this is a finite group of rank at most $n-1$. Moreover, it is shown in \cite{ra} that a random bipartite graph $G(n,\alpha,q)$ is connected with probability $1-O(e^{-Kn})$ for some $K>0$ depending only on $q$ and $\alpha$. This will allow us to consider the rank of the cokernel of the reduced laplacian rather than the rank of the sandpile group directly, as they are equal with probability $1-O(e^{-Kn})$. \subsection{Binomial and Normal Distributions} We use $B(n,q)$ for the binomial distribution, the sum of $n$ independent Bernoulli random variables equal to $1$ with probability $q$ and $0$ otherwise. Recall that ${\mathbb E}(B(n,q))=qn$, where ${\mathbb E}(X)$ is the expected value of $X$. We will make repeated use of Hoeffding's inequality: \begin{thm}[Hoeffding's inequality] \label{thm:hoeffding} Let $B(n,q)$ be the binomial distribution, and let $\epsilon>0$. Then there exists a constant $K>0$, depending only on $q$ and $\epsilon$, such that ${\mathbb P}\left(\left|B(n,q)-qn\right|>\epsilon n\right)<e^{-Kn}$. \end{thm} For the proof, see for example \cite{hf}. \subsection{Schur Complements} Finally, we recall the basics of Schur complements, which will be a central tool in our proof. For a more thorough introduction to the subject, see \cite[Chapter 1]{zs}. \begin{defn} Let $A$ be an $n\times n$ matrix. Let $S$ be a subset of ${1,\dots,n}$, and let $T$ be the complement of $S$. We write $A_{S,S}$ for the submatrix given by restricting $A$ to the rows and columns whose indices are in $S$, $A_{T,T}$ for the submatrix of rows and columns with indices in $T$, and $A_{S,T}$ for the submatrix of rows in $A$ and columns in $T$. \end{defn} \noindent For example, if $S=\{1,\dots,k\}$, then $A=\begin{pmatrix} A_{S,S} & A_{S,T}\\ A_{T,S} & A_{T,T} \end{pmatrix} $ \begin{defn} Let $A,S,T$ as above. If $A_{S,S}$ is invertible, then we define the \textbf{Schur complement} $A/S$ (or $A/A_{S,S}$) by $A/S=A_{T,T}-A_{T,S}A_{T,T}^{-1}A_{S,T}$. \end{defn} \noindent Note that $A/S$ is a $|T|\times |T|$ matrix. Recall that the \textbf{corank} of an $n\times m$ matrix $A$ is defined as $\min(n,m)-{\operatorname{rank}}(A)$. We will use the following theorem several times: \begin{thm} Let $A$ and $S$ as above such that $A_{S,S}$ is invertible. Then ${\operatorname{corank}}(A/S)={\operatorname{corank}}(A)$. \end{thm} \begin{proof} Assume that $S$ is composed of the first $k$ entries for some $k\leq n$. It can be seen that \[A=\begin{pmatrix} A_{S,S} & A_{S,T}\\ A_{T,S} & A_{T,T} \end{pmatrix} =\begin{pmatrix} I_{k} & 0 \\ A_{T,S}A_{S,S}^{-1} & I_{n-k} \end{pmatrix} \begin{pmatrix} A_{S,S} & A_{S,T}\\ 0 & A/S \end{pmatrix}. \] \noindent Where $I_k$ is the $k\times k$ identity matrix. Since the matrix on the left is invertible, we get that \[{\operatorname{corank}}(A)={\operatorname{corank}} \begin{pmatrix} A_{S,S} & A_{S,T}\\ 0 & A/S \end{pmatrix}. \] As $A_{S,S}$ is invertible, row reduction gives us \[{\operatorname{corank}} \begin{pmatrix} A_{S,S} & A_{S,T}\\ 0 & A/S \end{pmatrix} = {\operatorname{corank}}(A/S), \] which completes the proof. \end{proof} \section{Closeness of Random Variables} \label{sec:closeness} In this section, we define when random variables are usually within small distance. This describes the ``closeness'' of random variables in a useful way. \begin{defn} \label{defn:closeness} Let $X_{n},Y_{n}$ be two sequences of random variables. We say $X_{n}$ and $Y_{n}$ are \textbf{usually within small distance} if there exist constants $c,K>0$ such that for every $n,m>0$, ${\mathbb P}(|X_{n}-Y_{n}|\geq m)\leq Ke^{-cm}$. \end{defn} We will use the following properties. \begin{lemma} \label{lemma:closeness} \begin{enumerate} \item If $X_{n},Y_{n}$ and $Y_{n},Z_{n}$ are pairs of sequences of random variables which are usually within small distance, then so are $X_{n},Z_{n}$. Hence being usually within small distance is an equivalence relation for sequences of random variables. \item If $X_{n},Y_{n}$ are sequences of random variables which usually within small distance, then $\left|{\mathbb E}(X_{n})-{\mathbb E}(Y_{n})\right|=O(1)$. \item If ${\mathbb P}(X_{n}\neq Y_{n})=1-O(e^{-cn})$ for some constant $c>0$ and $X_{n},Y_{n}$ are bounded by $O(n)$, then $X_n,Y_n$ are usually within small distance. \item If $X_{n},Y_{n}$ are sequences of random variables and $\max|X_{n}-Y_{n}|<K$ for some constant $K$, then $X_{n},Y_{n}$ are usually within small distance. \item If $X_{n},Y_{n},Z_{n}$ are sequences of random variables such that $X_{n}\leq Y_{n}\leq Z_{n}$ and $Z_{n}$ is usually within small distance of $X_{n}$, then so is $Y_{n}$. \item If $X_{n},Y_{n}$ are usually within small distance of $X'_{n},Y'_{n}$ respectively, and $C$ is constant, then $X_{n}+Y_{n},CX_{n},\max(X_{n},Y_{n})$, and $\min(X_{n},Y_{n})$ are usually within small distance of $X'_{n}+Y'_{n},CX'_{n},\max(X'_{n},Y'_{n})$, and $\min(X'_{n},Y'_{n})$ respectively. \end{enumerate} \end{lemma} \noindent The proofs are straightforward. Using the above definition, we can now state the main theorem about the distribution of the $p$-rank of the sandpile group, from which we will deduce Theorem~\ref{thm:main}: \begin{thm} \label{thm:strongMain} Let $G=G(n,\alpha,q)$ be a random bipartite graph, and $p$ a prime. Let $X_{n}=X(n,\alpha,q,p)$ be the $p$-rank of $\Gamma(G)$, and recall that $B(n,q)$ denotes a binomial random variable. Then $X_{n}$ is usually within small distance of $\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)$, where $B\left(n,\frac{1}{p}\right)$ is the binomial distribution. \end{thm} We will prove Theorem~\ref{thm:strongMain} in Section~\ref{sec:proof}. First, we show: \begin{prop} \label{prop:reduction} Theorem~\ref{thm:strongMain} implies Theorem~\ref{thm:main}. \end{prop} \begin{proof} By Lemma~\ref{lemma:closeness}, Theorem~\ref{thm:strongMain} implies that \[{\mathbb E}(X_{n})={\mathbb E}\left(\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)\right)+O(1).\] Hence it suffices to calculate ${\mathbb E}\left(\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)\right)$. We will split into three cases, depending on whether $\alpha<\frac{1}{p},\alpha>\frac{1}{p}$, or $\alpha=\frac{1}{p}$. \noindent{\bf The case $\alpha < \frac{1}{p}.$} Note that by Hoeffding's inequality ${\mathbb P}\left(B\left(n,\frac{1}{p}\right)-\alpha n>0\right)=1-O(e^{-cn}) $ for some constant $c>0$. Hence by lemma~\ref{lemma:closeness}, $B\left(n,\frac{1}{p}\right)-\alpha n$ is usually within small distance of $\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)$. Because of this, it suffices to calculate ${\mathbb E}\left(B\left(n,\frac{1}{p}\right)-\alpha n\right)$. Using the additivity of the expected value, we see that \[{\mathbb E}\left(B\left(n,\frac{1}{p}\right)-\alpha n\right)={\mathbb E}\left(B\left(n,\frac{1}{p}\right)\right)-\alpha n = \frac{1}{p}n-\alpha n=\left(\frac{1}{p}-\alpha\right)n.\] \noindent{\bf The case $\alpha > \frac{1}{p}.$} This case is similar. Again by Hoeffding's inequality, we get that ${\mathbb P}\left(B\left(n,\frac{1}{p}\right)-\alpha n>0\right)=O(e^{-cn})$ and hence $\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)$ is equal to $0$ with probability $1-O(e^{-cn})$. Hence $\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)$ is usually within small distance of $0$, which has expected value $O(1)$. \noindent{\bf The case $\alpha = \frac{1}{p}.$} Finally, the case where $\alpha =\frac{1}{p}$. In this case, we wish to calculate ${\mathbb E}\left(\max\left(B\left(n,\alpha\right)-\alpha n,0\right)\right)$. We will rely on the following claim: \begin{claim} \label{claim:estimation} Let $B(n,\alpha)$ be the binomial distribution, $s$ a positive integer. Then \[{\mathbb E}(B(n,\alpha)|B(n,\alpha)>s)=\alpha n+\alpha(1-\alpha)n\frac{{\mathbb P}(B(n-1,\alpha)=s)}{{\mathbb P}(B(n,\alpha)>s)}.\] \end{claim} \begin{proof} Let $Y=B(n,\alpha)$. We wish to calculate ${\mathbb E}(Y|Y>s)=\frac{\sum_{k>s}k{\mathbb P}(Y=k)}{\sum_{k>s}{\mathbb P}(Y=k)}$. Since we expect the main term in the expectation to be ${\mathbb E}\left(B(n,\alpha)\right)= \alpha n$, we wish to estimate \[{\mathbb E}(Y|Y>s)-\alpha n=\frac{\sum_{k>s}k{\mathbb P}(Y=k)}{\sum_{k>s}{\mathbb P}(Y=k)}-\alpha n = \frac{\sum_{k>s}k{\mathbb P}(Y=k)-\sum_{k>s} \alpha n{\mathbb P}(Y=k)}{\sum_{k>s}{\mathbb P}(Y=k)}.\] Now consider the two sums \begin{align} & \sum_{k>s} k{\mathbb P}(Y=k) =\sum_{k>s} k\alpha^k(1-\alpha)^{n-k} \binom{n}{k} \\ & \sum_{k>s} \alpha n{\mathbb P}(Y=k) = \alpha n \sum_{k>s} \alpha^k(1-\alpha)^{n-k} \binom{n}{k}. \end{align} We manipulate the sums as follows: In sum (1), replace $k\binom{n}{k}$ with the equal $n\binom{n-1}{k-1}$ and take $\alpha n$ out, so that it becomes $\alpha n\sum_{k>s}\alpha^{k-1}(1-\alpha)^{n-k}\binom{n-1}{k-1}$. Now, multiply by $\alpha+(1-\alpha)=1$, and expand, to obtain the two sums $(1)=(1a)+(1b)$, where \begin{align*} &(1a)\ &&\alpha n\sum_{k>s} \alpha^{k}(1-\alpha)^{n-k}\binom{n-1}{k-1} \\ &(1b)\ &&\alpha n\sum_{k>s} \alpha^{k-1}(1-\alpha)^{n-k+1}\binom{n-1}{k-1}. \end{align*} For sum (2), use $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ to obtain $(2)=(2a)+(2b)$, where \begin{align*} &(2a)\ &&\alpha n\sum_{k>s} \alpha^{k}(1-\alpha)^{n-k}\binom{n-1}{k} \\ &(2b)\ &&\alpha n\sum_{k>s} \alpha^{k}(1-\alpha)^{n-k}\binom{n-1}{k-1}. \end{align*} Now the difference $(1)-(2)$ cancels out! Observe that $(1a)=(2b)$, whereas $(1b)$ and $(2a)$ are just shifts of each other, so the difference cancels out in a telescopic sum, and we obtain \[(1)-(2)=(1b)-(2a)=\alpha n \left(\alpha^{s}(1-\alpha)^{n-s}\binom{n-1}{s}\right)=n\alpha (1-\alpha){\mathbb P}(B(n-1,\alpha)=s). \] Finally, putting our expression for $(1)-(2)$ back in our equation for the expectation, we get \begin{align*} {\mathbb E}(Y|Y>s)-\alpha n &= \frac{\sum_{k>s}k{\mathbb P}(Y=k)-\sum_{k>s} \alpha n{\mathbb P}(Y=k)}{\sum_{k>s}{\mathbb P}(Y=k)} \\ &= \frac{n\alpha (1-\alpha){\mathbb P}(B(n-1,\alpha)=s)}{{\mathbb P}(Y>s)}, \end{align*} \noindent which completes the proof of the claim. \end{proof} For estimating ${\mathbb E}\left(\max\left(B\left(n,\alpha\right)-\alpha n,0\right)\right)$, the following version of the De Moivre-Laplace theorem will be useful. \begin{thm}[{\cite[Theorem 2]{mt}}] \label{thm:dml} Let $s$ be an integer such that $|\alpha n-s|<\sqrt{n}$. Then \[{\mathbb P}(B(n,\alpha)=s)=\frac{1}{\sqrt{2\pi \alpha (1-\alpha)n}} e^{-\frac{(s-\alpha n)^{2}}{2\alpha(1-\alpha)n}}\left(1+O\left(\frac{1}{\sqrt{n}}\right)\right).\] \end{thm} In our calculation, we will need to estimate ${\mathbb P}(B(n-1,\alpha)=s)$ for $s=\lfloor\alpha n\rfloor$. As $(s-\alpha n)^{2}\leq 1$, we get that \[\left|e^{-\frac{(s-\alpha (n-1))^{2}}{2\alpha(1-\alpha)(n-1)}}-1\right|\leq\left|e^{-\frac{1}{2\alpha(1-\alpha)(n-1)}}-1\right|=O\left(\frac{1}{n}\right),\] and hence $ e^{-\frac{(s-\alpha (n-1))^{2}}{2\alpha(1-\alpha)(n-1)}}=1+O\left(\frac{1}{n}\right)$. As $\frac{1}{\sqrt{n-1}}=\frac{1}{\sqrt{n}}\left(1+O\left(\frac{1}{\sqrt{n}}\right)\right)$ we have by Theorem~\ref{thm:dml}: \begin{align} \nonumber {\mathbb P}(B(n-1,\alpha)=s)&=\frac{1}{\sqrt{2\pi \alpha (1-\alpha)(n-1)}} e^{-\frac{(s-\alpha (n-1))^{2}}{2\alpha(1-\alpha)(n-1)}}\left(1+O\left(\frac{1}{\sqrt{n}}\right)\right) \\ \nonumber &=\frac{1}{\sqrt{2\pi \alpha (1-\alpha)n}} \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right) \left( 1+O\left(\frac{1}{n}\right) \right) \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right)\\ &=\frac{1}{\sqrt{2\pi \alpha (1-\alpha)n}} \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right). \label{eqn:test} \end{align} Recall that we wish to estimate ${\mathbb E}\left(\max\left(B\left(n,\alpha\right)-\alpha n,0\right)\right)$. \begin{align*} {\mathbb E}(\max(B(n,\alpha)-\alpha n,0)&={\mathbb P}(B(n,\alpha)>\lfloor\alpha n\rfloor){\mathbb E}(B(n,\alpha)-\alpha n|B(n,\alpha)>\lfloor\alpha n\rfloor)\\ &={\mathbb P}(B(n,\alpha)>\lfloor\alpha n\rfloor)({\mathbb E}(B(n,\alpha)|B(n,\alpha)>\lfloor\alpha n\rfloor)-\alpha n). \end{align*} Using Claim~\ref{claim:estimation} with $s=\lfloor\alpha n\rfloor$, we get: \begin{align*} {\mathbb P}(B(n,\alpha)>\lfloor\alpha n\rfloor)&({\mathbb E}(B(n,\alpha)|B(n,\alpha)>\lfloor\alpha n\rfloor)-\alpha n) \\ &={\mathbb P}(B(n,\alpha)>\lfloor\alpha n\rfloor)\left(\alpha n+\alpha(1-\alpha)n\frac{{\mathbb P}(B(n-1,\alpha)=\lfloor\alpha n\rfloor)}{{\mathbb P}(B(n,\alpha)>\lfloor\alpha n\rfloor)}-\alpha n\right)\\ &=\alpha(1-\alpha)n{\mathbb P}(B(n-1,\alpha)=\lfloor\alpha n\rfloor). \end{align*} Finally, using \ref{eqn:test}, we get: \begin{align*} \alpha(1-\alpha)n{\mathbb P}(B(n-1,\alpha)=\lfloor\alpha n\rfloor) &=\alpha(1-\alpha)n\frac{1}{\sqrt{2\pi \alpha (1-\alpha)n}} \left(1+O\left(\frac{1}{\sqrt{n}}\right)\right)\\ &=\sqrt{\frac{\alpha(1-\alpha)n}{2\pi}}\left(1+O\left(\frac{1}{\sqrt{n}}\right)\right)\\ &=\sqrt{\frac{\alpha(1-\alpha)n}{2\pi}}+O(1). \end{align*} and substituting $\alpha=\frac{1}{p}$ gives us the expression from Theorem~\ref{thm:main}. \end{proof} \section{Min-Entropy and Random Matrix Rank} \label{sec:entropy} In this section, we define our notion of min-entropy, which is a variant on the definition given by Maples in \cite{km} and use it to prove some lemmas which will be useful in the proof of Theorem~\ref{thm:strongMain}. \begin{defn} Let $A$ be a random matrix over ${\mathbb Z}/p{\mathbb Z}$. Let $\beta>0$, and let $I$ be a set of entries in $A$. We say that \textbf{an entry $A_{i_{0}j_{0}}\in A$ has min-entropy at least $\beta$ with respect to $I$} if, for any choice of values $a_{ij}$ for the entries in $I$ that can occur with nonzero probability, and every $a\in {\mathbb Z}/p{\mathbb Z}$, the probability ${\mathbb P}(A_{i_{0}j_{0}}=a|A_{ij}=a_{ij}\forall (i,j)\in I)$ is at most $1-\beta$. We say that \textbf{the matrix $A$ has min-entropy at least $\beta$} if every entry of $A$ has min-entropy at least $\beta$ with respect to the set of all other entries. \end{defn} In other words, $A_{ij}$ has min-entropy greater than $\beta$ relative to a set of entries if fixing them cannot control $A_{ij}$, in the sense that it still has probability at most $1-\beta$ of being any specific value. We can think of min-entropy as a bound on how much fixing some entries of a matrix can influence other entries. We illustrate this notion of min-entropy with the following examples. If all the entries of $A$ are independent, the min-entropy of $A_{ij}$ is simply $\min_{x}(1-\mathbb{P}(A_{ij}=x))$. In particular, if the entries of $A_{ij}$ are all independent and uniformly distributed in $\mathbb{Z}/p{\mathbb Z}$, this min-entropy is $1-\frac{1}{p}$, which is the highest possible. For another example, consider $\Delta=\Delta(n,\alpha,q)$, the laplacian matrix of a random bipartite graph. Since every row in $\Delta$ sums to zero, for any entry $\Delta_{ij}$, fixing the rest of the entries in row $i$ determines $\Delta_{ij}$. Hence $\Delta_{ij}$ has zero min-entropy with respect to the rest of the entries in row $i$. \begin{thm} \label{thm:rectrank} Let $A$ be an $n\times m$ random matrix over ${\mathbb Z}/p{\mathbb Z}$, for $m\geq n$, with min-entropy at least $\beta$ for some $\beta>0$. Then the probability that $A$ has rank $n$ is at least $1-\frac{1}{\beta^{2}}(1-\beta)^{m+1-n}$. In particular, there exists a constant $K>0$ depending only on $\beta$ such that ${\mathbb P}({\operatorname{rank}}(A)=n)\geq 1-e^{-K(m-n)}$. \end{thm} \begin{proof} Let $v_{1},\dots,v_{n}$ be the rows of $A$. Then $A$ has rank $n$ only if the $v_{i}$ are independent, so the probability ${\mathbb P}({\operatorname{rank}}(A)=n)$ is equal to the product \[\prod_{i=1}^{n}\mathbb{P}(v_{i}\text{ is independent of }\{v_{1},\dots,v_{i-1}\}|\{v_{1},\dots,v_{i-1}\} \text{ are independent}).\] We now note that for each $i$, \[{\mathbb P}(v_{i}\text{ is independent of }\{v_{1},\dots,v_{i-1}\}|\{v_{1},\dots,v_{i-1}\} \text{ are independent})\geq (1-\beta)^{m-(i-1)}. \] To see this, assume that $\{v_{1},\dots,v_{i-1}\}$ are independent. Then there exists a subset $J=\{j_{1},\dots,j_{i-1}\}\subset [m]$ such that the restrictions of the $\{v_{j}\}_{j<i}$ to the entries in $J$ are independent. Assume that $J=\{1,\dots,i-1\}$. by the independence of the $v_{l}|_{J}$, there exist unique coefficients $a_{1},\dots,a_{i-1}$ such that for all $j<i$, $(v_{i})_{j}=\sum_{l<i} a_{l}(v_{l})_{j}$. $v_{i}$ is dependent on $\{v_{1},\dots,v_{i-1}\}$ only if there exists a linear combination of them that sums to $v_{i}$. By the uniqueness of the coefficients $a_{1},\dots,a_{i-1}$, this happens only if $v_{i}=\sum_{l<i}a_{l}v_{l}$. In particular, $v_{i}$ is dependent on the previous row vectors only if for all $j\geq i$, $(v_{i})_{j}=\sum_{l<i} a_{l}(v_{l})_{j}$. However, by the min-entropy assumption, this happens for each $j$ with probability at most $1-\beta$. As there are $m-(i-1)$ such entries, the probability that this equality holds for all of them is at most $(1-\beta)^{m-(i-1)}$. Hence the probability that $v_{i}$ is independent of $\{v_{1},\dots,v_{i-1}\}$ is at least $1-(1-\beta)^{m-(i-1)}$. Using this, we get the following bound \begin{align} \prod_{i=1}^{n}\mathbb{P}(v_{i}\text{ is independent of }\{v_{1},\dots,v_{i-1}\}|\{v_{1},\dots,v_{i-1}\} \text{ are independent}) \nonumber \\ \geq \prod_{i=1}^{n}(1-(1-\beta)^{m-(i-1)})=(1-(1-\beta)^{m})\cdots(1-(1-\beta)^{m+1-n}).\label{eqn:betaprod} \end{align} We now wish to bound (\ref{eqn:betaprod}) from below. \begin{claim} The product $(1-(1-\beta)^{m})\cdots(1-(1-\beta)^{m+1-n})$ is at least $1-\frac{1}{\beta^{2}}(1-\beta)^{m+1-n}$. \end{claim} Write $\gamma=1-\beta,r=m+1-n$. We need to find a lower bound on the product $(1-\gamma^{m})\cdots(1-\gamma^{r})$. We will rely on the fact that $0<\gamma<1$. First, recall that for any positive $x$, $x\geq\log(x)+1$. Using this for $x=(1-\gamma^{m})\cdots(1-\gamma^{r})$, we get: \[(1-\gamma^{m})\cdots(1-\gamma^{r})\geq 1+\log\left((1-\gamma^{m})\cdots(1-\gamma^{r})\right)\] Now split the product to get: \[1+\log((1-\gamma^{m})\cdots(1-\gamma^{r}))=1+\sum_{i=r}^{m}\log(1-\gamma^{i})\] For any $0<x<1$, $\log(1-x)>-\frac{x}{1-x}$. To see this, let $h(t)=\frac{1}{1-x}(t-(1-x))+\log(1-x)$ be the tangent line to $\log(t)$ at $t=1-x$. Then as $\log(t)$ is concave, $h(1)=\frac{x}{1-x}+\log(1-x)>\log(1)=0$, so $\log(1-x)>-\frac{x}{1-x}$. Using this for $x=1-\gamma^{i}$, we get: \[1+\sum_{i=r}^{m}\log(1-\gamma^{i})\geq 1+\sum_{i=r}^{m}\frac{-\gamma^{i}}{1-\gamma^{i}}\] As $\gamma<1$, we have: \[1+\sum_{i=r}^{m}-\frac{\gamma^{i}}{1-\gamma^{i}}\geq 1+\sum_{i=r}^{m}\frac{-\gamma^{i}}{1-\gamma}=1-\frac{1}{1-\gamma}\sum_{i=r}^{m}\gamma^{i}\] We will now bound this by the sum of the infinite series: \[1-\frac{1}{1-\gamma}\sum_{i=r}^{m}\gamma^{i}\geq1-\frac{1}{1-\gamma}\sum_{i=r}^{\infty}\gamma^{i}=1-\frac{\gamma^{r}}{(1-\gamma)^{2}}\]. Translating back through $\gamma=1-\beta$,$r=m+1-n$, this is $1-\frac{1}{\beta^{2}}(1-\beta)^{m+1-n}$, which proves the claim, and the theorem follows. \end{proof} \begin{cor} \label{cor:rectrank} Let $A_{n}$ be an $n\times m$ random matrix over ${\mathbb Z}/p{\mathbb Z}$ with min-entropy at least $\beta$ for some $\beta>0$ independent of $n$. Then ${\operatorname{corank}}(A)$ is usually within small distance of $0$. \end{cor} \begin{proof} Let $s>0$, and assume that $m\geq n$. We wish to show that ${\mathbb P}({\operatorname{corank}}(A)>s)=O(e^{-Ks})$, where $K>0$ is independent of $n$. Let $A'$ be the submatrix of $A$ given by taking the first $n-s$ rows. Then $A'$ is an $n-s\times m$ matrix, so by Theorem~\ref{thm:rectrank}, its rows are independent with probability a probability at least $1-e^{-K(m-(n-s))}=1-(e^{-K(m-n)})e^{-Ks}\geq 1-e^{-Ks}$, where $K$ depends only on $\beta$. But if $A'$ has rank $n-s$, the corank of $A$ is at most $s$, so ${\mathbb P}({\operatorname{corank}}(A)>s)\leq e^{-Ks}$. \end{proof} \section{Proof of Theorem~\ref{thm:strongMain}} \label{sec:proof} For this section, we fix a prime $p$, as well as constants $0<\alpha,q<1$. We will now prove Theorem~\ref{thm:strongMain}. We do this in two stages. First, we reduce the laplacian mod $p$, remove the first and last $p$ rows and columns, and set the diagonal entries to be uniformly distributed mod $p$. We call the resulting matrix $M$. We show that ${\operatorname{corank}}(M)$ is usually within small distance of the $p$-rank of the sandpile group, which reduces Theorem~\ref{thm:strongMain} to calculating the distribution of ${\operatorname{corank}}(M)$. In the second stage, we calculate the distribution of ${\operatorname{corank}}(M)$. Removing some of the rows and columns of the laplacian will allow the upper triangular entries of $M$ to have positive min-entropy with respect to the other upper triangular matrix, which will allow us to use Corollary~\ref{cor:rectrank} to compute ${\operatorname{corank}}(M)$. \subsection{Reduction to $M$} Let $(L,R)$ be the vertices of our random bipartite graph $G$, and let $\Delta$ be the laplacian of $G$. Note that $\Delta$ is of the form $\bigl(\begin{smallmatrix} D_{0,1} & -A_{0} \\ -A_{0}^{T} & D_{0,2} \end{smallmatrix}\bigr)$, where $A_{0}$ is the adjacency matrix between $L$ and $R$ and $D_{0,1}$ and $D_{0,2}$ are diagonal matrices. Since we wish to work over ${\mathbb Z}/p{\mathbb Z}$, we will consider $\Delta\otimes {\mathbb Z}/p{\mathbb Z}=\Delta/p$. As we saw earlier, $\Delta/p$ has min-entropy $0$. We resolve this issue by using the submatrix $\Delta_{1}$, which has positive min-entropy. \begin{defn} Let $G$ be a bipartite graph with laplacian $\Delta$, $p$ prime. We define the matrix $\Delta_{1}=\Delta_{1}(G)=\Delta_{1}(n,\alpha,q,p)$ over ${\mathbb Z}/p{\mathbb Z}$ to be the submatrix of $\Delta/p$ given by removing the first $p$ rows, the first $p$ columns, the last $p$ rows, and the last $p$ columns. \end{defn} \begin{lemma} \label{lemma:properties} Let $G=G(n,\alpha,q)$ be as above, and let $\Delta_{1}=\Delta_{1}(G)$. Write $\Delta_{1}=\bigl(\begin{smallmatrix} D_{1,1} & -A_{1} \\ -A_{1}^{T} & D_{1,2} \end{smallmatrix}\bigr)$. $\Delta_{1}$ has the following properties: \begin{enumerate} \item The diagonal values of $D_{1,1}$ are independent of each other, as well as of entries of $A_{1}$ outside of their row. \item The diagonal values of $D_{1,2}$ are independent of each other, as well as of entries of $A_{1}$ outside of their column. \item There exists $\beta>0$ depending only on $p,q,$ and $\alpha$ such that every non-constant entry in or above the diagonal in $\Delta_{1}$ has min-entropy at least $\beta$ with respect to the set of the entries in or above the diagonal. \item For any $a\in{\mathbb Z}/p{\mathbb Z}$, and any diagonal entry $x$ in $D_{1,1}$ or $D_{1,2}$, ${\mathbb P}(x=a)= \frac{1}{p}+O(e^{-cn})$ for some constant $c$. \end{enumerate} \end{lemma} \begin{proof} We first show $(1)$. Note that the value of the diagonal entry $(D_{1,1})_{ii}$ depends only on the $i$\textsuperscript{th} row of $A_{0}$. Hence the $(D_{1,1})_{ii}$ are independent of each other and of any entry outside of the $i$\textsuperscript{th} row of $A_{0}$, which in particular includes the entries of $\Delta_{1}$ outside the $i$\textsuperscript{th} row. The proof of $(2)$ is similar. We will now prove $(3)$. Let $x$ be an entry in the upper triangle of $\Delta_{1}$. If $x\in D_{1,1}$, then as $x$ is non-constant, it must be on the diagonal. As we saw above, $x$ depends only on the values in the $i$\textsuperscript{th} row of $A_{0}$. Fix the rest of the entries of the $i$\textsuperscript{th} row of $\Delta_{1}$. There are still $2p$ entries of the $i$\textsuperscript{th} row of $A_{0}$ not in $\Delta_{1}$, which are left undetermined. For any choice of the first $2p-1$ of these, the last entry can be either $-1$ with probability $q$ or $0$ otherwise, which would change the value of $x$. Hence $x$ has min-entropy at least $\min(q,1-q)$ with respect to the rest of the upper triangular entries. The case where $x\in D_{1,2}$ is similar. Now, assume $x\in -A_{1}$. Fix all the other entries of $\Delta_{1}$. The only ones of which $x$ is not independent are those in the row and column of $x$. The row sum (in $A_{0}$) must be equal to the corresponding row entry, and the column sum must be equal to the corresponding column entry. There are $p$ unfixed entries in the row that are in $A_{0}$ but not in $\Delta_{1}$, and the sum of these entries can be equal to any value in ${\mathbb Z}/p{\mathbb Z}$ with probability at least $\min(q,1-q)^{p+1}$.The same goes for the column sum. In particular, the probability that both the row and the column sum allow $x$ to be zero is at least $\min(q,1-q)^{2(p+1)}$. Similarly, the probability that both allow $x=-1$ is at least $\min(q,1-q)^{2(p+1)}$. Hence $x$ has min-entropy at least $\min(q,1-q)^{2(p+1)}$ with respect to the rest of the upper triangular entries. Finally, we prove $(4)$. Let $a\in{\mathbb Z}/p{\mathbb Z}$. To see that each entry of $D_{1,1}$ is equal to $a$ with probability $\frac{1}{p}+O(e^{-cn})$, note that it is equal to $a$ when the sum of the corresponding row in $A_{0}$ is equal to $a$. Since this row has $\alpha n$ independent entries equal to $1$ with probability $q$ and zero otherwise, its sum is uniformly distributed in ${\mathbb Z}/p{\mathbb Z}$ up to an $O(e^{-cn})$ error term, where $c$ is a constant depending only on $q,\alpha$ and $p$. \end{proof} We will take $M$ to be equal to $\Delta_{1}(n+2p,\alpha,q,p)$, then adjust the probability space so that the diagonal values of $M$ are equidistributed in $Z/p{\mathbb Z}$. Since this changes only an exponentially small part of the probability space, ${\operatorname{corank}}(M)$ is usually within small distance of ${\operatorname{corank}}(\Delta_{1}(n+2p,\alpha,q,p))$. But \[\left|{\operatorname{corank}}(\Delta_{1}(n+2p,\alpha,q,p))-{\operatorname{corank}}(\Delta_{1}(n,\alpha,q,p))\right|<4p,\] so by transitivity ${\operatorname{corank}}(M)$ is usually within small distance of ${\operatorname{corank}}(\Delta_{1}(n,\alpha,q,p))$. Hence we have: \begin{prop} \label{prop:mtomain} The $p$-rank of $\Gamma(G)$ is usually within small distance of ${\operatorname{corank}}(M)$. \end{prop} We will also assume that $\lfloor\alpha n\rfloor=\alpha n$, so that $M$ is a $(1+\alpha)n\times (1+\alpha)n$ matrix. We will write: \[M=\begin{pmatrix} D_{1} & A\\ A^{T} & D_{2} \end{pmatrix}\] \subsection{Calculating the corank of $M$} In this section, we prove the following statement about $M$: \begin{prop} \label{prop:mrank} Let $M=M(n,\alpha,q,p)=\begin{pmatrix} D_{1} & A\\ A^{T} & D_{2} \end{pmatrix}$ be the matrix described above. Then ${\operatorname{corank}}(M)$ is usually within small distance of $\max\left(B\left(n,\frac{1}{p}\right)-\alpha n,0\right)$. \end{prop} \noindent Together with Proposition~\ref{prop:mtomain}, this implies Theorem~\ref{thm:strongMain}. Throughout the proof, we will use ${\operatorname{height}}(A)$ and ${\operatorname{width}}(A)$ to denote the number of rows and columns of $A$ respectively. If $A$ is a square matrix, we use $\dim(A)$ for both of these. \begin{proof} Let $r$ be the number of zero entries on the diagonal of $D_{1}$, that is, $r={\operatorname{corank}}(D_{1})$. Since the diagonal values of $D_{1}$ are independent and uniformly distributed, it is easy to see that $r=B\left(n,\frac{1}{p}\right)$. Hence, it suffices to show that ${\operatorname{corank}}(M)$ is usually within small distance of $\max\left(r-\alpha n,0\right)$. Our proof will rely on finding nonsingular submatrices of $M$, and taking the Schur complement with respect to them. This will allow us to reduce the problem of finding ${\operatorname{corank}}(M)$ to finding the coranks of matrices which are either nonsingular (in the case where $r-\alpha n< 0$), or have a large block of zeros which makes finding the corank straightforwards (in the case where $r-\alpha n\geq 0$). Assume that the first $n-r$ entries of $D_{1}$ are the nonzero entries, so that $D_{1}$ is of the form $\begin{pmatrix} D_{1}' & 0\\ 0 & 0 \end{pmatrix}$ Where $D_{1}'$ is invertible. Hence we can write \[M=\begin{pmatrix} D_{1}' &0 &B_{1} \\ 0 &0 &B_{2} \\ B_{1}^{T} &B_{2}^{T} &D_{2} \end{pmatrix} \] where $B_{1},B_{2}$ are random matrices of dimension $\alpha n\times (n-r)$ and $\alpha n\times r$ respectively. Taking the Schur Complement of $M$ with respect to $D_{1}'$, we get: \[M/D_{1}'=\begin{pmatrix} 0 & B_{2}\\ B_{2}^{T} & D_{2}-B_{1}^{T}D_{1}'^{-1}B_{1} \end{pmatrix}. \] We will now split into cases: \noindent{\bf The case $r \ge \alpha n.$} In this case, we want to show that ${\operatorname{corank}}(M)$ is usually within small distance of $r-\alpha n$. Now, \[{\operatorname{height}}(B_{2})=r\geq\alpha n={\operatorname{width}}(B_{2}).\] As ${\operatorname{rank}}(B_{2})={\operatorname{rank}}(B_{2}^{T})$, it is easy to see that \[{\operatorname{rank}}(M/D_{1}')\geq {\operatorname{rank}}(B_{2})+{\operatorname{rank}}(B_{2}^{T})=2{\operatorname{rank}}(B_{2}),\] and thus \[{\operatorname{corank}}(M)={\operatorname{corank}}(M/D_{1}')\leq \dim(M/D_{1}')-{\operatorname{rank}}(M/D_{1}')=(\alpha n+r)-(2{\operatorname{rank}}(B_{2})).\] Conversely, The corank of $M/D_{1}'$ is at least the corank of the submatrix of the top $n-r$ rows, given by $\begin{pmatrix} 0 & B_{2} \end{pmatrix}. $ The rank of this submatrix is equal to ${\operatorname{rank}}(B_{2})$, so the corank is $r-{\operatorname{rank}}(B_{2})$. Since $B_{2}$ has min-entropy at least $\beta$ for some positive constant $\beta$, by Corollary~\ref{cor:rectrank}, ${\operatorname{rank}}(B_{2})$ is usually within small distance of \[\min({\operatorname{height}}(B_{2}),{\operatorname{width}}(B_{2}))=\min(r,\alpha n)=\alpha n.\] Applying this to our lower and upper bounds for ${\operatorname{corank}}(M)$, we get that the upper bound is usually within small distance of $(\alpha n+r)-(2\alpha n)=r-\alpha n$. Similarly, our lower bound is usually within small distance of $r-\alpha n$. Hence by Lemma~\ref{lemma:properties}, ${\operatorname{rank}}(M)$ is usually within small distance of $r-\alpha n$. \noindent{\bf The case $r <\alpha n.$} In this case, we need to show that ${\operatorname{corank}}(M)={\operatorname{corank}}(M/D_{1}')$ is usually within small distance of zero. Write $C=D_{2}-B_{1}^{T}D_{1}'^{-1}B_{1}$ for the bottom-right $\alpha n\times \alpha n$ submatrix of $M/D_{1}'$. We will use the following claim: \begin{claim} Let $s$ be the size of the largest set of indices $J\subseteq \{1,\dots,\alpha n\}$ with the property that $C_{J}=(C_{ij|i,j\in J})$ is nonsingular. Then for any constant $\epsilon>0$, $s\geq \left(\alpha(1 -\frac{1}{p})-\epsilon\right) n$ with probability $1-O(e^{-cn})$ for some constant $c>0$. \end{claim} \begin{proof} To see this, we build up a set $J$ by going through the indices $i\in\{1,\dots,\alpha n\}$. For each $i$, we add $i$ to $J$ if $C_{J\cup\{i\}}$ is nonsingular. We will show that for each $i$, we add $i$ with probability at least $1-\frac{1}{p}-\delta$, where $\delta>0$ is arbitrarily small as $n$ grows. Since $J$ is the sum of $n$ Bernoulli random variables, each equal to $1$ with probability at least $1-\frac{1}{p}-\delta$ independently of the previous values, we can say that $|J|\geq B\left(\alpha n, 1-\frac{1}{p}-\delta\right)$. By Hoeffding's inequality, \[B\left(\alpha n, 1-\frac{1}{p}-\delta\right)>(1-\delta)\alpha\left(1-\frac{1}{p}-\delta\right)n> \alpha\left(1 -\frac{1}{p}-\epsilon\right) n\] with probability $1-O(e^{-cn})$ (the second inequality holds for all sufficiently small $\delta$). To see that each $i$ can be added with probability at least $1-\frac{1}{p}-\delta$, note that the diagonal entries of $D_{2}$ are the sums of entries in $B_{1}$ with entries of $B_{2}$, which are independent of them. There are $r=B\left(n,\frac{1}{p}\right)$ entries in each column of $B_{2}$, so by Hoeffding's inequality the number of entries in each column of $B_{2}$ is greater than $\frac{1}{2p}n$ with probability $1-O(e^{-cn})$. Hence we can assume that the entries of $D_{2}$ are exponentially close to being uniformly distributed in ${\mathbb Z}/p{\mathbb Z}$, given any condition on $B_{1},D_{1}'$, and the previous diagonal entries of $D_{2}$. This means that for any $x\in Z/p{\mathbb Z}$ and any conditions on the rest of the entries of $C$, ${\mathbb P}(C_{ii}=x)\leq \frac{1}{p}+\delta$, where $\delta$ can be exponentially small in $n$. Let $J$ be the set of indices we obtain from taking the above process on $\{1,\dots,i-1\}$. We need to show that we add $i$ to $J$ with probability at least $1-\delta-\frac{1}{p}$. We add $i$ to $J$ unless $C_{J\cup\{i\}}$ becomes singular. But this happens only if the last column of $C_{J\cup\{i\}}$ is dependent on the first $|J|$ columns. Write $C_{J\cup\{i\}}=\begin{pmatrix} u_{1}\cdots u_{|J|} & u_{i}\\ C_{ij_{0}}\dots C_{ij_{|J|}} & C_{ii} \end{pmatrix}$. Since $C_{J}$ is nonsingular, there exist unique coefficients $a_{j}\in {\mathbb Z}/p{\mathbb Z}$ such that $\sum a_{j}u_{j}=u_{i}$. But $C_{J\cup\{i\}}$ only if its columns are dependent, which happens only if $\sum a_{j}C_{ij}=C_{ii}$. From the above statement with $x=\sum a_{j}C_{ij}$, this happens with probability at most $1-\delta-\frac{1}{p}$. This completes the proof of the claim. \end{proof} Getting back to the proof, assume that $J$ is composed of the last $s$ indices of $C$. Then the claim implies that, with probability $1-O(e^{-cn})$, we can write \[ M/D_{1}' = \begin{pmatrix} 0 &B_{3} &B_{4} \\ B_{3}^{T} &C_{1} &C_{2} \\ B_{4}^{T} &C_{2}^{T} & C_{3} \end{pmatrix}, \] Where $C_3$ is nonsingular and $\dim(C_3)=s\geq\left(\alpha(1 -\frac{1}{p})-\epsilon\right) n$. Taking $\epsilon$ to be sufficiently small so that $\alpha\left( \frac{1}{p}+\epsilon\right)<\frac{1}{p}-\epsilon$, we can assume that with probability $1-O(e^{-cn})$, \[r>\left(\frac{1}{p}-\epsilon\right)n>\alpha \left( \frac{1}{p}+\epsilon\right) n>\alpha n - s.\] Since $C_{1}$ is $(\alpha n-s)\times (\alpha n-s)$ and ${\operatorname{height}}(B_{3})=r$, we can assume that ${\operatorname{width}}(B_{3})=\dim(C_{1})=\alpha n-s<r={\operatorname{height}}(B_{3})$. Note that we can drop rows and columns from $C_3$ if necessary, thus increasing the width of $B_{3}$, up to a maximum of ${\operatorname{width}}(B_{3})+{\operatorname{width}}(B_{4})=\alpha n> r$. In particular, we can assume that $s=\alpha n-r$, so that $B_{3}$ is an $r\times r$ square matrix. We now wish to shows that ${\operatorname{rank}}(M/D_{1}')$ is usually within small distance of zero. To do this, we will split the rows into three sets, and successively show that that most of the rows are independent: First, let $u_{1},\dots,u_{\alpha n-r}$ be the bottom $\alpha n-r$ rows (those with elements in $C_{3}$). Since they contain as subrows the rows of $C_{3}$ (which we know are independent), they are independent. Secondly, let $v_{1},\dots,v_{r}$ be the top $r$ rows. By Corollary~\ref{cor:rectrank}, ${\operatorname{corank}}(B_{3})$ is usually within small distance of zero. In fact, we can make a stronger claim: We claim that the corank of the $\alpha n\times\alpha n$ matrix $\begin{pmatrix} B_{3} & B_{4}\\ C_{2}^{T} & C_{3} \end{pmatrix}$ is usually within small distance of zero. Let $v_{i}'$ be the top $r$ rows of this matrix, and $u_{i}'$ be the bottom $\alpha n-r$ rows. As before, the $u_{i}'$ are independent since their tails are the rows of $C_{3}$. Now assume that the first $k$ of the $v_{i}'$ are independent both of each other and of the $u_{i}'$ (that is, the set $\{u_{1}',\dots,u_{\alpha n-r}',v_{1}',\dots,v_{k}'\}$ is independent. We claim that the probability that $v_{k+1}$ is dependent on $\{u_{1}',\dots,u_{\alpha n-r}',v_{1}',\dots,v_{k}'\}$ is at most $(1-\beta)^{r-k}$. To see this, first choose a set $J$ of $\alpha n-r+k$ indices so that $\{u_{1}',\dots,u_{\alpha n-r}',v_{1}',\dots,v_{k}'\}$ are still independent when restricted to the entries in $J$. If $v_{k+1}'$ is dependent on $\{u_{1}',\dots,u_{\alpha n-r}',v_{1}',\dots,v_{k}'\}$, then we can write $v_{k+1}'=\sum_{i\leq k} a_{i}v_{i}'+\sum b_{i}u_{i}'$, where the $a_{i}$ and the $b_{i}$ are determined by the entries in $J$. This leaves $n-r$ undetermined coefficients in $v_{k+1}$, all of which must be equal to the corresponding entry of $\sum_{i\leq k} a_{i}v_{i}'+\sum b_{i}u_{i}'$. But the entries of $v_{k+1}$ all have min-entropy at least $\beta$ with respect to the other vectors, so each of them is equal to the corresponding entry of $\sum_{i\leq k} a_{i}v_{i}'+\sum b_{i}u_{i}'$ with conditional probability at most $1-\beta$, hence the probability that all $r-k$ of them satisfy this equality is at most $(1-\beta)^{r-k}$. From here, we can conclude that the corank of the matrix is usually within small distance of zero by following the same reasoning as the proof of Theorem~\ref{thm:rectrank}. Finally, it remains to show that the middle $r$ rows of $M/D_{1}'$, labeled $w_{1},\dots,w_{r}$, cannot add much to the corank. That is, we need to find a set of independent rows of $M/D_{1}'$ whose size is usually within small distance of $r+\alpha n$. We will assume that the $u_{i}'$ and $v_{i}'$ are all independent (otherwise we only have to drop $k$ of them, where $k$ is usually within small distance of zero). We proceed in a similar manner to before. For the first $w_{1}$, we let $J$ be the set of the last $\alpha n$ indices. Since the $u_{i}'$ and $v_{i}'$ are all independent, there exists a unique set of indices $a_{i},b_{i}$ so that $w_{1}=\sum a_{i}u_{i}+\sum b_{i}v_{i}$ holds when restricted to the last $\alpha n$ indices. As the first $r$ entries of $w_{1}$ have min-entropy at least $\beta>0$ with respect to the rest of the matrix, they all match the corresponding entries of $\sum a_{i}u_{i}+\sum b_{i}v_{i}$ with probability at most $(1-\beta)^{r}$. We proceed similarly, showing that for each $w_{k+1}$ such that the set $\{u_{1},\dots,u_{\alpha n-r},v_{1},\dots,v_{r},w_{1},\dots,w_{k}\}$ is independent, the probability that $w_{k+1}$ is dependent on is is at most $(1-\beta)^{r-k}$. As before, this shows that the number of independent $w_{i}$ is usually within small distance of $r$. Putting this all together, we get a set of independent rows whose size is usually within small distance of the height of $M/D_{1}'$. The corank of $M/D_{1}'$ is at most the number of rows not in our set, which is usually within small distance of zero. This completes the proof. \end{proof} \section{Proofs of the Corollaries} \label{sec:details} In this section, we prove the corollaries of Theorem~\ref{thm:main}. We begin by proving Corollary~\ref{cor:balanced}: \begin{proof}[Proof of Corollary~\ref{cor:balanced}] Let $\epsilon>0$, and let $X=X(n,p)$ be the $p$-rank of $G$. We need to show that as $n\rightarrow\infty$, ${\mathbb E}(X)<\epsilon n$. Assume that $\epsilon<\frac{1}{2}$, and remove $\frac{\epsilon}{2} n$ vertices from the right side of the graph. By Theorem~\ref{thm:main}, the expected $p$-rank of the resulting graph is $O(1)$. Since removing a vertex changes the $p$-rank of the sandpile group by at most $1$, removing $\frac{\epsilon}{2} n$ vertices changes it by at most $\frac{\epsilon}{2} n$. Hence $X\leq \frac{\epsilon}{2}n+O(1)<\epsilon n$ for large $n$, which completes the proof. \end{proof} We now prove Corollary~\ref{cor:cyclic}. To do this, we show that the $2$-rank of $\Gamma(G)$ when $\alpha<\frac{1}{2}$ has low probability of being $\leq 1$, so the $2$-part of the group has low probability of being cyclic. \begin{proof}[Proof of Corollary~\ref{cor:cyclic}] Consider the $2-$rank of $\Gamma(G)$. As we saw in Theorem~\ref{thm:strongMain}, the $2$-rank of $\Gamma(G)$ is usually within small distance of $\max\left(B\left(n,\frac{1}{2}\right)-\alpha n,0\right)$. As $\alpha<\frac{1}{2}$, we have that by Hoeffding's inequality, \[B\left(n,\frac{1}{2}\right)>\left(\frac{1}{2}-\epsilon\right)n>\alpha n+\epsilon n\] holds with probability $1-O(e^{-cn})$ for all $\epsilon>0$, where the second inequality will hold when $\epsilon<\frac{1}{2}\left(\frac{1}{2}-\alpha\right)$. Hence $\max\left(B\left(n,\frac{1}{2}\right)-\alpha n,0\right)>\epsilon n$ with probability $1-O(e^{-cn})$. But the $2$-rank of $\Gamma(G)$ is usually within small distance of $\max\left(B\left(n,\frac{1}{2}\right)-\alpha n,0\right)$. Hence the $2$-rank of $\Gamma(G)$ is larger than $\frac{1}{2}\epsilon n$ with probability $1-O(e^{-cn})$ for some $c>0$, and in particular will be at least $2$ with probability $1-O(e^{-cn})$. But if the $2$-rank of $\Gamma(G)$ is at least $2$, $\Gamma(G)$ cannot be cyclic. Hence the probability that $\Gamma(G)$ is cyclic is bounded by $O(e^{-cn})$ for some constant $c>0$. \end{proof} \end{document}

\begin{document} \title[On smooth divisors of a projective hypersurface.]{On smooth divisors of a projective hypersurface.} \author{Ellia Ph.} \address{Dipartimento di Matematica, via Machiavelli 35, 44100 Ferrara (Italy)} \email{[email protected]} \author{Franco D.} \address{Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Univ. Napoli "Federico II", Ple Tecchio 80, 80125 Napoli (Italy)} \email{[email protected]} \date{16/06/2004} \maketitle \hskip7cm{\it Dedicated to Christian Peskine.} \section*{Introduction.} This paper deals with the existence of smooth divisors of a projective hypersurface $\Sigma \subset\mathbb{P}^n $ (projective space over an algebraically closed field of characteristic zero). According to a celebrated conjecture of Hartshorne, at least when $n\geq 7$, any such a variety should be a complete intersection. Since the existence of smooth, non complete intersection, subcanonical $X \subset \mathbb{P}^n$ of codimension two is equivalent, via the correspondance of Serre, to the existence of indecomposable rank two vector bundles on $\mathbb{P}^n$ and since no indecomposable vector bundle of $\mathbb{P}^n $, $n\geq 5$, is presently known, it is widely believed that any smooth, subcanonical subvariety of $\mathbb{P}^n $, $n\ge5$, of codimension two is a complete intersection. Furthermore recall that, by a theorem of Barth, the subcanonical condition is automatically satisfied if $n \geq 6$. This in turn implies that a smooth (subcanonical if $n=5$) divisor of a projective hypersurface $\Sigma \subset\mathbb{P}^n $, $n\geq 5$, is a complete intersection too. \par In this paper we show that, roughly speaking, for any $\Sigma \subset \mathbb{P}^n$ there can be at most $\textit{finitely many}$ exceptions to the last statement. Indeed our main result is: \begin{theorem} \label{mainthm} Let $\Sigma \subset \mathbb{P}^n$, $n \geq 5$ be an integral hypersurface of degree $s$. Let $X \subset \Sigma$ be a smooth variety with $dim(X)=n-2$. If $n=5$, assume $X$ subcanonical. If $X$ is not a complete intersection in $\mathbb{P}^n$, then: $$d(X) \leq \frac{s(s-1)[(s-1)^2-n+1]}{n-1}+1.$$ \end{theorem} In other words a smooth codimension two subvariety of $\mathbb{P}^n$, $n \geq 5$ (if $n=5$, we assume $X$ subcanonical) which is not a complete intersection cannot lie on a hypersurface of too low degree (too low with respect to its own degree) and, {\it on a fixed hypersurface}, Hartshorne's conjecture in codimension two is "asymptotically" true. \par The starting point is Severi-Lefschetz theorem which states that if $n \geq 4$ and if $X$ is a Cartier divisor on $\Sigma$, then $X$ is the complete intersection of $\Sigma$ with another hypersurface. For instance if $\Sigma $ is either smooth or singular in a finite set of points and if $n \geq 5$, the picture is very clear: \begin{enumerate} \item there $\textit{exists}$ smooth $X\subset \Sigma$ with $dim(X)=n-2$ and with degree arbitrarily large; \item $\textit{any}$ smooth $X\subset \Sigma$ with $dim(X)=n-2$ $\textit{is}$ a complete intersection of $\Sigma$ with another hypersurface \item $\textit{no}$ smooth $X\subset \Sigma$ with $dim(X)=n-2$ can meet the singular locus of $\Sigma$. \end{enumerate} \vskip1cm Using Theorem \ref{mainthm} we get (the first statement comes again from an easy application of the Theorem of Severi-Lefschetz-Grothendieck): \begin{theorem} \label{sigma} Let $\Sigma \subset \mathbb{P}^n$, $n\geq 5$, be an integral hypersurface of degree $s$ with \par $dimSing(\Sigma)\geq1$. \begin{enumerate} \item If $n\geq 6$ and $dimSing(\Sigma)\leq n-5$ then $\Sigma $ does not contain any smooth variety of dimension $n-2$. \item Suppose $dimSing(\Sigma)\geq n-4$. If $X\subset \Sigma $ is smooth, subcanonical, with $dim(X)=n-2$ then $d(X)\leq s\frac{(s-1)((s-1)^2-n+1)}{n-1}+1$. \end{enumerate} \end{theorem} We point out a consequence of this result. \begin{corollary} \label{sigmahilb} Let $\Sigma \subset \mathbb{P}^n$, $n\geq 5$, be an integral hypersurface s.t. $dimSing(\Sigma)\geq 1$. \begin{enumerate} \item If $n\geq 6$ and $dimSing(\Sigma)\leq n-5$ then $\Sigma $ does not contain any smooth variety of dimension $n-2$. \item Suppose $dimSing(\Sigma)\geq n-4$. Then there are only finitely many components of $\mathcal{H}ilb(\Sigma)$ containing smooth, subcanonical varieties of dimension $n-2$. \end{enumerate} \end{corollary} Last but not least, at the end of the paper we show how this circle of ideas allows to improve the main results of \cite{EF} about subcanonical varieties of $\mathbb{P}^5 $ and $\mathbb{P}^6$: \begin{theorem} \label{n=s=5} Let $X \subset \mathbb{P}^5$ be a smooth threefold with $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$. If $h^0(\ensuremath{\mathcal{I}} _X(5)) \neq 0$, then $X$ is a complete intersection. \end{theorem} \begin{theorem} \label{n=s=6} Let $X \subset \mathbb{P}^6$ be a smooth fourfold. If $h^0(\ensuremath{\mathcal{I}} _X(6)) \neq 0$, then $X$ is a complete intersection. \end{theorem} Theorem \ref{mainthm} follows, thanks to a crucial remark essentially proved in \cite{EP} (see Lemma \ref{l1}), from a bound of $e$ (where $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$), see Theorem \ref{thmSpec}, which can be viewed as a strong (since the degree is not involved) generalization of the "Speciality theorem" of Gruson-Peskine \cite{GP}. The proof of this bound is quite simple if $X \cap Sing(\Sigma )$ has the right dimension. This is done in the first section where a weaker version of Theorem \ref{thmSpec} and hence of Theorem \ref{mainthm} is proved (if $n=5$ we assume $Pic(X) \simeq \mathbb{Z} .H$). In the second section we show how a refinement of the proof yields our final result. Finally let's observe that our approach doesn't apply to the case $n=4$. \vskip1cm \textbf{Acknowledgment:} It is a pleasure to thank Enzo Di Gennaro who explained to one of us (D.F.) some of the deep results of \cite{K}. \section{Reduction and the speciality theorem, weak version.} \begin{notations} Given a projective scheme $Y \subset \mathbb{P}^n $ we denote by $d(Y)$ the \textit{degree} of $Y$. \end{notations} \begin{notations} \label{not2} In this section, $X \subset \mathbb{P}^n, n \geq 5$, will denote a smooth, non degenerate, codimension two subvariety which is not a complete intersection. We will always assume $X$ subcanonical: $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$; notice that this condition is fullfilled if $Pic(X) \simeq \mathbb{Z} .H$; finally, thanks to a theorem of Barth, this last condition is automatically fullfilled if $n \geq 6$.\par\noindent By Serre's construction we may associate to $X$ a rank two vector bundle: $$ 0 \to \ensuremath{\mathcal{O}} \to E \to \ensuremath{\mathcal{I}} _X(e+n+1) \to 0 $$ The Chern classes of $E$ are: $c_1(E)=e+n+1,c_2(E)=d(X)=:d$.\par\noindent Let $\Sigma$ be an hypersurface of degree $s $ containing $X$. Then $\Sigma $ gives a section of $\ensuremath{\mathcal{I}} _X(s)$ which lifts to a section $\sigma _{\Sigma}\in H^0(E(-e-n-1+s))$ (notice that $\sigma_{\Sigma }$ is uniquely defined if $e+n+1-s<0$). {\it Assume} that $Z$, the zero-locus of $\sigma_{\Sigma }$, has codimension two. Notice that since $X$ is not a complete intersection, this certainly holds if $s = min\{t\:|\:h^0\ensuremath{\mathcal{I}} _X(t)) \neq 0\}$. Anyway, if $Z$ has codimension two, then $d(Z)=c_2(E(-e-n-1+s))=d-s(e+n+1-s)$ and $\omega _Z \simeq \ensuremath{\mathcal{O}} _Z(-e-2n-2+2s)$. \end{notations} \begin{remark} \label{snb} By \cite{R}, if $X\subset \Sigma \subset \mathbb{P}^n $, $n\geq 3$, with $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$ and $d(\Sigma)\leq n-2$ then $X$ is complete intersection, hence in the remainder of this paper we will assume $s\geq n-1$. \end{remark} \begin{remark} \label{jac} Notice that $E(-e-n-1)\mid _X\simeq \ensuremath{\mathcal{N}} ^*_X$. It is well known that the scheme $X\cap Z$ is the base locus of the jacobian system of $\Sigma $ on $X$: $X\cap Z=X\cap Jac(\Sigma)$. So, the \textit{fundamental cycle} (\cite{F} 1.5) of $Z$ in $\mathcal{A}_*(X)$ is $c_2(\ensuremath{\mathcal{N}} ^*_X(s))$ as soon as $X$ and $Z$ intersect in the expected codimension. \end{remark} The main goal of this section is to prove: \begin{theorem}[Speciality theorem, weak version] \label{thmSpecW} Let $X \subset \mathbb{P}^n$, $n \geq 5$ be a smooth codimension two subvariety. If $n=5$ assume $Pic(X) \simeq \mathbb{Z} .H$. Let $\Sigma$ be an hypersurface of degree $s$ containing $X$. If $X$ is not a complete intersection, then: $$e \leq \frac{(s-1)[(s-1)^2-n+1]}{n-1}-n+1$$ where $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$. \end{theorem} Let's see how this is related with a bound of the degree. First recall the following: \begin{lemma} \label{l1} Let $X \subset \mathbb{P}^n$, $n \geq 4$, be a smooth codimension two subvariety which is not a complete intersection. Let $\Sigma$ be an hypersurface of minimal degree containing $X$. Set $s:=d(\Sigma )$. \begin{enumerate} \item $n-4 \leq dim(X \cap Sing(\Sigma )) \leq n-3$. \item If $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$, then $d(X) \leq s(n-1+e)+1$. \item If $dim(X \cap Sing(\Sigma ))=n-3$ and if $Pic(X) \simeq \mathbb{Z} .H$, then $d(X) \leq (s-2)(n-1+e)+1$. \end{enumerate} \end{lemma} \begin{proof} The first item is \cite{EF}, Lemma 2.1; 2) is \cite{EF} Lemma 2.2 (i) and the last item is \cite{EF} Lemma 2.2 (ii) with $l=2$ (thanks to Severi and Zak theorems $h^1(\ensuremath{\mathcal{I}} _X(1))=0$, \cite{Z}). \end{proof} Theorem \ref{thmSpecW} and the second item of this lemma give us immediately: \begin{theorem} \label{thmA} Let $\Sigma \subset \mathbb{P}^n$, $n \geq 5$, be an integral hypersurface of degree $s$. Let $X \subset \Sigma$ be a smooth subvariety with $dim(X)=n-2$. If $n=5$ assume $Pic(X) \simeq \mathbb{Z} .H$. If $X$ is not a complete intersection, then $d(X) < \frac{s(s-1)[(s-1)^2-n+1]}{n-1}+1$. \end{theorem} In order to prove Theorem \ref{thmSpecW} we need some preliminary results. \begin{lemma} \label{Ysubcanonical} Let $\Sigma$ denote an hypersurface of degree $s$ containing $X$. With assumptions ($codim(\sigma _{\Sigma})_0=2$) and notations as in \ref{not2}, assume $dim(X\cap Z)=n-4$. Then $Y:=X\cap Z$ is a subcanonical, l.c.i. scheme with $\omega_Y\simeq\ensuremath{\mathcal{O}}_Y(2s-n-1)$. Moreover $Y$ is the base locus of the jacobian system of $\Sigma$ in $X$. \end{lemma} \begin{proof} We are assuming that $Y$ is a proper intersection between $X$ and $Z$ hence $$ 0 \to \ensuremath{\mathcal{O}} \to E\mid_X(-e-n-1+s) \to \ensuremath{\mathcal{I}} _{Y, X}(-e-n-1+2s) \to 0 $$ so $\ensuremath{\mathcal{N}}^*_{Y,X}\simeq E\mid_X(-s)$ and the first statement follows by adjunction. For the last statement, use \ref{jac}. \end{proof} \begin{notations} Keep the assumptions of Lemma \ref{Ysubcanonical} and denote by $\Sigma_1$ and $\Sigma_2$ two general partials of $\Sigma$. Since $dim(X\cap Z)=n-4$, $C:= X\cap \Sigma_1\cap \Sigma_2$ is a subcanonical, l.c.i. scheme containing $Y$ such that $\ensuremath{\mathcal{N}}_{C,X}\simeq \ensuremath{\mathcal{O}} _X(s-1) \oplus \ensuremath{\mathcal{O}}_X(s-1)$. We have $\omega_C\simeq \ensuremath{\mathcal{O}}_C(e+2s-2)$. The scheme $C$ is a complete intersection in $X$ which links $Y$ to another subscheme.\\ \end{notations} \begin{lemma} \label{R} With notations as in Lemma \ref{Ysubcanonical}, denote by $R$ the residual to $Y$ with respect to $C$. Then $C=Y\cup R$ is a geometric linkage and $\Delta:= R\cap Y$ is a Cartier divisor of $Y$ such that: $\ensuremath{\mathcal{I}} _{\Delta , Y}\simeq \ensuremath{\mathcal{O}}_Y (-e-n+1)$.\\ Furthermore: $d(\Delta)\leq (s-1)d(X)((s-1)^2-d(Z))$ and:\\ $d(Z)(e+n+1) \leq (s-1)[(s-1)^2-d(Z)]$. \end{lemma} \begin{proof} Denote by $Y_{red}$ the support of $Y$ and set $Y_{red}=Y_1 \cup \dots \cup Y_r$ where $Y_i$, $1\leq i \leq r$, are the irreducible components of $Y_{red}$. Furthermore, denote by $P_i$ the general point of $Y_i$. Since $Y$ is l.c.i. in $X$ and since $\ensuremath{\mathcal{I}} _{Y, X}(s-1)$ is globally generated by the partials of $\Sigma$, we can find two general elements in $Jac(\Sigma)$ generating the fibers of $\ensuremath{\mathcal{N}}^* _{Y, X}(s-1)$ at each $P_i$, $1\leq i\leq r$. This implies that $R\cup Y$ is a geometric linkage. \par Now consider the local Noether sequence (exact sequence of liaison): $$ 0\to \ensuremath{\mathcal{I}}_C \to \ensuremath{\mathcal{I}} _R \to \omega_Y \otimes \omega_C^{-1}\to 0. $$ we get $$ \omega_Y \otimes \omega_C^{-1}\simeq\frac{\ensuremath{\mathcal{I}}_R}{\ensuremath{\mathcal{I}}_C}\simeq \frac{\ensuremath{\mathcal{I}}_R +\ensuremath{\mathcal{I}}_Y}{\ensuremath{\mathcal{I}}_C+\ensuremath{\mathcal{I}} _Y}\simeq \frac{\ensuremath{\mathcal{I}}_{\Delta}}{\ensuremath{\mathcal{I}}_Y}\simeq \ensuremath{\mathcal{I}} _{\Delta , Y}$$ (the second isomorphism follow by geometric linkage, since $\ensuremath{\mathcal{I}}_R\cap\ensuremath{\mathcal{I}}_Y =\ensuremath{\mathcal{I}}_C$) hence $\omega_Y \otimes \omega_C^{-1}\simeq \ensuremath{\mathcal{O}}_Y(-e-n+1)\simeq \ensuremath{\mathcal{I}} _{\Delta , Y}$ and we are done.\\ For the last statement, the scheme $\Delta \subset R $ is the base locus of the jacobian system of $\Sigma $ in $R$, hence $\Delta \subset \tilde{\Sigma}\cap R $ with $\tilde{\Sigma }$ a general element of $Jac(\Sigma)$ and $d(\Delta )\leq d(R)\cdot (s-1)$. We conclude since $d(R)\cdot (s-1)=(d(C)- d(Z))\cdot (s-1)= ((s-1)^2d(X)-d(Z)d(X))\cdot (s-1)$. The last inequality follows from $d(\Delta )=d(Y)\cdot (e+n+1)=d(X)\cdot d(Z) \cdot (e+n+1)$. \end{proof} Now we can conclude the proof of Theorem \ref{thmSpecW} (and hence of Theorem \ref{thmA}). \begin{proof}[Proof of Theorem \ref{thmSpecW}] It is enough to prove the theorem for $s$ minimal. Let $\Sigma$ be an hypersurface of minimal degree containing $X$, we set $s:=d(\Sigma )$ and $d:=d(X)$. According to Lemma \ref{l1} we distinguish two cases.\\ 1) $dim(X \cap Sing(\Sigma ))=n-3$. In this case, by Lemma \ref{l1}, we have $d \leq (s-2)(n-1+e)+1$. On the other hand $d(Z) = d-s(e+n+1-s)$ (see \ref{not2}). It follows that: $d(Z) \leq (s-1)^2 -2(n-1+e)$. Since $d(Z) \geq n-1$ by \cite{R}, we get: $\frac{(s-1)^2-n+1}{2}-n+1 \geq e$. One checks (using $s \geq n-1$) that this implies the bound of Theorem \ref{thmSpecW}.\\ 2) $dim(X \cap Sing(\Sigma ))=n-4$. By the last inequality of Lemma \ref{R}, $e \leq (s-1)[\frac{(s-1)^2}{d(Z)}-1]-n+1$. Since $d(Z) \geq n-1$ by \cite{R}, we get the result. \end{proof} \section{The speciality theorem.} In this section we will refine the proof of Theorem \ref{thmSpecW} for $n=5$ in order to prove Theorem \ref{mainthm} of the introduction. For this we have to assume only that $X$ is subcanonical, which, of course, is weaker than assuming $Pic(X) \simeq \mathbb{Z} .H$. The assumption $Pic(X) \simeq \mathbb{Z} . H$ is used just to apply the last statement of Lemma \ref{l1} in order to settle the case $dim(X \cap Sing(\Sigma ))=n-3$. Here instead we will argue like in the proof of the case $dim(X \cap Sing(\Sigma ))=n-4$, but working modulo the divisorial part (in $X$) of $X \cap Sing(\Sigma )$; this will introduce some technical complications, but conceptually, the proof runs as before. Since the proof works for every $n \geq 5$ we will state it in this generality giving thus an alternative proof of Theorem \ref{thmSpecW}. \begin{notations} \label{not3} In this section, with assumptions and notations as in \ref{not2}, we will assume furthermore that $dim(X\cap Z)=n-3$ and will denote by $L$ the dimension $n-3$ part of $X\cap Z\subset X$; moreover we set $\ensuremath{\mathcal{L}} = \ensuremath{\mathcal{O}} _X(L) $. \par\noindent Set $Y':=res_L(X\cap Z)$, we have $\ensuremath{\mathcal{I}} _{Y', X}:=(\ensuremath{\mathcal{I}} _{X\cap Z , X}:\ensuremath{\mathcal{I}} _{L , X})$. Since we have: $$0 \to \ensuremath{\mathcal{O}} \to E\mid_X(-e-n-1+s)\otimes \ensuremath{\mathcal{L}} ^* \to \ensuremath{\mathcal{I}} _{Y' , X}(-e-n-1+2s)\otimes (\ensuremath{\mathcal{L}} ^*)^2 \to 0$$ it follows that $\ensuremath{\mathcal{N}}^*_{Y',X}\simeq E\mid_X(-s)\otimes \ensuremath{\mathcal{L}}$ and $Y'$ is a l.c.i. scheme with $\omega_{Y'}\simeq\ensuremath{\mathcal{O}}_Y(2s-n-1)\otimes (\ensuremath{\mathcal{L}}^*)^2$. \par \noindent Denote by $\Sigma_1$ and $\Sigma_2$ two general partials of $\Sigma$. Since $X \cap Z = X \cap Sing(\Sigma )$, $\Sigma_1$ and $\Sigma_2$ both contain $L$. Let $C':=res_L(X\cap \Sigma_1\cap \Sigma_2)$. Since $\ensuremath{\mathcal{N}}_{C',X}\simeq (\ensuremath{\mathcal{O}} _{C'}(s-1) \oplus \ensuremath{\mathcal{O}}_{C'}(s-1))\otimes \ensuremath{\mathcal{L}}^*$. We have $\omega_{C'}\simeq \ensuremath{\mathcal{O}}_{C'}(e+2s-2)\otimes (\ensuremath{\mathcal{L}}^*)^2$. \end{notations} \begin{lemma} \label{R2} Denote by $R'$ the residual to $Y'$ with respect to $C'$. Then $C'=Y'\cup R'$ is a geometric linkage and $\Delta':= R'\cap Y'$ is a Cartier divisor of $Y'$ such that: $\ensuremath{\mathcal{I}} _{\Delta' , Y'}\simeq \ensuremath{\mathcal{O}}_{Y'}(-e-n+1)$. \end{lemma} \begin{proof} We argue as in the proof of Lemma \ref{R}: denote by $Y_{red}'$ the support of $Y'$, set $Y_{red}'=Y_1' \cup \dots \cup Y_r'$, where $Y_i'$, $1\leq i \leq r$, are the irreducible components of $Y_{red}'$, and denote by $P_i$ the general point of $Y_i'$. Choose the partials $\Sigma_1$ and $\Sigma _2$ in such a way that they generate the ideal sheaf of $X\cap Z$ at each $P_i$, $1\leq i\leq r$. In order to check that $R'\cup Y'$ is a geometric linkage we only need to consider the components contained in $L$. Consider a point $P_i\in L$. Since $L\subset X\cap Z \subset \Sigma_1 \cap \Sigma _2$, the local equations of $X\cap Z$ in $(\ensuremath{\mathcal{I}} _{Y, X}(s-1))_{P_i}$ have the form $(lf,lg)$ where $l$ is the equation of $L$, $lf$ is the equation of $\Sigma_1$ and $lg$ the equation of $\Sigma_2$. Since $Y':=res_L(X\cap Z)$ and $C':=res_L(X\cap \Sigma_1\cap \Sigma_2)$ then the ideals of both $Y'$ and $C'$ at $P_i$ are equal to $(f,g)\subset (\ensuremath{\mathcal{I}} _{Y, X}(s-1))_{P_i}$. This implies that $R'\cup Y'$ is a geometric linkage and the remainder of the proof is similar as above. \end{proof} \begin{lemma} \label{lemmaN-3} Let $\Sigma \subset \mathbb{P}^n$, $n\geq 5$, be an hypersurface of degree $s$ containing $X$, a smooth variety with $dim(X)=n-2$ and $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$. Assume $\sigma _{\Sigma}$ vanishes in codimension two and $dim(X \cap Sing(\Sigma ))=n-3$ (see \ref{not2}). Then $e < s-n$ or $d(Z)\cdot (e+n+1) \leq (s-1)[(s-1)^2-d(Z)]$. \end{lemma} \begin{proof} We keep back the notations of \ref{not3}. Notice that the fundamental cycle of $Y'$ in $\textbf{A} _{n-4}(X)$ is $$c_2(E\mid_X(-e-n-1+s)\otimes\ensuremath{\mathcal{L}}^*)=d(Z)H^2 + (e+n+1-2s)H\cap L +L^2\:\:(+)$$ ($H$ represents the hyperplane class and $\cap $ denotes the \textit{cap product} in $\textbf{A}_*(X)$. By abuse of notations, for any $A\in \textbf{A} _{i}(X)\subset \textbf{A}_*(X)$ we denote by $d(A)\in \mathbb{Z}$ the \textit{degree} of $A$: $d(A):= d(A\cap H^i)$, $A\cap H^i\in A_0(\mathbb{P}^n )\simeq \mathbb{Z}$. \item For any closed subscheme $\Gamma \subset X$ we still denote by $\Gamma \in \textbf{A} _{*}(X)$ the \textit{fundamental cycle} of $\Gamma $ (\cite{F} 1.5).\\ We claim that: $$d(\Delta')\leq (s-1)d(X)((s-1)^2-d(Z))-[(s-1)(e+n-1)+(s-1)^2-d(Z)]d(H^2\cap L)+$$ $$+(e+n-1)d(H\cap L^2)\:\:(*) $$ Assume the claim for a while and let's show how to conclude the proof. Combining \ref{R2} with $(*)$ we get $$ d(\Delta')=d(Y')(e+n-1)\leq $$ $$ \leq (s-1)d(X)((s-1)^2-d(Z))-[(s-1)(e+n-1)+(s-1)^2-d(Z)]d(H^2\cap L)+$$ $$+(e+n-1)d(H\cap L^2) $$ and by $(+)$ above $$ d(\Delta')=(e+n-1)d(H\cap(d(Z)H^2 + (e+n+1-2s)H\cap L +L^2))\leq $$ $$ \leq (s-1)d(X)((s-1)^2-d(Z))-[(s-1)(e+n-1)+(s-1)^2-d(Z)]d(H^2\cap L)+$$ $$+(e+n-1)d(H\cap L^2). $$ If $e<s-n$ we are done, so we can assume $e+n\geq s$. We have $$d(X)d(Z)(e+n-1)\leq (s-1)d(X)((s-1)^2-d(Z))+$$ $$+[(e+n-1)(s-e-n)-(s-1)^2+d(Z)]d(L)$$ To conclude it is enough to check that $(e+n-1)(s-e-n)-(s-1)^2+d(Z)\leq 0$. Since $d(Z)=d-s(e+n+1-s)$ (see \ref{not2}) and since $d \leq s(n-1+e)+1$ by Lemma \ref{l1}, this follows from: $s(n-1+e)+1 \leq s(e+n+1-s)+(s-1)^2+(e+n-s)(e+n-1)$. A short computation shows that this is equivalent to $0 \leq (e+n-s)(e+n-1)$, which holds thanks to our assumption $e+n\geq s$.\\ {\it Proof of the claim:}\\ Denote by $\mid M \mid $ the moving part of the Jacobian system of $\Sigma $ in $X$ and by $\ensuremath{\mathcal{M}} $ the corresponding line bundle. The scheme $\Delta '$ is the base locus of $\mid M \mid_{R'}$ hence $\Delta '\subset \tilde{M}\cap R'$ where $\tilde{M}$ is a general element of $\mid M \mid $. We have $$ d(\Delta ')\leq d(\tilde{M}\cap R')=d(c_1(\ensuremath{\mathcal{M}} _{R'})). $$ \par In order to prove the statement we need to calculate the cycle $c_1(\ensuremath{\mathcal{M}} _{R'})\in \textbf{A} _{n-5}(X)$. First of all we calculate the fundamental cycle of $R'$ in $\textbf{A} _{n-4}(X)$: $$R'\sim C'-Y'\sim ((s-1)H-L)^2-(d(Z)H^2 + (e+n+1-2s)H\cap L +L^2)= $$ $$ =((s-1)^2-d(Z))H^2-(e+n-1)H\cap L.$$ Finally, the cycle $c_1(\ensuremath{\mathcal{M}} _{R'})\in \textbf{A} _{n-5}(X)$ is: $$c_1(\ensuremath{\mathcal{M}} _{R'})\sim ((s-1)H-L)\cap R'\sim $$ $$ \sim (s-1)((s-1)^2-d(Z))H^3-((s-1)(e+n-1)+(s-1)^2-d(Z))H^2\cap L+(e+n-1)H\cap L^2.$$ The claim follows from: $$ d(\Delta')\leq d(c_1(\ensuremath{\mathcal{M}} _{R'}))= $$ $$ d((s-1)((s-1)^2-d(Z))H^3-((s-1)(e+n-1)+(s-1)^2-d(Z))H^2\cap L+(e+n-1)H\cap L^2) $$ \end{proof} Now we can state the improved version of Theorem \ref{thmSpecW}: \begin{theorem}[Speciality theorem] \label{thmSpec} Let $X \subset \mathbb{P}^n$, $n \geq 5$, be a smooth variety with $dim(X)=n-2$ and $\omega _X \simeq \ensuremath{\mathcal{O}} _X(e)$. Let $\Sigma \subset \mathbb{P}^n$ denote an hypersurface of degree $s$ containing $X$. If $X$ is not a complete intersection, then: $$e \leq \frac{(s-1)[(s-1)^2-n+1]}{n-1}-n+1.$$ \end{theorem} \begin{proof} It is sufficient to prove the theorem for $s$ minimal. We distinguish two cases (see Lemma \ref{l1}).\\ If $dim(X \cap Sing(\Sigma ))=n-4$, then we argue exactly as in the proof of Theorem \ref{thmSpecW}.\\ If $dim(X \cap Sing(\Sigma ))=n-3$, then by Lemma \ref{lemmaN-3} we have $e < s-n$ or $d(Z)\cdot (e+n+1) \leq (s-1)[(s-1)^2-d(Z)]$. In the first case we conclude using $s \geq n-1$ (Remark \ref{snb}) and, in the second case, we conclude using the fact that $d(Z) \geq n-1$ by \cite{R}. \end{proof} \begin{proof}[Proof of Theorem \ref{mainthm}] As explained in the Section 1, it follows from Theorem \ref{thmSpec} and Lemma \ref{l1}. \end{proof} \section{Proofs of \ref{sigma} and of \ref{sigmahilb}.} \begin{proof}[Proof of Theorem \ref{sigma}] If $X$ is not a complete intersection, this follows from Theorem \ref{mainthm}. Assume $X$ is a complete intersection. Let $F$ and $G$ ($d(F)=f,d(G)=g$) be two generators of the ideal of $X$. Then the equation of $\Sigma$ has the form $PF+QG$. But since $\Sigma$ is irreducible and since $X \cap Sing(\Sigma )\neq \emptyset$, then both $P$ and $Q$ have degree $>0$. This implies $s-1 \geq f$ and $s-1 \geq g$ hence $d=fg \leq (s-1)^2 < s\frac{(s-1)((s-1)^2-n+1}{n-1}+1$. \end{proof} \begin{proof}[Proof of Corollary \ref{sigmahilb}] The argument goes as in the proof of \cite{CDG} Lemma 4.3: by \cite{K} the coefficients of the Hilbert polynomial of $X$ can be bounded in terms of the degree $d$ hence in terms of $s$, by \ref{sigma}, and there are finitely many components of $\mathcal{H}ilb(\Sigma)$ containing smooth varieties of dimension $n-2$. \end{proof} \section{Proof of \ref{n=s=5} and \ref{n=s=6}} \begin{notations} By \cite{EF}, we may assume that $X$ lies on an irreducible hypersurface $\Sigma$ of degree $n$, $5 \leq n \leq 6$ and that $h^0(\ensuremath{\mathcal{I}} _X(n-1))=0$. The assumption of \ref{not2} is satisfied and by Lemma \ref{R} and Lemma \ref{lemmaN-3}, we get: $e <s-n$ or $d(Z)\cdot (e+n-1) \leq (s-1)[(s-1)^2-d(Z)]$. The first case cannot occur in our situation since we may assume $e \geq 3$ if $n=5$ by \cite{BC} (resp. $e \geq 8$ if $n=6$ by \cite{HS} Cor. 6.2). So we may assume $d(Z)\cdot (e+n+1) \leq (s-1)[(s-1)^2-d(Z)]\: (*)$. Now if $e \geq E$, from $(*)$ we get: $d(Z) \leq \frac{(s-1)^3}{E+n+s}\:(+)$. \end{notations} \begin{proof}[Proof of Theorem \ref{n=s=5}] Applying $(+)$ with $n=s=5$ and $E=3$ we get $d(Z) \leq 4$, hence $d(Z)=4$ (\cite{R}). Arguing as in \cite{EF} Lemma 2.6, every irreducible component of $Z_{red}$ appears with multiplicity, so $Z$ is either a multiplicity four structure on a linear space or a double structure on a quadric. In both cases it is a complete intersection: in the first case this follows from \cite{Mano} and in the second one, from the fact that $Z$ is given by the Ferrand construction since $emdim(Z_{red})\leq 4$. \end{proof} \begin{proof}[Proof of Theorem \ref{n=s=6}] Applying $(+)$ with $n=s=6$ and $E=8$, we get $d(Z) \leq 6$. If $d(Z)=6$, $(*)$ implies $e \leq 8$. So $e=8$ and $6=d(Z)=d-6e-6$. It follows that $d=60$ and we conclude with \cite{EF} Theorem 1.1. So $d(Z) \leq 5$, hence (\cite{R}), $d(Z)=5$. Now $(*)$ yields $e \leq 13$. Moreover $5=d(Z)=d-6e-6$ yields $d=6e+11$. If $e \leq 10$, again, we conclude with Theorem 1.1 of \cite{EF}. We are left with the following possibilities: $(d,e)=(77,11),(83,12),(89,13)$. We conclude with \cite{HS} (list on page 216). \end{proof} \end{document}

\begin{document} \title{Weighted Branching Simulation Distance for Parametric Weighted Kripke Structures} \begin{abstract} This paper concerns branching simulation for weighted Kripke structures with parametric weights. Concretely, we consider a weighted extension of branching simulation where a single transition can be matched by a sequence of transitions while preserving the branching behavior. We relax this notion to allow for a small degree of deviation in the matching of weights, inducing a directed distance on states. The distance between two states can be used directly to relate properties of the states within a sub-fragment of weighted CTL. The problem of relating systems thus changes to minimizing the distance which, in the general parametric case, corresponds to finding suitable parameter valuations such that one system can approximately simulate another. Although the distance considers a potentially infinite set of transition sequences we demonstrate that there exists an upper bound on the length of relevant sequences, thereby establishing the computability of the distance. \end{abstract} \section{Introduction} In recent years within the area of embedded and distributed systems, a significant effort has been made to develop various formalisms for modeling and specification that address non-functional properties. Examples include extensions of classical Timed Automata \cite{DBLP:conf/icalp/AlurD90} with cost and resource consumption/production in Priced Timed Automata \cite{DBLP:conf/hybrid/BehrmannFHLPRV01} and Energy Automata \cite{DBLP:conf/formats/BouyerFLMS08}. For quantitative analysis of these systems, a generalization of bisimulation equivalence by Milner \cite{DBLP:books/daglib/0067019} and Park \cite{Park1981} as behavioral distances \cite{DBLP:journals/jlp/ThraneFL10,DBLP:journals/tcs/LarsenFT11,DBLP:journals/tse/AlfaroFS09} between system, has been studied. In parallel, \emph{parametric} extensions of various formalism have been intensively studied. Instead of requiring exact specification of e.g probabilities, cost or timing constraints, these formalisms allow for the use of \emph{parameters} representing unknown or unspecified values. This can be used to encode multiple configurations of the same system as a system being parametric in the configurable quantities. The problem is then to find ``good'' parameter values such that the instantiated system (configuration) performs as expected. For real-time systems, Parametric Timed Automata \cite{DBLP:conf/stoc/AlurHV93,DBLP:journals/ijfcs/AndreCFE09} and Parametric Stateful Timed CSP \cite{DBLP:journals/rts/Andre00D14} have been developed. Parametric probabilistic models \cite{DBLP:conf/cav/HahnHWZ10, DBLP:conf/nfm/HahnHZ11} have also been developed as well as parametric analysis for weighted Kripke structures \cite{christoffersen_et_al:OASIcs:2015:5611, DBLP:conf/lics/EmersonT99,DBLP:journals/fuin/KnapikP14}. \cite{DBLP:conf/lics/EmersonT99} provides an efficient model-checking algorithm for a parametric extension of real-time CTL on timed Kripke structures. \cite{DBLP:journals/fuin/KnapikP14} extends \cite{DBLP:conf/lics/EmersonT99} to full parameter synthesis by demonstrating that model-checking a finite subset of the entire set of parameter values is sufficient. In this paper we revisit (parametric) weighted Kripke structures with the purpose of lifting the behavioral distance defined in \cite{WCTL_logic} to the parametric setting, demonstrate its fixed point characterization and prove computability of the distance between any two systems. The distance is a generalization of a weighted extension of branching simulation \cite{branching_bisim}. Consider the following two processes $s,t$ both ending in the inactive process 0: \[ s \to_5 0 \text{ and } t \to_3 t_1 \to_2 0 \] If $s,t,t_1$ satisfy the same atomic proposition, $t_1$ may be deemed unobservable and $t$ may simulate $s$ as they both evolve into the process 0 with the same overall weight. \cite{WCTL_logic} captures this situation in generality by extending branching simulation with weights. Consider a similar scenario, where the process $t$ is now parametrized by the parameter $p$: \[ s \to_5 0 \text{ and } t \to_p t_1 \to_2 0 \] If $p \neq 3$ we know that $t$ can no longer simulate $s$. However, it should be intuitive that $p = 6$ is somehow worse than $p = 2$ as the latter is closer to 3. Thus, instead of considering pre-orders and Boolean answers we develop a parametric distance between states such that as the value of $p$ approaches $3$, the distance between $s$ and $t$ decreases towards 0. The distance will also give us a direct relation between the properties satisfied by $s$ and $t$ and a distance of 0 implies that any formula satisfied by $s$ is satisfied by $t$. In this way one can reason about how ``close'' a given implementation is to the specification and compare different configurations that are not necessarily able to fully simulate $s$. The structure of this paper is as follows: in \autoref{sec:prelim} we introduce preliminaries and recall results from \cite{WCTL_logic}, \autoref{sec:WKS_sim_dist} concerns the fixed point characterization of the distance for weighted systems, \autoref{sec:PWKS_dist} lifts the distance to the parametric setting and finally \autoref{sec:conc_future} concludes the paper and describes future work. \section{Preliminaries}\label{sec:prelim} A weighted Kripke Structure (WKS) extends the classical Kripke structure by associating to each transition a non-negative rational transition weight. \begin{definition}[Weighted Kripke Structure] A weighted Kripke Structure is a tuple $\mcal{K}=(S,AP,\mcal{L},\to)$ where $S$ is a finite set of states, $AP$ is a set of atomic propositions, $\mcal{L}: S \to \mathcal{P}(AP)$ is a labelling function, associating to each state a set of atomic propositions and $\to \subseteq S \times \Q_{\geq 0} \times S$ is the finite transition relation. \end{definition} A transition from $s$ to $s'$ with weight $w$ will be denoted by $s \to_w s'$ instead of $(s,w,s') \in \to$. \begin{example} \autoref{fig:WKSex} depicts the WKS $\mcal{K}=(S,AP,\mcal{L},\to)$ where $S = \{s,s_1,s_2,s_3,s_4,t,t_1,t_2\}$, $AP = \{a,b\}$, $\mcal{L}(s) = \mcal{L}(s_1) = \mcal{L}(s_2) = \mcal{L}(t) = \mcal{L}(t_2) = \{a\}$, $\mcal{L}(s_3) = \mcal{L}(s_4) = \mcal{L}(t_1) = \{b\}$ and\\ $\to = \{(s,1,s_1),(s,2,s_2),(s_1,2,s_2),(s_1,1,s_3),(s_1,3,s_4),(s_2,5,s_4),(t,2,t_1),(t,1,t_2),(t_2,2,t_2),(t_2,1,t_1)\}$. \begin{figure} \caption{WKS $\mathcal{K}$ where $s \not\leq t$ and $t \not\leq s$ but $s \simeps[0.5] t$.} \label{fig:WKSex} \end{figure} \end{example} To reason about behavior of WKSs, we introduce a weighted variant of the classical notion of branching simulation \cite{branching_bisim}. The basic idea is to let a transition $s \to_5 s'$ be matched by a sequence of transitions $t \to_{2} t_1 \to_{2} t_2 \to_{1} t_3$, if $t_3$ can simulate $s'$, as the accumulated weight equals 5. In addition, each intermediate state passed through in the matching transition sequence must be able to simulate $s$. In this way the branching structure of systems is preserved. Instead of always requiring exact weight matching we allow small relative deviations. These small deviations will in \autoref{sec:WKS_sim_dist} induce a directed distance between WKS states. \begin{definition}[Weighted Branching $\varepsilon$-Simulation \cite{WCTL_logic}] Given a WKS $\mcal{K}=(S,AP,\mcal{L},\to)$ and an $\varepsilon \in \R_{\geq 0}$, a binary relation $R^{\varepsilon} \subseteq S \times S$ is a weighted branching $\varepsilon$-simulation relation if whenever $(s,t) \in R^{\varepsilon}$: \begin{itemize} \item $\mcal{L}(s) = \mcal{L}(t)$ \item for all $s \to_w s'$ there exists $t \to_{v_1} t_1 \to_{v_2} \cdots \to_{v_k} t_k$ such that $\sum_{i=1}^k v_i \in [w(1-\varepsilon), w(1+\varepsilon)], (s',t_k) \in R^{\varepsilon}$ and $\forall i<k. (s,t_i) \in R^{\varepsilon}$. \end{itemize} \end{definition} If there exists a weighted branching $\varepsilon$-simulation relating $s$ to $t$ we write $s \simeps t$. If $\varepsilon = 0$ we write $s \leq t$ instead of $s \simeps[0] t$. Note that in this case $\sum_{i=1}^k v_i = w$. \begin{example}\label{ex:sim} Consider again \autoref{fig:WKSex} and the pair $(s,t)$. It is clear that $t \not\leq s$ because of the loop $t_2 \to_2 t_2$. We can also observe that $s \not\leq t$ as the transition $s \to_2 s_2$ can only be matched by $t \to_2 t_1$ but $s_2 \not\leq t_1$ as $\mcal{L}(s_2) \neq \mcal{L}(t_1)$. If we relax the matching requirements by 50\%, we get that $s$ can be simulated by $t$ i.e $s \simeps[0.5] t$; $s \to_2 s_2$ can be matched by $t \to_1 t_2$ as $[2(1-0.5),2(1+0.5)] = [1,3]$ and $1 \in [1,3]$ (another legal match would be $t \to_1 t_2 \to_2 t_2$). Now, $s_2 \to_5 s_4$ can be matched exactly by $t_2 \to_2 t_2 \to_2 t_2 \to_1 t_1$. It follows that $\varepsilon \geq 0.5 \iff s \simeps[\varepsilon] t$. \end{example} If we restrict weighted CTL to only encompass the existential quantifier and remove the next-operator and we know that $s \simeps t$, then for any property $\phi$ of $s$, there exists a related property $\phi^{\varepsilon}$ of $t$. \begin{definition}[Existential Fragment of Weighted CTL without next] The syntax of $EWCTL_{-X}$ is given by the following abstract syntax: \[ \phi ::= a \mid \neg a \mid \phi_1 \land \phi_2 \mid \phi_1 \lor \phi_2 \mid E(\phi_1U_I\phi_2), \] where $a \in AP$, $I =[l,u]$ and $l,u \in \Q_{\geq 0}$ such that $l \leq u$. For a WKS $\mcal{K}=(S,AP,\mcal{L},\to)$ and an arbitrary state $s \in S$, the semantics of $EWCTCL_{-X}$ formulae is given by a satisfiability relation, inductively defined on the structure of formulae in $EWCTL_{-X}$. For existential until; $\mathcal{K},s \models E(\phi_1 U_I \phi_2) \iff$ there exists a sequence $s \to_{w_1} s_1 \to_{w_2} \cdots \to_{w_k} s_k \to_{w_{k+1}} \ldots$ where $s_k \models \phi_2, \forall i < k. s_i \models \phi_1$ and $\sum_{i=1}^k w_i \in I$. Let the \emph{$\varepsilon$-expansion} of a formula $\phi = E(\phi_1U_{[l,u]}\phi_2)$ be given by $\phi^{\varepsilon} = E(\phi_1^{\varepsilon}U_{[l(1-\varepsilon),u(1+\varepsilon)]}\phi_2^{\varepsilon})$ where $\phi_1^{\varepsilon}$ and $\phi_2^{\varepsilon}$ are defined inductively by relaxing any interval by $\varepsilon$ percent in both directions (just as for $[l,u]$). \end{definition} \begin{theorem}\label{thm:sim_logic_relation} \cite{WCTL_logic} Let $\mcal{K}=(S,AP,\mcal{L},\to)$ be a WKS. Then for all $s,t \in S, \varepsilon \in \R_{\geq 0}$: \[ s \simeps t \quad \text{iff} \quad \forall \varepsilon' \in \Q_{\geq 0}, \varepsilon \leq \varepsilon'. [\forall \phi \in EWCTL_{-X}. s \models \phi \implies t \models \phi^{\varepsilon'}]. \] \end{theorem} \section{Weighted Branching Simulation Distance for WKSs}\label{sec:WKS_sim_dist} We now define a directed distance between WKS states as a least fixed point to a set of equations. The distance from $s$ to $t$, $d(s,t)$, represents the minimal $\varepsilon$ such that $s \simeps t$. Thus, if $d(s,t) = 0$ then $s \leq t$. As the distance is based upon weighted branching $\varepsilon$-similarity and its relative deviation in weight matching, it will not satisfy the triangle inequality and is therefore not a hemi-metric. The distance definition follows intuitively weighted branching $\varepsilon$-simulation. If $s \simeps t$ then no matter what transition $s$ chooses, $t$ has a matching transition sequence with a relative difference of at most $\varepsilon$. In order words, for a given transition $s \to_w s'$, the goal of $t$ is to find a matching sequence $t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n$ that \emph{minimizes} the relative difference $\left|\frac{\sum_{i=1}^n v_i}{w}-1\right|$ as well as ensuring that any intermediate state $t_i$ has as small a distance to $s$ as possible. The strategy of $s$ is then to find a \emph{maximal} move, given the minimization strategy of $t$. In the remainder of this section we assume a fixed WKS $\mcal{K}=(S,AP,\mcal{L},\to)$. \begin{definition}[Weighted Branching Simulation Distance] For an arbitrary pair of states $s,t \in S$ we define the weighted branching simulation distance from $s$ to $t$, $d(s,t)$, as the least fixed point ($\minfix$) of the following set of equations: \[ d(s,t) \minfix \left\{\begin{array}{ll} \infty & \text{ if } \mcal{L}(s) \neq \mcal{L}(t)\\ \max_{s \rightarrow_w s'} \left\{\min_{t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,d(s',t_n),\\ \max \{d(s,t_i) | \,i < n\} \end{array} \right\} \right\}\right\} &\text{ o.w} \end{array}\right. \] \end{definition} We assume the empty transition sequence to have accumulated weight 0 and let $\R_{\geq 0} = \{w\,|\, w \in \mathbb{R}, w \geq 0\} \cup \{\infty\}$ denote the extended set of non-negative reals. For any $d_1,d_2 \in \Rpos^{S \times S}$ let $d_1 \leq d_2$ iff $\forall (s,t) \in S \times S. d_1(s,t) \leq d_2(s,t)$. Then $(\Rpos^{S \times S}, \leq)$ constitutes a complete lattice. We now define a monotone function on $(\Rpos^{S \times S},\leq)$ that iteratively refines the distance: \begin{definition}\label{def:f} Let $\mcal{F} : \Rpos^{S \times S} \to \Rpos^{S \times S}$ be defined for any $d \in \Rpos^{S \times S}$: \[ \mcal{F}(d)(s,t) = \left\{\begin{array}{ll} \infty & \text{ if } \mcal{L}(s) \neq \mcal{L}(t)\\ \max_{s \rightarrow_w s'} \left\{\min_{t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,d(s',t_n),\\ \max \{d(s,t_i) | \,i < n\} \end{array} \right\} \right\}\right\} & \text{ o.w} \end{array}\right. \] \end{definition} By Tarski's fixed point theorem \cite{tarski} we are guaranteed the existence of a least (pre-)fixed point. Thus, the weighted branching simulation distance is well-defined. Note that any transition $s \to_w s'$, $t$ may have an infinite set of possible transition sequence matches in the presence of cycles in the system. To this end we demonstrate an upper bound, $N$, on the length of relevant matching sequences. As the set of sequences of length at most $N$ is finite (the WKS is finite) computability of the distance follows. The first step is proving that any sequence exercising a loop with accumulated weight 0 can be ignored. We refer to these cycles as \emph{0-cycles}. \begin{lemma}\label{lem:zerocycle} For a given move $s \to_w s'$, any transition sequence $t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n$ with a 0-cycle can be removed without affecting the distance $d(s,t)$. \end{lemma} \begin{proof} A transition sequence with one or more 0-cycles has the exact same accumulating weight as the corresponding sequence with no 0-cycles. Furthermore, exercising the loop (once) can only introduce new states, leading to a potentially larger value of $\max \{d(s,t_i) | \,i < n\}$. Thus, 0-cycles can be ignored. \end{proof} Given that 0-cycles can be removed, we now prove an upper bound $N$ on the length of sequences that affect the distance $d(s,t)$. Thus, any sequence longer than $N$ can be safely ignored. \begin{lemma}\label{lem:finsequence} Given that $\mathcal{K}$ has no 0-cycles, it is the case that whenever $s \to_w s'$: \begin{align*} \exists N. &\forall \pi = t\rightarrow_{v_1}t_1 \ldots \rightarrow_{v_n} t_n, n \geq N.\\ &\exists \pi^* = t\rightarrow_{u_1} t_1' \ldots \rightarrow_{u_m} t_m', m \leq N.\\ &t_n = t_m' \,\land\, \left|\frac{\sum_{i=1}^m u_i}{w}-1\right| \leq \left|\frac{\sum_{i=1}^n v_i}{w}-1\right| \land\\ & \{t_1', \ldots, t_{m-1}'\} \subseteq \{t_1,\ldots,t_{n-1}\} \end{align*} \end{lemma} \begin{proof} Let $w_{\min} = \min\{w\,|\,s \to_w s'\}$ be the minimum weight in the WKS and let $s_{w_{\max}} = \max\{w\,|\,s \to_w s'\}$ be the maximum weight out of $s$. We now demonstrate that $N \geq \frac{2 \cdot {s_{w_{\max}}}}{w_{\min}} \cdot |S|$ is sufficient. Any sequence of length $|S|$ must have a loop which, by assumption, cannot have accumulated weight 0. Thus, after $|S|$ transitions, the accumulated weight must be at least $w_{\min}$. Without loss of generality, assume that it is \emph{exactly} $w_{\min}$. If the sequence exercises the loop a number of time, the accumulated weight will at some point reach $2 \cdot s_{w_{\max}}$. Let this sequence be $\pi = t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_k} t_k$ and let $x$ denote the number of times the loop is exercised i.e $x \cdot w_{\min} \geq 2 \cdot s_{w_{\max}}$. Consider now the corresponding sequence $\pi^* = t\rightarrow_{u_1} t_1' \cdots \rightarrow_{u_l} t_l'$ where the loop is removed. As $\sum_{i=1}^k v_i \geq 2 \cdot s_{w_{\max}}$ it follows that $\left|\frac{\sum_{i=1}^k v_i}{s_{w_{\max}}}-1\right| > 1$. By assumption, removing the loop results in a strictly lower accumulated weight implying $\left|\frac{\sum_{i=1}^l u_i}{s_{w_{\max}}}-1\right| < \left|\frac{\sum_{i=1}^k v_i}{s_{w_{\max}}}-1\right|$. We also directly have $t_k = t_l'$ and $\{t_1,\ldots,t_l'\} \subseteq \{t_1,\ldots,t_k\}$. We will now derive $N$ from the inequality $x \cdot w_{\min} \geq 2 \cdot s_{w_{\max}}$. The number of times the loops is exercised must be equal to the length of the entire sequence divided by $|S|$ as we are sure to exercise the loop every $|S|$ states. Thus, $x = \frac{N}{|S|} \implies \frac{N}{|S|} \cdot w_{\min} \geq 2 \cdot s_{w_{\max}}$ and finally, \[ N \geq \frac{2 \cdot s_{w_{\max}}}{w_{\min}} \cdot |S|. \] \end{proof} \begin{theorem}[Computability]\label{thm:wdistcomputable} For two states $s,t \in S$, the weighted branching simulation distance is computable. \end{theorem} \begin{proof} \autoref{lem:finsequence} provides an upper bound on the length of transition sequence that we need to consider in the computation of $d(s,t)$ for any states $s,t \in S$ under the assumption that there are no 0-cycles. By \autoref{lem:zerocycle} we know that any 0-cycles can be removed without affecting the distance. Thus when computing the distance we know for the sub-expression \[ \min_{t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,d(s',t_n),\\ \max \{d(s,t_i) | \,i < n\} \end{array} \right\} \right\} \] that $n \leq \frac{2 \cdot s_{w_{\max}}}{w_{\min}} \cdot |S|$. As the WKS has a finite number of states and a finite transition relation, only a finite number of sequences of finite length exist. Thus we can modify the distance function to only consider these without affecting the computed distance. Thus, the distance must at some point converge as only a finite number of relative distances on the form $\left|\frac{\sum_{i=1}^n v_i}{w}-1\right|$ exists. \end{proof} We leave the exact complexity of computing $d(s,t)$ open but note that deciding $d(s,t) = 0$ is NP-complete \cite{WCTL_logic}. \begin{example} Consider again \autoref{fig:WKSex} and the computation of $d(s,t)$. For the transition $s \to_1 s_1$ only one sequence is considered instead of the entire infinite set arising from the loop; $t \to_1 t_2$. As $\left|\frac{3}{1}-1\right| > \left|\frac{1}{1}-1\right|$, even the sequence that only exercises the loop once is worse than just transitioning to $t_2$ directly. This happens because the accumulated matching weight exceeds the weight being matched and the same states are involved in both sequences. Therefore any sequence involving the loop can be ignored. Note that we in this example consider fewer sequences than implied by the upper bound given in \autoref{lem:finsequence}. For $s \to_1 s_1$ the bound would be $\frac{2 \cdot 2}{2} \cdot 8 = 16$ but it should be clear that the loop can be safely ignored. For the transition $s \to_2 s_2$, there are two relevant matching sequences; $t \to_1 t_2$ and $t \to_1 t_2 \to_2 t_2$. Thus, \[ d(s,t) \minfix \max\left\{ \begin{array}{l} \max\left\{\left|\frac{1}{1}-1\right|,d(s_1,t_2)\right\},\\[0.1cm] \min\left\{\begin{array}{l} \max\left\{\left|\frac{1}{2}-1\right|,d(s_2,t_2)\right\},\\[0.1cm] \max\left\{\left|\frac{3}{2}-1\right|,d(s_2,t_2),d(s,t_2)\right\} \end{array}\right\} \end{array}\right\} \] It is easily shown that $d(s_2,t_2) = 0$ as $s_2 \to_5 s_4$ can be matched exactly by $t_2 \to_2 t_2 \to_2 t_2 \to_1 t_1$. Thus, \[ d(s,t) \minfix \max\left\{\frac{1}{2}, d(s_1,t_2),d(s,t_2)\right\} \] where \begin{align*} &d(s_1, t_2) \minfix \max\left\{\!\!\!\!\!\begin{array}{ll} \max\left\{\left|\frac{2}{2}-1\right|,d(s_2,t_2)\right\},\\[0.1cm] \max\left\{\left|\frac{1}{1}-1\right|,d(s_3,t_1)\right\},\\[0.1cm] \max\left\{\left|\frac{3}{3}-1\right|,d(s_1,t_2),d(s_4,t_1)\right\} \end{array}\!\!\!\!\!\!\right\} \,\text{ and}\! &d(s,t_2) \minfix \max\left\{\!\!\!\!\begin{array}{ll} \max\left\{\left|\frac{2}{1}-1\right|, d(s_2,t_2)\right\},\\[0.1cm] \max\left\{\left|\frac{2}{2}-1\right|, d(s_2,t_2)\right\} \end{array}\!\!\!\!\!\right\}. \end{align*} As $s_4 \not\to$, $s_3 \not\to$ and $t_1 \not\to$ it follows that $d(s_4,t_1) = d(s_3,t_1) = 0$, hence \[ d(s_1,t_2) \minfix \max\left\{\frac{1}{2},d(s_1,t_2)\right\}. \] The least solution to this equation is $\frac{1}{2}$ hence $d(s_1,t_2) = d(s,t) = \frac{1}{2}$. From \autoref{ex:sim} we know that $s \simeps t$ for any $\varepsilon \geq 0.5$ i.e for any $\varepsilon \geq d(s,t)$. \end{example} Now that we have established the computability of the distance we prove its relation to weighted branching $\varepsilon$-simulation. \begin{theorem}\label{lem:sim_dist_relation} For two states $s,t \in S$ and $\varepsilon \in \R_{\geq 0}$: \[ d(s,t) \leq \varepsilon \text{ iff } s \simeps t \] \end{theorem} \begin{proof} $(\implies)$ For this direction we prove that $R^{\varepsilon} = \{(s,t) \,|\, s,t \in S, d(s,t) \leq \varepsilon\}$ is a weighted branching $\varepsilon$-simulation relation. Suppose $(s,t) \in R^{\varepsilon}$. Then $d(s,t) \leq \varepsilon$ and by the fixed point property of $d$, \[ d(s,t) = \max_{s \rightarrow_w s'} \left\{\min_{t\rightarrow_{v_1}t_0 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,\\ \max \{d(s',t_n)\} \cup \{d(s,t_i) | i < n\} \end{array} \right\} \right\}\right\} \] We immediately have that for any transition $s \to_w s'$ there exists a matching transitions sequence $t\rightarrow_{v_1}t_0 \cdots \rightarrow_{v_n} t_n$ such that $\left|\frac{\sum_{i=1}^n v_i}{w}-1\right| \leq \varepsilon$, $d(s',t_n) \leq \varepsilon$ and $\forall i < n. d(s,t_i) \leq \varepsilon$. Thus, by definition of $R^{\varepsilon}$, for any transition $s \to_w s'$ there exists a sufficient matching sequence from $t$ such that $(s',t_n) \in R^{\varepsilon}$ and $(s,t_i) \in R^{\varepsilon}$ for any $i < n$. $(\impliedby)$ Let \[ d^*(s,t) = \left\{\begin{array}{ll} \varepsilon &\text{ if } s \simeps t\\ \infty &\text{ otherwise} \end{array}\right. \] We now prove that $d$ is a pre-fixed point of $\mcal{F}$ i.e $\mcal{F}(d^*)(s,t) \leq d^*(s,t)$ for any pair $(s,t) \in S$. If $s \not\simeps t$ then $d^*(s,t) = \infty$ and there is nothing to prove. If $s \simeps t$ then for any transition $s \to_w s'$ there exists a matching sequence $t\rightarrow_{v_1}t_0 \cdots \rightarrow_{v_n} t_n$ such that $\sum_{i=1}^n v_i \in [w(1-\varepsilon), w(1+\varepsilon)]$, $s' \simeps t_n$ and $s \simeps t_i$ for any $i < n$. We can now argue that \[ \max_{s \rightarrow_w s'} \left\{\min_{t\rightarrow_{v_1}t_0 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,\\ \max \{d^*(s',t_n)\} \cup \{d^*(s,t_i) | i < n\} \end{array} \right\} \right\}\right\} \leq \varepsilon \] as $\sum_{i=1}^n v_i \in [w(1-\varepsilon), w(1+\varepsilon)]$ is equivalent to $\left|\frac{\sum_{i=1}^n v_i}{w}-1\right| \leq \varepsilon$, $s' \simeps t_n$ implies $d^*(s',t_n) = \varepsilon$ and similarly $d^*(s,t_i) = \varepsilon$ for any $i < n$. As $d^*$ is a pre-fixed point of $\mcal{F}$ and $d^*(s,t) = \varepsilon$ it must be the case that $d(s,t) \leq \varepsilon$ as $d$ is the \emph{smallest} pre-fixed point of $\mcal{F}$. \end{proof} Combining \autoref{thm:sim_logic_relation} and \autoref{lem:sim_dist_relation} we immediate get a relation between the distance from one state $s$ to another state $t$ and their $EWCTL_{-X}$ properties: \[ d(s,t) \leq \varepsilon \quad \text{iff} \quad \forall \varepsilon' \in \Q_{\geq 0}, \varepsilon \leq \varepsilon'. [\forall \phi \in EWCTL_{-X}. s \models \phi \implies t \models \phi^{\varepsilon'}. \] \section{Weighted Branching Simulation Distances for Parametric WKSs}\label{sec:PWKS_dist} We now extend WKS with parametric weights. The lifted parametric distance will be from a WKS to a parametric system and is represented as a parametric expression that can be evaluated to a rational by a \emph{parameter valuation}. If one abstracts multiple configurations of the same system as one parametric system and calculate the parametric distance, evaluating the distance with respect to a parameter valuation then corresponds to calculating the exact distance from a specific configuration (given by the valuation) to the WKS. Thus, instead of working with multiple WKS configurations, one can use a parametric system and compute the parametric distance once. A parametric weighted Kripke structure (PWKS) extends WKS by allowing transitions to have parametric weights. Let $\mcal{P} = \{p_1,\ldots,p_n\}$ be a fixed finite set of parameters. A \emph{parameter valuation} is a function mapping each parameter to a non-negative rational; $v : \mcal{P} \to \Q_{\geq 0}$. The set of all such valuation will be denote by $\mcal{V}$. \begin{definition}[Parametric Weighted Kripke Structure] A \emph{parametric weighted Kripke structure} is a tuple $\mcal{K_\mcal{P}}=(S,AP,\mcal{L},\to)$, where $S$ is a finite set of states, $AP$ is a set of atomic propositions, $\mcal{L}: S\to \mathcal{P}(AP)$ is a mapping from states to sets of atomic propositions and $\to \subseteq S \times \mcal{P} \cup \Q_{\geq 0} \times S$ the finite transition relation. \end{definition} Unless otherwise specified, we assume a fixed PWKS $\mcal{K_\mcal{P}}=(S,AP,\mcal{L},\to)$ in the remainder of this section. One can instantiate a PWKS to a WKS by applying a parameter valuation. A PWKS thus represents an infinite set of WKSs. \begin{definition} Given a parameter valuation $v \in \mcal{V}$, we define the \emph{instantiated WKS} of $\mathcal{K}_\mcal{P}$ under $v$ to be $\mcal{K}_{\mcal{P}}^{v}=(S,AP,\mcal{L},\to_v)$ where \[ \to_v = \{(s,v(p),s')\mid (s,p,s')\in \to, p \in \mcal{P}\} \cup \{(s,w,s')\mid (s,w,s')\in \to, w \in \Q_{\geq 0}\} \] \end{definition} For a state $s$ in $\mathcal{K}_\mcal{P}$ let $s[v]$ be the corresponding state in the WKS $\mathcal{K}_\mcal{P}^v$ and let $\simeps$ be lifted to disjoint unions of WKSs in the natural way. Given a WKS state $s$, a PWKS state $t$ and $\varepsilon \geq 0$ we now state three interesting problems: \begin{enumerate} \item Does there exist a $v \in \mcal{V}$ such that $s \simeps t[v]$? \item Can we characterize the set of ``good'' parameter valuation $V = \{v \,|\, v \in \mcal{V}, s \simeps t[v]\}$? \item Can we synthesize a valuation $v \in \mcal{V}$ that minimizes $\varepsilon$ for $s \simeps t[v]$? \end{enumerate} We will show how to solve (2) by fixed point computations. The result will be a set of linear inequalities over parameters and $\varepsilon$ which has as solution a set of parameter valuations. Instead of considering a concrete $\varepsilon \in \R_{\geq 0}$, one can let $\varepsilon$ be an extra parameter. Thus, (1) and (3) can be solved by first solving (2) and applying e.g $Z3$ \cite{DBLP:conf/tacas/MouraB08} and $\nu Z$ \cite{DBLP:conf/tacas/BjornerPF15} or similar tools to solve the inequalities and search for solutions that minimize $\varepsilon$. \begin{example} Consider \autoref{fig:PWKSex}. From \autoref{ex:sim} we know that $s \leq_{0.5} t[v]$ if $v(p) = 1$. Both $v(p) = 0$ and $v(p) = 2$ imply $s \leq_{1} t[v]$. It turns out that $v(p) = 1$ is the valuation that minimizes $\varepsilon$ for $s \simeps t[v]$. \begin{figure} \caption{A WKS (left) and a PWKS (right)} \label{fig:PWKSex} \end{figure} \end{example} When lifting the distance to the parametric setting, we consider disjoint unions of systems and require that only the simulating system can be parametric. Let $\mathcal{K_\mcal{P}}=(S_\mcal{P},AP,\mcal{L}_\mcal{P},\to^\mcal{P})$ be a PWKS and $\mcal{K}=(S,AP,\mcal{L},\to)$ a WKS. If we were to validate a given parameter valuation we could simply apply the valuation to the PWKS and use $\mcal{F}$ directly to decide if the distance is below some $\varepsilon$. As we want a full characterization of the good parameter valuation we will instead represent the distance as a function from a pair of states to a function that returns a weighted distance when a parameter valuation is applied; $d: S \times S_\mcal{P} \to (\mcal{V} \to \R_{\geq 0})$. We let the set of such function be denoted by $\mcal{D}$ and define an ordering as follows; for any $d^1,d^2 \in \mcal{D}$ let $d^1 \leq d^2$ iff $\forall s \in S, t \in S_\mcal{P}, v \in \mcal{V}: d^1(s,t)(v) \leq d^2(s,t)(v)$. Let $\equiv$ denote the set of pairs of semantic equivalent states. Then $(\mcal{D},\leq)$ constitutes a complete lattice and we can define a monotone function on $(\mcal{D}, \leq)$ that iteratively refines the distance: \begin{definition}\label{def:fpar} Let $\mcal{F} : \mcal{D} \to \mcal{D}$ be defined for any $d \in \mcal{D}$: \[ \mcal{F}(d)(s,t) = \left\{\begin{array}{ll} \infty & \text{ if } \mcal{L}(s) \neq \mcal{L}(t)\\ \max_{s \rightarrow_w s'} \left\{\min_{t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n} \left\{\max\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,d(s',t_n),\\ \max \{d(s,t_i) | \,i < n\} \end{array} \right\} \right\}\right\} & \text{ o.w} \end{array}\right. \] \end{definition} Again, by Tarski's fixed point theorem \cite{tarski} we are guaranteed a least fixed point, denoted by $d_{\min}$. The problem is now that the ordering $\leq$ implies a universal quantification over the entire infinite set of parameter valuations; thus, checking if a fixed point is reached is highly impractical. Instead of representing the distance as a function in valuations we will define it as a \emph{parametric expression} that captures the distance function syntactically. For any two states $s,t$ we associate an syntactic expression $E_{s,t}$ such that the solution set to the inequality $E_{s,t} \leq \varepsilon$ characterizes the set of good parameter valuations i.e applying a parameter valuation to $E_{s,t}$ yields a concrete weighted distance. The syntactic elements for the expressions can be derived directly from $\mcal{F}$; we need syntax for describing minimums of maximums of basic elements $\left|\frac{v}{w}-1\right|$ and $\infty$ where $w$ is rational and $v$ a linear expression in the parameters. Hence, we define the following abstract syntax: \[ E_1,E_2 ::= \infty \mid \left|\frac{v}{w}-1\right| \mid \texttt{MIN}\{E_1,E_2\} \mid \texttt{MAX}\{E_1,E_2\} \] where $w \in \Q_{\geq 0}$ and $v$ is on the form $\sum^{n}_{i=0} a_i p_i + b$ s.t $a_i \in \mathbb{N}$ for all $i < n$ and $b \in \Q_{\geq 0}$. We extend parameter valuations to expressions in the obvious way and denote by $\llbracket E \rrbracket (v)$ the value of $E$ under $v \in \mcal{V}$. Similar to disjunctive normal form for logical formulae, we assume all expression to be a $\texttt{MIN}$ of $\texttt{MAX}$'s of basic elements $\left|\frac{v}{w}-1\right|$or $\infty$. To convert an expression, note that for any $v \in \mcal{V}$ \[ \llbracket \texttt{MAX}\{\texttt{MIN}\{E_1,E_2\},E_3\} \rrbracket (v) = \llbracket \texttt{MIN}\{\texttt{MAX}\{E_1,E_3\},\texttt{MAX}\{E_2,E_3\}\} \rrbracket (v) \] The set of expression on this normal form will be denoted by $\mcal{E}$. Now the distance functions can be defined as functions associating to a pair of states a parametric expression; $d_\mcal{E}: S \times S_\mcal{P} \to \mcal{E}$. The set of syntactic distance function will be denoted by $\dfuncparsem_{\mcal{E}}$ and the syntactic iterator capturing $d_{\min}$ is defined as follows: \begin{definition}\label{def:fparsyn} Let $\ffunc_{\expr} : \dfuncparsem_{\mcal{E}} \to \dfuncparsem_{\mcal{E}}$ be defined for any $d_\mcal{E} \in \dfuncparsem_{\mcal{E}}$: \[ \ffunc_{\expr}(d_\mcal{E})(s,t) = \left\{\begin{array}{ll} \infty & \text{ if } \mcal{L}(s) \neq \mcal{L}(t)\\ \texttt{MAX}_{s \rightarrow_w s'} \left\{\texttt{MIN}_{t\rightarrow_{v_1}t_1 \cdots \rightarrow_{v_n} t_n} \left\{\texttt{MAX}\left\{\begin{array}{l} \left|\frac{\sum_{i=1}^n v_i}{w}-1\right|,d_\mcal{E}(s',t_n),\\ \texttt{MAX} \{d_\mcal{E}(s,t_i) | \,i < n\} \end{array} \right\} \right\}\right\} & \text{ o.w} \end{array}\right. \] \end{definition} We will now define an ordering on elements from $\dfuncparsem_{\mcal{E}}$, by first ordering elements from $\mcal{E}$. \begin{definition}\label{def:exporder} The syntactic ordering $\distsynorder \subseteq \mcal{E} \times \mcal{E}$ is defined inductively on the structure of $\mcal{E}$: \begin{alignat*}{2} \left|\frac{\sum_{i=1}^n a_ip_i + b}{w}-1\right| \distsynorder \infty \quad\quad &\text{always}\\ \left|\frac{\sum_{i=1}^n a_ip_i + b}{w}-1\right| \distsynorder \left|\frac{\sum_{i=1}^n a_i'p_i + b'}{w}-1\right| \quad & \text{iff} \quad && \left\{\begin{array}{lc} \forall i.a_i \leq a_i' \land b \leq b' &\text{ if } \frac{b'}{w},\frac{b}{w} \geq 1\\ \forall i.a_i=a_i' \land b=b' \quad &\text{otherwise} \end{array}\right.\\ \texttt{MAX}\{E_{1.1},\ldots,E_{1.n}\} \distsynorder \texttt{MAX}\{E_{2.1},\ldots,E_{2.m}\} \quad & \text{iff} \quad && \forall i. \exists j. E_{1.i} \distsynorder E_{2.j}\\ \texttt{MIN}\{E_{1.1},\ldots,E_{1.n}\} \distsynorder \texttt{MIN}\{E_{2.1},\ldots,E_{2.m}\} \quad & \text{iff} \quad && \forall j. \exists i. E_{1.i} \distsynorder E_{2.j} \end{alignat*} \end{definition} Let $\equiv_\mcal{E}$ be the set of pairs of syntactically equivalent expressions. We now extend the ordering to distance functions: \begin{definition} The \emph{syntactic} ordering on distance functions $\sqsubseteq_{\mcal{E}}$ is defined for any $d_{\mcal{E}}^1, d_{\mcal{E}}^2 \in \dfuncparsem_{\mcal{E}}$: \[ d_{\mcal{E}}^1 \sqsubseteq_{\mcal{E}} d_{\mcal{E}}^2 \quad \text{ iff } \quad \forall s,t \in S. d_{\mcal{E}}^1(s,t) \distsynorder d_{\mcal{E}}^2(s,t). \] \end{definition} As the syntactic expression computed by $\ffunc_{\expr}$ for any pair of states $(s,t)$ is merely syntactically representing the functions computed by $\mcal{F}$ for the same pair of states, the two concepts are closely related. For any expression $d_\mcal{E} \in \dfuncparsem_{\mcal{E}}$ let $d \in \mcal{D}$ be the associated semantic function. Then it is the case that the syntactic ordering of expressions implies the same semantic ordering of the associated semantic functions. Furthermore, iteratively updating the distances as parametric expressions by $\ffunc_{\expr}$ is semantically equivalent to computing the distances as functions by $\mcal{F}$. \begin{lemma}\label{lem:synsemrelation} For any $d_\mcal{E}^1,d_\mcal{E}^2 \in \dfuncparsem_{\mcal{E}}$ and $n \in \mathbb{N}$: \begin{enumerate} \item $d_\mcal{E}^1 \sqsubseteq_{\mcal{E}} d_\mcal{E}^2 \implies d^1 \leq d^2$. \item $\llbracket \ffunc_{\expr}^n(d_\mcal{E}^1)(s,t) \rrbracket (v) = \mcal{F}^n(d^1)(s,t)(v)$. \end{enumerate} \end{lemma} We will now demonstrate an upper bound on the relevant matching transition sequences for the syntactic computations in $\ffunc_{\expr}$, given that all loops have at least one strictly positive non-parametric weight. This is similar to assuming no 0-cycles in the weighted case (\autoref{lem:finsequence}). \begin{lemma}\label{lem:finsequence_par} Let $\mcal{K}=(S,AP,\mcal{L},\to)$ be a WKS with state $s \in S$ such that $s \to_w s'$ and let $\mathcal{K_\mcal{P}}=(S_\mcal{P},AP,\mcal{L}_\mcal{P},\to^\mcal{P})$ be a PWKS with the following property: \begin{itemize} \item There exists a $w_{\min} > 0$ such that for any valuation, the accumulated weight of every loop in $\mathcal{K}_\mcal{P}$ is at least $w_{\min}$ (strongly cost non-zeno). \end{itemize} Then for any $t \in S_\mcal{P}$: \begin{align*} \exists N. &\forall \pi = t\rightarrow_{v_1}^\mcal{P} t_1 \ldots \rightarrow_{v_n}^\mcal{P} t_n, n \geq N.\\ &\exists \pi^* = t\rightarrow_{u_1}^\mcal{P} t_1' \ldots \rightarrow_{u_m}^\mcal{P} t_m', m \leq N.\\ &t_n = t_m' \,\land\, \left|\frac{\sum_{i=1}^m u_i}{w}-1\right| \distsynorder \left|\frac{\sum_{i=1}^n v_i}{w}-1\right| \land\\ & \{t_1', \ldots, t_{m-1}'\} \subseteq \{t_1,\ldots,t_{n-1}\} \end{align*} \end{lemma} \begin{proof} Let the maximum weight out of $s$ be $s_{w_{\max}}$. Any sequence of length $|S_\mcal{P}|$ must have a loop which, by assumption, cannot have accumulated weight 0 w.r.t any parameter valuation. Thus, the accumulated weight w.r.t any valuation is at least $w_{\min}$. Without loss of generality we assume it to be exactly $w_{\min}$. Exercising the loop a number of times will at some point result in the accumulated weight being greater than $2 \cdot s_{w_{\max}}$ w.r.t any valuation. Let this sequence be $\pi^* = t\rightarrow_{v_1}^\mcal{P} t_1 \cdots \rightarrow_{v_k}^\mcal{P} t_k$ and let $x$ denote the number of times the loop is exercised i.e $x \cdot w_{\min} \geq 2 \cdot s_{w_{\max}}$. Let $\sum_{i=1}^k v_i= \sum_{i=1}^n a_ip_i+b$. Then it is clear that $\frac{b}{s_{w_{\max}}} > 1$. Now consider the corresponding non-looping sequence $\pi_1= t\rightarrow_{u_1}^\mcal{P} t_1' \cdots \rightarrow_{u_l}^\mcal{P} t_l'$ and let $\sum_{i=1}^l u_i= \sum_{i=1}^n a_i'p_i+b'$. We would like it to be the case that \[ \left|\frac{\sum_{i=1}^n a'_ip_i + b'}{w}-1\right| \distsynorder \left|\frac{\sum_{i=1}^n a_ip_i + b}{w}-1\right| \] but it might be the case that $\frac{b'}{s_{w_{\max}}} < 1$. Consider a third sequence $\pi = t\rightarrow_{x_1}^\mcal{P} t_1'' \cdots \rightarrow_{x_m}^\mcal{P} t_m''$, being $\pi^*$ modified to exercise the loop one more time and let $\sum_{i=1}^m x_i = \sum_{i=1}^n a_i''p_i+b''$. Now we know that $\frac{b''}{s_{w_{\max}}} > 1$ as $b'' > b'$ and furthermore $\left|\frac{\sum_{i=1}^n a_i'p_i + b'}{w}-1\right| \distsynorder \left|\frac{\sum_{i=1}^n a_i''p_i + b''}{w}-1\right|, t_k = t_m''$ and $\{t_1',\ldots,t_k\} \subseteq \{t_1'',\ldots,t_m''\}$. We can now derive $N$. For $\pi^*$ we have the inequality $x \cdot w_{\min} \geq 2 \cdot s_{w_{\max}}$ and by \autoref{lem:finsequence} this leads to the bound $\frac{2 \cdot s_{w_{\max}}}{w_{\min}} \cdot |S_\mcal{P}|$. As $\pi$ is at most $|S_\mcal{P}|$ longer than $\pi^*$ we get \[ N \geq \frac{2 \cdot s_{w_{\max}}}{w_{\min}} \cdot |S_\mcal{P}| + |S_\mcal{P}| \] \end{proof} Note that the bound also holds for the semantic function $\mcal{F}$ as the syntactic ordering implies the semantic ordering (\autoref{lem:synsemrelation}). We can now limit $\ffunc_{\expr}$ to only consider sequences of length $N$, assuming that the PWKS is strongly cost non-zeno. We apply this fact to prove that we will after a finite number of iterations of $\ffunc_{\expr}$ have discovered two syntactically equivalent expressions. As syntactic equivalence implies semantic equivalence of the associated functions, we get by \autoref{lem:synsemrelation} that $d_{\min}$ can be computed by repeated application of both $\mcal{F}$ and $\ffunc_{\expr}$ is a finite number of steps. \begin{lemma}\label{lem:synequiv} There exists $n < m$ such that $\ffunc_{\expr}^n(d_{\mcal{E}}^0) \equiv_\expr \ffunc_{\expr}^m(d_{\mcal{E}}^0)$. \end{lemma} \begin{proof} Let \[ \ffunc_{\expr}^n(d_{\mcal{E}}^0)(s,t) = \texttt{MIN}\left\{\texttt{MAX}\left\{E_{1.1},\ldots,E_{1.k}\right\},\ldots,\texttt{MAX}\left\{E_{m.1},\ldots,E_{m.n}\right\}\right\}. \] From the definition of $\equiv_\expr$ we directly get $\texttt{MAX}$ and $\texttt{MIN}$ expressions behave like sets. Duplicates can be ignored i.e $\texttt{MAX}\{E_1,E_2,E_2\} \equiv_\expr \texttt{MAX}\{E_1,E_2\}$, $\texttt{MIN}\{\texttt{MAX}\{E_1,E_2\},\texttt{MAX}\{E_1,E_2\}\} \equiv_\expr \texttt{MIN}\{\texttt{MAX}\{E_1,E_2\}\}$ and the ordering of elements does not matter; $\texttt{MAX}\left\{E_1,E_2\right\} \equiv_\expr \texttt{MAX}\left\{E_2,E_1\right\}$. By \autoref{lem:finsequence_par} we can limit the transition sequences to length $N$. This implies that only a finite number of basic elements $\left|\frac{v}{w}-1\right|$ exist when iteratively applying $\ffunc_{\expr}$. As one can only construct a finite number of unique sets from a finite set of elements, the number of syntactically unique expressions (w.r.t $\equiv_\expr$) is finite. Therefore, there must exist a $m > n$ such that $\ffunc_{\expr}^n(d_{\mcal{E}}^0) \equiv_\expr \ffunc_{\expr}^m(d_{\mcal{E}}^0)$. \end{proof} We can now demonstrate computability of the distance. \begin{theorem}[Computability] There exists a natural number $n$ such that for all states $s \in S, t \in S_\mcal{P}$ and all valuations $v \in \mcal{V}$ \[ \llbracket \mcal{F}^n(d_{\mcal{E}}^0)(s,t) \rrbracket (v) = d_{\min}(s,t)(v). \] \end{theorem} \begin{proof} By \autoref{lem:synequiv}, there exists $n < m$ such that $\ffunc_{\expr}^n(d_{\mcal{E}}^0) \equiv_\expr \ffunc_{\expr}^m(d_{\mcal{E}}^0)$. By \autoref{lem:synsemrelation} we thus get semantic equivalence $\mcal{F}^n(d^0) \equiv \mcal{F}^m(d^0)$ and as $\mcal{F}$ is monotonic on $(\mcal{D}, \leq)$ we have for all $i$ s.t $n \leq i \leq m$ that $\mcal{F}^i(d^0) \equiv \mcal{F}^m(d^0)$. Thus, $\mcal{F}^n(d^0)$ is a fixed point found after a finite number of steps and is captured syntactically by $\ffunc_{\expr}^n(d_{\mcal{E}}^0)$. The check for equivalence ($\equiv_\expr$) can therefore be used to capture a semantic fixed point syntactically. The fixed point must also be the least fixed point. To see this, suppose towards a contradiction that it is not the least fixed point. Then there exists a $k < n$ such that $\mcal{F}^k(d^0) = d_{\min}$ but by the fixed point property of $d_{\min}$ and the monotonicity of $\mcal{F}$ we immediately get $\mcal{F}^k(d^0) \equiv \mcal{F}^n(d^0)$ which contradicts our assumption that $\mcal{F}^n(d^0)$ is not the least fixed point of $\mcal{F}$. \end{proof} By computing the syntactic fixed point we thus get a syntactic expression $\ffunc_{\expr}^n(d_{\mcal{E}}^0)(s,t) = E_{s,t}$ for each pair of states $s,t$ such that the solution set to $E_{s,t} \leq \varepsilon$ characterizes the set of ``good'' parameter valuations. \begin{example} Consider the WKS and PWKS from \autoref{fig:PWKSex}. To compute $E_{s,t}$, let $d_{\mcal{E}}^i(s,t) = \ffunc_{\expr}^i(d_{\mcal{E}}^0)(s,t)$. We now show how the distance from $s$ to $t$ is updated after each iteration. \begin{alignat*}{2} &d_{\mcal{E}}^1(s,t) &&= \texttt{MAX}\left\{\begin{array}{ll} \texttt{MAX}\left\{\left|\frac{1}{1}-1\right|,d_{\mcal{E}}^0(s_1,t_2)\right\},\\[0.15cm] \texttt{MAX}\left\{\left|\frac{3}{2}-1\right|,d_{\mcal{E}}^0(s_1,t_2),d^0_\mcal{E}(s,t_2)\right\} \end{array}\right\}\\ &d_{\mcal{E}}^2(s,t) &&= \texttt{MAX}\left\{\begin{array}{ll} \texttt{MAX}\left\{\left|\frac{1}{1}-1\right|,0\right\},\\[0.15cm] \texttt{MAX}\left\{\left|\frac{3}{2}-1\right|,0,\frac{1}{2}\right\} \end{array}\right\}\\ &d_{\mcal{E}}^3(s,t) &&= \texttt{MAX}\left\{\begin{array}{l} \frac{1}{2},\left|\frac{p}{1}-1\right|,\\ \texttt{MIN}\left\{\left|\frac{p}{5}-1\right|,\left|\frac{p+2}{5}-1\right|,\left|\frac{p+4}{5}-1\right|\right\},\\[0.15cm] \texttt{MIN}\left\{\left|\frac{p}{3}-1\right|,\left|\frac{p+2}{3}-1\right|\right\} \end{array}\right\}\\ &d_{\mcal{E}}^4(s,t) && = d_{\mcal{E}}^3(s,t) \end{alignat*} We immediately see that any solution to $E_{s,t} \leq \varepsilon$ is bounded from below by $\frac{1}{2}$. This implies that there exists no valuation $v \in \mcal{V}$ such that $s \simeps t[v]$ for $\varepsilon < \frac{1}{2}$. If we consider the valuation $v_{\min}(p) = 1$ we get that $\llbracket E_{s,t} \rrbracket (v_{\min}) = \frac{1}{2}$ i.e $v_{\min}$ is the valuation that induces the minimal distance $d(s,t[v_{\min}]) = \frac{1}{2}$. \end{example} \section{Conclusion and Future Work}\label{sec:conc_future} We have characterized the distance from \cite{WCTL_logic} between weighted Kripke structures (WKS) as a least fixed point. The distance between any pair of states can thus be computed by first assuming the distance between any pair to be 0 and then applying a step-wise refinement of the distance. The computability of the distance is guaranteed as a finite number of the (potentially) infinite transition sequences of the system is sufficient. This we proved by demonstrating an upper bound on the relevant sequences. We furthermore lifted the distance to parametric WKS (PWKS), where transition weights can be parametric. The parameters can be used to abstract multiple configurations of the same system as one parametric system. In this case the distance is from a WKS to a PWKS and is concretely a parametric expression that one can evaluate to get an exact distance from the WKS to a specific WKS instance of the PWKS. The question is then which configuration (parameter valuation) is ``best'' i.e minimizes the induced distance. For computability we again demonstrate an upper bound on the length of relevant distances. To do this we assume all cycles to be cost non-zeno i.e any loop must include a transition with a positive rational weight. For future work, the actual complexity of computing the distance is open. From \cite{WCTL_logic} we know that checking whether the distance is 0 is NP-complete but the general complexity of checking whether the distance is less than some $\varepsilon \in \R_{\geq 0}$ is open. One could also investigate whether the distance has a polynomial approximation scheme. \end{document}

\begin{document} \title{High-order stroboscopic averaging methods for highly oscillatory delay problems } \author{ M. P. Calvo \and J. M. Sanz-Serna \and Beibei Zhu } \institute{M. P. Calvo\at Departamento de Matem\'atica Aplicada e IMUVA, Facultad de Ciencias, Universidad de Valladolid, Spain\\ \email{[email protected]} \and J. M. Sanz-Serna \at Departamento de Matem\'aticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911 Legan\'es (Madrid), Spain \\ \email{[email protected]} \and Beibei Zhu \at National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China\\ \email{[email protected]} } \date{} \maketitle \begin{abstract} We introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of arbitrarily high accuracy. \end{abstract} \noindent\textbf{Mathematical Subject Classification (2010)} 65L03, 34C29 \noindent\textbf{Keywords} Delay differential equations, stroboscopic averaging, highly oscillatory problems \section{Introduction} This paper suggests and analyzes heterogeneous multiscale methods \cite{E2003,EEngquist,Engquist,E2007,Li, Ariel,arieh,CS2} for the numerical solution of highly oscillatory systems of delay differential equations (DDEs) with constant delays. The methods may achieve arbitrarily high orders of convergence and are based on the idea of the stroboscopic averaging method (SAM) \cite{CCMS1,CCMS2} for highly oscillatory ordinary differential equations (ODEs). We are interested in integrating highly oscillatory delay differential systems of the form \begin{eqnarray}\label{eq:dde1} \frac{d}{dt}x(t)&=&f(x(t),x(t-\tau),t,\Omega t;\Omega),\qquad 0 \leq t \leq t_{max},\\ x(t)&=&\varphi(t),\qquad -\tau\le t \leq 0.\label{eq:dde2} \end{eqnarray} Here \(\tau>0\) is the constant delay, the angular frequency \(\Omega\gg 1\) is a large parameter and \(f\) is smooth, takes values in \(\mathbb{R}^D\) and is \(2\pi\)-periodic in its fourth argument. Note that, in addition to its fast periodic dependence on time through the combination \(\Omega t\), the function \(f\) depends (slowly) on \(t\) through its third argument. An example is given by \begin{eqnarray}\label{eq:toggle} \dot{x}_1(t)&=&\frac{\alpha}{1+x_2^{\beta}(t)}-x_1(t-\tau)+A\sin(\omega t)+B\sin(\Omega t),\\ \dot{x}_2(t)&=&\frac{\alpha}{1+x_1^{\beta}(t)}-x_2(t-\tau), \nonumber \end{eqnarray} where \(\alpha\), \(\beta\), \(A\), \(B\), \(\omega\) are constants. The term \(B\sin(\Omega t)\) induces fast oscillations in the solution and \(A\sin \omega t\) is a slow forcing. In the absence of external slow and fast forcing (\(A=B = 0\)) the system represents a delayed genetic toggle switch, a synthetic gene regulatory network \cite{gardner}. The paper \cite{Daza} studies the phenomenon of \emph{vibrational resonance} \cite{vr,guirao} of the switch, i.e. the enhancement of the response to the slow forcing created by the presence of the high frequency forcing. For additional examples of problems of the form \eqref{eq:dde1} see \cite{beibei}. The application of standard software to the integration of \eqref{eq:dde1} may be very expensive because accuracy typically requires that the step length be smaller than the small period \(T=2\pi/\Omega\). The difficulties increase in cases, such as \eqref{eq:toggle}, that have to be simulated over long time intervals for many choices of the parameters and constants that appear in the system. The algorithms suggested in this paper integrate an averaged version of \eqref{eq:dde1} that does not lead to solutions with fast oscillations. As other heterogeneous multiscale methods, the required information on the underlying averaged system is obtained on the fly by integrating \eqref{eq:dde1} in narrow time-windows. Here we follow the SAM technique \cite{CCMS1,CCMS2} where the right hand-side of the averaged system is retrieved by using finite differences. A similar approach has been used in \cite{beibei} but there are important differences between the algorithm in that reference and the integrators in this paper: \begin{itemize} \item The integrators suggested here are constructed by rewriting \eqref{eq:dde1}--\eqref{eq:dde2} as an ODE problem which is then solved by using the ODE SAM algorithms in \cite{CCMS1,CCMS2}. Reference \cite{beibei} borrows the main ideas of \cite{CCMS1,CCMS2} and adjusts them to the delay problem \eqref{eq:dde1}--\eqref{eq:dde2}. \item A single algorithm is introduced in \cite{beibei}; it is based on integrating the averaged problem with the second-order Adams-Basforth formula. In fact the approach in \cite{beibei} would be difficult to generalize to higher-order methods due to the lack of regularity of the solutions of DDEs. The methodology in this paper makes it possible to construct integrators of arbitrarily high orders. \item The analysis of the algorithms in this paper only uses techniques that are standard in the theories of the numerical integration and averaging of ODEs. The analysis in \cite{beibei} requires to develop special averaging results for DDEs. \end{itemize} This paper has seven sections. Section~\ref{sec:review} presents background material on SAM integrators for ODEs. Section~\ref{sec:ddes} explains the reformulation of \eqref{eq:dde1}--\eqref{eq:dde2} as an ODE problem. The new algorithms are described and analyzed in Sections~\ref{algorithms} and \ref{errorbounds} respectively. Numerical experiments are reported in Section~\ref{sec:experiments} and the final Section contains some proofs and some extensions. \section{A review of SAM for ODEs} \label{sec:review} The reader is referred to \cite{CCMS1,CCMS2} for a detailed description and analysis of SAM; here we restrict ourselves to those aspects of the method that are needed to present the algorithms in Section~\ref{algorithms}. SAM is a heterogeneous multiscale technique for the numerical integration of highly oscillatory systems of the form \begin{equation}\label{eq:sam1} \frac{d}{dt}y = f(y,\Omega t;\Omega), \end{equation} where the sufficiently smooth\footnote{ The exact number of derivatives that \(f\) must possess depends on the specific SAM algorithm, on the choice of \(J\) below, etc. In order to simplify the exposition we prefer not to keep track of that number. } function \(f:\mathbb{R}^D\times \mathbb{R}\times(0,\infty)\rightarrow \mathbb{R}^D\) depends \(2\pi\)-periodically on its second argument \(\Omega t\) and \(\Omega\gg 1\) is a large parameter. It is assumed that \(\Omega^{-1}f\) and its derivatives remain bounded as \(\Omega \uparrow \infty\). The solutions of \eqref{eq:sam1} are sought in an integration interval \(t_0\leq t\leq t_{max}\) assumed to be independent of \(\Omega\). SAM is applicable whenever, over one period, the solution change \(y(t_0+T)-y(t_0)\) is \(\mathcal{O}(\Omega^{-1})\) as \(\Omega \uparrow \infty\), see \cite{CCMS1}. Then, given an arbitrarily large integer \(J\), there exists a stroboscopic averaged system \cite{Chartier,Murua,Sanz-Serna, kurusch} \begin{equation}\label{eq:sam2} \frac{d}{dt} Y = F(Y;\Omega) \end{equation} such that, if \(y(t)\) and \(Y(t)\) are solutions of \eqref{eq:sam1} and \eqref{eq:sam2} that share a common value at time \(t_0\), then \(Y\) interpolates \(y\) with (small) \(\mathcal{O}(\Omega^{-J})\) errors at \emph{stroboscopic times}, i.e.\ at values of \(t\) of the form \(t_n=t_0+nT\), \(n\) an integer. The constant implied in the \(\mathcal{O}\) notation is independent of \(n\) for \(t_n\) ranging in a compact time interval. \begin{rem} \label{rem:depends}\em In \eqref{eq:sam2}, the function \(F\) may be chosen to be a polynomial in \(\Omega^{-1}\) whose degree increases with \(J\) (the dependence of \(F\) on \(J\) is not reflected in the notation). The coefficients of this polynomial are smooth functions of \(Y\) that depend on \(t_0\) (again this dependence has not been incorporated to the notation). Explicit formulas for the construction of \(F\) may be seen in \cite{Murua,kurusch}. \end{rem} Since \(F\) does not depend on the rapidly varying phase \(\Omega t\), the system \eqref{eq:sam2} is non-oscillatory and its numerical integration may be performed with step sizes that are not restricted in terms of the small period \(T\). This consideration, by itself, is not sufficient to construct a viable numerical algorithm because, for large \(J\), finding the analytic expression of \(F\) may be extremely expensive even with the help of a symbolic manipulator \cite{abel}. SAM is a technique for the numerical integration of \eqref{eq:sam1} based on integrating numerically \eqref{eq:sam2} without using that analytic expression; the required information on \(F\) is collected on the fly by means of numerical integrations of \eqref{eq:sam1}. In its crudest variant, SAM approximately evaluates \(F\) at a given vector \(w\in\mathbb{R}^D\) by using the finite difference formula \begin{equation} \label{eq:fw} F(w;\Omega) \approx \frac{1}{T}[\Phi_T(w)-w], \end{equation} where \(\Phi_T(w)\) is the value at time \(t_0+T\) of the solution of \eqref{eq:sam1} with value \(w\) at time \(t_0\). This makes sense because, up to a small \(\mathcal{O}(\Omega^{-J})\) error, \(\Phi_T(w)\) coincides with the value at \(t_0+T\) of the solution of \eqref{eq:sam2} with \(Y(t_0)=w\) and the slope of this solution at time \(t_0\) is \( F(w;\Omega)\). SAM consists of three parts: the macrointegrator, the numerical differentiation formula, and the microintegrator. These will be discussed presently. There is much freedom in the choice of each of these three elements. The macrointegrator is the algorithm used to integrate \eqref{eq:sam2}; it may be e.g.\ a Runge-Kutta or a linear multistep method. For simplicity we shall assume throughout that the macrointegrator uses a constant step size \(H\); however variable step sizes may be equally applied within SAM. It is not necessary that the step points used by the macrointegrator be stroboscopic times. If the macrointegration is arranged in such a way that output is produced at stroboscopic times, then that output provides approximations to the oscillatory solution \(y\). If, on the other hand, one needs to obtain an approximation to \(y(t)\) at a time \(t\) that is not stroboscopic, then one may use SAM to approximate \(y(t_n)\) at the largest stroboscopic time \(t_n\) less than \(t\) and then integrate \eqref{eq:sam1} in the short interval \([t_n,t]\) with length \(< T\). Instead of the crude differentiation formula \eqref{eq:fw} with \(\mathcal{O}(T)\), i.e.\ \(\mathcal{O}(\Omega^{-1})\), errors, one may use the familiar second-order central difference formula \begin{equation} \label{eq:cw} F(w;\Omega) \approx \frac{1}{2T}[\Phi_T(w)-\Phi_{-T}(w)], \end{equation} with \(\mathcal{O}(\Omega^{-2})\) errors (\(\Phi_{-T}(w)\) is the value at time \(t_0-T\) of the solution of \eqref{eq:sam1} with value \(w\) at time \(t_0\)), the fourth-order formula based on function values at \(t_0\pm T\), \(t_0\pm 2T\), etc. The microintegrator is the algorithm used to integrate \eqref{eq:sam1} to approximately obtain the values \(\Phi_{\pm kT}(w)\) required by the numerical differentiation formula being employed. The microintegrator may be e.g. a Runge-Kutta or a linear multistep method and need not coincide with the scheme used as a macrointegrator. It may use constant or variable step sizes; for simplicity we will restrict the attention to the case where the step size \(h\) is constant. When \eqref{eq:fw} is used, each evaluation of \(F\) requires a microintegration of the oscillatory system \eqref{eq:sam1} in the interval \(t_0\leq t\leq t_0+T\). As \(\Omega\) increases the microintegration step size \(h\) has to be reduced on accuracy grounds, but this is compensated by the fact that the microintegration interval length \(T\) shrinks correspondingly. The central difference formula \eqref{eq:cw} needs two microintegrations per evaluation of \(F\), one of them operates forward in time and finds \(\Phi_T(w)\) and the other goes backwards to find \(\Phi_{-T}(w)\). More involved differentiation formulas require forward microintegrations in longer intervals of the form \([t_0,t_0+kT]\) and/or backward integrations in intervals \([t_0-k^\prime T,t_0]\) (\(k\), \(k^\prime\) are small positive integers). \begin{rem} \label{rem:micro} \em It is important to note that each microintegration starts from an initial condition that is specified at time \(t_0\), regardless of the point of the time axis the macrointegration may have reached when the microintegration is carried out. This is a consequence of the fact, pointed out in Remark \ref{rem:depends}, that the averaged system \eqref{eq:sam2} depends on \(t_0\) (see \cite{CCMS1} for a detailed explanation). \end{rem} \begin{rem} \label{rem:slowtime} \em The presentation so far has been restricted to the format \eqref{eq:sam1}. It is also possible to apply SAM to problems \begin{equation}\label{eq:sam3} \frac{d}{dt}y = f(y,t,\Omega t;\Omega), \end{equation} where now \(f\) has an additional dependence on \(t\), \(t_0\leq t\leq t_{max}\), in addition to the \emph{fast} dependence through \(\Omega t\). In fact the case \eqref{eq:sam3} may be reduced to the format \eqref{eq:sam2} by the standard device of considering the second argument of \(f\) as a new dependent variable \(y^0\) and appending to the system the additional equation \(dy^0/dt = 1\). \end{rem} Error bounds for SAM are presented in Section~\ref{errorbounds}. \section{Highly-oscillatory DDEs} \label{sec:ddes} We are interested in integrating the highly oscillatory problem \eqref{eq:dde1}--\eqref{eq:dde2} under the hypothesis that \(\Omega^{-1} f\) and its derivatives \emph{remain bounded as \(\Omega\uparrow \infty\)}. Without losing generality \cite{beibei}, we assume that the (known) function \(\varphi\) that specifies the values of \(x\) in the interval \([-\tau,0]\) is \(\Omega\) independent. The assumption that the integration of \eqref{eq:dde1} starts at \(t=0\) does not reduce the generality either, as one may always make a translation along the time axis. In order to simplify the exposition, we shall also assume hereafter that the \(\Omega\)-independent end-point \(t_{max}\) of the integration interval is an integer multiple of \(\tau\), i.e.\ \(t_{max}=L\tau\). When this is not the case we may apply the integrators below after increasing \(t_{max}\) up to the smallest integer multiple of \(\tau\) larger than \(t_{max}\). Alternatively one may integrate with the algorithms described below up to the largest integer multiple \(L^\prime \tau\) of \(\tau\) smaller than \(t_{max}\) and then complete the integration by using a conventional integrator for \eqref{eq:dde1} in the short interval \([L^\prime\tau, t_{max}]\). The algorithms in this paper are based in the introduction of the functions \begin{eqnarray} x^{(0)}(t)&=&\varphi(t-\tau), \qquad 0\leq t \leq \tau, \label{eq:def1}\\ x^{(\ell)}(t)&=&x(t+(\ell-1)\tau), \qquad 0\leq t \leq \tau,\qquad \ell=1, \ldots, L;\label{eq:def2} \end{eqnarray} determining these functions is clearly equivalent to determining the solution \(x(t)\) of \eqref{eq:dde1}--\eqref{eq:dde2}. An illustration is given in Figure~\ref{fig:A}. \begin{figure} \caption{The top subplot gives a solution \(x\) of the oscillatory problem \eqref{eq:dde1} in the interval \(-\tau \leq t\leq 3\tau\), \(\tau=0.5\). The other subplots give the functions \(x^{(\ell)}\) for \(\ell = 0, 1, 2, 3\); these obviously provide all the information contained in \(x\). The discontinuous lines in the last three panels depict the solution of the averaged system of ODEs \eqref{eq:aver}. By definition of stroboscopic averaging, each \(X^{(\ell)}\) exactly coincides with the corresponding \(x^{(\ell)}\) at the initial time \(t=0\). An unrealistically low value of the frequency \(\Omega\) is used here so as not to clatter the plots. } \label{fig:A} \end{figure} In terms of the \(x^{(\ell)}\), the problem \eqref{eq:dde1}--\eqref{eq:dde2} is given by \begin{eqnarray} \label{eq:ode1} &&\frac{d}{dt}x^{(\ell)}(t)=\\&&f(x^{(\ell)}(t),x^{(\ell-1)}(t),t+(\ell-1)\tau,\Omega (t+(\ell-1)\tau);\Omega),\: 0\leq t \leq \tau, \: 1 \leq \ell \leq L,\nonumber \end{eqnarray} in tandem with the conditions \begin{equation} \label{eq:ode2} x^{(\ell)}(0)=x^{(\ell-1)}(\tau),\qquad 1 \leq \ell \leq L. \end{equation} \begin{rem}\label{rem:seq} \em Note that \(x^{(0)}\) is known from \eqref{eq:def1}. The unknown function \(x^{(1)}\) is determined from \eqref{eq:ode1} with \(\ell=1\) and the initial condition \(x^{(1)}(0) =\varphi(0)\); once \(x^{(1)}\) is known, \(x^{(2)}\) is determined from \eqref{eq:ode1} with \(\ell=2\) and the initial condition \(x^{(2)}(0) =x^{(1)}(\tau)\), etc. Thus, even though, in view of \eqref{eq:ode2}, the problem \eqref{eq:ode1}--\eqref{eq:ode2} has the appearance of a two-point boundary problem, we are really dealing with an initial-value problem (this was to be expected as \eqref{eq:ode1}--\eqref{eq:ode2} is just a way of writing \eqref{eq:dde1}--\eqref{eq:dde2}). \end{rem} Obviously \eqref{eq:ode1} is a highly-oscillatory system \emph{of ODEs} (rather than DDEs) \begin{equation}\label{eq:odebf} \frac{d}{dt}{\bf x} ={\bf f}({\bf x}, t,\Omega t;\Omega),\qquad 0\leq t \leq \tau, \end{equation} for the unknown function \[ {\bf x}(t) = [x^{(1)}(t), \dots, x^{(L)}(t)],\qquad 0\leq t\leq \tau, \] with values in \(\mathbb{R}^{LD}\) ($x^{(0)}$ is known, see \eqref{eq:def1}). The algorithms to be described below are based on the integration of \eqref{eq:odebf} with the help of SAM as described in the preceding section. If we denote by \(X^{(\ell)}\) the averaged version of \(x^{(\ell)}\), \(\ell = 1,\dots, L\), (\(X^{(0)}(t) =\varphi(t-\tau)\), \(0\leq t\leq \tau\)) the averaged system for \[ {\bf X}(t) = [X^{(1)}(t), \dots, X^{(L)}(t)],\qquad 0\leq t\leq \tau, \] is of the form \begin{eqnarray} \label{eq:aver} &&\frac{d}{dt}X^{(\ell)}(t)=\\&&F^{(\ell)}(X^{(\ell)}(t),X^{(\ell-1)}(t),\dots, X^{(0)}(t),t;\Omega),\quad 0\leq t \leq \tau, \quad 1 \leq \ell \leq L,\nonumber \end{eqnarray} where we note that \(X^{(\ell+1)}\), \dots , \(X^{(L)}\) do not appear in the right-hand side because \(x^{(\ell)}\) (and therefore its averaged version \(X^{(\ell)}\)) does not depend on the values of the solution \(x\) for \(t> \ell \tau\). \begin{rem}\label{rem:triangular} \em With a terminology borrowed from linear algebra, we may say that the system \eqref{eq:ode1} has a lower bidiagonal structure, while \eqref{eq:aver} is only lower triangular. The explicit formulas for the averaged system in \cite{beibei} show that in fact for \(J\) large, \(X^{(\ell-2)}\),\dots, \(X^{(0)}\) appear in the right-hand side of \eqref{eq:aver} in addition to \(X^{(\ell-1)}\) and \(X^{(\ell)}\). \end{rem} \section{Algorithms} \label{algorithms} We now introduce algorithms for the solution of \eqref{eq:dde1}--\eqref{eq:dde2}. \subsection{Case I: the delay is an integer multiple of the period} We study first the particular case where the delay \(\tau\) is an integer multiple of the period. The general situation requires algorithms with additional complications. We note that in some applications there is some freedom in choosing the exact value of the large angular frequency \(\Omega\); one may then use that freedom to ensure that \(\tau/T=\tau\Omega/(2\pi)\) is an integer and thus avoid the extra complications. \begin{figure} \caption{\footnotesize As Fig.~\ref{fig:A} in the particular case where the delay \(\tau\) is an integer multiple of the period \(T\). The numerical approximation \(X^{(\ell)}_N\) to \(X^{(\ell)}(\tau)\) approximates \(x^{(\ell)}(\tau)= x^{(\ell+1)}(0)\) and may be used as initial value to compute approximately \(X^{(\ell+1)}(t)\), \(0\leq t\leq \tau\), \(\ell=1,\dots, L-1\). } \label{fig:B} \end{figure} We apply SAM, based on a macro step size \(H\) of the form \(\tau/N\) with \(N\) a positive integer, to the integration of the \(LD\) dimensional ODE system \eqref{eq:odebf}. While at the outset the initial condition \({\bf x}(0)\) is not known (see \eqref{eq:ode2}), the triangular structure of the averaged system \eqref{eq:aver} noted in Remark~\ref{rem:triangular} makes it possible to complete the application of SAM by successively computing, for \(\ell = 1, 2,\dots, L\), the numerical approximations to the functions \(X^{(\ell)}(t)\), \(0\leq t\leq T\), very much as in Remark~\ref{rem:seq}. One first applies SAM to the oscillatory problem for \(x^{(1)}\). Because \(H\) is a submultiple of \(\tau\), the macrointegrator will produce an approximation \(X^{(1)}_N\) to \(X^{(1)}(\tau)=X^{(1)}(NH)\), see Figure~\ref{fig:B}. In addition we are assuming that \(\tau/T\in\mathbb{N}\), so that the final time \(t=\tau\) is stroboscopic and then \(X^{(1)}(\tau)\) is a very accurate approximation to \(x^{(1)}(\tau)\), i.e.\ to \(x^{(2)}(0)\). Therefore \(X^{(1)}_N\) provides an approximation to the missing initial value \(x^{(2)}(0)\) and it is then possible to approximate with SAM the solution \(x^{(2)}(t)\), \(0\leq t\leq \tau\). Iterating this procedure one approximates all the \(x^{(\ell)}(t)\), or, equivalently, the oscillatory solution \(x(t)\), \(0\leq t\leq t_{max}\). Since coding SAM algorithms requires some care, it may be helpful to provide a detailed description of SAM for a particular choice of integrators and differentiation formula. This is done in Table~\ref{tab:algorithm} that refers to the case where the macro and microintegrators are chosen to be the familiar second-order formula of Runge that for \((d/dt) z = g(z,t)\) reads \[ z_{j+1/2} = z_j + \frac{\Delta t}{2}\, g(z_j,t_j),\qquad z_{j+1} = z_j+\Delta t\, g(z_{j+1/2},t_j+\Delta t/2). \] The formula needs two function evaluations per step. The algorithm uses the central difference formula \eqref{eq:cw} except when approximating \(F^{(\ell)}\), \(\ell = 1,\dots, L\), in \eqref{eq:aver}, at \(t=0\) where the forward formula \eqref{eq:fw} is applied. At \(t=0\) central difference are not applicable: backward microintegrations cannot be performed because the system \eqref{eq:ode1} is only defined for \(t\geq 0\) (\(x^{(0)}\) is not defined for \(t<0\), see \eqref{eq:def1}). The algorithm consists of an initialization block followed by a loop for the successive computation of the approximations to \(X^{(\ell)}(t)\), \(\ell = 1,\dots, L\). Note the different treatment given at all the microintegrations to the third (slow time \(t\)) and fourth (fast rotating phase \(\Omega t\)) arguments of \(f\); this is in agreement with Remark~\ref{rem:micro}. \begin{table} \caption{SAM-RK2 Algorithm} \begin{center} \begin{tabular}{lcccc} \hline $X^{(1)}_0 = \varphi(0)$ \% \texttt{initial condition}\\ \texttt{Load history}\\ For $\nu=0:\nu_{max}$ \\ ~~~~ $x^{(0)}_{0,\nu} = \varphi(-\tau+\nu h)$, $x^{(0)}_{0,\nu+1/2} = \varphi(-\tau+(\nu+1/2)h)$, \\ end \\ For $n=1:N-1$ \\ ~~~~ For $\nu=-\nu_{max}:\nu_{max}$ \\ ~~~~~~~~ $x^{(0)}_{n,\nu} = \varphi(-\tau+nH+\nu h)$, $x^{(0)}_{n,\nu+1/2} = \varphi(-\tau+nH+(\nu+1/2)h)$, \\ ~~~~ end \\ end \\ \texttt{Integration starts}\\ For $\ell=1:L$ \\ ~~~~ For $n=0:N-1$\\ ~~~~~~~~ \texttt{Compute $F^{(\ell)}_{n}$}\\ ~~~~~~~~ $t^{(\ell)}_n = nH + (\ell-1)\tau$ \% \texttt{initial time}\\ ~~~~~~~~ $x^{(\ell)}_{n,0} = X^{(\ell)}_n$ \% \texttt{initial value}\\ ~~~~~~~~ If $n=0$ \\ ~~~~~~~~~~~~ \texttt{Forward micro-integration} \\ ~~~~~~~~~~~~ For $\nu=0:\nu_{max}-1$ \\ ~~~~~~~~~~~~~~~~ $t^{(\ell)}_{0,\nu} = t_0^{(\ell)}+\nu h$, $t^{(\ell)}_{0,\nu+1/2} = t_0^{(\ell)}+(\nu +1/2)h$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{0,\nu+1/2} = x^{(\ell)}_{0,\nu}+(h/2) f(x^{(\ell)}_{0,\nu},x^{(\ell-1)}_{0,\nu},t^{(\ell)}_{0,\nu},\Omega\nu h;\Omega)$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{0,\nu+1} = x^{(\ell)}_{0,\nu}+h f(x^{(\ell)}_{0,\nu+1/2},x^{(\ell-1)}_{0,\nu+1/2},t^{(\ell)}_{0,\nu+1/2},\Omega(\nu+1/2) h;\Omega)$, \\ ~~~~~~~~~~~~ end\\ ~~~~~~~~~~~~ $F^{(\ell)}_0=(x^{(\ell)}_{0,\nu_{max}}\hspace{-8pt}-x^{(\ell)}_{0,0})/T$ \% \texttt{slope at 1st stage }\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\texttt{of 1st macro-step for \(X^{(\ell)}\)}\\ ~~~~~~~~ else \\ ~~~~~~~~~~~~ \texttt{Forward micro-integration}\\ ~~~~~~~~~~~~ For $\nu=0:\nu_{max}-1$, \\ ~~~~~~~~~~~~~~~~ $t^{(\ell)}_{n,\nu} = t_n^{(\ell)}+\nu h$, $t^{(\ell)}_{n,\nu+1/2} = t_n^{(\ell)}+(\nu +1/2)h$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{n,\nu+1/2} = x^{(\ell)}_{n,\nu}+(h/2) f(x^{(\ell)}_{n,\nu},x^{(\ell-1)}_{n,\nu},t^{(\ell)}_{n,\nu},\Omega \nu h;\Omega)$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{n,\nu+1} = x^{(\ell)}_{n,\nu}+h f(x^{(\ell)}_{n,\nu+1/2},x^{(\ell-1)}_{n,\nu+1/2},t^{(\ell)}_{n,\nu+1/2},\Omega(\nu+1/2) h;\Omega)$, \\ ~~~~~~~~~~~~ end\\ ~~~~~~~~~~~~ \texttt{Backward micro-integration}\\ ~~~~~~~~~~~~ For $\nu=0:\nu_{max}-1$, \\ ~~~~~~~~~~~~~~~~ $t^{(\ell)}_{n,-\nu} = t_n^{(\ell)}-\nu h$, $t^{(\ell)}_{n,-(\nu+1/2)} = t_n^{(\ell)}-(\nu +1/2)h$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{n,-(\nu+1/2)} = x^{(\ell)}_{n,-\nu}-(h/2) f(x^{(\ell)}_{n,-\nu},x^{(\ell-1)}_{n,-\nu},t^{(\ell)}_{n,-\nu},-\Omega\nu h;\Omega)$ \\ ~~~~~~~~~~~~~~~~ $x^{(\ell)}_{n,-(\nu+1)} = x^{(\ell)}_{n,-\nu}$\\ ~~~~~~~~~~~~~~~~ $-hf(x^{(\ell)}_{n,-(\nu+1/2)},x^{(\ell-1)}_{n,-(\nu+1/2)}, t^{(\ell)}_{n,-(\nu+1/2)},-\Omega(\nu+1/2) h;\Omega)$, \\ ~~~~~~~~~~~~ end\\ ~~~~~~~~~~~~ $F^{(\ell)}_{n}=(x^{(\ell)}_{n,\nu_{max}}\hspace{-5pt}-x^{(\ell)}_{n,-\nu_{max}})/(2T)$ \% \texttt{slope at 1st stage} \\ ~~~~~~~~ end\\ ~~~~~~~~ \texttt{Macro-integration}\\ ~~~~~~~~ $X^{(\ell)}_{n+1/2}=X^{(\ell)}_{n}+(H/2)F^{(\ell)}_n$, \% \texttt{2nd stage of n-th macro-step} \\ ~~~~~~~~ \texttt{Compute $F^{(\ell)}_{n+1/2}$}\\ ~~~~~~~~ $x^{(\ell)}_{n+1/2,0} = X^{(\ell)}_{n+1/2}$ \% \texttt{initial value}\\ ~~~~~~~~ $t^{(\ell)}_{n+1/2} = (n+1/2)H + (\ell-1)\tau$ \% \texttt{initial time}\\ \end{tabular} \end{center} \label{tab:algorithm} \end{table} \begin{table*} \begin{center} \begin{tabular}{lcccc} ~~~~~~~~ \texttt{Forward micro-integration}\\ ~~~~~~~~ For $\nu=0:\nu_{max}-1$, \\ ~~~~~~~~~~~~ $t^{(\ell)}_{n+1/2,\nu} = t_{n+1/2}^{(\ell)}+\nu h, t^{(\ell)}_{n+1/2,\nu+1/2}=t_{n+1/2}^{(\ell)}+(\nu +1/2)h $ \\ ~~~~~~~~~~~~ $x^{(\ell)}_{n+1/2,\nu+1/2} = x^{(\ell)}_{n+1/2,\nu}+(h/2)$\\ ~~~~~~~~~~~~ $f(x^{(\ell)}_{n+1/2,\nu},x^{(\ell-1)}_{n+1/2,\nu},t^{(\ell)}_{n+1/2,\nu},\Omega \nu h;\Omega)$ \\% ~~~~~~~~~~~~ $x^{(\ell)}_{n+1/2,\nu+1} = x^{(\ell)}_{n+1/2,\nu}$\\ ~~~~~~~~~~~~ $+h f(x^{(\ell)}_{n+1/2,\nu+1/2},x^{(\ell-1)}_{n+1/2,\nu+1/2},t^{(\ell)}_{n+1/2,\nu+1/2},\Omega(\nu+1/2) h;\Omega)$, \\ ~~~~~~~~ end\\ ~~~~~~~~ \texttt{Backward micro-integration}\\ ~~~~~~~~ For $\nu=0:\nu_{max}-1$, \\ ~~~~~~~~~~~~ $t^{(\ell)}_{n+1/2,-\nu} = t_{n+1/2}^{(\ell)}-\nu h, t^{(\ell)}_{n+1/2,-(\nu+1/2)}=t_{n+1/2}^{(\ell)}-(\nu +1/2)h $ \\ ~~~~~~~~~~~~ $ x^{(\ell)}_{n+1/2,-(\nu+1/2)} = x^{(\ell)}_{n+1/2,\nu}$\\ ~~~~~~~~~~~~ $-(h/2) f(x^{(\ell)}_{n+1/2,\nu},x^{(\ell-1)}_{n+1/2,\nu},t^{(\ell)}_{n+1/2,-\nu},-\Omega\nu h;\Omega)$ \\ ~~~~~~~~~~~~ $x^{(\ell)}_{n+1/2,-(\nu+1)} = x^{(\ell)}_{n+1/2,-\nu}-h\times$\\% ~~~~~~~~~~~~ $f(x^{(\ell)}_{n+1/2, -(\nu+1/2)}, x^{(\ell-1)}_{n+1/2, -(\nu+1/2)}, t^{(\ell)}_{n+1/2,-(\nu+1/2)}, -\Omega(\nu+1/2) h; \Omega)$ \\ ~~~~~~~~ end\\ ~~~~~~~~ $F^{(\ell)}_{n+1/2} = (x^{(\ell)}_{n+1/2,\nu_{max}}\hspace{-8pt}-x^{(\ell)}_{n+1/2,-\nu_{max}})/(2T)$ \% \texttt{slope at 2nd stage}\\ ~~~~~~~~ \texttt{Macro-step with RK2}\\ ~~~~~~~~ $X^{(\ell)}_{n+1}=X^{(\ell)}_{n}+HF^{(\ell)}_{n+1/2}$ \\ ~~~~ end \\ ~~~~ if $\ell < L$ \\ ~~~~~~~~ $X^{(\ell+1)}_0 = X^{(\ell)}_N$ \\ ~~~~ end \\ end\\ \hline \end{tabular} \end{center} \end{table*} \begin{rem}\label{rem:fixednumber} \em The first-order forward difference formula is used \(L\) times per run of the algorithm, regardless of the value of \(H\) (or, equivalently, regardless of the number of macrosteps needed to span an interval of length \(\tau\)). \end{rem} \subsection{Case II: the delay is not an integer multiple of the period} \label{sec:case2} In this case we apply SAM to the ODE system in \eqref{eq:def1} in the interval \(0\leq t \leq MT\) with $M=\lfloor \tau/T \rfloor$ (see Figure~\ref{fig:A}). In the short final interval \(MT\leq t\leq \tau\) (whose length is \(<T\)) we integrate numerically the oscillatory problem itself and for this purpose we choose the integrator and step length being used for the microintegrations. In this way the initial value \(X^{(\ell)}(0)\), \(\ell=2,\dots,L\), required by SAM is computed approximately as the numerical result at \(t=\tau\) of the integration of the oscillatory equation for \( x^{(\ell-1)}(t)\) in \eqref{eq:def1} that starts at \(t=M\tau\) from the SAM approximation to \(X^{(\ell-1)}(NT)\approx x^{(\ell-1)}(NT)\). The integration of the equation for \(x^{(L)}(t)\) in \(MT\leq t\leq \tau\) yields an approximation to \(x(t_{max})\). \begin{rem}\em While the hypothesis \(\tau/T\in\mathbb{N}\) obviously simplifies the algorithm, it is not necessary for the methods to perform satisfactorily, as the numerical experiments below will show. This should be compared with the situation for the integrator in \cite{beibei}, where, for systems of the general form \eqref{eq:dde1}, there is a degradation in the error behaviour if the hypothesis \(\tau/T\in\mathbb{N}\) does not hold (see Remark 3 in \cite{beibei}). \end{rem} \section{Error bounds}\label{errorbounds} In this section we provide error bounds for the algorithms described above. We work first under the hypothesis that \(\tau\) is a multiple of the period (Case I in the preceding section) and then consider the general situation. The section concludes with the presentation of some refinements. For simplicity, we assume that the macro and micro integrators are (consistent) Runge-Kutta methods. \subsection{Basic estimate: Case I} As explained above, if the delay is a multiple of the period, the integrators to be analyzed are just SAM algorithms for the system of ODEs \eqref{eq:odebf}. For \(X^{(1)}\) the results given by our algorithms are those of macrointegrating \eqref{eq:sam2} with inaccurate values of \(F\). For \(X^{(\ell)}\), \(\ell>1\), we apply the macrointegrator with inaccurate values of \(F\) and in addition with a starting value \(X^{(\ell)}(0)\) that is itself not exact. Classical results of the theory of convergence of numerical ODEs show then that SAM solutions have an error bound\footnote{ Classical numerical analysis texts used to provide global error bounds for integrations subject to inaccuracies in the computation of the numerical solution at each step, see e.g. \cite[Chapter 8, Section 5, Theorem 3]{ik}. Such inaccuracies may be due to rounding errors or, as it is the case here, to other reasons. The importance of rounding errors has diminished over the years and accordingly modern texts assume that those inaccuracies do not exist. In fact the study of the impact of the inaccuracies on the global error is exactly the same as that of the impact of the local truncation error, see e.g. \cite[Remark 2]{granada}. } \begin{equation}\label{eq:mainbound} \mathcal{O}\left( H^P+\delta+\Omega\:\mu\right) \end{equation} where \begin{itemize} \item The contribution \(H^P\) (\(P\) is the order of the macrointegrator) arises from the global error of the macrointegrator and would remain even if \(F\) were known exactly rather than evaluated via finite differences. This contribution is uniform in \(\Omega\) as \(\Omega\) increases, because the stroboscopically averaged system being integrated is a polynomial in \(\Omega^{-1}\) (Remark~\ref{rem:depends}). \item \(\delta= \delta(H,\Omega)\) is a bound for the error due to the finite-difference formula used to compute \(F\). \item \(\mu(h,\Omega)\) is a bound for the microintegrating error when computing approximately \(\Phi_T(w)\) in \eqref{eq:fw} (or \(\Phi_{\pm T}(w)\) in \eqref{eq:cw}, etc.). The (large) factor \(\Omega\) in front of \(\mu\) in \eqref{eq:mainbound} is due to the denominator in the finite difference formulas \eqref{eq:fw}, \eqref{eq:cw}, etc. \end{itemize} We study \(\mu\) assuming that the oscillatory system is written in the format \eqref{eq:sam1}. It is best to introduce the slow time \(s=\Omega t\) that transforms \eqref{eq:sam1} into \begin{equation}\label{eq:slow} \frac{d}{ds}y = \Omega^{-1}f(y,s;\Omega). \end{equation} This system has to be integrated over a forward period \(0\leq s\leq 2\pi\) (or over a forward period and a backward period, etc. depending on the finite difference formula being used to recover \(F\)). Note that the rescaling of time is compatible with the RK discretization in the sense that the \(y\) vectors produced by the algorithm when the system is integrated in the variable \(t\) with step size \(h\) coincide with those obtained when the system is integrated in the variable \(s\) with step size \(\Delta s = \Omega h\). Since we assumed at the outset that \(\Omega^{-1}f\) is smooth and remains bounded together with its derivatives as \(\Omega\uparrow \infty\), the microintegration errors for \eqref{eq:slow} may be estimated, uniformly in \(\Omega\) as \(\mu=\mathcal{O}((\Delta s)^p)\), i.e. \begin{equation} \qquad \mu=\mathcal{O}(\Omega^ph^p), \label{eq:mu} \end{equation} where \(p\) is the order of the microintegrator. As an example we look at the algorithm in Table~\ref{tab:algorithm} with \(P=p=2\). Second order differentiation contribute to \(\delta\) with an \(\Omega^{-2}\) term. Since the first-order difference formula is only used at a number of macrosteps that is fixed as \(H\rightarrow 0\) (see Remark~\ref{rem:fixednumber}), its contribution to \(\delta\) is \(H\Omega^{-1}\). Thus we have the bound \[ \mathcal{O}\left( H^2+\Omega^{-2}+H\Omega^{-1}+\Omega^3h^2\right). \] \subsection{Basic estimate: Case II} In this case \eqref{eq:mainbound} has to be replaced by \begin{equation}\label{eq:mainbound2} \mathcal{O}\left( H^P+\delta+\Omega\:\mu+\nu\right), \end{equation} where \(\nu\) bounds the error introduced by the integrations of the oscillatory system in the final short interval \(MT\leq t\leq \tau\). Since these are carried out in intervals of length \(< T\) and there is a number \(L\) of them independent of the problem parameters, from \eqref{eq:mu}, we obtain the bounds \begin{equation} \nu=\mathcal{O}((\Delta s)^p)\qquad {\rm i.e.}\qquad \nu=\mathcal{O}(\Omega^ph^p). \label{eq:nu} \end{equation} \subsection{Refined microintegration estimates: \(\mathcal{O}(\Omega^{-1})\) microintegration errors} There are numerous circumstances where \eqref{eq:mu} is pessimistic. An instance is given by the case where \eqref{eq:sam1} is of the form \[ \frac{d}{dt}y = \Omega\: \Lambda(\Omega t)+f(y,\Omega t;\Omega), \] with \(\Lambda\) a (vector-valued) trigonometric polynomial and \(f\) and its derivatives are bounded as \(\Omega\uparrow \infty\). In terms of the slow time \(s\) we have \begin{equation}\label{eq:perturbed} \frac{d}{ds}y =\Lambda(s) +\Omega^{-1}f(y,s;\Omega), \end{equation} a system that may be seen as a perturbation of \((d/ds) y = \Lambda(s)\). For the unperturbed problem we have the following result that will be proved in Section~\ref{proofs1}. \begin{proposition} \label{prop} Let \(\Lambda\) be a (vector-valued) trigonometric polynomial. A Runge-Kutta scheme applied to the initial-value problem \((d/ds) y = \Lambda(s)\), \(y(0)=y_0\), with a constant stepsize \(\Delta s=2\pi/M\) (\(M\) a positive integer) gives approximations that are exact at \(s = \pm 2\pi\) provided that \(\Delta s\) is suffiently small. \end{proposition} Note that, by implication, the integrator also yields exact approximations at \(s=\pm 4\pi\), \(s=\pm 6\pi\), etc. Thus the computation of the values of \(\Phi_{\pm k T}\) used in the finite-difference formulas will be free from error and, for the unperturbed problem, \(\mu =0\). From the proposition it may be expected that for the perturbed system \eqref{eq:perturbed} the microintegration error after a whole number of periods will approach 0 as \(\Omega\uparrow \infty\) with \(\Delta s\) fixed. In fact in this case \eqref{eq:mu} may be replaced by \begin{equation}\label{eq:mu2} \mu=\Omega^{-1}\mathcal{O}((\Delta s)^p)\qquad {\rm or}\qquad \mu=\mathcal{O}(\Omega^{p-1}h^p), \end{equation} an estimate that will be established in Section~\ref{proofs2}. \subsection{Refined microintegration estimates: \(\mathcal{O}(\Omega^{-2})\) microintegration errors} \label{sec:omegados} An even more favourable situation holds when in \eqref{eq:sam1} \(f\) and its derivatives remain bounded as \(\Omega\uparrow \infty\) and as a function of its second argument is a trigonometric polynomial. According to \eqref{eq:slow}, for \(0\leq s \leq 2\pi\), \(y(s)-y(0) = \mathcal{O}(\Omega^{-1})\) and we may consider a decomposition \[ \frac{d}{ds}y = \Omega^{-1}f(y(0),s;\Omega)+\Omega^{-1} \Big(f(y,s;\Omega)-f(y(0),s;\Omega) \Big). \] For the unperturbed system \((d/ds)y = \Omega^{-1}f(y(0),s;\Omega)\) the output of the microintegrations is exact in view of the preceding proposition; the perturbation is \(\mathcal{O}(\Omega^{-2})\) for \(0\leq s\leq 2\pi\) and \eqref{eq:mu} may be replaced by the improved estimate (Section~\ref{proofs}) \begin{equation}\label{eq:mu3} \mu=\Omega^{-2}\mathcal{O}((\Delta s)^p)\qquad {\rm or}\qquad \mu=\mathcal{O}(\Omega^{p-2}h^p). \end{equation} We emphasize that the improved bounds for \(\mu\) we have just discussed hold because the integration of the unperturbed problem is exact \emph{after a whole number of periods}. The bound \eqref{eq:nu} cannot be improved similarly because there the integration is not carried out for a whole number of periods. \section{Numerical experiments} \label{sec:experiments} We now report numerical experiments based on SAM. They are based on the following algorithms: \begin{table} \caption{Coefficients of methods RK3 (left) and RK4 (right).} \begin{center} $\begin{array}{c|ccc} & & & \\ [6pt] 0& & & \\ [6pt] \frac{1}{3}& \frac{1}{3} & & \\ [6pt] \frac{2}{3}& 0 & \frac{2}{3} & \\ [6pt] \hline \\[-6pt] & \frac{1}{4} & 0 & \frac{3}{4} \end{array} \qquad\qquad \begin{array}{c|cccc} 0& & & & \\ [6pt] \frac{1}{2}& \frac{1}{2} & & & \\ [6pt] \frac{1}{2}& 0 & \frac{1}{2} & & \\ [6pt] 1& 0 & 0 & 1 & \\ [6pt]\hline \\[-6pt] & \frac{1}{6} & \frac{2}{6} & \frac{2}{6} & \frac{1}{6} \end{array}$ \end{center} \label{tab:coeff} \end{table} \begin{enumerate} \item SAM-RK3. This is a SAM algorithm, similar to that in Table~\ref{tab:algorithm}, that uses the well-known third order RK method in Table~\ref{tab:coeff} as macro and microintegrator. We approximate \(F\) by means of the differentiation formula with \(\mathcal{O}(\Omega^{-3})\) errors based on function values at \(-2T\), \(-T\), \(0\), \( T\). At \(t=0\), where backward microintegrations are not possible, we use the \(\mathcal{O}(\Omega^{-3})\) forward differentiation formula based on function values at \(0\), \(T\), \(2T\), \(3T\). \item SAM-RK4. A SAM algorithm, similar to that in Table~\ref{tab:algorithm}, that uses the \lq classical\rq\ order four RK method (see Table~\ref{tab:coeff}) as macro and microintegrator. We approximate \(F\) by means of the well-known differentiation formula based on function values at \(\pm T\), \(\pm 2T\) (\(\mathcal{O}(\Omega^{-4})\) errors). For the first-stage of the formula at \(t=0\), where backward microintegrations are not possible we use the \(\mathcal{O}(\Omega^{-4})\) formula based on function values at \(0\), \(T\), \(2T\), \(3T\), \(4T\). In addition the fourth stage requires values of \(F\) at the end point \(t=\tau\) and for those we use the \(\mathcal{O}(\Omega^{-4})\) formula based on \(-4T\), \(-3T\), \(-2T\), \(-T\), \(0\). \item SS-Z. This is the integrator introduced in \citep{beibei} that is not based on rewriting the system as an ODE. \end{enumerate} Experiments using the method in Table~\ref{tab:algorithm} were also conducted, but will not be reported as its performance is very similar to that of SS-Z. In fact, the number of possible combinations of integrators and differentiation formulas is bewildering. The choices used here are meant to illustrate the possibilities of the SAM idea and we have not attempted to identify the most efficient combinations. \subsection{Test problems} We have integrated the two test problems used in \citep{beibei}. The first is given by \eqref{eq:toggle} together with the history information $x_1(t) = 0.5$, $x_2(t) = 2.0$, for $-\tau \leq t\leq 0$. The constants in the model have the values $\alpha=2.5$, $\beta=2$, $A=0.1$, $\omega=0.1$, $B=4.0$, $\tau=0.5$. This leads to an ODE system that satisfies the hypotheses in Section~\ref{sec:omegados} so that the estimate \eqref{eq:mu3} holds. The second test problem is the following more demanding variant of \eqref{eq:toggle}: \begin{eqnarray}\label{eq:geneproblem} \frac{dx_1}{dt}&=&\frac{\alpha}{1+x_2^{\beta}}-x_1(t-\tau)+A\sin(\omega t)+\hat{B}\Omega\sin(\Omega t),\\ \nonumber \frac{dx_2}{dt}&=&\frac{\alpha}{1+x_1^{\beta}}-x_2(t-\tau), \end{eqnarray} with $\hat B=0.1$ and all other constants and the initial history as for \eqref{eq:toggle}. Now the amplitude of the fast forcing grows linearly with \(\Omega\) and, as a result, the solution undergoes fast oscillations of amplitude $O(1)$, as \(\Omega \rightarrow \infty\) (rather than $O(\Omega^{-1})$ as it is the case for \eqref{eq:toggle}). Clearly \eqref{eq:geneproblem} leads to a system of ODEs of the form \eqref{eq:perturbed} and estimate \eqref{eq:mu2} holds. \subsection{Results: case I} We first set $\Omega = 8\pi, 16\pi, \ldots$, so that the delay \(\tau = 0.5\) is an integer multiple of the fast period $T=2\pi/\Omega$. \begin{table}[t] \caption{Maximum errors at stroboscopic times in $x_1$ for SAM-RK4 with respect to the reference solution for problem \eqref{eq:toggle}} \footnotesize \begin{center} \resizebox{\textwidth}{!}{ \begin{tabular}{rcccccccccc} \hline N &$\Omega=16\pi$ & $\Omega=32\pi$ & $\Omega=64\pi$ & $\Omega=128\pi$ & $\Omega=256\pi$ & $\Omega=512\pi$ & $\Omega=1024\pi$ \\ \hline 1&1.18(-3)&6.17(-4)&3.48(-4)&1.86(-4)&9.41(-5)&4.50(-5)&1.95(-5)\\ 2&***&3.01(-5)&1.70(-5)&9.09(-6)&4.62(-6)&2.23(-6)&9.98(-7)\\ 4&***&***&1.00(-6)&5.40(-7)&2.77(-7)&1.35(-7)&6.18(-8)\\ 8&***&***&***&3.34(-8)&1.72(-8)&8.44(-9)&3.89(-9)\\ 16&***&***&***&***&1.12(-9)&5.26(-10)&2.23(-10)\\ 32&***&***&***&***&***&2.93(-11)&1.87(-11)\\ 64&***&***&***&***&***&***&2.30(-11)\\ \hline \end{tabular}} \end{center} \label{tab:3} \end{table} \subsubsection{Test problem \eqref{eq:toggle}} For each value of $\Omega$, we have first computed a reference solution of the problem in the interval $[0, 2]$ using the Matlab function dde23 with relative and absolute tolerances equal to $10^{-11}$; errors have been measured with respect to this reference solution. Notice that the interval $[0, 2]$ includes the locations $t=\ell \times \tau$ for $0 \leq \ell \leq 4$. When studying vibrational resonances in \eqref{eq:toggle} the interest lies in much longer time intervals, but we have not used them in our study due to the extremely high cost of finding the reference solution with dde23 when \(\Omega\) is large. We have run the algorithms with macro-stepsize $H=\tau/N$ and micro-stepsize $h=T/(2N)$ for $N=1, 2, 4, \ldots$ This implies that when $N$ is doubled, both the macro stepsize and the micro stepsize are divided by two and, consequently, the computational cost, which is independent of \(\Omega\), is multiplied by four. \begin{figure} \caption{Maximum error at stroboscopic times, with respect to the reference solution, in the first component of SAM-RK4 (diamonds), SAM-RK3 (squares) and SS-Z (triangles) versus CPU time for $\Omega =1024 \pi$. Errors for SAM-RK4 come from the last column in Table~\ref{tab:3}.} \label{fig:ff1} \end{figure} Table~\ref{tab:3} shows, for the first component of the solution, maximum errors at stroboscopic times in the interval \(0\leq t\leq 2\) when the integration is performed with SAM-RK4. Stars denote combinations $(N, \Omega)$ for which the numerical solution has not been computed because the macrostepsize $H$ is not significantly larger than the period $T$ and the heterogeneous multiscale approach does not make sense. Note that entries in the table below, say, $10^{-11}$ may not be reliable due to the accuracy we used in computing the reference solution. According to the estimates in the preceding section, for SAM-RK4 there is an \(H^4\), i.e. \(N^{-4}\), contribution to the error bound \eqref{eq:mainbound} arising from the macrointegrator, an \(\Omega^{-4}\) contribution arising from the use of finite differences and \(\Omega \mu\) may be bounded by \(\Omega^3h^4\), or \(\Omega^{-1}N^{-4}\). The numbers in the table have a clear \(\Omega^{-1}N^{-4}\) behaviour, which shows that the error is mainly due to the microintegrations. For the values of \(\Omega\) under consideration the finite differences employed are virtually exact and, in addition, the error arising from the macrointegrator is also negligible (the averaged solution varies very little in the short integration interval). For SAM-RK3 the contributions to \eqref{eq:mainbound} are respectively \(H^3\), \(\Omega^{-3}\) and \(\Omega^2h^3\). The results show an \(\Omega^2h^3\), i.e. \(\Omega^{-1}N^{-3}\), behaviour (which corresponds to the error being dominated by the microintegrations) and will not be reproduced here. Error bounds and numerical results for SS-Z may be seen in \citep{beibei}. We plot in Figure~\ref{fig:ff1} an efficiency diagram comparing these three integrators. The figure represents, in doubly-logarithmic scale, the maximum error in $x_1$ at stroboscopic times versus the CPU time when $\Omega=1024 \pi$. We first observe that the slopes of the different lines are close to $-1$ (triangles), $-3/2$ (squares) and $-2$ (diamonds). As mentioned above, due to our choice of $H$ and $h$ (i.e.\ $H=\tau/N$, $h=T/(2N)$, $N=1, 2, \ldots, 128$), the computational cost is multiplied by $2^2$ when $N$ is doubled and, consequently, the slopes observed in Figure~\ref{fig:ff1} correspond to a dependence on \(N\) of the form \(N^{-2}\), \(N^{-3}\), \(N^{-4}\), in agreement with the bounds of the preceding section (and those for SS-Z presented in \cite{beibei}). Comparing the three integrators, we also conclude that for errors larger than $10^{-4}$, SS-Z is the most efficient, for errors between $10^{-6}$ and $10^{-4}$ SAM-RK3 is preferable, and for errors smaller than $10^{-6}$ the more accurate SAM-RK4 requires the smallest CPU time. Similar conclusions may be drawn for other values of $\Omega$ (but the range of errors where one method is better than the others varies slightly with \(\Omega\)). \subsubsection{Test problem \eqref{eq:geneproblem}} \begin{table}[t] \caption{Maximum errors at stroboscopic times in $x_1$ for SAM-RK4 with respect to the reference solution for problem \eqref{eq:geneproblem}} \footnotesize \begin{center} \resizebox{\textwidth}{!}{ \begin{tabular}{rcccccccccc} \hline N &$\Omega=16\pi$ & $\Omega=32\pi$ & $\Omega=64\pi$ & $\Omega=128\pi$ & $\Omega=256\pi$ & $\Omega=512\pi$ \\ \hline 1&1.62(-3)&1.64(-3)&1.65(-3)&1.65(-3)&1.65(-3)&1.65(-3)\\ 2&***&8.26(-5)&8.29(-5)&8.29(-5)&8.29(-5)&8.29(-5)\\ 4&***&***&4.72(-6)&4.73(-6)&4.73(-6)&4.73(-6)\\ 8&***&***&***&2.93(-7)&2.93(-7)&2.93(-7)\\ 16&***&***&***&***&1.83(-8)&1.83(-8)\\ 32&***&***&***&***&***&1.15(-9)\\ \hline \end{tabular}} \end{center} \label{tab:4} \end{table} Table~\ref{tab:4} corresponds to \eqref{eq:geneproblem} integrated with SAM-RK4. Errors for $\Omega=1024\pi$ are not reported because, with our facilities, the computation of the reference solution with dde23 would take several days. The error bounds are different from those for \eqref{eq:toggle} because for this tougher problem the microintegration bound is as in \eqref{eq:mu2} so that the impact \(\Omega\mu\) of the microintegration is now \(\Omega^4h^4\) or \(N^{-4}\); this impact is then \(\Omega\) independent. In fact, the main difference observed when comparing Tables~\ref{tab:3} and ~\ref{tab:4}, is that in Table~\ref{tab:4} errors along each row stay constant while in Table~\ref{tab:3} they decrease as $\Omega$ increases as discussed above. On the other hand, we observe that the same macro stepsizes used to integrate \eqref{eq:toggle} can be successfully used in this new, more challenging problem and lead to errors that are not widely different; this should be compared with the direct integration of the oscillatory problem with dde23 where the costs for \eqref{eq:geneproblem} are much higher than those for \eqref{eq:toggle}. \subsection{Results: case II} We consider again the integration of \eqref{eq:toggle} and \eqref{eq:geneproblem} but now set $\Omega = 25, 50, \ldots$ These values are not very different from the values used in the preceding section, but now the delay $\tau=0.5$ is not an integer multiple of the period $T=2\pi/\Omega$ of the fast oscillations. The integrators SAM-RK3, SAM-RK4 and SS-Z have been run with macro-stepsize $H=H_{max}/N$, $N=1, 2, 4, \ldots$ with $H_{max}=M T$, $M=\lfloor \tau/T \rfloor$. The micro-stepsize is again $h=T/(2N)$. As explained in Section~\ref{sec:case2}, in order to get solution values at integer multiples of $\tau$, each macrointegration from $0$ to $MT$ is followed by a short integration of the oscillatory problem from $MT$ to $\tau$. We only report a representative small sample of the experiments we performed. \begin{table}[t] \caption{Errors at \(t_{max}\) in $x_1$ for SAM-RK4 with respect to the reference solution for problem \eqref{eq:toggle}} \footnotesize \begin{center} \resizebox{\textwidth}{!}{ \begin{tabular}{rccccccccc} \hline N &$\Omega=50$ & $\Omega=100$ & $\Omega=200$ & $\Omega=400$ & $\Omega=800$ & $\Omega=1600$ \\ \hline 1&3.98(-3)&3.93(-3)&2.27(-3)&3.91(-4)&3.99(-4)&4.82(-5)\\ 2&***&2.16(-4)&1.55(-4)&2.21(-5)&1.84(-5)&3.37(-6)\\ 4&***&***&5.14(-6)&1.32(-6)&9.01(-7)&2.07(-7)\\ 8&***&***&***&8.79(-8)&5.46(-8)&1.71(-8)\\ 16&***&***&***&***&3.10(-9)&1.05(-9)\\ 32&***&***&***&***&***&5.56(-11)\\ \hline \end{tabular}} \end{center} \label{tab:5} \end{table} Table~\ref{tab:5} contains the errors in the first component of the solution of \eqref{eq:toggle} at the final time $t_{max}=4\tau$, with respect to the reference dde23 solution when the integration is performed with SAM-RK4. In \eqref{eq:mainbound2}, \(P\), \(\delta\) and \(\mu\) are as in Case I and the additional contribution \(\nu\) from the short integrations is \((\Omega h)^4\), i.e. \(N^{-4}\). The errors displayed in this table show a clear \(N^{-4}\) behaviour along the columns. However the variation with \(\Omega\) is now not so regular as we found in Table~\ref{tab:3} for Case I, no doubt because now changing \(\Omega\) changes the phase of the oscillation at the final time, where errors are measured. Finally, we report in Table~\ref{tab:6} errors in the first component of the solution of the challenging problem \eqref{eq:geneproblem}. This is to be compared with Table~\ref{tab:4}; again the error behaviour as a function of \(\Omega\) is now more irregular, but the methodology outlined in this paper finds no difficulty in accurately integrating the problem. \begin{table}[t] \caption{Errors at \(t_{max}\) in $x_1$ for SAM-RK4 with respect to the reference solution for problem \eqref{eq:geneproblem}} \footnotesize \begin{center} \resizebox{\textwidth}{!}{ \begin{tabular}{crcccccccccc} \hline &N &$\Omega=50$ & $\Omega=100$ & $\Omega=200$ & $\Omega=400$ & $\Omega=800$\\ \hline \hphantom{$\Omega$}&1&4.86(-3)&9.97(-3)&1.20(-2)&3.19(-3)&8.30(-3)&\hphantom{$1600$} \\ &2&***&5.46(-4)&8.01(-4)&2.46(-4)&3.80(-4)&\\ &4&***&***&2.63(-5)&1.45(-5)&1.89(-5)&\\ &8&***&***&***&9.33(-7)&1.15(-6)&\\ &16&***&***&***&***&6.56(-8)&\\ \hline \end{tabular}} \end{center} \label{tab:6} \end{table} \section{Proofs and additional results} \label{proofs} We conclude the paper by supplying the proofs of some results presented in Section~\ref{errorbounds}. We also present some extensions of those results. \subsection{Proof of Proposition~\ref{prop}} \label{proofs1} It is clearly sufficient to carry out the proof for the particular case of the scalar differential equation \(dy/ds = \exp(iks)\), with \(k\neq 0\) an integer. The true solution has the value \(y_0\) at \(s=2\pi\). If \(\{b_j\}_{j=1}^\sigma\) and \(\{c_j\}_{j=1}^\sigma\) are the weights and abscissas of the RK formula and \(M\Delta s=2\pi\), the numerical solution at \(s= 2\pi\) is \[ y_M = y_0 + \Delta s\sum_{m=0}^{M-1} \sum_{j=1}^\sigma b_j \exp(ik ( (m+c_j)\Delta s)). \] Hence \[ y_M-y_0 =\Delta s \sum_{j=1}^\sigma b_j \exp(ik c_j\Delta s) \sum_{m=0}^{M-1} \exp(ik m\Delta s ) . \] If \(\Delta s\) is sufficiently small \(\exp(ik\Delta s)\neq 1\) and the inner sum takes the value \[ \frac{\exp(ikM\Delta s) - 1}{\exp(ik\Delta s)- 1} =\frac {\exp(i2k\pi) - 1}{\exp(ik\Delta s)- 1} =0. \] As a result \(y_M = y_0\), i.e. \(y_M\) coincides with the true solution. \begin{rem} \label{rem:alias} \em If \(\exp(ik\Delta s)= 1\) with \(k\neq 0\), then \(\exp(iks)=1\) at all mesh points \( s= 0, \Delta s, 2\Delta s, \dots\), i.e.\ the oscillatory function \(\exp(iks)\) is an \emph{alias} of the constant function \(1\). In that case, the inner sum equals \(M\) and \[ y_M -y_0 = 2\pi \sum_{j=1}^\sigma b_j \exp(ik c_j\Delta s). \] Thus the RK solution is not exact at \(s=2\pi\). \end{rem} \subsection{Proof of the improved micro-integration estimates} \label{proofs2} Let us prove the error bound \eqref{eq:mu2}; the proof of \eqref{eq:mu3} follows the same pattern and will not be given. We start by noting that the solution of the initial value problem given by \(y(0) = y_0\) and \eqref{eq:perturbed} may be written as \(y = v+z\), where the pair \((v,z)\) is the solution of the extended problem \begin{eqnarray} \label{eq:estimate1} &&\frac{dv}{ds} = \Lambda(s),\qquad v(0) = 0,\\ \label{eq:estimate2} &&\frac{dz}{ds} = \Omega^{-1} f(v+z,s;\Omega), \qquad z(0) = y_0. \end{eqnarray} By writing the equations that define the RK solution, it is straightforward to check that, similarly, the RK trajectory \(y_0\), \(y_1\), \dots, \(y_M\) is given by \(y_m=v_m+z_m\), \(m= 0,\dots, M\), where \((v_0,z_0)\),\dots, \((v_M,z_M)\) is the RK trajectory for the initial value problem \eqref{eq:estimate1}--\eqref{eq:estimate2}. From the proposition we know that, for \(\Delta s\) small, the RK approximation to the \(v\) component of the extended solution is exact at \(s=2\pi\), i.e. \(v_M=v(2\pi)\) and the proof concludes by showing that the RK errors in the \(z\) component \(z_M-z(2\pi)\) possesses an \(\Omega^{-1} \mathcal{O}((\Delta s)^p))\) bound. The RK discretization of \eqref{eq:estimate1}--\eqref{eq:estimate2} is of the form \begin{eqnarray} \label{eq:estimate3} v_{m+1} & = &v_m + \Delta s F(m\Delta s,\Delta s), \\ \label{eq:estimate4} z_{m+1} & = & z_m + \Delta s \Omega^{-1}G(v_m+z_m, m\Delta s, \Delta s; \Omega), \end{eqnarray} where \(F\) and \(G\) are suitable increment functions; \(G\) and its derivatives are bounded as \(\Omega\uparrow \infty\). Clearly, for the quadrature in \eqref{eq:estimate1}, \(\max_m |v_m - v(m\Delta s)| = \mathcal{O}((\Delta s)^p)\), with the constant implied in the \(\mathcal{O}\) notation independent of \(\Omega\). For the \(z\) component we define the local error \(\eta_m\) by \[ z((m+1)\Delta s) = z(m\Delta s) + \Delta s \Omega^{-1}G(v(m\Delta s)+z(m\Delta s), m\Delta s, \Delta s; \Omega)+ \eta_m. \] Since the right hand-side of the equation in \eqref{eq:estimate2} has a prefactor \(\Omega^{-1}\), the same happens for all the associated elementary differentials \cite{Butcher,hlw} in the expansion of \(z\) and as a consequence \(\max_m |\eta_m|= \Omega^{-1}\mathcal{O}((\Delta s)^{p+1})\) (again the implied constant is \(\Omega\)-independent). Subtraction of the last display from \eqref{eq:estimate4} leads to (\(C\) denotes an \(\Omega\)-independent Lipschitz constant) \begin{eqnarray*} |z_{m+1}-z((m+1)\Delta s)| &\leq& |z_{m}-z(m\Delta s)|\\ &&+\Delta s \Omega^{-1} C \Big(|v_{m}-v(m\Delta s)| +|z_{m}-z(m\Delta s)|\Big)\\&&+ |\eta_{m}|\\ &=& (1+\Delta s \Omega^{-1} C)|z_{m}-z(m\Delta s)| +\Delta s \Omega^{-1} \mathcal{O}((\Delta s)^p), \end{eqnarray*} and recursively we arrive at \(\max_m |z_{m}-z(m\Delta s)| = \Omega^{-1} \mathcal{O}((\Delta s)^p)\) and the proof is ready. \subsection{Extensions} The improved bounds \eqref{eq:mu2} and \eqref{eq:mu3} are based on Proposition~\ref{prop}. This proposition does not hold if \(\Lambda(s)\) is merely a smooth \(2\pi\)-periodic function rather than a trigonometric polynomial. In fact, if \(\Lambda\) contains infinitely many Fourier modes, then for each choice of \(\Delta s =2\pi/M\) there will be modes \(\exp(iks)\) that are alias of the function \(1 = \exp(i0s)\) and therefore are not exactly integrated as we know from Remark~\ref{rem:alias}. However for \(\Lambda\) smooth and \(2\pi\)-periodic it is still possible to derive superconvergence results that show that the RK solution with \(\Delta s=2\pi/M\) is more accurate at \(s=2\pi\) than it is for \(s<2\pi\). Those results are derived by decomposing the solution in a Fourier series. If \(\Lambda\) has derivatives of all orders, then the RK error at the final point may be proved to be \(\mathcal{O}((\Delta s)^q)\) for arbitrary \(q>0\). Under analyticity assumptions, the error may decrease exponentially. The situation is very similar to that of the trapezoidal rule studied in \cite{Trefethen}. (In fact, due to the periodicity, the sum \(\sum_{m=0}^{M-1} \exp(ik m\Delta s )\) we encountered in Section~\ref{proofs1} may be written in trapezoidal form \({\sum_{m=0}^{\prime\prime M}} \exp(ik m\Delta s )\), where the double prime indicates that the first and last terms are halved.) By using the technique in Section~\ref{proofs2} the superconvergence results for \((d/ds)y = \Lambda(s)\) give rise to improved micro-integration bounds for problems of the form \eqref{eq:perturbed} with \(f\) bounded and \(\Lambda\) \(2\pi\)-periodic or for the case where in \eqref{eq:sam1} \(f\) and its derivatives remain bounded as \(\Omega\) increases. \end{document}

\begin{document} \title[] {Instantaneously complete Chern-Ricci flow and K\"ahler-Einstein metrics } \author{Shaochuang Huang$^1$} \address[Shaochuang Huang]{Yau Mathematical Sciences Center, Tsinghua University, Beijing, China.} \email{[email protected]} \thanks{$^1$Research partially supported by China Postdoctoral Science Foundation \#2017T100059} \author{Man-Chun Lee} \address[Man-Chun Lee]{Department of Mathematics, University of British Columbia, Canada} \email{[email protected]} \author{Luen-Fai Tam$^2$} \address[Luen-Fai Tam]{The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China.} \email{[email protected]} \thanks{$^2$Research partially supported by Hong Kong RGC General Research Fund \#CUHK 14301517} \renewcommand{\subjclassname}{ \textup{2010} Mathematics Subject Classification} \subjclass[2010]{Primary 32Q15; Secondary 53C44 } \date{February 2019} \begin{abstract} In this work, we obtain some existence results of {\it Chern-Ricci Flows} and the corresponding {\it Potential Flows} on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\to 0$. These results can be viewed as a generalization of an existence result of Ricci flow by Giesen and Topping for surfaces of hyperbolic type to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of K\"ahler-Einstein metric on complete non-compact Hermitian manifolds, which generalizes the work of Lott-Zhang and Tosatti-Weinkove to complete non-compact Hermitian manifolds with possibly unbounded curvature. \end{abstract} \keywords{Chern-Ricci flow, instantaneous completeness, K\"ahler-Einstein metric} \maketitle \markboth{Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam}{Instantaneously complete Chern-Ricci flow and K\"ahler-Einstein metrics } \section{Introduction} In this work, we will discuss conditions on the existence of {\it Chern-Ricci Flows} and the corresponding {\it Potential Flows} on complex manifolds with possibly incomplete initial data. The flows will be described later. We will also discuss conditions on long-time existence and convergence to K\"ahler-Einstein metrics. We begin with the definitions of Chern-Ricci flow and the corresponding potential flow. Let $M^n$ be a complex manifold with complex dimension $n$. Let $h$ be a Hermitian metric on $M$ and let $\theta_0$ be the K\"ahler form of $h$: $$ \theta_0=\sqrt{-1} h_{i\bar{j}} dz^i\wedge d\bar z^j $$ where $h=h_{i\bar{j}} dz^i\otimes d\bar z^j$ in local holomorphic coordinates. {In this work, Einstein summation convention is enforced.} In general, suppose $\omega$ is a real (1,1) form on $M$, if $\omega=\sqrt{-1} g_{i\bar{j}} dz^i\wedge d\bar z^j$ in local holomorphic coordinates then the corresponding Hermitian form $g$ is given by $$ g=g_{i\bar{j}} dz^i\otimes d\bar z^j. $$ In case $\omega$ is only nonnegative, we call $g$ to be the Hermitian form of $\omega$ and $\omega$ is still called the K\"ahler form of $g$. Now if $(M^n,h)$ is a Hermitian manifold with K\"ahler form $\theta_0$, let $\nabla$ be the Chern connection $\nabla $ of $h$ and $\text{\rm Ric}(h)$ be the Chern-Ricci tensor of $h$ (or the first Ricci curvature). In holomorphic local coordinates such that $h=h_{i\bar{j}} dz^i\otimes d\bar z^j$, the Chern Ricci form is given by $$ \text{\rm Ric}(h)=-\sqrt{-1} \partial\bar\partial \log \det(h_{i\bar{j}}). $$ For the basic facts on Chern connection and Chern curvature, we refer readers to \cite[section 2]{ TosattiWeinkove2015}, see also \cite[Appendix A]{Lee-Tam} for example. Let $\omega_0$ be another nonnegative real (1,1) form on $M$. Define \begin{equation}\label{e-alpha} {\alpha}:=-\text{\rm Ric}(\theta_0)+e^{-t}\left(\text{\rm Ric}(\theta_0)+\omega_0\right) \end{equation} where $\text{\rm Ric}(\theta_0)$ is the Chern-Ricci curvature of $h$. We want to study the following parabolic complex Monge-Amp\`ere equation: \begin{equation}\label{e-MP-1} \left\{ \begin{array}{ll} {\displaystyle \frac{\partial u}{\partial t}}&=\displaystyle{\log\left(\frac{({\alpha}+\sqrt{-1}\partial\bar\partial u)^n}{\theta_0^n}\right)}-u\ \ \text{in $M\times(0,S]$} \\ u(0)&=0 \end{array} \right. \end{equation} so that ${\alpha}+\sqrt{-1}\partial\bar\partial u>0$ for $t>0$. When $M$ is compact and $\omega_0=\theta_0$ is smooth metric, it was first studied by Gill in \cite{Gill}. Here we are interested in the case when $\omega_0$ is possibly an incomplete metric on a complete non-compact Hermitian manifold $(M,h)$. Following \cite{Lott-Zhang}, \eqref{e-MP-1} will be called the {\it potential flow} of the following normalized Chern-Ricci flow: \begin{equation}\label{e-NKRF} \left\{ \begin{array}{ll} {\displaystyle \frac{\partial}{\partial t}\omega(t)} &= -\text{\rm Ric}(\omega(t))-\omega(t); \\ \omega(0)&= \omega_0. \end{array} \right. \end{equation} It is easy to see that the normalized Chern-Ricci flow will coincide with the normalized K\"ahler-Ricci flow if $\omega_0$ is K\"ahler. It is well-known that if $\omega_0$ is a Hermitian metric and $\omega(t)$ is Hermitian and a solution to \eqref{e-NKRF} which is smooth up to $t=0$, then \begin{equation}\label{e-potential} u(t)=e^{-t}\int_0^te^s\log \frac{(\omega (s))^n}{\theta_0^n}ds. \end{equation} satisfies \eqref{e-MP-1}. Moreover, $u(t)\to0$ in $C^\infty$ norm in any compact set as $t\to0$. On the other hand, if $u$ is a solution to \eqref{e-MP-1} so that ${\alpha}+\sqrt{-1}\partial\bar\partial u>0$ for $t>0$, then \begin{equation}\label{e-potential-1} \omega(t)={\alpha}+\sqrt{-1}\partial\bar\partial u \end{equation} is a solution to \eqref{e-NKRF} on $M\times(0,S]$. However, even if we know $u(t)\to0$ as $t\to0$ uniformly on $M$, it is still unclear that $\omega(t)\to\omega_0$ in general. The first motivation is to study Ricci flows starting from metrics which are possibly incomplete and with unbounded curvature. In complex dimension one, the existence of Ricci flow starting from an arbitrary metric has been studied in details by Giesen and Topping \cite{GiesenTopping-1,GiesenTopping-2, GiesenTopping,Topping}. In particular, the following was proved in \cite{GiesenTopping}: {\it If a surface admits a complete metric $H$ with constant negative curvature, then any initial data which may be incomplete can be deformed through the normalized Ricci flow for long time and converges to $H$. Moreover, the solution is instantaneously complete for $t>0$.} In higher dimensions, recently it is proved by Ge-Lin-Shen \cite{Ge-Lin-Shen} that on a complete non-compact K\"ahler manifold $(M,h)$ with $\text{\rm Ric}(h)\leq -h$ and bounded curvature, if $\omega_0$ is a K\"ahler metric, not necessarily complete, but with bounded $C^k$ norm with respect to $h$ for $k\ge 0$, then \eqref{e-NKRF} has a long time solution which converges to the unique K\"ahler-Einstein metric with negative scalar curvature, by solving \eqref{e-MP-1}. Moreover, the solution is instantaneously complete after it evolves. Motivated by the above mentioned works, we first study the short time existence of the potential flow and the normalized Chern-Ricci flow. Our first result is the following: \begin{thm}\label{main-instant-complete} Let $(M^n,h)$ be a complete non-compact Hermitian manifold with complex dimension $n$. Suppose there is $K>0$ such that the following hold. \begin{enumerate} \item There is a {proper} exhaustion function $\rho(x)$ on $M$ such that $$|\partial\rho|^2_h +|\sqrt{-1}\partial\bar\partial \rho|_h \leq K.$$ \item $\mathrm{BK}_h\geq -K$; \item The torsion of $h$, $T_h=\partial \omega_h$ satisfies $$|T_h|^2_h +|\nabla^h_{\bar\partial} T_h |\leq K.$$ \end{enumerate} Let $\omega_0$ be a nonnegative real (1,1) form with corresponding Hermitian form $g_0$ on $M$ (possibly incomplete {or degenerate}) such that \begin{enumerate} \item[(a)] $g_0\le h$ and $$|T_{g_0}|_h^2+|\nabla^h_{\bar\partial} T_{g_0}|_h+ |\nabla^{h}g_0|_h\leq K.$$ \item[(b)] There exist $f\in C^\infty(M)\cap L^\infty(M),\beta>0$ and $s>0$ so that $$-\text{\rm Ric}(\theta_0)+e^{-s}(\omega_0+\text{\rm Ric}(\theta_0))+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0.$$ \end{enumerate} Then \eqref{e-MP-1} has a solution on $M\times(0, s)$ so that $u(t)\to 0$ as $t\to0$ uniformly on $M$. Moreover, for any $0<s_0<s_1<s$, $\omega(t)={\alpha}+\sqrt{-1}\partial\bar\partial u$ is the K\"ahler form of a complete Hermitian metric which is uniformly equivalent to $h$ on $M\times[s_0, s_1]$. In particular, $g(t)$ is complete for $t>0$. \end{thm} Here $\mathrm{BK}_h\geq -K$ means that for any unitary frame $\{e_k\}$ of $h$, we have $R(h)_{i\bar ij\bar j}\geq -K$ for all $i,j$. \begin{rem} It is well-known that when $(M,h)$ is K\"ahler with bounded curvature, then condition (1) will be satisfied, \cite{Shi1989,Tam2010}. See also \cite{NiTam2013,Huang2018} for related results under various assumptions. \end{rem} Condition (b) was used in \cite{Lott-Zhang,TosattiWeinkove2015, Lee-Tam} with $\omega_0$ replaced by $\theta_0$ and is motivated as pointed out in \cite{Lott-Zhang} as follows. If we are considering cohomological class instead, in case that $\omega(t)$ is closed, then \eqref{e-NKRF} is: $$ \partial_t[\omega(t)]=-[\text{\rm Ric}(\omega(t)]-[\omega(t)] $$ and so $$ [\omega(t)]=-(1-e^{-t})[\text{\rm Ric}(\theta_0)]+e^{-t}[\omega_0]. $$ Condition (b) is used to guarantee that $\omega(t)>0$. In our case $\omega_0,\theta_0, \omega(t)$ may not be closed and $\omega_0$ may degenerate. These may cause some difficulties. Indeed, the result is analogous to running K\"ahler-Ricci flow from a rough initial data. When $M$ is compact, the potential flow from a rough initial data had already been studied by several authors, see for example \cite{BG2013,SongTian2017,To2017} and the references therein. On the other hand, a solution of \eqref{e-MP-1} gives rise to a solution of \eqref{e-NKRF} when $t>0$. It is rather delicate to see if the corresponding solution of \eqref{e-NKRF} will attain the initial Hermitian form $\omega_0$. In this respect, we will prove the following: \begin{thm}\label{t-initial-Kahler-1} With the same notation and assumptions as in Theorem \ref{main-instant-complete}. Let $\omega(t)$ be the solution of \eqref{e-NKRF} obtained in the theorem. If in addition $h$ is K\"ahler and $d\omega_0=0$. Let $U=\{\omega_0>0\}$. Then $\omega(t)\rightarrow \omega_0$ in $C^\infty(U)$ as $t\rightarrow 0$, {uniformly on compact subsets of $U$}. \end{thm} We should remark that if in addition $h$ has bounded curvature, then the theorem follows easily from pseudo-locality. The theorem can be applied to the cases studied in \cite{Ge-Lin-Shen} and to the case that $-\text{\rm Ric}(h)\ge {\beta}\theta_0$ outside a compact set $V$ and $\omega_0>0$ on $V$ with $\omega_0$ and its first covariant derivative are bounded. In particular, when $\Omega$ is a bounded strictly pseudoconvex domain of another manifold $M$ with defining function $\varphi$, then the $\Omega$ with the metric $h_{i\bar j}=-\partial_i \partial_{\bar j}\log(-\varphi)$ will satisfy the above, see \cite[(1.22)]{ChengYau1982}. Another motivation here is to study the existence of K\"ahler-Einstein metric with negative scalar curvature on complex manifolds using geometric flows. In \cite{Aubin, Yau1978-2}, Aubin and Yau proved that if $M$ is a compact K\"ahler manifold with negative first Chern class $c_1(M)<0$, then it admits a unique K\"ahler-Einstein metric with negative scalar curvature by studying the elliptic complex Monge-Amp\`ere equation. Later, Cao \cite{Cao} reproved the above result using the K\"ahler-Ricci flow by showing that one can deform a suitable initial K\"ahler metric through normalized K\"ahler-Ricci flow to the K\"ahler-Einstein metric. Recently, Tosatti and Weinkove \cite{TosattiWeinkove2015} proved that under the same condition that $c_1(M)<0$ on a compact complex manifold, the normalized Chern-Ricci flow \eqref{e-NKRF} with an arbitrary Hermitian initial metric also has a long time solution and converges to the K\"ahler-Einstein metric with negative scalar curvature. In \cite{ChengYau1982}, Cheng and Yau proved that if $M$ is a complete non-compact K\"ahler manifold with Ricci curvature bounded above by a negative constant, injectivity radius bounded below by a positive constant and curvature tensor with its covariant derivatives are bounded, then $M$ admits a unique complete K\"ahler-Einstein metric with negative scalar curvature. In \cite{Chau04}, Chau used K\"ahler-Ricci flow to prove that if $(M, g)$ is a complete non-compact K\"ahler manifold with bounded curvature and $\text{\rm Ric}(g)+g=\sqrt{-1}\partial\bar\partial f $ for some smooth bounded function $f$, then it also admits a complete K\"ahler-Einstein metric with negative scalar curvature. Later, Lott and Zhang \cite{Lott-Zhang} generalized Chau's result by assuming $$-\text{\rm Ric}(g)+\sqrt{-1}\partial\bar\partial f\ge{\beta} g$$ for some smooth function $f$ with bounded $k$th covariant derivatives for each $k\geq0$ and positive constant ${\beta}$. In this work, we will generalize the results in \cite{Lott-Zhang,TosattiWeinkove2015} to complete non-compact Hermitian manifolds with possibly unbounded curvature. For the long time existence and convergence, we will prove the following: \begin{thm}\label{main-longtime} Under the assumption of Theorem \ref{main-instant-complete}, if in addition, $$-\text{\rm Ric}(h)+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0$$ for some $f\in C^\infty(M)\cap L^\infty(M)$, $\beta>0$. Then the solution constructed from Theorem \ref{main-instant-complete} is a longtime solution and converges to a unique complete K\"ahler Einstein metric with negative scalar curvature on $M$. \end{thm} As a consequence, we see that if $h$ satisfies the conditions in the theorem, then $M$ supports a complete K\"ahler-Einstein metric with negative scalar curvature, generalizing the results in \cite{Lott-Zhang,TosattiWeinkove2015}. The paper is organized as follows: In section 2, we will derive a priori estimates along the potential flow and apply it in section 3 to prove Theorem \ref{main-instant-complete}. Furthermore, we will study the short time behaviour of the constructed solution. In section 4, we will prove the Theorem \ref{main-longtime} and discuss longtime behaviour for general K\"ahler-Ricci flow if the initial data satisfies some extra condition. In Appendix A, we will collect some information about the relation between normalized Chern-Ricci flow and unnormalized one {together with some useful differential inequalities. In Appendix B, we will state a maximum principle which will be used in this work.} \section{a priori estimates for the potential flow}\label{s-aprior} We will study the short time existence of the potential flow \eqref{e-MP-1} with $\omega_0$ only being assumed to be nonnegative. We need some a priori estimates for the flow. In this section, we always assume the following: \begin{enumerate} \item There is a {proper} exhaustion function $\rho(x)$ on $M$ such that $$|\partial\rho|^2_h +|\sqrt{-1}\partial\bar\partial \rho|_h \leq K.$$ \item $\mathrm{BK}_h\geq -K$. \item The torsion of $h$, $T_h=\partial \omega_h$ satisfies $$|T_h|^2_h +|\nabla^h_{\bar\partial} T_h |\leq K.$$ \end{enumerate} Here $K$ is some positive constant. On the other hand, let $\omega_0$ be a real (1,1) form with corresponding Hermitian form $g_0$. We always assume the following: \begin{enumerate} \item[(a)] $g_0\le h$ and $$|T_{g_0}|_h^2+|\nabla^h_{\bar\partial} T_{g_0}|_h+ |\nabla^{h}g_0|_h\leq K.$$ \item[(b)] There exist $f\in C^\infty(M)\cap L^\infty(M),\beta>0$ and $s>0$ so that $$-\text{\rm Ric}(\theta_0)+e^{-s}(\omega_0+\text{\rm Ric}(\theta_0))+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0.$$ \end{enumerate} Note that if $g_0\le Ch$, then we can replace $h$ by $Ch$, then (b) is still satisfied with a possibly smaller $\beta$. Since $g_0$ can be degenerate, we perturb $g_0$ in the following way: Let $1\ge \eta\ge 0$ be a smooth function on $\mathbb{R}$ such that $\eta(s)=1$ for $s\le 1$ and $\eta(s)=0$ for $s\ge 2$ so that $|\eta'|+|\eta''|\le c_1$, say. For $\epsilon>0$ and $\rho_0>>1$, let $\eta_{0}(x)=\eta(\rho(x)/\rho_0)$. Consider the metric: \begin{equation} \gamma_0=\gamma_0(\rho_0,\epsilon)=\eta_0\omega_0+(1-\eta_0)\theta_0+\epsilon\theta_0. \end{equation} Then \begin{itemize} \item $\gamma_0$ is the K\"ahler form of a complete Hermitian metric, which is uniformly equivalent to $h$; \item $\mathrm{BK}(\gamma_0 )\ge -C$ for some $C$ which may depend on $\rho_0, \epsilon$; \item The torsion $|T_{\gamma_0} |_{\gamma_0}+|\nabla^{\gamma_0}_{\bar \partial} T_{\gamma_0}|_{\gamma_0}$ is uniformly bounded by a constant which may depend on $\rho_0, \epsilon$. \end{itemize} We will obtain a short time existence for the potential flow starting with $\gamma_0$: \begin{lma}\label{l-perturbed-1} \eqref{e-MP-1} has a solution $u(t)$ on $M\times[0, s)$ with ${\alpha}=-\text{\rm Ric}(\theta_0)+e^{-t}\left(\text{\rm Ric}(\theta_0)+\gamma_0\right)$ and $\omega(t)={\alpha}+\sqrt{-1}\partial\bar\partial u$ such that $\omega(t)$ satisfies \eqref{e-NKRF} with initial data $\gamma_0$, where $\omega(t)$ is the K\"ahler form of $g(t)$. Moreover, $g(t)$ is uniformly equivalent to $h$ on $M\times[0, s_1]$ for all $s_1<s$. \end{lma} \begin{proof} By the proof of \cite[Theorem 4.1]{Lee-Tam}, it is sufficient to prove that for any $0<s_1<s$, $$ -\text{\rm Ric}(\gamma_0)+e^{-s_1}(\gamma_0+\text{\rm Ric}(\gamma_0))+\sqrt{-1}\partial\bar\partial f_1\ge \beta_1\gamma_0 $$ for some smooth bounded function $f_1$ and some constant $\beta_1>0$. To simplify the notations, if $\eta, \zeta$ are real (1,1) forms, we write $\eta \succeq \zeta$ if $\eta+\sqrt{-1}\partial\bar\partial \phi\ge \zeta$ for some smooth and bounded function $\phi$. We compute: \begin{equation*}\begin{split} -\text{\rm Ric}(\gamma_0)+e^{-s_1}(\gamma_0+\text{\rm Ric}(\gamma_0)) =&-(1-e^{-s_1})\text{\rm Ric}(\gamma_0)+e^{-s_1}\gamma_0\\ \succeq&-(1-e^{-s_1})\text{\rm Ric}(\theta_0)+e^{-s_1}\gamma_0\\ \succeq&\frac{1-e^{-s_1}}{1-e^{-s}}(\beta \theta_0-e^{-s}\omega_0)+e^{-s_1}\gamma_0 \\ \ge&\frac{1-e^{-s_1}}{1-e^{-s}} \beta \theta_0 \\ \ge& \beta_1\gamma_0\end{split}\end{equation*} for some $\beta_1>0$ because $0<s_1<s$ and $\gamma_0\ge \omega_0$. Here we have used {condition (b) above}, the fact that $\gamma_0^n =\theta_0^ne^H$ for some smooth bounded function $H$ and the definition of Chern-Ricci curvature. \end{proof} Let $\omega(t)$ be the solution in the lemma and let $u(t)$ be the potential as in \eqref{e-potential}. Since we want to prove that \eqref{e-MP-1} has a solution $u(t)$ on $M\times(0, s)$ with ${\alpha}=-\text{\rm Ric}(\theta_0)+e^{-t}\left(\text{\rm Ric}(\theta_0)+\omega_0\right)$ in next section, we need to obtain some uniform estimates of $u, \dot u$ and $\omega(t)$ which is independent of $\rho_0$ and $\epsilon$. The estimates are more delicate because the initial data $\omega_0$ maybe degenerate. For later applications, we need to obtain estimates on $(0,1]$ and $[1,s)$ if $s>1$. Note that for fixed $\rho_0, \epsilon$, $u(t)$ is smooth up to $t=0$. Moreover, $u, \dot u=:\frac{\partial}{\partial t}u$ are uniformly bounded on $M\times[0,s_1]$ for all $0<s_1<s$. \subsection{a priori estimates for $u$ and $\dot u$}\label{ss-uudot} We first give estimates for upper bound of $u$ and $\dot u$. \begin{lma}\label{l-uudot-upper-1} There is a constant $C$ depending only on $n$ and $K$ such that $$ u\le C\min\{t,1\}, \ \ \dot u\le \frac{Ct}{e^t-1} $$ on $M\times[0, s)$, provided $0<\epsilon<1$. \end{lma} \begin{proof}The proofs here follow almost verbatim from the K\"ahler case \cite{TianZhang2006}, but we include brief arguments for the reader's convenience. For notational convenience, we use $\Delta=g^{i\bar j} \partial_i \partial_{\bar j}$ to denote the Chern Laplacian associated to $g(t)$. Since $-\text{\rm Ric}(\theta_0)=\omega(t)-e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)-\sqrt{-1}\partial\bar\partial u$ by \eqref{e-potential-1}, we have \begin{equation}\label{e-udot-1} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) (e^t\dot u)=&e^t\dot u-e^t \operatorname{tr}_{\omega}\text{\rm Ric}(\theta_0)-e^t\lf(\frac{\p}{\p t}-\Delta\ri) u-n e^t\\ =&e^t\operatorname{tr}_\omega \left(- \text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial u\right)-ne^t\\ =&e^t\operatorname{tr}_\omega\left(\omega-e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)\right)-ne^t\\ =& -\operatorname{tr}_\omega (\text{\rm Ric}(\theta_0)+\gamma_0)\\ =&\lf(\frac{\p}{\p t}-\Delta\ri) (\dot u+u)+n -\operatorname{tr}_\omega(\gamma_0). \end{split} \end{equation} Hence \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri)(\dot u+u+nt-e^t\dot u)=\operatorname{tr}_\omega\gamma_0\ge0. \end{equation*} At $t=0$, $\dot u+u+nt-e^t\dot u=0$. By maximum principle Lemma \ref{max}, we have \begin{equation}\label{e-uudot-upper-1} (e^t-1)\dot u\le nt+u. \end{equation} Next consider \begin{equation*} F=u-At-\kappa\rho \end{equation*} on $M\times[0, s_1]$ for any fixed $s_1<s$. Here $\kappa>0$ is a constant. Suppose $\sup\limits_{M\times[0, s_1]}F>0$, then there exists $(x_0, t_0)\in M\times(0, s_1]$ such that $F\leq F(x_0, t_0)$ on $M\times[0, s_1]$, and at this point, \begin{equation*}\begin{split} 0\leq& \dot u -A=\log \left(\frac{\omega^n(t)}{\theta_0^n}\right)-u-A. \end{split} \end{equation*} Also, $\sqrt{-1}\partial\bar\partial u\le \kappa\sqrt{-1}\partial\bar\partial \rho\le \kappa K\theta_0$. Hence at $(x_0,t_0)$, \begin{equation*} \begin{split} \omega(t)=&-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)+\sqrt{-1}\partial\bar\partial u\\ \le&(-1+e^{-t})\text{\rm Ric}(\theta_0)+e^{-t}\gamma_0+\kappa K\theta_0\\ \le &(L+2+\kappa K)\theta_0, \end{split} \end{equation*} here $\text{\rm Ric}(\theta_0)\ge -L(n, K)\theta_0$. Hence at $(x_0,t_0)$ we have \begin{equation*} \begin{split} u\le & n\log(L+2+\kappa K)-A\\ \le &0 \end{split} \end{equation*} if $A=n\log(L+2)+1$ and $\kappa>0$ is small enough. Hence $F(x_0,t_0)<0$. This is a contradiction. Hence $F\le 0$ on $M\times[0, s_1]$ provided $A=A(n,K)=n\log(L+2)+1$ and we have \begin{equation}\label{e-uudot-upper-2} u\le At \end{equation} by letting $\kappa\to0$. Combining this with \eqref{e-uudot-upper-1}, we conclude that $$ \dot u\le \frac{(A+n)t}{e^t-1}. $$ Combining this with \eqref{e-uudot-upper-2}, we conclude that $u\le C$ for some constant $C$ depending only on $n, K$. Since $s_1$ is arbitrary, we complete the proof of Lemma \ref{l-uudot-upper-1}. \end{proof} Next, we will estimate the lower bound of $u$ and $\dot u$. \begin{lma}\label{l-all-u}\begin{enumerate} \item[(i)] $u(x,t)\geq - \frac{C}{1-e^{-s}} t+nt\log(1-e^{-t})$ on $M\times[0, s)$ for some constant $C>0$ depending only on $ n, \beta, K, ||f||_\infty$. \item [(ii)] For $0<s_1\leq 1$ and $s_1<s$, \begin{equation*} \dot u+u\ge\frac1{1-e^{s_1-s}}\left(n\log t-\frac{C}{1-e^{-s}}\right) \end{equation*} some constant $C>0$ depending only on $ n, \beta, K, ||f||_\infty$ on $M\times(0, s_1]$. \item [(iii)] For $0<s_1\leq 1$ and $s_1<s$, $$\dot u+u\geq -C$$ on $M\times[0, s_1]$ for some constant $C>$ depending only on $ n, \beta$, $K, ||f||_\infty, s_1, s$ and $\epsilon$. \item [(iv)] Suppose $s>1$, then for $1<s_1<s$, $$\dot u+u\ge -\frac{C(1+s_1e^{s_1-s})}{1-e^{s_1-s}}$$ on $M\times[1,s_1]$ for some constant $C(n, \beta, ||f||_\infty, K)>0$. \item[(v)] For $0<s_1<s$, $$u\ge -\frac{C(1+s_1e^{s_1-s})}{1-e^{s_1-s}}$$ on $M\times[0,s_1]$ for some constant $C(n, \beta, ||f||_\infty, K)>0$. \end{enumerate} \end{lma} \begin{proof} In the following, $C_i$ will denote positive constants depending only on $n, \beta, ||f||_\infty, K$ and $D_i$ will denote positive constants which may also depend on $\rho_0, \epsilon$ but not on $\kappa$. To prove (i): Consider \begin{equation*} F=u(x,t)-\frac{1-e^{-t}}{1-e^{-s}}f(x)+A\cdot t-nt\log(1-e^{-t})+\kappa\rho(x) .\end{equation*} Suppose $\inf\limits_{M\times[0, s_1]}F<0$. Then there exists $(x_0, t_0)\in M\times(0, s_1]$ such that $F\geq F(x_0, t_0)$ on $M\times[0, s_1]$. At this point, we have \begin{equation*}\begin{split} 0\geq & \frac{\p}{\p t} F \\=&\dot u+A-\frac{e^{-t}}{1-e^{-s}}f(x)-n\log(1-e^{-t})-\frac{nt}{e^t-1}.\\ =&\log\frac{(-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)+\sqrt{-1}\partial\bar\partial u)^n}{\theta_0^n}-u +A\\ &-n\log(1-e^{-t})-\frac{nt}{e^t-1}-\frac{e^{-t}}{1-e^{-s}}f\\ \geq& \log\frac{(-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)+ \frac{1-e^{-t}}{1-e^{-s}}\sqrt{-1}\partial\bar\partial f-\kappa\sqrt{-1}\partial\bar\partial\rho)^n}{\theta_0^n}\\ &-C(n, K)-\frac{e^{-t}}{1-e^{-s}}f+A-n\log(1-e^{-t})-\frac{nt}{e^t-1},\\ \end{split} \end{equation*} where we have used the fact that $u\leq C(n, K)$, and $\sqrt{-1}\partial\bar\partial u\ge\frac{1-e^{-t}}{1-e^{-s}}\sqrt{-1}\partial\bar\partial f-\kappa\sqrt{-1}\partial\bar\partial \rho$. Note that $$ -\text{\rm Ric}(\theta_0)\ge\frac1{1-e^{-s}}\left(\beta\theta_0-e^{-s}\omega_0-\sqrt{-1}\partial\bar\partial f\right), $$ hence \begin{equation*} \begin{split} &-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\gamma_0)+\frac{1-e^{-t}}{1-e^{-s}}\sqrt{-1}\partial\bar\partial f-\kappa\sqrt{-1}\partial\bar\partial\rho\\ \ge&e^{-t}\gamma_0+\frac{1-e^{-t}}{1-e^{-s}}\left(\beta\theta_0-e^{-s}\omega_0 \right) -\kappa K\theta_0 \\ \ge& \frac{1}{2}\frac{1-e^{-t}}{1-e^{-s}} \beta\theta_0 \end{split} \end{equation*} if $\kappa $ is small enough. Here we have used the fact that $0<t<s$ and $\gamma_0\ge \omega_0$. Hence at $(x_0,t_0)$, \begin{equation*} \begin{split} 0\geq& n\log(1-e^{-t}) -C_1 \\ &-\frac{e^{-t}}{1-e^{-s}}f+A-n\log(1-e^{-t})-\frac{nt}{e^t-1}\\ \geq& -\frac{1}{1-e^{-s}}||f||_\infty+A-C_2 \\ >&0 \end{split} \end{equation*} if $A=\frac{1}{1-e^{-s}}||f||_\infty+C_2+1$. Hence for such $A$, $F\ge 0$ and for all $\kappa>0$ small enough, we conclude that $$ u(x,t)\ge -At+nt\log(1-e^{-t}). $$ To prove (ii), we have \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri)(\dot u+u)=-\operatorname{tr}_\omega(\text{\rm Ric}(\theta_0))-n. \end{equation*} On the other hand, by \eqref{e-udot-1}, we also have \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) (e^t\dot u)=-\operatorname{tr}_\omega(\text{\rm Ric}(\theta_0)+\gamma_0). \end{equation*} Hence \begin{equation}\label{e-udot-2} \begin{split} & \lf(\frac{\p}{\p t}-\Delta\ri)\left((1-e^{t-s})\dot u+u\right)\\ =&\operatorname{tr}_\omega(-\text{\rm Ric}(\theta_0)+e^{-s}(\text{\rm Ric}(\theta_0)+\gamma_0))-n\\ \ge&\beta\operatorname{tr}_\omega(\theta_0)-\Delta f-n. \end{split} \end{equation} Let $F=(1-e^{t-s})\dot u+u-f-A\log t+\kappa\rho$, where $A>0$ is a constant to be determined. Since $\log t\to-\infty$ as $t\rightarrow 0$, we conclude that for $0<s_1<s$, if $\inf_{M\times[0,s_1]}F\le 0$, then there is $(x_0,t_0)\in M\times(0,s_1]$ so that $F(x_0,t_0)=\inf_{M\times[0,s_1]}F$. By \eqref{e-udot-2}, at $(x_0,t_0)$ we have \begin{equation*} \begin{split} 0\ge&\lf(\frac{\p}{\p t}-\Delta\ri) F\\ \ge&\beta\operatorname{tr}_\omega(\theta_0)-n-\frac At-\kappa D_1\\ \ge& n\beta\exp(-\frac1n(\dot u+u))-n-\frac At-\kappa D_1 \end{split} \end{equation*} where $D_1>0$ is a constant independent of $\kappa$. Hence at this point, $$ \dot u+u\ge -n\log\left(\frac1{n\beta}(n+\frac At+\kappa D_1)\right). $$ Hence at $(x_0,t_0)$, noting that $0<t_0\le s_1<s$ and $s_1\le 1$, \begin{equation*} \begin{split} F\geq& (1-e^{t-s})(\dot u+u)+e^{t-s}u-f-A\log t\\ \ge&-(1-e^{t-s})n\log\left(\frac1{n\beta}(n+\frac At+\kappa D_1)\right)-\sup_M f-A\log t\\ &- \frac{C_3}{1-e^{-s}}+nt\log(1-e^{-t})\\ \ge&[(1-e^{t-s})n-A]\log t-(1-e^{t-s})n\log\left(\frac1{n\beta}(nt+A+\kappa t D_1)\right)\\ &-||f||_\infty-\frac{C_4}{1-e^{-s}} \\ \ge &- n\log\left(\frac1{n\beta}(2n+\kappa D_1)\right)-||f||_\infty-\frac{C_4}{1-e^{-s}}\end{split} \end{equation*} if $A=n$. Here we may assume that $\beta>0$ is small enough so that $2/\beta>1$. Hence we have \begin{align*} F\ge - n\log\left(\frac1{n\beta}(2n+\kappa D_1)\right)-||f||_\infty-\frac{C_4}{1-e^{-s}}. \end{align*} on $M\times(0,s_1]$. Let $\kappa\to0$, we conclude that \begin{equation*} \begin{split} (1-e^{t-s})\left(\dot u+u\right)=& (1-e^{t-s})\dot u+u-e^{t-s}u \\ \ge&n\log t-\frac{C_5}{1-e^{-s}}, \end{split} \end{equation*} where we have used the upper bound of $u$ in Lemma \ref{l-uudot-upper-1}. From this (ii) follows because $t\le s_1$. The proof of (iii) is similar to the proof of (ii) by letting $A=0$. Note that in this case, the infimum of $F$ may be attained at $t=0$ which depends also on $\epsilon$. To prove (iv), let $F$ as in the proof of (ii) with $A=0$. Suppose $\inf_{M\times[\frac 12,s_1]}F=\inf_{M\times\{\frac 12\} }F$, then by (i) and (ii), we have $$ F\ge -C_6. $$ Suppose $\inf_{M\times[\frac 12,s_1]}F<\inf_{M\times\{\frac 12\}}F$, then we can find $(x_0,t_0)\in M\times(\frac 12,s_1]$ such that $F(x_0,t_0)$ attains the infimum. As in the proof of (ii), at this point, \begin{equation*} \begin{split} \dot u+u\ge-n\log\left(\frac1{n\beta}(n+\kappa D_2)\right) \end{split} \end{equation*} where $D_2>0$ is a constant independent of $\kappa$. Hence as in the proof of (ii), \begin{equation*} \begin{split} F(x_0,t_0)\ge &(1-e^{t_0-s})(\dot u+u)+e^{t_0-s}u-f\\ \geq&-n(1-e^{t_0-s}) \log\left(\frac1{n\beta}(n+\kappa D_2)\right)- \frac{C_7s_1e^{s_1-s}}{1-e^{-s}} -C_8\\ \ge&-n \log\left(\frac1{n\beta}(n+\kappa D_2)\right)- \frac{C_7s_1e^{s_1-s}}{1-e^{-s}} -C_8 \end{split} \end{equation*} because $t_0\le s_1$, where we have used (i) and we may assume that $\beta<1$. Let $\kappa\to0$, we conclude that on $M\times[\frac 12, s_1]$, \begin{equation*}\begin{split} &(1-e^{t-s})(\dot u+u)+e^{t-s}u-f\ge n \log\beta- \frac{C_7s_1e^{s_1-s}}{1-e^{-s}} -C_8.\end{split} \end{equation*} By Lemma \ref{l-uudot-upper-1}, we have \begin{equation*} \dot u+u\ge -\frac{C_9(1+s_1e^{s_1-s})}{1-e^{s_1-s}} \end{equation*} on $M\times[\frac 12,s_1]$ for some constant because $s>1$. Finally, (v) follows from (i), Lemma \ref{l-uudot-upper-1} and (iv) by integration. \end{proof} \subsection{a priori estimates for $\omega(t)$}\label{ss-trace} Next we will estimate the uniform upper bound of $g(t)$. Before we do this, we first give uniform estimates for the evolution of the key quantity $\log \operatorname{tr}_hg(t)$. Let $\hat T$ and $T_0$ be the torsions of $h, \gamma_0$ respectively. Note that $\gamma_0$ depends on $\rho_0, \epsilon$. Let $\hat\nabla$ be the Chern connection of $h$. Recall that $T_{ij\bar l}=\partial_ig_{j\bar l}-\partial_j g_{i\bar l}$ etc. Let $\widetilde g$ be such that $g(t)=e^{-t}\widetilde g(e^t-1)$. Let $s=e^t-1$. Then \begin{equation*} \begin{split} -\text{\rm Ric} (\widetilde g(s))-g(t)=&-\text{\rm Ric} (g(t))-g(t)\\ =&\frac{\partial}{\partial t}g(t)\\ =&-e^{-t}\widetilde g(e^t-1)+\frac{\partial}{\partial s}\widetilde g(s)\\ =&-g(t)+\frac{\partial }{\partial s}\widetilde g(s). \end{split} \end{equation*} So \begin{equation*} \frac{\partial }{\partial s}\widetilde g(s)=-\text{\rm Ric}(\widetilde g(s)) \end{equation*} and $\widetilde g(0)=\gamma_0$. Let $\Upsilon(t)=\operatorname{tr}_{h}g(t)$ and $\widetilde\Upsilon(s)=\operatorname{tr}_{h}\widetilde g(s)$. By Lemma \ref{l-a-1}, we have \begin{equation*} \lf(\frac{\p}{\p s}-\wt\Delta\ri) \log \widetilde\Upsilon=\mathrm{I+II+III} \end{equation*} where \begin{equation*} \begin{split} \mathrm{I}\le &2\widetilde\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} \widetilde g^{k\bar q} (T_0)_{ki\bar l}\hat \nabla_{\bar q}\widetilde\Upsilon\right). \end{split} \end{equation*} \begin{equation*} \begin{split} \mathrm{II}=&\widetilde\Upsilon^{-1} \widetilde g^{i\bar{j}} h^{k\bar l}\widetilde g_{k\bar q} \left(\hat \nabla_i \overline{(\hat T)_{jl}^p}- h^{p\bar q}\hat R_{i\bar lp\bar j}\right)\\ \end{split} \end{equation*} and \begin{equation*} \begin{split} \mathrm{III}=&-\widetilde\Upsilon^{-1} \widetilde g^{{i\bar{j}}} h^{k\bar l}\left(\hat \nabla_i\left(\overline{( T_0)_{jl\bar k} } \right) +\hat \nabla_{\bar l}\left( ( T_0)_{ik\bar j} \right)-\overline{ (\hat T)_{jl}^q}( T_0)_{ik\bar q} \right) \end{split} \end{equation*} Now $$ \widetilde \Upsilon(s)=e^t\Upsilon(t). $$ So \begin{equation*} \lf(\frac{\p}{\p s}-\wt\Delta\ri) \log\widetilde\Upsilon(s)=e^{-t}\left(\lf(\frac{\p}{\p t}-\Delta\ri) \log\Upsilon+1\right) \end{equation*} \begin{equation*} \begin{split} \mathrm{I}\le &2e^{- 2t}\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q} (T_0)_{ki\bar l}\hat \nabla_{\bar q} \Upsilon\right). \end{split} \end{equation*} \begin{equation*} \begin{split} \mathrm{II}=&e^{-t}\Upsilon^{-1} g^{i\bar{j}} h^{k\bar l} g_{k\bar q} \left(\hat \nabla_i \overline{(\hat T)_{jl}^q}- h^{p\bar q}\hat R_{i\bar lp\bar j}\right)\\ \end{split} \end{equation*} and \begin{equation*} \begin{split} \mathrm{III}=&-e^{-2t}\Upsilon^{-1} g^{{i\bar{j}}} h^{k\bar l}\left(\hat \nabla_i\left(\overline{( T_0)_{jl\bar k} } \right) +\hat \nabla_{\bar l}\left( ( T_0)_{ik\bar j} \right)-\overline{ (\hat T)_{jl}^q}( T_0)_{ik\bar q} \right) \end{split} \end{equation*} Hence \begin{equation}\label{e-logY} \lf(\frac{\p}{\p t}-\Delta\ri)\log \Upsilon=\mathrm{I}'+\mathrm{II}'+\mathrm{III}'-1 \end{equation} where \begin{equation*} \begin{split} \mathrm{I}'\le &2e^{-t}\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q} (T_0)_{ki\bar l}\hat \nabla_{\bar q} \Upsilon\right). \end{split} \end{equation*} \begin{equation*} \begin{split} \mathrm{II}'=& \Upsilon^{-1} g^{i\bar{j}} h^{k\bar l} g_{k\bar q} \left(\hat \nabla_i \overline{(\hat T)_{jl}^q}- h^{p\bar q}\hat R_{i\bar lp\bar j}\right)\\ \end{split} \end{equation*} and \begin{equation*} \begin{split} \mathrm{III}'=&-e^{-t}\Upsilon^{-1} g^{{i\bar{j}}} h^{k\bar l}\left(\hat \nabla_i\left(\overline{( T_0)_{jl\bar k} } \right) +\hat \nabla_{\bar l}\left( ( T_0)_{ik\bar j} \right)-\overline{ (\hat T)_{jl}^q}( T_0)_{ik\bar q} \right) \end{split} \end{equation*} Now we want to estimate the terms in the above differential inequality. \underline{\it Estimate of $\mathrm{II}'$} Choose an frame unitary with respect to $h$ so that $g_{{i\bar{j}}}=\lambda_i\delta_{ij}$. Then \begin{equation}\label{e-logY-1} \begin{split} \mathrm{II}'=& (\sum_l\lambda_l)^{-1}\lambda_i^{-1}\lambda_k\left(\hat\nabla_i\overline{(\hat T)_{ik}^k}-\hat R_{i\bar kk\bar i}\right)\\ \le &C(n,K)\operatorname{tr}_{g}h. \end{split} \end{equation} \underline{\it Estimate of $\mathrm{III}'$} Next, we compute the torsion of $\gamma_0$, $T_0=T_{\gamma_0}$, where $\gamma_0=\eta(\frac{\rho(x)}{\rho_0})g_0+(1-\eta(\frac{\rho(x)}{\rho_0}))h+\epsilon h$:\begin{equation*}\begin{split} (T_0)_{ik\bar q}=&\partial_i(\gamma_0)_{k\bar q}-\partial_k(\gamma_0)_{i\bar q}\\ =&\eta'\frac{1}{\rho_0}[\rho_i(x)(g_0)_{k\bar q}-\rho_k(x)(g_0)_{i\bar q}]+\eta[\partial_i(g_0)_{k\bar q}-\partial_k(g_0)_{i\bar q}]\\ &+(1-\eta+\epsilon)[\partial_ih_{k\bar q}-\partial_kh_{i\bar q}]-\eta'\frac{1}{\rho_0}[\rho_ih_{k\bar q}-\rho_kh_{i\bar q}]. \end{split} \end{equation*} By the assumptions, all terms above are bounded by $C(n, K)$ for all $\rho_0\geq 1$ and for all $\epsilon\leq 1$. It remains to control $\hat \nabla_{\bar l}\left( ( T(\gamma_0))_{ik\bar j} \right)$. We may compute $\hat \nabla_{\bar l}\left( ( T(\gamma_0))_{ik\bar j} \right)$ directly. \begin{equation*}\begin{split} & \hat \nabla_{\bar l}\left( ( T(\gamma_0))_{ik\bar j} \right)\\=&\hat \nabla_{\bar l}(\partial_i(\gamma_0)_{k\bar j}-\partial_k(\gamma_0)_{i\bar j})\\ =&\hat \nabla_{\bar l}\{\eta'\frac{1}{\rho_0}[\rho_i(x)(g_0)_{k\bar j}-\rho_k(x)(g_0)_{i\bar j}]+\eta[\partial_i(g_0)_{k\bar j}-\partial_k(g_0)_{i\bar j}]\\ &+(1-\eta+\epsilon)[\partial_ih_{k\bar j}-\partial_kh_{i\bar j}]-\eta'\frac{1}{\rho_0}[\rho_ih_{k\bar q}-\rho_kh_{i\bar q}]\}\\ =&\eta''\rho_{\bar l}\frac{1}{\rho^2_0}[\rho_i(g_0)_{k\bar j}-\rho_k(g_0)_{i\bar j}]+\eta'\frac{1}{\rho_0}[\rho_{i\bar l}(g_0)_{k\bar j}-\rho_{k\bar l}(g_0)_{i\bar j}]\\ &+\eta'\frac{1}{\rho_0}[\rho_i\hat \nabla_{\bar l}(g_0)_{k\bar j}-\rho_k\hat \nabla_{\bar l}(g_0)_{i\bar j}]+\eta_{\bar l}[\partial_i(g_0)_{k\bar j}-\partial_k(g_0)_{i\bar j}]\\ &+\eta \hat\nabla_{\bar l} T(g_0)_{ik\bar q}+(1-\eta+\epsilon)\hat \nabla_{\bar l} T(h)_{ik\bar j}-\eta'\frac{\rho_{\bar l}}{\rho_0}T(h)_{ik\bar j}\\ &-\eta'\frac{1}{\rho_0}[\rho_{i\bar l}h_{k\bar q}-\rho_{k\bar l}h_{i\bar q}]-\eta''\frac{1}{\rho^2_0}[\rho_{\bar l}\rho_{i}h_{k\bar q}-\rho_{\bar l}\rho_{k}h_{i\bar q}]. \end{split} \end{equation*} Since we can control every term of the above equation by $C(n, K)$. Therefore, $|\hat \nabla_{\bar l}\left( ( T(\gamma_0))_{ik\bar j} \right)|\leq C(n, K)$. Therefore, if $0<\epsilon<1,\rho_0>1$ \begin{equation}\label{e-logY-2} \mathrm{III}'\leq C(n, K)\cdot e^{-t}\Upsilon^{-1} \Lambda. \end{equation} where $\Lambda=\operatorname{tr}_{g}h$. Now we will prove the uniform upper bound of $g(t)$. \begin{lma}\label{l-trace-2} \begin{enumerate} \item [(i)] For $0<s_1<s$, $$ \operatorname{tr}_{h}g(x,t)\le \exp\left(\frac{C(E-\log(1-e^{-s}))}{1-e^{-t}}\right) $$ on $M\times(0,s_1]$ for some constant $C>0$ depending only on $n,K, \beta, ||f||_\infty$ provided such that if $0<\epsilon<1$, $\rho_0>1$, where $$ E=\frac{(1+s_1e^{s_1-s})}{(1-e^{-s})(1-e^{s_1-s})}. $$ \item [(ii)] For $0<s_1<s$, there is a constant $C$ depending only on $n,K, \beta, ||f||_\infty, s, s_1$ and also on $\epsilon$, but independent of $\rho_0$ such that $$ \operatorname{tr}_{h}g\le C $$ on $M\times[0, s_1]$. \end{enumerate} \end{lma} \begin{proof} In the following, $C_i$ will denote constants depending only on $n,K, {\beta}$ and $||f||_\infty$, but not $\rho_0$ and $\epsilon$. $D_i$ will denote constants which may also depend on $\epsilon, \rho_0$, but not $\kappa$. We always assume $0<\epsilon<1<\rho_0$. Let $v(x,t)\ge1$ be a smooth bounded function. As before, let $\Upsilon=\operatorname{tr}_{h}g$ and $\Lambda=\operatorname{tr}_gh$ and let $\lambda=0$ or 1. For $\kappa>0$, consider the function $$F=(1-\lambda e^{-t})\log \Upsilon-Av+\frac 1v-\kappa\rho+Bt\log (1-\lambda e^{-t}) $$ on $M\times[0, s_1]$, where $A, B>0$ are constants to be chosen. We want to estimate $F$ from above. Let $$ \mathfrak{M}=\sup_{M\times[0,s_1]}F. $$ Either (i) $\mathfrak{M}\le 0$; (ii) $\mathfrak{M}=\sup_{M\times\{0\}}F$; or (iii) there is $(x_0,t_0)$ with $t_0>0$ such that $F(x_0,t_0)=\mathfrak{M}$. If (ii) is true, then \begin{equation}\label{e-tr-1} \mathfrak{M}\le C_1(n). \end{equation} because $g(0)=\gamma_0\le (1+\epsilon)h$. Suppose (iii) is true. If at this point $\Upsilon(x_0,t_0)\le 1$. Then \eqref{e-tr-1} is true with a possibly larger $C_1$. So let us assume that $\Upsilon(x_0,t_0)>1$. By \eqref{e-logY}, \eqref{e-logY-1} and \eqref{e-logY-2}, at $(x_0,t_0)$ we have: \begin{equation*} \begin{split} 0\le&\lf(\frac{\p}{\p t}-\Delta\ri) F\\ =&(1-\lambda e^{-t})\lf(\frac{\p}{\p t}-\Delta\ri)\log \Upsilon+\lambda e^{-t}\log \Upsilon-(\frac{1}{v^2}+A)\lf(\frac{\p}{\p t}-\Delta\ri) v\\ &-\frac{2}{v^3}|\nabla v|^2+\kappa \Delta \rho+B\left(\log(1-\lambda e^{-t})+\frac{\lambda t}{e^t-\lambda}\right)\\ \le &(1-\lambda e^{-t})C_2\Lambda \left( 1 +e^{-t}\Upsilon^{-1} \right)\\& + 2(1-\lambda e^{-t})e^{-t}\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q} (T_0)_{ki\bar l}\hat \nabla_{\bar q} \Upsilon\right)\\ &+\lambda e^{-t}\log \Upsilon- (\frac 1{v^2}+A)\lf(\frac{\p}{\p t}-\Delta\ri) v -\frac{2|\nabla v|^2}{v^3}\\ &+B\left(\log(1-\lambda e^{-t})+\frac{\lambda t}{e^t-\lambda}\right)+\kappa D_1. \end{split} \end{equation*} At $(x_0,t_0)$, we also have: $$(1-\lambda e^{-t}) \Upsilon^{-1}\hat \nabla \Upsilon-(\frac 1{v^2}+A)\hat\nabla v- \kappa\hat \nabla \rho=0.$$ Hence \begin{equation*}\begin{split} &2(1-\lambda e^{-t})e^{-t} \Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q} (T_0)_{ki\bar l}\hat \nabla_{\bar q} \Upsilon\right)\\ =& \frac{2e^{-t}}{\Upsilon}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q} (T_0)_{ki\bar l}((\frac 1{v^2}+A)\hat\nabla_{\bar q} v- \kappa\hat \nabla_{\bar q} \rho)\right) \\ \leq&\frac{1}{v^3}|\nabla v|^2+\frac{C_3(A+1+\frac{1}{v^2})^2\cdot v^3 \Lambda}{\Upsilon^2}+\kappa D_2\\ \end{split}\end{equation*} Using the fact that $\Upsilon(x_0,t_0)>1$, we have at $(x_0,t_0)$: Hence \begin{equation}\label{e-g-1}\begin{split} 0\le& C_2(1-\lambda e^{-t})\Lambda + \frac{C_3(A+\frac{1}{v^2})^2\cdot v^3 \Lambda}{\Upsilon} +\lambda e^{-t}\log \Upsilon\\ &- (\frac 1{v^2}+A)\lf(\frac{\p}{\p t}-\Delta\ri) v +B\left(\log(1-\lambda e^{-t})+\frac{\lambda t}{e^t-\lambda}\right)+\kappa D_3.\end{split} \end{equation} Now let $$ v=u-\frac{1- e^{-t}}{1-e^{-s}}f+\frac{C_4(1+s_1e^{s_1-s})}{(1-e^{-s})(1-e^{s_1-s})} $$ By Lemmas \ref{l-uudot-upper-1} and \ref{l-all-u}, we can find $C_4>0$ so that $v\ge 1$, and there is $C_5>0$ so that $$v\le \frac{C_5(1+s_1e^{s_1-s})}{(1-e^{-s})(1-e^{s_1-s})}. $$ Let \begin{equation}\label{e-E} E:=\frac{(1+s_1e^{s_1-s})}{(1-e^{-s})(1-e^{s_1-s})}. \end{equation} Note that \begin{equation*} \begin{split} & \lf(\frac{\p}{\p t}-\Delta\ri) u\\ =&\dot u-\Delta u\\ =&\dot u-n+\operatorname{tr}_g\left(-(1-e^{-t}) \text{\rm Ric}(\theta_0)+e^{-t}\gamma_0 \right)\\ \ge&\dot u-n+\operatorname{tr}_g\left(\frac{1-e^{-t}}{1-e^{-s}}\left(\beta\theta_0-e^{-s}\omega_0-\sqrt{-1}\partial\bar\partial f\right)+ e^{-t}\gamma_0\right)\\ \ge&\dot u+\left[\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right]\Lambda-\frac{1-e^{-t}}{1-e^{-s}}\Delta f -n\\ \ge&\dot u+\left[\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right]\Lambda+\lf(\frac{\p}{\p t}-\Delta\ri) \left(\frac{1- e^{-t}}{1-e^{-s}} f\right)-\frac{ e^{-t}}{1-e^{-s}}f-n\\ \ge&\dot u+u+\left[\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right]\Lambda+\lf(\frac{\p}{\p t}-\Delta\ri) \left(\frac{1- e^{-t}}{1-e^{-s}} f\right)-\frac{C_6}{1-e^{-s}}. \end{split} \end{equation*} because $\gamma_0\ge \omega_0+\epsilon\theta_0$ and $t<s$. On the other hand, $$ -\dot u-u=\log\left(\frac{\det h}{\det g}\right)\le c(n)+n\log \Lambda. $$ Hence \begin{equation}\label{e-g-2} \lf(\frac{\p}{\p t}-\Delta\ri) v\ge -n\log \Lambda+ \left[\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right]\Lambda -\frac{C_7}{1-e^{-s}}. \end{equation} On the other hand, in a unitary frame with respect to $h$ so that $g_{i\bar{j}}=\lambda_i\delta_{ij}$, then \begin{equation}\label{e-tr-2} \begin{split} \Upsilon=&\sum_i\lambda_i\\ =&\frac{\det g}{\det h}\sum_{i} (\lambda_1\dots\hat\lambda_i\dots\lambda_n)^{-1}\\ \le &C_{8}\Lambda^{n-1}. \end{split} \end{equation} where we have used the upper bound of $\dot u+u=\log\frac{\det g}{\det h} $ in Lemma \ref{l-uudot-upper-1}. Combining \eqref{e-g-1}, \eqref{e-g-2} and \eqref{e-tr-2}, at $(x_0,t_0)$ we have \begin{equation} \label{e-tr-revised} \begin{split} 0\le& C_2(1-\lambda e^{-t})\Lambda\left(1 + \frac{C_9 E^3(A+1)^2}{(1-\lambda e^{-t})\Upsilon}\right) +\lambda e^{-t}\left(\log C_8+(n-1)\log\Lambda\right)\\ &+ (\frac 1{v^2}+A)\left( n\log \Lambda- \left[\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right]\Lambda+\frac{C_7}{1-e^{-s}}\right) \\ &+B\left(\log(1-\lambda e^{-t})+\frac{\lambda t}{e^t-\lambda}\right)+\kappa D_3\\ \le&\Lambda\left[C_2(1-\lambda e^{-t}) \left(1 + \frac{C_9E^3(A+1)^2}{(1-\lambda e^{-t})\Upsilon}\right)-\frac{A+1}{C^2_5E^2}\left(\frac{\beta(1-e^{-t})}{1-e^{-s}}+\epsilon e^{-t}\right)\right] \\ &+[n(1+A)+\lambda(n-1)] \log \Lambda+ \frac{C_{10}(A+1) }{1-e^{-s}}\\&+B\left(\log(1-\lambda e^{-t})+\frac{\lambda t}{e^t-\lambda}\right)+\kappa D_3+\lambda\log C_8 \end{split} \end{equation} where we have used the fact that $1\le v\le C_5E$. {\bf Case 1}: Let $\lambda=1$. Suppose at $(x_0,t_0)$, $$ \frac{C_2C_9E^3(A+1)^2}{(1- e^{-t})\Upsilon}\ge \frac12\frac{1}{C^2_5E^2}\cdot(A+1)\cdot\beta\cdot\frac{1}{1-e^{-s}} $$ Then \begin{equation*} (1- e^{-t})\Upsilon\le \frac{2C_2C_9C^2_5E^5(1-e^{-s})(A+1)}{\beta}\leq C_{11}E^5(A+1). \end{equation*} Hence, \begin{equation*} (1- e^{-t})\log \Upsilon\le (1- e^{-t})\log(C_{11}E^5(A+1)) -(1-e^{-t})\log(1- e^{-t}) . \end{equation*} Therefore, \begin{equation}\label{e-tr-1-2} \mathfrak{M}\le C(1+\log E)+\log (A+1). \end{equation} for some $C(n,\beta, K,||f||_\infty)>0$. Suppose at $(x_0,t_0)$, $$ \frac{C_2C_9E^3(A+1)^2}{(1- e^{-t})\Upsilon}< \frac12\frac{1}{C^2_5E^2}\cdot(A+1)\cdot\beta\cdot\frac{1}{1-e^{-s}}, $$ then at $(x_0,t_0)$ we have \begin{equation*} \begin{split} 0\le & (1- e^{-t}) \Lambda \left(C_2 - \frac12\frac{1}{C^2_5E^2}\cdot(A+1)\cdot\beta\cdot\frac{1}{1-e^{-s}}\right) +n(A+2)\log \Lambda\\ &+ \frac{C_{10}(A+1) }{1-e^{-s}} +B\left(\log(1- e^{-t})+\frac{ t}{e^t-1}\right)+\kappa D_3+\log C_8\\ =&\Lambda\left[ (1- e^{-t}) \left(C_2 - \frac12\frac{1}{C^2_5E^2}\cdot(A+1)\cdot\beta\cdot\frac{1}{1-e^{-s}} \right)\right]\\ &+n(A+2)\log ((1-e^{-t})\Lambda)+\frac{C_{10}(A+1) }{1-e^{-s}}-n(A+2)\log(1-e^{-t})\\ &+B\left(\log(1- e^{-t})+\frac{ t}{e^t-1}\right)+\kappa D_3+\log C_8\\ \le &-(1-e^{-t})\Lambda +n(A+2)\log ((1-e^{-t})\Lambda)+\frac{C_{12}E^2}{1-e^{-s}}, \end{split} \end{equation*} provided $A=C_{13}E^2$ so that $$ \frac12\frac{1}{C^2_5E^2}\cdot(A+1)\cdot\beta\cdot\frac{1}{1-e^{-s}} \ge (C_2+1) $$ and $B$ is chosen so that $B=n(A+2)$ and $\kappa$ is small enough so that $\kappa D_2\le 1$. Hence using $1+\frac{1}{2}\log x\leq \sqrt{x},\;\forall x>0$, we have at $(x_0,t_0)$, $$ (1-e^{-t})\Lambda\le \frac{C_{14}E^4}{1-e^{-s}}, $$ and so $$ \log\Lambda\le \log\frac{C_{14}E^4}{1-e^{-s}}-\log(1-e^{-t}). $$ By \eqref{e-tr-2}, we have \begin{equation}\label{e-tr-4} \begin{split} &(1-e^{-t})\log\Upsilon\\ \leq&(1-e^{-t})\left( \log C_8+(n-1)\log \Lambda\right) \\ \le&(1-e^{-t})\left( \log C_8+(n-1)\left(\log\frac{C_{14}E^4}{1-e^{-s}}-\log(1-e^{-t})\right)\right)\\ \le &(n-1)\log(\frac{1}{1-e^{-s}})+C_{15}(1+\log E). \end{split} \end{equation} Hence $\mathfrak{M}\le (n-1)\log(\frac{1}{1-e^{-s}})+C_{16}(1+\log E).$ By combining \eqref{e-tr-1}, \eqref{e-tr-1-2} and using the choice of $A$, we may let $\kappa\rightarrow 0$ to conclude that on $ M\times(0,s_1]$, $$ (1-e^{-t})\log \Upsilon\le (n-1)\log(\frac{1}{1-e^{-s}})+C_{17}(1+E). $$ and hence (i) in the lemma is true. Here we have used the fact that $E\geq \log E+1$. {\bf Case 2}: Let $\lambda=0$, then \eqref{e-tr-revised} becomes: \begin{equation*} \begin{split} 0\le&\Lambda\left[C_2\left(1 + \frac{C_9E^3(A+1)^2}{\Upsilon}\right)-\frac{1}{C^2_5E^2}(A+1)\epsilon e^{-t}\right] \\ &+n(1+A) \log \Lambda+ \frac{C_{10}(A+1) }{1-e^{-s}}+\kappa D_3. \end{split} \end{equation*} We can argue as before to conclude that (ii) is true. \end{proof} Combining the lower bound of $\dot u+u$, we obtain: \begin{cor}\label{eq-g} For any $0<s_0<s_1<s$, there is a constant $C$ depending only on $n,K, {\beta}, ||f||_\infty$ and $s_0, s_1, s$ but independent of $\epsilon,\rho_0$ such that if $0<\epsilon<1$, $\rho_0>1$, we have \begin{equation*} C^{-1}h\leq g(t)\leq Ch \end{equation*} on $M\times[s_0, s_1]$. There is also a constant $\widetilde C(\epsilon)>0$ which may also depend on $\epsilon$ such that \begin{equation*} \widetilde C^{-1}h\leq g(t)\leq \widetilde Ch \end{equation*} on $M\times[0, s_1]$. \end{cor} \section{Short time existence for the potential flow and the normalized Chern-Ricci flow} Using the a priori estimates in previous section, we are ready to discuss short time existence for the the potential flow and the Chern-Ricci flow. We begin with the short time existence of the potential flow. We have the following: \begin{thm}\label{t-instant-complete} Let $(M,h)$ be a complete non-compact Hermitian metric { with K\"ahler form $\theta_0$.} Suppose there is $K>0$ such that the following holds. \begin{enumerate} \item There is a {proper} exhaustion function $\rho(x)$ on $M$ such that $$|\partial\rho|^2_h +|\sqrt{-1}\partial\bar\partial \rho|_h \leq K.$$ \item $\mathrm{BK}_h\geq -K$; \item The torsion of $h$, $T_h=\partial \omega_h$ satisfies $$|T_h|^2_h +|\nabla^h_{\bar\partial} T_h |\leq K.$$ \end{enumerate} Let $\omega_0$ be a nonnegative real (1,1) form with corresponding Hermitian form $g_0$ on $M$ (possibly incomplete or degenerate) such that \begin{enumerate} \item[(a)] $g_0\le h$ and $$|T_{g_0}|_h^2+|\nabla^h_{\bar\partial} T_{g_0}|_h+ |\nabla^{h}g_0|_h\leq K.$$ \item[(b)] There exist $f\in C^\infty(M)\cap L^\infty(M),\beta>0$ and $s>0$ so that $$-\text{\rm Ric}(\theta_0)+e^{-s}(\omega_0+\text{\rm Ric}(\theta_0))+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0.$$ \end{enumerate} Then \eqref{e-MP-1} has a solution on $M\times(0, s)$ so that $u(t)\to 0$ as $t\to0$ uniformly on $M$. Moreover, for any $0<s_0<s_1<s$, { let $$ {\alpha}(t)=-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\omega_0) $$} then $$\omega(t)={\alpha}+\sqrt{-1}\partial\bar\partial u$$ is the K\"ahler form of a complete Hermitian metric which is uniformly equivalent to $h$ on $M\times[s_0, s_1]$. \end{thm} \begin{proof}[Proof of Theorem \ref{t-instant-complete}] For later application, we construct the solution in the following way. Combining the local higher order estimate of Chern-Ricci flow (See \cite{ShermanWeinkove2013} for example) with Corollary \ref{eq-g} for any $1>\epsilon>0$, using diagonal argument as $\rho_0\to \infty$ we obtain a solution $u_\epsilon(t)$ to \eqref{e-MP-1} with initial data $\omega_0+\epsilon \theta_0$ on $M\times[0,s)$ which is smooth up to $t=0$, so that the corresponding solution $g_\epsilon(t)$ of \eqref{e-NKRF} has smooth solution on $M\times[0,s)$ with initial metric $g_\epsilon(0)=g_0+\epsilon h$. Moreover, $g_\epsilon$ is uniformly equivalent to $h$ on $M\times[0,s_1]$ for all $0<s_1<s$ and for any $0<s_0<s_1<s$, there is a constant $C>0$ independent of $\epsilon$ such that $$ C^{-1}h\le g_\epsilon\le Ch $$ on $M\times[s_0,s_1]$. Using the local higher order estimate of Chern-Ricci flow \cite{ShermanWeinkove2013} again, we can find $\epsilon_i\to0$ such that $u_{\epsilon_i}$ converge locally uniformly on any compact subsets of $M\times(0,s)$ to a solution $u$ of \eqref{e-MP-1}. By Lemmas \ref{l-uudot-upper-1}, \ref{l-all-u}, we see that $u(t)\to 0$ as $t\to0$ uniformly $M$. Moreover, for any $0<s_0<s_1<s$, $\omega(t)={\alpha}+\sqrt{-1}\partial\bar\partial u$ is the K\"ahler form of the solution to \eqref{e-NKRF}. Also, the corresponding Hermitian metric $g(t)$ is a complete Hermitian metric which is uniformly equivalent to $h$ in $M\times[s_0, s_1]$ for any $0<s_0<s_1<1$. \end{proof} Next we want to discuss { the short time existence of the Chern-Ricci flow. The solution $\omega(t)$ obtained from the Theorem \ref{t-instant-complete} satisfies the normalized Chern-Ricci flow on $M\times(0,s)$. Hence we concentrate on the discussion of the behaviour of $\omega(t)$ as $t\to0$ for the solution obtained in Theorem \ref{t-instant-complete}}. In case that $h$ is K\"ahler and $\omega_0$ is closed, we have the following: \begin{thm}\label{t-initial-Kahler-1} With the same notation and assumptions as in Theorem \ref{t-instant-complete}. Let $\omega(t)$ be the solution of \eqref{e-NKRF} obtained in the theorem. If in addition $h$ is K\"ahler and $d\omega_0=0$. Let $U=\{\omega_0>0\}$. Then $\omega(t)\rightarrow \omega_0$ in $C^\infty(U)$ as $t\rightarrow 0$, {uniformly in compact sets of $U$}. \end{thm} \begin{rem} If in addition $h$ has bounded curvature, then one can use Shi's K\"ahler-Ricci flow \cite{Shi1989,Shi1997} and the argument in \cite{ShermanWeinkove2012} to show that the K\"ahler-Ricci flow $g_i(t)$ starting from $g_0+\epsilon_i h$ has bounded curvature when $t>0$. The uniform local $C^k$ estimates will follow from the pseudo-locality theorem \cite[Corollary 3.1]{HeLee2018} and the modified Shi's local estimate \cite[Theorem 14.16]{Chow2}. \end{rem} By Theorem \ref{t-instant-complete} we have the following: \begin{cor}\label{c-shorttime} Let $(M,h)$ be a complete non-compact K\"ahler manifold with bounded curvature. Let $\theta_0$ be the K\"ahler form of $h$. Suppose there is a compact set $V$ such that outside $V$, $-\text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial f\ge\beta \theta_0$ for some $\beta>0$ for some bounded smooth function $f$. Then for any closed nonnegative real (1,1) form $\omega_0$ such that $\omega_0\le \theta_0$, $|\nabla_h\omega_0|$ is bounded, and $\omega_0>0$ on $V$, there is $s>0$ such that \eqref{e-NKRF} has a solution $\omega(t)$ on $M\times(0,s)$ so that $\omega(t)$ is uniformly equivalent to $h$ on $M\times[s_0,s_1]$ for any $0<s_0<s_1<s$ and $\omega(t)$ attains initial data $\omega_0$ in the set where $\omega_0>0$. \end{cor} \begin{proof} Let $s>0$, then $$ -(1-e^{-s})\text{\rm Ric}(\theta_0)+(1-e^{-s})\sqrt{-1}\partial\bar\partial f\ge (1-e^{-s})\beta \theta_0 $$ outside $V$. On $V$, $$ \omega_0 -(1-e^{-s})\text{\rm Ric}(\theta_0)+(1-e^{-s})\sqrt{-1}\partial\bar\partial f\ge \beta'\theta_0 $$ for some $\beta'>0$, provided $s$ is small enough. The Corollary then follows from Theorems \ref{t-instant-complete} and \ref{t-initial-Kahler-1}. \end{proof} \begin{rem}\label{r-shorttime} Suppose $\Omega$ is a bounded strictly pseudoconvex domain in $\mathbb{C}^n$ with smooth boundary, then there is a complete K\"ahler metric with Ricci curvature bounded above by the negative constant near infinity by \cite{ChengYau1982}. Hence Corollary \ref{c-shorttime} can be applied to this case, which has been studied by Ge-Lin-Shen \cite{Ge-Lin-Shen}. \end{rem} To prove the Theorem \ref{t-initial-Kahler-1}, suppose $h$ is K\"ahler and $d\omega_0=0$, then solution in Theorem \ref{t-instant-complete} is the limit of solutions $g_i(t)$ of the normalized K\"ahler-Ricci flow on $M\times[0, s)$ with initial data $g_0+\epsilon_i h$, where $\epsilon_i\to 0$. Here we may assume $s\leq 1$. By Lemma \ref{l-all-u} (iii) and Lemma \ref{l-trace-2} (ii), each $g_i(t)$ is uniformly equivalent to $h$, the uniform constant here may depend on $\epsilon_i$. In this section, we will use $\tilde g_i(t)=(t+1)g_i( \log (t+1))$ to denote the unnormalized K\"ahler-Ricci flow and $\phi_i$ be the corresponding potential flow to the unnormalized K\"ahler-Ricci flow $\tilde g_i(t)$, see appendix. We want to prove the following: \begin{lma}\label{l-initial-Kahler-1} With the same notation and assumptions as in Theorem \ref{t-initial-Kahler-1}, for any precompact open subset $\Omega$ of $U$, there is $S_1>0$ and $C>0$, $$ C^{-1}h\le \tilde g_i(t)\le Ch $$ for all $i$ in $\Omega\times[0,S_1]$. \end{lma} {\begin{proof}[Proof of Theorem \ref{t-initial-Kahler-1}] Suppose the lemma is true, then Theorem \ref{t-initial-Kahler-1} will follow from the local estimates in \cite{ShermanWeinkove2012}. \end{proof} It remains to prove Lemma \ref{l-initial-Kahler-1}.} \begin{lma}\label{slma-1} We have $|\phi_i|\leq C_0,\;\dot\phi_i\leq C_0$ on $M\times[0, e^s-1)$ for some positive constant $C_0$ independent of $i$. \begin{proof} By Lemma \ref{l-uudot-upper-1}, we have \begin{equation*} \log\frac{{\omega_i}^n(s)}{\theta_0^n}=\dot u_i+u_i\leq C. \end{equation*} Here $C$ is a positive constant independent of $i$ and ${\widetilde\omega_i}(s)$ is the corresponding normalized flow with the relation \begin{equation*} \widetilde g_i(t)e^{-s}= g_i(s), t=e^s-1. \end{equation*} Then by the equation $\dot\phi_i=\log\frac{\widetilde\omega^n_i(t)}{\theta_0^n}$, we obtain the upper bound on $\dot\phi_i(t)$. The lower bound on $\phi_i$ follows from Lemma \ref{l-all-u}. \end{proof} \end{lma} Before we state the next lemma, we fix some notations. Let $p\in U$. By scaling, we may assume that there is a holomorphic coordinate neighbourhood of $p$ which can be identified with $B_0(2)\subset \mathbb{C}^n$ with $p$ being the origin and $B_0(r)$ is the Euclidean ball with radius $r$. Moreover, $B_0(2)\Subset U$. We may further assume $\frac14h\le h_E\le 4h$ in $B_0(2)$ where $h_E$ is the Euclidean metric. Since $\omega_0>0$, there is $\sigma>0$ such that $B_{g_i(0)}(p,2\sigma)\subset B_0(2)$ and $$ g_i(0)\ge 4\sigma^2h $$ in $B_0(2)$ for some $0<\sigma<1$. This is because $g_i(0)=\omega_0+\epsilon_i h$. Here we use $h_E$ because we want to use the estimates in \cite{ShermanWeinkove2012} explicitly. Let $\tau=e^{s}-1$, where $s$ is the constant in assumption in Theorem \ref{t-instant-complete}, { and let $\dot \phi$ be as in the proof of Lemma \ref{slma-1}. It is easy to see that Lemma \ref{l-initial-Kahler-1} follows from the following:} \begin{lma}\label{local-bound} With the same notation and assumptions as in Theorem \ref{t-initial-Kahler-1} and with the above set up. There exist positive constants $1>\gamma_1, \gamma_2>0$ with $\gamma_2<\tau$ which are independent of $i$ such that { $$\gamma_1^{-2}h\ge \widetilde g_i(t)\geq \gamma_1^2 h$$} on $B_{\widetilde g_i(t)}(p,\sigma),\; t\in [0,\gamma_2\gamma_1^{8(n-1)}]$. \end{lma} \begin{proof} The lower bound in lemma will follow from the following:\vskip .1cm \noindent\underline{\it Claim}: There are constants $1>\gamma_1, \gamma_2>0$ independent of ${\alpha}>0$ and $i$ with $\gamma_2<\tau$ such that if $\widetilde g_i(t)\ge {\alpha}^2h$ on $B_{\widetilde g_i(t)}(p,\sigma)$, $t\in [0, \gamma_2{\alpha}^{8(n-1)}]$, then $\widetilde g_i(t)\ge \gamma_1^2 h$ on $B_{\widetilde g_i(t)}(p,\sigma)$ for $t\in [0, \gamma_2{\alpha}^{8(n-1)}]$. \vskip .1cm The main point is that $\gamma_1$ does not depend on ${\alpha}$. Suppose the claim is true. Fix $i$, let ${\alpha}\le \gamma_1$ be the supremum of $\widetilde{\alpha}$ so that $\widetilde g_i(t)\ge \widetilde{\alpha}^2h$ on $ B_{\widetilde g_i(t)}(p,\sigma)$, $t\in [0, \gamma_2\widetilde{\alpha}^{8(n-1)}]$. Since $\widetilde g_i(0)\ge \sigma^2h$ in $U$, we see that ${\alpha}>0$. Suppose ${\alpha}<\gamma_1$. By continuity, there is $\epsilon>0$ such that ${\alpha}+\epsilon<\gamma_1$. Then $\gamma_2 {\alpha}^{8(n-1)} \le \gamma_2 <\tau$. By the claim, we can conclude that $$ \widetilde g_i(t)\ge \gamma_1^2 h\geq ({\alpha}+\epsilon)^2h $$ in $B_{\widetilde g_i(t)}(p,\sigma)$, $t\in [0,\gamma_2{\alpha}^{8(n-1)}]$. By choosing a possibly smaller $\epsilon$ and by continuity, the above inequality is also true for $t\in [0,\gamma_2({\alpha}+\epsilon)^{8(n-1)}]$. This is a contradiction. To prove the claim, let $\gamma_1$ and $\gamma_2>0$ be two constants to be determined later and are independent of ${\alpha}$ and $i$. In the following, $C_k$ will denote a positive constant independent of ${\alpha}$ and $i$. In the following, for simplicity in notation, we suppress the index $i$ and simply write $\widetilde g_i$ as $g$. Suppose ${\alpha}\le \gamma_1$ is such that $$ g(t)\ge {\alpha}^2 h $$ in $ B_{g(t)}(p,\sigma),\;t\in[0,\gamma_2 {\alpha}^{8(n-1)}]$. By Lemma \ref{slma-1}, $\det(g(t))/\det(h)\le C_1$ for some $C_1>1$. Hence we have \begin{equation*} {\alpha}^2h\le g(t)\le C_1{\alpha}^{-2(n-1)}h \end{equation*} on $B_{g(t)}(p,\sigma),\;t\in[0,\gamma_2 {\alpha}^{8(n-1)}]$ and hence on $B_h(p,C_1^{-1/2}{\alpha}^{n-1}\sigma)\times [0,\gamma_2 {\alpha}^{8(n-1)}]$ because $B_h(p,C_1^{-1/2}{\alpha}^{n-1}\sigma)\subset B_{g(t)}(p,\sigma)$ for $t\in[0,\gamma_2 {\alpha}^{8(n-1)}]$. This can be seen by considering the maximal $h$-geodesic inside $B_t(p,\sigma)$. Together with the fact that $\frac14h \le h_E\le 4h$ on $B_0(2)$, we conclude that \begin{equation}\label{e-alpha-2} {\alpha}_1^2h_E\le g(t)\le {\alpha}_1^{-2}h_E \end{equation} on $ B_0(\frac1{2\sqrt{C_1}}{\alpha}^{n-1}\sigma)\times[0, \gamma_2{\alpha}^{8(n-1)}]$, where ${\alpha}_1>0$ is given by \begin{equation}\label{e-alpha-1} {\alpha}_1^2=\frac1{4C_1}{\alpha}^{2(n-1)}. \end{equation} By \cite[Theorem 1.1]{ShermanWeinkove2012}, we conclude that \begin{equation}\label{e-Rm} |\text{\rm Rm}(g(t))|\le \frac{C_2}{{\alpha}_1^8t} \end{equation} on $ B_0(\frac\sigma 2{\alpha}_1)\times[0, \gamma_2{\alpha}^{8(n-1)}]$. From the proof in \cite{ShermanWeinkove2012}, the constant $C_2$ depends on an upper bound of the existence time but not its precise value. In particular, it is independent of ${\alpha}$ here. By \eqref{e-alpha-2}, we conclude that \eqref{e-Rm} is true on { $ B_{g(t)}(p, \frac\sigma2{\alpha}_1^2)$, $t\in[0, \gamma_2{\alpha}^{8(n-1)}]$}. By \cite[Lemma 8.3]{Perelman2003} (see also \cite[Chapter 18, Theorem 18.7]{Chow}), we have: \begin{equation}\label{e-distance-1} \lf(\frac{\p}{\p t}-\Delta\ri) (d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12)\ge0 \end{equation} in the sense of barrier (see the definition in Appendix \ref{s-max}) outside $B_{g(t)}(p,{\alpha}_1^4\sqrt t)$, provided \begin{equation}\label{e-t-1} t^\frac12\le \frac\sigma2{\alpha}_1^{-2}. \end{equation} Let $\xi\ge0$ be smooth with $\xi=1$ on $[0,\frac 43]$ and is zero outside $[0,2]$, with $\xi'\le 0, |\xi'|^2/\xi+ |\xi''|\le C $. Let $$ \Phi(x,t)=\xi( \sigma^{-1}\eta(x,t)) $$ where $\eta(x,t)=d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12$. For any $\epsilon>0$, for $t>0$ satisfying \eqref{e-t-1}, if $d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12<\frac43\sigma$, then $\Phi(x,t)=1$ near $x$ and so \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri)(\log(\Phi+\epsilon))=0. \end{equation*} If $d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12\ge\frac43\sigma$ and $d_t(p,x)\ge {\alpha}_1^4t^\frac12$, then in the sense of barrier we have: \begin{equation}\label{e-Phi-1} \begin{split} & \lf(\frac{\p}{\p t}-\Delta\ri) \log (\Phi+\epsilon)\\ =& \left(\frac{\xi'}{\xi} \sigma^{-1}\lf(\frac{\p}{\p t}-\Delta\ri)\eta-\frac{\xi''}{\xi} \sigma^{-2}|\nabla\eta|^2+\frac{(\xi')^2}{\xi^2} \sigma^{-2}|\nabla \eta|^2\right)\\ \le & C_4(\Phi+\epsilon)^{-1}. \end{split} \end{equation} by the choice of $\xi$ and \eqref{e-distance-1}. Hence there exists $C_5>0$ such that if \begin{equation}\label{e-t-2} t^\frac12\le C_5{\alpha}_1^4 \end{equation} then $t$ also satisfies \eqref{e-t-1} and $C_3{\alpha}_1^{-4}t^{1/2}<\frac\sigma 3$. Moreover, $C_5$ can be chosen so that either $d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12<\frac43\sigma$ or $d_t(p,x)+C_3{\alpha}_1^{-4}t^\frac12\ge\frac43\sigma$ and $d_t(p,x)\ge {\alpha}_1^4t^\frac12$. Hence \eqref{e-Phi-1} is true in the sense of barrier for $t\in (0, C_5^2{\alpha}_1^8]$. Consider the function $$ F=\log \operatorname{tr}_hg -Lv+m\log (\Phi+\epsilon) $$ where $v=(\tau-t)\dot\phi+\phi-f+nt$, $\tau=e^s-1$. Here $L, m>0$ are constants to be chosen later which are independent of $i,\ {\alpha}$. Recall that $v$ satisfies \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) v=\operatorname{tr}_g (\omega_0-\tau \text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial f) \geq {\beta} \operatorname{tr}_g h. \end{equation*} and \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) \log \operatorname{tr}_h g\le C_6\operatorname{tr}_g h \end{equation*} by Lemma \ref{l-a-1} with vanishing torsion terms here. Let \begin{equation}\label{e-L} L\beta= C_6+1+\tau^{-1}. \end{equation} Note that by the A.M.-G.M. inequality and the definition of $\dot \phi$, we have \begin{equation}\label{e-AMGM} -\dot \phi \le n\log \operatorname{tr}_gh;\ \ \log \operatorname{tr}_hg\le \dot\phi +(n-1)\log\operatorname{tr}_gh. \end{equation} So \begin{equation*} \log \operatorname{tr}_gh\ge \frac{1}{ n(\tau L-1)+(n-1)}(\log \operatorname{tr}_hg-\tau L\dot\phi) \end{equation*} Then in the sense of barrier \begin{equation*} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) F\le & -\operatorname{tr}_gh+m C_4 (\Phi+\epsilon)^{-1}\\ \le &-\exp\left(C_7 (\log \operatorname{tr}_hg-\tau L\dot\phi)\right)+m C_4 (\Phi+\epsilon)^{-1}\\ \le &-\exp\left(C_7F-C_8-C_7m\log(\Phi+\epsilon)\right)+m C_4 (\Phi+\epsilon)^{-1}\\ =&-(\Phi+\epsilon)^{-1}mC_4\left[\exp (C_7F-C_8-\log(mC_4))-1\right]\\ \end{split} \end{equation*} if $mC_7=1$, where we have used the upper bound of $\dot \phi$ and the bound of $\phi$ in Lemmas \ref{slma-1}. So \begin{equation*} \begin{split} &\lf(\frac{\p}{\p t}-\Delta\ri) (C_7F-C_8-\log(mC_4))\\ \le& -\frac{mC_4C_7}{ \Phi+\epsilon}\left[\exp (C_7F-C_8-\log(mC_4))-1\right]\\ \le&0 \end{split} \end{equation*} in the sense of barrier whenever $C_7F-C_8-\log(mC_4)>0$. Then by the maximum principle Lemma \ref{max}, we conclude that $$ C_7F-C_8-\log(mC_4)\le\sup_{t=0}\left(C_7F-C_8-\log(mC_4)\right). $$ Let $\epsilon\to0$, using the definition of $\Phi$, the choice of $C_5$ and the bound of $|\phi|$, we conclude that in $ B_{g(t)}(p,\sigma)$, \begin{equation}\label{e-trace-lower} \log \operatorname{tr}_hg-L(\tau-t)\dot \phi \le C_9 \end{equation} provided $t\in [0, C_5^2{\alpha}_1^8]$. On the other hand, as in \eqref{e-AMGM}, we have \begin{equation*} \begin{split} \log \operatorname{tr}_gh\le& -\dot \phi+(n-1)\log\operatorname{tr}_hg\\ =&(n-1)\left(\log\operatorname{tr}_hg-L(\tau-t)\dot \phi\right)+(n-1)(L(\tau-t)-1)\dot\phi\\ \le&C_{10} \end{split} \end{equation*} provided \begin{equation}\label{e-t-3} Lt\le L\tau-1. \end{equation} Here we have used the upper bound of $\dot \phi$ in Lemma \ref{slma-1}. Hence there is $\gamma_1>0$ independent of ${\alpha}$ and $i$ such that if $t$ satisfies \eqref{e-t-2} and \eqref{e-t-3}, then $$ g_i(t)\ge \gamma_1^2h $$ on $B_{g_i(t)}(p,\sigma)$. Let $\gamma_2<\tau$ be such that $$ \gamma_2=\min\{C_5^2,L^{-1}(L\tau-1)\}\times (4C_1)^{-4} $$ where $C_1, C_5$ are the constants in \eqref{e-alpha-1} and \eqref{e-t-2} respectively and $L$ is given by \eqref{e-L}. If $t\in[0,\gamma_2{\alpha}^{8(n-1)}]$, then $t$ will satisfy \eqref{e-t-2}. One can see that the claim this true. { By \eqref{e-trace-lower} and Lemma \ref{slma-1}, we conclude that $$ \widetilde g_i(t)\le C_{11}h $$ on $B_{\widetilde g_i(t)}(p,\sigma)$ for $t\in[0,\gamma_2{\alpha}^{8(n-1)}]$. The upper bound in the Lemma follows by choosing a possibly smaller $\gamma_1$.} \end{proof} For the case of Chern-Ricci flow, the result is less satisfactory because the property of $d(x,t)$ does not behave as nice as in the K\"ahler case. As before, under the assumptions of Theorem \ref{t-instant-complete}, let $g(t)$ be the Chern-Ricci flow $g(t)$ constructed in the theorem. We have the following: \begin{prop}\label{p-initial-CR} With the same notation and assumptions as in Theorem \ref{t-instant-complete}. Suppose $\operatorname{tr}_{g_0}h=o(\rho)$. Then $g(t)\rightarrow g_0$ as $t\rightarrow 0$ in $M$. The convergence is in $C^\infty$ topology and is uniform in compact subsets of $M$. \end{prop} Note that $g_0$ may still be complete. But it may not be equivalent to $h$ and { the curvature of $g_0$ may be unbounded}. As before, $g(t)$ is the limit of solutions $g_i(t)$ of the unnormalized Chern-Ricci flow on $M\times[0, s)$ with initial data $g_0+\epsilon_i h$ with $\epsilon_i\to 0$. Here we may assume $s\leq 1$. We want to prove the following: \begin{lma}\label{l-initial-CR-1} With the same notation and assumptions as in Proposition \ref{p-initial-CR} and let $S<\tau:=e^s-1$, for any precompact open subset $\Omega$ of $M$, there is $C>0$, $$ C^{-1}h\le g_i(t)\le Cg $$ for all $i$ in $\Omega\times[0, S]$. \end{lma} Suppose the lemma is true, then Proposition \ref{p-initial-CR} will follows from the local estimates in \cite{ShermanWeinkove2013} for Chern-Ricci flow. To prove the lemma, first we prove the following. Let $\phi_i$ be the potential for $g_i$. \begin{sublma}\label{sl-initial-CR-1} Suppose $$\liminf_{\rho\to\infty}\rho^{-1}\log\frac{\omega_0^n}{\theta_0^n}\ge0. $$ Then for any $\sigma>0$ (small enough independent of $i$), there is a constant $C>0 $ independent of $i$ such that \begin{equation*} \dot\phi_i\ge -C -\sigma\rho \end{equation*} on $M\times[0, S]$. \end{sublma} \begin{proof} In the following, we will denote $\phi_i$ simply by $\phi$ and $g_i(t)$ simply by $g(t)$ if there is no confusion arisen. Note that $g(t)$ is uniformly equivalent to $h$. Let $\sigma>0$. Let $F=-(\tau-t)\dot \phi-\phi+f-nt-\sigma \rho $. By \eqref{e-a-1} and \eqref{e-a-2}, for $0\le t\le S$, we have \begin{equation*} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri)(-(\tau-t)\dot \phi-\phi) =& (\tau-t)\operatorname{tr}_g\text{\rm Ric}(\theta_0)+\dot\phi-\dot\phi+\operatorname{tr}_g(\sqrt{-1}\partial\bar\partial \phi)\\ =&(\tau-t)\operatorname{tr}_g\text{\rm Ric}(\theta_0)+\left(n+t\operatorname{tr}_g(\text{\rm Ric}(\theta_0))-\operatorname{tr}_g(\theta_0)\right)\\ =&\tau\operatorname{tr}_g\text{\rm Ric}(\theta_0)+n-\operatorname{tr}_g\theta_0 \\ \end{split} \end{equation*} Hence by the fact that: \begin{equation*} \omega_0-\tau\text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial f\ge \beta\theta_0, \end{equation*} we have \begin{equation*} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) F\le &\tau\operatorname{tr}_g\text{\rm Ric}(\theta_0)-\operatorname{tr}_g\theta_0-\Delta f+\sigma \Delta \rho\\ \le& (-\beta+ \sigma C_1)\operatorname{tr}_g\theta_0\\ <&0 \end{split} \end{equation*} for some constant $C_1$ independent of $\sigma$ and $i$ for $\sigma$ with $C_1\sigma<\beta$. Since $F$ is bounded from above, by the maximum principle Lemma \ref{max}, we conclude that $$ \sup_{M\times[0, S]}F\le \sup_{M\times\{0\}}F. $$ At $t=0$, $$ F=-\tau\dot\phi-\sigma\rho+f. $$ By the assumption, we conclude that $F\le C(\sigma)$ at $t=0$. Hence we have $$ F\le C(\sigma) $$ on $M\times[0, S]$. Since $\phi, f$ are bounded, the sublemma follows. \end{proof} \begin{sublma}\label{sl-initial-CR-2} With the same notations as in Sublemma \ref{sl-initial-CR-1}. Suppose $\operatorname{tr}_{g_0}h=o(\rho)$. Then $$ \operatorname{tr}_hg_i\le C\exp(C'\rho) $$ on $M\times[0, S]$ for some positive constants $C, C'$ independent of $i$. \end{sublma} \begin{proof} We will denote $g_i$ by $g$ again and $\omega_{0}$ to be the K\"ahler form of the initial metric $g_i(0)=g_0+\epsilon_ih$. Note that \begin{equation*} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri)\phi=&\dot\phi-\Delta \phi\\ =&\dot \phi-(n-\operatorname{tr}_g\omega_{0}+t\operatorname{tr}_g(\text{\rm Ric}(\theta_0)))\\ \ge& \dot \phi-n+ \operatorname{tr}_g\omega_{0}+\frac{t\beta}{\tau}\operatorname{tr}_gh-\frac{t}{\tau}\operatorname{tr}_g\omega_0-\frac t\tau\Delta f\\ \ge &\dot \phi- n-\frac t\tau\Delta f+(1-\frac S\tau)\operatorname{tr}_g\omega_{0}. \end{split} \end{equation*} Then we have: \begin{equation}\label{e-initial-CR-1} \lf(\frac{\p}{\p t}-\Delta\ri) (\phi+ nt-\frac t\tau f)\ge \dot\phi+(1-\frac S\tau)\operatorname{tr}_g\omega_{0}-C_0. \end{equation} Since $|\phi|$ is bounded by a constant independent of $i$ on $M\times[0, S]$, see Lemma \ref{l-uudot-upper-1} and Lemma \ref{l-all-u}, there is a constant $C_1, C_2>0$ so that $\xi:=\phi +nt-\frac t\tau f+C_1\ge 1$ and $\xi\le C_2$ on $M\times[0, S]$. Here and below $C_j$ will denote positive constants independent of $i$. Let $\Phi( \varsigma)=2-e^{-\varsigma}$ for $\varsigma\in \mathbb{R}$. Then for $\xi:=\phi +nt-\frac t\tau f+C_1\ge 1$, we have \begin{equation}\label{e-initia-CR-2} \left\{ \begin{array}{ll} \Phi(\xi)\ge & 1 \\ \Phi'(\xi)\ge & e^{-C_2}\\ \Phi''(\xi)\le &-e^{-C_2} \end{array} \right. \end{equation} on $M\times[0, S]$. Next, let $P(\varsigma)$ be a positive function on $\mathbb{R}$ so that $P'>0$. Define $$ F(x,t)=\Phi(\xi)P(\rho). $$ Let $\Upsilon=tr_hg$, here $g=g_i$. Let $F\to\infty$ near infinity be a smooth function of $x, t$. Then by Lemma \ref{l-a-1}, we have \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) (\log \Upsilon-F)=\mathrm{I+II+III}-\lf(\frac{\p}{\p t}-\Delta\ri) F \end{equation*} where \begin{equation*} \begin{split} \mathrm{I}\le &2\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q}( T_0)_{ki\bar l} \hat \nabla_{\bar q}\Upsilon\right), \end{split} \end{equation*} \begin{equation*} \begin{split} \mathrm{II}=&\Upsilon^{-1} g^{i\bar{j}} h^{k\bar l}g_{k\bar q} \left(\hat \nabla_i \overline{(\hat T)_{jl}^q}- \hat h^{p\bar q}\hat R_{i\bar lp\bar j}\right),\\ \end{split} \end{equation*} and \begin{equation*} \begin{split} \mathrm{III}=&-\Upsilon^{-1} g^{{i\bar{j}}} h^{k\bar l}\left(\hat \nabla_i\left(\overline{( T_0)_{jl\bar k}} \right) +\hat \nabla_{\bar l}\left( {( T_0)_{ik\bar j} }\right)-\overline{ (\hat T)_{jl}^q}( T_0)_{ik\bar q}^p \right). \end{split} \end{equation*} Let $\Theta=\operatorname{tr}_gh$. Suppose $\log \Upsilon-F$ attains a positive maximum at $(x_0,t_0)$ with $t_0>0$, then at this point, $$ \Upsilon^{-1}\hat\nabla \Upsilon=\hat\nabla F, $$ and so \begin{equation*} \begin{split} \mathrm{I}\le &2\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q}( T_0)_{ki\bar l} \hat \nabla_{\bar q}\Upsilon\right)\\ \le &C\Upsilon^{-1}\Theta^\frac12|\nabla F|\\ \le&C'\Upsilon^{-1}\Theta^\frac12\left(P|\nabla \xi|+P'\Theta^\frac12\right). \end{split} \end{equation*} because $|\partial\rho|_h$ is bounded. Here we use the norm with respect to the evolving metric $g(t)$. $$ \mathrm{II}\le C\Theta, $$ $$ \mathrm{III}\le C\Upsilon^{-1}\Theta. $$ Here $C, C'$ are positive constants independent of $i$. On the other hand, \begin{equation*} \begin{split} &\lf(\frac{\p}{\p t}-\Delta\ri) F\\ =&P\lf(\frac{\p}{\p t}-\Delta\ri)\Phi+2{\bf Re}\left(g^{i\bar{j}}\partial_i\Phi\partial_{\bar j}P\right)+\Phi\lf(\frac{\p}{\p t}-\Delta\ri) P\\ \ge&P\left(\Phi'\lf(\frac{\p}{\p t}-\Delta\ri) \xi -\Phi''|\nabla\xi|^2\right)-C_4\Phi'P'\Theta^\frac12|\nabla\xi|-C_4\Theta\Phi (P'+|P''|)\\ \ge&P\Phi'\dot\phi+e^{-C_2}P(1-\frac S\tau)\operatorname{tr}_g\omega_0-C_0P+e^{-C_2}P|\nabla\xi|^2-\frac12e^{-C_2}P|\nabla\xi|^2 \\ &-C_5\frac{(P')^2}{P}\Theta-C_4\Theta(P'+|P''|). \end{split} \end{equation*} Here we have used the fact that $|\partial\rho|_h, |\partial\bar\partial\rho|_h$ are bounded $\Phi(\xi)\le 2$ and \eqref{e-initia-CR-2}. So at $(x_0,t_0)$, \begin{equation*} \begin{split} &\lf(\frac{\p}{\p t}-\Delta\ri) (\log \Upsilon-F)\\ \le& C_3\left( \Upsilon^{-1}\Theta^\frac12\left(P|\nabla \xi|+P'\Theta^\frac12\right) + \Upsilon^{-1}\Theta+ \Theta\right)\\ &- P\Phi'\dot\phi-e^{-C_2}P(1-\frac S\tau)\operatorname{tr}_g\omega_0+C_0P-\frac12e^{-C_2}P|\nabla\xi|^2\\ &+\Theta\left(C_5\frac{(P')^2}{P}+ C_4(P'+|P''|)\right)\\ \le&- P\Phi'\dot\phi-e^{-C_2}P(1-\frac S\tau)\operatorname{tr}_g\omega_0+C_0P+\left(-\frac12e^{-C_2} +\Upsilon^{-1}\right)P|\nabla\xi|^2 \\ &+C_6\Theta\left(\Upsilon^{-1}+1+\Upsilon^{-1}P'+P'+\Upsilon^{-1}P+\frac{(P')^2}{p}+|P''|\right). \end{split} \end{equation*} Now \begin{equation*} -\dot\phi\le c(n)\log \Theta. \end{equation*} Suppose $\omega_0\ge \frac1{Q(\rho)}\theta_0$ with $Q>0$ and suppose $\Upsilon^{-1}\le \frac12e^{-C_2}$ at $(x_0,t_0)$, then at $(x_0,t_0)$, we have \begin{equation*} \begin{split} &\lf(\frac{\p}{\p t}-\Delta\ri) (\log \Upsilon-F)\\ \le&C_7P(\log\Theta+1)+\Theta\left[-C_8PQ^{-1}+C_9 \left( 1+ P'+\frac{(P')^2}{P}+|P''|\right)\right]. \end{split} \end{equation*} By the assumption on $\operatorname{tr}_{g_0}h$, for any $\sigma>0$ there is $\rho_0>0$ such that if $\rho\ge \rho_0$, then $\operatorname{tr}_{g_0}h\le \sigma\rho.$ Hence we can find $C=C(\sigma)$ such that $$ g_0\ge \frac{1}{\sigma(\rho+C(\sigma))}h $$ and $\rho+C(\sigma)\ge 1$ on $M$. Let $Q(\rho)=\sigma(\rho+C(\sigma)), P(\rho)=\rho+C(\sigma)$, then above inequality becomes \begin{equation*} \begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) (\log \Upsilon-F) \le&C_7P\log (e\Theta)+\Theta\left(-C_8\sigma^{-1}+3C_9\right)\\ \le& C_7P\log (e\Theta)-\frac12C_8\Theta \end{split} \end{equation*} if we choose $\sigma$ small enough independent of $i$. Since $\log\Upsilon-F\to-\infty$ near infinity and uniform in $t\in [0, S]$, and $\log\Upsilon-F<0$ at $t=0$, by maximum principle, either $\log\Upsilon-F\le 0$ on $M\times[0, S]$ or there is $t_0>0$, $x_0\in M$ such that $\log\Upsilon-F$ attains a positive maximum at $(x_0,t_0)$. Suppose at this point $\Upsilon^{-1}\ge\frac12 e^{-C_2}$, then $$ \log\Upsilon-F\le C_{10}. $$ Otherwise, at $(x_0,t_0)$ we have \begin{equation*} 0\le C_7P\log (e\Theta)-\frac12C_8\Theta. \end{equation*} Hence we have at this point $\Theta\le C_{11}$ which implies $\Upsilon\le C_{12}$ because $\dot\phi\le C$ for some constant independent of $i$. So $$ \log\Upsilon-F\le\log C_{12}. $$ Or $$ \Theta\le C_{13}P^2. $$ This implies $\log \Upsilon\le C_{14}(1+\log P)$. Hence \begin{equation*} \log \Upsilon-F\le C_{14}. \end{equation*} From these considerations, we conclude that the sublemma is true. \end{proof} \begin{proof}[Proof of Lemma \ref{l-initial-CR-1}] The lemma follows from Sublemmas \ref{sl-initial-CR-1} and \ref{sl-initial-CR-2}. \end{proof} \section{Long time behaviour and convergence} In this section, we will first study the longtime behaviour for the solution constructed in Theorem \ref{t-instant-complete}. Namely, we will show the following theorem: \begin{thm}\label{longtime} Under the assumption of Theorem \ref{t-instant-complete}, if in addition, $$-\text{\rm Ric}(h)+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0$$ for some $f\in C^\infty(M)\cap L^\infty(M)$, $\beta>0$. Then the solution constructed from Theorem \ref{t-instant-complete} is a longtime solution and converges to a unique complete K\"ahler Einstein metric with negative scalar curvature on $M$. \end{thm} Before we prove Theorem \ref{longtime}, let us prove a lower bound of $\dot u$ which will be used in the argument of convergence. Once we have uniform equivalence of metrics, we can obtain a better lower bound of $\dot{u}$. \begin{lma}\label{du-convergence} Assume the solution constructed from Theorem \ref{t-instant-complete} is a longtime solution, then there is a positive constant $C$ such that \begin{equation*} \dot{u}\geq-Ce^{-\frac t2} \end{equation*} on $M\times[2, \infty)$. \begin{proof} Since we do not have upper bound of $g(t)$ as $t\to 0$, we shift the initial time of the flow to $t=1$. Note that \begin{equation*}\begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) (e^t\dot{u}-f)=&-tr_{ g}(\text{\rm Ric}(h)+g(1))+\Delta f\\ \geq&-tr_{ g}g(1)\geq-C_1. \end{split}\end{equation*} Consider $Q=e^t\dot{u}-f+(C_1+1)t$. Then we can use maximum principle argument as before to obtain $Q(x, t)\geq \inf\limits_MQ(0)$. Then we have\begin{equation*} e^t\dot{u}\geq -C_2-(C_1+1)t \end{equation*} which implies \begin{equation*} \dot{u}\geq -C_3e^{-\frac t2} \end{equation*} on $M\times[1, \infty)$. We shift the time back, we obtain the result. \end{proof} \end{lma} \begin{proof}[Proof of Theorem \ref{longtime}] The assumption $-\text{\rm Ric}(h)+\sqrt{-1}\partial\bar\partial f\geq \beta \theta_0$ implies that for all $s$ large enough, \begin{equation*} -Ric(h)+e^{-s}(\omega_0+Ric(h))+\sqrt{-1}\partial\bar\partial \hat f\geq \frac\beta2 \theta_0. \end{equation*} Here $\hat f=(1-e^{-s})f$ is a bounded function on $M$. By Theorem \ref{t-instant-complete} and Lemma \ref{l-trace-2}, \eqref{e-MP-1} has a smooth solution on $M\times(0, \infty)$ with $g(t)$ uniformly equivalent to $h$ on any $[a, \infty)\subset(0, \infty)$. Combining the local higher order estimate of Chern-Ricci flow (See \cite{ShermanWeinkove2013} for example) and Lemma \ref{du-convergence}, we can conclude that $u(t)$ converges smoothly and locally to a smooth function $u_\infty$ as $t\to\infty$ and $\log\frac{\omega^n_\infty}{\theta_0^n}=u_\infty$. Taking $\sqrt{-1}\partial\bar\partial$ on both sides, we have \begin{equation*} -\text{\rm Ric}(g_\infty)+\text{\rm Ric}(h)=\sqrt{-1}\partial\bar\partial u_\infty, \end{equation*} which implies $-\text{\rm Ric}(g_\infty)=g_\infty$. Obviously, $g_\infty$ is K\"ahler. Uniqueness follows from \cite[Theorem 3]{Yau1978} (see also Proposition 5.1 in \cite{HLTT}). \end{proof} Taking $g_0=h$ in the theorem, we have \begin{cor} Let $(M,h)$ be a complete Hermitian manifold satisfying the assumptions in Theorem \ref{longtime}. Then the Chern-Ricci flow with initial data $h$ exists on $M\times[0,\infty)$ and converge uniformly on any compact subsets to the unique complete K\"ahler-Einstein metric with negative scalar curvature on $M$. \end{cor} For K\"ahler-Ricci flow, we have the following general phenomena related to Theorem \ref{longtime}. \begin{thm}\label{convergence-krf} Let $(M,h)$ be a smooth complete Hermitian manifold with $\mathrm{BK}(h) \geq -K_0$ and $|\nabla^h_{\bar\partial}T_h|_h\leq K_0$ for some constant $K_0\geq 0$. Moreover, assume \begin{equation*} -\text{\rm Ric}(h)+\sqrt{-1}\partial\bar\partial f \geq k h \end{equation*} for some constant $k>0$ and function $f\in C^\infty(M)\cap L^\infty(M)$. Suppose $g(t)$ is a smooth complete solution to the normalized K\"ahler-Ricci flow on $M\times[0,+\infty)$ with $g(0)=g_0$ which satisfies \begin{equation*} \frac{\det g_0}{\det h}\leq \Lambda \end{equation*} and \begin{equation*} R(g_0)\geq -L \end{equation*} for some $\Lambda,L>0$. Then $g(t)$ satisfies \begin{equation*} C^{-1}h\leq g(t)\leq C h \end{equation*} on $M\times[1, \infty)$ for some constant $C=C(n, K_0, k, ||f||_\infty, \Lambda, L)>0$. In particular, $ g(t)$ converges to the unique smooth complete K\"ahler-Einstein metric with negative scalar curvature. \end{thm} \begin{proof} We can assume $k=1$, otherwise we rescale $h$. We consider the corresponding unnormalized K\"ahler-Ricci flow $\widetilde g(s)=e^{t}g(t)$ with $s=e^t-1$. Then the corresponding Monge-Amp\`ere equation to the unnormalized K\"ahler-Ricci flow is: \begin{equation*}\left\{\begin{array}{l} \frac{\partial}{\partial s}\phi=\displaystyle{\log\frac{(\omega_0-s\text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial\phi)^n}{\theta_0^n}} \\ \phi(0)=0. \end{array} \right. \end{equation*} Here $\theta_0$ is the K\"ahler form of $h$. By the assumption $R(g_0)\geq -L $, Proposition 2.1 in \cite{Chen2009} and Lemma 5.1 in \cite{HLTT} with the fact \begin{equation*} \lf(\frac{\p}{\p s}-\wt\Delta\ri) \widetilde R\geq \frac{1}{n}\widetilde R^2,\end{equation*} we conclude that $\widetilde R:=R(\widetilde g(s))\geq \max\{-L, -\frac ns\}$ on $M\times[0, \infty)$. Note that $\ddot{\phi}=-R(\widetilde g(s))$, we have on $M\times[0, 1]$, $\dot{\phi}\leq C(L, \Lambda)$; on $M\times[1, \infty)$, $\dot{\phi}\leq C(L, \Lambda)+n\log s$. For lower bound of $\dot{\phi}$, we consider $Q=-\dot{\phi}+f$. We compute: \begin{equation*}\begin{split} \lf(\frac{\p}{\p s}-\wt\Delta\ri) Q=&-\lf(\frac{\p}{\p s}-\wt\Delta\ri) \dot{\phi}-\Delta f \\ =&tr_{\widetilde g}[\text{\rm Ric}(\theta_0)-\sqrt{-1}\partial\bar\partial f]\\ \leq&-tr_{\widetilde g}h\\ \leq&-ne^{-\frac{\dot{\phi}}{n}}\\ \leq&-ne^{\frac{1}{n}(Q-f)}\\ \leq&-C(n, ||f||_\infty)e^{\frac{Q}{n}}\\ \leq&-C(n, ||f||_\infty)Q^2, \end{split}\end{equation*} whenever $Q>0$. Then by the same argument as in the proof of Proposition 2.1 in \cite{Chen2009}, we conclude that $\dot{\phi}\geq -C(n, \lambda, ||f||_\infty)$ on $M\times[0, \infty)$. Here $\lambda$ is the lower bound of $\frac{\det g_0}{\det h}$. However, this estimate is not enough for later applications. We consider $F=-\dot{\phi}+f+n\log s$. Then we similarly obtain \begin{equation*} \lf(\frac{\p}{\p s}-\wt\Delta\ri) F\leq -C(n, ||f||_\infty)F^2, \end{equation*} whenever $F>0$. By Lemma 5.1 in \cite{HLTT}, we conclude that $F\leq\frac{C(n, ||f||_\infty)}{s}$ on $M\times[0, \infty)$. Therefore, we obtain \begin{equation*} \dot{\phi}\geq -C(n, ||f||_\infty)+n\log s \end{equation*} on $M\times[1, \infty)$. To sum up, for the bound of $\dot{\phi}$, we have: (i) On $M\times[0, 1]$, $-C(n, \lambda, ||f||_\infty)\leq \dot{\phi}\leq C(L, \Lambda)$; (ii) On $M\times[1, \infty)$, $-C(n, ||f||_\infty)+n\log s\leq \dot{\phi}\leq C(L, \Lambda)+n\log s$. Then we consider back to the normalized K\"ahler-Ricci flow $g(t)$. Since \begin{equation*} \log\frac{\det g(t)}{\det h}=-n\log (s+1)+\frac{\partial}{\partial s}\phi(s), \end{equation*} where $s=e^t-1$, we obtain: \begin{equation*} -C(n, ||f||_\infty)\leq\dot{u}(t)+u(t)\leq C(L, \Lambda) \end{equation*} on $M\times[\log 2, \infty)$. Here $u$ solves \eqref{e-MP-1}. Next, we consider $G(x, t)=\log tr_h g(t)-A(\dot{u}(t)+u(t)+f)$. Here $A$ is a large constant to be chosen. As in Section 1, we have \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) \log tr_h g(t)\leq C(n, K_0)tr_{ g(t)}h-1. \end{equation*} Therefore, \begin{equation*}\begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) G\leq& C(n, K_0)tr_{ g(t)}h-1+An+A(tr_{ g}\text{\rm Ric}(h)+tr_{ g}\sqrt{-1}\partial\bar\partial f)\\ \leq& (-A+C(n, K_0))tr_{g(t)}h-1+An\\ \leq& -tr_{g(t)}h+An. \end{split}\end{equation*} Here we take $A=C(n, K_0)+1$. On the other hand, \begin{equation*} tr_h g(t)\leq\frac{1}{(n-1)!}\cdot(tr_{ g(t)}h)^{n-1}\cdot\frac{\det g}{\det h}\leq C(n, L, \Lambda)(tr_{ g(t)}h)^{n-1}. \end{equation*} Then we have \begin{equation*}\begin{split} \lf(\frac{\p}{\p t}-\Delta\ri) G\leq&-C(n, L, \Lambda)(tr_h g(t))^{\frac{1}{n-1}}+C(n,K_0)\\ =&-C(n, L, \Lambda)e^{\frac{1}{n-1}\log tr_hg(t)}+C(n,K_0)\\ =&-C(n, L, \Lambda)e^{\frac{1}{n-1}[G+A(\dot{u}(t)+u(t)+f)]}+C(n,K_0)\\ \leq&-C(n, L, \Lambda, ||f||_\infty)e^{\frac{1}{n-1}G}+C(n,K_0)\\ \leq&-C(n, L, \Lambda, ||f||_\infty)G^2+C(n,K_0), \end{split}\end{equation*} whenever $G>0$. By similar argument as in the proof of Lemma 5.1 in \cite{HLTT}, we conclude that $G\leq C(n, L, \Lambda, ||f||_\infty, K_0)$ on $M\times[1, \infty)$. The difference here is that we consider the normalized K\"ahler-Ricci flow instead of K\"ahler-Ricci flow. The Perelman's distance distortion lemma for normalized K\"ahler-Ricci flow is the following:\begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) d_t(x_0, x)\geq -\frac{5(n-1)}{3}r_0^{-1}-d_t(x_0, x). \end{equation*} We then consider $t\cdot\phi(\frac{1}{Ar_0}[e^t\cdot d_t(x_0, x)+\frac{5(n-1)e^t}{3}r_0^{-1}])\cdot G(x,t)$, the results follows from the same argument as in the proof of Lemma 5.1 in \cite{HLTT}. This implies \begin{equation*} g(t)\leq C(n, L, \Lambda, ||f||_\infty, K_0)h \end{equation*} on $M\times[1, \infty)$. For lower bound, combining with $e^{\dot{u}(t)+u(t)}=\frac{\det g}{\det h}$, we have \begin{equation*} g(t)\geq C^{-1}(n, L, \Lambda, ||f||_\infty, K_0)h \end{equation*} on $M\times[1, \infty)$. Once we obtain the uniform equivalence of metrics of the normalized K\"ahler-Ricci flow, the convergence follows from the same argument as in the proof of Theorem 5.1 in \cite{HLTT}. This completes the proof of Theorem \ref{convergence-krf}. \end{proof} \appendix \section{Some basic relations} Let $g(t)$ be a solution to the Chern-Ricci flow, $$ \partial_tg=-\text{\rm Ric}(g) $$ and $h$ is another Hermitian metric. Let $\omega(t)$ be the K\"ahler form of $g(t)$, $\theta_0$ be the K\"ahler form of $h$. Let $$ \phi(t)=\int_0^t\log \frac{\omega^n(s)}{\theta_0^n}ds. $$ \begin{equation}\label{e-a-1} \omega(t)=\omega(0)-t\text{\rm Ric}(\theta_0)+\sqrt{-1}\partial\bar\partial\phi. \end{equation} Let $\dot\phi=\frac{\partial}{\partial t}\phi$. Then \begin{equation}\label{e-a-2} \lf(\frac{\p}{\p t}-\Delta\ri)\dot\phi=-\operatorname{tr}_g(\text{\rm Ric}(\theta_0)), \end{equation} where $ \Delta$ is the Chern Laplacian with respect to $ g$. On the other hand, if $g$ is as above, the solution $\widetilde g$ of the corresponding normalized Chern-Ricci flow with the same initial data $$ \partial_t\widetilde g=-\text{\rm Ric}(\widetilde g)-\widetilde g $$ is given by $$ \widetilde g(x,t)=e^{-t}g(x,e^{t}-1). $$ The corresponding potential $u$ is given by $$ u(t)=e^{-t}\int_0^te^s\log \frac{\widetilde \omega^n(s)}{\theta_0^n}ds $$ where $\widetilde \omega(s)$ is the K\"ahler form of $\widetilde g(s)$. Also, \begin{equation}\label{e-a-3} \widetilde\omega(t)=-\text{\rm Ric}(\theta_0)+e^{-t}(\text{\rm Ric}(\theta_0)+\omega(0))+\sqrt{-1}\partial\bar\partial u. \end{equation} \begin{equation}\label{e-a-5} \left(\frac{\partial}{\partial t}-\widetilde \Delta\right)(\dot u+u)=-\operatorname{tr}_{\widetilde g}\text{\rm Ric}(\theta_0)-n, \end{equation} where $\widetilde \Delta$ is the Chern Laplacian with respect to $\widetilde g$. \begin{lma}[See \cite{TosattiWeinkove2015,Lee-Tam}]\label{l-a-1} Let $g(t)$ be a solution to the Chern-Ricci flow and let $\Upsilon=\operatorname{tr}_{ h}g$, and $\Theta=\operatorname{tr}_g h$. \begin{equation*} \lf(\frac{\p}{\p t}-\Delta\ri) \log \Upsilon=\mathrm{I+II+III} \end{equation*} where \begin{equation*} \begin{split} \mathrm{I}\le &2\Upsilon^{-2}\text{\bf Re}\left( h^{i\bar l} g^{k\bar q}( T_0)_{ki\bar l} \hat \nabla_{\bar q}\Upsilon\right). \end{split} \end{equation*} \begin{equation*} \begin{split} \mathrm{II}=&\Upsilon^{-1} g^{i\bar{j}} \hat h^{k\bar l}g_{k\bar q} \left(\hat \nabla_i \overline{(\hat T)_{jl}^q}- \hat h^{p\bar q}\hat R_{i\bar lp\bar j}\right)\\ \end{split} \end{equation*} and \begin{equation*} \begin{split} \mathrm{III}=&-\Upsilon^{-1} g^{{i\bar{j}}} h^{k\bar l}\left(\hat \nabla_i\left(\overline{( T_0)_{jl\bar k}} \right) +\hat \nabla_{\bar l}\left( {( T_0)_{ik\bar j} }\right)-\overline{ (\hat T)_{jl}^q}( T_0)_{ik\bar q}^p \right). \end{split} \end{equation*} where $T_0$ is the torsion of $g_0=g(0)$, $\hat T$ is the torsion of $h$ and $\hat\nabla$ is the derivative with respect to the Chern connection of $h$. \end{lma} \section{A maximum principle}\label{s-max} We have the following maximum principle, see \cite{HLTT} for example. \begin{lma}\label{max} Let $(M^n,h)$ be a complete non-compact Hermitian manifold satisfying condition: There exists a smooth positive real exhaustion function $\rho$ such that $|\partial \rho|^2_h+|\sqrt{-1}\partial\bar\partial \rho|_h\leq C_1$. Suppose $g(t)$ is a solution to the Chern-Ricci flow on $M\times[0,S)$. Assume for any $0<S_1<S$, there is $C_2>0$ such that $$ C_2^{-1}h\le g(t) $$ for $0\leq t\le S_1$. Let $f$ be a smooth function on $M\times[0,S)$ which is bounded from above such that $$ \lf(\frac{\p}{\p t}-\Delta\ri) f\le0 $$ on $\{f>0\}$ in the sense of barrier. Suppose $f\le 0$ at $t=0$, then $f\le 0$ on $M\times[0,S)$. \end{lma} {We say that $$ \lf(\frac{\p}{\p t}-\Delta\ri) f\le \phi $$ in the sense of barrier means that for fixed $t_1>0$ and $x_1$, for any $\epsilon>0$, there is a smooth function $\sigma(x)$ near $x$ such that $\sigma(x_1)=f(x_1,t_1)$, $\sigma(x)\le f(x,t_1)$ near $x_1$, such that $\sigma$ is $C^2$ and at $(x_1,t_1)$ \begin{equation*} \frac{\partial_-}{\partial t}f(x,t)-\Delta \sigma(x)\le \phi(x)+\epsilon. \end{equation*} Here \begin{equation*} \frac{\partial_-}{\partial t}f(x,t)=\liminf_{h\to 0^+}\frac{f(x,t)-f(x,t-h)}h. \end{equation*} for a function $f(x,t)$.} \end{document}

\begin{document} \title{Initial-boundary value problems for nearly incompressible vector fields, and applications to the Keyfitz and Kranzer system} \author{Anupam Pal Choudhury\footnote{APC: Departement Mathematik und Informatik, Universit\"at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland. Email: [email protected]}, Gianluca Crippa\footnote{GC: Departement Mathematik und Informatik, Universit\"at Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland. Email: [email protected]}, Laura V. Spinolo\footnote{LVS: IMATI-CNR, via Ferrata 1, I-27100 Pavia, Italy. Email: [email protected]} } \date{} \maketitle \begin{abstract} We establish existence and uniqueness results for initial boundary value problems with nearly incompressible vector fields. We then apply our results to establish well-posedness of the initial-boundary value problem for the Keyfitz and Kranzer system of conservation laws in several space dimensions. \end{abstract} \section{Introduction} The Keyfitz and Kranzer system is a system of conservation laws in several space dimensions that was introduced in~\cite{KK} and takes the form \begin{equation} \label{e:KK} \partial_{t} U+\sum_{i=1}^{d} \partial_{x_{i}} (f^{i}(\vert U \vert) U) =0. \notag \end{equation} The unknown is $U: \mathbb R^d \to \mathbb R^N$ and $|U|$ denotes its modulus. Also, for every $i=1, \dots, d$ the function $f^i: \mathbb R \to \mathbb R^N$ is smooth. In this work we establish existence and uniqueness results for the initial-boundary value problem associated to~\eqref{e:KK}. The well-posedness of the Cauchy problem associated to~\eqref{e:KK} was established by Ambrosio, Bouchut and De Lellis in~\cite{ABD,AD} by relying on a strategy suggested by Bressan in~\cite{Br}. Note that the results in~\cite{ABD,AD} are one of the very few well-posedness results that apply to systems of conservation laws in several spaces dimensions. Indeed, establishing either existence or uniqueness for a general system of conservation laws in several space dimensions is presently a completely open problem, see~\cite{Daf,Serre1,Serre2} for an extended discussion on this topic. The basic idea underpinning the argument in~\cite{ABD,AD} is that~\eqref{e:KK} can be (formally) written as the coupling between a \emph{scalar} conservation law and a transport equation with very irregular coefficients. The scalar conservation law is solved by using the foundamental work by Kru{\v{z}}kov~\cite{Kr}, while the transport equation is handled by relying on Ambrosio's celebrated extension of the DiPerna-Lions' well-posendess theory, see~\cite{A} and~\cite{Diperna-Lions}, respectively, and~\cite{AC,Delellis2} for an overview. Note, however, that Ambrosio's theory~\cite{A} does not directly apply to~\eqref{e:KK} owing to a lack of control on the divergence of the vector fields. In order to tackle this issue, a theory of \textit{nearly incompressible vector fields} was developed, see~\cite{Delellis1} for an extended discussion. Since we will need it in the following, we recall the definition here. \begin{definition}\label{near-incom} Let $\Omega \subseteq \mathbb{R}^{d} $ be an open set and $T>0 $. We say that a vector field $b \in L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d})$ is \textbf{nearly incompressible} if there are a density function $\rho \in L^{\infty}((0,T) \times \Omega) $ and a constant $C > 0 $ such that \begin{itemize} \item[i.] $0 \leq \rho \leq C, \ \mathcal{L}^{d+1}-a.e. \ \text{in} \ (0,T) \times \Omega, $ and \item[ii.] the equation \begin{equation} \label{e:continuityrho} \partial_{t}\rho +\mathrm{div}(\rho b)=0 \end{equation} holds in the sense of distributions in $(0, T) \times \Omega$. \end{itemize} \end{definition} The analysis in~\cite{ABD, AD, Delellis1} ensures that, if $b \in L^\infty ((0, T) \times \mathbb R^d; \mathbb R^d) \cap BV ((0, T) \times \mathbb R^d; \mathbb R^d)$ is a nearly incompressible vector field with density $\rho \in BV ((0, T) \times \mathbb R^d)$, then the Cauchy problem $$ \left\{ \begin{array}{ll} \partial_{t}[ \rho u] +\mathrm{div}[ \rho bu ]=0 & \text{in $(0, T) \times \mathbb R^d$}\\ u = \overline{u} & \text{at $t=0$}\\ \end{array} \right. $$ is well-posed for every initial datum $\overline{u} \in L^\infty (\mathbb R^d)$. This result is pivotal to the proof of the well-posedness of the Cauchy problem for the Keyfitz and Kranzer system~\eqref{e:KK}. See also~\cite{ACFS} for applications of nearly incompressible vector fields to the so-called chromatography system of conservation laws. Note, furthermore, that here and in the following we denote by $BV$ the space of functions with \emph{bounded variation}, see~\cite{AFP} for an extended introduction. The present paper aims at extending the analysis in~\cite{ABD, AD, Delellis1} to the case of initial-boundary value problems. First, we establish the well-posedness of initial-boundary value problems with $BV$, nearly incompressible vector fields, see Theorem~\ref{IBVP-NC} below for the precise statement. In doing so, we rely on well-posedness results for continuity and transport equations with weakly differentiable vector fields established in~\cite{CDS1}, see also~\cite{CDS2} for related results. Next, we discuss the applications to the Keyfitz and Kranzer system~\eqref{e:KK}. We now provide a more precise description of our results concerning nearly incompressible vector fields. We fix an open, bounded set $\Omega$ and a nearly incompressible vector field $b$ with density $\rho$ and we consider the initial-boundary value problem \begin{equation} \left\{ \begin{array}{ll} \partial_{t} [\rho u]+ \text{div}[\rho u b]=0 & \text{in $(0,T)\times \Omega$}\\ u=\overline{u} & \text{at $t=0$}\\ u= \overline{g} & \text{on $ \Gamma^{-}$}, \end{array} \right. \label{prob-2} \end{equation} where $\Gamma^{-}$ is the part of the boundary $(0,T) \times \partial \Omega $ where the characteristic lines of the vector field $\rho b $ are \emph{inward pointing}. Note that, in general, if $b$ and $\rho$ are only weakly differentiable, one cannot expect that the solution $u$ is a regular function. Since $\Gamma^-$ will in general be negligible, then assigning the value of $u$ on $\Gamma^-$ is in general not possible. In~\S~\ref{s:formu} we provide the rigorous (distributional) formulation of the initial-boundary value problem~\eqref{prob-2} by relying on the theory of normal traces for low regularity vector fields, see~\cite{ACM,Anz,CF,CTZ}. We can now state our well-posedness result concerning~\eqref{prob-2}. \begin{theorem}\label{IBVP-NC} Let $T > 0 $ and $\Omega \subseteq \mathbb{R}^{d}$ be an open, bounded set with $C^2 $ boundary. Also, let $b \in BV((0,T) \times \Omega; \mathbb{R}^{d}) \cap L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d})$ be a nearly incompressible vector field with density $\rho \in BV((0,T) \times \Omega) \cap L^{\infty}((0,T) \times \Omega)$, see Definition~\ref{near-incom}. Further, assume that $\overline{u} \in L^{\infty}(\Omega) $ and $\overline{g} \in L^{\infty}(\Gamma^{-}) $. Then there is a distributional solution $u \in L^{\infty}((0,T) \times \Omega) $ to \eqref{prob-2} satisfying the maximum principle \begin{equation} \label{e:maxprin} \| u \|_{L^\infty} \leq \max \{ \| \overline u \|_{L^\infty}, \| \overline g \|_{L^\infty} \}. \end{equation} Also, if $u_1, \; u_2 \in L^\infty ((0, T) \times \Omega)$ are two different distributional solutions of the same initial-boundary value problem, then $\rho u_1 = \rho u_2$ $a.e.$ in $(0,T) \times \Omega$. \end{theorem} Note that the reason why we do not exactly obtain uniqueness of the function $u$ is because $\rho$ can attain the value $0$. If $\rho$ is bounded away from $0$, then the distributional solution $u$ of~\eqref{prob-2} is unique. Also, we refer to~\cite{Bar,Boyer,CDS1,CDS2,GS,Mis} for related results on the well-posedness of initial-boundary value problems for continuity and transport equation with weakly differentiable vector fields. In~\S~\ref{s:KK} we discuss the applications of Theorem~\ref{IBVP-NC} to the Keyfitz and Kranzer system and our main well-posedness result is Theorem~\ref{t:KK}. Note that the proof of Theorem~\ref{t:KK} combines Theorem~\ref{IBVP-NC}, the analysis in~\cite{Delellis1}, and well-posedness results for the initial-boundary value problems for scalar conservation laws established in~\cite{BLN, CR, Serre2}. \subsection*{Paper outline} In~\S~\ref{s:prel} we go over some preliminary results concerning normal traces of weakly differentiable vector fields. By relying on these results, in~\S~\ref{s:formu} we provide the rigorous formulation of the initial-boundary value problem~\eqref{prob-2}. In~\S~\ref{s:exi} we establish the existence part of Theorem~\ref{IBVP-NC}, and in~\S~\ref{s:uni} the uniqueness. In~\S~\ref{s:ssc} we establish some stability and space continuity property results. Finally, in~\S~\ref{s:KK} we discuss the applications to the Keyfitz and Kranzer system. \subsection*{Notation} For the reader's convenience, we collect here the main notation used in the present paper. \begin{itemize} \item $\mathrm{div}$: the divergence, computed with respect to the $x$ variable only. \item $\mathrm{Div}$: the complete divergence, i.e. the divergence computed with respect to the $(t, x)$ variables. \item $\mathrm{Tr} ( B, \partial \Lambda)$: the normal trace of the bounded, measure-divergence vector field $B$ on the boundary of the set $\Lambda$, see \S~\ref{s:prel}. \item $(\rho u)_0$, $\rho_0$: the initial datum of the functions $\rho u$ and $\rho$, see Lemma~\ref{trace-existence} and Remark~\ref{r} . \item $ T (f)$: the trace of the $BV$ function $f$, see Theorem~\ref{bv-trace}. \item $\mathcal H^s$: the $s$-dimensional Hausdorff measure. \item $f_{|_E}$: the restriction of the function $f$ to the set $E$. \item $\mu \llcorner E$: the restriction of the measure $\mu$ to the measurable set $E$. \item $a.e.$: almost everywhere. \item $|\mu|$: the total variation of the measure $\mu$. \item $a \cdot b$: the (Euclidean) scalar product between $a$ and $b$. \item $\mathbf{1}_E:$ the characteristic function of the measurable set $E$. \item $\Gamma, \Gamma^-, \Gamma^+, \Gamma^0$: see~\eqref{e:gamma}. \item $\vec n$: the outward pointing, unit normal vector to $\Gamma$. \end{itemize} \section{Preliminary results} \label{s:prel} In this section, we briefly recall some notions and results that shall be used in the sequel. First, we discuss the notion of normal trace for weakly differentiable vector fields, see~\cite{ACM,Anz,CF,CTZ}. Our presentation here closely follows that of \cite{ACM}. Let $\Lambda \subseteq \mathbb{R}^{N} $ be an open set and let us denote by $\mathcal{M}_{\infty}(\Lambda) $, the family of bounded, measure-divergence vector fields. The space $\mathcal{M}_{\infty}(\Lambda) $, therefore, consists of bounded functions $B \in L^{\infty}(\Lambda;\mathbb{R}^{N})$ such that the distributional divergence of $B$ (denoted by $\text{Div} B $) is a locally bounded Radon measure on $\Lambda$. The normal trace of $B \in \mathcal{M}_{\infty}(\Lambda)$ on the boundary $\partial \Lambda $ can be defined as follows. \begin{definition} Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with Lipschitz continuous boundary and let $B \in \mathcal{M}_{\infty}(\Lambda)$. The normal trace of $B$ on $\partial \Lambda $ is a distribution defined by the identity \begin{equation} \Big\langle \emph{Tr}(B,\partial \Lambda), \psi \Big\rangle = \int_{\Lambda} \nabla \psi \cdot B \ dy + \int_{\Lambda} \psi\ d(\emph{Div} B) , \qquad \forall \ \psi \in C^{\infty}_{c}(\mathbb{R}^{N}). \label{prel-1} \end{equation} Here $\emph{Div} B $ denotes the distributional divergence of $B$ and is a bounded Radon measure on $\Lambda $. \end{definition} Note that, owing to the Gauss-Green formula, if $B$ is a smooth vector field, then $\mathrm{Tr}(B,\partial \Lambda) = B \cdot \vec n$, where $\vec n$ denotes the outward pointing, unit normal vector to $\partial \Lambda$. Note, furthermore, that the analysis in~\cite{ACM} shows that the normal trace distribution satisfies the following properties. \begin{itemize} \item[(a)] The normal trace distribution is induced by an $L^{\infty}$ function on $\partial \Lambda $, which we shall continue to refer to as $\mathrm{Tr}(B,\partial \Lambda) $. The bounded function $\mathrm{Tr}(B,\partial \Lambda) $ satisfies \[\Vert \mathrm{Tr}(B,\partial \Lambda) \Vert_{L^{\infty}(\partial \Lambda)} \leq \Vert B \Vert_{L^{\infty}(\Lambda)}. \] \item[(b)] Let $\Sigma $ be a Borel set contained in $\partial \Lambda_{1} \cap \partial \Lambda_{2} $, and let $\vec{n}_{1}=\vec{n}_{2} \ \text{on}\ \Sigma$ (here $\vec{n}_{1},\vec{n}_{2} $ denote the outward pointing, unit normal vectors to $\partial \Lambda_{1},\partial \Lambda_{2} $ respectively). Then \begin{equation} \mathrm{Tr}(B, \partial \Lambda_{1}) = \mathrm{Tr}(B, \partial \Lambda_{2}) \qquad \text{$\mathcal{H}^{N-1}$-a.e.~on $\Sigma$.} \label{prel-2} \end{equation} \end{itemize} In the following we will use several times the following renormalization result, which was established in~\cite{ACM}. \begin{theorem}\label{trace-renorm} Let $B \in BV (\Lambda;\mathbb{R}^{N})\cap L^{\infty}(\Lambda;\mathbb{R}^{N}) $ and $w \in L^{\infty}(\Lambda) $ be such that $\emph{Div} (wB )$ is a Radon measure. If $\Lambda' \subset \subset \Lambda $ is an open set with bounded and Lipschitz continuous boundary and $h \in C^{1}(\mathbb{R})$, then \begin{equation} \emph{Tr}(h(w)B,\partial \Lambda')=h\left(\frac{\emph{Tr}(wB,\partial \Lambda')}{\emph{Tr}(B,\partial \Lambda')}\right) \emph{Tr}(B,\partial \Lambda') \qquad \text{$\mathcal{H}^{N-1}$-a.e.~on~$\partial \Lambda'$,} \notag \end{equation} where the ratio $\displaystyle{\frac{\emph{Tr}(wB,\partial \Lambda')}{\emph{Tr}(B,\partial \Lambda')} }$ is arbitrarily defined at points where the trace $\emph{Tr}(B,\partial \Lambda') $ vanishes. \end{theorem} We can now introduce the notion of normal trace on a general bounded, Lipschitz continuous, oriented hypersurface $\Sigma \subseteq \mathbb{R}^{N}$ in the following manner. Since $\Sigma $ is oriented, an orientation of the normal vector $\vec{n}_{\Sigma} $ is given. We can then find a domain $\Lambda_{1} \subseteq \mathbb{R}^{N} $ such that $\Sigma \subseteq \partial \Lambda_{1} $ and the normal vectors $\vec{n}_{\Sigma}, \vec{n}_{1} $ coincide. Using \eqref{prel-2}, we can then define \[\text{Tr}^{-}(B,\Sigma):= \text{Tr}(B,\partial \Lambda_{1}). \] Similarly, if $\Lambda_{2} \subseteq \mathbb{R}^{N} $ is an open set satisfying $\Sigma \subseteq \partial \Lambda_{2} $, and $\vec{n}_{2}=-\vec{n}_{\Sigma} $, we can define \[\text{Tr}^{+}(B,\Sigma):=- \text{Tr}(B,\partial \Lambda_{2}). \] Furthermore we have the formula \[(\text{Div} B)\llcorner \Sigma= \Big( \text{Tr}^{+}(B,\Sigma)-\text{Tr}^{-}(B,\Sigma) \Big) \mathcal{H}^{N-1} \llcorner \Sigma. \] Thus $\text{Tr}^{+} $ and $\text{Tr}^{-} $ coincide $\mathcal{H}^{N-1}- $a.e. on $\Sigma$ if and only if $\Sigma $ is a $(\text{Div} B)$-negligible set.\\ We next recall some results from \cite{ACM} concerning space continuity. \begin{definition}\label{graph} A family of oriented surfaces $\{\Sigma_{r} \}_{r \in I} \subseteq \mathbb{R}^{N} $ (where $I \subseteq \mathbb{R}$ is an open interval) is called a family of graphs if there exist \begin{itemize} \item a bounded open set $D \subseteq \mathbb{R}^{N-1}$, \item a Lipschitz function $f:D \rightarrow \mathbb{R}$, \item a system of coordinates $(x_{1},\cdots,x_{N})$ \end{itemize} such that the following holds true: For each $r \in I$, we can write \begin{equation} \Sigma_{r}=\big\{(x_{1},\cdots,x_{N}): f(x_{1},\cdots,x_{N-1})-x_{N}=r \big\}, \label{ACM-99} \end{equation} and the orientation of $\Sigma_{r}$ is determined by the normal $\displaystyle{\frac{(-\nabla f,1)}{\sqrt{1+\vert \nabla f \vert^{2}}} }$. \end{definition} We now quote a space continuity result. \begin{theorem}[see \cite{ACM}]\label{Weak-continuity} Let $B \in \mathcal{M}_{\infty}(\mathbb{R}^{N})$ and let $\{\Sigma_{r} \}_{r \in I} $ be a family of graphs as above. For a fixed $r_{0} \in I$, let us define the functions $\alpha_{0}, \alpha_{r}: D \rightarrow \mathbb{R} $ as \begin{equation} \begin{aligned} \alpha_{0}(x_{1},\cdots,x_{N-1})&:=\emph{Tr}^{-}(B,\Sigma_{r_{0}})(x_{1},\cdots,x_{N-1},f(x_{1},\cdots,x_{N-1})-r_{0}), \ \text{and} \\ \alpha_{r}(x_{1},\cdots,x_{N-1})&:=\emph{Tr}^{+}(B,\Sigma_{r})(x_{1},\cdots,x_{N-1},f(x_{1},\cdots,x_{N-1})-r) . \end{aligned} \label{ACM-100} \end{equation} Then $\alpha_{r} \stackrel{*}{\rightharpoonup} \alpha_{0} $ weakly$^{*}$ in $L^{\infty}(D,\mathcal{L}^{N-1} \llcorner D) $ as $r \rightarrow r^{+}_{0}$. \end{theorem} We will also need the following result, which was originally established in~\cite{CDS1}. \begin{lemma}\label{extension} Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with bounded and Lipschitz continuous boundary and let $B$ belong to $\mathcal{M}_{\infty}(\Lambda)$. Then the vector field \begin{equation} \tilde{B}(z):= \left\{\begin{array}{ll} B(z) & z \in \Lambda \\ 0 & \text{otherwise} \end{array}\right. \notag \end{equation} belongs to $\mathcal{M}_{\infty}(\mathbb{R}^{N})$. \end{lemma} We conclude by recalling some results concerning traces of $BV$ functions and we refer to~\cite[\S 3]{AFP} for a more extended discussion. \begin{theorem} \label{bv-trace} Let $\Lambda \subseteq \mathbb{R}^{N}$ be an open and bounded set with bounded and Lipschitz continuous boundary. There exists a bounded linear mapping \begin{equation} T: BV(\Lambda) \rightarrow L^{1}(\partial \Lambda;\mathcal{H}^{N-1}) \label{ACM-101} \end{equation} such that $T (f) = f_{|_{\partial \Lambda}}$ if $f$ is continuous up to the boundary. Also, \begin{equation} \int_{\Lambda} \nabla \psi \cdot f \ dy = - \int_{\Lambda} \psi \ d(\emph{\text{Div}} f) + \int_{\partial \Lambda} \psi \ Tf \cdot \vec n \ d\mathcal{H}^{N-1}, \label{ACM-102} \end{equation} for all $f \in BV(\Lambda)$ and $\psi \in C^{\infty}_{c}(\mathbb{R}^{N})$. In the above expression, $\vec n$ denotes the outward pointing, unit normal vector to $\partial \Lambda$. \end{theorem} By comparing~\eqref{prel-1} and~\eqref{ACM-102} we conclude that \begin{equation} \label{e:equal} \mathrm{Tr} (f, \partial \Lambda) = T (f) \cdot \vec n, \quad \text{for every $f \in BV (\Lambda)$}. \end{equation} By combining Theorems~3.9 and~3.88 in~\cite{AFP} we get the following result. \begin{theorem}[\cite{AFP}] \label{t:traceafp} Assume $\Lambda \subseteq \mathbb R^N$ is an open set with bounded and Lipschitz continuous boundary. If $f \in BV (\Lambda; \mathbb R^m)$, then there is a sequence $\{\tilde f_m \} \subseteq C^\infty (\Lambda)$ such that \begin{equation} \label{e:tracefp} \tilde f_m \to f \; \text{ strongly in $L^1 (\Lambda; \mathbb R^m)$}, \qquad T (\tilde f_m) \to T(f) \text{ strongly in $L^1 (\partial \Lambda; \mathbb R^m)$}. \end{equation} Also, we can choose $\tilde f_m$ in such a way that \begin{itemize} \item $\tilde f_m \ge 0$ if $f \ge 0$, \item if $f \in L^\infty (\Lambda; \mathbb R^m)$, then \begin{equation} \label{e:four} \| \tilde f_m \|_{L^\infty} \leq 4 \| f \|_{L^\infty}. \end{equation} \end{itemize} \end{theorem} A sketch of the proof of Theorem~\ref{t:traceafp} is provided in~\S~\ref{s:proof1}. \section{Distributional formulation of the problem} \label{s:formu} In this section, we follow~\cite{Boyer,CDS1} and we provide the distributional formulation of the problem \eqref{prob-2}. We first establish a preliminary result. \begin{lemma}\label{trace-existence} Let $\Omega \subseteq \mathbb{R}^{d}$ be an open bounded set with $C^2$ boundary and let $T > 0 $. We assume that $b \in L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}) $ is a nearly incompressible vector field with density $\rho \in L^{\infty}((0,T) \times \Omega) $, see Definition \ref{near-incom}. If $u \in L^{\infty}((0,T) \times \Omega)$ satisfies \begin{equation} \int_{0}^{T} \int_{\Omega} \rho u (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt= 0, \quad \forall \ \phi \in \mathcal{C}^{\infty}_{c} ((0,T) \times \Omega), \label{iden-2} \end{equation} then there are two unique functions, which we henceforth denote by $\emph{Tr}(\rho u b) \in L^{\infty}((0,T) \times \partial \Omega) $ and $(\rho u)_{0} \in L^{\infty}(\Omega)$, that satisfy \begin{equation} \int_{0}^{T} \! \! \int_{\Omega} \rho u (\partial_{t} \psi+ b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \! \! \int_{\partial \Omega} \emph{Tr}(\rho u b) \psi \ d\mathcal{H}^{d-1}\ dt - \int_{\Omega} \psi(0,\cdot) (\rho u)_{0}\ dx, \quad \forall \psi \in \mathcal{C}^{\infty}_{c} ([0,T) \times \mathbb{R}^{d}). \label{iden-3} \end{equation} Also, we have the bounds \begin{equation} \label{e:maxprintraces2} \| \emph{Tr}(\rho u b) \|_{L^\infty((0,T) \times \partial \Omega) } , \; \| (\rho u)_{0} \|_{ L^{\infty}(\Omega)} \leq \max\{ \| \rho u \|_{L^\infty((0,T) \times \Omega) } ; \| \rho u b \|_{L^\infty((0,T) \times \Omega) } \}. \end{equation} \end{lemma} \begin{proof} First of all, let us note that the uniqueness of such functions follow from the liberty in choosing the test functions $\psi$. Therefore it is enough to discuss the existence of the functions with the above properties. Let us define \begin{equation} B(t,x):= \left\{ \begin{array}{ll} (u \rho, u \rho b) & (t,x) \in (0,T)\times \Omega \\ 0 &\text{elsewhere in}\ \mathbb{R}^{d+1}.\\ \end{array} \right. \label{e:extend} \end{equation} Then $B \in L^{\infty}(\mathbb{R}^{d+1})$ and from \eqref{iden-2}, it also follows that $\big[\text{Div} B \llcorner {(0,T) \times \Omega} \big]=0 $. We can now apply Lemma \ref{extension} with $\Lambda= (0,T) \times \Omega $ to conclude that $B \in \mathcal{M}_{\infty}(\mathbb{R}^{d+1}).$ Hence $B$ induces the existence of normal trace on $\partial \Lambda$. Let \begin{equation} \text{Tr}(\rho u b):= \text{Tr} (B,\partial \Lambda) \Big\vert_{(0,T) \times \partial \Omega}, \ \ (\rho u)_{0}:= -\text{Tr}(B,\partial \Lambda) \Big\vert_{\{0 \} \times \Omega}. \notag \end{equation} The identity \eqref{iden-3} then follows from \eqref{prel-1} by virtue of the fact that $\text{Div}B=0 \ \text{in}\ (0,T)\times \Omega $. \end{proof} \begin{remark} \label{r} We define the vector field $P:=(\rho,\rho b) $ and we point out that $P \in {L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d+1})}$ since $\rho$ and $b$ are both bounded functions. By introducing the same extension as in~\eqref{e:extend} and using the fact that \begin{equation} \int_{0}^{T} \int_{\Omega} \rho (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt= 0, \quad \forall \ \phi \in \mathcal{C}^{\infty}_{c} ((0,T) \times \Omega), \notag \end{equation} we can argue as in the proof of the above lemma to establish the existence of unique functions $\emph{Tr}(\rho b) \in L^{\infty}((0,T) \times \partial \Omega) $ and $\rho_0 \in L^\infty(\Omega)$ defined as $$ \emph{Tr}(\rho b):= \emph{Tr}(P, \partial \Lambda) \Big\vert_{(0,T) \times \partial \Omega}, \quad \rho_0 : = - \emph{Tr}(P, \partial \Lambda) \Big\vert_{\{ 0 \} \times \Omega}. $$ In this way, we can give a meaning to the normal trace $\mathrm{Tr} (\rho b)$ and to the initial datum $\rho_0$. Also, we have the bounds \begin{equation} \label{e:maxprintraces1} \| \emph{Tr}(\rho b) \|_{L^\infty((0,T) \times \partial \Omega) } , \; \| \rho_{0} \|_{ L^{\infty}(\Omega)} \leq \max\{ \| \rho \|_{L^\infty((0,T) \times \Omega) } ; \| \rho b \|_{L^\infty((0,T) \times \Omega) }\}. \end{equation} \end{remark} We can now introduce the distributional formulation to the problem \eqref{prob-2} by using Lemma \ref{trace-existence}. We introduce the following notation: \begin{equation} \left. \begin{array}{ll} \Gamma: = (0, T) \times \partial \Omega, & \Gamma^{-}:= \{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x)<0 \},\\ \Gamma^{+}:=\{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x) > 0 \}, & \Gamma^0:=\{(t,x) \in \Gamma: \ \text{Tr} (\rho b)(t,x) = 0 \}. \\ \end{array} \right. \label{e:gamma} \end{equation} \begin{definition} \label{d:distrsol} Let $\Omega \subseteq \mathbb{R}^{d}$ be an open bounded set with $C^2$ boundary and let $T > 0 $. Let $b \in L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}) $ be a nearly incompressible vector field with density $\rho $, see Definition~\ref{near-incom}. Fix $\overline{u} \in L^\infty (\Omega)$ and $\overline{g} \in L^\infty (\Gamma^-)$. We say that a function $u \in L^{\infty}((0,T)\times \Omega)$ is a distributional solution of \eqref{prob-2} if the following conditions are satisfied: \begin{itemize} \item[i.] $u$ satisfies \eqref{iden-2}; \item[ii.] $(\rho u)_{0}= \overline{u} \rho_0 $; \item[iii.] $\emph{Tr}(\rho u b)= \overline{g} \emph{Tr}(\rho b) $ on the set $\Gamma^{-}$. \end{itemize} \end{definition} \section{Proof of Theorem~\ref{IBVP-NC}: existence of solution} \label{s:exi} In this section we establish the existence part of Theorem~\ref{IBVP-NC}, namely we prove the existence of functions $u \in L^{\infty}((0,T)\times \Omega) $ and $w \in L^{\infty}(\Gamma^{0}\cup \Gamma^+ ) $ such that for every $\psi \in C^{\infty}_{c}([0,T)\times \mathbb{R}^{d})$, \begin{equation} \int_{0}^{T} \int_{\Omega} \rho u (\partial_{t} \psi+b \cdot \nabla \psi) \ dx dt= \int_{\Gamma^{-}} \overline{g} \text{Tr}(\rho b) \psi \ d\mathcal{H}^{d-1} dt +\int_{\Gamma^{+}\cup \Gamma^0} \! \!\text{Tr}(\rho b) \psi w \ d\mathcal{H}^{d-1} dt-\int_{\Omega} \rho_0 \ \overline{u}\ \psi(0,\cdot)\ dx . \label{weak-exist1} \end{equation} We proceed as follows: first, in~\S~\ref{ss:as} we introduce an approximation scheme. Next, in~\S~\ref{ss:limit} we pass to the limit and establish existence. \subsection{Approximation scheme} \label{ss:as} In this section we rely on the analysis in~\cite[\S~3.3]{Delellis1}, but we employ a more refined approximation scheme which guarantees strong convergence of the traces. We set $\Lambda: =(0, T) \times \Omega$ and we recall that by assumption $\rho \in BV(\Lambda) \cap L^\infty (\Lambda).$ We apply Theorem~\ref{t:traceafp} and we select a sequence $\{ \tilde \rho_m \} \subseteq C^\infty (\Lambda)$ satisfying~\eqref{e:tracefp} and~\eqref{e:four}. Next, we set \begin{equation} \label{e:rhoenne} \rho_m: = \frac{1}{m} + \tilde \rho_m \ge \frac{1}{m}. \end{equation} We then apply Theorem~\ref{t:traceafp} to the function $b \rho$ and we set \begin{equation} \label{e:benne} b_m : = \frac{\widetilde{(b \rho)}_m}{\rho_m}. \end{equation} Owing to Theorem~\ref{t:traceafp} we have \begin{equation} \label{e:elle1conv} \rho_m \to \rho \; \text{strongly in $L^1 ((0, T) \times \Omega)$}, \quad b_m \rho_m \to b \rho \; \text{strongly in $L^1 ((0, T) \times \Omega;\mathbb R^d)$}. \end{equation} and, by using the identity~\eqref{e:equal}, \begin{equation} \label{e:traceconv} \begin{split} \mathrm{Tr} (\rho_m) \to \mathrm{Tr} (\rho)& \; \text{strongly in $L^1 (\Gamma)$}, \quad \mathrm{Tr} (\rho_mb_m) \to \mathrm{Tr} (\rho b) \; \text{strongly in $L^1 (\Gamma)$}, \\ & \quad \rho_{m0} \to \rho_0 \; \text{strongly in $L^1 (\Omega)$}. \end{split} \end{equation} Note, furthermore, that \begin{equation} \label{e:linftytraces} \| \mathrm{Tr} (b_m \rho_m ) \|_{L^\infty} \stackrel{\eqref{e:maxprintraces1}}{\leq} \| b_m \rho_m \|_{L^\infty} \stackrel{\eqref{e:four}}{\leq} 4 \| b \rho \|_{L^\infty}. \end{equation} In the following, we will use the notation \begin{equation} \label{e:gammadef} \Gamma_m^- : = \big\{(t, x) \in \Gamma: \; \mathrm{Tr} (\rho_m b_m) < 0 \big\}, \qquad \Gamma_m^+ : = \big\{(t, x) \in \Gamma: \; \mathrm{Tr} (\rho_m b_m) > 0 \big\} \end{equation} Finally, we extend the function $\overline{g}$ to the whole $\Gamma$ by setting it equal to $0$ outside $\Gamma^-$ and we construct two sequences $\{ \overline{g}_m \} \subseteq C^1 (\Gamma)$ and $\{\overline{u}_m \} \subseteq C^\infty (\Omega)$ such that \begin{equation} \label{e:convbdata} \overline{g}_m \to \overline{g} \; \text{strongly in $L^1 (\Gamma)$}, \quad \overline{u}_m \to \overline{u} \; \text{strongly in $L^1 (\Omega)$} \end{equation} and \begin{equation} \label{e:tomaxprin} \| \overline{g}_m \|_{L^\infty} \leq \| \overline{g} \|_{L^\infty}, \quad \| \overline{u}_m \|_{L^\infty} \leq \| \overline{u} \|_{L^\infty}. \end{equation} We can now define the function $u_m$ as the solution of the initial-boundary value problem \begin{equation} \left\{ \begin{array}{ll} \partial_{t} u_m+b_m \cdot \nabla u_m=0 & \text{on $(0, T) \times \Omega$} \\ u_m=\overline{u}_m & \text{at $t=0$}\\ u_m= \overline{g}_m & \text{on} \; \tilde \Gamma^{-}_m, \end{array} \right. \label{exist3} \end{equation} where $\tilde \Gamma^-_m$ is the subset of $\Gamma$ such that the characteristic lines of $b_m$ starting at a point in $\tilde \Gamma^-_m$ are entering $(0, T) \times \Omega$. We recall~\eqref{e:gammadef} and we point out that $$ \Gamma^-_m \subseteq \tilde \Gamma^-_m \subseteq \big\{ (t, x) \in \Gamma: \; b_m \cdot \vec n \leq 0 \big\}. $$ In the previous expression, $\vec n$ denotes as the outward pointing, unit normal vector to $\partial \Omega$. By using the classical method of characteristics (see also~\cite{Bar}) we establish the existence of a solution $u_m$ satisfying \begin{equation} \Vert u_m\Vert_{\infty} \leq \max\{\Vert \overline{u}_m \Vert_{\infty}, \Vert \overline{g}_m \Vert_{\infty} \} \stackrel{\eqref{e:tomaxprin}}{\leq} \max \{\Vert \overline{u} \Vert_{\infty}, \Vert \overline{g} \Vert_{\infty} \}. \label{mp} \end{equation} We now introduce the function $h_m$ by setting \begin{equation} \label{e:accaenne} h_m : = \partial_t \rho_m + \mathrm{div} (b_m \rho_m) \end{equation} and by using the equation at the first line of~\eqref{exist3} we get that $$ \partial_t (\rho_m u_m ) + \mathrm{div} (b_m \rho_m u_m ) = h_m u_m. $$ Owing to the Gauss-Green formula, this implies that, for every $\psi \in C^\infty_c ([0, T) \times \mathbb R^d)$, \begin{equation} \begin{aligned} &\int_{0}^{T} \int_{\Omega} \rho_m u_m [\partial_{t} \psi+ b_m \cdot \nabla \psi ] \ dx dt + \int_0^T \int_\Omega h_m u_m \psi \, dx dt \\ &\quad = -\int_{\Omega} \psi(0,x) \overline{\rho}_{m0} \overline{u}_{m} \ dx- \int_{0}^{T} \! \! \int_{\partial \Omega} \psi u_m \rho_m b_m \cdot \vec n \, d\mathcal{H}^{d-1} dt \\ & \quad = -\int_{\Omega} \psi(0,x) \overline{\rho}_{m0} \overline{u}_{m} \ dx- \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^-} \overline{g}_{m} \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt - \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^+} u_m \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt. \end{aligned} \label{weak-exist2} \end{equation} In the above expression, we have used the notation introduced in~\eqref{e:gammadef} and the fact that $\mathrm{Tr} (\rho_m b_m )=0$ on~${\Gamma \setminus (\Gamma^-_m \cup \Gamma^+_m)}$. \subsection{Passage to the limit} \label{ss:limit} Owing to the uniform bound~\eqref{mp}, there are a subsequence of $\{ u_m \}$ (which, to simplify notation, we do not relabel) and a function $u \in L^\infty ((0, T) \times \Omega$ such that \begin{equation} \label{e:uweaks} u_m \weaks u \; \text{weakly$^\ast$ in $L^\infty ((0, T) \times \Omega)$. } \end{equation} The goal of this paragraph is to show that the function $u$ in~\eqref{e:uweaks} is a distributional solution of~\eqref{prob-2} by passing to the limit in~\eqref{weak-exist2}. We first introduce a technical lemma. \begin{lemma} \label{l:meyerserrin} We can construct the approximating sequences $\{ \rho_m \}$ and $\{ b_m \}$ in such a way that the sequence $\{ h_m \}$ defined as in~\eqref{e:accaenne} satisfies \begin{equation} \label{e:convaccaemme} h_m \to 0 \; \text{strongly in $L^1 ((0, T) \times \Omega)$}. \end{equation} \end{lemma} The proof of Lemma~\ref{l:meyerserrin} is deferred to~\S~\ref{s:proof1} . For future reference, we state the next simple convergence result as a lemma. \begin{lemma} \label{l:traces} Assume that \begin{equation} \label{e:hyp} \mathrm{Tr} (\rho_m b_m )\to \mathrm{Tr}(\rho b) \; \text{strongly in $L^1 (\Gamma)$}. \end{equation} Let $\Gamma^-_m$ and $\Gamma^+_m$ as in~\eqref{e:gammadef} and $\Gamma^-$ and $\Gamma^+$ as in~\eqref{e:gamma}, respectively. Then, up to subsequences, \begin{equation} \label{e:convchar1} \mathbf{1}_{\Gamma^-_m} \to \mathbf{1}_{\Gamma^-} + \mathbf{1}_{\Gamma'} \; \text{strongly in $L^1 (\Gamma)$} \end{equation} and \begin{equation} \label{e:convchar2} \mathbf{1}_{\Gamma^+_m} \to \mathbf{1}_{\Gamma^+} + \mathbf{1}_{\Gamma''} \; \text{strongly in $L^1 (\Gamma)$}, \end{equation} where $\Gamma'$ and $\Gamma''$ are (possibly empty) measurable sets satisfying \begin{equation} \label{e:subsetgamma0} \Gamma', \Gamma'' \subseteq \Gamma^0. \end{equation} \end{lemma} \begin{proof}[Proof of Lemma~\ref{l:traces}] Owing to~\eqref{e:hyp} we have that, up to subsequences, the sequence $\{ \mathrm{Tr} (\rho_m b_m) \}$ satisfies $$ \mathrm{Tr} (\rho_m b_m) (t, x) \to \mathrm{Tr} (\rho b)(t, x), \quad \text{for $\mathcal{L}^1 \otimes \mathcal{H}^{d-1}$-almost every $(t, x) \in \Gamma.$} $$ Owing to the Lebesgue Dominated Convergence Theorem, this implies~\eqref{e:convchar1} and~\eqref{e:convchar2}. \end{proof} We can now pass to the limit in all the terms in~\eqref{weak-exist2}. First, by combining~\eqref{e:elle1conv},~\eqref{mp},~\eqref{e:uweaks} and~\eqref{e:convaccaemme} we get that \begin{equation} \label{e:conv11} \int_{0}^{T} \! \! \int_{\Omega} \rho_m u_m [\partial_{t} \psi+ b_m \cdot \nabla \psi ] \ dx dt + \int_0^T \! \! \int_\Omega h_m u_m \psi \, dx dt \to \int_{0}^{T} \! \! \int_{\Omega} \rho u [\partial_{t} \psi+ b \cdot \nabla \psi ] \ dx dt, \end{equation} for every $ \psi \in C^\infty_c ([0, T) \times \mathbb R^d)$. Also, by combining the second line of~\eqref{e:traceconv} with~\eqref{e:convbdata} and~\eqref{e:tomaxprin} we arrive at \begin{equation} \label{e:conv21} \int_{\Omega} \psi(0,x) {\rho}_{m0} \overline{u}_{m} \ dx \to \int_{\Omega} \psi(0,x) {\rho}_{0} \overline{u} \ dx , \end{equation} for every $ \psi \in C^\infty_c ([0, T) \times \mathbb R^d). $ Next, we combine~\eqref{e:traceconv},~\eqref{e:convbdata},~\eqref{e:tomaxprin},~\eqref{e:convchar1},~\eqref{e:subsetgamma0} and the fact that $\mathrm{Tr}(\rho b) =0 $ on $\Gamma^0$ to get that \begin{equation} \label{e:conv4} \begin{split} \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^-} \overline{g}_{m} \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt \to & \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma^-} \overline{g} \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt \\ & = \int_{0}^{T} \! \! \int_{\Gamma^-} \overline{g} \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt, \end{split} \end{equation} for every $\psi \in C^\infty_c ([0, T) \times \Omega; \mathbb R^d)$. We are left with the last term in~\eqref{weak-exist2}: first, we denote by $u_{m{|_\Gamma}}$ the restriction of $u_m$ to $\Gamma$. Since $u_m$ is a smooth function, then $$ \| u_{m{|_\Gamma}} \|_{L^\infty (\Gamma)} \leq \| u_m \|_{L^\infty ((0, T) \times \Omega)} \stackrel{\eqref{mp}}{\leq} \max \big\{ \| \bar u \|_{L^\infty}, \| \bar g \|_{L^\infty} \big\} $$ and hence there is a function $w \in L^\infty (\Gamma)$ such that, up to subsequences, \begin{equation} \label{e:convw} u_{m{|_\Gamma}} \weaks w \; \text{weakly$^\ast$ in $L^\infty (\Gamma)$}. \end{equation} By combining~\eqref{e:traceconv},~\eqref{e:convchar2},~\eqref{e:convw} and the fact that $\mathrm{Tr} (\rho b) =0$ on $\Gamma^0$ we get that \begin{equation} \begin{split} \label{e:conv5} \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma_m^+} u_m \psi \mathrm{Tr} (\rho_m b_m ) d\mathcal{H}^{d-1} dt \to & \int_{0}^{T} \! \! \int_{\partial \Omega} \mathbf{1}_{\Gamma^+} w \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt \\ & = \! \! \int_{\Gamma^+ \cup \Gamma^0} w \psi \mathrm{Tr} (\rho b ) d\mathcal{H}^{d-1} dt . \end{split} \end{equation} By combining~\eqref{e:conv11},~\eqref{e:conv21},~\eqref{e:conv4} and~\eqref{e:conv5} we get that $u$ satisfies~\eqref{weak-exist1} and this establishes existence of a distributional solution of~\eqref{prob-2}. \subsection{Proof of Lemma~\ref{l:meyerserrin}} \label{s:proof1} To ensure that~\eqref{e:convaccaemme} holds we use the same approximation \emph{\`a la} Meyers-Serrin as in~\cite[pp.122-123]{AFP}. We now recall some details of the construction. First, we fix a countable family of open sets $\big\{ \Lambda_h \big\}$ such that \begin{itemize} \item[i.] $\Lambda_h$ is compactly contained in $\Lambda$, for every $h$; \item[ii.] $\big\{ \Lambda_h \big\}$ is a covering of $\Lambda$, namely $$ \bigcup_{h=1}^\infty \Lambda_h = \Lambda; $$ \item[iii.] every point in $\Lambda$ is contained in at most $4$ sets $\Lambda_h$. \end{itemize} Next, we consider a partition of unity associated to $\big\{ \Lambda_h \big\}$, namely a countably family of smooth, nonnegative functions $\{ \zeta_h \}$ such that \begin{itemize} \item[iv.] we have \begin{equation} \label{e:isone} \sum_{h=1}^\infty \zeta_h \equiv1 \quad \text{in $\Omega$} ; \end{equation} \item[v.] for every $h>0$, the support of $\zeta_h$ is contained in $\Lambda_h$. \end{itemize} Finally, we fix a convolution kernel $\eta: \mathbb R^{d+1} \to \mathbb R^+$ and we define $\eta_\ee$ by setting $$ \eta_\ee (z) : = \frac{1}{\ee^{d+1}} \eta \left( \frac{z}{\ee} \right) $$ For every $m>0$ and $h>0$ we can choose $\ee_{mh}$ in such a way that $(\rho \zeta_h) \ast \eta_{\ee_{mh}} $ is supported in $\Lambda_h$ and furthermore \begin{equation} \label{e:ms2} \int_0^T \! \! \int_\Omega | \rho \zeta_h - ( \rho \zeta_h) \ast \eta_{\ee_{mh}} |+ | \rho \, \partial_t \zeta_h - ( \rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}}| + | \rho b \cdot \nabla \zeta_h - ( \rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}}| dx dt \leq \frac{2^{-h}}{m}. \end{equation} We then define $\tilde \rho_m$ by setting \begin{equation} \label{e:ms3} \tilde \rho_m : = \sum_{h=1}^\infty (\rho \zeta_h) \ast \eta_{\ee_{mh}} . \end{equation} The function $(\widetilde{\rho b})_m$ is defined analogously. Next, we proceed as in~\cite[p.123]{AFP} and we point out that \begin{equation*} \begin{split} h_m \stackrel{\eqref{e:accaenne}}{=} & \partial_t \rho_m + \mathrm{div} ({\rho_m b_m}) \stackrel{\eqref{e:accaenne}}{=} \underbrace{\sum_{h=1}^\infty (\partial_t \rho \zeta_h) \ast \eta_{\ee_{mh}} + \sum_{h=1}^\infty (\mathrm{div} (\rho b) \zeta_h) \ast \eta_{\ee_{mh}}}_{= 0 \; \text{by~\eqref{e:continuityrho} } } \\ &\quad + \sum_{h=1}^\infty (\rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}} + \sum_{h=1}^\infty (\rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}} \\ & \stackrel{\eqref{e:isone}}{=} \sum_{h=1}^\infty (\rho \, \partial_t \zeta_h) \ast \eta_{\ee_{mh}} - \rho \sum_{h=1}^\infty \partial_t \zeta_h \quad + \sum_{h=1}^\infty (\rho b \cdot \nabla \zeta_h) \ast \eta_{\ee_{mh}}- \rho b \cdot \sum_{h=1}^\infty \nabla \zeta_h \end{split} \end{equation*} By using~\eqref{e:ms2} we then get that $$ \int_0^T \! \! \int_\Omega |h_m| dx dt \leq \sum_{h=1}^\infty \frac{2^{-h}}{m} = \frac{1}{m} $$ and this establishes~\eqref{e:convaccaemme}. \label{s:proof2} \section{Proof Theorem~\ref{IBVP-NC}: comparison principle and uniqueness} \label{s:uni} In this section we complete the proof of Theorem~\ref{IBVP-NC}. More precisely, we establish the following comparison principle. \begin{lemma} \label{l:uni} Let $\Omega$, $b$ and $\rho$ as in the statement of Theorem~\ref{IBVP-NC}. Assume $u_1$ and $u_2 \in L^{\infty}((0,T) \times \Omega)$ are distributional solutions (in the sense of Definition~\ref{d:distrsol}) of the initial-boundary value problem~\eqref{prob-2} corresponding to the initial and boundary data $\overline{u}_{1} \in L^{\infty}(\Omega)$, $\overline{g}_1 \in L^\infty(\Gamma^-)$ and $\overline{u}_2 \in L^{\infty}(\Omega)$, $\overline{g}_2 \in L^\infty(\Gamma^-)$, respectively. If $\overline{u}_1 \ge \overline{u}_2$ and $\overline{g}_1 \ge \overline{g}_2$, then \begin{equation} \label{e:compa} \rho u_1 \ge \rho u_2 \quad a.e. \; \text{in} \; (0, T) \times \Omega. \end{equation} \end{lemma} Note that the uniqueness of $\rho u$, where $u$ is a distributional solution of the initial-boundary value problem~\eqref{prob-2}, immediately follows from the above result. \begin{proof} [Proof of Lemma~\ref{l:uni}] Let us define the function $$ \tilde{\beta}(u)= \left\{ \begin{array}{ll} u^2 & u \geq 0 \\ 0 & u<0. \end{array}\right. $$ In what follows, we shall prove that the identity $\rho\ \tilde{\beta}(u_{2}-u_{1})=0 $ holds almost everywhere, whence the comparison principle follows. To see this, we proceed as described below. First, we point out that, since the equation at the first line of~\eqref{prob-2} is linear, then $u_2-u_1$ is a distributional solution of the initial boundary value problem with data $\overline{u}_2 - \overline{u}_1$, $\overline{g}_2 - \overline{g}_1$. In particular, for every $ \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d} )$ we have \begin{equation} \int_{0}^{T} \int_{\Omega} \rho (u_{2}-u_{1}) (\partial_{t} \psi +b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \int_{\partial \Omega} [\text{Tr}(\rho u_2 b) - \text{Tr}(\rho u_{1} b)] \ \psi \ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) {\rho}_0 (\overline{u}_{2}-\overline{u}_{1}) \ dx \label{e7} \end{equation} and \begin{equation} \label{e:ntraces} \text{Tr}(\rho u_2 b) = \overline{g}_2 \text{Tr}(\rho b), \quad \text{Tr}(\rho u_1 b) = \overline{g}_1 \text{Tr}(\rho b) \quad \text{on $\Gamma^-$}. \end{equation} Note that~\eqref{e7} implies that \begin{equation} \int_{0}^{T} \int_{\Omega} \rho (u_{2}-u_{1}) (\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt=0, \quad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega). \label{e5} \end{equation} By using~\cite[Lemma 5.10]{Delellis1} (renormalization property inside the domain), we get \begin{equation} \int_{0}^{T} \int_{\Omega} \rho \ \tilde{\beta}(u_{2}-u_{1})(\partial_{t} \phi+b \cdot \nabla \phi) \ dx dt=0, \qquad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega). \label{e10} \end{equation} We next apply Lemma \ref{trace-existence} to the function $\tilde{\beta}(u_{2}-u_{1})$ to infer that there are bounded functions $\text{Tr}(\rho \tilde{\beta}(u_{2}-u_{1}) b)$ and $(\rho \tilde{\beta}(u_{2}-u_{1}))_{0} $ such that, for every $ \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d} ),$ we have \begin{equation} \int_{0}^{T} \int_{\Omega} \rho \ \tilde{\beta}(u_{2}-u_{1}) (\partial_{t} \psi +b \cdot \nabla \psi) \ dx dt= \int_{0}^{T} \int_{\partial \Omega} \text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) \ \psi \ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) (\rho \ \tilde{\beta}(u_{2}-u_{1}))_{0} \ dx. \label{e11} \end{equation} We recall~\eqref{e7} and we apply Lemma \ref{trace-renorm} (trace renormalization property) with $w= u_2 -u_1$, $h= \tilde \beta$, $B=(\rho,\rho b) $, $\Lambda = \mathbb R^{d+1}$ and $\Lambda'=(0,T)\times \Omega$. We recall that the vector field $P$ is defined by setting $P:= (\rho, \rho b)$ and we get \begin{equation} \begin{aligned} (\rho\ \tilde{\beta} (u_{2}-u_{1}))_{0}=- \text{Tr}(\tilde{\beta}(u_{2}-u_{1}) P,\partial \Lambda') \Big\vert_{\{0\} \times \Omega}&= - \tilde{\beta}\left(\frac{(\rho (u_{2}-u_{1}))_{0}}{\text{Tr}(P,\partial \Lambda')\Big\vert_{\{0\}\times \Omega}} \right) \text{Tr}(P,\partial \Lambda')\Big\vert_{\{ 0\} \times \Omega}\\ &=-\tilde{\beta}\left( \frac{\rho_0 (\overline{u}_{2}-\overline{u}_{1})}{\overline{\rho}} \right) \rho_0 \\ & =0, \; \text{since} \ \overline{u}_{1} \geq \overline{u}_{2} \phantom{\int} \end{aligned} \label{e12} \end{equation} and \begin{equation} \begin{aligned} \text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) &= \text{Tr}(\tilde{\beta}(u_{2}-u_{1}) \rho, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega} = \tilde{\beta} \left( \frac{\text{Tr}((u_{2}-u_{1})\rho, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}}{\text{Tr}(P, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}} \right) \text{Tr}(P, \partial \Lambda')\Big\vert_{(0,T) \times \partial \Omega}\\ &=\tilde{\beta}\left(\frac{\text{Tr}(\rho (u_{2}-u_{1}) b)}{\text{Tr}(\rho b)} \right) \text{Tr}(\rho b). \end{aligned} \notag \end{equation} By recalling~\eqref{e:ntraces} and the inequality $\bar g_1 \ge \bar g_2$, we conclude that $$ \text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) = 0 \quad \text{on $\Gamma^-$} $$ and, since $\tilde \beta \ge 0$, that \begin{equation} \label{e:ntracein} \text{Tr}(\rho \ \tilde{\beta}(u_{2}-u_{1}) b) \ge 0 \quad \text{on $\Gamma$.} \end{equation} We now choose a test function $\nu \in C^\infty_c (\mathbb R^d)$ in such a way that $\nu \equiv 1$ on the bounded set $\Omega$. Note that \begin{equation} \label{e:zerozero} \partial_t \nu + b \cdot \nabla \nu =0 \quad \text{on $(0, T) \times \Omega$.} \end{equation} Next we choose a sequence of functions $\chi_{n} \in {C}^{\infty}_{c}([0,+\infty))$ that satisfy \[\chi_{n} \equiv 1 \ \text{on}\ [0,\bar{t}],\ \chi_{n}\equiv 0\ \text{on}\ [\bar{t}+\frac{1}{n},+\infty),\ \chi'_{n} \leq 0, \] and we define \[\psi_{n}(t,x):= \chi_{n}(t) \nu(x), \ (t,x)\in [0,T)\times \mathbb{R}^{d}.\] Note that $\psi$ is smooth, non-negative and compactly supported in $[0,T)\times \mathbb{R}^{d}$. By combining the identities \eqref{e11},~\eqref{e12} and the inequality~\eqref{e:ntracein}, we get \begin{equation} \begin{aligned} 0 &\leq \int_{0}^{T} \int_{\Omega} \rho\ \tilde{\beta}(u_{2}-u_{1}) [\partial_{t}(\chi_{n} \nu)+b \cdot \nabla (\chi_{n} \nu)] \ dx dt \\ & = \int_{0}^{T} \int_{\Omega} \nu \rho\ \tilde{\beta}(u_{2}-u_{1}) \chi'_{n} \ dx dt+ \int_{0}^{T} \int_{\Omega} \chi_{n} \rho \ \tilde{\beta}(u_{2}-u_{1}) (\partial_{t} \nu +b \cdot \nabla \nu) \ dx dt \\ & \stackrel{\eqref{e:zerozero}}{=} \int_{0}^{T} \int_{\Omega} \nu \rho \ \chi'_{n} \ \tilde{\beta}(u_{2}-u_{1}) \ dx dt. \\ \end{aligned} \notag \end{equation} Passing to the limit as $n \rightarrow +\infty $ and noting that $\chi'_{n} \rightarrow -\delta_{\bar{t}} $ as $n \rightarrow \infty $ in the sense of distributions and recalling that $\nu \equiv 1$ on $\Omega$ we obtain \begin{equation} \int_{\Omega} \rho(\bar{t},\cdot) \tilde{\beta}(u_{2}-u_{1})(\bar{t},\cdot) \leq 0. \notag \end{equation} Since the above inequality is true for arbitrary $\bar t \in [0, T]$, we can conclude that \begin{equation} \begin{aligned} \rho \ \tilde{\beta}(u_2-u_1)=0,\ \text{for almost every}\ (t,x) \Rightarrow \rho u_{1} \geq \rho u_{2}, \ \text{for almost every}\ (t,x). \end{aligned} \label{e14} \end{equation} This concludes the proof of Lemma~\ref{l:uni}. \end{proof} \section{Stability and space continuity properties} \label{s:ssc} In this section, we discuss some qualitative properties of solutions of the initial-boundary value problem~\eqref{prob-2}. First, we establish Theorem~\ref{stability-weak}, which establishes (weak) stability of solutions with respect to perturbations in the vector fields and the data. Theorem~\ref{stability-strong} implies that, under stronger hypotheses, we can establish strong stability. Finally, Theorem~\ref{space-continuity} establishes space continuity properties. \begin{theorem}\label{stability-weak} Let $T>0$ and let $\Omega \subseteq \mathbb R^d$ be an open and bounded set with $C^2$ boundary. Assume that $$ b_{n}, b \in BV((0,T) \times \Omega; \mathbb{R}^{d}) \cap L^{\infty}((0,T) \times \Omega; \mathbb{R}^{d}), \qquad \rho_{n},\rho \in BV((0,T) \times \Omega) \cap L^{\infty}([0,T) \times \Omega) $$ satisfy \begin{equation} \begin{aligned} \partial_{t} \rho_{n}+\mathrm{div} (b_{n}\rho_{n})=0,\\ \partial_{t} \rho+\mathrm{div} (b \rho)=0, \end{aligned} \label{stability-1} \end{equation} in the sense of distributions on $(0, T) \times \Omega$. Assume furthermore that \begin{equation} 0 \leq \rho_{n}, \rho \leq C \; \text{and} \; \Vert b_{n} \Vert_{\infty}\ \text{is uniformly bounded}, \label{stability-2} \end{equation} \begin{equation} (b_{n},\rho_{n}) \xrightarrow[n \rightarrow \infty]{} (b,\rho) \ \text{strongly in} \ L^{1}((0,T) \times \Omega; \mathbb R^{d+1}), \label{stability-3} \end{equation} \begin{equation} \rho_{n0} \xrightarrow[n \rightarrow \infty]{} \rho_0 \; \text{strongly in}\ L^{1}(\Omega), \label{stability-4} \end{equation} \begin{equation} \emph{Tr}(\rho_{n} b_n) \xrightarrow[n \rightarrow \infty]{} \emph{Tr}(\rho b)\ \text{strongly in}\ L^{1}(\Gamma), \label{stability-5} \end{equation} Let $u_{n} \in L^{\infty}((0,T) \times \Omega) $ be a distributional solution (in the sense of Definition~\ref{d:distrsol}) of the initial-boundary value problem \begin{equation} \label{e:ibvpapp} \left\{ \begin{array}{lll} \partial_{t}(\rho_{n} u_{n})+\mathrm{div}(\rho_{n} u_{n} b_{n})=0 & \text{in} \ (0,T)\times \Omega \\ u_{n}=\overline{u}_{n} & \text{at $t=0$}\\ u_{n} =\overline{g}_{n} & \text{on} \ \Gamma_{n}^{-} \\ \end{array} \right. \end{equation} and $u \in L^{\infty}((0,T) \times \Omega) $ be a distributional solution of the equation \begin{equation} \label{e:ibvplimit} \left\{ \begin{array}{lll} \partial_{t}(\rho u)+\mathrm{div}(\rho u b)=0 & \text{in} \ (0,T)\times \Omega \\ u=\overline{u} & \text{at $t=0$}\\ u=\overline{g} & \text{on}\ \Gamma^{-}. \end{array} \right. \end{equation} If $u_m, \bar u \in L^\infty (\Omega)$ and $ \overline{g}_{n}, \bar g \in L^\infty (\Gamma)$ satisfy \begin{equation} \overline{u}_{n} \stackrel{\ast}{\rightharpoonup} \overline{u}\ \text{weak-$^\ast$ in}\ L^{\infty}(\Omega), \label{stability-7} \end{equation} \begin{equation} \overline{g}_{n} \stackrel{\ast}{\rightharpoonup} \overline{g} \; \text{weak-$^\ast$ in}\ L^{\infty}(\Gamma) , \label{stability-8} \end{equation} then \begin{equation} \rho_{n} u_{n} \stackrel{*}{\rightharpoonup} \rho u \ \text{weak-* in}\ L^{\infty}((0,T) \times \Omega) \label{stability-10} \end{equation} and \begin{equation} \emph{Tr}(\rho_{n} u_{n} b_{n}) \stackrel{*}{\rightharpoonup} \emph{Tr}(\rho u b) \ \text{weak-* in}\ L^{\infty}(\Gamma). \label{stability-9} \end{equation} \end{theorem} Note that in the statement of the above theorem $\overline{g}_m$ and $\overline{g}$ are functions defined on the whole $\Gamma$, although the values of $\rho_m u_m$ and $\rho u$ are only determined by their values on $\Gamma^-_m$ and $\Gamma^-$, respectively. \begin{proof} We proceed according to the following steps. \\ {\sc Step 1:} we apply Theorem~\ref{IBVP-NC} and we infer that the function $\rho_n u_n$ satisfying~\eqref{e:ibvpapp} is unique. Also, without loss of generality, we can redefine the function $u_n$ on the set $\{\rho_n=0\}$ in such a way that $u_n$ satisfies the maximum principle~\eqref{e:maxprin}. Owing to~\eqref{stability-9}, the sequences $\| \overline{u}_m \|_{L^\infty}$ and $\| \overline{g}_m \|_{L^\infty}$ are both uniformly bounded and by the maximum principle so is $\| u_m \|_{L^\infty}$. Also, by combining~\eqref{e:maxprintraces2} and~\eqref{stability-2} we infer that the sequence $\Vert \text{Tr}(\rho_{n} b_{n} u_{n}) \Vert_{\infty} $ is also uniformly bounded. We conclude that, up to subsequences (which we do not label to simplify the notation), we have \begin{comment} We begin with the preliminary observation that if we are able to prove that \begin{equation} u_{n} \stackrel{*}{\rightharpoonup} u \ \text{weak-* in}\ L^{\infty}((0,T) \times \Omega), \label{stability-10} \end{equation} we can combine the fact that $\Vert u_{n} \Vert_{\infty} $ are uniformly bounded (this follows from the maximum principle and \eqref{stability-7}-\eqref{stability-8}) with \eqref{stability-3} to infer \eqref{stability-9}. Therefore it is sufficient to establish \eqref{stability-10} which we pursue next. We note that since $\Vert u_{n} \Vert_{\infty} $ and $\Vert \text{Tr}(\rho_{n} b_{n} u_{n}) \Vert_{\infty} $ are uniformly bounded (see Remark $2.2 (a)$), there exist $R_{1} \in L^{\infty}((0,T) \times \Omega)$ and $R_{2} \in L^{\infty}((0,T)\times \partial \Omega)$ such that as $n \rightarrow \infty $, \end{comment} \begin{equation} \begin{aligned} & u_{n} \stackrel{*}{\rightharpoonup} r_{1} \ \text{weak-* in} \ L^{\infty}((0,T) \times \Omega),\\ & \text{Tr}(\rho_{n} u_{n} b_{n}) \stackrel{*}{\rightharpoonup} r_{2} \ \text{weak-* in} \ L^{\infty}(\Gamma) \end{aligned} \label{stability-11} \end{equation} for some $r_{1} \in L^{\infty}((0,T) \times \Omega)$ and $r_{2} \in L^{\infty}(\Gamma)$. By using \eqref{iden-2} and \eqref{iden-3}, we get that \begin{equation} \int_{0}^{T} \int_{\Omega} \rho r_{1} (\partial_{t} \phi+b \cdot \nabla \phi)\ dx dt=0, \quad \forall \phi \in C^{\infty}_{c}((0,T) \times \Omega), \label{stability-12} \end{equation} and \begin{equation} \int_{0}^{T} \int_{\Omega} \rho r_{1} (\partial_{t} \psi+b \nabla \psi)\ dx dt= \int_{0}^{T} \int_{\partial \Omega} r_{2} \psi\ d\mathcal{H}^{d-1} dt -\int_{\Omega} \psi(0,\cdot) {\rho}_0\ \overline{u}\ dx ,\ \forall \psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d}). \label{stability-13} \end{equation} From Lemma \ref{trace-existence}, it also follows that \begin{equation} r_{2}= \text{Tr}(\rho r_{1} b). \label{stability-14} \end{equation} Assume for the time being that we have established the equality \begin{equation} \label{e:whatww} r_{2}=\overline{g} \text{Tr}(\rho b), \quad \text{on}\ \Gamma^{-} , \end{equation} then by recalling~\eqref{stability-14} and the uniqueness part in Theorem~\ref{IBVP-NC} we conclude that $r_1= \rho u$ and $r_2 = \text{Tr}(\rho bu) $. Owing to~\eqref{stability-11}, this concludes the proof of the theorem. \\ {\sc Step 2:} we establish~\eqref{e:whatww}. First, we decompose $\text{Tr}(\rho_m u_m b_m) $ as \begin{equation} \label{e:decompo} \begin{split} \text{Tr}(\rho_n u_n b_n) &= \text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{-}_{n}}+\text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{+}_{n}} + \text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{0}_{n}} \\ & = \overline{g}_n \text{Tr}(\rho_n b_n) \mathbf{1}_{\Gamma^{-}_{n}}+\text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{+}_{n}} + \text{Tr}(\rho_n u_n b_n) \mathbf{1}_{\Gamma^{0}_{n}} , \end{split} \end{equation} where $\Gamma^-_n$, $\Gamma^+_n$ and $\Gamma^0_n$ are defined as in~\eqref{e:gamma}. By using Lemma~\ref{trace-renorm} (trace renormalization), one could actually prove that the last term in the above expression is actually $0$. This is actually not needed here. Indeed, it suffices to recall~\eqref{stability-5} and Lemma~\ref{l:traces} and point out that by combining~\eqref{e:convchar1} and~\eqref{e:convchar2} we get \begin{equation} \label{e:conchar3} \mathbf{1}_{\Gamma^{0}_{n}} \to \mathbf{1}_{\Gamma^0} - \mathbf{1}_{\Gamma'} - \mathbf{1}_{\Gamma''}. \end{equation} Next, we recall that the sequence $\| \text{Tr}(\rho_n u_n b_n)\|_{L^\infty}$ is uniformly bounded owing to the uniform bounds on $\| \rho_n \|_{L^\infty}$ and $\| u_n \|_{L^\infty}$. By recalling~\eqref{stability-8}, we conclude that \begin{equation} \label{e:conv1} \overline{g}_n \text{Tr}(\rho_n b_n) \mathbf{1}_{\Gamma^{-}_{n}} \stackrel{*}{\rightharpoonup} \overline{g} \, \text{Tr}(\rho b) \Big( \mathbf{1}_{\Gamma^{-}} + \mathbf{1}_{\Gamma'} \Big) \qquad \text{weak-* in $L^\infty (\Gamma)$.} \end{equation} By recalling that $\Gamma' \subseteq \Gamma^0$ we get that $\text{Tr}(\rho b) \mathbf{1}_{\Gamma'} =0$. We now pass to the weak star limit in~\eqref{e:decompo} and using~\eqref{e:convchar1},~\eqref{e:convchar2},~\eqref{stability-11},~\eqref{stability-8} and~\eqref{e:conv1} we get \begin{equation} \label{e:conv2} r_2 = \overline{g} \text{Tr}(\rho b) \mathbf{1}_{\Gamma^{-}} + r_2 \Big( \mathbf{1}_{\Gamma^{+}} + \mathbf{1}_{\Gamma'} \Big)+ r_2 \Big( \mathbf{1}_{\Gamma^{0}} - \mathbf{1}_{\Gamma'} -\mathbf{1}_{\Gamma''} \Big), \end{equation} which owing to the properties $$ \Gamma^- \cap \Gamma^{0}= \emptyset, \quad \Gamma^- \cap \Gamma'=\emptyset, \quad \Gamma^- \cap \Gamma''= \emptyset $$ implies~\eqref{e:whatww}. This concludes the proof Theorem~\ref{stability-weak}. \end{proof} \begin{theorem}\label{stability-strong} Under the same assumptions as in Theorem~\ref{stability-weak}, if we furthermore assume that \begin{equation} \overline{u}_{n} \xrightarrow[n \rightarrow \infty]{} \overline{u}\ \text{strongly in}\ L^{1}(\Omega), \label{stability-35} \end{equation} \begin{equation} \overline{g}_{n} \xrightarrow[n \rightarrow \infty]{} \overline{g} \ \text{strongly in}\ L^{1}(\Gamma) , \label{stability-36} \end{equation} then we get \begin{equation} \begin{aligned} &\rho_{n} u_{n} \xrightarrow[n \rightarrow \infty]{} \rho u \ \text{strongly in}\ L^{1}((0,T) \times \Omega),\\ &\emph{Tr}(\rho_{n} u_{n} b_{n}) \xrightarrow[n \rightarrow \infty]{} \emph{Tr}(\rho u b) \ \text{strongly in}\ L^{1}(\Gamma). \end{aligned} \label{stability-37} \end{equation} \end{theorem} \begin{proof} First, we point out that the first equation in~\eqref{stability-9} implies that \begin{equation} \label{e:ell2} \rho_n u_m {\rightharpoonup} \rho {u}\ \text{weakly in}\ L^{2}((0, T) \times \Omega ). \end{equation} Next, by using Lemma \ref{trace-renorm} (trace-renormalization property), we get that $\rho_m u^{2}_{n}$ and $\rho u^{2}$ satisfy (in the sense of distributions) \begin{equation} \left\{ \begin{array}{lll} \partial_{t}(\rho_{n} u_{n}^{2})+\text{div}(\rho_{n} u_{n}^{2} b_{n})=0 & \text{in} \ (0,T)\times \Omega \\ u_{n}^{2}=\overline{u}_{n}^{2} & \text{at $t=0$}\\ u^{2}_{n} =\overline{g}^{2}_{n} & \text{on}\ \Gamma_{n}^{-}, \\ \end{array} \right. \notag \end{equation} and \begin{equation} \left\{ \begin{array}{lll} \partial_{t}(\rho u^{2})+\text{div}(\rho u^{2} b)=0 & \text{in} \ (0,T)\times \Omega \\ u^{2}=\overline{u}^{2} & \text{at $t=0$} \\ u^{2} =\overline{g}^{2} & \text{on}\ \Gamma^{-}, \\ \end{array} \right. \notag \end{equation} respectively. Also, by combinig~\eqref{stability-7},\eqref{stability-8}, \eqref{stability-35} and \eqref{stability-36}, we get that \begin{equation} \overline{u}^2_{n} \stackrel{*}{\rightharpoonup} \overline{u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}(\Omega), \qquad \overline{g}^2_{n} \stackrel{*}{\rightharpoonup} \overline{g}^2 \; \text{weak-$^\ast$ in}\ L^{\infty}(\Gamma) \notag \end{equation} and by applying Theorem~\ref{stability-weak} to $\rho_m u_m^2$ we conclude that $$ \rho_m u_m^2 \stackrel{*}{\rightharpoonup} \rho {u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}((0, T) \times \Omega ) $$ and that \begin{equation} \label{e:convbur} \text{Tr}(\rho_{n} u^2_{n} b_{n}) \stackrel{*}{\rightharpoonup} \text{Tr}(\rho u^2 b) \ \text{weak-$^\ast$ in}\ L^{\infty}(\Gamma ). \end{equation} Since the sequence $\| \rho_m \|_{L^\infty}$ is uniformly bounded, then by recalling~\eqref{stability-3} we get $$ \rho^2_m u_m^2 \stackrel{*}{\rightharpoonup} \rho^2 {u}^2\ \text{weak-$^\ast$ in}\ L^{\infty}((0, T) \times \Omega ) $$ and hence \begin{equation} \label{e:square} \rho^2_m u_m^2 {\rightharpoonup} \rho^2 {u}^2\ \text{weakly in}\ L^2((0, T) \times \Omega ). \end{equation} By combining~\eqref{e:ell2} and~\eqref{e:square} we get that $\rho^2_m u_m^2 \longrightarrow \rho u$ strongly in $L^2((0, T) \times \Omega )$ and this implies the first convergence in~\eqref{stability-37}. Next, we establish the second convergence in $L^2((0, T) \times \Omega )$. Since $\Gamma $ is a set of finite measure, from \eqref{stability-9} and \eqref{e:convbur} we can infer that \begin{equation} \begin{aligned} & \text{Tr}(\rho_{n} u_{n} b_{n}) \rightharpoonup \text{Tr}(\rho u b) \ \text{weakly in} \ L^{2}(\Gamma),\\ & \text{Tr}(\rho_{n} u^{2}_{n} b_{n}) \rightharpoonup \text{Tr}(\rho u^{2} b) \ \text{weakly in} \ L^{2}(\Gamma). \end{aligned} \label{stability-39} \end{equation} By using the uniform bounds for $\Vert \text{Tr}(\rho_{n} b_{n})\Vert_{\infty} $, we infer from the $L^{1}$ convergence of $\text{Tr}(\rho_{n} b_{n}) $ to $\text{Tr}(\rho b)$ that \begin{equation} \text{Tr}(\rho_{n} b_{n}) \xrightarrow[n \rightarrow \infty]{} \text{Tr}(\rho b) \ \text{strongly in}\ L^{2}(\Gamma). \label{stability-40} \end{equation} Next, we apply Lemma \ref{trace-renorm} (trace renormalization property) and we get that \begin{equation} [\text{Tr}(\rho_{n} u_{n} b_{n})]^{2}= \left[\frac{\text{Tr}(\rho_{n} u_{n} b_{n})}{\text{Tr}(\rho_{n} b_{n})} \right]^{2} [\text{Tr}(\rho_{n} b_{n})]^{2}= \text{Tr}(\rho_{n} u_{n}^{2} b_{n}) \text{Tr}(\rho_{n}b_{n}) \notag \end{equation} and \begin{equation} [\text{Tr}(\rho u b)]^{2}= \left[\frac{\text{Tr}(\rho u b)}{\text{Tr}(\rho b)} \right]^{2} [\text{Tr}(\rho b)]^{2}= \text{Tr}(\rho u^{2} b) \text{Tr}(\rho b). \notag \end{equation} From \eqref{stability-39} and \eqref{stability-40}, we can then conclude that \begin{equation} [\text{Tr}(\rho_{n} u_{n} b_{n})]^{2} \rightharpoonup [\text{Tr}(\rho u b)]^{2} \ \text{weakly in}\ L^{2}(\Gamma), \label{stability-41} \end{equation} and by recalling~\eqref{stability-39} the second convergence in \eqref{stability-37} follows. \end{proof} Finally, we establish space-continuity properties of the vector field $(\rho u, \rho u b)$ similar to those established in~\cite{Boyer,CDS1}. \begin{theorem}\label{space-continuity} Under the same assumptions as in Theorem~\ref{IBVP-NC}, let $P$ be the vector field $P : = ( \rho, \rho b)$, $u$ be a distributional solution of~\eqref{prob-2} and $\{\Sigma_{r} \}_{r \in I} \subseteq \mathbb R^d$ be a family of graphs as in Definition \ref{graph}. Also, fix $r_{0} \in I $ and let $\gamma_{0}, \gamma_{r}: (0,T) \times D \rightarrow \mathbb{R} $ be defined by \begin{equation} \begin{aligned} \gamma_{0}(t,x_{1},\cdots,x_{d-1})&:= \emph{Tr}^{-}(uP,(0,T)\times \Sigma_{r_{0}})(t,x_{1},\cdots,x_{d-1},f(x_{1},\cdots,x_{d-1})-r_{0}),\\ \gamma_{r}(t,x_{1},\cdots,x_{d-1})&:=\emph{Tr}^{+}(uP,(0,T) \times \Sigma_{r})(t,x_{1},\cdots,x_{d-1},f(x_{1},\cdots,x_{d-1})-r) . \end{aligned} \label{space1} \end{equation} Then $\gamma_{r} \rightarrow \gamma_{0}$ strongly in $L^{1}((0,T)\times D) $ as $r \rightarrow r^{+}_{0} $. \end{theorem} The proof of the above result follows the same strategy as the proof of~\cite[Proposition 3.5]{CDS1} and is therefore omitted. \section{Applications to the Keyfitz and Kranzer system} \label{s:KK} In this section, we consider the initial-boundary value problem for the Keyfitz and Kranzer system~\cite{KK} of conservation laws in several space dimensions, namely \begin{equation} \left\{ \begin{array}{lll} \partial_{t} U+\displaystyle{ \sum_{i=1}^{d} \partial_{x_{i}} (f^{i}(\vert U \vert) U)=0} & \text{in}\ (0,T) \times \Omega \\ U = U_{0} & \text{at $t=0$} \\ U = U_{b} & \text{on} \ \Gamma. \displaystyle{\phantom{\int}} \\ \end{array} \right. \label{KK1} \end{equation} Note that, in general, we cannot expect that the boundary datum is pointwise attained on the whole boundary $\Gamma$. We come back to this point in the following. We follow the same approach as in~\cite{ABD,AD,Br,Delellis2} and we formally split the equation at the first line of~\eqref{KK1} as the coupling between a scalar conservation law and a linear transport equation. More precisely, we set $F:=(f^{1},\cdots,f^{d})$ and we point out that the modulus $\rho: = |U|$ formally solves the initial-boundary value problem \begin{equation} \left\{ \begin{array}{ll} \partial_{t} \rho+\text{div} (F(\rho) \rho)=0 & \text{in}\ (0,T) \times \Omega\\ \rho =\vert U_{0} \vert & \text{at $t=0$}\\ \rho= \vert U_{b} \vert & \text{on}\, \Gamma. \end{array} \right. \label{KK2} \end{equation} We follow~\cite{BLN,CR,Serre2} and we extend notion of \emph{entropy admissible} solution (see~~\cite{Kr}) to initial boundary value problems. \begin{definition} A function $\rho \in L^{\infty}((0,T) \times \Omega) \cap BV((0,T) \times \Omega) $ is an entropy solution of \eqref{KK2} if for all $k \in \mathbb{R}$, \begin{equation} \begin{aligned} &\int_{0}^{T} \int_{\Omega} \Big\{\vert \rho(t,x)-k \vert\ \partial_{t} \psi + \emph{sgn}(\rho-k)[F(\rho)-F(k)] \cdot \nabla \psi \Big\} \ dx dt \\ &+ \int_{\Omega} \vert \rho_{0}-k \vert\ \psi(0, \cdot) \ dx -\int_{0}^{T} \int_{\partial \Omega} \emph{sgn}(\vert U_{b} \vert(t,x)-k)\ [F(T(\rho))-F(k)]\cdot \vec n \ \psi \ dx dt \geq 0, \end{aligned} \notag \end{equation} for any positive test function $\psi \in C^{\infty}_{c}([0,T) \times \mathbb{R}^{d}; \mathbb{R}^{+}).$ In the above expression $T(\rho)$ denotes the trace of the function $\rho$ on the boundary $\Gamma$ and $\vec n$ is the outward pointing, unit normal vector to $\Gamma$. \end{definition} Existence and uniqueness results for entropy admissible solutions of the above systems were obtained by Bardos, le Roux and N{\'e}d{\'e}lec~\cite{BLN} by extending the analysis by Kru{\v{z}}kov to initial-boundary value problems (see also~\cite{CR,Serre2} for a more recent discussion). Note, however, that one cannot expect that the boundary value $|U_b|$ is pointwise attained on the whole boundary $\Gamma$, see again~\cite{BLN,CR,Serre2} for a more extended discussion. Next, we introduce the equation for the \emph{angular part} of the solution of~\eqref{KK1}. We recall that, if $|U_b|$ and $|U_0|$ are of bounded variation, then so is $\rho$ and hence the trace of $F(\rho) \rho$ on $\Gamma$ is well defined. As usual, we denote it by $T(F(\rho) \rho)$. In particular, we can introduce the set $$ \Gamma^- : = \big\{ (t, x) \in \Gamma: \; T(F(\rho) \rho) \cdot \vec n <0 \big\}, $$ where as usual $\vec n$ denotes the outward pointing, unit normal vector to $\Gamma$. We consider the vector $\theta=(\theta_{1},\cdots,\theta_{N}) $ and we impose \begin{equation} \left\{ \begin{array}{llll} \partial_{t}(\rho \theta)+\text{div}(F(\rho) \rho \theta)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\ \theta=\displaystyle{\frac{U_{0}}{\vert U_{0} \vert}}& \text{at $t=0$}\\ \theta=\displaystyle{\frac{U_{b}}{\vert U_{b} \vert}} & \text{on} \ \Gamma^{-}, \end{array} \label{KK5} \right. \end{equation} where the ratios $U_0 / |U_0|$ and $U_b / |U_b|$ are defined to be an arbitrary unit vector when $|U_0|=0$ and $|U_b|=0$, respectively. Note that the product $U=\theta \rho$ formally satisfies the equation at the first line of~\eqref{KK1}. We now extend the notion of \emph{renormalized entropy solution} given in~\cite{ABD,AD,Delellis2} to initial-boundary value problems. \begin{definition} \label{d:res} A renormalized entropy solution of~\eqref{KK1} is a function $U \in L^\infty ( (0, T) \times \Omega; \mathbb R^N)$ such that $U = \rho \theta$, where \begin{itemize} \item $\rho = |U|$ and $\rho$ is an entropy admissible solution of~\eqref{KK2}. \item $\theta = (\theta_1, \dots, \theta_N)$ is a distributional solution, in the sense of Definition~\ref{d:distrsol}, of~\eqref{KK5}. \end{itemize} \end{definition} Some remarks are here in order. First, we can repeat the proof of \cite[Proposition 5.7]{Delellis1} and conclude that, under fairly general assumptions, any renormalized entropy solution is an entropy solution. More precisely, let us fix a renormalized entropy solution $U$ and an \emph{entropy-entropy flux pair} $(\eta, Q)$, namely a couple of functions $\eta: \mathbb R^N \to \mathbb R$, $Q: \mathbb R^N \to \mathbb R^d$ such that $$ \nabla \eta D f^i = \nabla Q^i, \quad \text{for every $i=1, \dots, d$.} $$ Assume that $$ \mathcal L^1 \big\{ r \in \mathbb R: \; (f^1)'(r) = \dots = (f^d)' (r)=0 \big\}=0. $$ By arguing as in~\cite{Delellis1} we conclude that, if $\eta$ is convex, then $$ \int_0^T \int_\Omega \eta (U) \partial_t \phi + Q(U)\cdot \nabla \phi \, dx dt \ge 0 $$ for every \emph{entropy-entropy flux pair} $(\eta, Q)$ and for every nonnegative test function $\phi \in C^\infty_c ((0, T) \times \Omega)$. Second, we point out that, as the Bardos, le Roux and N{\'e}d{\'e}lec~\cite{BLN} solutions of scalar initial-boundary value problems, renormalized entropy solutions of the Keyfitz and Kranzer system do not, in general pointwise attain the boundary datum $U_0$ on the whole boundary $\Gamma$. We now state our well-posedness result. \begin{theorem} \label{t:KK} Assume $\Omega$ is a bounded open set with $C^2$ boundary. Also, assume that $U_0 \in L^\infty (\Omega; \mathbb R^N)$ and $U_b \in L^\infty (\Gamma; \mathbb R^N)$ satisfy $|U_0| \in BV ( \Omega)$, $|U_b| \in BV (\Gamma).$ Then there is a unique renormalized entropy solution of~\eqref{KK1} that satisfies $U \in L^\infty ((0, T)\times \Omega; \mathbb R^N)$. \end{theorem} \begin{proof} We first establish existence, next uniqueness. \\ {\sc Existence:} first, we point out that the results in~\cite{BLN,CR,Serre2} imply that there is an entropy admissible solution of~\eqref{KK2} satisfying $\rho \in L^\infty ((0, T) \times \Omega) \cap BV ((0, T) \times \Omega).$ Also, $\rho$ satisfies the maximum principle, namely \begin{equation} \label{e:rhomaxprin} 0 \leq \rho \leq \max \big\{ \| U_0 \|_{L^\infty}, \|U_b \|_{L^\infty} \big\}. \end{equation} For every $j=1, \dots, N$ we consider the initial-boundary value problem \begin{equation} \left\{ \begin{array}{llll} \partial_{t}(\rho \theta_j)+\text{div}(F(\rho) \rho \theta_j)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\ \theta_j=\displaystyle{\frac{U_{0j}}{\vert U_{0} \vert}}& \text{at $t=0$}\\ \theta_j=\displaystyle{\frac{U_{bj}}{\vert U_{b} \vert}} & \text{on} \ \Gamma^{-}, \end{array} \label{KK6} \right. \end{equation} where $U_{0j}$ and $U_{bj}$ is the $j$-th component of $U_0$ and $U_b$, respectively. The existence of a distributional solution $\theta_j$ follows from the existence part in Theorem~\ref{IBVP-NC}. We now set $U: = \rho \theta$, where $\theta = (\theta_1, \dots, \theta_N)$. To conclude the existence part we are left to show that $|U|=\rho$. To this end, we point out that, by combining~\cite[Lemma 5.10]{Delellis1} (renormalization property inside the domain) with Theorem~\ref{trace-renorm} (trace renormalization property) and by arguing as in \S~\ref{s:uni}, we conclude that, for every $j=1, \dots, N$, $\theta^2_j$ is a distributional solution, in the sense of Definition~\ref{d:distrsol}, of the initial-boundary value problem \begin{equation*} \left\{ \begin{array}{llll} \partial_{t}(\rho \theta^2_j)+\text{div}(F(\rho) \rho \theta^2_j)=0 & \text{in}\ (0,T) \times \Omega \phantom{\displaystyle{\int}}\\ \theta_j=\displaystyle{\frac{U^2_{0j}}{\vert U_{0} \vert^2}}& \text{at $t=0$}\\ \theta=\displaystyle{\frac{U^2_{bj}}{\vert U_{b} \vert^2}} & \text{on} \ \Gamma^{-}. \end{array} \right. \end{equation*} By adding from $1$ to $N$, we conclude that $|\theta|^2$ is a distributional solution of \begin{equation*} \left\{ \begin{array}{llll} \partial_{t}(\rho |\theta|^2)+\text{div}(F(\rho) \rho |\theta|^2)=0 & \text{in}\ (0,T) \times \Omega \\ \theta_j=1& \text{at $t=0$}\\ \theta=1 & \text{on} \ \Gamma^{-}. \end{array} \right. \end{equation*} By recalling the equation at the first line of~\eqref{KK2} we infer that $|\theta|^2 =1$ is a solution of the above initial-boundary value problem. By the uniqueness part of Theorem~\ref{IBVP-NC}, we then deduce that $\rho |\theta|^2= \rho$ and this concludes the proof of the existence part. \\ {\sc Uniqueness:} assume $U_1$ and $U_2$ are two renormalized entropy solutions, in the sense of Definition~\ref{d:res}, of the initial-boundary value problem~\eqref{KK1}. Then $\rho_1: = |U_1|$ and $\rho_2 : =|U_2|$ are two entropy admissible solutions of the initial-boundary value problem~\eqref{KK2} and hence $\rho_1=\rho_2$. By applying the uniqueness part of Theorem~\ref{IBVP-NC} to the initial-boundary value problem~\eqref{KK6}, for every $j=1, \dots, N$, we can then conclude that $U_1 =U_2$. \end{proof} \section*{Acknowledgments} This paper has been written while APC was a postdoctoral fellow at the University of Basel supported by a ``Swiss Government Excellence Scholarship'' funded by the State Secretariat for Education, Research and Innovation (SERI). APC would like to thank the SERI for the support and the Department of Mathematics and Computer Science of the University of Basel for the kind hospitality. GC was partially supported by the Swiss National Science Foundation (Grant 156112). LVS is a member of the GNAMPA group of INdAM (``Istituto Nazionale di Alta Matematica"). Also, she would like to thank the Department of Mathematics and Computer Science of the University of Basel for the kind hospitality during her visit, during which part of this work was done. \small \end{document}

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Partially supported by KBN grant 1 P03A 025 27.} \footnotetext[5]{Partially supported by the Berkman Faculty Development Fund, CMU.} \footnotetext[6]{Courant Institute, New York University, New York, NY 10012, USA.} \footnotetext[7]{Institut f\"ur Informatik, Humboldt Universit\"at, Berlin 10099, Germany. Supported by an Alexander von Humboldt fellowship.} \renewcommand{\arabic{footnote}}{\arabic{footnote}} \begin{abstract} Let $D(G)$ be the smallest quantifier depth of a first order formula which is true for a graph $G$ but false for any other non-isomorphic graph. This can be viewed as a measure for the first order descriptive complexity of $G$. We will show that almost surely $D(G)=\Theta(\frac{\ln n}{\ln\ln n})$, where $G$ is a random tree of order $n$ or the giant component of a random graph $\C G(n,\frac cn)$ with constant $c>1$. These results rely on computing the maximum of $D(T)$ for a tree $T$ of order $n$ and maximum degree $l$, so we study this problem as well.\end{abstract} \section{Introduction} This paper deals with graph properties expressible in first order logic. The vocabulary consists of variables, connectives ($\vee$, $\wedge$ and $\neg$), quantifiers ($\exists$ and $\forall$), and two binary relations: the equality and the graph adjacency ($=$ and $\sim$ respectively). The variables denote vertices only so we are not allowed to quantify over sets or relations. The notation $G\models A$ means that a graph $G$ is a model for a \emph{sentence} $A$ (a first order formula without free variables); in other words, $A$ is true for the graph $G$. All sentences and graphs are assumed to be finite. The Reader is referred to Spencer's book~\cite{spencer:slrg} (or to~\cite{kim+pikhurko+spencer+verbitsky:03rsa}) for more details. A first order sentence $A$ \emph{distinguishes} $G$ from $H$ if $G\models A$ but $H\not\models A$. Further, we say that $A$ \emph{defines} $G$ if $A$ distinguishes $G$ from any non-isomorphic graph $H$. In other words, $G$ is the unique (up to an isomorphism) finite model for $A$. The \emph{quantifier depth} (or simply \emph{depth}) $D(A)$ is the largest number of nested quantifiers in $A$. This parameter is closely related to the complexity of checking whether $G\models A$. The main parameter we will study is $D(G)$, the smallest quantifier depth of a first order formula defining $G$. It was first systematically studied by Pikhurko, Veith and Verbitsky~\cite{pikhurko+veith+verbitsky:03} (see also~\cite{pikhurko+verbitsky:03}). In a sense, a defining formula $A$ can be viewed as the canonical form for $G$ (except that $A$ is not unique): in order to check whether $G\cong H$ it suffices to check whether $H\models A$. Unfortunately, this approach does not seem to lead to better isomorphism algorithms but this notion, being on the borderline of combinatorics, logic and computer science, is interesting on its own and might find unforeseen applications. Within a short time-span various results on the values of $D(G)$ for order-$n$ graphs appeared. The initial papers~\cite{pikhurko+veith+verbitsky:03,pikhurko+verbitsky:03} studied the maximum of $D(G)$ (the `worst' case). The `best' case is considered by Pikhurko, Spencer, and Verbitsky~\cite{pikhurko+spencer+verbitsky:04} while Kim, Pikhurko, Spencer and Verbitsky~\cite{kim+pikhurko+spencer+verbitsky:03rsa} obtained various results for random graphs. Here we study these questions for trees and sparse random structures. Namely, the three main questions we consider are: \begin{description} \item[Section~\ref{general}:] What is $D^{\mathrm{tree}}(n,l)$, the maximum of $D(T)$ over all trees of order at most $n$ and maximum degree at most $l$? \item[Section~\ref{giant}:] What is $D(G)$, where $G$ is the giant component of a random graph $\C G(n,\frac{c}{n})$ for constant $c>1$? \item[Section~\ref{random}:] What is $D(T)$ for a random tree $T$ of order $n$? \end{description} In all cases we determine the order of magnitude of the studied function. Namely, we prove that $D^{\mathrm{tree}}(n,l)=\Theta(\frac{l\ln n}{\ln l})$, and whp we have $D(G)=\Theta(\frac{\ln n}{\ln\ln n})$, whenever $G$ is a random tree of order $n$ or the giant component of a random graph $\C G(n,\frac cn)$ with constant $c>1$. (The acronym \emph{whp} stands for `with high probability', i.e.,\ with probability $1-o(1)$.) Moreover, for some cases involving trees we estimate the smallest quantifier depth of a first order formula defining $G$ up to a factor of $1+o(1)$. For instance, we show that for a random tree $T$ of order $n$ we have whp $D(T)=(1+o(1))\frac{\ln n}{\ln\ln n}$. \comment{ also we prove that $D^{\mathrm{tree}}(n,l)=(1/2+o(1))\frac{l\ln n}{\ln l}$ whenever both $l=l(n)$ and $\ln n/\ln l$ tends to infinity as $n\to\infty$. } \section{Further Notation and Terminology} Our main tool in the study of $D(G)$ is the \emph{Ehrenfeucht game}. Its description can be found in Spencer's book~\cite{spencer:slrg} whose terminology we follow (or see~\cite[Section~2]{kim+pikhurko+spencer+verbitsky:03rsa}), so here we will be very brief. Given two graphs $G$ and $G'$, the \emph{Ehrenfeucht game} $\mbox{\sc Ehr}_k(G,G')$ is a perfect information game played by two players, called \emph{Spoiler} and \emph{Duplicator}, and consists of $k$ rounds, where $k$ is known in advance to both players. For brevity, let us refer to Spoiler as `him' and to Duplicator as `her'. In the $i$-th round, $i=1,\dots,k$, Spoiler selects one of the graphs $G$ and $G'$ and marks one of its vertices by $i$; Duplicator must put the same label $i$ on a vertex in the other graph. At the end of the game let $x_1,\dots,x_k$ be the vertices of $G$ marked $1,\dots,k$ respectively, regardless of who put the label there; let $x_1',\dots,x_k'$ be the corresponding vertices in $G'$. Duplicator wins if the correspondence $x_i\leftrightarrow x_i'$ is a partial isomorphism, that is, we require that $\{x_i,x_j\}\in E(G)$ iff $\{x_i',x_j'\}\in E(G')$ as well as that $x_i=x_j$ iff $x_i'=x_j'$. Otherwise, Spoiler wins. The key relation is that $D(G,G')$, the smallest depth of a first order sentence $A$ distinguishing $G$ from $G'$, is equal to the smallest $k$ such that Spoiler can win $\mbox{\sc Ehr}_k(G,G')$. Also, \beq{D} D(G)=\max_{G'\not\cong G} D(G,G'), \end{equation} see e.g.~\cite[Lemma~1]{kim+pikhurko+spencer+verbitsky:03rsa}. Sometimes it will be notationally more convenient to prove the bounds on $D(G,G')$ for colored graphs which generalize the usual (uncolored) graphs. Graphs $G,G'$ are \emph{colored} if we have unary relations $U_i:V(G)\cup V(G')\to\{0,1\}$, $i\in I$. We say that the vertices in the set $U_i^{-1}(1)$ have color $i$. Note that some vertices may be uncolored and some may have more than one color. There are no restrictions on a color class, i.e.,\ it does not have to be an independent set. When the Ehrenfeucht game is played on colored graphs, Duplicator must additionally preserve the colors of vertices. Colorings can be useful even if we prove results for uncolored graphs. For example, if $x\in V(G)$ and $x'\in V(G')$ were selected in some round, then, without changing the outcome of the remaining game, we can remove $x$ and $x'$ from $G$ and $G'$ respectively, provided we color their neighbors with a new color. (Note that in an optimal strategy of Spoiler, there is no need to select the same vertex twice.) We will also use the following fact, which can be easily deduced from the general theory of the Ehrenfeucht game. Let $x,y\in V(G)$ be distinct vertices. Then the smallest quantifier depth of a first order formula $\Phi(z)$ with one free variable $z$ such that $G\models \Phi(x)$ but $G\not\models \Phi(y)$ is equal to the minimum $k$ such that Spoiler can win the $(k+1)$-round game $\mbox{\sc Ehr}_{k+1}(G,G)$, where the vertices $x_1=x$ and $x_1'=y$ have been selected in the first round. In this paper $\ln$ denotes the natural logarithm, while the logarithm base $2$ is written as $\log_2$. \section{General Trees}\label{general} Let $D^{\mathrm{tree}}(n,l)$ be the maximum of $D(T)$ over all colored trees of order at most $n$ and maximum degree at most $l$. We split the possible range of $l,n$ into a few cases. \bth{MaxDeg} Let both $l$ and $\ln n/\ln l$ tend to the infinity. Then \beq{MaxDeg} D^{\mathrm{tree}}(n,l)= \left(\frac12+o(1)\right)\, \frac{ l\ln n}{\ln l}. \end{equation} In fact, the lower bound can be achieved by uncolored trees. \end{theorem} In order to prove Theorem~\ref{th:MaxDeg} we need some preliminary results. Let $\mathrm{dist}_G(x,y)$ denote the distance in $G$ between $x,y\in V(G)$. \blm{Distance} Suppose $x,y\in V(G)$ at distance $k$ were selected while their counterparts $x',y'\in V(G')$ are at a strictly larger distance (possibly infinity). Then Spoiler can win in at most $\log_2k+1$ extra moves, playing all of the time inside $G$.\end{lemma} \bpf We prove the claim by induction on $k$. Assume $k\ge 2$ and choose an appropriate $xy$-path $P$. Spoiler selects a vertex $w\in V(G)$ which is a \emph{middle vertex} of $P$, that is, $k_1=\mathrm{dist}_P(x,w)$ and $k_2=\mathrm{dist}_P(y,w)$ differ at most by one. Suppose that Duplicator responds with $w'\in G'$. It is impossible that $G'-z'$ contains both an $x'w'$-path of length at most $k_1$ and a $y'w'$-path of length at most $k_2$. If, for example, the latter does not exist, then we apply induction to $y,w\in G$. The required bound follows by observing that $k_1,k_2\le \ceil{\frac k2}$.\qed The same method gives the following lemma. \blm{path} Let $G,G'$ be colored graphs. Suppose that $x,y\in V(G)$ and $x',y'\in V(G')$ have been selected such that $G$ contains some $xy$-path $P$ of length at most $k$ such that some vertex of $P$ has color $c$ while this is not true with respect to $G'$. Then Spoiler can win in at most $\log_2 k +1$ moves playing all of the time inside $G$. The same conclusion holds if all internal vertices of $P$ have colors from some fixed set $A$ while any $x'y'$-path of length at most $k$ has a color not in $A$.\qed\end{lemma} \blm{Tree} Let $T$ be a tree of order $n$ and let $T'$ be a graph which is not a tree. Then $D(T,T')\le \log_2n+3$.\end{lemma} \bpf If $T'$ is not connected, Spoiler selects two vertices $x',y'\in T'$ from different components. Then he switches to $G$ and applies Lemma~\ref{lm:Distance}, winning in at most $\log_2 n+3$ moves in total. Otherwise, let $C'\subset T'$ be a cycle of the shortest length $l$. If $l>2n+1$, then Spoiler picks two vertices $x',y'$ at distance at least $n$ in $C'$ (or equivalently in $T'$). But the diameter of $T$ is at most $n-1$, Spoiler switches to $T$ and starts halving the $xy$-path, making at most $\log_2 n+3$ moves in total, cf.\ Lemma~\ref{lm:Distance}. If $l\le 2n+1$, then Spoiler selects some three adjacent vertices of $C'$, say $x',z',y'$ in this order. Now, he applies Lemma~\ref{lm:path} with respect to $k=l-2$.\qed \bpf[Proof of Theorem~\ref{th:MaxDeg}.] Let us prove the upper bound first. Let $T$ be any tree of order at most $n$ and maximum degree at most $l$. Let $T'$ be an arbitrary colored graph not isomorphic to $T$. By Lemma~\ref{lm:Tree} we can assume that $T'$ is a tree. In fact, we will be proving the upper bound on the version of the $(T,T')$-game, wherein some distinguished vertex, called the \emph{root}, is given and all graph isomorphisms must additionally preserve the root. (This can be achieved by introducing a new color $U_0$ which is assigned to the root only.) The obtained upper bound, if increased by $1$, applies to the original function $D(T,T')$ because we can regard $x_1$ and $x_1'$, the first two moves of the Ehrenfeucht game, as the given roots. It is easy to show that $T$ contains a vertex $x\in T$ such that any component of $T-x$ has order at most $\frac n2$. We call such a vertex a \emph{median} of $T$. Spoiler selects this vertex $x$; let Duplicator reply with $x'$. We can assume that the degrees of $x$ and $x'$ are the same: otherwise Spoiler can exhibit this discrepancy in at most $l+1$ extra moves. \comment{Alternating sides at most once.} We view the components of $T-x$ and $T'-x'$ as colored rooted graphs with the neighbors of $x$ and $x'$ being the roots. As $T\not\cong T'$, some component $C_1$ has different multiplicities $m_1$ and $m_1'$ in $T-x$ and $T'-x'$. As $d(x)=d(x')$, we have at least two such components. Assume that for $C_1$ and $C_2$ we have $m_1>m_1'$ and $m_2<m_2'$. By the condition on the maximum degree, $m_1'+m_2\le l-1$. Hence, $\min(m_1',m_2)\le \frac{l-1}2$. Let us assume, for example, that $m_1'\le \frac{l-1}2$. Spoiler chooses the roots of any $m_1'+1$ $C_1$-components of $T-x$. It must be the case that some vertices $y\in V(T)$ and $y'\in V(T')$ have been selected, so that $y$ lies in a $C_1$-component $F\subset T-x$ while $y'$ lies in a component $F'\subset T'-x$ not isomorphic to $C_1$. Let $n_1$ be the number of vertices in $F$. By the choice of $x$, $n_1\le \frac n2$. Now, Spoiler restricts his moves to $V(F)\cup V(F')$. If Duplicator moves outside this set, then Spoiler uses Lemma~\ref{lm:path}, winning in at most $\log_2n+O(1)$ moves. Otherwise Spoiler uses the recursion applied to $F$. Let $f(n,l)$ denote the largest number of moves (over all trees $T,T'$ with $v(T)\le n$, $\Delta(T)\le l$, and $T\not\cong T'$) that Duplicator can survive against the above strategy with the additional restriction that a situation where Lemma~\ref{lm:path} can be applied never occurs and we always have that $d(x)=d(x')$. Clearly, \beq{DTf} D^{\mathrm{tree}}(n,l)\le f(n,l) + \log_2n + l +O(1). \end{equation} As $m_1\le \frac{n-1}{n_1}$, we get the following recursive bound on $f$. \beq{DT} \textstyle f(n,l)\le \max\Big\{2 + \min(\frac{l-1}2,\frac{n-1}{n_1}) + f(n_1,l): 1\le n_1\le \frac n2\Big\}. \end{equation} Denoting $n_0=n$ and unfolding~\req{DT} as long as $n_i\ge 1$, say $s$ times, we obtain that $f(n,l)$ is bounded by the maximum of \beq{f} 2s + \sum_{i=1}^s \min\left(\frac{l-1}2,\frac{n_{i-1}}{n_i}\right), \end{equation} over all sequences $n_1,\dots,n_s$ such that \beq{n} 1\le n_i \le \frac{n_{i-1}}2,\quad i\in[s]. \end{equation} Note that the restrictions~\req{n} force $s$ to be at most $\log_2 n$. Let us maximize~\req{f} over all $s\in\I N$ and real $n_i$'s satisfying~\req{n}. It is routine to see that for the optimal sequence we have $2\le \frac{n_{i-1}}{n_i}\le \frac{l-1}2$, $i\in[s]$; moreover, both these inequalities can be simultaneously strict for at most one index $i$. \comment{Indeed, suppose on the contrary that for two indexes $1\le i<j< s$ we have $2<n_i/n_{i+1}<\frac{l-1}2$ and $2<n_j/n_{j+1}<\frac{l-1}2$. Redefine a new sequence: $n_h'=n_h$ if $h\le i$ or $h>j$, while $n_h'=xn_h$ for $i<h\le j$. If $x=1$, then we obtain the same sequence. Note that $\frac{n_h'}{n_{h+1}'}=\frac{n_h}{n_{h+1}}$ for any $h$ except $h=i$ or $h=j$. So, we can slightly perturb $x$ either way, without violating~\req{n}. The right-hand side of~\req{f}, as a function of $x$ in a small neighborhood of $x=1$, is of the form $ax+\frac bx+c$ with $a,b>0$. But this function is strictly convex, so it cannot attain its maximum at $x=1$, a contradiction.} Let $t$ be the number of times we have $n_{i-1}=2n_i$. The bound~\req{f} reads \beq{st} f(n,l)- 2 \log_2 n \le 2t + (s-t)\, \frac{l-1}2. \end{equation} Given that $2^t(\frac{l-1}2)^{s-t-1}\le n$, the right hand side of~\req{st} is maximized for $t=O(\log l)$ and $s=(1+o(1))\, \frac{\ln n}{\ln l}$, implying the upper bound~\req{MaxDeg} by~\req{DTf}. Let us prove the lower bound. Let $k=\floor{l/2}$. Define $G_0=K_{1,l-1}$ and $G_0'=K_{1,l-2}$. Let $r_0\in V(G_0)$, $r_0'\in V(G_0')$ be their roots. Define inductively on $i$ the following graphs. $G_{i}$ is obtained by taking $k$ copies of $G_{i-1}$ and $k-1$ copies of $G_{i-1}'$, pairwise vertex-disjoint, plus the root $r_i$ connected to the root of each copy of $G_{i-1}$ and $G_{i-1}'$. We have $d(r_i)\le l-1$. The graph $G_{i}'$ is defined in a similar way except that we take $k-1$ copies of $G_{i-1}$ and $k$ copies of $G_{i-1}'$. Let $i$ be the largest index such that $\max(v(G_i),v(G_i'))\le n$. Let us disregard all roots, i.e.,\ view $G_j$ and $G_j'$ as usual (uncolored) graphs. Note that the trees $G_i$ and $G_i'$ are non-isomorphic as for every $j$ we can identify the level-$j$ roots as the vertices at distance $j+1$ from some leaf. Define $g_j=(k-1)j+l-2$, $j\in[0,i]$. Let us show by induction on $j$ that Duplicator can survive at least $g_j$ rounds in the $(G_j,G_j')$-game. This is clearly true for $j=0$. Let $j\ge 1$. If Spoiler claims one of $r_j,r_j'$ then Duplicator selects the other. If Spoiler selects a vertex in a graph from the ``previous'' level, for example $F\subset G_j$ with $F\cong G_{j-1}'$, then Duplicator chooses an $F'\subset G_i'$, $F'\cong G_{j-1}'$ and keeps the isomorphism between $F$ and $F'$. So any moves of Spoiler inside $V(F)\cup V(F')$ will be useless and we can ignore $F$ and $F'$. Thus it takes Spoiler at least $k-1$ moves before we are down to the pair $(G_{j-1},G_{j-1}')$, which proves the claim. Thus we have $D(G_i) \ge D(G_i,G_i') \ge g_i=(\frac12+o(1))\, \frac{l\ln n}{\ln l}$, finishing the proof.\qed \noindent{\bf Remark.} Verbitsky~\cite{verbitsky:04} proposed a different argument to estimate $D^{\mathrm{tree}}(n,l)$ which gives a weaker bounds than those in Theorem~\ref{th:MaxDeg} but can be applied to other classes of graphs with small separators. Let us study $D^{\mathrm{tree}}(n,l)$ for other $l,n$. The methods have much in common with the proof of Theorem~\ref{th:MaxDeg} so our explanations are shorter. \bth{l=n^C} Let an integer $t\ge1$ be fixed. Suppose that $l,n\to\infty$ so that $n\ge l^t$ but $n=o(l^{t+1})$. Then $D^{\mathrm{tree}}(n,l)=(\frac{t+1}2+o(1))\, l$. In fact, the lower bound can be achieved by uncolored trees.\end{theorem} \bpf The lower bound is proved by the induction on $t$. If $t=1$, take $T_1=K_{1,l-2}$. One needs at least $l-1$ moves to distinguish it from $T_1'=K_{1,l-1}$. Let $a=\floor{l/2}$ and $b=\ceil{l/2}$. Suppose we have already constructed $T_{t-1}$ and $T_{t-1}'$, rooted trees with $\le l^{t-1}$ vertices such that the root has degree at most $l-1$. To construct $T_t$ take $a$ copies of $T_{t-1}$ and $b-1$ copies of $T_{t-1}'$ and connect them to the common root. For $T_t'$ we take $a-1$ and $b$ copies respectively. The degree of the main root is $a+b-1= l-1$ as required. The order of $T_t$ is at most $(a+b-1)l^{t-1}+1\le l^t$. Also, Spoiler needs at least $a$ moves before reducing the game to $(T_{t-1},T_{t-1}')$ (while, for $t=1$, $l$ moves are needed to finish the game), giving the required bound. Let us turn to the upper bound. Spoiler uses the same strategy as before. Namely, he chooses a median $x\in T$ and of two possible multiplicities, summing up to $l$, chooses the smaller. Let $m_1+1,m_2+1,\dots,m_k+1$ be the number of moves per each selected median. We have $n\ge \prod_{i=1}^k m_i$. Also, we have $k\le \log_2n$ because we always choose a median. Given these restrictions, the inequalities $m_i\le l/2$, $i\in[k-1]$, and $m_k\le l-1$, the sum $\sum_{i=1}^k m_i$ is maximized if $m_k=l-1$ and as many as possible $m_j=l/2$ are maximum possible. We thus factor out $l/2$ at most $t-1$ times until the remaining terms have the product (and so the sum) $o(l)$. Thus, $$ \sum_{i=1}^k (m_i+1)\le \log_2n+\sum_{i=1}^km_i\le l+\frac{(t-1)l}2+o(l), $$ completing the proof.\qed Theorems~\ref{th:MaxDeg} and~\ref{th:l=n^C} do not cover all the possibilities for $n,l$. The asymptotic computation in the remaining cases seems rather messy. However, the order of magnitude of $D^{\mathrm{tree}}(n,l)$ is easy to compute with what we already have. Namely, Theorem~\ref{th:l=n^C} implies that for $l=\Theta(n^t)$ with fixed $t\in \I N$ we have $D^{\mathrm{tree}}(n,l)=\Theta(l)$. Also, if $l\ge 2$ is constant, then $D^{\mathrm{tree}}(n,l)=\Theta(\ln n)$, where the lower bound follows from considering the order-$n$ path and the upper bound is obtained by using the method of Theorem~\ref{th:MaxDeg}. \section{The Giant Component}\label{giant} Let $c>1$ be a constant, $p=\frac cn$, and $G$ be the giant component of a random graph $\C G(n,p)$. \comment{ Kim, Pikhurko, Spencer and Verbitsky~\cite{kim+pikhurko+spencer+verbitsky:03rsa} conjectured that whp $D(G)=O(\ln n)$. } Here we show the following result. \bth{giant} Let $c>1$ be a constant, $p=c/n$, and $G$ be the giant component of $\C G(n,p)$. Then whp \beq{giant} D(G)=\Theta\left(\frac{\ln n}{\ln \ln n}\right) \end{equation} \end{theorem} This result allows us to conclude that for any $p=O(n^{-1})$ a random graph $H\in \C G(n,p)$ satisfies whp \beq{d/n} D(H)=({\mathrm e}^{-np}+o(1))\, n. \end{equation} The proof is an easy modification of that in~\cite{kim+pikhurko+spencer+verbitsky:03rsa} where the validity of \req{d/n} was established for $p\le (1.19...+o(1))\, n^{-1}$. The lower bound in~\req{d/n} comes from considering the graph $H'$ obtained from $H$ by adding an isolated vertex (and noting that whp $H$ has $({\mathrm e}^{-np}+o(1))\, n$ isolated vertices). The method in~\cite{kim+pikhurko+spencer+verbitsky:03rsa} shows that the upper bound~\req{d/n} can fail only if $D(G)>({\mathrm e}^{-np}+o(1))\, n$, where $G$ is the giant component of $H$. (And $p/n\approx 1.19...$ is the moment when $v(G)\approx {\mathrm e}^{-np}$.) \subsection{Upper Bound} The structure of the giant component is often characterized using its core and kernel (e.g., see Janson, \L uczak, and Ruci\'nski~\cite[Section~5]{janson+luczak+rucinski:rg}). We follow this approach in the proof of the upper bound in \req{giant}. Thus, we first bound $D(G)$ from above for a graph $G$ with small diameter whose kernel fulfills some ``sparsness'' conditions. Then, we show that these conditions hold whp for the kernel of the giant component of a random graph. \subsubsection{Bounding $D(G)$ Using the Kernel of $G$}\label{DKernel} The \emph{core} $C$ of a graph $G$ is obtained by removing, consecutively and as long as possible, vertices of degree at most $1$. If $G$ is not a forest, then $C$ is non-empty and $\delta(C)\ge 2$. First we need an auxiliary lemma which is easily proved, similarly to the auxiliary lemmas in Section~\ref{general}, by the path-halving argument. \blm{cycle} Let $G,G'$ be graphs. Suppose $x\in V(G)$ and $x'\in V(G')$ have been selected such that $G$ contains some cycle $P\ni x$ of length at most $k$ while $G'$ does not. Then Spoiler can win in at most $\log_2 k+O(1)$ moves, playing all time inside $G$.\qed\end{lemma} \blm{DCore} Let $G,G'$ be graphs and $C,C'$ be their cores. If Duplicator does not preserve the core, then Spoiler can win in at most $\log_2d+O(1)$ extra moves, where $d$ is the diameter of $G$.\end{lemma} \bpf Assume that $\mathrm{diam}(G')=\mathrm{diam}(G)$ for otherwise we are easily done. Suppose that, for example, some vertices $x\in C$ and $x'\in \O {C'}$ have been selected. If $x$ lies on a cycle $C_1\subset C$, then we can find such a cycle of length at most $2d+1$. Of course, $G'$ cannot have a cycle containing $x'$, so Spoiler wins by Lemma~\ref{lm:cycle} in $\log_2(2d+1)+O(1)$ moves, as required. Suppose that $x$ does not belong to a cycle. Then $G$ contains two vertex-disjoint cycles $C_1,C_2$ connected by a path $P$ containing $x$. Choose such a configuration which minimizes the length of $P\ni x$. Then the length of $P$ is at most $d$. Spoiler selects the branching vertices $y_1\in V(C_1)\cap V(P)$ and $y_2\in V(C_2)\cap V(P)$. If some Duplicator's reply $y_i'$ is not on a cycle, we done again by Lemma~\ref{lm:cycle}. So assume there are cycles $C_i'\ni y_i'$. In $G$ we have \beq{dist} \mathrm{dist}(y_1,y_2)= \mathrm{dist}(y_1,x) + \mathrm{dist}(y_2,x). \end{equation} As $x'\not\in C'$, any shortest $x'y_1'$-path and $x'y_2'$-path enter $x'$ via the same edge $\{x',z'\}$. But then \beq{distp} \mathrm{dist}(y_1',y_2')\le \mathrm{dist}(y_1',z')+\mathrm{dist}(y_2',z')= \mathrm{dist}(y_1',x') + \mathrm{dist}(y_2',x')-2. \end{equation} By~\req{dist} and~\req{distp}, the distances between $x,y_1,y_2$ cannot be all equal to the distances between $x',y_1',y_2'$. Spoiler can demonstrate this in at most $\log_2 (\mathrm{dist}(y_1,y_2)) +O(1)$, as required.\qed In order to state our upper bound on $D(G)$ we have to define a number of parameters of $G$. In outline, we try to show that any distict $x,y\in V(C)$ can be distinguished by Spoiler reasonably fast. This would mean that each vertex of $C$ can be identified by a first order formula of small depth. Note that $G$ can be decomposed into the core and a number of trees $T_x$, $x\in V(C)$, rooted at vertices of $C$. Thus, by specifying which pairs of vertices of $C$ are connected and describing each $T_x$, $x\in V(C)$, we completely define $G$. However, we have one unpleasant difficulty that not all pairs of points of $C$ can be distinguished from one another. For example, we may have a pendant triangle on $\{x,y,z\}$ with $d(x)=d(y)=2$, in which case the vertices $x$ and $y$ are indistinguishable. However, we will show that whp we can distinguish any two vertices of degree $3$ or more in $C$, which suffices for our purposes. Let us give all the details. For $x\in V(C)$, let $T_x\subset G$ denote the tree rooted at $x$, i.e., $T_x$ is a component containing $x$ in the forest obtained from $G$ by removing all edges of $C$. Let $$ t=\max\{D(T_x): x\in V(C)\}, $$ where $D(T_x)$ is taken with respect to the class of graphs with one root. Let the \emph{kernel} $K$ of $G$ be obtained from $C$ by the \emph{serial reduction} where we repeat as long as possible the following step: if $C$ contains a vertex $x$ of degree $2$, then remove $x$ from $V(C)$ but add the edge $\{y,z\}$ to $E(C)$ where $y,z$ are the two neighbors of $x$. Note that $K$ may contain loops and multiple edges. We agree that each loop contributes $2$ to the degree. Then we have $\delta(K)\ge 3$. Let $u=\Delta(G)$ and $d$ be the diameter of $G$. It follows that each edge of $K$ corresponds to the path $P$ in $C$ of length at most $2d$. \comment{(For otherwise any two vertices of $P$ at distance $d+1$ contradict the definition of $d$.)} Let $l$ be an integer such that every set of $v\le 6 l$ vertices of $K$ spans at most $v$ edges in $K$. (Roughly speaking, we do not have two short cycles close together.) For $\{x,y\}\in E(K)$ let $A_{x,y}$ be the set of vertices obtained by doing breadth first search in $K-x$ starting with $y$ until the process dies or, after we have added a whole level, we reach at least $k=2^{l-2}$ vertices. Let $K_{x,y}=K[A_{x,y}\cup \{x\}]$. The \emph{height} of $z\in V(K_{x,y})$ is the distance in $K-x$ between $z$ and $y$. It is easy to deduce from the condition on short cycles that each $K_{x,y}\subset K-x$ has at most one cycle and the maximum height is at most $l$. In fact, the process dies only in the case if $y$ is an isolated loop in $K-x$. For $xy\in E(K)$ let $G_{x,y}$ be a subgraph of $G$ corresponding to $K_{x,y}$. We view $K_{x,y}$ and $G_{x,y}$ as having two special \emph{roots} $x$ and $y$. Here is another assumption about $G$ and $l$ we make. Suppose that for any $xx',yy'\in E(K)$ if $K_{x,x'}$ and $K_{y,y'}$ have both order at least $k$ and $A_{x,x'}\cap A_{y,y'}=\emptyset$, then the rooted graphs $G_{x,x}$ and $G_{y,y'}$ are not isomorphic. Let \begin{eqnarray} b_0&=&\frac{l(\ln u+\ln \ln n + l)}{\ln l} +2u+\log_2d,\label{eq:b0}\\ b&=& b_0 + t+ u +2\log_2d.\label{eq:b} \end{eqnarray} \blm{a} Under the above assumptions on $G$, we have $D(G)\le b+O(1)$. \end{lemma} \bpf Let $G'\not\cong G$. Let $C',K'$ be its core and kernel. We can assume that $\Delta(G')=u$ and its diameter is $d$ for otherwise Spoiler easily wins in $u+2$ or $\log_2d+O(1)$ moves. By Lemma~\ref{lm:DCore} it is enough to show that Spoiler can win the Ehrenfeucht $(G,G')$-game in at most $b-\log_2d+O(1)$ moves provided Duplicator always respects $C$ and $K$. Call this game $\C C$. Color $V(K)\cup E(K)$ and $V(C)$ by the isomorphism type of the subgraphs of $G$ which sit on a vertex/edge. We have a slight problem with the edges of $K$ as the color of an unordered edge may depend in which direction we traverse it. So, more precisely, every edge of $K$ is considered as a pair of ordered edges each getting its own color. Do the same in $G'$. As $G\not\cong G'$, the obtained colored digraphs $K$ and $K'$ cannot be isomorphic. Call the corresponding digraph game $\C K$. \claim1 If Spoiler can win the game $\C K$ in $m$ moves, then he can win $\C C$ in at most $m+t+u+\log_2d+O(1)$ moves. \bpf[Proof of Claim.] We can assume that each edge of $K'$ corresponds to a path in $G'$ of length at most $2d+1$: otherwise Spoiler selects a vertex of $C'$ at the $C'$-distance at least $d+1$ from any vertex of $K'$ and wins in $\log_2d+ O(1)$ moves. Spoiler plays according to his $\C K$-strategy by making moves inside $V(K)\subset V(G)$ or $V(K')\subset V(G')$. Duplicator's reply are inside $V(K')$, so they correspond to replies in the $\C K$-game. In at most $m$ moves, Spoiler can achieve that the set of colored edges between some selected vertices $x,y\in K$ and $x',y'\in K'$ are different. (Or loops if $x=y$.) In at most $u+1$ moves, Spoiler can either win or select a vertex $z$ inside a colored $xy$-path $P$ (an edge of $K$) such that $z'$ either is not inside an $x'y'$-path (an edge of $K'$) or its path $P'\ni z'$ has a different coloring from $P$. In the former case, Spoiler wins by Lemma~\ref{lm:path}: in $G$ there is an $xy$-path containing $z$ and no vertex from $K$. Consider the latter case. Assume that $|P|=|P'|$, for otherwise we are done by Lemma~\ref{lm:path}. Spoiler selects $w\in P$ such that for the vertex $w'\in P'$ with $\mathrm{dist}_P(w,x)=\mathrm{dist}_{P'}(w',x')$ we have $T_w\not\cong T'_{w'}$. If Duplicator does not reply with $w'$, then she has violated distances. Otherwise Spoiler needs at most $t$ extra moves to win the game $\C T$ on $(T_w,T'_{w'})$ (and at most $\log_2d+O(1)$ extra moves to catch Duplicator if she does not respect $\C T$).\cqed It remains to bound $D(K)$, the colored digraph version. This requires a few preliminary results. \claim2 For any $\{x,x'\}\in K$ we have $D(K_{x,x})\le b_0+O(1)$ in the class of colored digraphs with two roots, where $b_0$ is defined by~\req{b0}. \bpf[Proof of Claim.] Let $T=K_{x,x}$ and $T'\not\cong T$. If $T$ is a tree, then we just apply a version of Theorem~\ref{th:MaxDeg} using the order ($\le\! u 2^{l}$) and maximum degree ($\le\! u$). Otherwise, Spoiler first selects a vertex $z\in T$ which lies on the (unique) cycle. We have at most $u-1$ components in $T-z$, viewing each as a colored tree where one extra color marks the neighbors of $z$. As $T\not\cong T'$, in at most $u+1$ moves we can restrict our game to one of the components. (If Duplicator does not respect components, she loses in at most $\log_2 d +O(1)$ moves.) Now, one of the graphs is a colored tree, and Theorem~\ref{th:MaxDeg} applies.\cqed \claim3 For every two distinct vertices $x,y\in V(K)$ there is a first order formula $\Phi_{x,y}(z)$ with one free variable and quantifier rank at most $b_0+\log_2d+O(1)$ such that $G\models \Phi_{x,y}(x)$ and $G\not\models \Phi_{x,y}(y)$. (Note that we have to find $\Phi_{x,y}$ for $x,y$ in the kernel only, but we evaluate $\Phi_{x,y}$ with respect to $G$.) \bpf[Proof of Claim.] To prove the existence of $\Phi_{x,y}$ we have to describe Spoiler's strategy, where he has to distinguish $(G,x)$ and $(G,y)$ for given distinct $x,y\in K$. If the multiset of isomorphism classes $K_{x,x'}$, over $\{x,x'\}\in E(K)$ is not equal to the multiset $\{ K_{y,y'}: \{y,y'\}\in E(K)\}$, then we are done by Claim~2. So let us assume that these multisets are equal. Note that an isomorphism $K_{x,x'}\cong K_{y,y'}$ implies an isomorphism $G_{x,x'}\cong G_{y,y'}$. Also, by our assumption on $l$, the isomorphism $G_{x,x'}\cong G_{y,y'}$ implies that $V(K_{x,x'})\cap V(K_{y,y'})\not=\emptyset$. At most one neighbor of $x$ can be an isolated loop for otherwise, we get 3 vertices spanning 4 edges. The same holds for $y$. As the height of any $K_{a,b}$ is at most $l$, we conclude that $\mathrm{dist}_K(x,y)\le 2l$. A moment's thought reveals that there must be a cycle of length at most $4l$ containing both $x$ and $y$. But this cycle rules out the possibility of a loop adjacent to $x$ or to $y$. Thus, in order to exclude $2$ short cycles in $K$ close to each other, it must be the case that $\mathrm{dist}(x,y)\le l-1$ and $d_K(x)=d_K(y)=3$. Moreover, let $x_1,x_2,x_3$ and $y_1,y_2,y_3$ be the neighbors of $x$ and $y$ such that $G_{x,x_i}\cong G_{y,y_i}$; then (up to a relabeling of indices), we have the following paths between $x$ and $y$: either $(x,x_1,\dots,y_1,y)$ and $(x,x_2,\dots,y_3,y)$ or $(x,x_1,\dots,y_3,y)$ and $(x,x_2,\dots,y_1,y)$ Now, $K_{x,x_3}$ is not isomorphic to $K_{x,x_1}$ nor to $K_{x,x_2}$ by the vertex-disjointness. (Note that it is not excluded that $K_{x,x_1}\cong K_{x,x_2}$: they may intersect, for example, in $y$.) But then $z=x$ is different from $z=y$ in the following respect: the (unique) short cycle of $K$ containing $z$ has its two edges entering $z$ from subgraphs isomorphic to $K_{x,x_1}$ and $K_{x,x_2}$ (while for $z=y$ the corresponding subgraphs are isomorphic to $K_{x,x_1}$ and $K_{x,x_3}$). This can be used by Spoiler as follows. Spoiler selects $x_1,x_2$. If Duplicator replies with $y_3$, then Spoiler can use Claims~2 and~3 because $K_{y,y_3}$ is not-isomorphic to $K_{x,x_1}$ nor to $K_{x,x_2}$. Otherwise, the edge $\{x,x_2\}$ is on a short cycle while $\{y,y_2\}$ is not. Spoiler uses Lemma~\ref{lm:cycle}.\cqed By Lemma~\ref{lm:DCore} we can find $\Phi_K(x)$, a formula of rank at most $\log_2d+O(1)$ which, with respect to $G$, evaluates to $1$ for all $x\in V(K)$ and to $0$ otherwise. More precisely, Lemma~\ref{lm:DCore} gives a formula $\Phi_C(x)$ testing for $x\in V(C)$. But $V(K)\subset V(C)$ are precisely the vertices of degree at least $3$ in $C$. \comment{So we can take $$ \Phi_K(x)= \Phi_C(x) \wedge \exists_{x_1,x_2,x_1} \left( \Phi_C(x_1)\wedge \Phi_C(x_2)\wedge \Phi_C(x_3)\wedge x\sim x_1\wedge x\sim x_2\wedge x\sim x_3\wedge_{i\not= j} x_i\not=x_j\right). $$ } Now, as it is easy to see, for any $x\in K$ the formula \beq{Phi} \Phi_x(v):= \Phi_K(v) \wedge \bigwedge_{y\in V(K)\setminus \{x\}} \Phi_{x,y}(v) \end{equation} identifies uniquely $x$ and has rank at most $\log_2d+b_0+ O(1)$. Take $x\in V(K)$. If there is no $x'\in V(K')$ such that $G'\models \Phi_{x}(x')$, then Spoiler selects $x$. Whatever Duplicator's reply $x'$ is, it evaluates differently from $x$ on $\Phi_{x}$. Spoiler can now win in at most $D(\Phi_{x})$ moves, as required. If there are two distinct $y',z'\in K'$ such that $G'\models \Phi_{x}(y')$ and $G'\models \Phi_{x}(z')$, then Spoiler selects both $y'$ and $z'$. At least one of Duplicator's replies is not equal to $x$, say, $y\not=x$. Again, the selected vertices $y\in V(K)$ and $y'\in V(K')$ are distinguished by $\Phi_x$, so Spoiler can win in at most extra $D(\Phi_x)$ moves. Therefore, let us assume that for every $x\in V(K)$ there is the unique vertex $x'=\phi(x)\in V(K')$ such that $G'\models \Phi_x(x')$. Clearly, $\phi$ is injective. Furthermore, $\phi$ is surjective for if $x'\not\in \phi(V(K))$, then Spoiler wins by selecting $x'\in V(K')$ and then using $\Phi_x$, where $x\in V(K)$ is Duplicator's reply. Moreover, we can assume that Duplicator always respects~$\phi$ for otherwise Spoiler wins in at most $\log_2d+b_0+O(1)$ extra moves. As $K\not\cong K'$, Spoiler can select $x,y\in V(K)$ such that the multisets of colored paths (or loops if $x=y$) between $x$ and $y$ and between $x'=\phi(x)$ and $y'=\phi(y)$ are distinct. Again, this means that some colored path has different multiplicities and Spoiler can highlight this in at most $u+1$ moves. Then in at most $\log_2l+O(1)$ moves he can ensure that some vertices $z\in V(K)$ and $z'\in V(K')$ are selected such that the removed trees $T_z$ and $T_{z'}$ rooted at $z$ and $z'$ are not isomorphic, compare with Lemma~\ref{lm:path}. Now, by the definition of $t$, at most $t$ moves are enough to distinguish $T_z$ from $T_{z'}'$ (plus possible $\log_2 d +O(1)$ moves to catch Duplicator if she replies outside $V(T_z)\cup V(T_{z'})$). This completes the proof of Lemma~\ref{lm:a}.\qed \subsubsection{Probabilistic Part} Here we estimate the parameters from the previous section. As before, let $G$ be the giant component of $\C G(n,\frac cn)$, let $C$ be its core, etc. It is well-known that whp $u=O(\frac{\ln n}{\ln\ln n})$ and $d=O(\ln n)$. \comment{Reference???} \blm{Shaved} Whp every edge of $K$ corresponds to at most $O(\ln n)$ vertices of $G$. Similarly, for any $x\in V(C)$ we have $v(T_x)=O(\ln n)$.\end{lemma} \bpf The expected number of $K$-edges, each corresponding to precisely $i=O(\ln n)$ vertices in $G$ is at most $$ \binom{n}{i}\binom{i}{2} p^{i-1} i^{i-2} (1-p)^{(i-2)(n-i)} \le n i^2\left(\frac{{\mathrm e} c}{{\mathrm e}^c}\right)^{i}. $$ But ${\mathrm e} c< {\mathrm e}^c$ for $c>1$, so if $i$ is large enough, $i>M\ln n$, then the expectation is $o(n^{-3})$. Similarly, the expected number of vertices $x$ with $v(T_x)=i=O(\ln n)$ is at most $$n\binom{n-1}{i-1}p^{i-1}i^{i-2}(1-p)^{(i-1)(n-i)}\leq 2n i\left(\frac{{\mathrm e} c}{{\mathrm e}^c}\right)^{i}.$$ \qed In particular, our results from Section~\ref{general} imply that whp $t=O(\frac{\ln n}{\ln \ln n})$. Let, for example, $l=2\ln \ln n$. Thus $k/\ln n\to\infty$, where $k=2^{l-2}$. It remains to prove that this choice of $l$ satisfies all the assumptions. \blm{ShortCycle} Whp any set of $s\le 6l$ vertices of $K$ spans at most $s$ edges.\end{lemma} \bpf A moment's thought reveals that it is enough to consider sets spanning connected subgraphs only. Let $L=M\ln n$ be given by Lemma~\ref{lm:Shaved}. The probability that there is a set $S$ such that $|S|=s\leq 6l$ and $K[S]$ is a connected graph with at least $s+1$ edges is at most \begin{align*} &o(1)+\sum_{s=4}^{6l}\binom{n}{s}\, s^{s-2}\, {s\choose 2}^2\sum_{0\leq \ell_1,\ldots,\ell_{s+1}\leq L} \prod_{i=1}^{s+1}\binom{n}{\ell_i}(\ell_i+2)^{\ell_i}p^{\ell_i+1}(1-p)^{\ell_i(n-\ell_i-2)}\\ &\leq o(1)+\sum_{s=4}^{6l}\bfrac{n{\mathrm e}}{s}^s s^{s+2}\sum_{0\leq \ell_1,\ldots,\ell_{s+1}\leq L} \prod_{i=1}^{s+1}\left(\frac{c{\mathrm e}^2}{n}\left(\frac{{\mathrm e} c}{{\mathrm e}^c}\right)^{\ell_i}\right)\ \le\ o(1)+ \sum_{s=4}^{6l}\frac{(O(1))^s}{n}\ =\ o(1). \end{align*} The lemma is proved.\qed \blm{Kab} Whp $K$ does not contain four vertices $x,x',y,y'$ such that $xx',yy'\in E(K)$, $v(K_{x,x'})\ge k$, $A_{x,x'}\cap A_{y,y'}=\emptyset$, and $G_{x,x'}\cong G_{y,y'}$.\end{lemma} \bpf Given $c$, choose the following constants in this order: small $\epsilon_1>0$, large $M_1$, large $M_2$, small $\epsilon_2>0$, and large $M_3$. Consider breadth-first search in $G-x$ starting with $x'$. Let $L_1=\{x'\}$, $L_2$, $L_3$, etc., be the levels. Let $T_i=\{x\}\cup (\cup_{j=1}^i L_i)$. Let $s$ be the smallest index such that $|T_s|\ge M_2\ln n$. Chernoff's bound implies that the probability of $|T_s|> 2cM_2 \ln n$ is $o(n^{-2})$. Indeed, this is at most the probability that the binomial random variable with parameters $(n, \frac cn \times M_2\ln n)$ exceeds $2cM_2\ln n$. Similarly, with probability $1-o(n^{-3})$ we have $|L_{i+1}|=(c\pm \epsilon_2)|L_i|$ provided $i\ge s$ and $|T_i|=o(n)$. Hence, we see that from the first time we reach $2M_2\ln n$ vertices, the levels increase proportionally with the coefficient close to $c$ for further $\Theta(\ln n)$ steps. Take some $i$ with $|T_i|=O(\ln n)$. The sizes of the first $\Theta(\ln n)$ levels of the breadth-first search from the vertices of $L_i$ can be bounded from below by independent branching processes with the number of children having the Poisson distribution with mean $c-\epsilon_2$. Indeed, for every active vertex $v$ choose a pool $P$ of $\ceil{(1-\frac{\epsilon_2}c)n}$ available vertices and let $v$ choose its neighbors from $P$, each with probability $c/n$. (The edges between $v$ and $\O P$ are ignored.) If $v$ claimed $r$ neighbors, then, when we take the next active vertex $u$, we add extra $r$ vertices to the pool, so that its size remains constant. With positive probability $p_1$ the ideal branching process survives infinitely long; in fact, $p_1$ is the positive root of $1-p_1={\mathrm e}^{-cp_1}$. Let $$ p_2=\max_{j\ge 0} \frac{c^j{\mathrm e}^{-c}}{j!} <1. $$ The numbers $p_1>0$ and $p_2<1$ are constants (depending on $c$ only). Take the smallest $i$ such that $|T_i|\ge 2cM_3\ln n$. The breadth-first search inside $G$ goes on for at least $M_1$ further rounds (after the $i$-th round) before we reach a vertex outside $G_{x,x'}$. We know that $|L_i|\ge (\frac{c-1}c-\epsilon_1)\,|T_i|$ because the levels grow proportionally from the $s$-th level. Let $Z$ consist of the vertices of $L_i$ for which the search process in $G-x$ goes on for at least $M_1$ further levels before dying out. By Chernoff's bound, with probability $1-o(n^{-2})$ we have $|Z|\ge \frac{p_1}2 |L_i|$. Let us fix any $K_{x,x'}$ having all the above properties and compute the expected number of copies of $K_{x,x'}$ in $G$. More precisely, we compute the expected number of subgraphs of $G$ isomorphic to $G[T_{i}]$ such that a specified $|Z|$-subset of the last level has specified trees, each of height at least $M_1$, sitting on it. The expected number of $G[T_i]$-subgraphs is at most $n^{|T_i|}\,p_1^{|T_i|-1}$. This has to be multiplied by $$ (p_2+o(1))^{M_1|Z|} \le p_2^{M_1(c-1)p_1\,|T_i|/4c}: $$ because if we want to get a given height-$M_1$ tree, then at least $t$ times we have to match the sum of degrees of a level, each coincidence having probability at most $p_2+o(1)$. As the constant $M_1$ can be arbitrarily large, we can make the total expectation $o(n^{-2})$. Markov's inequality implies the lemma.\qed Finally, putting all together we deduce the upper bound of Theorem~\ref{th:giant}. \subsection{Lower Bound} Let $l=(1-\epsilon) \frac{\ln n}{\ln \ln n}$ for some $\epsilon>0$. We claim that whp the core $C$ has a vertex $i$ adjacent to at least $l$ leaves of $G$. (Then we have $D(C)\ge l+1$: consider the graph obtained from $C$ by adding an extra leaf to $i$.) Let us first prove this claim for the whole random graph $H\in \C G(n,c/n)$ (rather than for the giant component $G\subset H$). For $i\in [n]$ let $X_i$ be the event that the vertex $i$ is incident to at least $l$ leaves. It is easy to estimate the expectation of $X=\sum_{i=1}^n X_i$: \begin{eqnarray*} E(X) &=& n \binom{n-1}{ l} p^l (1-p)^{\binom{l}{ 2} + l(n-l)} +O(1)\times n\binom{n}{ l+1} p^{l+1}(1-p)^{(l+1)n}\\ &=& (1+o(1)) \frac{nc^l{\mathrm e}^{-cl}}{l!}\ \to\ \infty. \end{eqnarray*} Also, for $i\not=j$, \begin{eqnarray*} E(X_i\wedge X_j) &=&(1+o(1))\,\binom{n-2}{ l} \binom{n-l-2}{ l}p^{2l} (1-p)^{\binom{2l}{ 2} +2l(n-2l-1)}\\ &=& (1+o(1))\, E(X_i)E(X_j). \end{eqnarray*} The second moment method gives that $X$ is concentrated around its mean. Now, let us reveal the vertex set $A$ of the $2$-core of the whole graph $H$. When we expose the stars demonstrating $X_i=1$ one by one, then for each $i$ the probability of $i\in A$ is $\frac{|A|}n+o(1)$. The sharper results of {\L}uczak~\cite{luczak:91} \comment{Or Pittel~\cite{pittel:90}?} imply that whp the core $C$ of the giant component has size $\Theta(n)$. Hence, whp at least one vertex $i$ with $X_i=1$ belongs to the $V(C)$, giving the required. \section{Random Trees}\label{random} We consider the probabilistic model $\C T(n)$, where a tree $T$ on the vertex set $[n]$ is selected uniformly at random among all $n^{n-2}$ trees. In this section we prove that whp $D(T)$ is close to the maximum degree of $T$. \bth{RandomTree} Let $T\in\C T(n)$. Whp $D(T)=(1+o(1))\Delta(T)=(1+o(1))\frac{\ln n}{\ln\ln n}$. \end{theorem} \newcommand{{\textrm{Var}}}{{\textrm{Var}}} \newcommand{{\textrm{Ch}}}{{\textrm{Ch}}} \newcommand{{\textrm{del}}}{{\textrm{del}}} Let ${\cal F}(n,k)$ be a forest chosen uniformly at random from the family of ${\cal F}_{n,k}$ of all forests with the vertex set $[n]$, which consist of $k$ trees rooted at vertices $1,2,\dots,k$. Note that a random tree $T\in {\cal T}(n)$ can be identified with ${\cal F}(n,1)$. We recall that $|{\cal F}_{n,k}|=kn^{n-k-1}$, see e.g.\ Stanley~\cite[Theorem~5.3.2]{stanley:ec}. We start with the following simple facts on ${\cal F}(n,k)$. \blm{forest} Let $k=k(n)\le \ln^4 n$. \begin{enumerate} \renewcommand{(\roman{enumi})}{(\roman{enumi})} \item The expected number of vertices in all trees of ${\cal F}(n,k)$, except for the largest one, is $O(k\sqrt n)$. \item The probability that ${\cal F}(n,k)$ contains precisely $\ell$, $\ell=0,\dots,k-1$, isolated vertices is given by $(1+O({k^2}/{n})) \binom{k-1}\ell {\mathrm e}^{-\ell}(1-{\mathrm e}^{-1})^{k-\ell-1}$. \item The probability that the roots of ${\cal F}(n,k)$ have more than $k(1+1/\ln n)+2\ln^2 n$ neighbors combined is $o(n^{-3})$. \item The probability that $\ell$ given roots of ${\cal F}(n,k)$ have degree at least $s\ge 4$ each is bounded from above by $(2/(s-1)!)^\ell$ \end{enumerate} \end{lemma} \bpf If $i\le n/2+1$, then the probability that a tree rooted at a vertex $j=1,2,\dots,k$ in the forest ${\cal F}(n,k)$ has precisely $i$ vertices is given by $$\binom {n-k}{i-1} i^{i-2} \frac{(k-1)(n-i)^{n-i-k}}{k n^{n-k-1}}=O(i^{-3/2})\,.$$ Consequently, the expectation of the sum of the orders of all components of ${\cal F}(n,k)$ with at most $n/2+1$ vertices is $O(k \sqrt n)$. In order to see (ii) note that from the generalized inclusion-exclusion principle the stated probability equals \begin{equation}\label{eqf1} \begin{aligned} \sum_{i=\ell}^k&\binom i\ell(-1)^{i-\ell}\binom ki\frac{(k-i)(n-i)^{n-k-1}}{kn^{n-k-1}}\\ =&\Big(1+O\Big(\frac{k^2}{n}\Big)\Big) \sum_{i=\ell}^k\frac{(k-1)!}{\ell!(i-\ell)!(k-1-i)!}(-1)^{i-\ell}{\mathrm e}^{-i}\\ =&\Big(1+O\Big(\frac{k^2}{n}\Big)\Big) \binom{k-1}\ell {\mathrm e}^{-\ell}(1-{\mathrm e}^{-1})^{k-\ell-1}\,. \end{aligned} \end{equation} For the probability that precisely $m$ ($\ge\! k$) vertices of ${\cal F}(n,k)$ are adjacent to the roots, Stirling's formula gives \begin{equation}\label{f1} \binom{n-k}{m}k^m\frac{m\,(n-k)^{n-k-m-1}}{k\,n^{n-k-1}} \le \Big(1+O\Big(\frac{k^2}n\Big)\Big)\Big(\frac{{\mathrm e}^ {1-k/m}k}{m}\Big)^{m}. \end{equation} For every $x$, $0<x<1$, we have $x{\mathrm e}^{1-x}\le {\mathrm e}^{-(1-x)^2/2}$, so the above formula is bounded from above by $\exp(-\frac{(m-k)^2}{2m})$. Since $$\sum_{m\ge k(1+1/\ln n)+2\ln ^2n}\exp\Big(-\frac{(m-k)^2}{2m}\Big)=o(n^{-3})\,,$$ the assertion follows. For $k=1$ the probability that a given root has degree at least $s$ is bounded from above by $$\sum_{t\ge s}\binom{n-1}{t}\frac{t(n-1)^{n-t-2}}{n^{n-2}}\le \sum_{t\ge s}\frac{1}{(t-1)!}\le \frac{2}{(s-1)!}\;.$$ If we fix some $\ell\ge 2$ roots, then if we condition on the vertex sets of the $\ell$ corresponding components, the obtained trees are independent and uniformly distributed, implying the required bound by the above calculation. \qed Using the above result one can estimate the number of vertices of $T\in {\cal T}(n)$ with a prescribed number of pendant neighbors. \blm{vert} Let $X_{\ell,m}$ denote the number of vertices in $T\in {\cal T}(n)$ with precisely $\ell$ neighbors of degree one and $m$ neighbors of degree larger than one. Let $$ A\subseteq\{(\ell,m)\colon\; 0\le \ell\le \ln n, \quad 1\le m\le \ln n \}\,,$$ be a set of pairs of natural numbers and $X_A=\sum_{(\ell,m)\in A} X_{\ell,m}$. Then, the expectation \begin{equation}\label{eqf2} E(X_A)=(1+o(1))\,n\sum_{(\ell,m)\in A} \frac{{\mathrm e}^{-\ell-1}}{\ell!}\frac{(1-{\mathrm e}^{-1})^{m-1}}{(m-1)!} \end{equation} and $E(X_A(X_A-1))=(1+o(1))\,(E(X_A))^2$. \end{lemma} \bpf Using Lemma~\ref{lm:forest}(ii) we get \begin{equation*} E(X_A)=(1+o(1))n\sum_{(\ell,m)\in A}\binom{n-1}{m+\ell}\binom{m+\ell-1}\ell {\mathrm e}^{-\ell} (1-{\mathrm e}^{-1})^{m-1}\frac{(m+\ell)(n-1)^{n-m-\ell-2}}{n^{n-2}} \end{equation*} which gives (\ref{eqf2}). In order to count the expected number of pairs of vertices with prescribed neighborhoods one needs first to choose $\ell+m$ neighbors of a vertex and then compute the expectation of the number of vertices of a given neighborhood in the random forest ${\cal F}(n,\ell+m)$ obtained in this way. However, the largest tree of ${\cal F}(n,\ell+m)$ has the expectation $n-O(\sqrt n \ln n)$ (Lemma~\ref{lm:forest}); one can easily observe that this fact implies that the expected number of vertices with a prescribed neighborhood in ${\cal F}(n,\ell+m)$ is $(1+o(1))\,E(X_A)$, and so $E(X_A(X_A-1))=(1+o(1))\,(E(X_A))^2$. \qed As an easy corollary of the above result we get a lower bound for $D({\cal T}(n))$. \bth{lower} Let $T\in\C T(n)$. Whp $D(T)\ge (1-o(1))\Delta(T)=(1-o(1))\, \frac{\ln n}{\ln \ln n}$. \end{theorem} \bpf Since whp the maximum degree is $(1-o(1)){\ln n}/{\ln\ln n}$, in order to prove the assertion it is enough to show that whp $T$ contains a vertex $v$ with \begin{equation}\label{eqf3} \ell_0=(1-o(1))\, \frac{\ln n}{\ln \ln n} \end{equation} neighbors of degree one; indeed, to characterize such a structure Spoiler needs at least $\ell_0+1$ moves. Using Lemma~\ref{lm:vert}, we infer that the for the number of vertices $X_{\ell}$ of $T$ with exactly $\ell$ neighbors of degree $1$ we have $E(X_\ell)=O({\mathrm e}^{-\ell}n/\ell!)$. Thus, one can choose $\ell_0$ so that (\ref{eqf3}) holds and $E(X_{\ell_0})\to\infty$. Then, due to Lemma~\ref{lm:vert}, ${\textrm{Var}}(X_{\ell_0})=o((E(X_{\ell_0}))^2)$, and Chebyshev's inequality implies that whp $X_{\ell_0}>0$.\qed Let us state another simple consequence of Lemma~\ref{lm:forest} which will be used in our proof of Theorem~\ref{th:RandomTree}. Here and below $N_r(v)$ denotes the $r$-neighborhood of $v$, i.e., the set of all vertices of a graph which are at the distance $r$ from $v$, and $N_{\le r}(v)=\bigcup_{i=0}^r N_i(r)$. \blm{largedegrees} Let $r_0=r_0(n)= \lceil 7 \ln n\rceil $. Then, whp the following holds for every vertex $v$ of $T\in {\cal T}(n)$: \begin{enumerate} \renewcommand{(\roman{enumi})}{(\roman{enumi})} \item $|N_{\le r_0}(v)|\le 10^8 \ln^4n\;,$ \item $N_{\le r_0}(v)$ contains fewer than $\ln n/(\ln\ln n)^2$ vertices of degree larger than $(\ln\ln n)^5$. \end{enumerate} \end{lemma} \bpf For $s\le r_0$ let $W_s=\cup_{i=0}^s N_i(v)$. Note that, conditioned on the structure of the subtree of $T$ induced by $W_s$ for some $s\le r_0$, the forest $T- W_{s-1}$ can be identified with the random forest on $n-|W_{s-1}|$ vertices, rooted at the set $W_s$. Thus, it follows from Lemma~\ref{lm:forest}(iii) that once for some $i$ we have $|N_i(v)|\ge 4 \ln ^3 n$ then $|N_{i+1}(v)|\le |N_i(v)|(1+2/\ln n)$, so that $$|N_{\le r_0}(v)|\le 4 r_0\ln ^3n (1+2/\ln n)^{r_0}\le 10^8 \ln^4n\;.$$ In order to show (ii) note that (i) and Lemma~\ref{lm:forest}(iv) imply that the probability that, for some vertex $v$, at least $\ell=\lfloor \ln n/(\ln\ln n)^2\rfloor$ vertices of $N_{\le r_0}(v)$ have degree larger than $m=(\ln\ln n)^5$ is bounded from above by $$n\binom {\ln^5 n}{\ell}\left(\frac{2}{(m-1)!}\right)^\ell \le n\left(\frac{2{\mathrm e} \ln^5n}{\ell(m-1)!}\right)^\ell\le n{\mathrm e}^{-m\ell}=o(1).$$ \comment{ Here is a small hole: we know that the probability of having at least $>m$ neighbors is at most $2/m!$ but why is the probability that $l$ given vertices each have degree $>m$ is at most $(2/(m-1)!)^l$? Proof: we expose levels one by one. Once we have exposed a level, we allow an adversary to choose any number of active vertices, provided he does not choose more than $\ell$ vertices in total. Then adversary succeeds (all his points have high degree) with probability at most $(2/(m-1)!)^\ell$. } \qed In our further argument we need some more definitions. Let $T$ be a tree and let $v$ be a vertex of $T$. For a vertex $w\in N_r(v)$ let $P_{vw}$ denote the unique path connecting $v$ to $w$ (of length $r$). Let the \emph{check} ${\textrm{Ch}}(v;P_{vw})$ be the binary sequence $b_0\cdots b_r$, in which, for $i=0,\dots, r$, $b_{i}$ is zero (resp.\ 1) if the $i$-th vertex of $P_{vw}$ is adjacent (resp.\ not adjacent) to a vertex of degree one. Finally, the \emph{$r$-checkbook} ${\textrm{Ch}}_r(v)$ is the set $$ {\textrm{Ch}}_r(v)=\{{\textrm{Ch}}(v;P_{vw})\colon w\in N_r\textrm{\ and }P_{vw} \textrm{ is a path of length $r$}\}. $$ Note that a checkbook is not a multiset, i.e., a check from ${\textrm{Ch}}_r(v)$ may correspond to more than one paths $P_{vw}$. Our proof of the upper bound for $D({\cal T}(n))$ is based on the following fact. \bth{checks} Let $r_0=\lceil 7 \ln n\rceil$. Whp for each pair $P_{vw}$, $P_{v'w'}$ of paths of length $r_0$ in $T\in {\cal T}(n)$ which share at most one vertex, the checks ${\textrm{Ch}}(v;P_{vw})$ and ${\textrm{Ch}}(v;P_{v'w'})$ are different. \end{theorem} \bpf Let $C={\textrm{del}}(T)$ denote the tree obtained from $T$ by removing all vertices of degree one. From Lemma~\ref{lm:vert} it follows that whp the tree $C$ has $(1-{\mathrm e}^{-1}-o(1))n$ vertices of which $$ (1+o(1))\,n \sum_{\ell>0} \frac{{\mathrm e}^{-\ell-1}}{\ell!} = (\exp({\mathrm e}^{-1}-1)-{\mathrm e}^{-1}+o(1))\,n$$ vertices have degree one and $$ \alpha n = (1-\exp({\mathrm e}^{-1}-1) +o(1))\, n. $$ vertices have degree greater than one. Moreover, among the set $B$ of $({\mathrm e}^{-1}+o(1))n$ vertices removed from $T$, $$ (1+o(1))n\sum_{l=0}^\infty \ell\frac{{\mathrm e}^{-\ell-1}}{\ell!}=(1+o(1))\exp({\mathrm e}^{-1}-2)n\,$$ were adjacent to vertices which became pendant in $C$. Let $B'$ denote the set of the remaining $$ ({\mathrm e}^{-1}-\exp({\mathrm e}^{-1}-2)+o(1))n=(\rho_0+o(1))n $$ vertices which are adjacent to vertices of degree at least two in $C$. Note that, given $C={\textrm{del}}(T)$, each attachment of vertices from $B\setminus B'$ to pendant vertices of $C$ such that each pendant vertex of $C$ get at least one vertex from $B\setminus B'$, as well as each attachment of vertices from $B'$ to vertices of degree at least two from $C$ is equally likely. Let $P_{vw}$, $P_{v'w'}$, be two paths of length $r_0$ in $T$ which share at most one vertex. Clearly, each vertex of $P_{vw}$, except, maybe, at most two vertices at each of the ends, belong to $C$ and have in it at least two neighbors; the same is true for $P_{v'w'}$. Since $(\rho_0+o(1))n$ vertices from $B'$ are attached to the $\alpha n$ vertices of degree at least two in $C$ at random, the probability that one such vertex gets no attachment is $$ p_0=(1+o(1))\, \left(1-\frac1{\alpha n}\right)^{\rho_0 n}= (1+o(1))\, {\mathrm e}^{-\rho_0/\alpha} = 0.692...+o(1). $$ Therefore, the probability that the checks ${\textrm{Ch}}(v,P_{vw})$ and ${\textrm{Ch}}(w,P_{v'w'})$ are identical is bounded from above by $$ \left(p_0^2+(1-p_0)^2 +o(1)\right)^{r_0}\le {\mathrm e}^{-3\ln n}=o(n^{-2})\,. $$ Since by Lemma~\ref{lm:largedegrees}(i) whp $T$ contains at most $O(n\ln^4 n)$ checks of length $r_0$, the assertion follows. \qed Now, let $r_0=\lceil 7 \ln n\rceil$. We call a tree $T$ on $n$ vertices \emph{typical} if: \begin{itemize} \item for each pair of paths $P_{vw}$, $P_{v'w'}$ of length $r_0$ which share at most one vertex, the checks ${\textrm{Ch}}(v;P_{vw})$, ${\textrm{Ch}}(v;P_{v'w'})$ are different, \item for the maximum degree $\Delta$ of $T$ we have $$\frac{\ln n}{2\ln\ln n}\le \Delta\le \frac{2\ln n}{\ln\ln n} \,,$$ \item $|N_{\le r_0}|\le 10^8\ln ^4 n$, for every vertex $v$, \item for every vertex $v$ at most $\ln n/(\ln\ln n)^2$ vertices of degree larger than $(\ln\ln n)^5$ lie within distance $r_0$ from $v$. \end{itemize} \bth{upper} For a typical tree $T\in\C T(n)$ we have $D(T)\le (1+o(1))\, \Delta$. \end{theorem} \bpf Let $T$ be a typical tree and $T'$ be any other graph which is not isomorphic to $T$. We shall show that then Spoiler can win the Ehrenfeucht game on $T$ and $T'$ in $(1+o(1))\Delta$ moves. Let us call a vertex $v$ of a graph a \emph{yuppie}, if there are two paths $P_{vw}$, $P_{vw'}$ of length $r_0$ starting at $v$ so that $V(P_{vw})\cap V(P_{vw'})=\{v\}$. Note that the set of all yuppies $Y$ spans a subtree in $T$, call it $K$. Our approach is similar to that for the giant component from Section~\ref{giant}. Let us view $K$ as a colored graph where the color of a vertex $x$ is the isomorphism type of the component of $T-(Y\setminus\{x\})$ rooted at $x$. Let $Y'$ be the set of yuppies of $T'$, and let $K'=T'[Y']$. We can assume that Duplicator preserves the subgraphs $K$ and $K'$, for otherwise Spoiler wins in extra $O(\ln \ln n)$ moves. \claim1 Any distinct $v,v'\in K$ can be distinguished (with respect to $G$) in $O(\ln\ln n)$ moves. \bpf[Proof of Claim.] Assume that the $r_0$-checkbooks of $v,v'$ are the same for otherwise Spoiler wins in $\log_2(r_0)+O(1)$ moves. (Please note that the checkbooks are viewed as sets, not as multisets, so the number of moves does not depend on the degrees of $v$ and $v'$.) Take a path $P_{vx}$ of length $r_0$, which shares with $P_{vv'}$ only vertex $v$. Spoiler selects $x$. Let Duplicator reply with $x'$. Assume that ${\textrm{Ch}}(w,P_{vx})={\textrm{Ch}}(v,P_{v'x'})$. The path $P_{v'x'}$ must intersect $P_{vx}$; thus $v\in P_{v'x'}$. Next, Spoiler selects the $P_{vx}$-neighbor $y$ of $v$; Duplicator's reply must be $y'\in P_{v'x'}$. Let $z\in T$ maximize $\mathrm{dist}(v,z)$ on the condition that ${\textrm{Ch}}(z)={\textrm{Ch}}(v)$ and $v$ lies between $y$ and $z$ in $T$. Define the analogous vertex $z'$, replacing $v,y$ in the definition by $v',y'$. We have $\mathrm{dist}(v,z)>\mathrm{dist}(v',z')$. Let Spoiler select $w=z$. If Duplicator's reply $w'$ satisfies ${\textrm{Ch}}(w')\not\cong {\textrm{Ch}}(w)$, then Spoiler quickly wins. Otherwise, $\mathrm{dist}(v,w)> \mathrm{dist}(v',w')$. Moreover, $\mathrm{dist}(v,w)\le 2r_0$ (because their $r_0$-checkbooks are non-empty and equal). Spoiler wins in $\log_2 r_0+O(1)$ extra moves. The claim has been proved.\cqed Similarly to the argument surrounding~\req{Phi}, one can agrue that for every vertex $x\in K$ there is a formula $\Phi_x(v)$ of rank $O(\ln \ln n)$ identifying $x$ (with respect to $T$). Moreover, we can assume that this gives us an isomorphism $\phi:K\to K'$ which is respected by Duplicator. As $T\not\cong T'$, there are two cases to consider. \case1 There is $x\in K$ such that $T_x\not\cong T'_{x'}$, where $x'=\phi(x)$ and $T_{x'}'$ is the component of $T'-(Y' \setminus\{x'\})$ rooted at $x'$. Since each vertex of $T$ is within distance at most $r_0$ from some yuppie, the tree $T_x$ has height at most $r_0$. If $T'_{x'}$ has a path of length greater than $2r_0$ or a cycle, then Spoiler easily wins, so assume that $T'$ is a tree. Now Spoiler should select all vertices of $T_x$ which are of degree larger than $(\ln\ln n)^5$, say $w_1,\dots,w_t$. Since $T$ is typical there are at most $\ln n/(\ln\ln n)^2$ such vertices in $T_v$. Suppose that, in responce to that, Duplicator chooses vertices $w'_1,\dots,w'_s$ in $T'_{x'}$. Then, $T_v\setminus \{w_1,\dots,w_s\}$ splits into a number of trees $F_1, \dots, F_u$, colored accordingly to their adjacencies to the $w_i$'s. Now, for some $i$ the multisets of colored trees adjacent to $w_i$ and $w_i'$ are different. Spoiler can highlight this by using at most $\Delta(T)+1$ moves. Now Spoiler plays inside some $F_i$ the strategy of Theorem~\ref{th:MaxDeg}. Note that $F_i$ has diameter at most $2r_0$ and maximum degree at most $(\ln\ln n)^5$. \case2 $T'$ is not connected. As $K'\cong K$ is connected, there is a component $C'$ of $T'$ without a yuppie. Spoiler chooses an $x'\in C'$. Now, any Duplicator's reply $x$ is within distance $r_0$ from a yuppie, which is not true for $x'$. Spoiler can win in $O(\ln \ln n)$ moves. Consequently, for a typical tree $T$, $$ D(T)\le \Delta(T)+\frac{\ln n}{(\ln\ln n)^2}+O((\ln\ln n)^6)\,, $$ and the assertion follows. \qed \noindent {\it Proof of Theorem~\ref{th:RandomTree}.} Theorem~\ref{th:RandomTree} is an immediate consequence of Theorems~\ref{th:lower} and~\ref{th:upper} and the fact that, due to Lemmas~\ref{lm:forest} and~\ref{lm:largedegrees}, whp a random tree $T\in {\cal T}(n)$ is typical. \qed \section{Restricting Alternations} If Spoiler can win the Ehrenfeucht game, alternating between the graphs $G$ and $G'$ at most $r$ times, then the corresponding sentence has the \emph{alternation number} at most $r$, that is, any chain of nested quantifiers has at most $r$ changes between $\exists$ and $\forall$. (To make this well-defined, we assume that no quantifier is within the range of a negation sign.) Let $D_r(G)$ be the smallest depth of a sentence which defines $G$ and has the alternation number at most $r$. It is not hard to see that $D_r(G)=\max\{D_r(G,G'): G'\not\cong G\}$, where $D_r(G,G')$ may be defined as the smallest $k$ such that Spoiler can win $\mbox{\sc Ehr}_k(G,G')$ with at most $r$ alternations. For small $r$, this is a considerable restriction on the structure of the corresponding formulas, so let us investigate the alternation number given by our strategies. Let $D^{\mathrm{tree}}_r(n,l)$ be the maximum of $D_r(T)$ over all colored trees of order at most $n$ and maximum degree at most $l$. Unfortunately, in Theorem~\ref{th:MaxDeg} we have hardly any control on the number of alternations. However, we can show that alternation number $0$ suffices if we are happy to increase the upper bound by a factor of $2$. \begin{lemma}\label{lem:treetree} Let $T$ and $T'$ be colored trees. Suppose that $T\not\cong T'$, where $\cong$ stands for the isomorphism relation for colored trees, i.e., the underlying (uncolored) trees of $T$ and $T'$ may be isomorphic. Furthermore, assume that $v(T)\ge v(T')$ and denote $n=v(T)$. Assume also that $\Delta(T)\le l$ and let both $l$ and $\ln n/\ln l$ tend to the infinity. Then Spoiler can win the Ehrenfeucht game on $(T,T')$ in at most \beq{D1} (1+o(1)) \frac{l \ln n}{\ln l}. \end{equation} moves playing all time in~$T$. \end{lemma} \bpf In the first move Spoiler selects a median $x\in T$; let $x'$ be Duplicator's reply. If $d(x)>d(x')$, then Spoiler wins in extra $l$ moves, which is negligible when compared to~(\ref{eq:D1}). So, suppose that $d(x')\ge d(x)$. Let $t=d(x)$ and $C_1,\dots,C_t$ be the (rooted) components of $T-x$ indexed so that $v(C_1)\ge v(C_2)\ge\ldots\ge v(C_t)$. Referring to the root of a component we mean the vertex of it which is adjacent to $x$. Spoiler starts selecting, one by one, the roots of $C_1,C_2,\ldots$. Duplicator is enforced to respond with roots of distinct components of $T'-x'$. Spoiler keeps doing so until the following situation occurs: he selects the root $y$ of a component $C=C'_i$ while Duplicator selects the root $y'$ of a component $C'$ such that $v(C)\ge v(C')$ and $C\not\cong C'$ (as rooted trees). Such a situation really must occur for some $i\le t$ due to the conditions that $v(T)\ge v(T')$, $d(x)\le d(x')$, and $T\not\cong T'$. We claim that if Spoiler selects a vertex $z$ inside $C$, then Duplicator must reply with some $z'\in C'$ for otherwise Spoiler wins in at most $\log_2 n$ moves. Indeed, suppose $z'\not\in C'$. Spoiler selects $z_1$ which is a middle point of the $yz$-path. Whatever the reply $z_1'$ is, the $z'z_1'$-path or $z_1'y'$-path contains the vertex $x'$. Suppose it is the $z'z_1$-path. Then Spoiler halves the $zz_1$-path. In at most $\log_2n$ times he wins. Thus making $i+1\le t+1\le l+1$ steps, we have reduced the game to two non-isomorphic (rooted) trees, $C$ and $C'$, with $v(C)\le \min(\frac1i,\frac12)\, v(T)$. In the game on $(C,C')$ Spoiler applies the same strategy recursively. Two ending conditions are possible: the root of $C$ has strictly larger degree than the root of $C'$ and Duplicator violates a color, the adjacency, or the equality relation. It is easy to argue, cf.\ the proof of Theorem~\ref{th:MaxDeg}, that the worst case for us is when we have $i=(1+o(1))\, l$ all the time, which gives the required bound~(\ref{eq:D1}). \qed \bth{DT0} Let both $l$ and $\ln n/\ln l$ tend to the infinity. Then \beq{} D^{\mathrm{tree}}_0(n,l)\le (1+o(1)) \frac{l \ln n}{\ln l}. \end{equation} \end{theorem} \bpf Let $T$ be a tree of order $n$ and maximum degree at most $l$ and let $G\not\cong T$. If $\Delta(T)\ne\Delta(G)$ then Spoiler wins the Ehrenfeucht game on $(T,G)$ in at most $l+2$ moves playing in the graph of the larger degree. We will therefore assume that $T$ and $G$ have the same maximum degree not exceeding~$l$. \case1 $G$ contains a cycle of length no more than $n+1$. Spoiler plays in $G$ proceeding as in the last paragraph of the proof of Lemma~\ref{lm:Tree}. \case2 $G$ is connected and has no cycle of length up to $n+1$. If $v(G)\le n$, then $G$ must be a tree. Lemma \ref{lem:treetree} applies. Let us assume $v(G)>n$. Let $A$ be a set of $n+1$ vertices spanning a connected subgraph in $G$. This subgraph must be a tree. Spoiler plays in $G$ staying all time within $A$. Lemma~\ref{lem:treetree} applies. \case3 $G$ is disconnected and has no cycle of length up to $n+1$. We can assume that every component $H$ of $G$ is a tree for otherwise Spoiler plays the game on $(T,H)$ staying in $H$, using the strategy described above. Suppose first that $G$ has a tree component $H$ such that $H\not\cong T$ and $v(H)\ge n$. If $v(H)=n$, let $T'=H$. Otherwise let $T'$ be a subtree of $H$ on $n+1$ vertices. Spoiler plays the game on $(T,T')$ staying in $T'$ and applying the strategy of Lemma \ref{lem:treetree} (with $T$ and $T'$ interchanged and perhaps with $n+1$ in place of~$n$). Suppose next that all components of $G$ are trees of order less than~$n$. In the first move Spoiler selects a median $x$ of $T$. Let Duplicator respond with a vertex $x'$ in a component $T'$ of $G$. If in the sequel Duplicator makes a move outside $T'$, then Spoiler wins by Lemma~\ref{lm:path}. As long as Duplicator stays in $T'$, Spoiler follows the strategy of Lemma \ref{lem:treetree}. Finally, it remains to consider the case that $G$ has a component $T'$ isomorphic to~$T$. Spoiler plays in $G$. In the first move he selects a vertex $x'$ outside $T'$. Let $x$ denote Duplicator's response in $T$. Starting from the second move Spoiler plays the game on $(T,T')$ according to Lemma \ref{lem:treetree}, where $x$ is considered colored in a color absent in $T'$. Our description of Spoiler's strategy is complete.\qed It is not clear what the asymptotics of $D^{\mathrm{tree}}_0(n,l)$ is. We could not even rule out the possibility that $D^{\mathrm{tree}}_0(n,l)=(\frac12+o(1))\, \frac{l\ln n}{\ln l}$. \comment{Also, it would be interesting to know $D^{\mathrm{tree}}_i$ for other small $i$, such as $i=1$ or $i=2$.} The similar method shows that $D^{\mathrm{tree}}_0(n,l)=\Theta(\ln n)$ if $l\ge 2$ is constant and $D^{\mathrm{tree}}_0(n,l)=\Theta(l)$ if $\frac{\ln n}{\ln l}=O(1)$ but the exact asymptotics seems difficult to compute. Using these results, one can show that the upper bounds in Theorems~\ref{th:RandomTree} and~\ref{th:giant} apply to $D_1(G)$, that is, there are strategies for Spoiler requiring at most one alternation. It is not clear whether 0 alternations is possible here. One of a few places that seem to require an alternation is establishing that $\phi$ is a bijection: Spoiler may be forced to start in one of the graphs, while later (for example, when showing that $T_x\not\cong T'_{x'}$) he may need to swap graphs. \end{document} \end{document}

\begin{document} \title{Approximating Turaev-Viro 3-manifold invariants \\is universal for quantum computation} \author{Gorjan Alagic} \affiliation{Institute for Quantum Computing, University of Waterloo} \author{Stephen P. Jordan} \affiliation{Institute for Quantum Information, California Institute of Technology} \author{Robert K\"onig} \affiliation{Institute for Quantum Information, California Institute of Technology} \author{Ben W. Reichardt} \affiliation{Institute for Quantum Computing, University of Waterloo} \date{\today} \begin{abstract} The Turaev-Viro invariants are scalar topological invariants of compact, orientable $3$-manifolds. We give a quantum algorithm for additively approximating Turaev-Viro invariants of a manifold presented by a Heegaard splitting. The algorithm is motivated by the relationship between topological quantum computers and $(2+1)$-D topological quantum field theories. Its accuracy is shown to be nontrivial, as the same algorithm, after efficient classical preprocessing, can solve any problem efficiently decidable by a quantum computer. Thus approximating certain Turaev-Viro invariants of manifolds presented by Heegaard splittings is a universal problem for quantum computation. This establishes a novel relation between the task of distinguishing non-homeomorphic $3$-manifolds and the power of a general quantum computer. \end{abstract} \pacs{03.67.-a, 05.30.Pr, 03.65.Vf} \maketitle The topological quantum computer is among the most striking examples of known relationships between topology and physics. In such a computer, quantum information is encoded in a quantum medium on a $2$-D surface, whose topology determines the ground space degeneracy. Surface deformations implement encoded operations. Topological quantum computers are universal, i.e., can implement arbitrary quantum circuits. It is natural to try to identify the topological origin of this computational power. One answer is that the power stems from the underlying $(2+1)$-D topological quantum field theory (TQFT)~\cite{FreedmanKitaevWang00}. The TQFT assigns a Hilbert space ${\mathcal H}_\Sigma$ to a $2$-D surface $\Sigma$, and a unitary map $U(f): {\mathcal H}_\Sigma \rightarrow {\mathcal H}_{\Sigma'}$ to every diffeomorphism $f: \Sigma \rightarrow \Sigma'$, subject to a number of axioms~\cite{Walker91}. However, this answer is not fully satisfactory; the definition of a TQFT is involved, and uses mathematics that appears in similar form in the theory of quantum computation. A second answer, arising in~\cite{AharonovJonesLandau06Jones, AharonovArad06Jones, GarneroneMarzuoliRasetti06Jones, WocjanYard07Jones}, is that quantum computers' power comes from their ability to approximate the evaluation, at certain points, of the Jones polynomial of the plat closure of a braid. Here we give an alternative topological description of the power of quantum computers, in terms of the Turaev-Viro $3$-manifold invariants. Observe that restricting TQFTs to closed manifolds results in scalar invariants. We show that approximating certain such invariants is equivalent to performing general quantum computations. That is, we give an efficient quantum algorithm for additively approximating Turaev-Viro invariants, and conversely we show that for any problem decidable in bounded-error, quantum polynomial time (BQP), there is an efficient classical reduction to the Turaev-Viro invariant approximation problem. The classical procedure outputs the description of a $3$-manifold whose certain Turaev-Viro invariant is either large or small depending on whether the original BQP algorithm outputs $1$ or $0$. Turaev and Viro~\cite{TuraevViro92} defined a family of invariants for compact, orientable $3$-manifolds. The original definition parameterized the invariants by the quantum groups $\SU(2)_k$, for $k \in \mathbb{N}$, but it was extended by Barrett and Westbury~\cite{BarrettWestbury96invariants} to give an invariant for any spherical tensor category~${\mathcal C}$. Any compact $3$-manifold $M$ is homeomorphic to a finite collection of tetrahedra glued along their faces~\cite{Moise52}. Beginning with such a triangulation, assign a certain rank-six tensor $F$ to each tetrahedron and a certain gluing tensor $d$ to every edge. The invariant $\TV_{\mathcal C}(M)$ is the contraction of the tensor network, which can be written out as \begin{equation} \label{eq:triang_invar} \hspace{-0.5ex} \TV_{\mathcal C}(M) = {\mathcal D}^{-2 \abs V}\hspace{-0.2ex} \sum_{\textrm{labelings}} \prod_{\text{edges}} d_e \prod_{\text{tetrahedra}} \frac{F^{ijm}_{kln}}{\sqrt{d_m d_n}} \enspace \end{equation} if~${\mathcal C}$ is multiplicity-free. Here, the sum is over edge labelings of the triangulation by particles from the category~${\mathcal C}$. The index $e$ on $d$ is the label of an edge, while the indices $i, \ldots, n$ are the labels of the six edges involved in a tetrahedron, ordered and oriented following certain rules. The fusion tensor $F$, the quantum dimensions~$d$ and the total quantum dimension ${\mathcal D}$ are parameters of~${\mathcal C}$. $\abs V$ is the number of vertices of the triangulation. The topological invariance of $\TV_{\mathcal C}(M)$ follows from the fact that any two triangulations of $M$ can be related by a finite sequence of local Pachner moves~\cite{Pachner91}, under which the above quantity is invariant. In this paper we consider multiplicity-free unitary modular tensor categories, which include the $\SU(2)_k$ case, but are not as general as spherical tensor categories. To formulate a BQP-complete problem~\cite{WocjanZhang06BQPcomplete} of estimating the Turaev-Viro invariant, we require a presentation of $3$-manifolds known as a Heegaard splitting. Consider two genus-$g$ handlebodies (e.g., the solid torus for $g=1$). They can be glued together, to give a $3$-manifold, using a self-homeomorphism of the genus-$g$ surface. The set of orientation-preserving self-homeomorphisms modulo those isotopic to the identity form the mapping class group $\MCG(g)$ of the surface. It is an infinite group generated by the $3g-1$ Dehn twists illustrated in \figref{fig:Dehn}. A Heegaard splitting thus consists of a natural number $g$ and an element $x \in \MCG(g)$, defining a manifold $M(g, x)$. Every compact, orientable $3$-manifold can be obtained in this way, up to homeomorphism. \begin{figure} \caption{A Dehn twist is a $2 \pi$ rotation about a closed curve. The Dehn twists about the $3g-1$ curves shown above generate the full mapping class group of the genus-$g$ surface~\cite{Lickorish64generators}.} \label{fig:Dehn} \end{figure} \begin{theorem} \label{t:TVproblem} For any fixed multiplicity-free unitary modular tensor category ${\mathcal C}$, there is a quantum algorithm that, given $\delta, \epsilon > 0$, $g \in \mathbb{N}$ and a length-$m$ word $x$ in the Dehn-twist generators of $\MCG(g)$ from \figref{fig:Dehn}, runs in time $\mathrm{poly}(g, m, \log 1/\delta, 1/\epsilon)$ and, except with probability at most $\delta$, outputs an approximation of $\TV_{\mathcal C}(M(g, x))$ to within~$\pm {\mathcal D}^{2(g-1)}\, \epsilon$. Conversely, for ${\mathcal C}$ the category associated to $\SU(2)_k$ or $\SO(3)_k$ for $k\geq 3$ such that $k+2$~is prime, it is BQP-hard to decide whether ${\mathcal D}^{2(1-g)} \, \TV_{\mathcal C}(M(g, x))$ is greater than $2/3$ or less than $1/3$. More precisely, given any quantum circuit $\Upsilon$ of $T$ two-qubit gates acting on $n$ qubits $\ket{0^n}$, with output either $0$ or $1$, one can classically find in polynomial time a word $x = x_1 \ldots x_m$ in the standard Dehn-twist generators of~$\MCG(g)$, with $g = n + 1$ and $m = \mathrm{poly}(T)$, such that \begin{equation} \big\lvert \Pr[\text{$\Upsilon$ outputs $1$}] - {\mathcal D}^{2(1-g)} \, \TV_{\mathcal C}(M(g, x)) \big\rvert < 1/6 \enspace . \end{equation} \end{theorem} The additive approximation error is exponential in~$g$. Complexity-theoretic reasons make it unlikely that quantum computers can efficiently obtain a multiplicative or otherwise presentation-independent error~\cite{Kuperberg09}. In fact, a similar statement to \thmref{t:TVproblem} also holds for approximating the Witten-Reshetikhin-Turaev (WRT) invariants~\cite{Witten89, ReshetikhinTuraev91}. For any $g$, a modular category~${\mathcal C}$ can be used to define a projective representation $\rho_{{\mathcal C}, g}: \MCG(g) \rightarrow \GL({\mathcal H}_{{\mathcal C}, g})$. This representation will be given below. The WRT invariant for a $3$-manifold~$M(g, x)$ is then given by a matrix element \begin{equation} \label{eq:wrtinvariant} \WRT_{\mathcal C}(M(g,x)) = {\mathcal D}^{g-1} \bra{v_{{\mathcal C}, g}} \rho_{{\mathcal C}, g}(x) \ket{v_{{\mathcal C}, g}}\ , \end{equation} where $\ket{v_{{\mathcal C}, g}} \in {\mathcal H}_{{\mathcal C}, g}$ is a certain unit-normalized vector. As the representation is projective,~$\WRT_{\mathcal C}$ is a 3-manifold invariant only up to a multiple of $e^{2\pi i c/24}$ where $c$ is called the central charge. (Eq.~\eqref{eq:wrtinvariant} is the Crane-Kohno-Kontsevich presentation~\cite{Crane91, Kohno92, Kontsevich88} of the WRT invariant, which is more commonly defined in terms of a Dehn surgery presentation of~$M$. Equivalence of these definitions for ${\mathcal C} = \SU(2)_k$ is shown in~\cite{Piunikhin93}; see also~\cite[Sec.~2.4]{Kohno02}.) The fact that Eq.~\eqnref{eq:wrtinvariant} indeed gives an invariant can be established by studying the problem of when two Heegaard splittings~$(g, x)$ and $(g', x')$ describe homeomorphic manifolds. Since taking the connected sum of a manifold $M$ with the $3$-sphere~$S^3$ does not change the manifold, i.e., $M\# S^3\cong M$, the standard Heegaard splitting of~$S^3$ into two genus-one handlebodies allows defining a ``stabilization'' map $(g, x) \mapsto (g+1, \tilde x)$ such that $M(g, x) \cong M(g+1, \tilde x)$. A general theorem of Reidemeister~\cite{Reidemeister33} and Singer~\cite{Singer33} asserts that $M(g, x) \cong M(g', x')$ if and only if $(g,x)$ and $(g',x')$ are equivalent under stabilization and the following algebraic equivalence relation for the case of equal genus~\cite{Funar95} \begin{equation} (g, x) \equiv (g, x') \; \textrm{ if $x = yx'z$ with $y, z \in \MCG^+(g)$} \enspace . \end{equation} Here $\MCG^+(g) \subset \MCG(g)$ is the subgroup of self-homeomorphisms (classes) of the genus-$g$ surface that extend to the genus-$g$ handlebody. Invariance of $\WRT_{\mathcal C}(M(g,x))$ now follows essentially from the fact that $\ket{v_{{\mathcal C}, g}}$ is invariant under the action of~$\MCG^+(g)$. The Turaev-Viro and WRT invariants are related by \begin{equation} \label{eq:tvwrtinvariant} \TV_{\mathcal C}(M) = \abs{\WRT_{\mathcal C}(M)}^2 \end{equation} as shown by Turaev~\cite{Turaev91} and Walker~\cite{Walker91} (see also~\cite{Turaev94book, Roberts95}). In~\cite{KoenigKuperbergReichardt10TVcode}, Eq.~\eqnref{eq:tvwrtinvariant} is discussed in the category-theoretic formalism used here. Identities~\eqnref{eq:wrtinvariant} and~\eqnref{eq:tvwrtinvariant}, together with density and locality properties of the representations~$\rho_{{\mathcal C}, g}$, are the basis of our BQP-completeness proof. Previously, a quantum algorithm for approximating the $\SU(2)_k$ Turaev-Viro and WRT invariants was given by Garnerone \emph{et al.}~\cite{GarneroneMarzuoliRasetti07}, assuming the manifold is specified by Dehn surgery rather than a Heegaard splitting. BQP-hardness of the approximation was left as an open problem. In unpublished work, Bravyi and Kitaev have proven the BQP-completeness of the problem of approximating the $\SU(2)_4$ WRT invariant of $3$-manifolds with boundary~\cite{BravyiKitaev00}, where the manifold is specified using Morse functions. We remark that one can use Arad and Landau's quantum algorithm for approximating tensor network contractions to compute the Turaev-Viro invariant of a triangulated manifold~\cite{AradLandau08tensor}. While this algorithm would run polynomially in the number of tetrahedra, its precision depends on the order in which tensors are contracted and may be trivial. We will only briefly describe the space ${\mathcal H}_{{\mathcal C}, g}$, the representation $\rho_{{\mathcal C}, g} : \MCG(g) \rightarrow \GL({\mathcal H}_{{\mathcal C}, g})$ and the state $\ket{v_{{\mathcal C}, g}} \in {\mathcal H}_{{\mathcal C}, g}$ from Eq.~\eqnref{eq:wrtinvariant}. Details are in~\cite{Crane91, Kohno92, Kontsevich88, KoenigKuperbergReichardt10TVcode}. Let ${\mathcal C}$ be a multiplicity-free unitary modular tensor category. It specifies a set of particles $i$ with quantum dimensions $d_i > 0$, and including a trivial particle~$\vac$. The total quantum dimension is ${\mathcal D} = \sqrt{\sum_i d_i^2}$. ${\mathcal C}$ additionally specifies a particle duality map $i \mapsto i^*$, fusion rules, $F$-symbols $F^{ijm}_{kln}$ and $R$-symbols $R_i^{jk}$. These tensors obey certain identities, such as the pentagon and hexagon equations, which can be found in, e.g., \cite{Preskill98notes, KoenigKuperbergReichardt10TVcode}. Let $g \in \mathbb{N}$, $g \geq 2$. The space ${\mathcal H}_{{\mathcal C}, g}$ can be defined by specifying an orthonormal basis. Decompose the genus-$g$ surface~$\Sigma_g$ into three-punctured spheres (or ``pants'') by cutting along $3g - 3$ noncontractible curves, as illustrated in \figref{fig:dualgraphexamples}. Dual to such a decomposition is a trivalent graph~$\Gamma$. Direct arbitrarily the edges of $\Gamma$. A basis vector $\ket \ell_\Gamma$ is a fusion-consistent labeling of the edges of $\Gamma$ by particles of the category~${\mathcal C}$. Fusion-consistency is defined by the fusion rules, i.e., a set of triples~$(i,j,k)$ that are allowed to meet at every vertex, and particle duality, which switches the direction of an edge, replacing a label~$i$ by the antiparticle~$i^*$. Define the states $\cB_{\Gamma}:=\{\ket{\ell}_\Gamma\}_{\ell}$ to be orthonormal, and their span to be ${\mathcal H}_{{\mathcal C}, g}$. Note that this definition gives a natural encoding of ${\mathcal H}_{{\mathcal C}, g}$ into qudits, with one qudit to store the label of each edge of~$\Gamma$. The directed graph $\Gamma$ can be stored in a classical register. The above definition depends on~$\Gamma$, but alternative pants decompositions simply represent different bases~$\cB_\Gamma$ for the same Hilbert space. To convert between all possible pants decompositions of~$\Sigma_g$ we need two moves, each corresponding to a local unitary operator. \begin{figure} \caption{Three examples of decompositions of the genus-two surface $\Sigma_2$ into three-punctured spheres. In each case, a trivalent adjacency graph of the punctured spheres is shown in red.} \label{fig:dualgraphexamples} \end{figure} The $F$ move relates bases that differ by a ``flip" of a cut between two three-punctured spheres. In the qudit encoding, it is a five-qudit unitary, with four control qudits. Its action is given by \begin{equation} \label{eq:Fmove} \raisebox{-1.3cm}{\includegraphics[scale=.5]{images/fmove1}} = \sum_n F^{i j m}_{k l n} \raisebox{-1.3cm}{\includegraphics[scale=.5]{images/fmove2}} \end{equation} The $S$ move applies when two boundaries of a single three-punctured sphere are connected. It is a two-qudit unitary, with one control qudit, and its action is given by \begin{equation} \label{eq:Smove} \raisebox{-1.3cm}{\includegraphics[scale=.5]{images/smove1}} = \sum_k S^i_{jk} \raisebox{-1.3cm}{\includegraphics[scale=.5]{images/smove2}} \end{equation} Most presentations of modular tensor categories do not explicitly provide values for $S^i_{jk}$. However, as discussed in~\cite{Walker91}, $S^i_{jk}$~can be calculated by the identity \begin{equation} {\mathcal D} S^i_{jk} = \sum_{\substack{l:\,(j,k^*,\ell)\\~\textrm{fusion-consistent}}} \hspace{-0.5cm} F^{i k^* k}_{l j^* j} \frac{d_l}{\sqrt {d_i}} R^{k j^*}_l R^{j k^*}_{l^*} = \raisebox{-.55cm}{\includegraphics[scale=1]{images/Sabc}} \end{equation} (The last expression uses ribbon graph notation.) The action~$\rho_{{\mathcal C}, g}$ of~$\MCG(g)$ on~${\mathcal H}_{{\mathcal C}, g}$ can now be specified by the action of the Dehn-twist generators on basis vectors. For a Dehn twist about a curve $\sigma$, apply a sequence of $F$ and $S$ moves to change into a basis $\cB_\Gamma$, i.e., a pants decomposition of $\Sigma_g$, in which $\sigma$ divides two three-punctured spheres. In such a basis, the Dehn twist acts diagonally: if the edge of~$\Gamma$ crossing $\sigma$ has label~$i$, the twist applies a phase shift of~$R^{ii^*}_0$. To complete the definition of~$\WRT_{\mathcal C}(M(g,x))$ from Eq.~\eqnref{eq:wrtinvariant}, it remains to define the state~$\ket{v_{{\mathcal C}, g}}$. As on the right-hand side of Eq.~\eqnref{eq:Smove}, decompose $\Sigma_g$ with a meridional cut through each handle. Then $\ket{v_{{\mathcal C}, g}}$ is the state in which every edge of~$\Gamma$ is labeled by~$\vac$, the trivial particle. Let us now prove \thmref{t:TVproblem}. Although not obvious from Eq.~\eqnref{eq:triang_invar}, the original tensor-network-contraction-based definition of the Turaev-Viro invariant, \thmref{t:TVproblem} is a straightforward consequence of the definition based on the representation~$\rho_{{\mathcal C}, g}$, and of known density results. The Turaev-Viro and WRT invariants for $M(g, x)$ can be approximated essentially by implementing $\rho_{{\mathcal C}, g}(x)$. The algorithm maintains a classical register storing the graph~$\Gamma$, together with a quantum register containing the current state in~${\mathcal H}_{{\mathcal C}, g}$ in the basis $\cB_\Gamma$. If~${\mathcal C}$ has~$N$ particle types, the algorithm uses an $N$-dimensional qudit for each edge of~$\Gamma$. Then~$\rho_{{\mathcal C}, g}(x_j)$ can be applied by using a sequence of $F$ and $S$ moves, i.e., certain local unitaries, to change to a basis in which $x_j$ acts diagonally. Since $x_j$~is one of the generators from \figref{fig:Dehn}, starting with the graph $\Gamma$ of \figref{fig:encoding} (for which every edge is labeled~$\vac$ in $\ket{v_{{\mathcal C}, g}}$) at most one $F$ and one $S$ move are needed. An estimate to within $\epsilon$ of the desired matrix element $\bra{v_{{\mathcal C}, g}} \rho_{{\mathcal C}, g}(x) \ket{v_{{\mathcal C}, g}}$ can be given, except with probability~$\delta$, using $O(\log (1/\delta) /\epsilon^2)$ Hadamard tests, as in~\cite{AharonovJonesLandau06Jones}. \begin{figure}\label{fig:encoding} \label{fig:local} \end{figure} \def\Upsilon{\Upsilon} To prove BQP-hardness we reduce from the BQP-complete problem of deciding whether $\lvert \bra{0^g} \Upsilon \ket{0^g}\rvert^2$ is larger than $5/6$ or less than $1/6$, given the $g$-qubit quantum circuit~$\Upsilon$~\cite{AharonovJonesLandau06Jones}. Let~${\mathcal C}$ be the modular tensor category associated with~$\SU(2)_k$ or $\SO(3)_k$, with $k \geq 3$ and $k+2$ prime. Given~$\Upsilon$ consisting of $T$ two-qubit gates, our aim is to construct efficiently the Heegaard splitting~$(g,x)$ of a manifold~$M=M(g,x)$ such that ${\mathcal D}^{2(1-g)} \TV_{\mathcal C}(M)$ approximates $\lvert \bra{0^g} \Upsilon \ket{0^g}\rvert^2$. As illustrated in \figref{fig:encoding}, we use one handle of a genus-$g$ handlebody to encode each qubit. Such a labeling is fusion-consistent, and the encoding of the initial state $\ket{0^g}$ is exactly $\ket{v_{{\mathcal C}, g}} \in {\mathcal H}_{{\mathcal C}, g}$. As shown in~\cite{FLW_dense, LarsenWang05dense}, for ${\mathcal C} = \SO(3)_k$ the representation $\rho_{{\mathcal C}, g}$ has a dense image, up to phases, in the group of unitaries on~${\mathcal H}_{{\mathcal C},g}$, for $g \geq 2$. By the density for $g = 2$ and the Solovay-Kitaev theorem~\cite{NielsenChuang00}, therefore any two-qubit gate can be approximated in the codespace to precision $1/(6T)$ by applying a $(\log T)^{O(1)}$-long sequence of the five Dehn twists shown in \figref{fig:local}. This holds also for ${\mathcal C} = \SU(2)_k$, as $\SO(3)_k$ is just the restriction of $\SU(2)_k$ to particles with integer spins. Thus we obtain a polynomial-length word~$x = x_1\cdots x_{poly(T)}$ in the Dehn-twist generators whose action approximates $\Upsilon$ on the codespace. Then $\bra{v_{{\mathcal C}, g}} \rho_{{\mathcal C}, g}(M(g, x)) \ket{v_{{\mathcal C}, g}}$ approximates~$\bra{0^g} \Upsilon \ket{0^g}$. This work demonstrates how quantum physics, in the form of TQFTs, can inspire new quantum algorithms for problems based on topology and tensor networks. The approach taken here realizes in a sense the traditional vision of quantum computers as universal simulators for physical systems, but with a different outcome: it provides a purely mathematical problem whose difficulty exactly captures the power of a quantum computer. S.J.\ acknowledges support from the Sherman Fairchild Foundation and NSF grant PHY-0803371. R.K.\ acknowledges support by the Swiss National Science Foundation (SNF) under grant PA00P2-126220. B.R.\ and G.A.\ acknowledge support from NSERC and ARO. Some of this research was conducted at the Kavli Institute for Theoretical Physics, supported by NSF grant PHY05-51164. \end{document}

\begin{document} \title{Geometric phase accumulated in a driven quantum system coupled to a structured environment} \author{Paula I. Villar} \affiliation{Departamento de F\'\i sica {\it Juan Jos\'e Giambiagi}, FCEyN UBA and IFIBA CONICET-UBA, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabell\' on I, 1428 Buenos Aires, Argentina.} \author{Alejandro Soba} \affiliation{ Centro At\'omico Constituyentes, Comisi\'on Nacional de Energ\'\i a At\'omica, Avenida General Paz 1499, San Mart\'\i n, Argentina} \date{\today} \begin{abstract} We study the role of driving in a two-level system evolving under the presence of a structured environment in different regimes. We find that adding a periodical modulation to the two-level system can greatly enhance the survival of the geometric phase for many time periods in an intermediate coupling to the environment. In this regime, where there are some non markovian features characterizing the dynamics, we have noted that adding driving to the system leads to a suppression of non-markovianity altogether, allowing for a smooth dynamical evolution and an enhancement of the robustness condition of the geometric phase. As the model studied herein is the one used to model experimental situations such as hybrid quantum classical systems feasible with current technologies, we positive believe this knowledge can aid the search for physical set-ups that best retain quantum properties under dissipative dynamics. \end{abstract} \maketitle The state of a point like discrete energy level quantum system interacting with a quantum field acquires a geometric phase (GP) that is independent of the state of the field \cite{berry}. The phase depends only on the system's path in parameter space, particularly the flux of some gauge field enclosed by that path. Due to its topological properties and close connection with gauge theories of quantum fields, the GP has recently become a fruitful venue of investigation to infer features of the quantum system. For pure field states, the GP is said to encode information about the number of particles in the field \cite{caridi}. If the field is in a thermal state, the GP encodes information about its temperature, and so it has been used in a proposal to measure the Unruh effect at low accelerations \cite{martinez1}. Furthermore, in \cite{martinez2}, it has been proposed as a high precision thermometer in order to infer the temperature of two atoms interacting with a known hot source and an unknown temperature cold cavity. In this context, the study of the GP in open quantum systems has been a subject of investigation lately. The definition of the geometric phase for non-unitary evolution was first stated in \cite{Tong}. This definition has been used to measure the corrections of the GP in a non-unitary evolution \cite{prl} and to explain the noise effects in the observation of the GP in a superconducting qubit \cite{leek,pra}. The geometric phase of a two-level system under the influence of an external environment has been studied in a wide variety of scenarios \cite{papers}. It has further been used to track traces of quantum friction in an experimentally viable scheme of a neutral particle traveling at constant velocity in front of a dielectric plate \cite{nature} and in a very simplistic analytical model of an atom coupled to a scalar quantum field \cite{epl}. The coupling of the quantum system to the environment is described by the spectral density function. If the system couples to all modes of the environment in an equal way the spectrum of the reservoir is flat. If, otherwise, the spectral density function strongly varies with the frequency of the environmental oscillators, the environment is said to be structured. In this type of environment the memory effects induce a feedback of information from the environment into the system. They are therefore called non-markovian \cite{breuer}. Numerous works have investigated the presence of non-markovianity in a variety of scenarios in quantum open systems so as to determine whether non-markovianity is a useful resource for quantum technologies. It has been studied how the presence of a driving field affects the non-markovian features of a quantum open system. For instance, studies which assessed the effectiveness of optimal control methods \cite{Zhu,Krotov} in open quantum system evolutions showed that non-markovianity allowed for an improved controllability \cite{Schmidt, Reich, Triana}. Likewise, the non-markovian effects were associated to the reduction of efficiency in dynamical decoupling schemes \cite{addis} and accounted for corrections to the GP acquired \cite{pra1,Luo,Oh}. In this work, we investigate to what extent external driving acting solely on the system can increase non-markovianity (and therefore modify the geometric phase) with respect to the undriven case. To this end, we consider a two-level system described by a time-periodic Hamiltonian interacting with a structured environment. It has been recently shown that the driving has a peculiar effect on the non-markovian character of the system dynamics: it can generate a large enhancement of the degree of non-markovianity with respect to the static case for a weak coupling between the system and environment \cite{Poggi}. The importance of the driven two-state model is especially pronounced in quantum computation and quantum technologies, where one or more driven qubits constitute the basic building block of quantum logic gates \cite{nielsen}. Geometric quantum computation exploits GPs to implement universal sets of one-qubit and two-qubit gates, whose realization finds versatile platforms in systems of trapped atoms \cite{Duan}, quantum dots \cite{solinas} and superconducting circuit-QED \cite{Faoro}. Different implementations of qubits for quantum logic gates are subjected to different types of environmental noise, i.e., to different environmental spectra. Since the model studied herein can be implemented in these experimental contexts, using real or artificial atoms, it is important to unveil the time behavior of the qubit geometric phase for driven systems. We shall only focus on weak or intermediate coupling since we try to track traces of the geometric phase, which is literally destroyed under a strong influence of the environment. This means that while there are non-markovian effects that induce a correction to the unitary GP, the system maintains its purity for several cycles, which allows the GP to be observed. It is important to note that if the noise effects induced on the system are of considerable magnitude, the coherence terms of the quantum system are rapidly destroyed and the GP literally disappears \cite{papers}. This knowledge can aid the search for physical set-ups that best retain quantum properties under dissipative dynamics. This paper is structured as follows. In Sec. \ref{modelo} we present the model consisting of a two-level system described by a time-periodic Hamiltonian interacting with a structured environment. In Sec. \ref{dinamica}, we numerically solve the dynamics of the system for different regimes through the hierarchy method beyond the rotating wave approximation. In Sec. \ref{fase} we compute the geometric phase for a two-level driven system and analyze its deviation from the unitary geometric phase under different regimes. Since we want to track traces of the geometric phase, which is literally destroyed under a strong influence of the environment, we shall restrict our study to two situations: (A) weakly coupling and (B) intermediate coupling. Therein, we analyze the robustness condition of the geometric phase acquired by the driven two level system and the best scenarios for its experimental detection. Finally, in Sec. \ref{conclusiones}, we summarize the results and present conclusions. \section{The Model} \label{modelo} We shall consider a two-level system described by a time-periodic Hamiltonian interacting with an environment. The total Hamiltonian which describes this model reads (we set $\hbar=1$ from here on) \begin{equation} H= \bar{\omega}_0 (t) \sigma_+\sigma_- + \sigma_x \sum_k (g_k b_k + g_k^*b_k^{\dagger}) + \sum_k \bar{\omega}_k b_k^{\dagger} b_k, \end{equation} where $\sigma_{\pm}= \sigma_x \pm i \sigma_y$ (with $\sigma_{\alpha}$ ($\alpha=x,y,z$) the Pauli matrices) and $b_k$, $b_k^{\dagger}$ the annihilation and creation operators corresponding to the $k-$th mode of the bath. The coupling constant is $g_k$ and $\bar{\omega}_0(t)$ is the time-dependent energy difference between the states $|0\rangle$ and $|1\rangle$ of the two-level system. We shall assume it has the following form: \begin{equation} \bar{\omega}_0(t)= \bar{\Omega} + \bar{\Delta} \cos({\bar {\omega}_D} t). \end{equation} The exact dynamics of the system in the interaction picture has been derived in \cite{Tanimura}. If the qubit and the bath are initially in a separable state, i.e. $\rho(0)=\rho_s(0)\otimes \rho_B$, the formal solution is: \begin{eqnarray} \tilde{\rho}_S(t) &=& {\cal T} \exp\bigg(-\int_0^t dt_2\int_0^{t_2} dt_1 \tilde{\sigma_x}^\times (t_2) \\ && [C^R(t_2-t_1)\tilde{\sigma_x}^{\times}(t_1) + i C^I(t_2-t_1)\tilde{\sigma_x}^{\circ}(t_1)] \bigg), \nonumber \label{rhoexacta} \end{eqnarray} where ${\cal T}$ implies the chronological time-ordering operator and $\tilde {o}$, denotes the expression of the operator $o$ in the interaction picture. We have further introduced the following notation $A ^{\times} B=[A,B]= AB-BA$ and $A^{\circ}B= \{A,B\}= A B+ B A$. $C^R(t_2-t_1)$ and $C^I(t_2-t_1)$ are the real and imaginary parts of the bath time-correlation function, defined as \begin{eqnarray} C(t_2-t_1) &\equiv& \langle B(t_2) B(t_1) \rangle = {\rm Tr}[B(t_2)B(t_1)\rho_B] \nonumber \\ &=& \int_0^{\infty} d\omega J(\omega) e^{-i \omega (t_2-t_1)} \end{eqnarray} and \begin{equation} B(t)=\sum_k \bigg(g_k b_k \exp(-i \omega_k t) + g_k^*b_k^{\dagger} \exp(i \omega_k t)\bigg). \nonumber \end{equation} Eq.(\ref{rhoexacta}) is difficult to solve directly. An effective method for obtaining a solution has been developed by defining a set of hierarchy equations \cite{Tanimura,Tanimura2,Sun}. The key condition in deriving the hierarchy equations is that the correlation function can be decomposed into a sum of exponential functions of time. At finite temperatures, the system-bath coupling can be described by the Drude spectrum, however, if we consider qubit devices, they are generally prepared in nearly zero temperatures. Then we shall consider a Lorentz type spectral density $J(\omega)$, \begin{equation} J(\omega)= \frac{{\bar{\gamma}_0}}{2 \pi} \frac{\lambda^2}{(\omega- \bar{\Omega})^2 + \lambda^2}, \end{equation} and the hierarchy method can also be applied \cite{Sun2}. As has been stated in \cite{Poggi}, this method can be used if i) the initial state of the system plus bath is separable, ii) the interaction Hamiltonian is bilinear, and iii) if the environmental correlation function can be cast in multi-exponential form. In this case, $\bar{\gamma_0}$ is the coupling strength between the system and the bath and $\lambda$ characterizes the broadening of the spectral peak, which is connected to the bath correlation time $\tau_c= \lambda^{-1}$. The relaxation time scale at which the state of the system changes is determined by $\tau_r=\bar{\gamma_0}^{-1}$. At zero-temperature, if we consider the bath in a vacuum state, the correlation function can be expressed as \begin{equation} C(t_2-t_1)= \frac{\lambda \bar{\gamma_0}}{2} \exp([-(\lambda + i \bar{\Omega})|t_2-t_1|]) \end{equation} which is the exponential form required for the hierarchy method. The advantage of solving the dynamics of the system by this method is that we can take an insight of different regimes of the dynamics. For example, in the limiting case $\bar{\gamma_0} \ll \lambda$, i.e. $\tau_c \ll \tau_r$, we have a flat spectrum and the correlation tends to $C(t_2-t_1)\rightarrow \bar{\gamma_0} \delta (t_2-t_1)$. This is the so called markovian limit. Therefore, we can study the full spectrum of behavior by solving the hierarchy method, which can be expressed as \begin{widetext} \begin{equation} \frac{d}{d\tau}\rho_{\vec{n}}(\tau)= -(i H_s[\tau]^{\times}+ \vec{n}.\vec{\nu})\rho_{\vec{n}}(\tau) - i \sum_{k=1}^2 \sigma_x^{\times}\rho_{\vec{n}+\vec{e}_k}(\tau)- i\frac{{\gamma_0}}{2} \sum_{k=1}^2 n_k [\sigma_x^{\times} + (-1)^k \sigma_x^{\circ}] \rho_{\vec{n}-\vec{e}_k}(\tau), \label{hierarchy} \end{equation} \end{widetext} where we have defined dimensionless parameters variables $\tau=\lambda t$ and $x=\bar{x}/\lambda$ where $x$ is any parameter with units of energy in the model described. The subscript $\vec{n}=(n_1,n_2)$ with integers numbers $n_{1(2)} \geq 0$, and $\rho_S(t) \equiv \rho_{(0,0)} (t)$. This means that the ``physical" solution is encoded in $\rho_{(0,0)} (t)$ and all other $\rho_{\vec{n}}(\tau)$ with $\vec{n} \neq (0,0)$ are auxiliary operators implemented for the sake of computation. We have defined the vector $\vec{\nu}=(\nu_1,\nu_2)=(1-i \Omega, 1 + i \Omega)$. The hierarchy equations are a set of linear differential equations, that can be solved by using a Runge Kutta rutine. For numerical computations, the hierarchy equations must be truncated for large $\vec{n}$. The hierarchy terminator equation is similar to that of Eq.(\ref{hierarchy}) for the term $\vec{N}$, and the corresponding term related to $\rho_{\vec{N}+\vec{e}_k}$ is dropped \cite{Tanimura}. The numerical results in this paper have been all tested and converged, using a maximum value of $\vec{N}=(25,25)$. We shall take advantage of this model, whose non-markovian properties has been studied in \cite{Poggi}, and set the scenario to study the corrections to the GP for a driven two-level system. \section{Environmentally induced Dynamics} \label{dinamica} We begin by studying the environmentally induced dynamics by considering a qubit with no driving at all ($\Delta=0$). In this case, we must consider a qubit and a dipolar coupling to the cavity mode, for example. This means, that the dynamics of the system contemplates decoherence and dissipation as well as variation of the population numbers (in contrast to the spin boson model). The density matrix for this case has a formal expression as \begin{equation} \rho_s(\tau)= \bigg( \begin{matrix} \rho_{11}|G(\tau)|^2 & \rho_{12} G(\tau) \\ \rho_{21} G^*(\tau) & 1-\rho_{11}|G(\tau)|^2 \end{matrix} \label{rho} \bigg) \end{equation} where $G(\tau)$ is a single-complex valued function that characterizes the dynamics of the system. We herein do not write its explicit form since we shall solve the problem numerically through the hierarchy approach. Decoherence time $\tau_D$ is mostly known as the timescale at which the quantum interferences are suppressed. This is formally true for a purely dephasing process where noise only affects the off-diagonal terms of the reduced density matrix. However, Eq.(\ref{rho}) describes a process where populations and off-diagonal terms are both affected by the presence of noise. Qualitatively, decoherence can be thought of as the deviation of probabilities measurements from the ideal intended outcome. Therefore decoherence can be understood as fluctuations in the Bloch vector $\vec{R}$ induced by noise. In a wider sense, we will represent decoherence as the change of $|\vec{R}(\tau)|$ in time, starting from $|R(0)| = 1$ for the initial pure state, and decreasing as long as the quantum state loses purity. The contributions of the bath to the dynamics of the system, including both dissipation and Lamb shift, are fully contained in the hierarchy equation. In Fig. \ref{Fig1} we present the absolute value of the Bloch vector of the state system $R(\tau)=\sqrt{x(\tau)^2+y(\tau)^2+z(\tau)^2}$ as a function of time measured in natural cycles $\omega_0 \tau= N 2\pi $ for different values of $\gamma_0$. In this case, we can note that the trajectory differs substantially from the unitary one, meaning the system's dynamics is affected by the noise effects. In the case the unitary dynamics is considered, $\gamma_0=0$ and $R=1$ for all times. \begin{figure} \caption{(Color online) We plot the loss of the quantum state purity $R(\tau)$ as function of time $N$($\omega_0 \tau= N 2\pi $ number of cycles). We can see that as the coupling constant with the bath increases for a fixed value of $\lambda=0.01$ ($\gamma_0=\bar{\gamma}_0/\lambda$), the dynamical behavior is modified. Orange dashed line is $\gamma_0=0.3$ and dot dashed purple line for $\gamma_0=0.1$ represent situations of $\tau_r > \tau_c$. Dot-dashed brown line is for $\gamma_0=1$, blue solid line for $\gamma_0=0.7$ are situations of $\tau_r \sim \tau_c$. In the inset we show different solutions for $\gamma_0=0.1$ by varying the order of truncation. We can state that from $\vec{N}=(10,10)$ we can obtain a converged positive reduced matrix $\rho(\tau)$. Parameters used: $\Delta=0$, $\vec{N}=(25,25)$, $\Omega=20$.} \label{Fig1} \end{figure} We can notice that the dynamical behavior is modified as the coupling constant $\gamma_0$ is increased. It is interesting to see the interplay between time and $\gamma_0$: a stronger bath can initially produce less damage on the dynamics but has a stronger effect in the renormalization of the frequency. A weak-coupling has a more ``adiabatic" modification of the dynamics in an equal period of time. In Fig. \ref{Fig1}, we have set $\lambda$ fixed. As $\gamma_0$ increases, the relaxation time $\tau_r$ of the system decreases and $\tau_r \sim \tau_c$. The presence of oscillations in the Bloch vector $R(\tau)$ for short times, as $\gamma_0$ becomes similar to $\lambda$ indicates non markovian dynamics induced by the reservoir memory and describing the feedback of information and/or energy from the reservoir into the system \cite{addis}. We can see that as long as $\bar{\gamma_0}/\lambda < 1$, the systems exhibits a markovian dynamics (orange-dashed line, purple dot-dashed line and red solid line). On the other side, if $\bar{\gamma_0}/\lambda \geq 1$, there are non markovian features in the system's dynamics. We can notice that as long as $\bar{\gamma_0} \leq \lambda$ and $\lambda \ll 1$, the behavior remains similar to that of $\lambda \rightarrow 0$ (markovian since $\tau_c \rightarrow \infty$). However, as $\lambda$ increases, the environmentally-induced dynamics is considerably modified, introducing oscillations again. So, with this kind of environment we can simulate different regimes by the solely selection of the $\bar{\gamma_0}$ and $\lambda$ parameters. In the inset of Fig. \ref{Fig1}, we show a simulation for different truncations of the system of equations for $\gamma_0=1$. We show that by setting the order of truncation in 25, we already obtain a converged positive reduced matrix $\rho(\tau)$. \begin{figure} \caption{(Color online) We plot different dynamics for a fixed value of $\lambda$, but different values of the $\bar{\gamma}_0$ parameter. The left column is for $\bar{\gamma_0}/\lambda <1$ and the right for $\bar{\gamma_0}/ \lambda \geq 1$. On top we show the behavior of $\rho_{11}(t)$ and the absolute value of $\rho_{12}(t)$ in each case. We can see some time revivals on the right reduced matrix elements. The lower plots show the trajectory ($\vec{R}=(x,y,z)$) of the two level system in the Bloch sphere. Parameters used: $\Delta=0$, $\vec{n}=(25,25)$, $\Omega=20$, $\tau_c=100$. } \label{Fig3} \end{figure} In Fig. \ref{Fig3}, we compare the dynamics of two different environmental situations: the left column is for $\bar{\gamma_0}/\lambda <1$, and the right one for $\bar{\gamma_0}/\lambda \geq 1$, both evolutions are simulated for fixed $\bar{\Omega}$ and zero driving ($\Delta=0$). In this example, we can see that when $\tau_c < \tau_r$, the system presents a markovian evolution. On the other hand, if $\tau_c > \tau_r$, non-markovian effects can be seen, for example by accelerating the transition between quantum states and revivals for longer times. For initial short times, the spontaneous decay of the atom can not only be suppressed or enhanced, but also partly reversed, when non-markovian oscillations induced by reservoir memory effects are present. As has been shown, by choosing the right set of parameters, we can simulate different type of environments and obtain the corresponding dynamics beyond the rotating wave approximation. \section{Correction to the Geometric Phase}\label{fase} In this section, we shall compute the geometric phase for the central spin and analyze its deviation from the unitary geometric phase for a two-level driven system. A proper generalization of the geometric phase for unitary evolution to a non-unitary evolution is crucial for practical implementations of geometric quantum computation. In \cite{Tong}, a quantum kinematic approach was proposed and the geometric phase (GP) for a mixed state under non-unitary evolution has been defined as \begin{eqnarray} \Phi & = & {\rm arg}\{\sum_k \sqrt{ \varepsilon_k (0) \varepsilon_k (T)} \langle\Psi_k(0)|\Psi_k(T)\rangle \nonumber \\ & & \times e^{-\int_0^{T} dt \langle\Psi_k| \frac{\partial}{\partial t}| {\Psi_k}\rangle}\}, \label{fasegeo} \end{eqnarray} where $\varepsilon_k(t)$ are the eigenvalues and $|\Psi_k\rangle$ the eigenstates of the reduced density matrix $\rho_{\rm s}$ (obtained after tracing over the reservoir degrees of freedom). In the last definition, $T$ denotes a time after the total system completes a cyclic evolution when it is isolated from the environment. Taking the effect of the environment into account, the system no longer undergoes a cyclic evolution. However, we will consider a quasi cyclic path ${\cal P}: T~\epsilon~[0,\tau_S]$ with $\tau_S=2 \pi/\omega_0$ ($\omega_0$ the system's dimensionless frequency). When the system is open, the original GP $\phi_u$, i.e. the one that would have been obtained if the system had been closed, is modified. This means, in a general case, the phase is $ \phi_g=\phi_u+ \delta \phi$, where $\delta \phi$ depends on the kind of environment coupled to the main system \cite{papers, zanardi}. For a spin-1/2 particle in $SU(2)$, the unitary GP is known to be $\phi_u= \pi(1+\cos(\theta_0))$. It is worth noticing that the proposed GP is gauge invariant and leads to the well known results when the evolution is unitary. As this method can be used when the initial state of the whole system is separable, we shall start by assuming $\rho(0)= \rho_s(0) \otimes \rho_{\cal E}(0)$. The initial state of the quantum system is supposed to be a pure state of the form: \begin{equation} |\Psi (0) \rangle=\cos(\theta_0/2) |0 \rangle + \sin(\theta_0/2) |1 \rangle . \nonumber\end{equation} We shall solve the master equation and then, compute the geometric phase acquired by the quantum system. If the environment is strong, then the unitary evolution is destroyed in a decoherence time $\tau_D$. Otherwise, we can imagine an scenario where the effect of the environment is not so drastic. In the following, we shall focus on how driving can affect (or even benefit) the measurement of the geometric phase under different regimes, both weakly coupling and intermediate coupling. In particular, we shall investigate to what extent external driving acting solely on the system can correct the geometric phase with respect to the undriven or unitary case. \subsection{Geometric phase under a weakly coupling} \begin{figure} \caption{(Color online) Comparison between the accumulated geometric phase $\phi_g$ for the unitary case (asterisk blue dot-dashed line) and a weak coupled environment in a markovian regime $\bar{\gamma_0 }\ll \lambda$ (circled red dot-dashed line) (corresponding to a similar situation to that of the left column of Fig. \ref{Fig3}). Parameters used: $\gamma_0=0.01$, $\Omega=20$, $\theta_0=\pi/4$, $\Delta=0$ and $\omega_D=0$.} \label{Fig4} \end{figure} The dynamics of the driven two-level system comprises of three different dynamical effects, occurring each at a different timescale. Dissipation and decoherence occur at the relaxation timescale $\tau_r$, non-markovian memory effects occur at times shorter or similar to the reservoir correlation timescale $\tau_c$ \cite{addis}. Finally, nonsecular terms cause oscillations in a timescale of the system $\tau_S=(\Omega^2+\Delta^2)^{-1/2}$. Generally, this nonsecular terms can be neglected when $\tau_c \ll \tau_S $. We shall consider the secular regime, by assuming $\tau_S \ll \tau_c $, and in the markovian regime $\tau_S \ll \tau_c \ll \tau_r$. As we are dealing with a structured environment, we shall start by studying a weakly coupled system, which leads to a markovian regime (i.e $\gamma_0<\lambda$). Firstly, we shall compare a two level undriven ($\Delta=0$) evolution to an unitary one in order to see how different the open evolution is and decide whether the geometric phase can be measured in such scenario. Hence, in Fig. \ref{Fig4}, we show the total geometric phase accumulated for the non-unitary (red circled line) and unitary (blue asterisk line) evolution as time evolves, being the number of cycles $N=\tau/\tau_S$. Therein, it is possible to see that initially the geometric phases are similar, with an estimate error of $2.5\%$ for 5 cycles and $10\%$ for 15 cycles when $\gamma_0=0.01$. As time evolves, the difference among both lines increases as expected, since for long times the loss of purity of the system would be considerable. \begin{figure} \caption{(Color online) We study the effect of adding $\Delta$ by computing the geometric phase accumulated in time: $\Delta=0$ black dotted line, $\Delta=0.3$ turquoise cross line, $\Delta=0.5$ blue triangle line and $\Delta=1$ circle orange line for $\gamma_0=0.1$. For $\gamma_0=0.01$ gray squares ($\Delta=1$) and magenta solid line ($\Delta=0$) are very similar, while the blue asterisk line is the unitary geometric phase for reference. In the inset, we show the $\phi_g/\phi_u$ for $\gamma_0=0.01$ with different values of $\Delta$. Parameters used: $\Omega=20$, $\theta_0=\pi/4$ and $\omega_D=0$.} \label{Fig5} \end{figure} In Fig. \ref{Fig5}, we show the geometric phase acquired when adding detuning frequencies to the two level system compared to the case when $\Delta \neq 0$, for different environments, say $\gamma_0=0.1$ and $\gamma_0=0.01$. When the coupling to the environment is very weak, the corrections to the geometric phase acquired are very small and one can expect to obtain very similar results to the unitary geometric phase for few cycles of evolution. Evolutions with $\Delta \neq 0$ are very similar to $\Delta =0$ if we compared the $\gamma_0=0.01$ results in Fig. \ref{Fig5} and Fig. \ref{Fig4}. In this case, the dominant correction to the geometric phase is given by the interaction with the environment, parametrized by the value of $\gamma_0$. However, for larger values of $\gamma_0$ ($\gamma_0=0.1$ but still weakly coupled), evolutions with bigger values of $\Delta$ acquire a geometric phase of considerable difference for long time evolutions. For the first few cycles, the geometric phases acquired are all very similar. As time evolves, different features as the magnitude of the coupling to the environment and the system's frequency (with $\Delta$ involved) have impact on the dynamics and therefore, in the geometric phase acquired. In the inset of Fig. \ref{Fig5}, we plot the normalized geometric phase ($\phi_g/\phi_u$) for $\gamma_0=0.01$. We can see the $\Delta=0$ geometric phase represented by a magenta solid line, $\Delta=0.3$ turquoise cross line, $\Delta=0.5$ orange circled dotted line, $\Delta=1$ gray circled line, $\Delta=3$ blue asterisk line and $\Delta=5$ cross red line. The distance from unity becomes relevant as the number of cycles increases. As expected, if $\Delta$ is added to the system, then the geometric phase acquired is different from that with $\Delta=0$, modifying the system's timescale involved and enhancing non-markovian effects as reported in \cite{Poggi}. This can be a severe experimental problem to overcome. However, for low-values of $\Delta$ considered here, the addition of a tunnel frequency does not considerably affect the geometric phase, obtaining $\phi_g/\phi_u \sim 1$ for many evolution cycles in a weakly coupled regime. \begin{figure} \caption{(Color online) We include driving in the model and compute the geometric phase acquired $\phi_g$. Blue asterisk line correspond to $\omega_D=0.1$ and $\Delta=3$. Magenta line is for $\Delta=0$, red dotted line is for $\Delta=5$ and $\omega_D=0.3$, the green circled solid line for $\Delta=2$ and $\omega_D=0.3$ and the orange circled dot-dashed line for $\Delta=1$ and $\omega_D=0.5$. Black dotted unitary geometric phase is included for a reference. Low-frequency driving corrects the geometric phase accumulated for short times. Parameters used: $\gamma_0=0.01$, $\Omega=20$, $\theta_0=\pi/4$.} \label{Fig6} \end{figure} We shall therefore study the interplay of adding driving to the two level system. In particular, we shall focus on the effect of driving when considering the possibility of measuring the geometric phase acquired by the two-state particle. In Fig. \ref{Fig6}, we show the geometric phase acquired when low-frequency driving is added: blue asterisk line correspond to $\Delta=3$ and $\omega_D=0.1$. Magenta line is for $\Delta=0$, red dotted line is for $\Delta=5$ and $\omega_D=0.3$, the green circled solid line for $\Delta=2$ and $\omega_D=0.3$ and the orange circled dot-dashed line for $\Delta=1$ and $\omega_D=0.5$. Black dotted unitary geometric phase is included for a reference. In the zoom plot we show the geometric phase acquired for $\Delta=3$ and $\omega_D=0$ (static) and compared it to $\Delta=3$ and $\omega_D=0.1$ (low-frequency field). We can see that the driven system acquires a geometric phase closer to the unitary one for longer periods of time, still the difference is very small. In the main plot of Fig. \ref{Fig6}, we note that other driven systems are closer to the unitary geometric phase for ten periods as well. Therefore, there are some set of parameters for which driving ``preserves purity". The geometric phase acquired is more similar to the unitary geometric phase acquired when there is low frequency driving added for low values of $\Delta$. This fact can easily be observed in the inset plot, where the lines with asterisks and circles are closer to the unitary one (black solid line) than the corresponding static ones. \begin{center} \begin{figure} \caption{(Color online) $\phi_g/\phi_u$ for different values of $\Delta$ and $\omega_D$ for (a) $N=4$ and (b) $N=8$ under weakly coupling. For short time evolution there is a wider set of parameters that yield $\phi_g/\phi_u \sim 1$. As time evolves, the set of parameters becomes smaller. Parameters used: $\gamma_0=0.01$, $\Omega=20$, $\theta_0=\pi/4$.} \label{Fig7} \end{figure} \end{center} In Fig. \ref{Fig7}, we further explore this result by representing the normalized geometric phase $\phi_g/\phi_u$ as function of $\Delta$ and $\omega_D$, for two time evolutions: (a) $N=4$ and (b) $N=8$. It is easy to note that for short times, several model's parameters yield a $\phi_g/\phi_u \sim 1$. This can be understood because, as explained in Fig. \ref{Fig5}, the main contribution of the correction to geometric phase is given by the magnitude of the coupling between the environment and the system (say, if we assume $\phi_g=\phi_u + \delta \phi $ when $\gamma_0=0$, $\delta \phi=0$ and the geometric phase obtained is the unitary geometric phase $\phi_u$). However, as time evolves the intrinsic dynamic of each set of values ($\Delta, \omega_D$) will gain more importance. This type of behavior in the correction to the geometric phase has been observed in other studies, yielding that for short time the main correction derives from the fact that the environment is present and the system performs an ``open" evolution (and only markovian effects where taken into account) \cite{nature,ludmila}. As time elapses, the values of $\omega_D$ that preserve the unitary of the GP are less. For example, it can be seen in Figs. \ref{Fig6} and \ref{Fig7}, that $\Delta=5$ and $\omega_D=0.1$ renders a value $\phi_g$ closer to $\phi_u$ than $\Delta=5$ and $\omega_D=0.3$. Likewise, adding a very low frequency driving and small detuning frequencies for short time evolutions renders a geometric phase similar to the unitary geometric phase, which leads to a good scenario of measuring the geometric phase in structured environments. It has been shown in \cite{Poggi} that $\omega_D/\Delta <1$ increases the degree of non markovianity (for a small coupling), and particularly, non markovianity increases with $\gamma_0$ and decreases with $\Delta$ for a given environment (fixed $\gamma_0$). We are not strictly in the regime reported in \cite{Poggi}, since we are studying the situation for different evolving times. However, we must say that if we want to maintain the markovian regime, we should only add low detuning frequencies (if any at all), because by adding detuning frequencies and driving frequencies we shall be modifying considerably the dynamics of the system and comparison with the undriven situation will be useless. However, we can still note that when adding a low-frequency driving, geometric phases acquired are very similar to the non-driven isolated geometric phases for bigger values of $\Delta$. This fact agrees with the result obtained in \cite{lofranco}, where authors state that for a small qubit classical field coupling, a non-resonant control ($\Delta \neq 0 $) is more convenient to stabilize the geometric phase of the open qubit. The use of driven systems can help the measurement of geometric phases under some set of parameters. This knowledge can aid the search for physical set-ups that best retain quantum properties under dissipative dynamics. As can be inferred for the different simulations done, for weak coupling, the better scenario would be to have a small detuning frequency and very low frequency driving field, so as to maintain the smoothness of a markovian evolution and acquire a geometric phase similar to the unitary geometric phase. \begin{figure*} \caption{(Color online) Top: Bloch vector ($R(\tau)$) temporal evolution for different set of model parameters. On the right corner we show some matrix elements: population $\rho_{11}(\tau)$ and the absolute value of the off-diagonal term $\rho_{12}(\tau)$ for three different set of parameters. Bottom: Trajectories in the Bloch sphere ($\vec{R}=(x,y,z)$) for three sets of parameters with $\gamma_0/\lambda \geq 1 $: in magenta, $\omega_D=0=\Delta$; orange for $\omega_D=0$ and $\Delta=5$ and light-blue for $\omega_D=4$ and $\Delta=7$. In all cases $\tau_c=100$ and $\Omega=20$.} \label{Fig8} \end{figure*} \subsection{Geometric phase under an intermediate coupling} \begin{figure} \caption{(Color online) Geometric phase accumulated $\phi_g$ as number of periods evolved for different set of parameters. Colors represent the parameters: black dotted line is the unitary geometric phase; the black squared line if for a markovian evolution as described above; in magenta non-markovian evolution for $\omega_D=0=\Delta$; dot-dashed diamond orange for $\Delta=5$ and $\omega_D=5$; light-blue asterisk line for $\Delta=7$ and $\omega_D=4$. Parameters used: $\gamma_0=1$, $\Omega=20$, $\theta_0=\pi/4$.} \label{Fig9} \end{figure} In the above section we showed the geometric phase acquired by the two-level driven system in a weakly coupled structured environment. The above selection of parameters rendered a markovian situation where one could still find evidence of a quasi-cyclic evolution, since the degradation of the pure state was done slowly and there were no revivals. In the following, we shall show what happens if the parameters are chosen so as to simulate a non-markovian environment by considering $\bar{\gamma_0}/\lambda> 0.25$, as it has been said in \cite{Poggi}. This situation can model for example a two-level emitter with a transition frequency driven by an external classical field of frequency $\omega_D$ embedded in a zero temperature reservoir formed by the quantized modes of a high-Q cavity. In such a case, the evolution is wildly modified and one can find revivals after a given number of periods. This shall help us to understand the role of driving in this type of environments. If we set parameters so as to see non-markovian behavior, then we must mention that finding tracks of the geometric phase can be much more difficult. In Fig. \ref{Fig8}, we show different scenarios by setting the model parameters. On top of Fig. \ref{Fig8}, we show the temporal evolution of the Bloch vector ($|R(\tau)|$) for different driven frequencies with $\bar{\gamma_0}$ and fixed $\lambda$. As can be seen, this type of environment starts to exhibit non-markovian environment though revivals are small in amplitude: the magenta line represents $\Delta=0.0=\omega_D$; the dot-dashed magenta line is for $\Delta=0.5$ and $\omega_D=0$. The orange line is for $\Delta=5$; while the red dot-dashed line is for $\Delta=5$ but $\omega_D=0.1$. The dotted green line represents $\Delta=5$ and $\omega_D=5$ and dot-dashed cyan line $\Delta=7$ and $\omega_D=4$. We have also included a markovian evolution just for reference (black dotted line for $\gamma_0=0.001$). We can easily note that the amount of driving changes considerably the evolution of the initial quantum state. On the top right corner we show the populations probability for different lines: magenta dashed line, represents the $\Delta=0.0=\omega_D$, the orange dotted lined $\Delta=5$, and the cyan dashed line $\omega_D=4$ and $\Delta=7$. We can see that by adding a frequency $\Delta$ and a driving frequency $\omega_D$, revivals disappear, recuperating the opportunity to track traces of a geometric phase. This fact can be easily observed in the Bloch sphere. At the bottom of Fig. \ref{Fig8}, we represent the trajectory ($\vec{R}=(x(\tau),y(\tau),z(\tau)$) in the Bloch sphere of the initial state of the three different sets of parameters for the same number of cycles evolved. We can see that the transition among states is done in a short time for the magenta line. The revivals stimulate the exploration of the south pole of the Bloch sphere, for another period of time until it finally decays. In such an evolution, one can only achieve a geometric phase during the revivals and compare it to the one the system would have acquired if it has started at that latitude of the Bloch sphere. In the case of the orange line, transition among states is delayed by the frequency change of the system's period $\tau_S=2 \pi/(\Omega+\Delta)$. In this case, the geometric phase can be measured for very short initial periods. Finally, for the cyan curve we can observe that the evolution remains ``frozen" at a latitude for almost 3 cycles before continuing the transition among states. \begin{figure} \caption{(Color online) Geometric phase accumulated $\phi_t/\phi_g$ in the ($\Delta$, $\omega_D$) plane for different number of periods evolved in a non markovian environment: $N=2$, $N=3$, $N=4$ and $N=5$. Parameters used: $\gamma_0=1$, $\Omega=20$, $\theta_0=\pi/4$. } \label{Fig10} \end{figure} We can therefore compute the geometric phase for these different situations in order to see if it is possible to track traces of an accumulated geometric phase during the evolutions. In Fig. \ref{Fig9} we show the geometric phase accumulated for different set of parameters (as those considered in Fig. \ref{Fig8}). The colors of the lines in Fig. \ref{Fig9} correspond to the same values of Fig. \ref{Fig8}. The magenta line (with dots) is the temporal evolution of an initial state under a structured environment in a non-markovian regime with $\Delta=0$. In this case after 4 periods, the evolution presents some revivals after having made a transition from the upper to the lower state (therefore revivals are done in the south pole sphere). This is easily understood with the information given in Fig. \ref{Fig8} where we see that transition is done at very short times. Therefore, the geometric phase acquired for $\Delta=0$ is very different to that the system would have acquired in a markovian regime (black squared solid line) or an isolated evolution (black dotted line). However, in Fig. \ref{Fig9}, we also present the geometric phase for driven systems under non-markovian regime. The diamond orange line represents a driven case of $\omega_D=5$ and $\Delta=5$. In such situation, we see that the evolution of the system initially recovers some ``unitarity", acquiring a geometric phase very similar to that of the unitary case. Finally, after some periods, it makes a transition and the evolution explores the south pole sphere. Finally, the light-blue asterisk line for $\omega_D=4$ and $\Delta=7$, acquires a geometric phase similar to the markovian one for longer time periods. In this last driven case, we see that adding driving has a relevant consequence: the geometric phase acquired is closer to the one acquired under a markovian evolution, and therefore closer to the unitary one for a short number of periods evolved ($\sim N=10$). For smaller time periods, we see that adding driving preserves the geometric phase: in all cases showed, the geometric phase is recovered compared to the case when $\Delta=0$. Finally, in Fig. \ref{Fig10} we show a general scenario of the situation described above for $\phi_g/\phi_u$ at different times: $N=2$, $N=3$, $N=4$ and $N=5$. We effectively notice regions of the $\omega_D-\Delta$ space where the accumulated geometric phase $\phi_g/\phi_u$ remains close to one, meaning that the geometric phase acquired is close to the unitary one. There are regions where $\phi_g$ departs enormously from $\phi_u$. As this situation exhibits non-markovian effects as revivals, it is not that easy to find a general rule so as to when it is more convenient to measure the geometric phase. However, we must say that there are some situations where driving enhances the ``robustness" condition of the geometric phase when $\Delta$ delays the revivals and $\omega_D$ is small. We can see that for some particular situations, the addition of a frequency $\Delta$ and driving $\omega_D$ becomes a useful scenario so as to get control of the geometric phase. These situations deal with a smoothening of the revivals as shown in Fig. \ref{Fig8} for the cyan curve. In \cite{Poggi}, authors show that there is a large region, corresponding to $\omega_D/\Delta \sim {\cal O}(1)$ where non-markovianity is suppressed altogether for an intermediate coupling. They even state that in a strong coupling regime ($\gamma_0>1$), the driving is unable to increase the degree of non-markovianity, contrary to what one can expect when adding driving to the system. On this aspect, authors in \cite{lofranco} state that intense classical fields strongly reduce non-markovianity of the system. To prevent this, they state that the larger the coupling, the higher the values of detuning required in order to maintain a given degree of non-markovianity when dealing with hybrid quantum-classical systems. Herein, we assume an intermediate coupling $\gamma_0=1$, where there are some parameters $\omega_D/\Delta \sim {\cal O}(1)$ that verify a suppression of revivals and assure a smooth evolution and an acquisition of a geometric phase more similar to the unitary one. This fact, in addition to some other features related to the initial quantum state explained below, contribute to a better understanding of driven systems and should be taken into consideration when designing experimental set-ups to measure geometric phases. \subsubsection{Dependence on $\rho_s(0)$} In this section we shall study the dependence upon the initial state of the quantum system. As explained above, we consider an initial pure state of the form $ |\Psi (0) \rangle=\cos(\theta_0/2) |0 \rangle + \sin(\theta_0/2) |1 \rangle,$ with $0\leq \theta_0 \leq \pi/2 $. This determines the initial values of the reduced density matrix $\rho_{11}(0)=\cos(\theta_0)^2$ and $\rho_{12}(0)=1/2 \sin(2 \theta_0)$. In the manuscript, we have always started with an initial $\theta_0=\pi/4$, so as to consider an initial average state (where the geometric phase is more ``stable"). In the following, we shall study how decoherence affects different initial states of the two-level system. We shall use the change in time of the absolute value of $|R(\tau)|=R(\tau)$ as a measure of decoherence. In Fig. \ref{fig1ap}, we show $R(\tau)$ as function of time: (a) is for a weakly coupling while (b) for an intermediate coupling. The curves start from an angle of $\theta_0=0.15$ radians (near the north pole of the Bloch sphere) to $\theta_0=1.5$ radians (near the Equator). The measurement of the ``robustness" of quantum states is that the loss of purity of the state vector is very small for many cycles. The dependence of this magnitude upon time (measured in cycles) depends on the initial quantum state for the same parameters of the model. As it can be seen there, the state is more affected for smaller initial angles. The purity of the state remains close to unity (isolated case) when the initial state is located near the equator of the Bloch Sphere ($\theta_0 \sim \pi/2$) for both environments. This means an initial state of the form $|+\rangle=\cos(\pi/4)|0\rangle + \sin(\pi/4)|1\rangle$. This can be understand by noting that the Interaction Hamiltonian is proportional to $\sigma_x$, which in turn is eigenstate of the Interaction Hamiltonian. As for the experimental detection of the geometric phase, we need to find a compromise between the loss of purity and the area enclosed in the path trajectory. As the natural evolution of the system would be to make a transition to the lower state of the quantum state, we need to ``control" this evolution so as to obtain small variations of the trajectory and still find traces of the geometric phase. The non-markovian evolution is the one that provides the more interesting results and the one that can be used to model experimental situations such as hybrid quantum classical systems feasible with current technologies, so we shall explore in detail the dependence upon the initial angles. \begin{widetext} \begin{center} \begin{figure} \caption{(Color online) Loss of purity $R(\tau)$ as function of time for (a) a weakly coupling ($\gamma_0=0.01$) and (b) an intermediate coupling ($\gamma_0=1$) for different initial angles $\theta_0$. We can see that as $\theta_0$ reaches $\pi/2$ (labeled as 1.5) the curves are closer to the unitary evolution.} \label{fig1ap} \end{figure} \end{center} \end{widetext} \begin{figure} \caption{(Color online) $R(\tau)$ as function of the number of cycles evolved for a non-markovian environment with driving for different values of the initial quantum state ($\theta_0$ labeled in degrees). For reference, we also included the static non-markovian evolution $\omega_D=0=\Delta$ with a black dotted line. Parameters used: $\gamma_0=1$, $\Omega=20$, $\omega_D=4$ and $\Delta=7$.} \label{fig2ap} \end{figure} In Fig. \ref{fig2ap}, we show the loss of purity $R(\tau)$ for a non-markovian evolution with driving considered in the manuscript (included in Fig. \ref{Fig9}) for different initial angles $\theta_0$: the lower initial angle considered is $23.5^\circ$ with a red solid line and a brown solid line for the bigger angle considered $\theta_0=84.5^\circ$. In between, we have considered several angles $\theta_0=35^\circ$, $\theta_0=40^\circ$, $\theta_0=45^\circ$, $\theta_0=50^\circ$, $\theta_0=62^\circ$ and $\theta_0=73^\circ$. For reference, we also included the static non-markovian evolution $\omega_D=0=\Delta$ with a black dotted line. \begin{figure} \caption{(Color online) Geometric phase accumulated normalized ($\phi_g/\phi_u$) by the unitary geometric phase accumulated for several cycles of the evolution under a non-markovian environment for different initial angles. Parameters used: $\gamma_0=1$, $\Omega=20$, $\omega_D=4$ and $\Delta=7$.} \label{fig3ap} \end{figure} We can see that all cases considered are qualitatively similar, however $\theta_0 \sim \pi/4$ is the one that maintains the degree of purity for several cycles. This fact can be fruitfully exploited for the detection of the geometric phase. In Fig. \ref{fig3ap}, we show the geometric phase accumulated normalized by the unitary geometric phase accumulated for several cycles of the evolution under a non-markovian environment for different initial angles. The lower angle considered is $23.5^\circ$ and we can see that the GP acquired is very different to the unitary one ($\phi_g/\phi_u \neq 1$). We considered increasing initial angles up to $45^\circ$ indicated by a magenta solid lined which gives a $\phi_g/\phi_u $ close to 1 for several cycles (in agreement to results shown in Fig. \ref{Fig9}). The next light blue-asterisk line is for $\theta_0=50^\circ$ and shows a similar behavior. Angles continue to increase up to the hexagram blue curve indicating $\theta_0=73^\circ$. Therein, we can see that as the angle increases the difference between $\phi_g$ and $\phi_u$ grows becoming considerable for large angles. That is the reason we believe that in order to experimentally detect the geometric phase we need to only consider the decoherence model of noise but the geometric aspects of $SU(2)$ as well. \section{Conclusions} \label{conclusiones} In this manuscript, we have focused on the hierarchy equations of motion method in order to study the interplay between driving and geometric phases. This method can be used if (i) the initial state of the system plus the bath is separable and (ii) the interaction Hamiltonian is bilinear. It results in an advantageous method since it provides a tool to simulate markovian and non markovian behavior in the structured spectrum. We have therefore studied the dynamics of the system and computed the geometric phase for different environment regimes defined by the relation among the model's parameters. In all cases we have focused on the effect of adding driving to the two-state system. By numerically studying the proposed model for various parameter regimes, we find a remarkable result: the driving can produce a large enhancement of non markovian effects, but only when the coupling between system and environment is small. We have seen for a weakly coupled configuration, when adding a low frequency driving to the quantum system's frequency, the system's dynamic tends to be corrected towards the undriven situation only for very small values of $\omega_D$. This can be understood that by adding a detuning frequency changes considerably the system's timescale and therefore, the geometric phase would be different from the unitary undriven one. More interesting, for a stronger coupling or non markovian regime, that there are some situations where driving enhances the ``robustness" condition of the geometric phase when $\Delta$ delays the revivals and $\omega_D$ is small, particularly when $\omega_D/\Delta \sim {\cal O}(1)$. As stated in the existing bibliography, in strong-coupling regime, on the other hand, the driving is unable to increase the degree of non markovianity. In this manuscript, we have further studied the intermediate coupling, since we try to track traces of the geometric phase, which is literally destroyed under a strong influence of the environment. In this regime, where there are some non markovian features characterizing the dynamics, we have noted a suppression of non markovianity altogether, allowing for a smooth dynamical evolution. We have further shown that for low-frequency driving, the driving fails to increase the degree of non markovianity with respect to the static case, recuperating in some cases a scenario where a geometric phase can still be measured ($\phi_g=\phi_u +\delta \phi$). This knowledge can aid the search for physical set-ups that best retain quantum properties under dissipative dynamics. As we have noticed that the non markovian evolution (with intermediate coupling) is the situation that provides the more interesting results, and further it is the one that can be used to model experimental situations such as hybrid quantum classical systems feasible with current technologies, we have explored in detail the dependence upon the initial angles for a better understanding of the results. We have found that exits a set of more ``stable" initial angles. This means that while there are dissipative and diffusive effects that induce a correction to the unitary GP, the system maintains its purity for several cycles, which allows the GP to be observed. It is important to note that if the noise effects induced on the system are of considerable magnitude, the coherence terms of the quantum system are rapidly destroyed and the GP literally disappears. It has been argued that the observation of GPs should be done in times long enough to obey the adiabatic approximation but short enough to prevent decoherence from deleting all phase information. As the geometric phase accumulates over time, its correction becomes relevant at a relative short timescale, while the system still preserves purity. All the above considerations lead to a scenario where the geometric phase can still be found and it can help us infer features of the quantum system that otherwise might be hidden to us. \\ We acknowledge UBA, CONICET and ANPCyT--Argentina. The authors wish to express their gratitude to the TUPAC cluster, where the calculations of this paper have been carried out. We thank F. Lombardo and P. Poggi for their warmhearted discussions and comments.\\ {} \end{document}

\begin{document} \title[Fully bounded noetherian rings]{Fully bounded noetherian rings and Frobenius extensions} \author{S. Caenepeel} \address{Faculty of Engineering, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium} \email{[email protected]} \urladdr{http://homepages.vub.ac.be/\~{}scaenepe/} \author{T. Gu\'ed\'enon} \address{Faculty of Engineering, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium} \email{[email protected], [email protected]} \thanks{Research supported by the project G.0278.01 ``Construction and applications of non-commutative geometry: from algebra to physics" from FWO Vlaanderen} \subjclass{16W30} \keywords{Frobenius extension, Fully bounded noetherian ring, coring, Hopf algebra action, quasi-projective module} \begin{abstract} Let $i:\ A\to R$ be a ring morphism, and $\chi:\ R\to A$ a right $R$-linear map with $\chi(\chi(r)s)=\chi(rs)$ and $\chi(1_R)=1_A$. If $R$ is a Frobenius $A$-ring, then we can define a trace map ${\rm tr}\,:\ A\to A^R$. If there exists an element of trace 1 in $A$, then $A$ is right FBN if and only if $A^R$ is right FBN and $A$ is right noetherian. The result can be generalized to the case where $R$ is an $I$-Frobenius $A$-ring. We recover results of Garc\'{\i}a and del R\'{\i}o and by D\v{a}sc\v{a}lescu, Kelarev and Torrecillas on actions of group and Hopf algebras on FBN rings as special cases. We also obtain applications to extensions of Frobenius algebras, and to Frobenius corings with a grouplike element. \end{abstract} \maketitle \section*{Introduction} A ring $A$ is called right bounded if every essential right ideal contains a non-zero two-sided ideal. $A$ is right fully bounded noetherian or right FBN if $A$ is noetherian, and $A/P$ is right bounded for every two-sided prime ideal $P$ of $A$. Obviously commutative noetherian rings are right FBN; more generally, noetherian PI-rings and artinian rings are FBN. A series of conjectures in classical ring theory can be proved in the case of rings with the FBN property, we refer to the introduction of \cite{Sorin} for a brief survey.\\ Assume that a finite group $G$ acts on $A$. Garc\'{\i}a and Del R\'{\i}o \cite{Garcia} investigated the relationship between the FBN property for $A$ and its subring of invariants $A^G$. The main result is that, in case $A$ is right noetherian, the right FBN property for $A$ is equivalent to the right FBN property for $A^G$, if there exists an element in $A$ having trace $1$. A similar statement was proved in \cite{Nasta} for rings graded by a finite group $G$. These results can be generalized to Hopf algebra actions (see \cite{Sorin,Guedenon}). \\ We have observed that the methods introduced in \cite{Garcia} can be applied in an apparently completely different situation. Let $S$ be a Frobenius algebra (with Frobenius system $(e=e^1\otimes e^2,\overline{\nu})$) and $j:\ S\to A$ an algebra map, with $A$ a right noetherian ring. If there exists $a\in A$ such that $j(e^1)aj(e^2)=1$, then $A$ is right FBN if and only if $C_S(A)$ is right FBN.\\ In this note, we propose a unified approach to these results, based on the concept of an $A$-ring with a grouplike character, as introduced in \cite{CVW}. Basically, this consists of a ring morphism $i:\ A\to R$, together with a right $A$-linear map $\chi:\ R\to A$ such that the formula $a\hbox{$\leftharpoonup$} r=\chi(ar)$ makes $A$ into a right $R$-module. The subring of invariants is defined as $B=\{b\in A~|~b\chi(r)=\chi(br)\}$. The main result is basically the following: if $R$ is a Frobenius $A$-ring, and $A$ is projective as a right $R$-module, then $A$ is right FBN if and only if $B$ is right FBN and $A$ is right noetherian. The methods of proof are essentially the same as in \cite{Garcia}. If $R$ is a Frobenius $A$-ring, then we can define a trace map ${\rm tr}\,:\ A\to B$, and $A$ is projective (and a fortiori quasi-projective) as a right $R$-module if and only if there exists an element of trace 1. The condition that $R$ is Frobenius can be relaxed in the sense that it suffices that $R$ is Frobenius of the second kind, with respect to a strict Morita context $(A,A,I,J,f,g)$. Then the trace map is a map ${\rm tr}\,:\ J\to B$.\\ The above mentioned results on group and Hopf algebra actions and extensions of Frobenius algebras can be obtained as special cases. We also present an application to Frobenius corings with a grouplike element. \section{Rings with a grouplike character}\selabel{1} Let $A$ be an associative ring with unit. The category of $A$-bimodules ${}_A\mathcal{M}_A$ is a monoidal category, and we can consider algebras in ${}_A\mathcal{M}_A$. Such an algebra $R$ is a ring $R$ together with a ring morphism $i:\ A\to R$. The bimodule structure on $A$ is then given by $arb=i(a)ri(b)$, for all $a,b\in A$ and $r\in R$. A {\sl right grouplike character} on $R$ is a right $A$-linear map $\chi:\ R\to A$ such that \begin{equation}\eqlabel{1.1.0} \chi(\chi(r)s)=\chi(rs)~~{\rm and}~~\chi(1_R)=1_A, \end{equation} for all $r,s\in R$. We then say that $(R,i,\chi)$ is an $A$-ring with a right grouplike character. Right grouplike characters were introduced in \cite{CVW}. The terminology is motivated by the fact that the dual of a coring with a grouplike element is a ring with a grouplike character (see \seref{7}). For all $a\in A$, we have that $$\chi(i(a))=\chi(1_R\cdot a)=\chi(1_R)a=1_Aa=a,$$ so $\chi\circ i={\rm Id}_A$, and $i$ is injective, $\chi$ is surjective. Sometimes we will regard $i$ as an inclusion. $A$ is a right $R$-module, with right $R$-action \begin{equation}\eqlabel{1.1.1} a\hbox{$\leftharpoonup$} r=\chi(ar). \end{equation} $A$ is a cyclic right $R$-module, since $$a=\chi(i(a))=\chi(1_Ai(a))=1_A\hbox{$\rightharpoonup$} i(a),$$ for all $a\in A$. For $M\in \mathcal{M}_R$, the submodule of invariants is defined as $$M^R=\{m\in M~|~mr=m\chi(r),~{\rm for~all~}r\in R\}.$$ Let $$B=A^R=\{b\in A~|~b\chi(r)=\chi(br),~{\rm for~all~}r\in R\}.$$ Then $B$ is a subring of $A$, $M^R$ is a right $B$-module, and we have the invariants functor $(-)^{R}:\ \mathcal{M}_R\to \mathcal{M}_B$. We will now present some elementary properties of $$Q=R^R=\{q\in R~|~qr=q\chi(r),~{\rm for~all~}r\in R\}.$$ \begin{lemma}\lelabel{1.1} Let $(R,i,\chi)$ be an $A$-ring with a right grouplike character. \begin{enumerate} \item $Q$ is a $(R,B)$-subbimodule of $R$; \item $\chi$ restricts to a $B$-bimodule map $\chi:\ Q\to B$; \item if $1_R\in Q$, then $i$ is an isomorphism of rings, with inverse $\chi$. \end{enumerate} \end{lemma} \begin{proof} 1) We refer to \cite[Prop. 2.2]{CVW}.\\ 2) For all $q\in Q$ and $r\in R$, we have $$\chi(q)\chi(r)=\chi(q\chi(r))=\chi(qr)=\chi(\chi(q)r),$$ hence $\chi(q)\in B$. $\chi$ is right $A$-linear, so its restriction to $Q$ is right $B$-linear. For all $q\in Q\subset R$ and $b\in B$, we have, by the definition of $A^R=B$ that $b\chi(q)=\chi(bq)$, so $\chi$ is also left $B$-linear.\\ 3) If $1_R\in Q$, then we have for all $r\in R$ that $$r=1_Rr=1_R\chi(r)=1_Ri(\chi(r))=i(\chi(r)).$$ It follows that $i$ is a left inverse of $\chi$. We have seen above that $i$ is always a right inverse of $\chi$, so it follows that $i$ is an isomorphism. \end{proof} If $M\in \mathcal{M}_R$, then ${\rm Hom}_R(A,M)\in \mathcal{M}_B$, with right $B$-action $(fb)(a)=f(ba)$, for all $b\in B$, $f\in {\rm Hom}_R(A,M)$ and $a\in A$.\\ ${\rm End}_R(A)$ is a $B$-bimodule, with left $B$-action $(bf)(a)=bf(a)$, for all $b\in B$, $f\in {\rm End}_R(A)$ and $a\in A$. \begin{lemma}\lelabel{1.2} Let $(R,i,\chi)$ be an $A$-ring with a right grouplike character, and $M$ a right $R$-module. \begin{enumerate} \item ${\rm Hom}_R(A,M)\cong M^R$ as right $B$-modules; \item ${\rm End}_R(A)\cong B$ as $B$-bimodules and as rings. \end{enumerate} \end{lemma} \begin{proof} 1) For $f\in {\rm Hom}_R(A,M)$ and $r\in R$, we have $$f(1_A)r=f(1_A\hbox{$\leftharpoonup$} r)=f(\chi(r))=f(1_A)\chi(r),$$ so $f(1_A)\in M^R$, and we have a well-defined map $$\phi:\ {\rm Hom}_R(A,M)\to M^R,~~\phi(f)=f(1_A).$$ $\phi$ is right $B$-linear since $$\phi(fb)=(fb)(1_A)=f(b1_A)=f(1_Ab)=f(1_A)b=\phi(f)b.$$ The inverse of $\phi$ is given by the formula $$\phi^{-1}(m)(a)=ma,$$ for all $m\in M^R$ and $a\in A$.\\ 2) If $M=A$, then $\phi$ is also left $B$-linear since $$\phi(bf)=(bf)(1_A)=bf(1_A)=\phi(f)b.$$ \end{proof} \section{Quasi-projective modules}\selabel{2} A right $R$-module $M$ is called {\sl quasi-projective} if the canonical map ${\rm Hom}_R(M,M)\to {\rm Hom}_R(M,M/N)$ is surjective, for every $R$-submodule $N$ of $M$. This means that every right $R$-linear map $f:\ M\to M/N$ factorizes through the canonical projection $p:\ M\to M/N$, that is, there exists a right $R$-linear map $g:\ M\to M$ such that $f=p\circ g$. \begin{proposition}\prlabel{2.1} Let $(R,i,\chi)$ be an $A$-ring with a right grouplike character. The following assertions are equivalent. \begin{enumerate} \item $A$ is quasi-projective as a right $R$-module; \item for every right $R$-submodule $I$ of $A$, and every $a+I\in (A/I)^R$, there exists $b\in B$ such that $b-a\in I$; \item for every right $R$-submodule $I$ of $A$, $(A/I)^R\cong (B+I)/I$. \end{enumerate} \end{proposition} \begin{proof} $\underline{1)\Rightarrow 2)}$. Observe that \begin{equation}\eqlabel{2.1.1} (A/I)^R=\{a+I\in A/I~|~a\chi(r)-\chi(ar)\in I,~{\rm for~all~}r\in R\}. \end{equation} For $a+I\in (A/I)^R$, we have a well-defined right $A$-linear map $$f:\ A\to A/I,~~f(a')=aa'+I.$$ $f$ is right $A$-linear since \begin{eqnarray*} &&\hspace*{-2cm} f(a'\hbox{$\leftharpoonup$} r)=a(a'\hbox{$\leftharpoonup$} r)+I=a\chi(a'r)+I\\ &=&\chi(aa'r)+I=((aa')\hbox{$\leftharpoonup$} r)+I=f(a')\hbox{$\leftharpoonup$} r. \end{eqnarray*} Let $p:\ A\to A/I$ be the canonical projection. Since $A$ is quasi-projective, there exists $g\in {\rm Hom}_R(A,A)$ such that $p\circ g= f$, that is $aa'+I=g(a')+I$ and, in particular, $a+I=g(1_A)+I$, or $g(1_A)-a\in I$. Let us show that $b=g(1_A)\in B$. Indeed, for all $r\in R$, we have \begin{eqnarray*} &&\hspace*{-2cm} \chi(br)-b\chi(r)=\chi(g(1_A)r)-g(1_A)\chi(r) = (g(1_A)\hbox{$\leftharpoonup$} r)-(g(1_A)\hbox{$\leftharpoonup$} (i\circ \chi)(r))\\ &=& g(1_A\hbox{$\leftharpoonup$} r)-g((\chi\circ i\circ \chi)(r)) = g(\chi(r))-g(\chi(r))=0. \end{eqnarray*} $\underline{2)\Rightarrow 3)}$. The map $B\to (A/I)^R$, $b\mapsto b+I$ induces a monomorphism $(B+I)/I\to (A/I)^R$. Condition 2) means precisely that this map is surjective.\\ $\underline{3)\Rightarrow 1)}$. Take a right $R$-linear map $f:\ A\to A/I$, with $I$ a right $R$-submodule of $A$. Then $$\chi(f(1_A)r)=f(1_A)\hbox{$\leftharpoonup$} r=f(1_A\hbox{$\leftharpoonup$} r)= f(\chi(1_Ar))=f(\chi(r))=f(1_A)\chi(r),$$ so $f(1_A)\in (A/I)^R\cong (B+I)/I$. Take $b\in B$ such that $f(1_A)=b+I$, and consider the map $g:\ A\to A$, $g(a)=ba$. $g$ is right $R$-linear since $$g(a\hbox{$\leftharpoonup$} r)=b(a\hbox{$\leftharpoonup$} r)=b\chi(ar)=\chi(bar)=(ba)\hbox{$\leftharpoonup$} r=g(a)r.$$ Finally $$(p\circ g)(a)=p(ba)=ba+I=f(1_A)a=f(a).$$ \end{proof} In \prref{2.1}, we characterize quasi-projectivity of $A$ as a right $R$-module. Projectivity has been characterized in \cite[Prop. 2.4]{CVW}: \begin{proposition}\prlabel{2.2} Let $(R,i,\chi)$ be an $A$-ring with a right grouplike character. The following assertions are equivalent. \begin{enumerate} \item $A$ is projective as a right $R$-module; \item there exists $q\in Q$ such that $\chi(q)=1$. \end{enumerate} We refer to \cite[Prop. 2.4]{CVW} for more equivalent properties. \end{proposition} \begin{proposition}\prlabel{2.3}\cite[4.11]{Albu} Let $R$ be a ring, $M$ a quasi-projective right $R$-module, and $N$ a noetherian right $R$-module. Then ${\rm Hom}_R(M,N)$ is a noetherian right ${\rm End}_R(M)$-module. \end{proposition} \section{$I$-Frobenius rings}\selabel{3} Let $(R,i)$ be an $A$-ring, and $I=(A,A,I,J,f,g)$ a strict Morita context connecting $A$ with itself. We say that $R$ is an $I$-Frobenius $A$-ring if there exist an element $e=e^1\otimes u^1\otimes e^2\in R\otimes_A I\otimes_A R$ (summation understood implicitely) and an $A$-bimodule map $\overline{\nu}:\ R\otimes_A I\to A$ such that the following conditions are satisfied, for all $r\in R$ and $u\in I$: \begin{eqnarray} &&re^1\otimes u^1\otimes e^2=e^1\otimes u^1\otimes e^2r;\eqlabel{3.1.1}\\ && \overline{\nu}(e^1\otimes_A u^1)e^2=1_R;\eqlabel{3.1.2}\\ && e^1\otimes_A u^1\overline{\nu}(e^2\otimes_A u)=r1_R\otimes_A u.\eqlabel{3.1.3} \end{eqnarray} If $I=(A,A,A,A,{\rm id}_A,{\rm id}_A)$, then the notion ``$I$-Frobenius" coincides with the classical Frobenius property. Equivalent definitions are given in \cite[Theorem 2.7]{CDM}.\\ $f:\ I\otimes_A J\to A$ and $g:\ J\otimes_A I\to A$ are $A$-bimodule isomorphisms, and \begin{equation}\eqlabel{3.1.4} f(u\otimes_A v)u'=ug(v\otimes_A u')~~;~~g(v\otimes_A u)v'=vf(u\otimes_A v'), \end{equation} for all $u,u'\in I$ and $v,v'\in J$. We will write $$f^{-1}(1_A)=\sum_i u_i\otimes v_i\in I\otimes_A J.$$ From the fact that $f$ is an $A$-bimodule isomorphism, it follows easily that \begin{equation}\eqlabel{3.1.5} \sum_i au_i\otimes v_i=\sum_i u_i\otimes v_ia, \end{equation} for all $a\in A$. We have the following generalization of \cite[Theorem 2.7]{CVW}. \begin{theorem}\thlabel{3.1} Let $(R,i,\chi)$ be an $I$-Frobenius $A$-ring with a right grouplike character. Then $J$ is an $(R,B)$-bimodule, with left $R$-action \begin{equation}\eqlabel{3.1.6} r\cdot v=\sum_i \overline{\nu}(rg(v\otimes \chi(e^1)u^1)e^2\otimes_A u_i)v_i, \end{equation} and we have an isomorphism $\alpha:\ J\to Q$ of $(R,B)$-bimodules. \end{theorem} \begin{proof} The map $\alpha$ is defined by the formula \begin{equation}\eqlabel{3.1.7} \alpha(v)=g(v\otimes_A\chi(e^1)u^1)e^2, \end{equation} for all $v\in J$. Let us first show that $\alpha(v)\in Q$. For all $r\in R$, we compute \begin{eqnarray*} &&\hspace*{-1cm} \alpha(v)r= g(v\otimes_A\chi(e^1)u^1)e^2r\equal{\equref{3.1.1}} g(v\otimes_A\chi(re^1)u^1)e^2\\ &\equal{\equref{1.1.0}}&g(v\otimes_A\chi(\chi(r)e^1)u^1)e^2 \equal{\equref{3.1.1}}g(v\otimes_A\chi(e^1)u^1)e^2\chi(r)=\alpha(v)\chi(r). \end{eqnarray*} $\alpha$ is right $B$-linear since \begin{eqnarray*} &&\hspace*{-2cm}\alpha(vb)=g(vb\otimes_A\chi(e^1)u^1)e^2= g(v\otimes_Ab\chi(e^1)u^1)e^2\\ &=& g(v\otimes_A\chi(be^1)u^1)e^2 g(v\otimes_A\chi(e^1)u^1)e^2b\equal{\equref{3.1.1}}\alpha(v)b, \end{eqnarray*} for all $b\in B$. The inverse $\beta$ of $\alpha$ is given by the composition $$Q\subset R\rTo^{R\otimes_A f^{-1}}R\otimes_AI\otimes_AJ\rTo^{\overline{\nu}\otimes_AJ} A\otimes_AJ\cong J,$$ or $$\beta(q)=\sum_i\overline{\nu}(q\otimes_A u_i)v_i,$$ for all $q\in Q$. Indeed, we compute for all $q\in Q$ that \begin{eqnarray*} &&\hspace*{-15mm} \alpha(\beta(q))= g(\sum_i\overline{\nu}(q\otimes_A u_i)v_i\otimes_A\chi(e^1)u^1)e^2\\ &=&\sum_ig(\overline{\nu}(q\otimes_A u_i)v_i\chi(e^1)\otimes_Au^1)e^2 \equal{\equref{3.1.5}} \sum_ig(\overline{\nu}(q\otimes_A \chi(e^1)u_i)v_i\otimes_Au^1)e^2\\ &=& \sum_ig(\overline{\nu}(q\chi(e^1)\otimes_A u_i)v_i\otimes_Au^1)e^2 \equal{\equref{3.1.1}} \sum_ig(\overline{\nu}(\chi(e^1)\otimes_A u_i)v_i\otimes_Au^1)e^2q\\ &=& \sum_i\overline{\nu}(\chi(e^1)\otimes_A u_i) g(v_i\otimes_Au^1)e^2q = \sum_i\overline{\nu}(\chi(e^1)\otimes_A u_i g(v_i\otimes_Au^1))e^2q\\ &\equal{\equref{3.1.4}}& \sum_i\overline{\nu}(\chi(e^1)\otimes_A f(u_i \otimes_Av_i)u^1)e^2q = \overline{\nu}(e^1\otimes_A e^1)e^2q=q. \end{eqnarray*} For all $v\in J$, we have that \begin{eqnarray*} &&\hspace*{-15mm} \beta(\alpha(v))= \sum_i\overline{\nu}(g(v\otimes_A\chi(e^1)u^1)e^2\otimes_A u_i)v_i= \sum_ig(v\otimes_A\chi(e^1)u^1)\overline{\nu}(e^2\otimes_A u_i)v_i\\ &=&\sum_ig(v\otimes_A\chi(e^1)u^1\overline{\nu}(e^2\otimes_A u_i))v_i \equal{\equref{3.1.3}} \sum_i g(v\otimes_A \chi(1_R)u_i)v_i\\ &=&\sum_i g(v\otimes_A u_i)v_i \equal{\equref{3.1.4}} \sum_i vf(u_i\otimes_A v_i)=v. \end{eqnarray*} This shows that $\alpha$ is an isomorphism of right $B$-modules. We can transport the left $B$-action on $Q$ to $J$ such that $\alpha$ becomes an $(R,B)$-bimodule map. This yields formula \equref{3.1.6}. \end{proof} The composition $${\rm tr}\,=\chi\circ \alpha:\ J\to Q\to B$$ is a $B$-bimodule map (see \leref{1.1}), and will be called the {\sl trace map}. It is given by the formula \begin{equation}\eqlabel{3.18} {\rm tr}\,(v)=\chi(g(v\otimes_A\chi(e^1)u^1)e^2). \end{equation} Combining \prref{2.2} and \thref{3.1}, we obtain the following result: \begin{proposition}\prlabel{3.2} Let $(R,i,\chi)$ be an $I$-Frobenius $A$-ring with a right grouplike character. The following assertions are equivalent. \begin{enumerate} \item $A$ is projective as a right $R$-module; \item there exists $v\in J$ such that ${\rm tr}\,(v)=1_B$. \end{enumerate} \end{proposition} Now assume that $R$ is Frobenius $A$-ring, that is, $I=A$. Then the above formulas simplify. $e=e^1\otimes e^2\in R\otimes_AR$, $\overline{\nu}:\ R\to A$ is an $A$-bimodule map, and the trace map ${\rm tr}\,:\ A\to B$ is given by $${\rm tr}\,(a)= \chi(a\chi(e^1)e^2).$$ \section{Fully bounded noetherian rings}\selabel{4} We recall some definitions and basic results from \cite{Garcia}. Let $R$ be a ring, and $M,P\in \mathcal{M}_R$. For a subset $X$ of ${\rm Hom}_R(P,M)$, we write $$r_P(X)=\cap\{{\rm Ker}\, f~|~f\in X\}.$$ In particular, for $X\subset M\cong {\rm Hom}_R(R,M)$, we have $$r_R(X)=\{r\in R~|~xr=0\}.$$ $M$ is called {\sl finitely} $P$-{\sl generated} if there exists an epimorphism of right $R$-modules $P^n\to M\to 0$.\\ $M$ is called $P$-{\sl faithful} if ${\rm Hom}_R(P,M')\neq 0$, for every nonzero submodule $M'\subset M$.\\ $R$ is called {\sl right bounded} if every essential right ideal contains a non-zero two-sided ideal. $R$ is called {\sl right fully bounded} if $R/P$ is right bounded, for every two-sided prime ideal $P$ of $R$. A ring $R$ that is right fully bounded and right noetherian is called a {\sl right fully bounded noetherian ring} or a {\sl FBN ring}. Characterizations of right FBN rings are given in \cite[Theorem 1.2]{Garcia}. For later use, we recall one of them. \begin{proposition}\prlabel{4.1} For a ring $R$, the following conditions are equivalent. \begin{enumerate} \item $R$ is right FBN; \item for every finitely generated right $R$-module $M$, there exists a finite subset $F\subset M$ such that $r_R(M)=r_R(F)$. \end{enumerate} \end{proposition} A right $R$-module $P$ is called a {\sl right FBN-module} if it is noetherian and for every finitely generated $P$-faithful right $R$-module $M$, there exists a finite subset $F\subset {\rm Hom}_R(P,M)$ such that $r_P(F)=r_P({\rm Hom}_R(P,M))$. We recall the following properties from \cite{Garcia}. \begin{proposition}\prlabel{4.2} \cite[Theorem 1.7]{Garcia} For a quasi-projective, noetherian right $R$-module $P$, the following assertions are equivalent: \begin{enumerate} \item ${\rm End}_R(P)$ is right FBN; \item $P$ is an FBN right $R$-module. \end{enumerate} \end{proposition} \begin{proposition}\prlabel{4.3} \cite[Corollary 1.8]{Garcia} Let $P$ be a quasi-projective FBN right $R$-module, $Q$ a finitely $P$-generated right $R$-module, and $M$ a finitely generated $Q$-faithful right $R$-module. For every $X\subset {\rm Hom}_R(Q,M)$, there exists a finite subset $F\subset X$ such that $r_Q(X)=r_Q(F)$. \end{proposition} \begin{proposition}\prlabel{4.4} \cite[Corollary 1.9]{Garcia} A right noetherian ring $R$ is right FBN if and only if every finitely generated right $R$-module is FBN. \end{proposition} We can now state the main result of this paper. \begin{theorem}\thlabel{4.5} Let $(R,i,\chi)$ be an $A$-ring with a right grouplike character, and consider the following statements. \begin{enumerate} \item $R\in \mathcal{M}_A$ is finitely generated and $A$ is right FBN; \item $R$ is right FBN and $A$ is right noetherian; \item $B$ is right FBN and $A$ is right noetherian. \end{enumerate} Then $1)\Rightarrow 2)$.\\ If $A$ is quasi-projective as a right $R$-module, then $2)\Rightarrow 3)$.\\ If $A$ is projective as a right $R$-module and $R$ is an $I$-Frobenius $A$-ring for some strict Morita context $I=(A,A,I,J,f,g)$, then $3)\Rightarrow 1)$ and the three conditions are equivalent. \end{theorem} \begin{proof} $1)\Rightarrow 2)$. It follows from \prref{4.4} that $R$ is an FBN right $R$-module. Let $M$ be a finitely generated right $R$-module; then $M$ is also finitely generated as a right $A$-module. We claim that $M$ is an $R$-faithful right $A$-module. Indeed, take a non-zero right $A$-module $M'\subset M$. Since $M'\cong {\rm Hom}_A(A,M')$, there exists a non-zero $f\in {\rm Hom}_A(A,M')$, and the composition $f\circ \chi:\ R\to M'$ is non-zero, since $\chi$ is surjective.\\ Now take $P=R$, $Q=A$ in \prref{4.3}, and consider the subset $M\cong {\rm Hom}_R(R,M)\subset {\rm Hom}_A(R,M)$. It follows that there exists a finite $F\subset M$ such that $r_A(F)=r_A(M)$. It then follows from \prref{4.1} that $R$ is right FBN.\\ $2)\Rightarrow 3)$. $A$ is a finitely generated (even cyclic) right $R$-module, so it follows from \prref{4.4} that $A$ is an FBN right $R$-module. It then follows from \prref{4.2} that ${\rm End}_R(A)\cong B$ is right FBN.\\ $3)\Rightarrow 1)$. We will apply \prref{2.3} with $M=A$ and $N=R$. By assumption, $A$ is quasi-projective as a right $R$-module. Since $R/A$ is $I$-Frobenius, $R$ is finitely generated projective as a right $R$-module. Since $A$ is right noetherian, $R$ is also right noetherian.\\ It follows from \leref{1.2}, \prref{2.3} and \thref{3.1} that ${\rm Hom}_R(A,R)\cong R^R=Q\cong J$ is noetherian as a right module over ${\rm End}_R(A)\cong A^R=B$. It then follows that $J$ is finitely generated as a right $B$-module. Let $\{e_1,\cdots,e_k\}$ be a set of generators of $J$ as a right $B$-module.\\ Recall that we have an $A$-bimodule isomorphism $f:\ I\otimes_A J\to A$. With notation as in \seref{3}, we have, for $a\in A$, $$f^{-1}(a)=\sum_{i=1}^n u_i\otimes_A v_ia\in I\otimes_A J.$$ For every $i$, we can find $b_{i1},\cdots,b_{ik_i}\in B$ such that $$v_ia=\sum_{j=1}^{k_i}e_jb_{ij}.$$ We then easily compute that \begin{eqnarray*} a&=& f\Bigl(\sum_{i=1}^n u_i\otimes_A v_ia\Bigr)= f\Bigl(\sum_{i=1}^n\sum_{j=1}^{k_i} u_i\otimes_A e_jb_{ij}\Bigr)=\ \sum_{i=1}^n\sum_{j=1}^{k_i} f(u_i\otimes_Ae_j)b_{ij}, \end{eqnarray*} and we conclude that $A$ is finitely generated as a right $B$-module.\\ Take $M\in \mathcal{M}_A$ finitely generated. Then $M$ is also finitely generated as a right $B$-module. We now show that $M$ is an $A$-faithful right $B$-module. Let $M'$ be a non-zero right $B$-submodule of $M$, and take $0\neq m'\in M'$. It follows from \prref{3.2} that there exists $v\in J$ such that ${\rm tr}\,(v)=1_B$. The map $f:\ A\to M$, $f(a)=m'{\rm tr}\,(va)$ is right $B$-linear, and different from $0$ since $f(1_A)=m'\neq 0$.\\ Observe now that \begin{itemize} \item $B$ is a quasi-projective FBN right $B$-module; \item $A$ is a finitely $B$-generated right $B$-module; \item $M$ is a finitely generated $A$-faithful right $B$-module. \end{itemize} Applying \prref{4.3} to $M\cong {\rm Hom}_A(A,M)\subset {\rm Hom}_B(A,M)$, we find that there exists a finite subset $F\subset M$ such that $r_A(F)= r_A(M)$. It then follows from \prref{4.1} that $A$ is right FBN. \end{proof} \begin{remark}\relabel{4.6} We do not know whether the implication $3)\Rightarrow 1)$ holds under the weaker assumption that $A\in \mathcal{M}_R$ is quasi-projective. The projectivity is used at the point where we applied \prref{4.3}. \end{remark} \section{Application to Frobenius algebras}\selabel{5} Let $k$ be a commutative ring, and consider two $k$-algebras $A$ and $S$, and an algebra map $j:\ S\to A$. All unadorned tensor products in this Section are over $k$. It is easy to establish that $(R=S^{\rm op}\otimes A, i,\chi)$ with $$i:\ A\to S^{\rm op} \otimes A,~~i(a)=1_S\otimes a,$$ $$\chi:\ S^{\rm op}\otimes A\to A,~~\chi(s\otimes a)=j(s)a$$ is an $A$-ring with a right grouplike character. Also observe that the categories $\mathcal{M}_R$ and ${}_S\mathcal{M}_A$ are isomorphic. For $M\in {}_S\mathcal{M}_A$, we have that $$M^R=\{m\in M~|~sm=mj(s),~{\rm for~all~}s\in S\}=C_S(M).$$ In particular, $B=A^R=C_S(A)$ and $$Q=\{\sum_i s_i\otimes a_i\in S^{\rm op} \otimes A~|~ \sum_i ts_i\otimes a_i=\sum_i s_i\otimes a_ij(t),~{\rm for~all~}t\in S\}.$$ Consequently $A$ is projective as a right $R$-module if and only if there exists $\sum_i s_i\otimes a_i\in Q$ such that $\sum_i j(s_i)a_i=1_A$.\\ From \prref{2.1}, it follows that $A$ is quasi-projective as a right $R$-module if and only if for every $(S,A)$-submodule $I$ of $A$ and $a\in A$ such that $as-sa\in I$, for all $s\in S$, there exists $b\in B$ such that $a-b\in I$.\\ Assume that $S$ is a Frobenius $k$-algebra, with Frobenius system $(e=e^1\otimes e^2,\overline{\nu})$. Then $S^{\rm op}$ is also a Frobenius algebra, with Frobenius system $(e=e^2\otimes e^1,\overline{\nu})$, and $S^{\rm op}\otimes A$ is a Frobenius $A$-ring, with Frobenius system $(E, N)$, with $E=(e^2\otimes 1_A)\otimes_A (e^1\otimes 1_A)$ and $$N:\ S^{\rm op}\otimes A\to A,~~N(s\otimes a)=\overline{\nu}(s)a.$$ We then have the isomorphism $$\alpha:\ A\to Q,~~\alpha(a)=e^1\otimes aj(e^2)$$ and the trace map $${\rm tr}\,:\ A\to B,~~{\rm tr}\,(a)=j(e^1)aj(e^2).$$ $A$ is projective as a right $R$-module if and only if there exists $a\in A$ such that ${\rm tr}\,(a)=1$. \begin{corollary}\colabel{5.1} Let $S$ be a Frobenius algebra over a commutative ring $k$, and $j:\ S\to A$ an algebra map. Furthermore, assume that there exists $a\in A$ such that ${\rm tr}\,(a)=1$. Then the following assertions are equivalent: \begin{enumerate} \item $A$ is right FBN; \item $S^{\rm op}\otimes A$ is right FBN and $A$ is right noetherian; \item $B=C_S(A)$ is right FBN and $A$ is right noetherian. \end{enumerate} \end{corollary} \section{Application to Hopf algeba actions}\selabel{6} Let $H$ be a finitely generated projective Hopf algebra over a commutative ring $k$, and $A$ a left $H$-module algebra. The smash product $R=A\# H$ is equal to $A\otimes H$ as a $k$-module, with multiplication given by the formula $$(a\# h)(b\# k)=a(h_{(1)}\cdot b)\# h_{(2)}k.$$ The unit is $1_A\# 1_H$. Consider the maps $$i:\ A\to A\#H,~~i(a)=a\#1_H,$$ $$\chi:\ A\# H\to A,~~\chi(a\# h)=a\varepsilon(h).$$ Straightforward computations show that $(A\# H,i,\chi)$ is an $A$-ring with a left grouplike character. It is also easy to prove that $$A^R=\{a\in A~|~h\cdot a=\varepsilon(h)a,~{\rm for~all~}h\in H\}=A^H$$ is the subalgebra of invariants of $R$.\\ In a similar way, we can associate an $A$-ring with right grouplike character to a right $H$-comodule algebra. We will discuss the left handed case here, in order to recover the results from \cite{Sorin,Garcia,Guedenon}. The results from the previous Sections can easily be restated for rings with a left grouplike character.\\ Let $I=\int_{H^*}^l$ and $J=\int_H^l$ be the spaces of left integrals on and in $H$. $I$ and $J$ are projective rank one $k$-modules, and $H/k$ is $I$-Frobenius (see for example \cite[Theorem 3.4]{CDM}). We need an explicit description of the Frobenius system. From the Fundamental Theorem, it follows that we have an isomorphism $$\phi:\ I\otimes H\to H^*,~~\phi(\varphi\otimes h)=h\cdot \varphi,$$ with $\langle h\cdot \varphi,k\rangle=\langle \varphi,kh\rangle$. If $t\in J$, then $$\phi(\varphi\otimes t)(h)=\langle \varphi,ht\rangle=\langle\varphi,t\rangle\varepsilon(h),$$ so $\phi$ restricts to a monomorphism $\tilde{\phi}:\ I\otimes J\to k\varepsilon$. If $I$ and $J$ are free of rank one, then $\tilde{\phi}$ is an isomorphism, as there exist $\varphi\in I$ and $t\in J$ such that $\langle\varphi,t\rangle=1$ (see for example \cite[Theorem 31]{CMZ}, \cite{Pareigis71}. Hence $\tilde{\phi}$ is an isomorphism after we localize at a prime ideal $p$ of $k$, and this implies that $\tilde{\phi}$ is itself an isomorphism. Consequently $J^*\cong I$. Consider $\tilde{\phi}^{-1}(\varepsilon)=\sum_i\varphi_i\otimes t_i\in I\otimes J$. Then \begin{equation}\eqlabel{6.1.1} \sum_i \langle\varphi_i,t_i\rangle=1. \end{equation} Furthermore $\{(\varphi_i,t_i)~|~i=1,\cdots,n\}$ is a finite dual basis for $I$, so we have $t=\sum_i \langle\varphi_i,t\rangle t_i$, $\varphi=\sum_i\langle\varphi,t_i\rangle \varphi$, for all $t\in J$ and $\varphi\in I$. $\phi$ induces an isomorphism $$\psi:\ H\to H^*\otimes J,~~\psi(h)=\sum_i h\cdot\varphi_i\otimes t_i.$$ The inverse of $\psi$ is given by the formula $$\psi^{-1}(h^*\otimes t)=\langle h^*,\overline{S}(t_{(1)})\rangle t_{(2)},$$ where $\overline{S}$ is the inverse of the antipode $S$; recall from \cite{Pareigis71} that the antipode of a finitely generated projective Hopf algebra is always bijective. Indeed, it is straightforward to show that $\psi^{-1}$ is a right inverse of $\psi$. First observe that $$\psi(\psi^{-1}(h^*\otimes t))=\sum_i\langle h^*,\overline{S}(t_{(1)})\rangle t_{(2)}\cdot \varphi_i\otimes t_i.$$ Now we compute for all $h\in H$ that \begin{eqnarray*} &&\hspace*{-2cm} \langle h^*,\overline{S}(t_{(1)})\rangle \langle t_{(2)}\cdot\varphi_i,h\rangle= \langle h^*,\overline{S}(t_{(1)})\overline{S}(h_{(2)})h_{(1)}\rangle \langle \varphi_i,h_{(3)}t_{(2)}\rangle\\ &=& \langle h^*,\overline{S}(h_{(2)}t_{(1)})h_{(1)}\rangle \langle \varphi_i,h_{(3)}t_{(2)}\rangle\\ &=& \langle h^*,\overline{S}(1_H)h_{(1)}\rangle \langle \varphi_i,h_{(2)}t\rangle = \langle h^*,h\rangle \langle \varphi_i,t\rangle, \end{eqnarray*} where we used the fact that $\varphi_i$ and $t$ are integrals. It follows that $$\psi(\psi^{-1}(h^*\otimes t))=\sum_i h^*\otimes \langle\varphi_i,t\rangle t_i=h^*\otimes t.$$ A right inverse of an invertible element is also a left inverse, so it follows that $$ 1_H=\psi(\psi^{-1}(1_H))=\sum_i\langle \varphi_i,\overline{S}(t_{i(1)})\rangle t_{i(2)}= \sum_i\langle \varphi_i\circ \overline{S},t_{i(1)}\rangle t_{i(2)}= \sum_i\langle \varphi_i\circ \overline{S},t_i\rangle 1_H,$$ where we used the fact that $\varphi_i\circ \overline{S}$ is a right integral on $H$. We conclude that \begin{equation}\eqlabel{6.1.2} \sum_i\langle \varphi_i,\overline{S}(t_i)\rangle=1. \end{equation} Consider the particular situation where $I$ and $J$ are free rank one modules. Then there exist free generators $\varphi_1$ of $ I$ and $t_1$ of $ J$ such that $\langle\varphi_1,t_1\rangle=1$. From \equref{6.1.2} it follows that $\langle\varphi_1,\overline{S}(t_1)\rangle=1$. For arbitrary $\varphi=x\varphi_1\in I$ and $t=yt_1\in J$, it then follows that $\langle \varphi,t\rangle=xy \langle\varphi_1,t_1\rangle=xy= xy\langle\varphi_1,\overline{S}(t_1)\rangle=\langle\varphi,\overline{S}(t)\rangle$. Consider the case where $I$ and $J$ are not necessarily free, and take $\varphi\in I$, $t\in J$ and a prime ideal $p$ of $k$. Then the images of $\langle \varphi,t\rangle$ and $\langle\varphi,\overline{S}(t)\rangle$ in the localized ring $k_p$ are equal, since the integral space of the Hopf $k_p$-algebra $H_p$ is free. So we can conclude that \begin{equation}\eqlabel{6.1.3} \langle \varphi,t\rangle=\langle\varphi,\overline{S}(t)\rangle. \end{equation} \begin{lemma}\lelabel{6.1} Let $H$ be a finitely generated projective Hopf algebra over a commutative ring $k$. There exist $t_i\in J=\int_H^l$ and $\varphi_i\in I=\int_{H^*}^l$ such that $\sum_i \langle\varphi_i,t_i\rangle=1$. $H$ is an $I$-Frobenius $k$-algebra, with Frobenius system $(e,\overline{\nu})$ with \begin{eqnarray*} &&e=\sum_i t_{i(2)}\otimes\varphi_i\otimes\overline{S}(t_{i(1)})\\ &&\overline{\nu}=\sum_j t_j\otimes \varphi_j\in (H\otimes I)^*\cong J\otimes H^* \end{eqnarray*} \end{lemma} \begin{proof} It is straightforward to show that $e\in C_H(H\otimes I\otimes H)$; this also follows from \cite[Prop. 3.3]{CDM}, taking into account that $e= i'(\varphi\otimes \overline{S}(t))$.\\ Write $e=e^1\otimes u^1\otimes e^2\in H\otimes I\otimes H$. We compute that \begin{eqnarray} &&\hspace*{-2cm} \overline{\nu}(e^1\otimes u^1\otimes e^2)= \sum_{i,j}\langle \varphi_j,t_{i(2)}\rangle \langle \varphi,t_j\rangle \overline{S}(t_{i(1)})\nonumber\\ &=& \sum_{i}\langle \varphi_i,t_{i(2)}\rangle \overline{S}(t_{i(1)}) = \sum_i\overline{S}(\langle \varphi_i,t_i\rangle 1_H)\equal{\equref{6.1.1}}1_H.\eqlabel{6.1.4} \end{eqnarray} For all $\varphi\in I$, we calculate \begin{eqnarray} &&\hspace*{-2cm} e^1\otimes u^1\overline{\nu}( e^2\otimes \varphi)= \sum_{i,j} t_{i(2)}\otimes\varphi_i\langle\varphi_j,\overline{S}(t_{i(1)})\rangle\langle\varphi,t_j\rangle\nonumber\\ &=& \sum_{i,j} 1_H\otimes\varphi_i\langle\varphi_j,\overline{S}(t_{i})\rangle\langle\varphi,t_j\rangle = \sum_{i} 1_H\otimes\varphi_i\langle\varphi,\overline{S}(t_{i})\rangle\nonumber\\ &\equal{\equref{6.1.3}}& \sum_{i} 1_H\otimes\varphi_i\langle\varphi,t_{i}\rangle=1_H\otimes \varphi. \eqlabel{6.1.5} \end{eqnarray} It now follows from \cite[Theorem 3.1]{CDM} that $(e,\overline{\nu})$ is a Frobenius system. \end{proof} \begin{proposition}\prlabel{6.2} Let $H$ be a finitely generated projective Hopf algebra over a commutative ring $k$, and $A$ a left $H$-module algebra. Then $A\otimes H$ is an $A\otimes I$-Frobenius $A$-algebra, with Frobenius system $(E,N)$, with \begin{eqnarray*} &&\hspace*{-2cm} E=E^1\otimes_AU^1\otimes_AE^2=(1_A\# e^1)\otimes_A(1_A\otimes u^1)\otimes_A(1_A\# e^1)\\ &=&\sum_i (1_A\# t_{i(2)})\otimes_A(1_A\otimes\varphi_i) \otimes_A(1_A\#\overline{S}(t_{i(1)})),\\ &&\hspace*{-2cm}N:\ (A\#H)\otimes_A (A\otimes I)\cong A\# H\otimes I\to A,\\ &&N(a\#h\otimes\varphi)=a\overline{\nu}(h\otimes\varphi)=\sum_j a\langle\varphi_j,h\rangle\langle \varphi,t_j\rangle. \end{eqnarray*} Here we used the notation introduced above. \end{proposition} \begin{proof} The proof is an adaptation of the proof of \cite[Proposition 5.1]{CVW}. Let us first show that $E$ satisfies \equref{3.1.1}. \begin{eqnarray*} &&\hspace*{-15mm} \sum_i (1_A\# t_{i(2)})\otimes_A(1_A\otimes\varphi_i)\otimes_A(1_A\#\overline{S}(t_{i(1)})(a\# h)\\ &=& \sum_i (1_A\# t_{i(3)})\otimes_A(1_A\otimes\varphi_i)\otimes_A(\overline{S}(t_{i(2)})\cdot a \# \overline{S}(t_{i(1)}) h)\\ &=& \sum_i (1_A\# t_{i(3)})\otimes_A(\overline{S}(t_{i(2)})\cdot a\otimes\varphi_i)\otimes_A(1_A \# \overline{S}(t_{i(1)}) h)\\ &=& \sum_i ((t_{i(3)}\overline{S}(t_{i(2)}))\cdot a\# t_{i(4)})\otimes_A(1_A\otimes\varphi)\otimes_A(1_A \# \overline{S}(t_{i(1)}) h)\\ &=& \sum_i ( a\# t_{i(2)})\otimes_A(1_A\otimes\varphi)\otimes_A(1_A \# \overline{S}(t_{i(1)}) h)\\ &=& \sum_i ( a\#h t_{i(2)})\otimes_A(1_A\otimes\varphi)\otimes_A(1_A \# \overline{S}(t_{i(1)}) )\\ &=& \sum_i ( a\#h)(1_A\# t_{i(2)})\otimes_A(1_A\otimes\varphi_i)\otimes_A(1_A\#\overline{S}(t_{i(1)}). \end{eqnarray*} Obviously $N$ is left $A$-linear. Right $A$-linearity can be proved as follows: \begin{eqnarray*} &&\hspace*{-2cm} N((1\# h\otimes \varphi)a)=N(h_{(1)}a\# h_{(2)}\otimes\varphi)\\ &=& \sum_j h_{(1)}\cdot a\langle\varphi_j,h_{(2)}\rangle\langle \varphi,t_j\rangle = N(1\# h\otimes \varphi)a. \end{eqnarray*} \equref{3.1.2} is satisfied since \begin{eqnarray*} &&\hspace*{-2cm} N(E^1\otimes_AU^1)E^2=1_A\overline{\nu}(e^1\otimes u^1)(1_A\# e^2)\\ &\equal{\equref{6.1.4}}& 1_A\#\overline{\nu}(e^1\otimes u^1) e^2=1_A\#1_H. \end{eqnarray*} Let us finally show that \equref{3.1.3} holds. For all $a\in A$ and $\varphi\in I$, we have \begin{eqnarray*} &&\hspace*{-2cm} E^1\otimes_AU^1N(E^2\otimes_A(a\otimes\varphi))\\ &=& \sum_i(1_A\#t_{i(2)})\otimes_A(1_A\otimes\varphi_i)N(a\#\overline{S}(t_{i(1)})\otimes\varphi)\\ &=&\sum_{i,j}(1_A\#t_{i(2)})\otimes_A(a\otimes\varphi_i)\langle\varphi_j,\overline{S}(t_{i(1)}) \langle\varphi,t_j\rangle\\ &\equal{\equref{6.1.4}}& (1_A\#1_H)\otimes_A (a\otimes\varphi) \end{eqnarray*} \end{proof} \begin{proposition}\prlabel{6.3} Let $H$ be a finitely generated projective Hopf algebra, and $A$ a left $H$-module algebra. The trace map ${\rm tr}\,:\ A\otimes J\to B=A^H$ is given by the formula $${\rm tr}\,(a\otimes t)=t\cdot a.$$ \end{proposition} \begin{proof} Observe that the map $g:\ (J\otimes A)\otimes_A(I\otimes A)$ in the Morita context associated to $I\otimes A$ is given by the formula $$g((t\otimes a)\otimes_A (\varphi\otimes b))=\langle \varphi,t\rangle ab.$$ Using the left handed version of \equref{3.18}, we compute, for $V=a\otimes t\in A\otimes J$ that \begin{eqnarray*} &&\hspace*{-15mm} {\rm tr}\,(a\otimes v)=\chi(E^1g(U^1\chi(E^2)\otimes_AV)) =\sum_i \chi((1_A\# t_i)g((1_A\otimes\varphi)\otimes (a\otimes t)))\\ &=&\sum_i \chi((1_A\# t_i)a\langle\varphi,t\rangle)=\chi((1_A\# t)a) =\chi(t_{(1)}\cdot a\# t_{(2)})=t\cdot a. \end{eqnarray*} \end{proof} We can now apply Propositions \ref{pr:2.1}, \ref{pr:2.2} and \ref{pr:3.2}, and \thref{4.5}, and obtain the following result. \begin{corollary}\colabel{6.4} Let $H$ be a finitely generated projective Hopf algebra, and $A$ a left $H$-module algebra. Assume that there exist $a_i\in A$ and $t_i\in \int_l^H$ such that $\sum_it_i\cdot a_i=1$.\\ Then the following assertions are equivalent; \begin{enumerate} \item $A$ is left FBN; \item $A\# H$ is left FBN and $A$ is left noetherian; \item $B$ is left FBN and $A$ is left noetherian. \end{enumerate} \end{corollary} We recover \cite[Theorem 2.3 and Corollary 2.4]{Garcia}, \cite[Theorem 8]{Sorin} and \cite[Theorem 2.4]{Guedenon}. If $H$ is Frobenius (e.g. if $k$ is a field, or $H=kG$ is a finite group algebra), then the space of left integrals is free. We can then take a free generator $t$ of $\int_H^l$ and the condition of the trace map means that there exists $a\in A$ such that $t\cdot a=1$. We observe that - in the case where the space of integrals is not free - the sufficient condition in \coref{6.4} that there exist $a_i\in A$ and $t_i\in \int_l^H$ such that $\sum_it_i\cdot a_i=1$ is weaker than the one given in \cite[Theorem 8]{Sorin}, where a single $t\in \int_l^H$ and $a\in A$ with $t\cdot a=1$ are needed.\\ In \cite{Garcia} and \cite{Guedenon}, it is stated that \coref{6.4} holds under the weaker assumption (called (C1)) that $A$ is $A\# H$-{\sl quasi}-projective. There seems to be a hole in the proofs in \cite{Garcia} and \cite{Guedenon}: the proof of the implication $3)\Longrightarrow 1)$ uses the projectivity of $A$ as an $A\# H$-module (see \reref{4.6}). \section{Application to corings}\selabel{7} Let $A$ be a ring. An $A$-coring is a coalgebra in the category of $A$-bimodules ${}_A\mathcal{M}_A$. This means that we have two $A$-bimodule maps $$\Delta_\mathcal{C}:\ \mathcal{C}\to \mathcal{C}\otimes_A\mathcal{C}~~{\rm and}~~\varepsilon_\mathcal{C}:\ \mathcal{C}\to A$$ satisfying some coassociativity and counit axioms. The maps $\Delta_\mathcal{C}$ and $\varepsilon_\mathcal{C}$ are called the comultiplication and counit, and we use the Sweedler notation $$\Delta_\mathcal{C}(c)=c_{(1)}\otimes_A c_{(2)},$$ where summation is understood implicitely. Corings were revived recently in \cite{Brzezinski02}, and we refer to \cite{BrzezinskiWisbauer} for a detailed discussion of all kinds of applications. The left dual $R={}^*\mathcal{C}={}_A{\rm Hom}(\mathcal{C},A)$ is an $A$-ring, with multiplication rule $$(f\# g)(c)=g(c_{(1)}f(c_{(2)})),$$ for all $c\in \mathcal{C}$ and $f,g\in {}^*\mathcal{C}$. The unit is $\varepsilon_\mathcal{C}$, and the ring morphism $i:\ A\to{}^* \mathcal{C}$ is given by $$i(a)(c)=\varepsilon_\mathcal{C}(c)a.$$ The $A$-bimodule structure on ${}^*\mathcal{C}$ is then given by the formula $$(afb)(c)=f(ca)b,$$ for all $a,b\in A$, $f\in \mathcal{C}^*$ and $c\in \mathcal{C}$.\\ $x\in \mathcal{C}$ is called grouplike if $\Delta_\mathcal{C}(x)=x\otimes_R x$ and $\varepsilon_\mathcal{C}(x)=1_R$. $(\mathcal{C},x)$ is then called an $R$-coring with a grouplike element. Now consider the map $\chi:\ {}^*\mathcal{C}\to A$, $\chi(f)=f(x)$. It can be shown easily (see \cite{CVW}) that $(\mathcal{C}^*,i,\chi)$ is an $A$-ring with a right grouplike character. We can also compute that $$B=A^R=\{a\in A~|~f(xa)=af(x),~{\rm for~all~}f\in \mathcal{C}^*\}.$$ Using the grouplike element $x$, we can define a right $\mathcal{C}$-coaction on $A$, namely $$\rho:\ \mathcal{C}\to A\otimes_A\mathcal{C}\cong \mathcal{C},~~\rho(r)=1_A\otimes_A xa=xa.$$ We can consider the subring of coinvariants $$A^{{\rm co}\mathcal{C}}=\{a\in A~|~ax=xa\}.$$ In general, $A^{{\rm co}\mathcal{C}}$ is a subring of $A^R$, and they are equal if $\mathcal{C}$ is finitely generated and projective as a right $A$-module.\\ An $A$-coring $\mathcal{C}$ is called Frobenius if there exist an $A$-bimodule map $\theta:\ \mathcal{C}\otimes_A\mathcal{C}\to A$ and $z\in C_A( \mathcal{C})$ (that is, $az=za$, for all $a\in A$) such that the following conditions hold, for all $c,d\in \mathcal{C}$: $$c_{(1)}\theta(c_{(2)}\otimes d)=\theta(c\otimes d_{(1)})d_{(2)},$$ $$\theta(z\otimes c)=\theta(c\otimes z)=1.$$ We refer to \cite[Theorem 35]{CMZ} for the explanation of this definition. If $\mathcal{C}$ is Frobenius, $\mathcal{C}$ is finitely generated and projective as a (left or right) $A$-module, and ${}^*\mathcal{C}/A$ is Frobenius (see \cite[Theorem 36]{CMZ}). Then we also have (see \cite[Sec. 3]{CVW}) that $$Q=\{q\in {}^*\mathcal{C}~|~c_{(1)}q(c_{(2)})=q(c)x\}.$$ It follows from \cite[Theorem 2.7]{CVW} or \thref{3.1} that we have an isomorphism of $({}^*\mathcal{C},B)$-bimodules $\alpha:\ A\to Q$, given by $$\alpha(a)(c)=\theta(ca\otimes_A x),$$ for all $a\in A$ and $c\in \mathcal{C}$. The inverse $\alpha^{-1}$ is given by $\alpha^{-1}(q)=q(z)$, and the left ${}^*\mathcal{C}$-action on $A$ is $$f\cdot a=\theta(z_{(1)}f(z_{(2)})\otimes_A x).$$ This can be verified directly as follows: $$\alpha(\alpha^{-1}(a))=\theta(za\otimes_A x)=\theta(az\otimes x)=a\theta(z\otimes x)=a,$$ and \begin{eqnarray*} &&\hspace*{-2cm} \alpha(\alpha^{-1}(q))(c)=\theta(cq(z)\otimes_A x)=\theta(c\otimes_A q(z)x) = \theta(c\otimes_A z_{(1)}q(z_{(2)})\\ &=& \theta(c\otimes_A z_{(1)})q(z_{(2)}) =q(\theta(c\otimes_A z_{(1)})z_{(2)})\\ &=&q(c_{(1)}\theta(c_{(2)}\otimes z)) = q(c_{(1)}\varepsilon(c_{(2)}))=q(c). \end{eqnarray*} The trace map ${\rm tr}\,:\ A\to B$ is given by $${\rm tr}\,(a)=\theta(xa\otimes_A x).$$ \begin{corollary}\colabel{7.1} Let $(\mathcal{C},x)$ be a Frobenius $A$-coring with a fixed grouplike element, and Frobenius system $(\theta, z)$, and assume that there exists $a\in A$ such that ${\rm tr}\,(a)=1$. Then the following assertions are equivalent. \begin{enumerate} \item $A$ is right FBN; \item ${}^*\mathcal{C}$ is right FBN and $A$ is right noetherian; \item $B=A^{{\rm co}\mathcal{C}}$ is right FBN and $A$ is right noetherian. \end{enumerate} \end{corollary} \begin{center} {\bf Acknowledgement} \end{center} We thank Angel del R\'{\i}o and Sorin D\v{a}sc\v{a}lescu for discussing with us the sufficiency of the quasi-projectivity assumption in the proof of $3)\Longrightarrow 1)$ in \thref{4.5}. \end{document}

\begin{document} \title{Absolute Concentration Robustness in Rank-One Kinetic Systems} \begin{abstract} A kinetic system has an absolute concentration robustness (ACR) for a molecular species if its concentration remains the same in every positive steady state of the system. Just recently, a condition that sufficiently guarantees the existence of an ACR in a rank-one mass-action kinetic system was found. In this paper, it will be shown that this ACR criterion does not extend in general to power-law kinetic systems. Moreover, we also discussed in this paper a necessary condition for ACR in multistationary rank-one kinetic system which can be used in ACR analysis. Finally, a concept of equilibria variation for kinetic systems which are based on the number of the system's ACR species will be introduced here. \end{abstract} \baselineskip=0.30in \section{Introduction} Robustness is the capacity of a system to maintain essential functions in the presence of internal or external stresses \cite{kitano}. It is an important characteristic that helps biological systems adapt to environmental changes and thrive. Several types of robustness have already been identified, but this paper focuses on the kind that requires the positive steady states of a system to possess certain qualities. As defined in \cite{shifein}, a biological system has an absolute concentration robustness (ACR) for a molecular species if in every positive steady state, the system admits, the concentration of the said species is the same. The identification of the steady states of a system is not an easy task, making this property difficult to determine. Shinar and Feinberg provided a sufficient condition that guarantees a deficiency-one mass-action kinetic (MAK) system exhibits an ACR \cite{shifein}. The said condition requires the corresponding network to have two nonterminal complexes that differ only in a specific species. This structural condition can be easily observed in most small networks and thus offers some advantages. In 2018, Fortun et al. showed that the result of Shinar and Feinberg can be readily extended to deficiency-one power-law kinetic systems with reactant-determined interactions (PL-RDK) \cite{fort4}. A PL-RDK system is a kinetic system that is more general than the MAK system which requires reactions with the same reactant complex to have identical kinetic order vectors. In a MAK system, the elements of a kinetic order vector are the stoichiometric coefficients in the corresponding reactant complex. The deficiency-one requirement of the results mentioned above is significantly limiting when it comes to analyzing the capacity of a system to admit an ACR. Fortunately, Fortun and Mendoza \cite{fort3} and Lao et al. \cite{lao} came up with more general results which do not require a network to have a specific deficiency. Their results utilized the concept of network decomposition which is done by partitioning the reaction set of the network such that each partition generates a network (called subnetwork) smaller than the given network. The analysis then focuses on the low-deficiency subnetworks (with a deficiency of at most one) that contain a Shinar-Feinberg pair (SF-pair). An SF-pair is a pair of reactions with kinetic order vectors that only differ in a particular species. This result allows the ACR determination to be confined to the identified subnetworks that are relatively easier to handle. Recently, Meshkat et al. \cite{mesh} provided a necessary and sufficient condition for the existence of stable ACR in a rank-one MAK system. A system has stable ACR if it has an ACR such that each of its positive steady states is stable. However, this ACR criterion does not extend in general to PL-RDK systems, in contrast to the sufficient conditions for low deficiency systems by Horn and Jackson in 1972 \cite{horn} (as generalized by Fortun and Mendoza in 2021 \cite{fort3}) and Shinar and Feinberg in 2010 \cite{shifein}. Counterexamples and examples of PL-RDK systems for the ACR criterion are discussed in this paper. This paper also discussed a necessary condition for ACR in multistationary rank-one kinetic systems. The use of this condition in ACR analysis is illustrated by examples from classes of multistationary rank-one mass action systems introduced by Pantea and Voitiuk in 2022 \cite{voitiuk}. In addition, a concept of equilibria variation for kinetic systems based on the number of ACR species is introduced here. Its basic properties are derived and a general low bound is computed for deficiency zero PL-RDK systems. For multistationary rank-one systems, however, the necessary condition leads to a much sharper lower bound. This is how this paper was organized. Fundamental concepts on chemical reaction networks, kinetic systems, and robustness are provided in Section 2. In Section 3, the problem of extending the result of Meshkat et al. in PLK sytem is presented. A necessary condition for ACR in rank-one multistationary kinetic system is given in Section 4. ACR and equilibria variation are discussed in Section 5. A summary and an outlook are provided in the last section. \section{Fundamentals of chemical reaction networks and kinetic systems} \subsection{Structure of chemical reaction networks} We review in this section some necessary concepts and results on chemical reaction network, the details of which can be found in \cite{fein2, arce, fort3}. First, we introduce some notations used in this paper. The sets of real number, nonnegative real numbers, and positive real numbers are denoted, respectively, by $\mathbb R, \mathbb R_{\geq 0}$, and $\mathbb R_{> 0}$. Given that $\mathscr I$ is a finite index set, $\mathbb R^{\mathscr I}$ denotes the usual vector space of real valued functions with domain $\mathscr I$ where addition, subtraction, and scalar multiplication are defined the usual way. For $x\in \mathbb R^{\mathscr I}$ and $i\in \mathscr I$, $x_i$ denotes the $i^{th}$ coordinate of $x$. Lastly, if $x\in \mathbb R^{\mathscr I}_{> 0}$ and $y\in \mathbb R^{\mathscr I}$, then $x^y\in \mathbb R^{\mathscr I}_{> 0}$ is defined to be $x^y = \displaystyle \prod_{i\in \mathscr I}x^{y^i}_i$. \begin{definition} A \textbf{chemical reaction network} is a triple $\mathscr N = (\mathscr S, \mathscr C, \mathscr R)$ of three non-empty finite sets: \begin{enumerate} \item a set \textbf{species} $\mathscr S$; \item a set $\mathscr C$ of \textbf{complexes}, which are nonnegative integer linear combinations of the species; and \item a set $\mathscr R \subseteq \mathscr C \times \mathscr C$ of reactions such that \begin{itemize} \item $(y,y)\notin \mathscr R$ for all $y\in \mathscr C$, and \item for each $y\in \mathscr C$, there exists a $y'\in \mathscr C$ such that $(y,y')\in \mathscr R$ or $(y',y)\in \mathscr R$. \end{itemize} \end{enumerate} \end{definition} \noindent The nonnegative coefficients of the species in a complex are referred to as \textbf{stoichiometric coefficients}. In this paper, a reaction $R_i=(y_i,y'_i)$ is also denoted by $R_i: y_i\rightarrow y'_i$ and $y_i$ and $y'_i$ are called the \textbf{reactant} and \textbf{product complexes} of $R_i$, respectively. Further, we use the symbols $\sigma, \kappa$, and $\rho$ to denote the numbers of species, complexes, and reactions, respectively. The following example shows that a CRN can be represented by a digraph where the complexes and reactions serve as the digraph's vertices and arcs, respectively. \subsection*{Running example 1} Chemical reaction networks (CRNs) can be represented as a directed graph. The vertices or nodes are the complexes and the reactions are the edges. The CRN is not unique and might not have a physical interpretation. Let us consider the following CRN: \\ \begin{center} \begin{tikzpicture}[baseline=(current bounding box.center)] \tikzset{vertex/.style = {draw=none,fill=none}} \tikzset{edge/.style = {bend left,->,> = latex', line width=0.20mm}} \node[vertex] (1) at (-4,1.5) {$X_1 + X_2 + X_3$}; \node[vertex] (2) at (0,1.5) {$X_3$}; \node[vertex] (3) at (-4,0) {$2X_1$}; \node[vertex] (4) at (0,0) {$3X_1 + X_2$}; \node[vertex] (5) at (4,0) {$4X_1 + 2X_2$}; \node[vertex] (6) at (0,-1.5) {$4X_1+X_2$}; \node[vertex] (7) at (4,-1.5) {$3X_1$}; \draw [edge] (1) to["$k_1$"] (2); \draw [edge] (4) to["$k_2$"] (3); \draw [edge] (4) to["$k_3$"] (5); \draw [edge] (7) to["$k_4$"] (6); \end{tikzpicture} \end{center} The $k_i$'s are called the reaction rate constants. We have $m=3$ (species), $n=7$ (complexes), $n_r=3$ (reactant complexes) and $r=4$ (reactions). Here, we can write $$\mathscr{S}=\left\{ X_1, X_2, X_3\right\}, \quad \mathscr{C}=\left\{X_1 + X_2 + X_3, X_3, 2X_1, 3X_1 + X_2, 4X_1 + 2X_2, 4X_1+X_2, 3X_1 \right\}.$$ On the other hand, the set of reaction $\mathscr{R}$ consists of the following: $$\begin{array}{l} R_{1}: X_1 + X_2 + X_3 \rightarrow X_3 \\ R_{2}: 3X_1+X_2 \rightarrow 2X_1\\ \end{array} \quad \begin{array}{l} R_{3}: 3X_1+X_2 \rightarrow 4X_1 + 2X_2 \\ R_{4}: 3X_1 \rightarrow 4X_1+X_2\\ \end{array}$$ We denote the CRN $\mathscr{N}$ as $\mathscr{N}= (\mathscr{S}, \mathscr{C}, \mathscr{R})$. The \textbf{linkage classes} of a CRN are the subgraphs of a reaction graph where for any complexes $C_i$, $C_j$ of the subgraph, there is a path between them. Thus, the number of linkage classes, denoted as $l$, of Running Example 1 is three ($l=3$). The linkage classes are: $$\mathscr{L}_1=\left\{ R_1 \right\}, \quad \mathscr{L}_2=\left\{R_2,R_3 \right\} \quad \mathscr{L}_3=\left\{R_4 \right\}.$$ A subset of a linkage class where any two vertices are connected by a directed path in each direction is said to be a \textbf{strong linkage class}. Considering Running Example 1, there are no strong linkage classes whose number is denoted by $sl$. We also identify the \textbf{terminal strong linkage classes}, the number denoted as $t$, to be the strong linkage classes where there is no reaction from a complex in the strong linkage class to a complex outside the same strong linkage class. The terminal strong linkage classes can be of two kinds: cycles (not necessarily simple) and singletons (which we call ``terminal points''). We now define important CRN classes. A CRN is \textbf{weakly reversible} if every linkage class is a strong linkage class. A CRN is \textbf{t-minimal} if $t = l$, i.e. each linkage class has only one terminal strong linkage class. Let $n_r$ be the number of reactant complexes of a CRN. Then $n - n_r$ is the number of terminal points. A CRN is called \textbf{cycle-terminal} if and only if $n - n_r = 0$, i.e., each complex is a reactant complex. Clearly, the CRN of the Running Example 1 is both not t-minimal and weakly reversible. With each reaction $y\rightarrow y'$, we associate a \textbf{reaction vector} obtained by subtracting the reactant complex $y$ from the product complex $y'$. The \textbf{stoichiometric subspace} $S$ of a CRN is the linear subspace of $\mathbb{R}^\mathscr{S}$ defined by $$S := \text{span } \left\lbrace y' - y \in \mathbb{R}^\mathscr{S} \mid y\rightarrow y' \in \mathscr{R}\right\rbrace.$$ The \textbf{map of complexes} $\displaystyle{Y: \mathbb{R}^\mathscr{C} \rightarrow \mathbb{R}^\mathscr{S}_{\geq}}$ maps the basis vector $\omega_y$ to the complex $ y \in \mathscr{C}$. The \textbf{incidence map} $\displaystyle{I_a : \mathbb{R}^\mathscr{R} \rightarrow \mathbb{R}^\mathscr{C}}$ is defined by mapping for each reaction $\displaystyle{R_i: y \rightarrow y' \in \mathscr{R}}$, the basis vector $\omega_{R_i}$ (or simply $\omega_i$) to the vector $\omega_{y'}-\omega_{y} \in \mathscr{C}$. The \textbf{stoichiometric map} $\displaystyle{N: \mathbb{R}^\mathscr{R} \rightarrow \mathbb{R}^\mathscr{S}}$ is defined as $N = Y \circ I_a$. In Running Example 1, the matrices $Y$ and $I_a$ are $$Y=\begin{blockarray}{cccccccc} C_1 & C_2 & C_3 & C_4 & C_5 & C_6 & C_7 \\ \begin{block}{[ccccccc]c} 1 & 0 & 2 & 3 & 4 & 4 & 3 & X_1 \\ 1 & 0 & 0 & 1 & 2 & 1 & 0 & X_2 \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & X_3 \\ \end{block} \end{blockarray}$$ $$I_a=\begin{blockarray}{ccccc} R_1 & R_2 & R_3 & R_4 \\ \begin{block}{[cccc]c} -1 & 0 & 0 & 0 & C_1 \\ 1 & 0 & 0 & 0 & C_2 \\ 0 & 1 & 0 & 0 & C_3 \\ 0 & -1 & -1 & 0 & C_4 \\ 0 & 0 & 1 & 0 & C_5 \\ 0 & 0 & 0 & 1 & C_6 \\ 0 & 0 & 0 & -1 & C_7 \\ \end{block} \end{blockarray}.$$ The \textbf{deficiency} $\delta$ is defined as $\delta = n - l - \dim S$. This non-negative integer is, as Shinar and Feinberg pointed out in \cite{shifein2}, essentially a measure of the linear dependency of the network's reactions. In Running Example 1, the deficiency of the network is 3. \subsection{Dynamics of chemical reaction networks} A \textbf{kinetics} is an assignment of a rate function to each reaction in a CRN. A network $\mathscr N$ together with a kinetics $K$ is called a \textbf{chemical kinetic system} (CKS) and is denoted here by $(\mathscr N,K)$. \textbf{Power law kinetics} (PLK) is identified by the \textbf{kinetic order matrix} which is an $\rho\times \sigma$ matrix $F=[F_{ij}]$, and vector $k\in \mathbb R^{\mathscr R}_{>0}$, called the \textbf{rate vector}. \begin{definition} A kinetics $K: \mathbb R^{\mathscr S}_{>0} \rightarrow \mathbb R^{\mathscr R}$ is a \textbf{power law kinetics} if \begin{center} $K_i(x)=k_ix^{F_{i,\cdot}}$ for $i=1,\dots, r$ \end{center} \noindent where $k_i\in \mathbb R_{>0}$, $F_{i,j} \in \mathbb R$, and $F_{i,\cdot}$ is the row of $F$ associated to reaction $R_i$. \end{definition} We can classify a PLK system based on the kinetic orders assigned to its \textbf{branching reactions} (i.e., reactions sharing a common reactant complex). \begin{definition} A PLK system has \textbf{reactant-determined kinetics} (of type PL-RDK) if for any two branching reactions $R_i, R_j\in \mathscr R$, the corresponding rows of kinetic orders in $F$ are identical, i.e., $F_{ih}=F_{jh}$ for $h=1, \dots,m$. Otherwise, a PLK system has \textbf{non-reactant-determined kinetics} (of type PL-NDK). \end{definition} Consider the CRN in Running example 1 with the following kinetic order matrix. \begin{equation} \nonumber F=\begin{blockarray}{cccc} X_1 & X_2 & X_3 \\ \begin{block}{[ccc]c} 0 & 0 & 2 & R_1 \\ 1 & 1 & 0 & R_2 \\ 1 & 1 & 0 & R_3 \\ 1 & 0 & 0 & R_4 \\ \end{block} \end{blockarray}. \end{equation} \noindent Observe that $R_2$ and $R_3$ are two branching reactions whose corresponding rows in $F$ (or \textbf{kinetic order vectors}) are the same. Hence, the CRN is a PL-RDK system. The well-known \textbf{mass action kinetic system} (MAK) forms a subset of PL-RDK systems. In particular, MAK is given by $K_i(x)=k_ix^{Y_{.,j}}$ for all reactions $R_i: y_i \rightarrow y'_i \in \mathscr R$ with $k_i\in \mathbb R_{>0}$ (called \textbf{rate constant}). The vector $Y_{.,j}$ contains the stoichiometric coefficients of a reactant complex $y_i\in \mathscr C$. \begin{definition} The \textbf{species formation rate function} of a chemical kinetic system is the vector field \begin{center} $f(c)=NK(c)=\displaystyle \sum_{y_i\rightarrow y'_i\in \mathscr R}K_i(c)(y'_i-y_i)$, where $c\in \mathbb R^{\mathscr S}_{\geq 0}$, \end{center} where $N$ is the $m \times r$ matrix, called \textbf{stoichiometric matrix}, whose columns are the reaction vectors of the system. \noindent The equation $dc/dt=f(c(t))$ is the \textbf{ODE or dynamical system} of the chemical kinetic system. An element $c^*$ of $\mathbb R^{\mathscr S}_{>0}$ such that $f(c^*)=0$ is called a \textbf{positive equilibrium} or \textbf{steady state} of the system. We use $E_+(\mathscr N,K)$ to denote the set of all positive equilibria of a CKS. \end{definition} Deficiency is one of the important parameters in CRNT to establish claims regarding the existence, multiplicity, finiteness and parametrization of the set of positive steady states. The dynamical system $f(x)$ (or species formation rate function (SFRF)) of the Running Example 1 can be written as $$\left[ \begin{array}{c} \dot{X_1} \\ \dot{X_2} \\ \dot{X_3} \\ \end{array} \right]= \begin{blockarray}{cccc} R_1 & R_2 & R_3 & R_4 \\ \begin{block}{[cccc]} -1 & -1 & 1 & 1 \\ -1 & -1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end{block} \end{blockarray} \left[ \begin{array}{c} k_1 X_3^2 \\ k_2 X_1 X_2 \\ k_3 X_1 X_2 \\ k_4 X_1 \\ \end{array} \right] =NK(x).$$ Analogous to the species formation rate function, we also have the complex formation rate function. \begin{definition} The \textbf{complex formation rate function} $g: \mathbb R^{\mathscr S}_{>0}\rightarrow \mathbb R^{\mathscr C}$ of a chemical kinetic system is the vector field \begin{center} $g(c)=I_aK(c)=\displaystyle \sum_{y_i\rightarrow y'_i\in \mathscr R}K_i(c)(\omega_{y'_i}-\omega_{y_i})$, where $c\in \mathbb R^{\mathscr S}_{\geq 0}$. \end{center} \noindent where $I_a$ is the incidence map. A CKS is \textbf{complex balanced} if it has complex balanced steady state, i.e., there is a composition $c^{**}\in \mathbb R_{>0}^{\mathscr S}$ such that $g(c^{**} )=0$. We denote by $Z_+(\mathscr N,K)$ the set of all complex balanced steady states of the system. \end{definition} \begin{theorem}[Corollary 4.8, \cite{fein3}] \label{1.5} If a CKS has deficiency 0, then its steady states are all complex balanced. \end{theorem} \subsection{A brief review of concentration robustness} The concept of \textbf{absolute concentration robustness (ACR)} was first introduced by Shinar and Feinberg in their well-cited paper published in \textit{Science} \cite{shifein}. ACR pertains to a phenomenon in which a species in a kinetic system carries the same value for any positive steady state the network may admit regardless of initial conditions. In particular, a PL-RDK system $(\mathscr{N},K)$ has ACR in a species $X \in \mathscr{S}$ if there exists $c^*\in E_+(\mathscr{N},K)$ and for every other $c^{**} \in E_+(\mathscr{N},K)$, we have $c^{**}_X =c^*_{X}$ where $c^{**}_X$ and $c^*_{X}$ denote the concentrations of $X$ in $c^*$ and $c^{**}$, respectively. Fortun and Mendoza \cite{fort3} emphasized that ACR as a dynamical property is conserved under dynamic equivalence, ie., they generate the same set of ordinary differential equations. They further investigated ACR in power law kinetic systems and derived novel results that guarantee ACR for some classes of PLK systems. For these PLK systems, the key property for ACR in a species $X$ is the presence of an SF-reaction pair. A pair of reactions in a PLK system is called a \textbf{Shinar-Feinberg pair} (or \textbf{SF-pair)} in a species $X$ if their kinetic order vectors differ only in $X$. A subnetwork of the PLK system is of \textbf{SF-type} if it contains an SF-pair in $X$. \section{The extension problem for the ACR criterion for rank-one systems} Just recently, Meshkat et al. gave a necessary and sufficient condition that guarantees the existence of stable ACR in rank-one MAK systems \cite{mesh}. A system is said to have a stable ACR if it has an ACR where each of its positive steady states is stable. The condition requires the existence of an embedded one-species network that follows some structures described using arrow diagrams. \subsection{A review of the key results of Meshkat et al.} A network $\mathscr N=\left(\mathscr S, \mathscr C, \mathscr R \right)$ is called one-species network if there is a species $A_i$ such that $(y,y')\in \mathscr R$ implies that $y_j= y_j'=0$ for all species $A_j\in S-\{A_i\}$. In other words, every complex in the network takes the form $kA_i$, where $k$ is a nonnegative integer. The following definition and example were taken from \cite{mesh}. \begin{definition} Let $\mathscr N=\left(\{A\}, \mathscr C, \mathscr R \right)$ be a one-species network with $|\mathscr R|\geq 1$. Let every reaction of $\mathscr N$ be of the form $aA\rightarrow bA$, where $a,b\geq 0$ and $a\neq b$. Suppose $\mathscr N$ has $m$ distinct reactant complexes with $a_1<a_2<\dots<a_m$ as their stoichiometric species. The \textbf{arrow diagram} of $\mathscr N$, denoted by $\rho=(\rho_1,\rho_2,\dots,\rho_m)$, is the element of $\{\rightarrow,\leftarrow, \longleftrightarrow\}^m$ given by: \begin{equation}\nonumber \rho_i = \left\{ \begin{array}{clrc@{\qquad}l} \rightarrow & \textnormal{if for all reactions\;} a_iA \rightarrow bA \textnormal{\;in\;} G, \textnormal{\;it is the case that\;} b > a_i \\ \leftarrow & \textnormal{if for all reactions\;} a_iA \rightarrow bA \textnormal{\;in\;} G, \textnormal{\;it is the case that\;} b < a_i\\ \longleftrightarrow & \textnormal{otherwise} \end{array} \right. \end{equation} \end{definition} \begin{example} Consider the network determined by $\{B\rightarrow A, 2A+B \rightarrow A+2B\}$. After removing species $A$, the embedded network has the following reaction set $\{0\leftarrow B \rightarrow 2B\}$ which has $(\longleftrightarrow)$ as arrow diagram. On the other hand, $\{0\rightarrow A, A\rightarrow 2A\}$ is the embedded network obtained after removing $B$ with arrow diagram $(\rightarrow,\leftarrow)$. \end{example} Here is the main result in \cite{mesh} that provides a necessary and sufficient condition that ensure the existence of ACR in some rank-one MAK systems. Their results are based on the idea of arrow diagrams. \begin{theorem}\label{meshkatthm} Let $\mathscr N$ be a rank-one network with species $A_1, A_2, \dots, A_m$. Then, the following are equivalent: \begin{enumerate} \item $\mathscr N$ has stable ACR and admits a positive state. \item There is a species $A_{i^*}$ such that the following holds: \begin{enumerate} \item for the embedded network of $\mathscr N$ obtained by removing all species except $A_{i^*}$, the arrow diagram has one of the following forms: \begin{equation} \begin{array}{rl} (\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow), & (\rightarrow, \rightarrow, \dots, \rightarrow,\longleftrightarrow) \\ (\rightarrow, \rightarrow, \dots, \rightarrow,\longleftrightarrow, \leftarrow,\leftarrow,\dots,\leftarrow), & (\rightarrow, \rightarrow, \dots, \rightarrow, \leftarrow,\leftarrow,\dots,\leftarrow) \end{array} \end{equation} \item the reactant complexes of $\mathscr N$ differ only in species $A_{i^*}$ (i.e., if $y$ and $\hat{y}$ are both reactant complexes, then $y_i=\hat{y_i}$ for all $i\neq i^*$). \end{enumerate} \end{enumerate} \end{theorem} Notice that in the above example, the embedded network obtained by removing species $A$ from $\mathscr N$ has $(\longleftrightarrow)$ as arrow diagram that falls under the enumerated arrow diagrams in the above theorem. Moreover, the reactant complexes in $\mathscr N$ differ only in $A$. It follows that $\mathscr N$ has a stable ACR. \subsection{Examples of PL-RDK systems for which the ACR critera do not hold} Here is the direct adaptation of the sufficient condition of Theorem \ref{meshkatthm} in PLK-systems and formalism. Let $(\mathscr N, K)$ be a rank-one PLK-system having species $A_1, A_2, \dots, A_m$. Also, let $F$ be the kinetic order matrix of the system. Then, $\mathscr N$ has an ACR if there is species $A_{j^*}$ such that the following holds \begin{enumerate} \item for the embedded network of $\mathscr N$ obtained by removing all species except $A_{j^*}$, the arrow diagram has one of the following forms: \begin{equation} \begin{array}{rl} (\longleftrightarrow,\leftarrow,\leftarrow,\dots,\leftarrow), & (\rightarrow, \rightarrow, \dots, \rightarrow,\longleftrightarrow) \\ (\rightarrow, \rightarrow, \dots, \rightarrow,\longleftrightarrow, \leftarrow,\leftarrow,\dots,\leftarrow), & (\rightarrow, \rightarrow, \dots, \rightarrow, \leftarrow,\leftarrow,\dots,\leftarrow) \end{array} \end{equation} \item the kinetic order vectors are pairwise SF-pairs in $A_{j^*}$ (i.e., $F_{ij}=F_{lj}$ for all $j\neq j^*$). \end{enumerate} The following counterexample shows that the statement above is not necessarily true. \begin{example} Consider $\mathscr N=(\{A,B\},\mathscr C, \mathscr R)$, where $\mathscr R$ consists of the following reactions and rate constants: \begin{equation}\nonumber \begin{array}{cl} R_1:& B {\xrightarrow[r_1]{}} A \\ R_2:& 2A+B {\xrightarrow[r_2]{}} 3A\\ R_3:& 3A+B {\xrightarrow[r_3]{}} 2A+2B\\ R_4:& 4A+B {\xrightarrow[r_4]{}} 3A+2B\\ \end{array} \end{equation} This is a rank-one network with $S=\textnormal{span\;}\{A-B\}$ as its stoichiometric subspace. Suppose that it is endowed with power law kinetics and has the following kinetic order matrix: \begin{equation}\nonumber F=\begin{blockarray}{ccc} A & B \\ \begin{block}{[cc]c} p_1 & q & R_1 \\ p_2 & q & R_2 \\ p_3 & q & R_3 \\ p_4 & q & R_4\\ \end{block} \end{blockarray} \end{equation} where $p_i$'s are integers. We obtained the following embedded network after removing $B$. \begin{equation}\nonumber \begin{array}{c} 0\rightarrow A\\ 2A\leftrightarrow 3A\leftarrow 4A \end{array} \end{equation} With the four distinct reactant complexes in this network, here is the corresponding arrow diagram $(\rightarrow, \rightarrow, \leftarrow, \leftarrow)$. Now, observe that $(\mathscr N, K)$ has the following ODEs: \begin{equation}\nonumber \begin{array}{cc} \dot{A}= & {r_1}A^{p_1}B^q + {r_2}A^{p_2}B^q - {r_3}A^{p_3}B^q - {r_4}A^{p_4}B^q \\ \dot{B}= & -{r_1}A^{p_1}B^q - {r_2}A^{p_2}B^q + {r_3}A^{p_3}B^q + {r_4}A^{p_4}B^q \end{array} \end{equation} Solving for the positive equilibria, one of these equations is equated to zero and get \begin{equation}\nonumber \begin{array}{cc} B^q({r_1}A^{p_1} + {r_2}A^{p_2} - {r_3}A^{p_3} - {r_4}A^{p_4} & = 0\\ \Leftrightarrow {r_1}A^{p_1} + {r_2}A^{p_2} - {r_3}A^{p_3} - {r_4}A^{p_4} & = 0 \end{array} \end{equation} Observe that when $p_1<p_2<p_3<p_4$, the polynomial \begin{equation}\label{poly} {r_1}A^{p_1} + {r_2}A^{p_2} - {r_3}A^{p_3} - {r_4}A^{p_4} \end{equation} has exactly one positive root by the Descarte's Rule of Signs. This means that the system has ACR in $A$. On the other hand, suppose that \begin{center} $ \begin{bmatrix} p_1 \\ p_2 \\ p_3 \\ p_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \\ 1 \\ 2 \end{bmatrix} \;$ and $ \begin{bmatrix} r_1 \\ r_2 \\ r_3 \\ r_4 \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ 2 \\ 3 \end{bmatrix} \;$ \end{center} Then, the polynomial in (\ref{poly}) becomes \begin{equation} 4+A^3-2A-3A^2=4-2A-3A^2+A^3 \end{equation} This polynomial has two positive roots namely, $1$ and $1 + \sqrt{5}$. This means that the system does not have an ACR in $A$ in this case. \end{example} It can be observed in the above counterexample the crucial role played by $p_i$'s. If these parameters are not carefully chosen, ACR may not be observed in the system. The problem becomes more complicated when $p_i$'s are non integers since the solvability of the polynomial in (\ref{poly}) may not be determined immediately. This negatively affects the possibility of extending the result of Meshkat et al. to PLK-systems where $p_i$'s are allowed to be real numbers. \subsection{The stable ACR criterion for homogenous PL-quotients of mass action systems} In this section, we introduce the set of homogeneous monomial quotients of mass action systems (PL-MMK) where the stable ACR criterion holds. To define PL-MMK, we recall some definition and result from \cite{ACB2022}. \begin{definition} \label{PFF} Two kinetics $K, K'$ in $\mathscr{K}_\Omega (\mathscr{N})$ are \textbf{positive function factor equivalent} (PPF-equivalent) if for all $x\in \mathbb{R}^{\mathscr{S}}_{>}$ and every reaction $q$, $\dfrac{K_q(x)}{K'_q(x)}$ is a positive function of $x$ only, i.e. independent of $q.$ \end{definition} A key property of PFF-equivalence is expressed in the following proposition: \begin{proposition}[\cite{ACB2022}] If $K$, $K'$ are PPF, then $E_+(\mathscr{N}, K) = E_+(\mathscr{N}, K')$. \label{PFFresult} \end{proposition} Let $(\mathscr{N}, K)$ be a MAK system with $m$ species and $r$ reactions and $(\beta_1,\cdots,\beta_m)$ any real vector. Let $F_{MAK}$ and $F_\beta$ be the $r \times m$ matrices defined as follows: \begin{itemize} \item $F_{MAK,q} = Y^\top_{\rho(q)}$, where $q$ is a reaction and $Y$ is the matrix of complexes of $(\mathscr{N}, K)$. \item $F_\beta$ with identical rows $\beta = (\beta_1,\cdots,\beta_m)$. \end{itemize} \begin{definition} A \textbf{homogeneous PL quotient} of a MAK system $(\mathscr{N}, K)$ is a PLK system $(\mathscr{N}, K_{PLK})$ with the same rate constants and a kinetic order matrix $F_{PLK} = F_{MAK} - F_\beta$. \end{definition} We have the following result: \begin{proposition} The stable ACR criterion holds for homogeneous PL-quotients of rank-one mass action systems. \end{proposition} \begin{proof} Let $(\mathscr{N}, K_{PLK})$ be a homogeneous PL quotient of a rank-one MAK system $(\mathscr{N}, K)$ where $(\mathscr{N}, K)$ satisfies the stable ACR criterion. For each species, write $X_i^{\alpha_i} = X_i^{\beta_i}(X_i^{\alpha_i - \beta_i})$. For a reaction $q$, each $K_q(x) = k_q \prod X_i^{\beta_i} \prod (X_i^{\alpha_i - \beta_i})$. Note that $\prod X_i^{\beta_i}$ is independent of the reaction $q$ and hence, from Definition \ref{PFF}, $K_{PLK}$ and $K$ are PFF-equivalent. From Proposition \ref{PFFresult}, $E_+(\mathscr{N}, K) = E_+(\mathscr{N}, K_{PLK})$. Thus, their ACR coincide and, hence the stable ACR criterion also holds for $(\mathscr{N}, K_{PLK})$. \end{proof} \section{A necessary condition for ACR in multistationary rank-one kinetic systems} In this section, we present a necessary condition for the occurrence of ACR species in any multistationary rank-one kinetic system. We introduce the class of co-conservative kinetic systems and show the multistationary rank-one systems of this kind do not have any ACR species. We further illustrate the condition for a number of multistationary rank-one mass action systems, using a recently presented classification of such systems by Pantea and Voitiuk \cite{voitiuk}. \subsection{The necessary condition for ACR in rank-one stationary kinetic systems} The necessary condition is the following fundamental observation: \begin{theorem} \label{4.1.1} If $\mathscr N$ has rank one and $\left (\mathscr N, K\right )$ is multistationary, then (the line) $S$ lies in the species hyperplane of every ACR species $X$. In other words, for any basis vector $v$ of $S$, its $X$-coordinate is $0$. \end{theorem} \begin{proof} Since $\left (\mathscr N, K\right )$ is multistationary, there is a stoichiometric class that contains two distinct positive equilibria $x_1$ and $x_2$. In other words, $x_1 - x_2 \in S$ and $x_1 - x_2 \neq 0$. For any ACR species $X$ of $\left (\mathscr N, K\right )$, it follows that the $X$-th coordinate of $x_1 - x_2 $ is $0$. Since $x_1 - x_2 \neq 0$, it is a basis vector for the one-dimensional subspace $S$ of the rank-one system, proving the claim. \end{proof} The necessary condition immediately leads to an upper bound for the number of ACR species in a multistationary rank-one system: \begin{corollary} \label{cor1} Let $m_{\textnormal{ACR}}$ be the number of $\textnormal{ACR}$ species of a rank-one multistationary system $\left (\mathscr N, K\right )$. Then $m_{\textnormal{ACR}} \leq m - |\textnormal{supp\;}v|$, where $\textnormal{supp\;}v$ is the support of any basis vector of $S$. \end{corollary} Note that the right-hand side of the inequality in Corollary \ref{cor1} is just the number of zeros in $v$. Now, recall that a network is called conservative if the orthogonal complement of $S$ contains a positive vector. We hence define the following term. \begin{definition} A network is called \textbf{co-conservative} if $S$ contains a positive vector. \end{definition} In general, a positive vector in $S$ can be a linear combination of reaction vectors with $0$ or negative coordinates. For example, for $m=3$, the reactions $X_1 \rightarrow X_1 + X_2$ and $2X_1 \rightarrow 3X_1 + 2X_3$ have the reaction vectors $(0,1,-1)$ and $(1, 0, 2)$ whose sum is $(1,1,1)$. In rank-one networks however, a positive vector requires the occurrence of a positive reaction vector $y'- y$. This in turn implies a reaction $y \rightarrow y'$ with the following characteristics: \begin{itemize} \item all species occur in the product complex; \item each species has a higher stoichiometric coefficient in the product complex compared to the corresponding species in the reactant complex; and \item no enzymatic regulation on a reaction. \end{itemize} \begin{corollary} If a rank-one and co-conservative network is multistationary, then it has no $\textnormal{ACR}$ species. \end{corollary} \begin{proof} Any non-zero vector $v$ in $S$ has only positive or negative coordinates, i.e., $|\textnormal{supp\;}v|=m$. Hence, $m_{\textnormal{ACR}}=0$. \end{proof} \subsection{Examples from the multistationary rank-one mass action systems} Pantea and Voitiuk introduced a complete classification of multistationary rank-one mass action systems in \cite{voitiuk}. The following table provides an overview of the eight classes that they identified. \begin{table}[H] \centering \begin{tabularx}{\linewidth}{|X|X|} \hline \textbf{Network} & \textbf{Definition} \\ \hline Class 1-alt$^c$: 1-alt complete networks & has a 1D projection containing both $(\leftarrow,\rightarrow)$ and $(\rightarrow,\leftarrow)$ patterns \\ \hline Class 2-alt: 2-alternating & has a 1D projection containing both $(\leftarrow,\rightarrow,\leftarrow)$ and $(\rightarrow,\leftarrow,\rightarrow)$ patterns \\ \hline Class $Z$: zigzag network & has a 2D projection containing a zigzag \\ \hline Class $S_1$: one-source networks & contains exactly one source complex and two reactions of opposite directions \\ \hline Class $S^z_2$ : two-source zigzag networks & has two species, contains exactly two source complexes and has a zigzag of slope -1 \\ \hline Class $L$: line networks & has two species and at least three source complexes satisfying some properties* \\ \hline Class $S^{nz}_2$ : two-source non-zigzag networks & an essential network that contains exactly two source complexes and $N \in$ 1-alt$^c - Z$. \\ \hline Class $C$: corner networks & an essential, 1-alt complete network that contains at least three source complexes satisfying some properties* \\ \hline \end{tabularx} \caption{Classes of rank-one networks (for details concerning the concepts and symbols used here, the readers are referred to \cite{voitiuk}).} \label{rankonenet} \end{table} Under mass-action, our running example is a rank-one zigzag network but not a line, corner and two-source zigzag network. We illustrate Theorem \ref{4.1.1} with two examples from different classes of multistationary rank-one systems identified in \cite{voitiuk}. \begin{example}[Class 1-alt$^c$] The rank-one network below was the focus of Example 4.2 in \cite{voitiuk}. It was shown in that paper that it has the capacity for multiple positive and non-degenerate equilibria. Moreover, the network has stoichiometric subspace generated only by the vector $v=(1,1,1,0,-1,-1)$. \begin{equation}\nonumber \begin{array}{cl} R_1:& B+2C+2E \rightarrow A+2B+3C+E \\ R_2:& 2A+2B+C+2D+E \rightarrow A+B+2D+2E\\ R_3:& A+3C+D+2E \rightarrow 2A+B+4C+D+E\\ R_4:& 3A+3B+C+E \rightarrow 2A+2B+2E\\ \end{array} \end{equation} \end{example} \begin{example}[Class 2-alt] The rank-one network below is a 2-alternating network with stoichiometric subspace generated by the vector $v=(2,1,0)$. From Corollary 4.3 in \cite{voitiuk}, we conclude that it has the capacity for multiple positive and non-degenerate equilibria. \begin{equation}\nonumber \begin{array}{cl} R_1:& 2X + 2Y \rightarrow Y \\ R_2:& 3X + Y \rightarrow 5X + 2Y\\ R_3:& 4X + 2Y + Z \rightarrow Z \\ R_4:& 4X + 2Y + 2Z \rightarrow 2X + Y + 2Z\\ \end{array} \end{equation} \end{example} Direct computations show that the preceding systems both have ACR in species $D$ and $Z$, respectively. That is, their basis vectors of $S$ have $0$ coordinate in species $D$ and $Z$, respectively. This illustrates Theorem \ref{4.1.1}. \section{ACR and equilibria variation in kinetic systems} In this section, we briefly discuss the relationship between concentration robustness and equilibria variability in general by introducing the concept of (positive) equilibria variation. We consider several examples to illustrate its use. We assume throughout the section that the kinetic system $(\mathscr N, K)$ is positively equilibrated, i.e., $E_+(\mathscr N, K) \neq \varnothing$. \subsection{The equilibria variation of a kinetic system} The motivation for the following definition comes from the observation that a kinetic system has a unique (positive) equilibrium in species space if and only if it possesses absolute concentration robustness in each of its species. \begin{definition} The \textbf{(positive) equilibria variation} of a kinetic system $(\mathscr N, K)$ is the number of non-ACR species divided by the number of species, i.e., \begin{equation*} v_+(\mathscr N, K) = \dfrac{m-m_{ACR}}{m}. \end{equation*} \end{definition} Clearly, the variation values lie between $0$ and $1$, and the following proposition characterizes the attainment of the extreme values: \begin{proposition} Let $E_+(\mathscr N, K)$ be a kinetic system. Then, \begin{enumerate}[i.] \item $v_+(\mathscr N, K) = 0 \Leftrightarrow |E_+(\mathscr N, K)| = 1$ and \item $v_+(\mathscr N, K) = 1 \Leftrightarrow (\mathscr N, K)$ has no ACR species \end{enumerate} \end{proposition} The proofs follow directly from the definition. We have the following corollary for any multistationary system: \begin{corollary} Let $(\mathscr N, K)$ be a kinetic system. \begin{enumerate} [a.] \item If $(\mathscr N, K)$ is multistationary, then $v_+(\mathscr N, K) \geq \dfrac{1}{m}$. In particular, if $v_+(\mathscr N, K) = 0$, then $(\mathscr N, K)$ is monostationary. \item If $\mathscr N$ is open and $v_+(\mathscr N, K) > 0$, then $(\mathscr N, K)$ is multistationary. \end{enumerate} \end{corollary} \begin{proof} For $(a)$: Multistationarity implies at least two distinct positive equilibria, hence $m_{ACR} \leq m - 1$, and the claims follow. For $(b)$: if the network is open, then there is only one stoichiometric class. Hence, multistationarity is equivalent to the occurrence of at least two distinct equilibria in species space. \end{proof} \begin{example} Schmitz's model of the earth's pre-industrial carbon cycle system was analyzed by Fortun et al. in \cite{fort3.5}. Below is its corresponding reaction network. \begin{equation} \label{schmitz} \begin{tikzcd} M_5 \arrow[dd, "R_1 "'] \arrow[rd, "R_2", shift left] & & M_2 \arrow[ld, "R_5"', shift right] \arrow[rd, "R_{11}"', shift right] \arrow[dd, "R_9"'] & \\ & M_1 \arrow[lu, "R_3", shift left] \arrow[ru, "R_6"', shift right] \arrow[rd, "R_8"', shift right] & & M_4 \arrow[lu, "R_{10}"', shift right] \arrow[ld, "R_{12}"', shift right] \\ M_6 \arrow[ru, "R_4"'] & & M_3 \arrow[lu, "R_7"', shift right] \arrow[ru, "R_{13}"', shift right] & \end{tikzcd} \end{equation} \noindent In this network, $M_1, M_2, M_3, M_4, M_5,$ and $M_6$ stand for atmosphere, warm ocean surface waters, cool ocean surface waters, deep ocean waters, terrestrial biota, and soil and detritus, respectively. Important numbers of the network as well as its kinetic order matrix are given in the following. \begin{figure} \caption{(a) Some numbers related to the reaction network of Schmitz's model of the earth's pre-industrial carbon cycle system; (b) kinetic order matrix of the network.} \label{schmitz1} \end{figure} \pagebreak Fortun and Mendoza \cite{fort3.5} showed that the system has no ACR species, i.e., $v_+(\mathscr N, K) = 1 > \dfrac{1}{6}$. Since the system has a conservative, concordant, and weakly reversible network and weakly monotonic kinetics, by Theorem 6.6 of \cite{shinar}, it has a unique positive equilibrium in each stoichiometric class and consequently monostationary. Note also that the network has rank $5 < 6$, hence it is a closed network. \end{example} If the multistationary kinetic system has rank one, Theorem \ref{4.1.1} provides a better lower bound for the equilibria variation. \begin{proposition} Let $(\mathscr N, K)$ be a multistationary rank-one kinetic system and $v$ a basis vector of $S$. Then, $v_+(\mathscr N, K) \geq \dfrac{|supp(v)|}{m}$. \end{proposition} \begin{proof} According to Theorem \ref{4.1.1}, $m_{ACR} \leq m - |supp(v)| \Leftrightarrow |supp(v)| \leq m - m_{ACR}$, leading to the new lower bound. Note that since $v$ is a basis vector, $|supp(v)|\geq 1$, confirming the improvement. \end{proof} One can derive a lower bound for equilibria variation for any kinetic system using the general species hyperplane criterion for ACR introduced by Hernandez and Mendoza in \cite{hernandez}. It is based on the following considerations: \begin{definition} For any species $X$, the $(m-1)$-dimensional subspace \begin{equation} \nonumber H_X:=\{x\in \mathbb R^{\mathscr S}| x_X=0\} \end{equation} is called the \textbf{species hyperplane} of $X$. \end{definition} For $U$ containing $\mathbb R_{>0}$, let $\phi: U\rightarrow \mathbb R$ be an injective map, i.e., $\phi: U\rightarrow \textnormal{Im}~\phi$ is a bijection. By component-wise application ($m$ times), we obtain a bijection $U^\mathscr S \rightarrow \mathbb R^{\mathscr S}$, which we also denote with $\phi$. We formulate our concepts for any subset $Y$ of $E_+(\mathscr N, K)$, although we are mainly interested in $Y=E_+(\mathscr N, K)$. \begin{definition} For a subset $Y$ of $E_+(\mathscr N, K)$, the set \begin{equation} \nonumber \Delta_{\phi}Y:=\{\phi(x)-\phi(x')| x,x'\in Y\} \end{equation} is called the \textbf{difference set of $\phi$-transformed equilibria} in $Y$, and its span $\langle \Delta_{\phi}Y \rangle$ the difference space of $\phi$-transformed equilibria in $Y$. \end{definition} In Proposition 6 of \cite{hernandez}, it is shown that \begin{equation} \nonumber m_{ACR} \leq m - \dim \langle \Delta_{\phi}E_+\rangle. \end{equation} \noindent It follows immediately that we have the following lower bound for the equilibria variation. \begin{proposition} Let $(\mathscr N, K)$ be a kinetic system. Then, $\dfrac {\dim \langle \Delta_{\phi}E_+\rangle}{m} \leq v_+(\mathscr N, K)$. \end{proposition} In general, it is challenging to compute $\dim \langle \Delta_{\phi}E_+\rangle$. However, for PLP systems, this can be done, and in the next section, we apply it to compute equilibria variation for weakly reversible, deficiency zero PL-RDK systems. \subsection{Equilibria variation in deficiency zero PL-RDK systems} A kinetic system $(\mathscr N, K)$ is a PLP (\textbf{positively equilibrated-log parametrized}) system if there is a reference equilibrium $x^*$ and a flux subspace $P_E$ of $\mathbb R^{\mathscr S}$ such that $E_+(\mathscr N, K) = \{ x \in \mathbb R^{\mathscr S}_{>} | \log x - \log x^* \in P_E^{\perp}\}$. For any PLP system, Lao et al. \cite{lao} showed that ACR is characterized by the species hyperplane criterion for PLP systems: $X$ is an ACR species if and only if it is contained in $H_X := \{x \in \mathbb R_{\mathscr S} | x_X = 0\}$. This implies that $m - \dim P_E \leq m - m_{ACR}$ and, hence, $1 - \dfrac{\dim P_E}{m} \leq v_+(\mathscr N, K)$. Jose et al. \cite{ACB2022} showed that a CLP (\textbf{complex balanced-log parametrized}) system, i.e., $Z_+(\mathscr N, K) = \{x\in \mathbb R^{\mathscr S}_> |\log x - \log x^* \in P_Z^{\perp}\}$ is absolutely complex balanced, i.e., $E_+(\mathscr N, K) = Z_+(\mathscr N, K)$ if and only if it is both CLP and PLP and $P_E = P_Z$. It was shown in \cite{muller} that any complex balanced PL-RDK system is a CLP system with $P_Z = \tilde{S}$. Since after Feinberg, any weakly reversible deficiency zero kinetic system is absolutely complex balanced, then a weakly reversible deficiency zero PL-RDK system is a PLP system with $P_E = \tilde{S}$. Hence, for any such system, $1 - \dfrac{\Tilde{s}}{m} \leq v_+(\mathscr N, K)$, where $\Tilde{s}$ is the kinetic rank of the system. \begin{example} Fortun and collaborators studied variants of the Anderies et al. \cite{anderies} model of the earth's pre-industrial carbon cycle (see the reaction network below) in \cite{noel} and \cite{fort3.5}. \end{example} \begin{equation} \label{anderies} \begin{tikzcd} A_1+2A_2 \arrow[r, "R_1"] & 2A_1+A_2 \\ A_1+A_2 \arrow[r, "R_2"] & 2A_2 \\ A_2 \arrow[r, "R_3 ", shift left] & A_3 \arrow[l, "R_4", shift left] \end{tikzcd} \end{equation} \noindent Note that $A_1$, $A_2$, and $A_3$ here stand for land, atmosphere, and ocean, respectively. Given below are some important numbers and the kinetic order matrix of the network in \ref{anderies}. \begin{figure} \caption{(a) Some numbers related to the reaction network of the model of the earth's pre-industrial carbon cycle system given by Anderies et al. in \cite{anderies}; (b) kinetic order matrix of the network.} \label{schmitz2} \end{figure} In \cite{fort3.5}, Fortun and Mendoza showed that the system is dynamically equivalent to a weakly reversible, deficiency zero system with $\Tilde{S}^{\perp} = \left \langle \begin{pmatrix} -1 & \dfrac{p_2-p_1}{q_2-q_1} & \dfrac{p_2-p_1}{q_2-q_1} \end{pmatrix} \right \rangle$. Let $R$ denote the ratio $\dfrac{p_2 - p_1}{q_2 - q_1}$. Based on the value of $R$, one obtains three classes of Anderies system: $AND_> (R > 0)$ consists of multistationary systems with no ACR species, $AND_0 (R = 0)$ contains only monostationary systems with two ACR species and $AND_<$ contains both injective and non-injective systems but also no ACR species. The variants studied in [10] and [14] belong to $AND_>$ and $AND_0$, respectively. Based on the previous results, we have $v_+(\mathscr N, K) = 1$ if $(\mathscr N, K)$ is contained in either $AND_>$ or $AND_<$, and $v_+(\mathscr N, K) = \dfrac{1}{3}$ if $(\mathscr N, K)$ is in $AND_0$. \subsection{Equilibria variation of independent subnetworks} In this section, we discuss equilibria variation for independent subnetworks. It is important in this regard to differentiate between the two concepts of subnetworks that we introduced in previous work: embedded and non-embedded. We hence begin with a review of these concepts and their properties relevant to absolute concentration robustness (ACR), which is the basis of our concept of equilibria variation. We then collect some relevant results from previous publications and add some new details. We conclude by using the results of the reaction network analysis of metabolic insulin signaling by Lubenia et al. \cite{lubenia} to illustrate the different concepts and relationships. We in fact show that the various inequalities between the different equilibria variation values are sharp, i.e., equality is achieved in various subnetworks of the insulin system. \subsubsection{Review of subnetwork properties} In a subnetwork in a decomposition $\mathscr N=\mathscr N_1\cup \mathscr N_2\cup\cdots \cup\mathscr N_k$, often, a smaller number of species occurs than in the whole network. If $\mathscr N=(\mathscr S, \mathscr C, \mathscr R)$, we call a subnetwork $(\mathscr C', \mathscr R')$, where $\mathscr R'\subseteq \mathscr R$ and $\mathscr C'=\mathscr C|_{\mathscr R'}$, \textbf{embedded} if its species space is $\mathscr S$ and \textbf{non-embedded} if it has $\mathscr S|_{\mathscr C'}$ as species space, which we denote with $\mathscr S'$. We use the embedded representation in a decomposition because it conveniently allows the set operations on equilibria sets. A basic fact is the following observation: \begin{proposition} If $X$ is an ACR species of a subnetwork, then $X$ is an element of $\mathscr S'$. \end{proposition} This means that we need only one count of ACR species in a subnetwork, $m'_{ACR}$. For non-ACR species and equilibria variation, we have the following relationships: \begin{proposition} Let $m'=|\mathscr S'|$ and $v_+(\mathscr N', K')$, $\Tilde{v}_+(\mathscr N', K')$ be the equilibria variations of the non-embedded and embedded subnetworks. Then, \begin{enumerate}[i.] \item $m-m'_{ACR}=(m'-m'_{ACR})+(m-m')$ and \item $0\leq \Tilde{v}_+(\mathscr N', K')-v_+(\mathscr N', K')\leq \dfrac{m-m'}{m}$ \end{enumerate} \end{proposition} \begin{proof} $(i.)$ should be read as: the number of non-ACR species in an embedded network is the number of non-ACR species in the non-embedded network plus the number of non-occurring species, and $(ii.)$: dividing the equation in $(i.)$ by $m$ and using the inequality $\dfrac{m'-m'_{ACR}}{m}\leq \dfrac{m'-m'_{ACR}}{m'}$, we obtain $\Tilde{v}_+(\mathscr N', K')-v_+(\mathscr N', K')\leq \dfrac{m-m'}{m}$. On the other hand, the left-hand side, after forming a common denominator, is now equal to $\dfrac{m'_{ACR}(m-m')}{mm'}\geq 0$, since all factors are non-negative. \end{proof} \subsubsection{Equilibria variation of independent subnetworks} In \cite{lao}, Proposition 4.4 states that if species $X$ has ACR in $\mathscr N_i$ and the decomposition is independent, then $X$ has ACR in $\mathscr N$, i.e., $\displaystyle |\mathscr S_{ACR, i}|\leq \left| \bigcup_{i=1}^{k} \mathscr S_{ACR,i} \right| \leq |\mathscr S_{ACR}|$. We hence have the following corollary: \begin{proposition} $v_+(\mathscr N, K)\leq \Tilde{v}_+(\mathscr N', K')$ for any independent subnetwork $\mathscr N'$. \end{proposition} \begin{proof} It follows that $m-m_{ACR}\leq m-m'_{ACR}$ and dividing both sides with $m$ results in the claim. \end{proof} \subsubsection{The example of metabolic insulin signaling in healthy cells} Lubenia et al. \cite{lubenia} constructed a mass action kinetic realization of the widely used model of metabolic insulin signaling (in healthy cells) by Sedaghat et al. \cite{sedaghat}. They used the kinetic system's finest independent decomposition (FID) to conduct an ACR analysis and showed that $m_{ACR} \geq 8$ for all rate constants (such that the system has positive equilibria) and $m_{ACR} = 8$ for some rate constants. Hernandez et al. [3] confirmed that $m_{ACR} = 8$ for all rate constants with a positively equilibrated system. Hence, $v_+(\mathscr N, K) = \dfrac{20 - 8}{20} = \dfrac{12}{20} =\dfrac{3}{5} = 0.60$. The table below (see Table \ref{lube}), which was taken from \cite{lubenia}, provides an overview of the characteristics of the FID subnetworks. Note that, with the exception of $\mathscr{N}_1$, all subnetworks are rank-one systems. Applying the Meshkat et al. criterion to these 9 subnetworks showed that only the one-species systems $\mathscr{N}_2$ and $\mathscr{N}_{10}$ had ACR for the species $X_6$ and $X_{20}$, respectively. All other subnetworks had no ACR species. Hence, we have: \begin{itemize} \item For $i=3,\dots,9, v_+(\mathscr N, K) = 0.60 < 1 = v_+(\mathscr N_i, K_i) = \Tilde{v}_+(\mathscr N_i, K_i)$ \item For $i=2,\dots,10, v_+(\mathscr N_i, K_i) = 0< 0.60 = v_+(\mathscr N, K) = \Tilde{v}_+(\mathscr N_i, K_i)=\dfrac{19}{20}=0.95$ \end{itemize} \begin{table}[H] \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|} \hline \textbf{Network numbers} & $\mathscr N_1$ & $\mathscr N_2$ & $\mathscr N_3$ & $\mathscr N_4$ & $\mathscr N_5$ & $\mathscr N_6$ & $\mathscr N_7$ & $\mathscr N_8$ & $\mathscr N_9$ & $\mathscr N_{10}$ \\ \hline Species & 7 & 1 & 4 & 3 & 3 & 2 & 3 & 3 & 4 & 1 \\ \hline Complexes & 7 & 2 & 6 & 2 & 4 & 2 & 4 & 4 & 6 & 2 \\ \hline Reactant complexes & 7 & 2 & 3 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ \hline Reversible reactions & 5 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\ \hline Irreversible reactions & 4 & 0 & 3 & 0 & 2 & 0 & 2 & 2 & 2 & 0 \\ \hline Reactions & 14 & 2 & 3 & 2 & 2 & 2 & 2 & 2 & 4 & 2 \\ \hline Linkage classes & 1 & 1 & 3 & 1 & 2 & 1 & 2 & 2 & 3 & 1 \\ \hline Strong linkage classes & 1 & 1 & 6 & 1 & 4 & 1 & 4 & 4 & 5 & 1 \\ \hline Terminal strong linkage classes & 1 & 1 & 3 & 1 & 2 & 1 & 2 & 2 & 3 & 1 \\ \hline Rank & 6 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline Reactant rank & 7 & 1 & 3 & 2 & 2 & 2 & 2 & 2 & 4 & 1 \\ \hline Deficiency & 0 & 0 & 2 & 0 & 1 & 0 & 1 & 1 & 2 & 0 \\ \hline Reactant deficiency & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \hline \end{tabular} \caption{Some numbers of FID subnetworks.} \label{lube} \end{table} The authors considered also the coarsening $\mathscr N_A \cup \mathscr N_B$, where $\mathscr N_A$ is the union of all deficiency zero subnetworks and $\mathscr N_B$ the union of all positive deficiency subnetworks. They also showed that all ACR species are contained solely in $\mathscr N_A$. Hence, in this case, we have: \begin{itemize} \item $v_+(\mathscr N_A, K_A) =\dfrac{13-8}{13}=0.38 < 0.6 = v_+(\mathscr N, K) = \Tilde{v}_+(\mathscr N_A, K_A)$ \item $v_+(\mathscr N, K) = 0.6<1 = v_+(\mathscr N_B, K_B) = \Tilde{v}_+(\mathscr N_B, K_B)$ \end{itemize} \noindent These examples show that all the inequalities in the propositions above are sharp. \section{Summary and outlook} This study was motivated by Meshkat et al.'s result which allows the analysis of rank-one MAK systems for possession of stable ACR. The result guarantees a MAK system such robustness if the two identified structural conditions are met. These conditions require the system to have an embedded network that follows certain structures and reactant complexes that differ in just one species. We attempted to extend this result to more general PLK systems but to no avail. We found an example illustrating how these conditions do not always work in a general setting. This means that, unlike other earlier results on ACR, the conditions in this result do not always insure the existence of ACR in a PLK system. On the other hand, we found a subclass of PLK systems where the stable ACR criterion of Meshkat et al. holds. We call this subclass homogenous monomial quotients of mass action systems or PL-MMK for short. This subclass is obtained by utilizing the set of rate constants of a given MAK system and its modified kinetic order matrix. We also discovered a property that is necessarily present in a multistationary rank-one system that possesses an ACR. Specifically, the corresponding result indicates that such system that has an ACR must have a basis vector generator of the stoichiometric subspace with 0 as a coordinate in the ACR species. We illustrated this result using a multistationary system that was given in the paper of Pantea and Voitiuk. Finally, we considered the concept of equilibria variation of independent subnetworks in this paper. We discussed some mathematical relationships of the equilibria variations of embedded and non-embedded subnetworks. These relationships were illustrated through the data shown in \cite{lubenia} that Lubenia et al. used for analyzing metabolic insulin signaling of healthy cells. It is important to note that ACR species of a subnetwork is always contained in the corresponding non-embedded subnetwork. For future studies, one can look at the extension problem of the rank-one ACR criterion, that is, the identification of further kinetic systems (beyond mass action) for which it holds. One can also consider the exploration of relationships between multistationarity classes and the necessary condition for rank-one mass action systems as well as an extension of the Pantea-Voitiuk classification beyond mass action. Further, it is also interesting to identify further kinetics sets for which the general low bound can be computed or a much sharper alternative as in rank-one multistationary systems can be derived. \noindent \textbf{Acknowledgement} \noindent D. Talabis, E. Jose, and L. Fontanil acknowledge the support of the University of the Philippines Los Ba\~{n}os through the Basic Research Program. \baselineskip=0.25in \end{document}

\begin{document} \maketitle \begin{abstract} In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$. \\ \noindent {\it Key Words and Phrases.} Schr\"odinger equation, well-posedness, Cauchy problem, scaling critical, multilinear estimate, bounded $p$-variation.\\ 2010 {\it Mathematics Subject Classification.} 35Q55, 35B65. \end{abstract} \section{Introduction\label{intro}} We consider the Cauchy problem of the fourth order nonlinear Schr\"odinger type equations: \begin{equation}\label{D4NLS} \begin{cases} \displaystyle (i\partial_{t}+\Delta ^2)u=\partial P_{m}(u,\overline{u}),\hspace{2ex}(t,x)\in (0,\infty )\times {\BBB R}^{d} \\ u(0,x)=u_{0}(x),\hspace{2ex}x\in {\BBB R}^{d} \end{cases} \end{equation} where $m\in {\BBB N}$, $m\geq 2$, $P_{m}$ is a polynomial which is written by \[ P_{m}(f,g)=\sum_{\substack{\alpha ,\beta \in {\BBB Z}_{\geq 0}\\ \alpha +\beta=m}}f^{\alpha}g^{\beta}, \] $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$ and the unknown function $u$ is ${\BBB C}$-valued. The fourth order Schr\"{o}dinger equation with $P_{m}(u,\overline{u})=|u|^{m-1}u$ appears in the study of deep water wave dynamics \cite{Dysthe}, solitary waves \cite{Karpman}, \cite{KS}, vortex filaments \cite{Fukumoto}, and so on. The equation (\ref{D4NLS}) is invariant under the following scaling transformation: \[ u_{\lambda}(t,x)=\lambda^{-3/(m-1)}u(\lambda^{-4}t,\lambda^{-1}x), \] and the scaling critical regularity is $s_{c}=d/2-3/(m-1)$. The aim of this paper is to prove the well-posedness and the scattering for the solution of (\ref{D4NLS}) in the scaling critical Sobolev space. There are many results for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearities (see \cite{S1}, \cite{S2}, \cite{HJ1}, \cite{HHW}, \cite{HHW2}, \cite{HJ3}, \cite{S3}, \cite{HJ2}, \cite{Y12}, \cite{HN15_1}, \cite{HN15_2}, and references cited therein). Especially, the one dimensional case is well studied. Wang (\cite{Y12}) considered (\ref{D4NLS}) for the case $d=1$, $m=2l+1$, $l\ge 2$, $P_{2l+1}(u,\overline{u})=|u|^{2l}u$ and proved the small data global in time well-posedness for $s=s_{c}$ by using Kato type smoothing effect. But he did not treat the cubic case. Actually, a technical difficulty appears in this case (see Theorem \ref{notC3} below). Hayashi and Naumkin (\cite{HN15_1}) considered (\ref{D4NLS}) for $d=1$ with the power type nonlineality $\partial_{x}(|u|^{\rho -1}u)$ ($\rho >4$) and proved the global existence of the solution and the scattering in the weighted Sobolev space. Moreover, they (\cite{HN15_2}) also proved that the large time asymptotics is determined by the self similar solution in the case $\rho =4$. Therefore, derivative quartic nonlinearity in the one spatial dimension is the critical in the sense of the asymptotic behavior of the solution. We firstly focus on the quartic nonlinearity $\partial _x (\overline{u}^4)$ in one space dimension. Since this nonlinearity has some good structure, the global solution scatters to a free solution in the scaling critical Sobolev space. Our argument does not apply to \eqref{D4NLS} with $P (u,\overline{u}) = |u|^3 u$ because we rely on the Fourier restriction norm method. Now, we give the first results in this paper. For a Banach space $H$ and $r>0$, we define $B_r(H):=\{ f\in H \,|\, \|f\|_H \le r \}$. \begin{thm}\label{wellposed_1} Let $d=1$, $m=4$ and $P_{4}(u,\overline{u})=\overline{u}^{4}$. Then the equation {\rm (\ref{D4NLS})} is globally well-posed for small data in $\dot{H}^{-1/2}$. More precisely, there exists $r>0$ such that for any $T>0$ and all initial data $u_{0}\in B_{r}(\dot{H}^{-1/2})$, there exists a solution \[ u\in \dot{Z}_{r}^{-1/2}([0,T))\subset C([0,T );\dot{H}^{-1/2}) \] of {\rm (\ref{D4NLS})} on $(0, T )$. Such solution is unique in $\dot{Z}_{r}^{-1/2}([0,T))$ which is a closed subset of $\dot{Z}^{-1/2}([0,T))$ {\rm (see Definition~\ref{YZ_space} and (\ref{Zr_norm}))}. Moreover, the flow map \[ S^{+}_{T}:B_{r}(\dot{H}^{-1/2})\ni u_{0}\mapsto u\in \dot{Z}^{-1/2}([0,T)) \] is Lipschitz continuous. \end{thm} \begin{rem} We note that $s=-1/2$ is the scaling critical exponent of (\ref{D4NLS}) for $d=1$, $m=4$. \end{rem} \begin{cor}\label{sccat} Let $r>0$ be as in Theorem~\ref{wellposed_1}. For all $u_{0}\in B_{r}(\dot{H}^{-1/2})$, there exists a solution $u\in C([0,\infty );\dot{H}^{s_{c}})$ of (\ref{D4NLS}) on $(0,\infty )$ and the solution scatters in $\dot{H}^{-1/2}$. More precisely, there exists $u^{+}\in \dot{H}^{-1/2}$ such that \[ u(t)-e^{it\Delta^2}u^{+} \rightarrow 0 \ {\rm in}\ \dot{H}^{-1/2}\ {\rm as}\ t\rightarrow + \infty. \] \end{cor} Moreover, we obtain the large data local in time well-posedness in the scaling critical Sobolev space. To state the result, we put \[ B_{\delta ,R} (H^s) := \{ u_0 \in H^s | \ u_0=v_0+w_0 , \, \| v_0 \| _{\dot{H}^{-1/2}} < \delta, \, \| w_0 \| _{L^2} <R \} \] for $s<0$. \begin{thm} \label{large-wp} Let $d=1$, $m=4$ and $P_{4}(u,\overline{u})=\overline{u}^{4}$. Then the equation {\rm (\ref{D4NLS})} is locally in time well-posed in $H^{-1/2}$. More precisely, there exists $\delta >0$ such that for all $R \ge \delta$ and $u_0 \in B_{\delta ,R} (H^{-1/2})$ there exists a solution \[ u \in Z^{-1/2}([0,T]) \subset C([0,T); H^{-1/2}) \] for $T=\delta ^{8} R^{-8}$ of \eqref{D4NLS}. Furthermore, the same statement remains valid if we replace $H^{-1/2}$ by $\dot{H}^{-1/2}$ as well as $Z^{-1/2}([0,T])$ by $\dot{Z}^{-1/2}([0,T])$. \end{thm} \begin{rem} For $s>-1/2$, the local in time well-posedness in $H^s$ follows from the usual Fourier restriction norm method, which covers for all initial data in $H^s$. It however is not of very much interest. On the other hand, since we focus on the scaling critical cases, which is the negative regularity, we have to impose that the $\dot{H}^{-1/2}$ part of initial data is small. But, Theorem \ref{large-wp} is a large data result because the $L^2$ part is not restricted. \end{rem} The main tools of the proof are the $U^{p}$ space and $V^{p}$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (\cite{HHK09}, \cite{HHK10}). We also consider the one dimensional cubic case and the high dimensional cases. The second result in this paper is as follows. \begin{thm}\label{wellposed_2} {\rm (i)}\ Let $d=1$ and $m=3$. Then the equation {\rm (\ref{D4NLS})} is locally well-posed in $H^{s}$ for $s\ge 0$. \\ {\rm (ii)}\ Let $d\geq 2$ and $(m-1)d\geq 4$. Then the equation {\rm (\ref{D4NLS})} is globally well-posed for small data in $\dot{H}^{s_{c}}$ (or $H^{s}$ for $s\ge s_{c}$) and the solution scatters in $\dot{H}^{s_{c}}$ (or $H^{s}$ for $s\ge s_{c}$). \end{thm} The smoothing effect of the linear part recovers derivative in higher dimensional case. Therefore, we do not use the $U^p$ and $V^p$ type spaces. More precisely, to establish Theorem \ref{wellposed_2}, we only use the Strichartz estimates and get the solution in $C([0,T);H^{s_c})\cap L^{p_m}([0,T); W^{q_m,s_{c}+1/(m-1)})$ with $p_m =2(m-1)$, $q_m =2(m-1)d/\{ (m-1)d-2\}$. Accordingly, the scattering follows from a standard argument. Since the condition $(m-1)d\geq 4$ is equivalent to $s_{c}+1/(m-1)\ge 0$, the solution space $L^{p_m}([0,T); W^{q_m,s_{c}+1/(m-1)})$ has nonnegative regularity even if the data belongs to $H^{s_{c}}$ with $-1/(m-1)\le s_c <0$. Our proof of Thorem~\ref{wellposed_2} {\rm (ii)} cannot applied for $d=1$ since the Schr\"odingier admissible $(a,b)$ in {\rm (\ref{admissible_ab})} does not exist. \begin{rem} For the case $d=1$, $m=4$ and $P_{4}(u,\overline{u})\ne \overline{u}^{4}$, we can obtain the local in time well-posedness of {\rm (\ref{D4NLS})} in $H^{s}$ for $s\ge 0$ by the same way of the proof of Theorem~\ref{wellposed_2}. Actually, we can get the solution in $C([0,T];H^s)\cap L^4 ([0,T];W^{s+1/2,\infty })$ for $s\ge 0$ by using the iteration argument since the fractional Leibnitz rule (see \cite{CW91}) and the H\"older inequality imply \[ \left\| |\nabla |^{s+\frac{1}{2}}\prod_{j=1}^{4}u_j \right\|_{L^{4/3}_{t}([0,T);L_{x}^{1})} \lesssim T^{1/4}\| |\nabla |^{s+\frac{1}{2}}u_1 \|_{L^{4}_{t}L_{x}^{\infty}}\| u_2 \|_{L^{4}_{t}L_{x}^{\infty}} \| u_3 \|_{L^{\infty}_{t}L_{x}^{2}}\| u_4 \|_{L^{\infty}_{t}L_{x}^{2}}. \] \end{rem} We give a remark on our problem, which shows that the standard iteration argument does not work. \begin{thm}\label{notC3} {\rm (i)}\ Let $d=1$, $m=3$, $s<0$ and $P_{3}(u,\overline{u})=|u|^{2}u$. Then the flow map of {\rm (\ref{D4NLS})} from $H^s$ to $C({\BBB R} ; H^s)$ is not smooth. \\ {\rm (ii)}\ Let $m\ge 2$, $s<s_c$ and $\partial =|\nabla |$ or $\frac{\partial}{\partial x_k}$ for some $1\le k \le d$. Then the flow map of {\rm (\ref{D4NLS})} from $H^s$ to $C({\BBB R} ; H^s)$ is not smooth. \end{thm} More precisely, we prove that the flow map is not $C^3$ if $d=1$, $m=3$, $s<0$ and $P_{3}(u,\overline{u})=|u|^{2}u$ or $C^m$ if $d \ge 1$, $m \ge 2$, and $s<s_c$. It leads that the standard iteration argument fails, because the flow map is smooth if it works. Of course, there is a gap between ill-posedness and absence of a smooth flow map. Since the resonance appears in the case $d=1$, $m=3$ and $P_{3}(u,\overline{u})=|u|^{2}u$, there exists an irregular flow map even for the subcritical Sobolev regularity. \text{} \\ \noindent {\bf Notation.} We denote the spatial Fourier transform by\ \ $\widehat{\cdot}$\ \ or $\mathcal{F}_{x}$, the Fourier transform in time by $\mathcal{F}_{t}$ and the Fourier transform in all variables by\ \ $\widetilde{\cdot}$\ \ or $\mathcal{F}_{tx}$. The free evolution $S(t):=e^{it\Delta^{2}}$ is given as a Fourier multiplier \[ \mathcal{F}_{x}[S(t)f](\xi )=e^{-it|\xi |^{4}}\widehat{f}(\xi ). \] We will use $A\lesssim B$ to denote an estimate of the form $A \le CB$ for some constant $C$ and write $A \sim B$ to mean $A \lesssim B$ and $B \lesssim A$. We will use the convention that capital letters denote dyadic numbers, e.g. $N=2^{n}$ for $n\in {\BBB Z}$ and for a dyadic summation we write $\sum_{N}a_{N}:=\sum_{n\in {\BBB Z}}a_{2^{n}}$ and $\sum_{N\geq M}a_{N}:=\sum_{n\in {\BBB Z}, 2^{n}\geq M}a_{2^{n}}$ for brevity. Let $\chi \in C^{\infty}_{0}((-2,2))$ be an even, non-negative function such that $\chi (t)=1$ for $|t|\leq 1$. We define $\psi (t):=\chi (t)-\chi (2t)$ and $\psi_{N}(t):=\psi (N^{-1}t)$. Then, $\sum_{N}\psi_{N}(t)=1$ whenever $t\neq 0$. We define frequency and modulation projections \[ \widehat{P_{N}u}(\xi ):=\psi_{N}(\xi )\widehat{u}(\xi ),\ \widetilde{Q_{M}^{S}u}(\tau ,\xi ):=\psi_{M}(\tau -|\xi|^{4})\widetilde{u}(\tau ,\xi ). \] Furthermore, we define $Q_{\geq M}^{S}:=\sum_{N\geq M}Q_{N}^{S}$ and $Q_{<M}^{S}:=Id -Q_{\geq M}^{S}$. The rest of this paper is planned as follows. In Section 2, we will give the definition and properties of the $U^{p}$ space and $V^{p}$ space. In Section 3, we will give the multilinear estimates which are main estimates to prove Theorems~\ref{wellposed_1} and \ref{large-wp}. In Section 4, we will give the proof of the well-posedness and the scattering (Theorem~\ref{wellposed_1}, Corollary~\ref{sccat}, and Theorem \ref{large-wp}). In Section 5, we will give the proof of Theorem~\ref{wellposed_2}. In Section 6, we will give the proof of Theorem~\ref{notC3}. \section{The $U^{p}$, $V^{p}$ spaces and their properties \label{func_sp}} In this section, we define the $U^{p}$ space and the $V^{p}$ space, and introduce the properties of these spaces which are proved by Hadac, Herr and Koch (\cite{HHK09}, \cite{HHK10}). We define the set of finite partitions $\mathcal{Z}$ as \[ \mathcal{Z} :=\left\{ \{t_{k}\}_{k=0}^{K}|K\in {\BBB N} , -\infty <t_{0}<t_{1}<\cdots <t_{K}\leq \infty \right\} \] and if $t_{K}=\infty$, we put $v(t_{K}):=0$ for all functions $v:{\BBB R} \rightarrow L^{2}$. \begin{defn}\label{upsp} Let $1\leq p <\infty$. For $\{t_{k}\}_{k=0}^{K}\in \mathcal{Z}$ and $\{\phi_{k}\}_{k=0}^{K-1}\subset L^{2}$ with $\sum_{k=0}^{K-1} \| \phi_{k} \| _{L^{2}}^{p}=1$ we call the function $a:{\BBB R}\rightarrow L^{2}$ given by \[ a(t)=\sum_{k=1}^{K}\mbox{\boldmath $1$}_{[t_{k-1},t_{k})}(t)\phi_{k-1} \] a ``$U^{p}${\rm -atom}''. Furthermore, we define the atomic space \[ U^{p}:=\left\{ \left. u=\sum_{j=1}^{\infty}\lambda_{j}a_{j} \right| a_{j}:U^{p}{\rm -atom},\ \lambda_{j}\in {\BBB C} \ {\rm such\ that}\ \sum_{j=1}^{\infty}|\lambda_{j}|<\infty \right\} \] with the norm \[ \| u \| _{U^{p}}:=\inf \left\{\sum_{j=1}^{\infty}|\lambda_{j}|\left|u=\sum_{j=1}^{\infty}\lambda_{j}a_{j},\ a_{j}:U^{p}{\rm -atom},\ \lambda_{j}\in {\BBB C}\right.\right\}. \] \end{defn} \begin{defn}\label{vpsp} Let $1\leq p <\infty$. We define the space of the bounded $p$-variation \[ V^{p}:=\{ v:{\BBB R}\rightarrow L^{2}|\ \| v \| _{V^{p}}<\infty \} \] with the norm \[ \| v \| _{V^{p}}:=\sup_{\{t_{k}\}_{k=0}^{K}\in \mathcal{Z}}\left(\sum_{k=1}^{K} \| v(t_{k})-v(t_{k-1}) \| _{L^{2}}^{p}\right)^{1/p}. \] Likewise, let $V^{p}_{-, rc}$ denote the closed subspace of all right-continuous functions $v\in V^{p}$ with $\lim_{t\rightarrow -\infty}v(t)=0$, endowed with the same norm $ \| \cdot \| _{V^{p}}$. \end{defn} \begin{prop}[\cite{HHK09} Proposition\ 2.2,\ 2.4,\ Corollary\ 2.6]\label{upvpprop} Let $1\leq p<q<\infty$. \\ {\rm (i)} $U^{p}$, $V^{p}$ and $V^{p}_{-, rc}$ are Banach spaces. \\ {\rm (ii)} For every $v\in V^{p}$, $\lim_{t\rightarrow -\infty}v(t)$ and $\lim_{t\rightarrow \infty}v(t)$ exist in $L^{2}$. \\ {\rm (iii)} The embeddings $U^{p}\hookrightarrow V^{p}_{-,rc}\hookrightarrow U^{q}\hookrightarrow L^{\infty}_{t}({\BBB R} ;L^{2}_{x}({\BBB R}^{d}))$ are continuous. \end{prop} \begin{thm}[\cite{HHK09} Proposition\ 2,10,\ Remark\ 2.12]\label{duality} Let $1<p<\infty$ and $1/p+1/p'=1$. If $u\in V^{1}_{-,rc}$ be absolutely continuous on every compact intervals, then \[ \| u \| _{U^{p}}=\sup_{v\in V^{p'}, \| v \| _{V^{p'}}=1}\left|\int_{-\infty}^{\infty}(u'(t),v(t))_{L^{2}({\BBB R}^{d})}dt\right|. \] \end{thm} \begin{defn} Let $1\leq p<\infty$. We define \[ U^{p}_{S}:=\{ u:{\BBB R}\rightarrow L^{2}|\ S(-\cdot )u\in U^{p}\} \] with the norm $ \| u \| _{U^{p}_{S}}:= \| S(-\cdot )u \| _{U^{p}}$, \[ V^{p}_{S}:=\{ v:{\BBB R}\rightarrow L^{2}|\ S(-\cdot )v\in V^{p}_{-,rc}\} \] with the norm $ \| v \| _{V^{p}_{S}}:= \| S(-\cdot )v \| _{V^{p}}$. \end{defn} \begin{rem} The embeddings $U^{p}_{S}\hookrightarrow V^{p}_{S}\hookrightarrow U^{q}_{S}\hookrightarrow L^{\infty}({\BBB R};L^{2})$ hold for $1\leq p<q<\infty$ by {\rm Proposition~\ref{upvpprop}}. \end{rem} \begin{prop}[\cite{HHK09} Corollary\ 2.18]\label{projest} Let $1< p<\infty$. We have \begin{align} & \| Q_{\geq M}^{S}u \| _{L_{tx}^{2}}\lesssim M^{-1/2} \| u \| _{V^{2}_{S}},\label{highMproj}\\ & \| Q_{<M}^{S}u \| _{V^{p}_{S}}\lesssim \| u \| _{V^{p}_{S}},\ \ \| Q_{\geq M}^{S}u \| _{V^{p}_{S}}\lesssim \| u \| _{V^{p}_{S}},\label{Vproj} \end{align} \end{prop} \begin{prop}[\cite{HHK09} Proposition\ 2.19]\label{multiest} Let \[ T_{0}:L^{2}({\BBB R}^{d})\times \cdots \times L^{2}({\BBB R}^{d})\rightarrow L^{1}_{loc}({\BBB R}^{d}) \] be a $m$-linear operator. Assume that for some $1\leq p, q< \infty$ \[ \| T_{0}(S(\cdot )\phi_{1},\cdots ,S(\cdot )\phi_{m}) \| _{L^{p}_{t}({\BBB R} :L^{q}_{x}({\BBB R}^{d}))}\lesssim \prod_{i=1}^{m} \| \phi_{i} \| _{L^{2}({\BBB R}^{d})}. \] Then, there exists $T:U^{p}_{S}\times \cdots \times U^{p}_{S}\rightarrow L^{p}_{t}({\BBB R} ;L^{q}_{x}({\BBB R}^{d}))$ satisfying \[ \| T(u_{1},\cdots ,u_{m}) \| _{L^{p}_{t}({\BBB R} ;L^{q}_{x}({\BBB R}^{d}))}\lesssim \prod_{i=1}^{m} \| u_{i} \| _{U^{p}_{S}} \] such that $T(u_{1},\cdots ,u_{m})(t)(x)=T_{0}(u_{1}(t),\cdots ,u_{m}(t))(x)$ a.e. \end{prop} Now we refer the Strichartz estimate for the fourth order Schr\"odinger equation proved by Pausader. We say that a pair $(p,q)$ is admissible if $2 \le p,q \le \infty$, $(p,q,d) \neq (2, \infty ,2)$, and \[ \frac{2}{p} + \frac{d}{q} = \frac{d}{2}. \] \begin{prop}[\cite{P07} Proposition\ 3.1]\label{Stri_est} Let $(p,q)$ and $(a,b)$ be admissible pairs. Then, we have \[ \begin{split} \| S(\cdot )\varphi \| _{L_{t}^{p}L_{x}^{q}}&\lesssim \| |\nabla|^{-2/p}\varphi \| _{L^{2}_{x}},\\ \left\| \int_{0}^{t}S(t-t' )F(t')dt'\varphi \right\| _{L_{t}^{p}L_{x}^{q}}&\lesssim \| |\nabla|^{-2/p-2/a}F \| _{L^{a'}_{t}L^{b'}_{x}}, \end{split} \] where $a'$ and $b'$ are conjugate exponents of $a$ and $b$ respectively. \end{prop} Propositions \ref{multiest} and ~\ref{Stri_est} imply the following. \begin{cor}\label{Up_Stri} Let $(p,q)$ be an admissible pair. \begin{equation}\label{U_Stri} \| u \| _{L_{t}^{p}L_{x}^{q}}\lesssim \| |\nabla|^{-2/p}u \| _{U_{S}^{p}},\ \ u\in U^{p}_{S}. \end{equation} \end{cor} Next, we define the function spaces which will be used to construct the solution. We define the projections $P_{>1}$ and $P_{<1}$ as \[ P_{>1}:=\sum_{N\ge 1}P_N,\ P_{<1}:=Id-P_{>1}. \] \begin{defn}\label{YZ_space} Let $s <0$.\\ {\rm (i)} We define $\dot{Z}^{s}:=\{u\in C({\BBB R} ; \dot{H}^{s}({\BBB R}^{d}))\cap U^{2}_{S}|\ \| u \| _{\dot{Z}^{s}}<\infty\}$ with the norm \[ \| u \| _{\dot{Z}^{s}}:=\left(\sum_{N}N^{2s} \| P_{N}u \| ^{2}_{U^{2}_{S}}\right)^{1/2}. \] {\rm (ii)} We define $Z^{s}:=\{u\in C({\BBB R} ; H^{s}({\BBB R}^{d})) |\ \| u \| _{Z^{s}}<\infty\}$ with the norm \[ \| u \| _{Z^{s}}:= \| P_{<1} u \| _{\dot{Z}^{0}}+ \| P_{>1} u \| _{\dot{Z}^{s}}. \] {\rm (iii)} We define $\dot{Y}^{s}:=\{u\in C({\BBB R} ; \dot{H}^{s}({\BBB R}^{d}))\cap V^{2}_{S}|\ \| u \| _{\dot{Y}^{s}}<\infty\}$ with the norm \[ \| u \| _{\dot{Y}^{s}}:=\left(\sum_{N}N^{2s} \| P_{N}u \| ^{2}_{V^{2}_{S}}\right)^{1/2}. \] {\rm (iv)} We define $Y^{s}:=\{u\in C({\BBB R} ; H^{s}({\BBB R}^{d})) |\ \| u \| _{Y^{s}}<\infty\}$ with the norm \[ \| u \| _{Y^{s}}:= \| P_{<1} u \| _{\dot{Y}^{0}}+ \| P_{>1 }u \| _{\dot{Y}^{s}}. \] \end{defn} \section{Multilinear estimate for $P_{4}(u,\overline{u})=\overline{u}^{4}$ in $1d$ \label{Multi_est}} In this section, we prove multilinear estimates for the nonlinearity $\partial_{x}(\overline{u}^{4})$ in $1d$, which plays a crucial role in the proof of Theorem \ref{wellposed_1}. \begin{lemm}\label{modul_est} We assume that $(\tau_{0},\xi_{0})$, $(\tau_{1}, \xi_{1})$, $\cdots$, $(\tau_{4}, \xi_{4})\in {\BBB R}\times {\BBB R}^{d}$ satisfy $\sum_{j=0}^{4}\tau_{j}=0$ and $\sum_{j=0}^{4}\xi_{j}=0$. Then, we have \begin{equation}\label{modulation_est} \max_{0\leq j\leq 4}|\tau_{j}-|\xi_{j}|^{4}| \geq \frac{1}{5}\max_{0\leq j\leq 4}|\xi_{j}|^{4}. \end{equation} \end{lemm} \begin{proof} By the triangle inequality, we obtain (\ref{modulation_est}). \end{proof} \subsection{The homogeneous case} \begin{prop}\label{HL_est_n} Let $d=1$ and $0<T\leq \infty$. For a dyadic number $N_{1}\in 2^{{\BBB Z}}$, we define the set $A_{1}(N_{1})$ as \[ A_{1}(N_{1}):=\{ (N_{2},N_{3},N_{4})\in (2^{{\BBB Z}})^{3}|N_{1}\gg N_{2}\geq N_{3} \geq N_{4}\}. \] If $N_{0}\sim N_{1}$, then we have \begin{equation}\label{hl} \begin{split} &\left|\sum_{A_{1}(N_{1})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right|\\ &\lesssim \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}\prod_{j=2}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}}. \end{split} \end{equation} \end{prop} \begin{proof} We define $u_{j,N_{j},T}:=\mbox{\boldmath $1$}_{[0,T)}P_{N_{j}}u_{j}$\ $(j=1,\cdots ,4)$ and put $M:=N_{0}^{4}/5$. We decompose $Id=Q^{S}_{<M}+Q^{S}_{\geq M}$. We divide the integrals on the left-hand side of (\ref{hl}) into $10$ pieces of the form \begin{equation}\label{piece_form_hl} \int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}Q_{j}^{S}u_{j,N_{j},T}\right) dxdt \end{equation} with $Q_{j}^{S}\in \{Q_{\geq M}^{S}, Q_{<M}^{S}\}$\ $(j=0,\cdots ,4)$. By the Plancherel's theorem, we have \[ (\ref{piece_form_hl}) = c\int_{\sum_{j=0}^{4}\tau_{j}=0}\int_{\sum_{j=0}^{4}\xi_{j}=0}N_{0}\prod_{j=0}^{4}\mathcal{F}[Q_{j}^{S}u_{j,N_{j},T}](\tau_{j},\xi_{j}), \] where $c$ is a constant. Therefore, Lemma~\ref{modul_est} implies that \[ \int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{m}Q_{<M}^{S}u_{j,N_{j},T}\right) dxdt=0. \] So, let us now consider the case that $Q_{j}^{S}=Q_{\geq M}^{S}$ for some $0\leq j\leq 4$. First, we consider the case $Q_{0}^{S}=Q_{\geq M}^{S}$. By the Cauchy-Schwartz inequality, we have \[ \begin{split} &\left|\sum_{A_{1}(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{0,N_{0},T}\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| Q_{\geq M}^{S}u_{0,N_{0},T} \| _{L^{2}_{tx}} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}}\prod_{j=2}^{4}\left\|\sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\right\|_{L^{12}_{t}L^{6}_{x}}. \end{split} \] Furthermore by (\ref{highMproj}) and $M\sim N_{0}^{4}$, we have \[ \| Q_{\geq M}^{S}u_{0,N_{0},T} \| _{L^{2}_{tx}} \lesssim N_{0}^{-2} \| u_{0,N_{0},T} \| _{V^{2}_{S}} \] and by (\ref{U_Stri}) and $V^{2}_{S}\hookrightarrow U^{4}_{S}$, we have \[ \begin{split} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L_{t}^{4}L_{x}^{\infty}} &\lesssim N_{1}^{-1/2} \| Q_{1}^{S}u_{1,N_{1},T} \| _{U^{4}_{S}} \lesssim N_{1}^{-1/2} \| Q_{1}^{S}u_{1,N_{1},T} \| _{V^{2}_{S}}. \end{split} \] While by the Sobolev inequality, (\ref{U_Stri}), $V^{2}_{S}\hookrightarrow U^{12}_{S}$ and the Cauchy-Schwartz inequality for the dyadic sum , we have \begin{equation}\label{L12L6_est} \begin{split} \left\|\sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\right\|_{L^{12}_{t}L^{6}_{x}} &\lesssim \left\| |\nabla |^{1/6}\sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \right\| _{L^{12}_{t}L^{3}_{x}} \lesssim \left\| \sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \right\| _{V^{2}_{S}}\\ & \lesssim N_1^{1/2} \left( \sum _{N_j \lesssim N_1} N_j^{-1} \| u_{j,N_j,T} \|_{V^2_S}^2 \right) ^{1/2} \lesssim N_{1}^{1/2} \| \mbox{\boldmath $1$}_{[0,T)}u_{j} \| _{\dot{Y}^{-1/2}} \end{split} \end{equation} for $2\leq j\leq 4$. Therefore, we obtain \[ \begin{split} &\left|\sum_{A_{1}(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{0,N_{0},T}\prod_{j=1}^{m}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\lesssim \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}\prod_{j=2}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}} \end{split} \] by (\ref{Vproj}) since $ \| \mbox{\boldmath $1$}_{[0,T)}u \| _{V^{2}_{S}}\lesssim \| u \| _{V^{2}_{S}}$ for any $T\in (0,\infty]$. For the case $Q_{1}^{S}=Q_{\geq M}^{S}$ is proved in same way. Next, we consider the case $Q_{i}^{S}=Q_{\geq M}^{S}$ for some $2\le i \le 4$. By the H\"older inequality, we have \[ \begin{split} &\left|\sum_{A_{1}(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{i,N_{i},T}\prod_{\substack{0\le j\le 4\\ j\neq i}}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\lesssim N_{0} \| Q_{0}^{S}u_{0,N_{0},T} \| _{L_{t}^{12}L_{x}^{6}} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L_{t}^{4}L_{x}^{\infty}}\\ &\ \ \ \ \times \left\| \sum_{N_{i}\lesssim N_{1}}Q_{\geq M}^{S}u_{i,N_{i},T} \right\|_{L_{tx}^{2}} \prod_{\substack{2\le j\le 4 \\ j\neq i}}\left\| \sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \right\| _{L_{t}^{12}L_{x}^{6}}. \end{split} \] By $L^{2}$ orthogonality and (\ref{highMproj}), we have \begin{equation}\label{hi_mod_234} \begin{split} \left\| \sum_{N_{i}\lesssim N_{1}}Q_{\geq M}^{S}u_{i,N_{i},T}\right\| _{L_{tx}^{2}} &\lesssim \left(\sum_{N_{2}} \| Q_{\geq M}^{S}u_{i,N_{i},T} \| _{L_{tx}^{2}}^{2}\right)^{1/2}\\ &\lesssim N_{1}^{-3/2} \| \mbox{\boldmath $1$}_{[0,T)}u_{i} \| _{\dot{Y}^{-1/2}} \end{split} \end{equation} since $M\sim N_{0}^{4}$. While, by the calculation way as the case $Q_{0}^{S}=Q_{\geq M}^{S}$, we have \[ \| Q_{0}^{S}u_{0,N_{0},T} \| _{L_{t}^{12}L_{x}^{6}}\lesssim \| Q_{0}^{S}u_{0,N_{0},T} \| _{V^{2}_{S}}, \] \[ \| Q_{1}^{S}u_{1,N_{1},T} \| _{L_{t}^{4}L_{x}^{\infty}}\lesssim N_{1}^{-1/2} \| Q_{1}^{S}u_{1,N_{1},T} \| _{V^{2}_{S}} \] and \[ \left\| \sum_{N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T} \right\|_{L_{t}^{12}L_{x}^{6}} \lesssim N_{1}^{1/2} \| \mbox{\boldmath $1$}_{[0,T)}u_{j} \| _{\dot{Y}^{-1/2}}. \] Therefore, we obtain \[ \begin{split} &\left|\sum_{A_{1}(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{i,N_{i},T}\prod_{\substack{0\le j\le 4\\ j\neq i}}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\lesssim \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}\prod_{j=2}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}} \end{split} \] by (\ref{Vproj}) since $ \| \mbox{\boldmath $1$}_{[0,T)}u \| _{V^{2}_{S}}\lesssim \| u \| _{V^{2}_{S}}$ for any $T\in (0,\infty]$. \end{proof} \begin{prop}\label{HH_est} Let $d=1$ and $0<T\leq \infty$. For a dyadic number $N_{2}\in 2^{{\BBB Z}}$, we define the set $A_{2}(N_{2})$ as \[ A_{2}(N_{2}):=\{ (N_{3}, N_{4})\in (2^{{\BBB Z}})^{4}|N_{2}\geq N_{3}\geq N_{4}\}. \] If $N_{0}\lesssim N_{1}\sim N_{2}$, then we have \begin{equation}\label{hh} \begin{split} &\left|\sum_{A_{2}(N_{2})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right|\\ &\lesssim \frac{N_{0}}{N_{1}} \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}N_{2}^{-1/2} \| P_{N_{2}}u_{2} \| _{V^{2}_{S}} \| u_{3} \| _{\dot{Y}^{-1/2}} \| u_{4} \| _{\dot{Y}^{-1/2}}. \end{split} \end{equation} \end{prop} The proof of Proposition~\ref{HH_est} is quite similar as the proof of Proposition~\ref{HL_est_n}. \subsection{The inhomogeneous case} \begin{prop}\label{HL_est_n-inh} Let $d=1$ and $0<T\leq 1$. For a dyadic number $N_{1}\in 2^{{\BBB Z}}$, we define the set $A_{1}'(N_{1})$ as \[ A_{1}'(N_{1}):=\{ (N_{2},N_{3},N_{4})\in (2^{{\BBB Z}})^{3}|N_{1}\gg N_{2}\geq N_{3} \ge N_{4}, \, N_4 \le 1 \}. \] If $N_{0}\sim N_{1}$, then we have \begin{equation}\label{hl-inh} \begin{split} \left|\sum_{A_{1}'(N_{1})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \lesssim T^{\frac{1}{6}} \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}} \prod_{j=2}^{4} \| u_{j} \| _{Y^{-1/2}}. \end{split} \end{equation} \end{prop} \begin{proof} We further divide $A_1'(N_1)$ into three pieces: \begin{align*} A_1'(N_1) & = \bigcup _{j=1}^3 A_{1,j}'(N_1), \\ A_{1,1}'(N_1) &:= \{ (N_{2},N_{3},N_{4}) \in A_1'(N_1) : N_3 \ge 1 \} ,\\ A_{1,2}'(N_2) &:= \{ (N_{2},N_{3},N_{4}) \in A_1'(N_1) : N_2 \ge 1 \ge N_3 \} ,\\ A_{1,3}'(N_2) &:= \{ (N_{2},N_{3},N_{4}) \in A_1'(N_1) : 1 \ge N_2 \} . \end{align*} We define $u_{j,N_{j}}:=P_{N_{j}}u_{j}$, $u_{j,T}:=\mbox{\boldmath $1$}_{[0,T)}u_{j}$ and $u_{j,N_{j},T}:=\mbox{\boldmath $1$}_{[0,T)}P_{N_{j}}u_{j}$\ $(j=1,\cdots ,4)$. We firstly consider the case $A_{1,1}'(N_1)$ In the case $T \le N_0^{-3}$, the H\"older inequality implies \begin{align*} & \left|\sum_{A_{1,1}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \\ & \le N_0 \| \mbox{\boldmath $1$}_{[0,T)}\|_{L^{2}_{t}} \| u_{0,N_0} \| _{L_t^4 L_x^{\infty}} \| u_{1,N_1} \| _{L_t^4 L_x^{\infty}} \prod _{j=2}^3 \left\| \sum_{1\le N_j \le N_{1}} u_{j,N_j} \right\| _{L_t^{\infty} L_x^2} \| P_{<1} u_{4} \| _{L_t^{\infty} L_x^{\infty}} \end{align*} Furthermore by (\ref{U_Stri}) and $V^{2}_{S}\hookrightarrow U^{4}_{S}$, we have \[ \begin{split} \| u_{0,N_0} \| _{L_t^4 L_x^{\infty}} \| u_{1,N_1} \| _{L_t^4 L_x^{\infty}} &\lesssim N_{0}^{-1/2}\| u_{0,N_0} \| _{U^{4}_{S}}N_{1}^{-1/2} \| Q_{1}^{S}u_{1,N_{1}} \| _{U^{4}_{S}}\\ &\lesssim N_{0}^{-1} \| u_{0,N_{0}} \| _{V^{2}_{S}} \| u_{1,N_{1}} \| _{V^{2}_{S}} \end{split} \] and by the Sobolev inequality, $V^{2}_{S}\hookrightarrow L^{\infty}_{t}L^{2}_{x}$ and the Cauchy-Schwartz inequality , we have \[ \| P_{<1} u_{4} \| _{L_t^{\infty} L_x^{\infty}}\lesssim \| P_{<1} u_{4} \| _{L_t^{\infty} L_x^{2}} \lesssim \left(\sum_{N\le 2}\|P_{N}P_{<1}u_{4}\|_{V^{2}_{S}}^{2}\right)^{1/2} \le \|P_{<1}u_4\|_{\dot{Y}^{0}} \] While by $L^{2}$ orthogonality and $V^{2}_{S}\hookrightarrow L^{\infty}_{t}L^{2}_{x}$, we have \[ \begin{split} \left\| \sum_{1\le N_j \le N_{1}} u_{j,N_j} \right\| _{L_t^{\infty} L_x^2} &\lesssim \left(\sum_{1\le N_j \le N_{1}} \| u_{j,N_{j}} \| _{V^{2}_{S}}^{2}\right)^{1/2} \lesssim N_{0}^{1/2} \| P_{>1}u_{j} \| _{\dot{Y}^{-1/2}} \end{split} \] Therefore, we obtain \[ \begin{split} &\left|\sum_{A_{1,1}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \\ &\lesssim T^{1/2}N_0 \| u_{0,N_{0}} \| _{V^{2}_{S}} \| u_{1,N_{1}} \| _{V^{2}_{S}}\prod_{j=2}^{3}\| P_{>1}u_{j} \| _{\dot{Y}^{-1/2}}\|P_{<1}u_4\|_{\dot{Y}^{0}} \end{split} \] and note that $T^{1/2}N_0\le T^{1/6}$. In the case $T \ge N_0^{-3}$, we divide the integrals on the left-hand side of (\ref{hl}) into $10$ pieces of the form \eqref{piece_form_hl} in the proof of Proposition \ref{HL_est_n}. Thanks to Lemma~\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\geq M}^{S}$ for some $0\leq j\leq 4$. First, we consider the case $Q_{0}^{S}=Q_{\geq M}^{S}$. By the same way as in the proof of Proposition \ref{HL_est_n} and using \[ \|Q_{4}^{S}P_{<1}u_{4,T}\|_{L^{12}_{t}L^{6}_{x}}\lesssim \|Q_{4}^{S}P_{<1}u_{4,T}\|_{V^{2}_{S}}\lesssim \|P_{<1}u_{4,T}\|_{\dot{Y}^{0}} \] instead of (\ref{L12L6_est}), we obtain \[ \begin{split} &\left|\sum_{A_{1,1}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{0,N_{0},T}\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| Q_{\geq M}^{S}u_{0,N_{0},T} \| _{L^{2}_{tx}} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \prod_{j=2}^{3} \left\|\sum_{1 \le N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\right\|_{L^{12}_{t}L^{6}_{x}} \|Q_{4}^{S}P_{<1}u_{4,T}\|_{L^{12}_{t}L^{6}_{x}}\\ & \lesssim N_0^{-\frac{1}{2}} \| P_{N_0} u_0 \| _{V^2_S} \| P_{N_1} u_1 \| _{V^2_S} \prod_{j=2}^{3} \left\| P_{>1} u_j \right\| _{\dot{Y}^{-1/2}} \| P_{<1} u_{4} \| _{\dot{Y}^0} \end{split} \] and note that $N_0^{-1/2}\le T^{1/6}$. Since the cases $Q_j^S = Q_{\ge M}^S$ ($j=1,2,3$) are similarly handled, we omit the details here. We focus on the case $Q_4^S = Q_{\ge M}^S$. By the same way as in the proof of Proposition \ref{HL_est_n} and using \[ \|Q_{\ge M}^{S}P_{<1}u_{4,T}\|_{L^{2}_{tx}}\lesssim N_{0}^{-2} \|P_{<1}u_{4,T}\|_{V^{2}_{S}}\lesssim N_{0}^{-2}\|P_{<1}u_{4,T}\|_{\dot{Y}^{0}} \] instead of (\ref{hi_mod_234}) with $j=4$, we obtain \[ \begin{split} &\left|\sum_{A_{1,1}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{4,N_{4},T}\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| u_{0,N_{0},T} \| _{L^{12}_{t}L_x^6} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \prod_{j=2}^{3} \left\|\sum_{1 \le N_{j}\lesssim N_{1}}Q_{j}^{S}u_{j,N_{j},T}\right\|_{L^{12}_{t}L^{6}_{x}} \|Q_{\geq M}^{S} P_{<1}u_{4,T}\|_{L^{2}_{tx}}\\ & \lesssim N_{0}^{-1/2}\| P_{N_0} u_0 \| _{V^2_S} \| P_{N_1} u_1 \| _{V^2_S} \prod_{j=2}^{3} \left\| P_{>1} u_j \right\| _{\dot{Y}^{-1/2}} \| P_{<1} u_4 \| _{\dot{Y}^0} \end{split} \] and note that $N_0^{-1/2}\le T^{1/6}$. We secondly consider the case $A_{1,2}'(N_1)$. In the case $T \le N_0^{-3}$, the H\"older inequality implies \[ \begin{split} & \left|\sum_{A_{1,2}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \\ & \le N_0 \| \mbox{\boldmath $1$}_{[0,T)}\|_{L^{2}_{t}} \| u_{0,N_0} \| _{L_t^4 L_x^{\infty}} \| u_{1,N_1} \| _{L_t^4 L_x^{\infty}} \left\| \sum _{1 \le N_2 \lesssim N_1} u_{2,N_2} \right\| _{L_t^{\infty} L_x^2} \prod_{j=3}^{4}\| P_{<1} u_{j} \| _{L_t^{\infty} L_x^4} . \end{split} \] By the same estimates as in the proof for the case $A_{1,1}'(N_1)$ and \[ \| P_{<1} u_{j} \| _{L_t^{\infty} L_x^4}\lesssim \| P_{<1} u_{j} \| _{L_t^{\infty} L_x^{2}} \lesssim \left(\sum_{N\le 2}\|P_{N}P_{<1}u_{j}\|_{V^{2}_{S}}^{2}\right)^{1/2} \le \|P_{<1}u_j\|_{\dot{Y}^{0}} \] for $j=3,4$, we obtain \[ \begin{split} &\left|\sum_{A_{1,2}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \\ &\lesssim T^{1/2}N_0^{1/2} \| u_{0,N_{0}} \| _{V^{2}_{S}} \| u_{1,N_{1}} \| _{V^{2}_{S}}\| P_{>1}u_{2} \| _{\dot{Y}^{-1/2}}\prod_{j=3}^{4}\|P_{<1}u_j\|_{\dot{Y}^{0}} \end{split} \] and note that $T^{1/2}N_0^{1/2}\le T^{1/3}$. In the case $T \ge N_0^{-3}$, we divide the integrals on the left-hand side of (\ref{hl}) into $10$ pieces of the form \eqref{piece_form_hl} in the proof of Proposition \ref{HL_est_n}. Thanks to Lemma~\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\geq M}^{S}$ for some $0\leq j\leq 4$. By the same argument as in the proof for the case $A_{1,1}'(N_1)$, we obtain \[ \begin{split} &\left|\sum_{A_{1,2}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{0,N_{0},T}\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| Q_{\geq M}^{S}u_{0,N_{0},T} \| _{L^{2}_{tx}} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \left\|\sum_{1 \le N_{2}\lesssim N_{1}}Q_{2}^{S}u_{2,N_{2},T}\right\|_{L^{12}_{t}L^{6}_{x}} \prod_{j=3}^{4} \| Q_{j}^{S}P_{<1}u_{j,T}\|_{L^{12}_{t}L^{6}_{x}}\\ & \lesssim N_0^{-1} \| P_{N_0} u_0 \| _{V^2_S} | P_{N_1} u_1 \| _{V^2_S} \left\| P_{>1} u_2 \right\| _{\dot{Y}^{-1/2}} \prod _{j=3}^4 \| P_{<1} v_j \| _{\dot{Y}^0} \end{split} \] if $Q_0 = Q_{\ge M}^S$ and \[ \begin{split} &\left|\sum_{A_{1,2}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{4}Q_{\geq M}^{S}u_{4,N_{4},T}\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| u_{0,N_{0},T} \| _{L^{12}_{t}L_x^6} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \left\|\sum_{1 \le N_{2}\lesssim N_{1}}Q_{2}^{S}u_{2,N_{2},T}\right\|_{L^{12}_{t}L^{6}_{x}} \\ &\hspace{21ex}\times \|Q_{3}^{S} P_{<1}u_{3,T}\|_{L^{12}_{t}L^{6}_{x}} \| Q_{\geq M}^{S} P_{<1}u_{4,T}\|_{L^{2}_{tx}}\\ & \lesssim N_0^{-1} \| P_{N_0} u_0 \| _{V^2_S} \| P_{N_1} u_1 \| _{V^2_S} \left\| P_{>1} u_2 \right\| _{\dot{Y}^{\frac{1}{2}}} \prod_{j=3}^{4}\| P_{<1} u_j \| _{\dot{Y}^0} \end{split} \] if $Q_4 = Q_{\ge M}^S$ Note that $N_0^{-1}\le T^{1/3}$. The remaining cases follow from the same argument as above. We thirdly consider the case $A_{1,3}'(N_1)$. In the case $T \le N_0^{-3}$, the H\"older inequality implies \[ \begin{split} & \left|\sum_{A_{1,3}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \\ & \le N_0 \| \mbox{\boldmath $1$}_{[0,T)}\|_{L^{2}_{t}}\| u_{0,N_0} \| _{L_t^4 L_x^{\infty}} \| u_{1,N_1} \| _{L_t^4 L_x^{\infty}} \prod_{j=2}^{4} \| P_{<1}u_{2} \| _{L_t^{\infty} L_x^3}. \end{split} \] By the same estimates as in the proof for the case $A_{1,1}'(N_1)$ and \[ \| P_{<1} u_{j} \| _{L_t^{\infty} L_x^3}\lesssim \| P_{<1} u_{j} \| _{L_t^{\infty} L_x^{2}} \lesssim \left(\sum_{N\le 2}\|P_{N}P_{<1}u_{j}\|_{V^{2}_{S}}^{2}\right)^{1/2} \le \|P_{<1}u_j\|_{\dot{Y}^{0}} \] for $j=2, 3,4$, we obtain \[ \begin{split} &\left|\sum_{A_{1,3}'(N_{1})} \int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right| \lesssim T^{1/2}\| u_{0,N_{0}} \| _{V^{2}_{S}} \| u_{1,N_{1}} \| _{V^{2}_{S}}\prod_{j=2}^{4}\| P_{<1}u_{j} \| _{\dot{Y}^{0}}. \end{split} \] In the case $T \ge N_0^{-3}$, we divide the integrals on the left-hand side of (\ref{hl}) into $10$ pieces of the form \eqref{piece_form_hl} in the proof of Proposition \ref{HL_est_n}. Thanks to Lemma~\ref{modul_est}, let us consider the case that $Q_{j}^{S}=Q_{\geq M}^{S}$ for some $0\leq j\leq 4$. By the same argument as in the proof for the case $A_{1,1}'(N_1)$, we obtain \[ \begin{split} &\left|\sum_{A_{1,3}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{0}Q_{\geq M}^{S}u_{0,N_{0},T}\prod_{j=1}^{4}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| Q_{\geq M}^{S}u_{0,N_{0},T} \| _{L^{2}_{tx}} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \prod_{j=2}^{4} \|Q_{j}^{S}P_{<1}u_{j,T}\|_{L^{12}_{t}L^{6}_{x}}\\ & \lesssim N_0^{-3/2} \| P_{N_0} u_0 \| _{V^2_S} \| P_{N_1} u_1 \| _{V^2_S} \left\| P_{<1} u_2 \right\| _{Y^{-1/2}} \prod _{j=3}^4 \| P_{<1} v_j \| _{\dot{Y}^0} \end{split} \] if $Q_0 = Q_{\ge M}^S$ and \[ \begin{split} &\left|\sum_{A_{1,3}'(N_{1})}\int_{{\BBB R}}\int_{{\BBB R}}\left(N_{4}Q_{\geq M}^{S}u_{4,N_{4},T}\prod_{j=0}^{3}Q_{j}^{S}u_{j,N_{j},T}\right)dxdt\right|\\ &\leq N_{0} \| u_{0,N_{0},T} \| _{L^{12}_{t}L_x^6} \| Q_{1}^{S}u_{1,N_{1},T} \| _{L^{4}_{t}L^{\infty}_{x}} \prod _{j=2}^3 \|Q_{j}^{S} P_{<1}u_{j,T}\|_{L^{12}_{t}L^{6}_{x}} \|Q_{\geq M}^{S} P_{<1}u_{4,T}\|_{L^2_{tx}}\\ & \lesssim N_0^{-3/2} \| P_{N_0} u_0 \| _{V^2_S} \| P_{N_1} u_1 \| _{V^2_S} \prod _{j=2}^4 \left\| P_{<1} u_j \right\| _{Y^{0}} \end{split} \] if $Q_4 = Q_{\ge M}^S$. Note that $N_0^{-3/2}\le T^{1/2}$. The cases $Q_j^S = Q_{\ge M}^S$ ($j=1,2,3$) are the same argument as above. \end{proof} Furthermore, we obtain the following estimate. \begin{prop}\label{HH_est-inh} Let $d=1$ and $0<T\leq 1$. For a dyadic number $N_{2}\in 2^{{\BBB Z}}$, we define the set $A_{2}'(N_{2})$ as \[ A_{2}'(N_{2}):=\{ (N_{3}, N_{4})\in (2^{{\BBB Z}})^{4}|N_{2}\geq N_{3}\ge N_{4} , \, N_4 \le 1 \}. \] If $N_{0}\lesssim N_{1}\sim N_{2}$, then we have \begin{equation}\label{hh-inh} \begin{split} &\left|\sum_{A_{2}'(N_{2})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right|\\ &\lesssim T^{\frac{1}{6}} \frac{N_{0}}{N_{1}} \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}N_{2}^{-1/2} \| P_{N_{2}}u_{2} \| _{V^{2}_{S}} \| u_{3} \| _{Y^{-1/2}} \| u_{4} \| _{Y^{-1/2}}. \end{split} \end{equation} \end{prop} Because the proof is similar as above, we skip the proof. \section{Proof of well-posedness \label{pf_wellposed_1}} \subsection{The small data case} In this section, we prove Theorem~\ref{wellposed_1} and Corollary~\ref{sccat}. We define the map $\Phi_{T, \varphi}$ as \[ \Phi_{T, \varphi}(u)(t):=S(t)\varphi -iI_{T}(u,\cdots, u)(t), \] where \[ I_{T}(u_{1},\cdots u_{4})(t):=\int_{0}^{t}\mbox{\boldmath $1$}_{[0,T)}(t')S(t-t')\partial_{x}\left(\prod_{j=1}^{4}\overline{u_{j}(t')}\right)dt'. \] To prove the well-posedness of (\ref{D4NLS}) in $\dot{H}^{-1/2}$, we prove that $\Phi_{T, \varphi}$ is a contraction map on a closed subset of $\dot{Z}^{-1/2}([0,T))$. Key estimate is the following: \begin{prop}\label{Duam_est} Let $d=1$. For any $0<T<\infty$, we have \begin{equation}\label{Duam_est_1} \| I_{T}(u_{1},\cdots u_{4}) \| _{\dot{Z}^{-1/2}}\lesssim \prod_{j=1}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}}. \end{equation} \end{prop} \begin{proof} We decompose \[ I_{T}(u_{1},\cdots u_{m})=\sum_{N_{1},\cdots ,N_{4}}I_{T}(P_{N_{1}}u_{1},\cdots P_{N_{4}}u_{4}). \] By symmetry, it is enough to consider the summation for $N_{1}\geq N_{2}\geq N_{3} \geq N_{4}$. We put \[ \begin{split} S_{1}&:=\{ (N_{1},\cdots ,N_{m})\in (2^{{\BBB Z}})^{m}|N_{1}\gg N_{2}\geq N_{3} \geq N_{4}\}\\ S_{2}&:=\{ (N_{1},\cdots ,N_{m})\in (2^{{\BBB Z}})^{m}|N_{1}\sim N_{2}\geq N_{3} \geq N_{4}\} \end{split} \] and \[ J_{k}:=\left\| \sum_{S_{k}}I_{T}(P_{N_{1}}u_{1},\cdots P_{N_{4}}u_{4}) \right\| _{\dot{Z}^{-1/2}}\ (k=1,2). \] First, we prove the estimate for $J_{1}$. By Theorem~\ref{duality} and the Plancherel's theorem, we have \[ \begin{split} J_{1}&\leq \left\{ \sum_{N_{0}}N_{0}^{-1}\left\| S(-\cdot )P_{N_{0}}\sum_{S_{1}}I_{T}(P_{N_{1}}u_{1},\cdots P_{N_{4}}u_{4})\right\|_{U^{2}}^{2}\right\}^{1/2}\\ &\lesssim \left\{\sum_{N_{0}}N_{0}^{-1}\sum_{N_{1}\sim N_{0}} \left( \sup_{ \| u_{0} \| _{V^{2}_{S}}=1}\left|\sum_{A_{1}(N_{1})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right|\right)^{2}\right\}^{1/2}, \end{split} \] where $A_{1}(N_{1})$ is defined in Proposition~\ref{HL_est_n}. Therefore by Proposition~\ref{HL_est_n}, we have \[ \begin{split} J_{1}&\lesssim \left\{\sum_{N_{0}}N_{0}^{-1}\sum_{N_{1}\sim N_{0}} \left( \sup_{ \| u_{0} \| _{V^{2}_{S}}=1} \| P_{N_{0}}u_{0} \| _{V^{2}_{S}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}\prod_{j=2}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}}\right)^{2}\right\}^{1/2}\\ &\lesssim \left(\sum_{N_{1}}N_{1}^{-1} \| P_{N_{1}}u_{1} \| _{V^{2}_{\Delta}}^{2}\right)^{1/2} \prod_{j=2}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}}\\ &=\prod_{j=1}^{4} \| u_{j} \| _{\dot{Y}^{-1/2}}. \end{split} \] Next, we prove the estimate for $J_{2}$. By Theorem~\ref{duality} and the Plancherel's theorem, we have \[ \begin{split} J_{2}&\leq \sum_{N_{1}}\sum_{N_{2}\sim N_{1}}\left(\sum_{N_{0}}N_{0}^{-1}\left\|S(-\cdot )P_{N_{0}}\sum_{A_{2}(N_{2})}I_{T}(P_{N_{1}}u_{1},\cdots P_{N_{4}}u_{4})\right\|_{U^{2}}^{2}\right)^{1/2}\\ &=\sum_{N_{1}}\sum_{N_{2}\sim N_{1}}\left(\sum_{N_{0}\lesssim N_{1}}N_{0}^{-1} \sup_{ \| u_{0} \| _{V^{2}_{S}}=1}\left| \sum_{A_{2}(N_{2})}\int_{0}^{T}\int_{{\BBB R}}\left(N_{0}\prod_{j=0}^{4}P_{N_{j}}u_{j}\right)dxdt\right|^{2}\right)^{1/2}, \end{split} \] where $A_{2}(N_{2})$ is defined in Proposition~\ref{HH_est}. Therefore by {\rm Proposition~\ref{HH_est}} and Cauchy-Schwartz inequality for the dyadic sum, we have \[ \begin{split} J_{2}&\lesssim \sum_{N_{1}}\sum_{N_{2}\sim N_{1}}\left(\sum_{N_{0}\lesssim N_{1}}N_{0}^{-1} \left(\frac{N_{0}}{N_{1}} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}N_{2}^{-1/2} \| P_{N_{2}}u_{2} \| _{V^{2}_{S}} \| u_{3} \| _{\dot{Y}^{-1/2}} \| u_{4} \| _{\dot{Y}^{-1/2}}\right)^{2}\right)^{1/2}\\ &\lesssim \left(\sum_{N_{1}}N_{1}^{-1} \| P_{N_{1}}u_{1} \| _{V^{2}_{S}}^{2}\right)^{1/2} \left(\sum_{N_{2}}N_{2}^{-1} \| P_{N_{2}}u_{2} \| _{V^{2}_{S}}^{2}\right)^{1/2} \| u_{3} \| _{\dot{Y}^{-1/2}} \| u_{4} \| _{\dot{Y}^{-1/2}}\\ &= \prod_{j=1}^{4} \| u_{j} \| _{\dot{Y}^{s_{c}}}. \end{split} \] \end{proof} \begin{proof}[\rm{\bf{Proof of Theorem~\ref{wellposed_1}.}}] For $r>0$, we define \begin{equation}\label{Zr_norm} \dot{Z}^{s}_{r}(I) :=\left\{u\in \dot{Z}^{s}(I)\left|\ \| u \| _{\dot{Z}^{s}(I)}\leq 2r \right.\right\} \end{equation} which is a closed subset of $\dot{Z}^{s}(I)$. Let $T>0$ and $u_{0}\in B_{r}(\dot{H}^{-1/2})$ are given. For $u\in \dot{Z}^{-1/2}_{r}([0,T))$, we have \[ \| \Phi_{T,u_{0}}(u) \| _{\dot{Z}^{-1/2}([0,T))}\leq \| u_{0} \| _{\dot{H}^{-1/2}} +C \| u \| _{\dot{Z}^{-1/2}([0,T))}^{4}\leq r(1+ 16 Cr^{3}) \] and \[ \begin{split} \| \Phi_{T,u_{0}}(u)-\Phi_{T,u_{0}}(v) \| _{\dot{Z}^{-1/2}([0,T))} &\leq C( \| u \| _{\dot{Z}^{-1/2}([0,T))}+ \| v \| _{\dot{Z}^{-1/2}([0,T))})^{3} \| u-v \| _{\dot{Z}^{-1/2}([0,T))}\\ &\leq 64Cr^{3} \| u-v \| _{\dot{Z}^{-1/2}([0,T))} \end{split} \] by Proposition~\ref{Duam_est} and \[ \| S(\cdot )u_{0} \| _{\dot{Z}^{-1/2}([0,T))}\leq \| \mbox{\boldmath $1$}_{[0,T)}S(\cdot )u_{0} \| _{\dot{Z}^{-1/2}}\leq \| u_{0} \| _{\dot{H}^{-1/2}}, \] where $C$ is an implicit constant in (\ref{Duam_est_1}). Therefore if we choose $r$ satisfying \[ r <(64C)^{-1/3}, \] then $\Phi_{T,u_{0}}$ is a contraction map on $\dot{Z}^{-1/2}_{r}([0,T))$. This implies the existence of the solution of (\ref{D4NLS}) and the uniqueness in the ball $\dot{Z}^{-1/2}_{r}([0,T))$. The Lipschitz continuously of the flow map is also proved by similar argument. \end{proof} Corollary~\ref{sccat} is obtained by the same way as the proof of Corollaty\ 1.2 in \cite{Hi}. \subsection{The large data case} In this subsection, we prove Theorem \ref{large-wp}. The following is the key estimate. \begin{prop}\label{Duam_est-inh} Let $d=1$. We have \begin{equation}\label{Duam_est_1-inh} \| I_{1}(u_{1},\cdots u_{4}) \| _{\dot{Z}^{-1/2}} \lesssim \prod_{j=1}^{4} \| u_{j} \| _{Y^{-1/2}}. \end{equation} \end{prop} \begin{proof} We decompose $u_j = v_j +w_j$ with $v_j = P_{>1}u_j \in \dot{Y}^{-1/2}$ and $w_j = P_{<1} u_j \in \dot{Y}^0$. >From Propositions \ref{HL_est_n-inh}, \ref{HH_est-inh}, and the same way as in the proof of Proposition~\ref{Duam_est}, it remains to prove that \[ \| I_{1}(w_{1},w_2,w_3,w_{4}) \| _{\dot{Z}^{-1/2}} \lesssim \prod_{j=1}^{4} \| u_{j} \| _{\dot{Y}^0}. \] By Theorem \ref{duality}, the Cauchy-Schwartz inequality, the H\"older inequality and the Sobolev inequality, we have \[ \| I_{1}(w_{1},w_2,w_3,w_{4}) \| _{\dot{Z}^{-1/2}} \lesssim \left\| \prod_{j=1}^{4}\overline{w_{j}} \right\|_{L^1([0,1];L^2)} \lesssim \prod _{j=1}^4 \| w_j \| _{L_t^{\infty} L_x^2} \lesssim \prod_{j=1}^{4} \| u_{j} \| _{\dot{Y}^{0}}, \] which completes the proof. \end{proof} \begin{proof}[\rm{\bf{Proof of Theorem \ref{large-wp}}}] Let $u_0 \in B_{\delta ,R}(H^{-1/2})$ with $u_0=v_0+w_0$, $v_0 \in \dot{H}^{-1/2}$, $w_0 \in L^2$. A direct calculation yields \[ \| S(t) u_0 \| _{Z^{-1/2}([0,1))} \le \delta +R. \] We start with the case $R=\delta = (4C+4)^{-4}$, where $C$ is the implicit constant in \eqref{Duam_est_1-inh}. Proposition \ref{Duam_est-inh} implies that for $u \in Z^{-1/2}_r([0,1])$ with $r=1/(4C+4)$ \begin{align*} \| \Phi_{1,u_{0}}(u) \| _{Z^{-1/2}([0,1))} & \leq \| S(t) u_0 \| _{Z^{-1/2}([0,1))} +C \| u \| _{Z^{-1/2}([0,1))}^{4} \\ & \leq 2r^4 + 16C r^4 = r^4 (16C+2) \le r \end{align*} and \begin{align*} \| \Phi_{1,u_{0}}(u)-\Phi_{1,u_{0}}(v) \| _{Z^{-1/2}([0,1))} &\leq C( \| u \| _{Z^{-1/2}([0,1))}+ \| v \| _{Z^{-1/2}([0,1))})^{3} \| u-v \| _{Z^{-1/2}([0,1))}\\ &\leq 64Cr^{3} \| u-v \| _{Z^{-1/2}([0,1))} < \| u-v \| _{Z^{-1/2}([0,1))} \end{align*} if we choose $C$ large enough (namely, $r$ is small enough). Accordingly, $\Phi_{1,u_{0}}$ is a contraction map on $\dot{Z}^{-1/2}_{r}([0,1))$. We note that all of the above remains valid if we exchange $Z^{-1/2}([0,1))$ by the smaller space $\dot{Z}^{-1/2}([0,1))$ since $\dot{Z}^{-1/2}([0,1)) \hookrightarrow Z^{-1/2}([0,1))$ and the left hand side of \eqref{Duam_est_1-inh} is the homogeneous norm. We now assume that $u_0 \in B_{\delta ,R}(H^{-1/2})$ for $R \ge \delta = (4C+4)^{-4}$. We define $u_{0, \lambda}(x) = \lambda ^{-1} u_0 (\lambda ^{-1}x)$. For $\lambda = \delta ^{-2} R^{2}$, we observe that $u_{0,\lambda} \in B_{\delta ,\delta}(H^{-1/2})$. We therefore find a solution $u_{\lambda} \in Z^{-1/2}([0,1))$ with $u_{\lambda}(0,x) = u_{0,\lambda}(x)$. By the scaling, we find a solution $u \in Z^{-1/2}([0, \delta ^8 R^{-8}))$. Thanks to Propositions \ref{HL_est_n-inh} and \ref{HH_est-inh}, the uniqueness follows from the same argument as in \cite{HHK10}. \end{proof} \section{Proof of Theorem~\ref{wellposed_2}}\label{pf_wellposed_2}\text{} In this section, we prove Theorem~\ref{wellposed_2}. We only prove for the homogeneous case since the proof for the inhomogeneous case is similar. We define the map $\Phi_{T, \varphi}^{m}$ as \[ \Phi_{T, \varphi}^{m}(u)(t):=S(t)\varphi -iI_{T}^{m}(u,\cdots, u)(t), \] where \[ I_{T}^{m}(u_{1},\cdots u_{m})(t):=\int_{0}^{t}\mbox{\boldmath $1$}_{[0,T)}(t')S(t-t')\partial \left(\prod_{j=1}^{m}u_{j}(t')\right)dt'. \] and the solution space $\dot{X}^{s}$ as \[ \dot{X}^{s}:=C({\BBB R};\dot{H}^{s})\cap L^{p_{m}}({\BBB R};\dot{W}^{s+1/(m-1),q_{m}}), \] where $p_{m}=2(m-1)$, $q_{m}=2(m-1)d/\{(m-1)d-2\}$ for $d \ge 2$ and $p_3=4$, $q_3=\infty$ for $d=1$. To prove the well-posedness of (\ref{D4NLS}) in $L^{2}({\BBB R} )$ or $H^{s_{c}}({\BBB R}^{d})$, we prove that $\Phi_{T, \varphi}$ is a contraction map on a closed subset of $\dot{X}^{s}$. The key estimate is the following: \begin{prop}\label{Duam_est_g} {\rm (i)}\ Let $d=1$ and $m=3$. For any $0<T<\infty$, we have \begin{equation}\label{Duam_est_1d} \| I_{T}^{3}(u_{1},u_{2}, u_{3}) \| _{\dot{X^{0}}}\lesssim T^{1/2}\prod_{j=1}^{3} \| u_{j} \| _{\dot{X}^{0}}. \end{equation} {\rm (ii)}\ Let $d\ge 2$, $(m-1)d\ge 4$ and $s_c=d/2-3/(m-1)$ For any $0<T\le \infty$, we have \begin{equation}\label{Duam_est_2} \| I_{T}^{m}(u_{1},\cdots, u_{m}) \| _{\dot{X^{s_c}}}\lesssim \prod_{j=1}^{m} \| u_{j} \| _{\dot{X}^{s_c}}. \end{equation} \end{prop} \begin{proof} {\rm (i)}\ By Proposition~\ref{Stri_est} with $(a,b)=\left( 4, \infty \right)$, we get \[ \| I_{T}^{3}(u_{1},u_{2}, u_{3}) \| _{L^{\infty}_{t}L^{2}_{x}} \lesssim \left\|\mbox{\boldmath $1$}_{[0,T)} |\nabla |^{-1/2}\partial \left(\prod_{j=1}^{3}u_{j}\right)\right\|_{L^{4/3}_{t}L^{1}_{x}} \] and \[ \| |\nabla |^{1/2}I_{T}^{3}(u_{1},u_{2}, u_{3}) \| _{L^{4}_{t}L^{\infty}_{x}} \lesssim \left\| \mbox{\boldmath $1$}_{[0,T)}|\nabla |^{1/2-1/2-1/2}\partial \left(\prod_{j=1}^{3}u_{j}\right)\right\|_{L^{4/3}_{t}L^{1}_{x}}. \] Therefore, thanks to the fractional Leibniz rule (see \cite{CW91}), we have \[ \begin{split} \| I_{T}^{3}(u_{1},\cdots, u_{3}) \| _{\dot{X^{0}}} & \lesssim \left\| \mbox{\boldmath $1$}_{[0,T)}|\nabla |^{1/2}\prod_{j=1}^{3}u_{j}\right\|_{L^{4/3}_{t}L^{1}_{x}} \\ & \lesssim \| \mbox{\boldmath $1$}_{[0,T)}\|_{L^{2}_{t}}\| |\nabla |^{1/2}u_{i} \| _{L^{4}_{t}L^{\infty}_{x}}\prod_{\substack{1\le j\le 3\\ j\neq i}} \| u_{j} \| _{L^{\infty}_{t}L^{2}_{x}}\\ &\lesssim T^{1/2}\prod_{j=1}^{3} \| u_{j} \| _{\dot{X}^{0}} \end{split} \] by the H\"older inequality. \\ {\rm (ii)}\ By Proposition~\ref{Stri_est} with \begin{equation}\label{admissible_ab} (a,b)=\left( \frac{2(m-1)}{m-2}, \frac{2(m-1)d}{(m-1)d-2(m-2)}\right), \end{equation} we get \[ \| |\nabla |^{s_c}I_{T}^{m}(u_{1},\cdots u_{m}) \| _{L^{\infty}_{t}L^{2}_{x}} \lesssim \left\| |\nabla |^{s_c-2/a}\partial \left(\prod_{j=1}^{m}u_{j}\right)\right\|_{L^{a'}_{t}L^{b'}_{x}} \] and \[ \| |\nabla |^{s_c+1/(m-1)}I_{T}^{m}(u_{1},\cdots u_{m}) \| _{L^{p_m}_{t}L^{q_m}_{x}} \lesssim \left\| |\nabla |^{s_c+1/(m-1)-2/p_m-2/a}\partial \left(\prod_{j=1}^{m}u_{j}\right)\right\|_{L^{a'}_{t}L^{b'}_{x}}. \] Therefore, thanks to the fractional Leibniz rule (see \cite{CW91}), we have \[ \begin{split} \| I_{T}^{m}(u_{1},\cdots u_{m}) \| _{\dot{X^{s_c}}} & \lesssim \left\| |\nabla |^{s_c+1/(m-1)}\prod_{j=1}^{m}u_{j}\right\|_{L^{a'}_{t}L^{b'}_{x}} \\ & \lesssim \sum_{i=1}^{m} \| |\nabla |^{s_c+1/(m-1)}u_{i} \| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\prod_{\substack{1\le j\le m\\ j\neq i}} \| u_{j} \| _{L^{p_{m}}_{t}L^{(m-1)d}_{x}}\\ &\lesssim \sum_{i=1}^{m} \| |\nabla |^{s_c+1/(m-1)}u_{i} \| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\prod_{\substack{1\le j\le m\\ j\neq i}} \| |\nabla |^{s_{c}+1/(m-1)}u_{j} \| _{L^{p_{m}}_{t}L^{q_{m}}_{x}}\\ &\lesssim \prod_{j=1}^{m} \| u_{j} \| _{\dot{X}^{s_c}} \end{split} \] by the H\"older inequality and the Sobolev inequality, where we used the condition $(m-1)d\ge 4$ which is equivalent to $s_{c}+1/(m-1)\ge 0$. \end{proof} The well-posedness can be proved by the same way as the proof of Theorem~\ref{wellposed_1} and the scattering follows from that the Strichartz estimate because the $\dot{X}^{s_c}$ norm of the nonlinear part is bounded by the norm of the $L^{p_m}L^{q_m}$ space (see for example \cite[Section 9]{P07}). \section{Proof of Theorem~\ref{notC3}}\label{pf_notC3} In this section we prove the flow of (\ref{D4NLS}) is not smooth. Let $u^{(m)}[u_0]$ be the $m$-th iteration of \eqref{D4NLS} with initial data $u_0$: \[ u^{(m)}[u_0] (t,x) := -i \int _0^t e^{i(t-t') \Delta ^2} \partial P_m( S(t') u_0, S(-t') \overline{u_0}) dt' . \] Firstly we consider the case $d=1$, $m=3$, $P_{3}(u,\overline{u})=|u|^{2}u$. For $N\gg 1$, we put \[ f_{N} = N^{-s+1/2} \mathcal{F}^{-1}[ \mbox{\boldmath $1$} _{[N-N^{-1}, N+N^{-1}]}] \] Let $u^{(3)}_{N}$ be the third iteration of (\ref{D4NLS}) for the data $f_{N}$. Namely, \[ u^{(3)}_{N}(t,x) = u^{(3)}[f_N] (t,x)= -i \int _0^t e^{i(t-t') \partial _x ^4} \partial _x \left( |e^{it' \partial _x^4} f_{N}| ^2 e^{it' \partial _x^4} f_{N} \right)(x) dt'. \] Note that $ \| f_{N} \| _{H^s} \sim 1$. Thorem~\ref{notC3} is implied by the following propositions. \begin{prop} If $s<0$, then for any $N\gg 1$, we have \[ \| u^{(3)}_{N} \| _{L^{\infty}([0,1]; H^s)} \rightarrow \infty \] as $N\rightarrow \infty$. \end{prop} \begin{proof} A direct calculation implies \[ \widehat{u^{(3)}_{N}} (t, \xi ) = e^{it \xi ^4} \xi \int _{\xi _1-\xi _2+\xi _3 =\xi} \int _0^t e^{it'(-\xi ^4 +\xi _1^4-\xi _2^4+\xi _3^4)} d t' \widehat{f_{N}}(\xi _1) \overline{\widehat{f_{N}}}(\xi _2) \widehat{f_{N}}(\xi _3) \] and \begin{equation} \label{modulation} \begin{split} &-(\xi _1-\xi _2+\xi _3)^4+\xi _1^4-\xi _2^4+\xi _3^4\\ &= 2 (\xi _1- \xi _2)(\xi _2-\xi _3) ( 2 \xi _1^2 +\xi _2^2+2\xi _3^2 -\xi _1 \xi _2 -\xi _2\xi _3 +3 \xi _3 \xi _1) . \end{split} \end{equation} >From $\xi _j \in [N-N^{-1}, N+N^{-1}]$ for $j=1,2,3$, we get \[ |-(\xi _1-\xi _2+\xi _3)^4+\xi _1^4-\xi _2^4+\xi _3^4| \lesssim 1. \] We therefore obtain for sufficiently small $t>0$ \begin{align*} |\widehat{u^{(3)}_{N}} (t,\xi ) | & \gtrsim t N^{-3s+5/2} \left| \int _{\xi _1-\xi _2+\xi _3 =\xi} \mbox{\boldmath $1$} _{[N-N^{-1}, N+N^{-1}]} (\xi _1) \mbox{\boldmath $1$} _{[N-N^{-1}, N+N^{-1}]} (\xi _2) \mbox{\boldmath $1$} _{[N-N^{-1}, N+N^{-1}]} (\xi _3) \right| \\ & \gtrsim t N^{-3s+1/2} \mbox{\boldmath $1$} _{[N-N^{-1},N+N^{-1} ]} (\xi ) . \end{align*} Hence, \[ \| u^{(3)}_{N} \| _{L^{\infty}([0,1]; H^s)} \gtrsim N^{-2s}. \] This lower bound goes to infinity as $N$ tends to infinity if $s<0$, which concludes the proof. \end{proof} Secondly, we show that absence of a smooth flow map for $d \ge 1$ and $m \ge 2$. Putting \[ g_N := N^{-s-d/2} \mathcal{F}^{-1}[ \mbox{\boldmath $1$} _{[-N,N]^d}] , \] we set $u_N^{(m)} := u^{(m)} [g_N]$. Note that $\| g_N \| _{H^s} \sim 1$. As above, we show the following. \begin{prop} If $s<s_c := d/2-3/(m-1)$ and $\partial =|\nabla |$ or $\frac{\partial}{\partial x_k}$ for some $1\le k\le d$, then for any $N \gg 1$, we have \[ \| u_N^{(m)} \| _{L^{\infty}([0,1];H^s)} \rightarrow \infty \] as $N \rightarrow \infty$. \end{prop} \begin{proof} We only prove for the case $\partial =|\nabla |$ since the proof for the case $\frac{\partial}{\partial x_k}$ is same. Let \[ \mathcal{A} := \{ (\pm _1, \dots , \pm _m) : \pm _j \in \{ +, - \} \, (j=1, \dots ,m) \} . \] Since $\mathcal{A}$ consists of $2^m$ elements, we write \[ \mathcal{A} = \bigcup _{\alpha}^{2^m} \{ \pm ^{(\alpha )} \} , \] where $\pm ^{( \alpha )}$ is a $m$-ple of signs $+$ and $-$. We denote by $\pm _{j}^{(\alpha )}$ the $j$-th component of $\pm ^{(\alpha )}$. A simple calculation shows that \[ \widehat{u_N^{(m)}} (t,\xi) = |\xi | \sum _{\alpha =0}^{2^m} e^{it |\xi |^4} \int _{\xi = \sum _{j=1}^m \pm _j^{(\alpha)} \xi _j} \int _0^t e^{it' (-|\xi|^4 + \sum _{j=1}^m \pm _j^{(\alpha )} |\xi _j|^4)} dt' \prod _{j=1}^m \widehat{g_N} (\xi _j) . \] From \[ \left| -|\xi|^4 + \sum _{j=1}^m \pm _j^{(\alpha )} |\xi _j|^4 \right| \lesssim N^4 \] for $|\xi _j| \le N$ ($j= 1, \dots , m$), we have \[ |\widehat{u_N^{(m)}} (t, \xi )| \gtrsim |\xi | N^{-4} N^{-m(s+d/2)} N^{(m-1)d} \mbox{\boldmath $1$} _{[-N.N]^d} (\xi ) \gtrsim N^{-3} N^{-m(s+d/2)} N^{(m-1)d} \mbox{\boldmath $1$} _{[N/2.N]^d} (\xi ) \] provided that $t \sim N^{-4}$. Accordingly, we obtain \[ \| u_N^{(m)} (N^{-4}) \| _{H^s} \gtrsim N^{-3} N^{-m(s+d/2)} N^{(m-1)d} N^{s+d/2} \sim N^{-(m-1)s+(m-1)d/2-3} , \] which conclude that $\limsup _{t \rightarrow 0} \| u^{(m)}_N(t) \| _{H^s} = \infty$ if $s<s_c$. \end{proof} \section*{Acknowledgment} The work of the second author was partially supported by JSPS KAKENHI Grant number 26887017. \end{document}

\begin{document} \fontsize{.5cm}{.5cm}\selectfont\sf \title[Schubert Draft]{Double Quantum Schubert Cells and\\ Quantum Mutations} \date{\today} \author{Hans P. Jakobsen} \address{ Department of Mathematical Sciences\\ University of Copenhagen\\Universitetsparken 5\\ DK-2100, Copenhagen, Denmark} \email{[email protected]} \begin{abstract}Let ${\mathfrak p}\subset {\mathfrak g}$ be a parabolic subalgebra of s simple finite dimensional Lie algebra over ${\mathbb C}$. To each pair $w^{\mathfrak a}\leq w^{\mathfrak c}$ of minimal left coset representatives in the quotient space $W_p\backslash W$ we construct explicitly a quantum seed ${\mathcal Q}_q({\mathfrak a},{\mathfrak c})$. We define Schubert creation and annihilation mutations and show that our seeds are related by such mutations. We also introduce more elaborate seeds to accommodate our mutations. The quantized Schubert Cell decomposition of the quantized generalized flag manifold can be viewed as the result of such mutations having their origins in the pair $({\mathfrak a},{\mathfrak c})= ({\mathfrak e},{\mathfrak p})$, where the empty string ${\mathfrak e}$ corresponds to the neutral element. This makes it possible to give simple proofs by induction. We exemplify this in three directions: Prime ideals, upper cluster algebras, and the diagonal of a quantized minor. \end{abstract} \subjclass[2010]{MSC 17B37 (primary),\ MSC 13F60, \ MSC 16T20 (primary), \ MSC 17A45 (secondary), \and MSC 20G42 (secondary)} \maketitle \section{Introduction} We study a class of quadratic algebras connected to quantum parabolics and double quantum Schubert cells. We begin by considering a finite-dimensional simple Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ and a parabolic sub-algebra ${\mathfrak p}\subset{\mathfrak g}$. Then we consider a fixed Levi decomposition \begin{equation} {\mathfrak p}={\mathfrak l}+{\mathfrak u}, \end{equation} with ${\mathfrak u}\neq 0$ and ${\mathfrak l}$ the Levi subalgebra. The main references for this study are the articles by A. Berenstein and A. Zelevinski \cite{bz} and by C. Geiss, B. Leclerc, J. Schr\"oer \cite{leclerc}. We also refer to \cite{jak-cen} for further background. Let, as usual, $W$ denote the Weyl group. Let $W_p=\{w\in W\mid w(\triangle^-)\cap \triangle^+\subseteq \triangle^+({\mathfrak l}) \}$ and $W^p$, by some called the Hasse Diagram of $G\backslash P$, denote the usual set of minimal length coset representatives of $W_p\backslash W$. Our primary input is a pair of Weyl group elements $w^{\mathfrak a},w^{\mathfrak c}\in W^p$ such that $w^{\mathfrak a}\leq w^{\mathfrak c}$. We will often, as here, label our elements $w$ by ``words'' ${\mathfrak a}$; $w=w^{\mathfrak a}$, in a fashion similar, though not completely identical, to that of \cite{bz}. Details follow in later sections, but we do mention here that the element $e$ in $W$ is labeled by ${\mathfrak e}$ corresponding to the empty string; $e=\omega^{\mathfrak e}$ while the longest elements in $W^p$ is labeled by ${\mathfrak p}$. To each pair $w^{\mathfrak a},w^{\mathfrak c}$ as above we construct explicitly a quantum seed \begin{equation}{\mathcal Q}_q({\mathfrak a},{\mathfrak c}):=({\mathcal C}_q({\mathfrak a},{\mathfrak c}), {\mathcal L}_q({\mathfrak a},{\mathfrak c}), {\mathcal B}_q({\mathfrak a},{\mathfrak c})).\end{equation} The cluster ${\mathcal C}_q({\mathfrak a},{\mathfrak c})$ generates a quadratic algebra ${\mathcal A}_q({\mathfrak a},{\mathfrak c})$ in the space of functions on ${\mathcal U}_q({\mathfrak n})$. After that we define transitions \begin{equation}{\mathcal Q}_q({\mathfrak a},{\mathfrak c})\rightarrow {\mathcal Q}_q({\mathfrak a}_1,{\mathfrak c}_1). \end{equation} We call our transitions quantum Schur (creation/annihilation) mutations and prove that they are indeed just (composites of) quantum mutations in the sense of Berenstein and Zelevinski. These actually have to be augmented by what we call creation/annihilation mutations which are necessary since we have to work inside a larger ambient space. To keep the full generality, we may also have to restrict our seeds to sub-seeds. The natural scene turns out to be \begin{equation}{\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}):=({\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal L}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})),\end{equation} which analogously is determined by a triple $w^{\mathfrak a},w^{\mathfrak b},w^{\mathfrak c}\in W^p$ such that $w^{\mathfrak a}\leq w^{\mathfrak b}\leq w^{\mathfrak c}$. Later we extend our construction to even \begin{equation} {\mathcal Q}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\textrm{ and }{\mathcal A}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n), \end{equation} though we do not use it here for anything specific. It is a major point of this study to establish how our seeds and algebras can be constructed, inside an ambient space, starting from a single variable (indeed: none). In this sense the quantized generalized flag manifold of $(G/P)_q$ as built from quantized Schubert Cells can be built from a single cell. Furthermore, we prove that we can pass between our seeds by Schubert creation and annihilation mutations inside a larger ambient space. This sets the stage for (simple) inductive arguments which is a major point of this article, and is what we will pursue here. We first prove by induction that the two-sided sided ideal $I({\det}_{s}^{{\mathfrak a},{\mathfrak c}})$ in ${\mathcal A}_q({\mathfrak a},{\mathfrak c})$ generated by the quantized minor ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}$ is prime. Then we prove that each upper cluster algebra ${\mathbb U}({\mathfrak a},{\mathfrak c})$ equals its quadratic algebra ${\mathcal A}_q({\mathfrak a},{\mathfrak c})$. There is a sizable overlap between these result and results previously obtained by K. Goodearl M. Yakimov (\cite{good},\cite{good1}). We further use our method to study the diagonal of a quantum minor. The idea of induction in this context was introduced in \cite{jz} and applications were studied in the case of a specific type of parabolic related to type $A_n$. Further ideas relating to explicit constructions of compatible pairs in special cases were studied in \cite{jp}. \section{A little about quantum groups and cluster algebras} \subsection{2.1 Quantum Groups} We consider quantized enveloping algebras $U={\mathcal U}_q({\mathfrak g})$ in the standard notation given either eg. by Jantzen (\cite{jan}) or by Berenstein and Zelevinsky (\cite{bz}), though their assumptions do not coincide completely. To be completely on the safe side, we state our assumptions and notation, where it may differ: Our algebra is a Hopf algebra defined in the usual fashion from a semi-simple finite-dimensional complex Lie algebra ${\mathfrak g}$. They are algebras over ${\mathbb Q}(q)$. $\Phi$ denotes a given set of roots and throughout, $\Pi=\{\alpha_1,\alpha_2,\dots,\alpha_R\}$ a fixed choice of simple roots. Our generators are then given as $$\{E_\alpha,F_\alpha,K^\alpha\}_{\alpha\in\Pi},$$ but we will allow elements of the form $K^\eta$ for any integer weight. $W$ denotes the Weyl group defined by $\Phi$. Finally we let $\{\Lambda_\alpha\mid\alpha\in\Pi\}$ denote the set of fundamental weights. We assume throughout that the diagonalizing elements $d_\alpha$ are determined by \begin{equation} \forall \alpha\in\Pi:(\Lambda_\alpha,\alpha)=d_\alpha. \end{equation} \begin{Lem}[(2.27) in \cite{fz}]\label{3.1} Let $\alpha_i\in \Phi$. Then $$(\sigma_i+1)(\Lambda_i)+\sum_{j\neq i}a_{ji}(\Lambda_j)=0.$$ \end{Lem} \subsection{Quantum Cluster Algebras} We take over without further ado the terminology and constructions of (\cite{bz}). Results from \cite{leclerc} are also put to good use. \begin{Def} We say that two elements $A,B$ in some algebra over ${\mathbb C}$ $q$-commute if, for some $r\in{\mathbb R}$: \begin{equation}AB=q^rBA. \end{equation} \end{Def} To distinguish between the original mutations and the more elaborate ones we need here, and to honor the founding fathers A. Berenstein, S. Fomin, and A. Zelevinski, we use the following terminology: \begin{Def}A quantum mutation as in \cite{bz} is called a BFZ-mutation. \end{Def} \subsection{A simple observation} If $\underline{a}=(a_1,a_2,\dots,a_{\mathfrak m})$ and $\underline{f}=(f_1,f_2,\dots,f_{\mathfrak m})$ are vectors then\begin{Lem}{(\cite{jz})}\label{2.22}\begin{equation} {\mathcal L}_q(\underline{a})^T=(\underline{f} )^T\Leftrightarrow\forall i:X_iX^{\underline{a}}=q^{f_i}X^{\underline{a}}X_i.\end{equation} In particular, if there exists a $j$ such that $\forall i: f_i=-\delta_{i,j}$ then the column vector $\underline{a}$ can be the $j$th column in the matrix ${\mathcal B}$ of a compatible pair. \end{Lem} \noindent However simple this actually is, it will have a great importance later on. \section{On Parabolics} The origin of the following lies in A. Borel \cite{borel}, and B. Kostant \cite{kos}. Other main contributors are \cite{bgg} and \cite{stein}. See also \cite{cap}. We have also found (\cite{sager}) useful. \begin{Def} Let $w\in W$. Set $$\Phi_\omega=\{\alpha\in \Delta^+\mid w^{-1}\alpha\in \Delta^-\}=w( \Delta^-)\cap \Delta^+.$$\end{Def} We have that $\ell(w)=\ell(w^{-1})=\vert\Phi_\omega\vert$. We set $\Phi_\omega=\Delta^+(w)$. From now on, we work with a fixed parabolic $\mathfrak p$ with a Levi decomposition \begin{equation} {\mathfrak p}={\mathfrak l}+{\mathfrak u}, \end{equation} where ${\mathfrak l}$ is the Levi subalgebra, and where we assume ${\mathfrak u}\neq 0$, Let \begin{Def} \begin{eqnarray*} W_p&=&\{w\in W\mid \Phi_\omega\subseteq \Delta^+({\mathfrak l})\},\\ W^p&=&\{w\in W\mid \Phi_\omega\subseteq \Delta^+({\mathfrak u})\}. \end{eqnarray*} $W^p$ is a set of distinguished representatives of the right coset space $W_p\backslash W$. \end{Def} It is well known (see eg (\cite{sager})) that any $w\in W$ can be written uniquely as $w=w_pw^p$ with $w_p\in W_p$ and $w^p\in W^p$. One defines, for each $w$ in the Weyl Group $W$, the Schubert cell $X_w$. This is a cell in ${\mathbb P}(V)$, the projective space over a specific finite-dimensional representation of ${\mathfrak g}$. The closure, ${X_w}$, is called a Schubert variety. The main classical theorems are \begin{Thm}[Kostant,\cite{kos}]$$G/P=\sqcup_{w\in W^p}X_w.$$\end{Thm} {\begin{Thm}[\cite{stein}]\label{stein} Let $w,w'\in W^p$. Then $$X_{w'}\subseteq {{X_{w}}}$$ if and only $w'\leq w$ in the usual Bruhat ordering. \end{Thm} If $\omega^{\mathfrak r}=\omega_m\tilde\omega$ and $\omega_{m}=\omega_n\hat\omega$ with $\omega_n,\omega_m\in W^P$ and all Weyl group elements reduced, we say that $\omega_n<_L\omega_m$ if $\hat\omega\neq e$. This is the weak left Bruhat order. \section{The quadratic algebras}\label{sec4} Let $\omega=s_{\alpha_1}s_{\alpha_2}\dots s_{\alpha_t}$ be an element of the Weyl group written in reduced form. Following Lusztig (\cite{luz}), we construct roots $\gamma_i=\omega_{i-1}(\alpha_i)$ and elements $Z_{\gamma_i}\in {\mathcal U}_q({\mathfrak n}_\omega)$. The following result is well known, but notice a change $q\to q^{-1}$ in relation to (\cite{jak-cen}). \begin{Thm}[\cite{lev},\cite{lev0}] \label{4.1}Suppose that $1\leq i<j\leq t$. Then $$Z_{i}Z_{j}=q^{-( \gamma_i,\gamma_j)}Z_{j}Z_{i} + {\mathcal R}_{ij},$$ where ${\mathcal R}_{ij}$ is of lower order in the sense that it involves only elements $Z_k$ with $i< k< j$. Furthermore, the elements $$Z_t^{a_t}\dots Z_2^{a_2}Z_1^{a_1}$$ with $a_1,a_2,\dots,a_t\in{\mathbb N}_0$ form a basis of ${\mathcal U}_q({\mathfrak n}_\omega)$. \end{Thm} Our statement follows \cite{jan},\cite{jan2}. Other authors, eg. \cite{lev}, \cite{leclerc} have used the other Lusztig braid operators. The result is just a difference between $q$ and $q^{-1}$. Proofs of this theorem which are more accessible are available (\cite{cp},\cite{jan2}). It is known that this algebra is isomorphic to the algebra of functions on ${\mathcal U}_q({\mathfrak n}_\omega)$ satisfying the usual finiteness condition. It is analogously equivalent to the algebra of functions on ${\mathcal U}^-_q({\mathfrak n}_\omega)$ satisfying a similar finiteness condition. See eg (\cite{leclerc}) and (\cite{jan}). \section{basic structure}\label{sec5} Let $\omega^{\mathfrak p}$ be the maximal element in $W^p$. It is the one which maps all roots in $\Delta^+({\mathfrak u})$ to $\Delta^-$. (Indeed: To $\Delta^-({\mathfrak u})$.) Let $w_0$ be the longest element in $W$ and $w_L$ the longest in the Weyl group of ${\mathfrak l}$, Then \begin{equation}w^{\mathfrak p}w_L=w_0.\end{equation} Let $\omega^{\mathfrak r}=\sigma_{i_1}\sigma_{i_2}\cdots\sigma_{i_r}\in W^p$ be written in a fixed reduced form. Then $\ell(\omega^{\mathfrak r})=r$. We assume here that $r\geq 1$. We set $e=\omega^{\mathfrak e}$ and $\ell(\omega^{\mathfrak e})=0$ where ${\mathfrak e}$ denotes the empty set, construed as the empty sequence. We also let ${\mathfrak r}$ denote the sequence $i_1,i_2,\dots,i_r$ if ${\mathfrak r}\neq {\mathfrak e}$. If a sequence ${\mathfrak s}$ corresponds to an analogous element $\omega^{\mathfrak s}\in W^p$ we define \begin{equation} {\mathfrak s}\leq {\mathfrak r}\Leftrightarrow \omega^{\mathfrak s}\leq_L \omega^{\mathfrak r}. \end{equation} Set \begin{equation}\Delta^+(\omega^{\mathfrak r})=\{ \beta_{i_1},\dots,\beta_{i_r}\}.\end{equation} \begin{Def} Let ${\mathbf b}$ denote the map $\Pi\to\{1,2,\dots,R\}$ defined by ${\mathbf b}(\alpha_i)=i$. Let $\overline\pi_{\mathfrak r}:\{1,2,\dots, r\}\to\Pi$ be given by \begin{equation}\overline\pi_{\mathfrak r}(j)=\alpha_{i_j}.\end{equation} If $\overline\pi_{\mathfrak r}(j)=\alpha$ we say that $\alpha$ (or $\sigma_\alpha$) occurs at position $j$ in $w^{\mathfrak r}$, and we say that $\overline\pi_{\mathfrak r}^{-1}(\alpha)$ are the positions at which $\alpha$ occurs in $w$. Set \begin{equation} {\pi}_{\mathfrak r}={\mathbf b}\circ\overline\pi_{\mathfrak r}. \end{equation} \end{Def} $\pi_{\mathfrak e}$ is construed as a map whose image is the empty set. Recall from (\cite{jak-cen}): \begin{Def}Let $\omega^{\mathfrak r}\in W^p$ be given and suppose $s\in Im(\pi_{\mathfrak r})$. Then $s=\pi_{\mathfrak r}(n)$ for some $n$ and we set $\omega_n:=\sigma_{i_1}\sigma_{i_2}\cdots\sigma_{i_n}$. Suppose $\omega_n=\omega_1 \sigma_{i_n}\omega_2\dots\omega_t \sigma_{i_n}$ and $\omega_i\in W\setminus\{e\}$ for $i>1$. Further assume that each $\omega_i$ is reduced and does not contain any $\sigma_{i_n}$. We denote this simply as $n\leftrightarrow (s,t)$. We further write $\beta_{n}\leftrightarrow \beta_{s,t}$ and \begin{equation}\omega_n\leftrightarrow \omega_{s,t}\end{equation} if $n,s,t$ are connected as above. It is convenient to set $\omega_{s,0}=e$ for all $s\in\{1,2,\dots, R\}$. For a fixed $s\in\{1,2,\dots,R\}$ we let $s_{\mathfrak r}$ denote the maximal such $t$. If there is no such decomposition we set $t=0$. So, in particular, $s_{\mathfrak e}=0$, and $s_{\mathfrak r}$ is the number of times $\sigma_s$ occurs in $\omega^{\mathfrak r}$. Finally we set (cf. (\cite{jak-cen})) \begin{equation} {\mathbb U}({\mathfrak r})=\{(s,t)\in {\mathbb N}\times {\mathbb N}_0\mid 1\leq s\leq R\textrm{ and }0\leq t\leq s_{\mathfrak r}\}. \end{equation} \end{Def} Notice that if $(s,t)\in{\mathbb U}({\mathfrak r})$ then we may construct a subset ${\mathbb U}({\mathbf s}, {\mathbf t})$ of ${\mathbb U}$ by the above recipe, replacing $\omega^{\mathfrak r}$ by $\omega_{s,t}$. In this subset $t$ is maximal. Likewise, if ${\mathfrak s}\leq {\mathfrak r}$ we have of course ${\mathbb U}({\mathfrak s})\subseteq {\mathbb U}({\mathfrak r})$ and may set ${\mathbb U}({\mathfrak r}\setminus {\mathfrak s})={\mathbb U}({\mathfrak r})\setminus {\mathbb U}({\mathfrak s})$. \section{Key structures and background results} \subsection{Quantized minors} Following a construction of classical minors by S. Fomin and A. Zelevinsky \cite{fz}, the last mentioned and A. Berenstein have introduced a family of quantized minors $\Delta_{u\cdot\lambda,v\cdot\lambda}$ in \cite{bz}. These are elements of the quantized coordinate ring ${\mathcal O}_q(G)$. The results by K. Brown and K. Goodearl (\cite{brown}) were important in this process. {The element $\Delta_{u\cdot\lambda,v\cdot\lambda}$ is determined by $u,v\in W$ and a positive weight $\lambda$. We will always assume that $u\leq_L v$. \subsection{Identifications} There is a well-known pairing between ${\mathcal U}^{\leq}$ and ${\mathcal U}^{\geq}$ (\cite{jan}) and there is a unique bilinear form on ${\mathcal U}_q({\mathfrak n})$. With this we can identify $({\mathcal U}^{\geq})^*$ with ${\mathcal U}^{\geq}$. One can even define a product in $({\mathcal U}_q({\mathfrak n}))^*$ that makes it isomorphic to ${\mathcal U}_q({\mathfrak n})$ \cite{leclerc}. We can in this way identify the elements $\Delta_{u\cdot\lambda,v\cdot\lambda}$ with elements of ${\mathcal U}^{\geq}$. \subsection{Key results from \cite{bz} and \cite{leclerc}} The quantized minors are by definitions functions on ${\mathcal U}_q({\mathfrak g})$ satisfying certain finiteness conditions. What is needed first are certain commutation relations that they satisfy. Besides this, they can be restricted to being functions on ${\mathcal U}_q({\mathfrak b})$ and even on ${\mathcal U}_q({\mathfrak n})$. Our main references here are (\cite{bz}) and (\cite{leclerc}); the details of the following can be found in the latter. \begin{Lem}[\cite{bz}]The element $\triangle_{u\lambda,v\lambda}$ indeed depends only on the weights $u\lambda,v\lambdaλ$, not on the choices of $u, v$ and their reduced words. \end{Lem} \begin{Thm}[A version of Theorem~10.2 in \cite{bz}] \label{10.2}For any $\lambda,\mu\in P^+$, and $s, s', t, t' \in W$ such that $$\ell(s's) = \ell(s') + \ell(s), \ell(t't) = \ell(t') + \ell(t) ,$$the following holds: $$ \triangle_{s's\lambda,t'\lambda} · \triangle_{s'\mu,t't\mu} =q^{(s\lambda | \mu) - (\lambda | t\mu)}\triangle_{s'\mu,t't\mu} · \triangle_{s's\lambda,t'\lambda}.$$ \end{Thm} It is very important for the following that the conditions essentially are on the Weyl group elements. The requirement on $\lambda,\mu$ is furthermore independent of those. An equally important fact we need is the following $q$-analogue of \cite[Theorem~1.17]{fz}: \begin{Thm}[\cite{leclerc}, Proposition~3.2]\label{3.2} Suppose that for $u,v\in W$ and $i\in I$ we have $l(us_i)=l(u)+1$ and $l(vs_i)=l(v)+1$. Then \begin{equation}\label{eq3.2} \Delta_{us_i(\Lambda_i),vs_i(\Lambda_i)}\,\Delta_{u(\Lambda_i),v(\Lambda_i)}= ({q^{-d_i}})\Delta_{us_i(\Lambda_i),v(\Lambda_i)}\,\Delta_{ u(\Lambda_i), vs_i(\Lambda_i)}+ \prod_{j\neq i}\Delta_{u(\Lambda_j),v(\Lambda_j)}^{-a_{ji}} \end{equation} holds in ${\mathcal O}_q(\frak g)$. \end{Thm} (That a factor $q^{-d_i}$ must be inserted for the general case is clear.) One considers in \cite{leclerc}, and transformed to our terminology, modified elements \begin{equation}\label{59}D_{\xi,\eta}=\triangle_{\xi,\eta}K^{-\eta}.\end{equation} We suppress here the restriction map $\rho$, and our $K^{-\eta}$ is denoted as $\triangle^\star_{\eta,\eta}$ in \cite{leclerc}. The crucial property is that \begin{equation} K^{-\eta}\triangle_{\xi_1,\eta_1}=q^{-(\eta,\xi_1-\eta_1)}\triangle_{\xi_1, \eta_1}K^{-\eta}. \end{equation} The family $D_{\xi,\eta}$ satisfies equations analogous to those in Theorem~\ref{10.2} subject to the same restrictions on the relations between the weights. The following result is important: \begin{Prop}[\cite{leclerc}] Up to a power of $q$, the following holds: \begin{equation}Z_{c,d}=D_{\omega^{\mathfrak r}_{c,d-1}(\Lambda_c),\omega^{\mathfrak r}_{c,d}(\Lambda_c)}. \end{equation} \end{Prop} We need a small modification of the elements $D_{\xi,\eta}$ of \cite{leclerc}: \begin{Def} \begin{equation}E_{\xi,\eta}:=q^{\frac14(\xi-\eta,\xi-\eta)+\frac12(\rho, \xi-\eta)}D_{\xi,\eta}.\end{equation} \end{Def} It is proved in (\cite{ki}), (\cite{re})) that $E_{\xi,\eta}$ is invariant under the dual bar anti-homomorphism augmented by $q\to q^{-1}$. Notice that this change does not affect commutators: \begin{equation} D_1D_2=q^\alpha D_2D_1\Leftrightarrow E_1E_2=q^\alpha E_2E_1 \end{equation} if $E_i=q^{x_i}D_i$ for $i=1,2$. \begin{Def}We say that \begin{equation}\label{less} E_{\xi,\eta}<E_{\xi_1,\eta_1} \end{equation} if $\xi=s's\lambda$, $\eta=t'\lambda$, $\xi_1=s'\mu$ and $\eta_1=t't\mu$ and the conditions of Theorem~\ref{10.2} are satisfied. \end{Def} The crucial equation is \begin{Cor} \begin{equation} E_{\xi,\eta}<E_{\xi_1,\eta_1}\Rightarrow E_{\xi,\eta}E_{\xi_1,\eta_1}=q^{(\xi-\eta,\xi_1+\eta_1)}E_{\xi_1,\eta_1}E_{\xi, \eta}. \end{equation}\end{Cor} \subsection{Connecting with the toric frames} \begin{Def}Suppose that $\triangle_i$, $i=1,\dots,r$ is a family of mutually $q$-commuting elements. Let $n_1,\dots,n_r\in{\mathbb Z}$. We then set \begin{equation}N(\prod_{i=1}^r \triangle_i^{n_i})=q^m\prod_{i=1}^r\triangle_i^{n_i}, \end{equation}where $q^m$ is determined by the requirement that \begin{equation}q^{-m}\triangle_r^{n_r}\dots \triangle_2^{n_2}\triangle_1^{n_1}= q^m \triangle_1^{n_1}\triangle_2^{n_2}\dots \triangle_r^{n_r}. \end{equation} \end{Def} It is easy to see that \begin{equation} \forall \mu\in S_r: N(\prod_{i=1}^r \triangle_{\mu(i)}^{n_{\mu(i)}})=N(\prod_{i=1}^r. \triangle_i^{n_i}) \end{equation} It is known through \cite{bz} that eg. the quantum minors are independent of the choices of the reduced form of $\omega^{\mathfrak r}_{\mathfrak p}$. Naturally, this carries over to $\omega^{\mathfrak r}$. The quadratic algebras we have encountered are independent of actual choices. In the coming definition we wish to maintain precisely the right amount of independence. Let us now formulate Theorem~\ref{3.2} in our language while using the language and notation of toric frames from \cite{bz}. In the following Theorem we first state a formula which uses our terminology, and then we reformulate it in the last two lines in terms of toric frames $M$. These frames are defined by a cluster made up by certain elements of the form $E_{\xi,\eta}$ to be made more precise later. \begin{Thm}\label{toric} \begin{eqnarray} E_{us_i\Lambda_i,vs_i\Lambda_i}&=&N\left( E_{us_i\Lambda_i,v\Lambda_i} E_{u\Lambda_i,vs_i\Lambda_i} E_{u\Lambda_i,v\Lambda_i}^{-1}\right) \\\nonumber&+&N\left((\prod_{j\neq i} E_{u(\Lambda_j),v(\Lambda_j)}^{-a_{ji}}) E_{u(\Lambda_i),v(\Lambda_i)}^{-1}\right)\\ &=&M(E_{us_i(\Lambda_i),v(\Lambda_i)}+E_{u(\Lambda_i), vs_i(\Lambda_i)}-E_{u(\Lambda_i),v(\Lambda_i)})\\&+& M(\sum_{j\neq i}-a_{ji}E_{u(\Lambda_j),v(\Lambda_j)}-E_{u(\Lambda_i),v(\Lambda_i)}). \end{eqnarray} \end{Thm} \noindent{\em Proof of Theorem~\ref{toric}:} We first state a lemma whose proof is omitted as it is straightforward. \begin{Lem} Let $\Delta_{\xi_k}$ be a family of $q$-commuting elements of weights $\xi_k$, $k=1,\dots,r$ in the sense that for any weight $b$:\begin{equation} \forall k=1,\dots,r: K^b\Delta_{\xi_k}=q^{(b,\xi_k)}\Delta_{\xi_k}K^b. \end{equation} Let $\alpha$ be defined by \begin{equation} \Delta_{\xi_r}\cdots\Delta_{\xi_1}=q^{-2\alpha} \Delta_{\xi_1}\cdots\Delta_{\xi_r} \end{equation}Furthermore, let $b_1,\dots,b_r$ be integer weights. Then \begin{eqnarray} &(\Delta_{\xi_1}\Delta_{\xi_2}\cdots\Delta_{\xi_r}) K^{b_1}K^{b_2}\cdots K^{b_r}=\\\nonumber &q^{\sum_{k<\ell}(b_k,\xi_\ell)}(\Delta_{\xi_1}K^{b_1})(\Delta_{\xi_2}K^{ b_2})\cdots(\Delta_{\xi_r}K^{b_r}),\textrm{ and,}\\\nonumber &(\Delta_{\xi_r}K^{b_r})\cdots(\Delta_{\xi_1}K^{b_1})=\\\nonumber &q^{-2\alpha} q^{(\sum_{k<\ell}-\sum_{\ell<k})(b_\ell,\xi_k)}(\Delta_{\xi_1}K^{b_1} )\cdots(\Delta_{\xi_r}K^{b_r}), \textrm{ so that}\\\nonumber &(\Delta_{\xi_1}K^{b_1})\cdots(\Delta_{\xi_r}K^{b_r})=\\\nonumber &q^{\alpha} q^{-\frac12(\sum_{k<\ell}-\sum_{\ell<k})(b_\ell,\xi_k)}N\left( (\Delta_{\xi_1}K^{b_1})\cdots(\Delta_{\xi_r}K^{b_r})\right). \end{eqnarray} Finally, \begin{eqnarray} &q^{-\alpha}(\Delta_{\xi_1}\Delta_{\xi_2}\cdots\Delta_{\xi_r}) K^{b_1}K^{b_2} \cdots K^{b_r}=\\\nonumber&q^{-\frac12(\sum_{\ell\neq k})(b_\ell,\xi_k)}N\left( (\Delta_{\xi_1}K^{b_1})\cdots(\Delta_{\xi_r}K^{b_r})\right). \end{eqnarray} \end{Lem} We apply this lemma first to the case where the elements $\xi_k$ are taken from the set $\{-\textrm{sign}(a_{ki})(u\Lambda_k-v\Lambda_k)\mid a_{ki}\neq0 \}$ and where each element corresponding to an $a_{ki}<0$ is taken $-a_{ki}$ times. Then $r=\sum_{k\neq i}\vert a_{ji}\vert+1$. The terms considered actually commute so that here, $\alpha=0$. The weights $b_k$ are chosen in the same fashion, but here $b_k=\textrm{sign}(a_{ki})(v\Lambda_k)$. We have that \begin{equation}\sum_{\ell\neq k}(b_\ell,\xi_k)=\left(\sum_{\ell}b_\ell, \sum_{k}\xi_k\right)-\sum_{k}(b_k,\xi_k). \end{equation} It follows from (\ref{3.1}) that $\sum_{\ell}b_\ell=-vs_i\lambda_i$ and $\sum_{k}\xi_k=(us_i\Lambda_i-vs_i\lambda_i)$. Now observe that for all $k$: $-(v\Lambda_k,(u-v)\Lambda_k)=\frac12(\xi_k,\xi_k)$. Let $\xi_0=(us_i-vs_i)\Lambda_i$. The individual summands in $\sum_k(b_k,\xi_k)$ can be treated analogously. Keeping track of the multiplicities and signs, it follows that \begin{eqnarray} q^{-\alpha}(\Delta_{\xi_1}\Delta_{\xi_2}\dots\Delta_{\xi_r})K^{b_1}K^{b_2}\dots K^{b_r}=\\\nonumber q^{-\frac14(\xi_0,\xi_0)+\frac14\sum_k\varepsilon_k(\xi_k,\xi_k)}N\left( (\Delta_{\xi_1}K^{b_1})\dots(\Delta_{\xi_r}K^{b_r})\right). \end{eqnarray} Let us turn to the term \begin{equation} q^{-d_i}\Delta_{us_i\Lambda_i,v\Lambda_i} \Delta_{u\Lambda_i,vs_i\Lambda_i}\Delta_{u\Lambda_i,v\Lambda_i}^{-1}K^{ -vs_i\Lambda_i}. \end{equation} We can of course set $K^{-vs_i\Lambda_i}=K^{-v\Lambda_i}K^{-vs_i\Lambda_i}K^{v\Lambda_i}$. Furthermore, it is known (and easy to see) that \begin{eqnarray} &\Delta_{u\Lambda_i,v\Lambda_i}^{-1}\Delta_{u\Lambda_i,vs_i\Lambda_i} \Delta_{us_i\Lambda_i,v\Lambda_i}=\\\nonumber&q^{-2d_i}\Delta_{us_i\Lambda_i, v\Lambda_i} \Delta_{u\Lambda_i,vs_i\Lambda_i}\Delta_{u\Lambda_i,v\Lambda_i}^{-1}, \end{eqnarray} so that $\alpha=d_i$ here. We easily get again that $\sum_{\ell}b_\ell=-vs_i\lambda_i$ and $\sum_{k}\xi_k=(us_i\Lambda_i-vs_i\lambda_i)$. Let us introduce elements $\tilde E_{\xi,\eta}=q^{\frac14(\xi-\eta,\xi-\eta)}\Delta_{\xi,\eta}K^{-\eta}$. It then follows that (c.f. Theorem~\ref{3.2}) \begin{eqnarray} \tilde E_{us_i\Lambda_i,vs_i\lambda_i}&=&N\left(\tilde E_{us_i\Lambda_i,v\Lambda_i} \tilde E_{u\Lambda_i,vs_i\Lambda_i}\tilde E_{u\Lambda_i,v\Lambda_i}^{-1}\right) \\\nonumber&+&N\left((\prod_{j\neq i}\tilde E_{u(\Lambda_j),v(\Lambda_j)}^{-a_{ji}})\tilde E_{u(\Lambda_i),v(\Lambda_i)}^{-1}\right). \end{eqnarray} The elements $E_{\xi,\eta}$ differ from the elements $\tilde E_{\xi,\eta}$ by a factor which is $q$ to an exponent which is linear in the weight $(\xi-\eta)$. Hence an equation identical to the above holds for these elements. \qed \section{Compatible pairs} We now construct some general families of quantum clusters and quantum seeds. The first, simplest, and most important, correspond to double Schubert Cells: Let ${\mathfrak e}\leq {\mathfrak s}<{\mathfrak t}<{\mathfrak v}\leq{\mathfrak p}$. Set \begin{eqnarray*}{\mathbb U}^{d,{\mathfrak t},{\mathfrak v}}&:=&\{(a,j)\in {\mathbb U}({\mathfrak p})\mid a_{\mathfrak t}<j\leq a_{\mathfrak v}\},\\ {\mathbb U}_{R<}^{d,{\mathfrak t},{\mathfrak v}}&:=&\{(a,j)\in {\mathbb U}({\mathfrak p})\mid a_{\mathfrak t}<j< a_{\mathfrak v}\},\\ {\mathbb U}^{u,{\mathfrak s},{\mathfrak t}}&:=&\{(a,j)\in {\mathbb U}({\mathfrak p})\mid a_{\mathfrak s}\leq j< a_{\mathfrak t}\},\\ {\mathbb U}_{L<}^{u,{\mathfrak s},{\mathfrak t}}&:=&\{(a,j)\in {\mathbb U}({\mathfrak p})\mid a_{\mathfrak s}< j< a_{\mathfrak t}\}. \end{eqnarray*} Further, set \begin{eqnarray} {\mathbb U}^{d,{\mathfrak t}}&=&{\mathbb U}^{d,{\mathfrak t},{\mathfrak p}},\\ {\mathbb U}^{u,{\mathfrak t}}&=&{\mathbb U}^{d,{\mathfrak e},{\mathfrak t}}. \end{eqnarray} It is also convenient to define \begin{Def} \begin{eqnarray}E_s(i,j)&:=&E_{\omega^{{\mathfrak p}}_{(s,i)}\Lambda_s,\omega^{\mathfrak p}_{(s,j)}\Lambda_s} \quad(0\leq i<j\leq s_{{\mathfrak p}}). \end{eqnarray} For $j'\geq s_{\mathfrak t}$ we set \begin{equation} E^d_{\mathfrak t}(s,j'):=E_s(s_{\mathfrak t},j'). \end{equation} For $j'\leq s_{\mathfrak t}$ we set \begin{equation} E^u_{\mathfrak t}(s,j'):=E_s(j',s_{\mathfrak t}). \end{equation} Finally, we set \begin{eqnarray} {\mathcal C}_q^d({\mathfrak t},{\mathfrak v})&=&\{E^d_{\mathfrak t}(s,j')\mid (s,j')\in {\mathbb U}^{d,{\mathfrak t},{\mathfrak v}}\},\\ {\mathcal C}_q^u({\mathfrak s},{\mathfrak t})&=&\{E^u_{\mathfrak t}(s,j'); (s,j')\in {\mathbb U}^{u,{\mathfrak s},{\mathfrak t}}\},\\ {\mathcal C}_q^d({\mathfrak t})&=&{\mathcal C}_q^d({\mathfrak t},{\mathfrak p}),\textrm{ and}\\ {\mathcal C}_q^u({\mathfrak t})&=&{\mathcal C}_q^u({\mathfrak s},{\mathfrak t}). \end{eqnarray} \end{Def} It is clear that ${\mathcal C}_q^d({\mathfrak t},{\mathfrak v})\subseteq {\mathcal C}_q^d({\mathfrak t})$ for any ${\mathfrak v}>{\mathfrak t}$ and ${\mathcal C}_q^u({\mathfrak s},{\mathfrak t})\subseteq {\mathcal C}_q^u({\mathfrak t})$ for any ${\mathfrak s}<{\mathfrak t}$. \begin{Lem}The elements in the set ${\mathcal C}_q^d({\mathfrak t})$ are $q$-commuting and the elements in the set ${\mathcal C}_q^u({\mathfrak t})$ are $q$-commuting.\label{above} \end{Lem} The proof is omitted as it is very similar to the proof of Proposition~\ref{7.13} which comes later. \begin{Def}${\mathcal A}_q^d({\mathfrak t}, {\mathfrak v})$ denotes the ${\mathbb C}$-algebra generated by ${\mathcal C}_q^d({\mathfrak t},{\mathfrak v})$ and ${\mathcal A}_q^u({\mathfrak s},{\mathfrak t})$ denotes the ${\mathbb C}$-algebra generated by ${\mathcal C}_q^u({\mathfrak s},{\mathfrak t})$. Further, ${\mathcal F}_q^d({\mathfrak t},{\mathfrak v})$ and ${\mathcal F}_q^u({\mathfrak s},{\mathfrak t})$ denote the corresponding skew-fields of fractions. Likewise, ${\mathbf L}_q^d({\mathfrak t},{\mathfrak v})$ and ${\mathbf L}_q^u({\mathfrak s},{\mathfrak t})$ denote the respective Laurent quasi-polynomial algebras. Finally, ${\mathcal L}_q^d({\mathfrak t},{\mathfrak v})$ and ${\mathcal L}_q^u({\mathfrak s},{\mathfrak t})$ denote the symplectic forms associated with the clusters ${\mathcal C}_q^d({\mathfrak t},{\mathfrak v})$, and ${\mathcal C}_q^u({\mathfrak s},{\mathfrak t})$, respectively. \end{Def} \begin{Def}Whenever ${\mathfrak a}<{\mathfrak b}$, we set \begin{equation}\forall s\in Im(\pi_{\mathfrak b}): {\det}_{s}^{{\mathfrak a},{\mathfrak b}}:=E_{\omega^{\mathfrak a}\Lambda_s,\omega^{\mathfrak b}\Lambda_s}. \end{equation} \end{Def} We conclude in particular that \begin{Prop}\label{quasipol}The elements ${\det}_{s}^{{\mathfrak t},{\mathfrak p}}$ $q$-commute with all elements in the algebra ${\mathcal A}_q^d({\mathfrak t})$ and the elements ${\det}_{s}^{{\mathfrak e},{\mathfrak t}}$ $q$-commute with all elements in the algebra ${\mathcal A}_q^d({\mathfrak t})$. \end{Prop} \begin{Def}An element $C$ in a quadratic algebra ${\mathcal A}$ that $q$-commutes with all the generating elements is said to be covariant. \end{Def} As a small aside, we mention the following easy generalization of the result in (\cite{jak-cen}): \begin{Prop}It ${\mathfrak a}<{\mathfrak b}$, then the spaces ${\mathcal A}_q^u({\mathfrak a},{\mathfrak b})$ and ${\mathcal A}_q^d({\mathfrak a},{\mathfrak b})$ are quadratic algebras. In both cases, the center is given by $Ker(\omega^{\mathfrak a}+\omega^{\mathfrak b})$. The semi-group of covariant elements in generated by $\{ {\det}_{s}^{{\mathfrak a},{\mathfrak b}}\mid s\in Im(\pi_{\mathfrak b})\}$. \end{Prop} We now construct some elements in ${\mathbf L}_q^d({\mathfrak t})$ and ${\mathbf L}_q^u({\mathfrak t})$ of fundamental importance. They are indeed monomials in the elements of $\left[{\mathcal C}_q^d({\mathfrak t})\right]^{\pm1}$ and $\left[{\mathcal C}_q^u({\mathfrak t})\right]^{\pm1}$, respectively. First a technical definition: \begin{Def} $p(a,j,k)$ denotes the largest non-negative integer for which $$\omega^{\mathfrak p}_{(k,p(a,j,k))}\Lambda_k=\omega^{\mathfrak p}_{(a,j)}\Lambda_k.$$ \end{Def} We also allow $E_a(j,j)$ which is defined to be $1$. Here are then the first building blocks: \begin{Def} \begin{eqnarray}\nonumber &\forall (a,j)\in {\mathbb U}^{d,{\mathfrak t}}:\\& H^d_{\mathfrak t}(a,j):=E_a(a_{\mathfrak t},j)E_a(a_{\mathfrak t},j-1) \prod_{a_{ka}<0}E_k(k_{\mathfrak t},p(a,j,k))^{a_{ka}} \\\nonumber &\forall (a,j)\in {\mathbb U}^{d,{\mathfrak t}}\textrm{ with }j<a_{\mathfrak p}:\\ &B^d_{\mathfrak t}(a,j):=H^d_{\mathfrak t}(a,j)(H^d_{\mathfrak t}(a,j+1))^{-1}. \end{eqnarray} \end{Def} The terms $E(k_{\mathfrak t},p(a,j,k))$ and $E_a(a_{\mathfrak t},j-1)$ are well-defined but may become equal to $1$. Also notice that, where defined, $H^d_{\mathfrak t}(a,j), B^d_{\mathfrak t}(a,j)\in {\mathbf L}_q^d({\mathfrak t})$. \begin{Lem}\label{7.10}If $E_{\xi,\eta}<H^d_{\mathfrak t}(a,j)$ in the sense that it is less than or equal to each factor $E_{\xi_1,\eta_1}$ of $H^d_{\mathfrak t}(a,j)$ (and $<$ is defined in (\ref{less})), then \begin{equation}\label{54} E_{\xi,\eta}H^d_{\mathfrak t}(a,j)=q^{(\xi-\eta,\omega^{\mathfrak t}(\alpha_a))}H^d_{\mathfrak t}(a,j)E_{\xi,\eta}. \end{equation} If $E_{\xi,\eta}\geq H^d_{\mathfrak t}(a,j)$, then \begin{equation}\label{55} E_{\xi,\eta}H^d_{\mathfrak t}(a,j)=q^{(-\xi-\eta,\omega^{\mathfrak t}(\alpha_a))}H^d_{\mathfrak t}(a,j)E_{\xi,\eta}. \end{equation} \end{Lem} \proof This follows from (\ref{less}) by observing that we have the following pairs $(\xi_1,\eta_1)$ occurring in $H^d_{\mathfrak t}(a,j)$: $$(\omega^{\mathfrak t}\Lambda_a,\omega(a,j)\Lambda_a),(\omega^{\mathfrak t}\Lambda_a,\omega(a,j)\sigma_a\Lambda_a),$$ and $$(-\omega^{\mathfrak t}\Lambda_k,-\omega(a,j)\Lambda_k) \textrm{ with multiplicity }(-a_{ka}).$$ Furthermore, as in (\ref{3.1}), $\Lambda_a+\sigma_a\Lambda_a+\sum_ka_{ka}\Lambda_k=0$ and, equivalently, $2\Lambda_a+\sum_ka_{ka}\Lambda_k=\alpha_a$ . \qed \begin{Prop}\label{7.10}$\forall (a,j),(b,j')\in {\mathbb U}^{d,{\mathfrak t}}, j<a_{\mathfrak p}$ the following holds: \begin{equation} E^d_{\mathfrak t}(b,j')B^d_{\mathfrak t}(a,j)=q^{-2(\Lambda_a,\alpha_a)\delta_{j,j'}\delta_{a,b}}B^d_{\mathfrak t}(a,j)E^d_{\mathfrak t}(b,j'). \end{equation} \end{Prop} \proof It is clear from the formulas (\ref{54}-\ref{55}) that if an element $E_{\xi,\eta}$ either is bigger than all factors in $B^d_{\mathfrak t}(s,j)$ or smaller than all factors, then it commutes with this element. The important fact now is that the ordering is independent of the fundamental weights $\Lambda_i$ - it depends only on the Weyl group elements. The factors in any $H_{\mathfrak t}^d$ are, with a fixed ${\mathfrak t}$, of the form $E_{\omega^{\mathfrak t}\Lambda_i,\omega\Lambda_i}$ or $E_{\omega^{\mathfrak t}\Lambda_a,\omega\circ\sigma_a\Lambda_a}$ for some $\omega\geq \omega^{\mathfrak t}$. The elements $E_{\xi,\eta}=E^d_{\mathfrak t}(b,j')$ we consider thus satisfy the first or the second case in Lemma~\ref{7.10} for either terms $H^d_{\mathfrak t}(a,j)$ and $H^d_{\mathfrak t}(a,j+1)$. Clearly, we then need only consider the in-between case $H^d_{\mathfrak t}(a,j)\leq E_{\xi,\eta}\leq H^d_{\mathfrak t}(a,j+1)$, and here there appears a factor $q^{-2(\xi,\omega^{\mathfrak t}(\alpha_a))}$ in the commutator with $\xi=\omega^{\mathfrak t}\Lambda_{b}$. This accounts for the term $-2(\Lambda_a,\alpha_a)\delta_{a,b}$. Finally, if $a=b$ the previous assumption forces $j=j'$. \qed Let us choose an enumeration \begin{equation} {\mathcal C}_q^d({\mathfrak t})=\{c_1,c_2,\dots, c_N\} \end{equation} so that each $(a,j)\leftrightarrow k$ and let us use the same enumeration of the elements $B^d_{\mathfrak t}(a,j)$. Set, for now $B^d_{\mathfrak t}(a,j)=b_k$ if $(a,j)\leftrightarrow k$. Let us also agree that the, say $n$, non-mutable elements $\det_s^{{\mathfrak t},{\mathfrak p}}$ of ${\mathcal C}_q^d({\mathfrak t})$ are written last, say numbers $N-n+1, N-n+2,\dots, N-n$. Then, as defined, \begin{equation} \forall j=1,\dots, N-n: b_j=q^{\alpha_j}\prod_kc_k^{b_{kj}} \end{equation} for some integers $b_{kj}$ and some, here inconsequential, factor $q^{\alpha_j}$. The symplectic form yields a matrix which we, abusing notation slightly, also denote ${\mathcal L}_q^d({\mathfrak t})$ such that \begin{equation} \forall i,j=1,\dots, N:\ \left({\mathcal L}_q^d({\mathfrak t})\right)_{ij}=\lambda_{ij} \end{equation} and \begin{equation} \forall i,j=1,\dots, N: c_jc_j=q^{\lambda_{ij}}c_jc_i.\end{equation} Similarly, we let ${\mathcal B}_q^d({\mathfrak t})$ denote the matrix \begin{equation} \forall i=1,\dots, N\ \forall j=1,\dots, N-n: \left({\mathcal B}_q^d({\mathfrak t})\right)_{ij}=b_{ij}. \end{equation} Then, where defined, \begin{equation} c_ib_j=\prod_kq^{\lambda_{ik}b_{kj}}b_jc_i, \end{equation} and Proposition~\ref{7.10} may then be restated as \begin{equation} \forall i=1,\dots, N\ \forall j=1,\dots, N-n: \sum_k\lambda_{ik}b_{kj}=-2(\Lambda_s,\alpha_s)\delta_{ij}, \end{equation} where we assume that $i\leftrightarrow (s,\ell)$. We have then established \begin{Thm} The pair $({\mathcal L}_q^d({\mathfrak t}),{\mathcal B}_q^d({\mathfrak t}))$ is a compatible pair and hence, \begin{equation} {\mathcal Q}_q^d({\mathfrak t}):=({\mathcal C}_q^d({\mathfrak t}),{\mathcal L}_q^d({\mathfrak t}),{\mathcal B}_q^d({\mathfrak t})) \end{equation} is a quantum seed with the $n$ non-mutable elements $\det_s^{{\mathfrak t},{\mathfrak p}}$, $(s,s_{\mathfrak p})\in {\mathbb U}^d({\mathfrak t})$. The entries of the diagonal of the matrix $\tilde D=({\mathcal B}_q^d({\mathfrak t}))^T{\mathcal L}_q^d({\mathfrak t})$ are in the set $\{2(\Lambda_s,\alpha_s)\mid s=1,\dots,R\}$. \end{Thm} It ${\mathfrak v}>{\mathfrak t}$, we let $({\mathcal L}_q^d({\mathfrak t},{\mathfrak v}),{\mathcal B}_q^d({\mathfrak t},{\mathfrak v}))$ denote the part of the compatible pair $({\mathcal L}_q^d({\mathfrak t}),{\mathcal B}_q^d({\mathfrak t}))$ that corresponds to the cluster ${\mathcal C}_q^d({\mathfrak t},{\mathfrak v})$ and we let ${\mathcal Q}_q^d({\mathfrak t},{\mathfrak v})$ be the corresponding triple. It is then obvious by simple restriction, that we in fact have obtained \begin{Thm} The pair $({\mathcal L}_q^d({\mathfrak t},{\mathfrak v}),{\mathcal B}_q^d({\mathfrak t},{\mathfrak v}))$ is a compatible pair and hence, \begin{equation} {\mathcal Q}_q^d({\mathfrak t},{\mathfrak v}):=({\mathcal C}_q^d({\mathfrak t},{\mathfrak v}),{\mathcal L}_q^d({\mathfrak t},{\mathfrak v}),{\mathcal B}_q^d({\mathfrak t},{\mathfrak v})) \end{equation} is a quantum seed with the $n$ non-mutable elements $\det_s^{{\mathfrak t},{\mathfrak v}}$, $(s,s_{\mathfrak v})\in {\mathbb U}^d({\mathfrak t},{\mathfrak v})$. \end{Thm} The case of ${\mathcal C}_q^u({\mathfrak t})$ is completely analogous: Define \begin{Def} \begin{eqnarray} H^u_{\mathfrak t}(a,j)&:=&E_a(j, a_{\mathfrak t})E_a(j-1,a_{\mathfrak t}) \prod_{a_{ka}<0}E_k(p(a,j,k), k_{\mathfrak t})^{a_{ka}} \ (1\leq j<a_{\mathfrak t}),\nonumber\\ B^u_{\mathfrak t}(a,j)&:=&H^u_{\mathfrak t}(a,j+1)(H^u_{\mathfrak t}(a,j))^{-1} \ (1\leq j<a_{\mathfrak t}). \end{eqnarray} The terms $E(p(a,j,k),k_{\mathfrak t})$ are well-defined but may become equal to $1$. Notice also the exponents on the terms $H^u_{\mathfrak t}$. \end{Def} The terms $E(p(a,j,k),k_{\mathfrak t})$ are well-defined but may become equal to $1$. As defined, $H^u_{\mathfrak t}(a,j)$, and $B^u_{\mathfrak t}(a,j)$ are in ${\mathbf L}_q^u({\mathfrak t})$. \begin{Prop}$\forall (a,j),(b,j')\in {\mathbb U}^{u,{\mathfrak t}}, 1\leq j$ the following holds: \begin{equation} E^u_{\mathfrak t}(b,j')B^u_{\mathfrak t}(a,j)=q^{2(\Lambda_a,\alpha_a)\delta_{j,j'}\delta_{a,b}}B^u_{\mathfrak t}(a,j)E^u_{\mathfrak t}(b,j'). \end{equation} \end{Prop} We then get in a similar way \begin{Thm} The pair $({\mathcal L}_q^u({\mathfrak t}),{\mathcal B}_q^u({\mathfrak t}))$ is a compatible pair and hence, \begin{equation} {\mathcal Q}_q^u({\mathfrak t}):=({\mathcal C}_q^u({\mathfrak t}),{\mathcal L}_q^u({\mathfrak t}),{\mathcal B}_q^u({\mathfrak t})) \end{equation} is a quantum seed with the $n$ non-mutable elements $\det_s^{{\mathfrak e},{\mathfrak t}}$, $(s,s_{\mathfrak t})\in {\mathbb U}^u({\mathfrak t})$. \end{Thm} Naturally, we even have \begin{Thm} The pair $({\mathcal L}_q^u({\mathfrak s},{\mathfrak t}),{\mathcal B}_q^u({\mathfrak s},{\mathfrak t}))$ is a compatible pair and hence, \begin{equation} {\mathcal Q}_q^u({\mathfrak s},{\mathfrak t}):=({\mathcal C}_q^u({\mathfrak s},{\mathfrak v}),{\mathcal L}_q^u({\mathfrak s},{\mathfrak t}),{\mathcal B}_q^u({\mathfrak s},{\mathfrak t})) \end{equation} is a quantum seed with the $n$ non-mutable elements $\det_s^{{\mathfrak s},{\mathfrak t}}$, $(s,s_{\mathfrak s})\in {\mathbb U}^u({\mathfrak s},{\mathfrak t})$. \end{Thm} We now wish to consider more elaborate seeds. The first generalization is the most important: Let \begin{equation} {\mathfrak e}\leq {\mathfrak a}\leq {\mathfrak b}\leq{\mathfrak c}\leq {\mathfrak p}, \textrm{ but } {\mathfrak a}\neq {\mathfrak c}.\end{equation} \begin{eqnarray}\label{l1} {\mathcal C}_q^d({\mathfrak a},{\mathfrak b},{\mathfrak c})&:=&\{E^d_{\mathfrak a}(s,j)\mid (a,j)\in ({\mathbb U}^{d,{\mathfrak b}}\setminus {\mathbb U}^{d,{\mathfrak c}})= {\mathbb U}^{d,{\mathfrak b},{\mathfrak c}}\},\\\label{l2}{\mathcal C}_q^u({\mathfrak a},{\mathfrak b},{\mathfrak c})&:=&\{E^u_{\mathfrak c}(s,j)\mid (s,j)\in ({\mathbb U}^{u,{\mathfrak b}}\setminus {\mathbb U}^{u,{\mathfrak a}})= {\mathbb U}^{u,{\mathfrak a},{\mathfrak b}}\}. \end{eqnarray} In (\ref{l1}), ${\mathfrak a}={\mathfrak b}$ is allowed, and in (\ref{l2}), ${\mathfrak b}={\mathfrak c}$ is allowed. \begin{Def} \begin{eqnarray*}{\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})&:=&{\mathcal C}_q^d({\mathfrak a},{\mathfrak b},{\mathfrak c})\cup{\mathcal C}_q^u({\mathfrak a},{\mathfrak b},{\mathfrak b}),\\ {\mathcal C}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})&:=&{\mathcal C}_q^u({\mathfrak a},{\mathfrak b},{\mathfrak c})\cup{\mathcal C}_q^d({\mathfrak b},{\mathfrak b},{\mathfrak c}). \end{eqnarray*} \end{Def} \begin{Prop}\label{7.13} The elements of ${\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})$ and ${\mathcal C}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})$, respectively, $q$-commute. \end{Prop} \proof The two cases are very similar, so we only prove it for the first case. We examine 3 cases, while using the following mild version of Theorem~\ref{10.2}: $ \triangle_{s's\lambda,t'\lambda}$ and $\triangle_{s'\mu,t't\mu}$ $q$ commute for any $\lambda,\mu\in P^+$, and $s, s', t, t' \in W$ for which $\ell(s's) = \ell(s') + \ell(s), \ell(t't) = \ell(t') + \ell(t)$. {\bf Case 1:} $E_{\mathfrak a}^d(s,t)$ and $E_{\mathfrak a}^d(s_1,t_1)$ for $(s,t)\in {\mathbb U}^{d, {\mathfrak b},{\mathfrak c}}$ and $(s,t)<(s_1,t_1)$: Set $\lambda=\Lambda_s,\mu=\Lambda_{s_1}$, $s=1,s'=\omega^{\mathfrak a}$, and $t'=\omega^{\mathfrak c}(s,t), t't=\omega^{\mathfrak c}(s_1,t_1)$. {\bf Case 2:} $E_{\mathfrak b}^u(s,t)$ and $E_{\mathfrak b}^u(s_1,t_1)$ for $(s,t)\in {\mathbb U}^{u, {\mathfrak a},{\mathfrak b}}$ and $(s,t)>(s_1,t_1)$: Set $\lambda=\Lambda_s,\mu=\Lambda_{s_1}$, $t=1$, $t'=\omega^{\mathfrak b}$ and $s'=\omega^{\mathfrak p}(s_1,t_1), s's=\omega^{\mathfrak r}(s,t)$. {\bf Case 3:} $E_{\mathfrak b}^u(s,t)$ and $E_{\mathfrak a}^d(s_1,t_1)$ for $(s,t)\in {\mathbb U}^{u, {\mathfrak a},{\mathfrak b}}$ and $(s_1,t_1)\in {\mathbb U}^{d, {\mathfrak b},{\mathfrak c}}$: Set $\lambda=\Lambda_s,\mu=\Lambda_{s_1}$, $s'=\omega^{{\mathfrak a}}$, $s=\omega^{\mathfrak p}(s,t)$, $t'=\omega^{\mathfrak b}$ and $t't=\omega^{\mathfrak p}(s_1,t_1)$. \qed Notice that the ordering in ${\mathbb U}^{u, {\mathfrak a},{\mathfrak b}}$ (Case 2) is the opposite of that of the two other cases. We also define, for ${\mathfrak a}<{\mathfrak b}$, \begin{eqnarray} {\mathcal C}_q^u({\mathfrak a},{\mathfrak b})&=&{\mathcal C}_q^u({\mathfrak a}, {\mathfrak b},{\mathfrak b}),\textrm{ and}\\\nonumber {\mathcal C}_q^d({\mathfrak a},{\mathfrak b})&=&{\mathcal C}_q^d({\mathfrak a}, {\mathfrak a},{\mathfrak b}). \end{eqnarray} We let ${\mathcal L}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})$ and ${\mathcal L}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})$ denote the corresponding symplectic matrices. We proceed to construct compatible pairs and give the details for just ${\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})$. We will be completely explicit except in the special cases $E^u_{\omega^{{\mathfrak a}}\Lambda_s,\omega^{{\mathfrak b}}\Lambda_s}={\det}_{s}^{{\mathfrak a},{\mathfrak b}}$ where we only give a recipe for $ {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}}) $. Notice, however, the remark following (\ref{77}). \begin{equation}\label{72}{B}_q^{{\mathfrak a}, {\mathfrak b},{\mathfrak c}}(s,j):=\left\{\begin{array}{lll}B^d_{{\mathfrak a}}(s,j)&\textrm{if }(s,j)\in {\mathbb U}_{R<}^{d, {\mathfrak b},{\mathfrak c}}\\ \ \\ B^u_{{\mathfrak b}}(s,j)&\textrm{if }(s,j)\in {\mathbb U}_{L<}^{u, {\mathfrak a},{\mathfrak b}} \end{array}\right..\end{equation} We easily get from the preceding propositions: \begin{Prop}\label{7.20} Let $E(b,j')\in {\mathcal C}_q({{\mathfrak a}, {\mathfrak b},{\mathfrak c}})$ and let ${B}_q^{{\mathfrak a}, {\mathfrak b},{\mathfrak c}}(s,j)$ be as in the previous equation. Then \begin{equation} E(b,j'){B}_q^{{\mathfrak a}, {\mathfrak b},{\mathfrak c}}(s,j)=q^{{-2(\Lambda_s,\alpha_s)\delta_{j,j'}\delta_{s,b}}}{B}_q^{{\mathfrak a}, {\mathfrak b},{\mathfrak c}}(s,j)E(b,j'), \end{equation} and ${B}_q^{{\mathfrak a}, {\mathfrak b},{\mathfrak c}}(s,j)$ is in the algebra ${\mathcal A}_q({{\mathfrak a}, {\mathfrak b},{\mathfrak c}})$ generated by the elements of ${\mathcal C}_q({{\mathfrak a}, {\mathfrak b},{\mathfrak c}})$. \end{Prop} This then leaves the positions $(s,s_{\mathfrak c})\in {\mathbb U}^{d,{\mathfrak b},{\mathfrak c}}$ and $(s,s_{\mathfrak a})\in {\mathbb U}^{u,{\mathfrak a},{\mathfrak b}}$ to be considered. Here, the first ones are considered as the non-mutable elements. In the ambient space ${\mathcal A}_q({{\mathfrak a}, {\mathfrak b},{\mathfrak c}})$, the positions in remaining cases define elements that are, in general, mutable. The elements in these cases are of the form $E_{\omega^{{\mathfrak a}}\Lambda_s,\omega^{{\mathfrak b}}\Lambda_s}$ for some $s$. To give a recipe we define the following elements in ${\mathcal A}_q({{\mathfrak a}, {\mathfrak b},{\mathfrak c}})$: \begin{eqnarray}&\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}}) :=\\&\left(H^u_{{\mathfrak b}}(s,s_{{\mathfrak a}}+1) H^d_{{\mathfrak a}}(s,s_{{\mathfrak b}}+1)\right)^{-1} E_s(s_{{\mathfrak a}},s_{{\mathfrak b}})^2\prod_{a_{ka}<0}E_k(k_{{\mathfrak a}},k_{{\mathfrak b}})^{a_{ks}}.\nonumber\end{eqnarray} If $\omega(s,s_{{\mathfrak a}}+1)=\omega^{{\mathfrak a}}\omega_x\sigma_s$ and $\omega(s,s_{{\mathfrak b}}+1)=\omega^{{\mathfrak b}}\omega_y\sigma_s$, and if we set $u=\omega^{{\mathfrak a}}\omega_x$, $v=\omega^{{\mathfrak b}}\omega_y$ this takes the simpler form \begin{equation}\label{75}\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}}) =E_{u\sigma_s\Lambda_s,v\Lambda_s}^{-1}E_{ u\Lambda_s,v\sigma_s\Lambda_s}^{-1}\prod_{a_{ks<0}} E_{\omega^{\mathfrak a}\Lambda_k,v\Lambda_k}^{-a_{ks}}\prod_{a_{ks<0}} E_{u\Lambda_k,\omega^{\mathfrak b}\Lambda_k}^{-a_{ks}}\prod_{a_{ks<0}} E_{\omega^{\mathfrak a}\Lambda_k,\omega^{\mathfrak b}\Lambda_k}^{a_{ks}}.\end{equation} \begin{Prop} \begin{equation} \forall\ell: E_{\omega^{\mathfrak a}\Lambda_\ell, \omega^{\mathfrak b}\Lambda_\ell}\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})=q^{-2\delta_{\ell.s}(\lambda_s,\alpha_s)} \tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})E_{\omega^{\mathfrak a}\Lambda_\ell, \omega^{\mathfrak b}\Lambda_\ell}. \end{equation} Besides this, $\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})$ commutes with everything in the cluster except possibly elements of the form $$ E_{\omega^{{\mathfrak a}}\Lambda_\ell,\omega^{{\mathfrak b}}\tilde\omega_y\Lambda_\ell}, \textrm{ and } E_{\omega^{{\mathfrak a}}\tilde\omega_x\Lambda_\ell,\omega^{{\mathfrak b}}\Lambda_\ell}, $$ with $1<\tilde\omega_x<\omega_x$ and $1<\tilde\omega_y<\omega_y$. \end{Prop} The exceptional terms above are covered by Proposition~\ref{7.20} which means that we can in principle make a modification $\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})\to {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})$ where the latter expression commutes with everything except $E_{\omega^{{\mathfrak a}}\Lambda_s,\omega^{{\mathfrak b}}\Lambda_s}$ where we get a factor $q^{-2(\Lambda_s,\alpha_s)}$. If $\omega_y=1$ we get a further simplification where now $u=\omega^{{\mathfrak a}}\omega_x$ and $v=\omega^{{\mathfrak b}}$: \begin{equation}\label{77}\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}}) =E_{u\sigma_s\Lambda_s,v\Lambda_s}^{-1}E_{ u\Lambda_s,v\sigma_s\Lambda_s}^{-1}\prod_{a_{ks<0}} E_{u\Lambda_k,v\Lambda_k}^{-a_{ks}}.\end{equation} Here we actually have $\tilde {B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}}) ={B}_q^{{\mathfrak a},{\mathfrak b},{\mathfrak c}}(s,s_{{\mathfrak a}})$, and this expression has the exact form needed for the purposes of the next section. We let ${\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})$ and ${\mathcal B}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})$ denote the corresponding symplectic matrices and can now finally define our quantum seeds: \begin{equation}{\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}):=({\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal L}_q({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})).\end{equation} \begin{Def}\begin{equation}{\mathcal Q}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c}):=({\mathcal C}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal L}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c}), {\mathcal B}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})).\end{equation}\end{Def} According to our analysis above we have established \begin{Thm}\label{seedth} They are indeed seeds. The non-mutable elements are in both cases the elements ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}; s\in Im(\pi_{\omega^{\mathfrak c}})$. \end{Thm} Let us finally consider a general situation where we are given a finite sequence of elements $\{\omega^{{\mathfrak r}_i}\}_{i=1}^n\in W^p$ such that \begin{equation}\label{genseq} {\mathfrak e}\leq {{\mathfrak r}_1}<\dots<{{\mathfrak r}_n}\leq {{\mathfrak p}}. \end{equation} Observe that \begin{equation}\forall(s,t)\in{\mathbb U}({\mathfrak r}_k):\omega^{{\mathfrak r}_k}_{(s,t)}=\omega^{{\mathfrak p}}_{(s,t)}. \end{equation} It may of course well happen that for some $a$, and some $ {{\mathfrak r}_i}<{{\mathfrak r}_j}$, \begin{equation}\omega^{{\mathfrak r}_i}\Lambda_a=\omega^{{\mathfrak r}_j}\Lambda_a. \end{equation} \begin{Def}Given (\ref{genseq}) we define \begin{eqnarray}{\mathcal C}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)&=&{\mathcal C}_q^d({\mathfrak r}_1, {\mathfrak r}_{n-1},{\mathfrak r}_n)\cup {\mathcal C}_q^u({\mathfrak r}_1,{\mathfrak r}_{2},{\mathfrak r}_{n-1})\cup\dots\\&=&\bigcup_{0<2i\leq n}{\mathcal C}_q^d({\mathfrak r}_i,{\mathfrak r}_{n-i},{\mathfrak r}_{n-i+1})\cup \bigcup_{0<2j\leq n-1}{\mathcal C}_q^u({\mathfrak r}_j,{\mathfrak r}_{j+1},{\mathfrak r}_{n-j}). \nonumber \end{eqnarray} It is also convenient to consider \begin{eqnarray}{\mathcal C}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)&=&{\mathcal C}_q^u({\mathfrak r}_1, {\mathfrak r}_{2},{\mathfrak r}_n)\cup {\mathcal C}_q^d({\mathfrak r}_2,{\mathfrak r}_{n-1},{\mathfrak r}_{n})\cup\dots\\&=&\bigcup_{0<2i\leq n}{\mathcal C}_q^u({\mathfrak r}_i,{\mathfrak r}_{i+1},{\mathfrak r}_{n-i+1})\cup \bigcup_{0<2j\leq n-1}{\mathcal C}_q^d({\mathfrak r}_{j+1},{\mathfrak r}_{n-j},{\mathfrak r}_{n-j+1}). \nonumber \end{eqnarray} \end{Def} Notice that \begin{eqnarray}{\mathcal C}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)&=&{\mathcal C}_q^d({\mathfrak r}_1, {\mathfrak r}_{n-1},{\mathfrak r}_n)\cup {\mathcal C}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-2},{\mathfrak r}_{n-1})\\\nonumber {\mathcal C}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)&=&{\mathcal C}_q^u({\mathfrak r}_1, {\mathfrak r}_{2},{\mathfrak r}_n)\cup {\mathcal C}_q({\mathfrak r}_2,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n) \end{eqnarray} For the last equations, notice that ${\mathcal C}_q^{d}({\mathfrak e},{\mathfrak r},{\mathfrak r}) =\emptyset={\mathcal C}_q^{d}({\mathfrak r},{\mathfrak r},{\mathfrak r})$. \begin{Prop}\label{qcom} The spaces \begin{equation} {\mathcal C}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\textrm{ and }{\mathcal C}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n) \end{equation} each consists of $q$-commuting elements. \end{Prop} \proof This is proved in the same way as Proposition~{\ref{7.13}. \qed Our goal is to construct seeds out of these clusters using (and then generalizing) Proposition~\ref{seedth}. With Proposition~\ref{qcom} at hand, we are immediately given the corresponding symplectic matrices \begin{equation} {\mathcal L}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\textrm{ and }{\mathcal L}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n). \end{equation} The construction of the accompanying $B$-matrices \begin{equation} {\mathcal B}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\textrm{ and }{\mathcal B}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n) \end{equation} takes a little more work, though in principle it is straightforward. The idea is in both cases to consider an element in the cluster as lying in a space \begin{eqnarray} {\mathcal C}_q^d({\mathfrak r}_i, {\mathfrak r}_{n-i},{\mathfrak r}_{n-i+1})\cup {\mathcal C}_q^u({\mathfrak r}_i, {\mathfrak r}_{i+1},{\mathfrak r}_{n-i})&\subseteq&{\mathcal C}_q({\mathfrak r}_i, {\mathfrak r}_{n-i},{\mathfrak r}_{n-i+1})\textrm{ or}\\ {\mathcal C}_q^u({\mathfrak r}_i, {\mathfrak r}_{i+1},{\mathfrak r}_{n-i+1})\cup {\mathcal C}_q^d({\mathfrak r}_{i+1}, {\mathfrak r}_{n-i},{\mathfrak r}_{n-i+1})&\subseteq&{\mathcal C}_q^o({\mathfrak r}_i, {\mathfrak r}_{i+1},{\mathfrak r}_{n-i+1})\quad \end{eqnarray} as appropriate. Then we can use the corresponding matrices ${\mathcal B}_q({\mathfrak r}_i, {\mathfrak r}_{n-i},{\mathfrak r}_{n-i+1})$ or ${\mathcal B}_q^o({\mathfrak r}_i, {\mathfrak r}_{i+1},{\mathfrak r}_{n-i+1})$ in the sense that one can extend these matrices to the full rank by inserting rows of zeros. In this way, we can construct columns even for the troublesome elements of the form $E(a_{{\mathfrak r}_i}, a_{{\mathfrak r}_j})$ that may belong to such spaces. Indeed, we may start by including $E(a_{{\mathfrak r}_{\frac{ n+0}2}}, a_{{\mathfrak r}_{\frac{n+2}2}})$ ($n$ even) or $E(a_{{\mathfrak r}_{\frac{n-1}2}}, a_{{\mathfrak r}_{\frac{n+1}2}})$ ($n$ odd) in a such space in which they may be seen as mutable. Then these spaces have new non-mutable elements which can be handled by viewing them in appropriate spaces. The only ones which we cannot capture are the elements ${\det}_{s}^{{\mathfrak r}_1,{\mathfrak r}_n}=E(s_{{\mathfrak r}_1}, s_{{\mathfrak r}_n})$. \begin{Def}In both cases, the elements ${\det}_{s}^{{\mathfrak r}_1,{\mathfrak r}_n}$, $s\in Im(\pi_{{\mathfrak r}_1})$ are the non-mutable elements. We let ${\mathcal N}_q({\mathfrak r}_1, {\mathfrak r}_n)$ denote the set of these. \end{Def} \begin{Prop} \begin{equation} {\mathcal Q}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\textrm{ and } {\mathcal Q}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n) \end{equation} are quantum seeds. \end{Prop} \section{Mutations} Here is the fundamental setup: Let $\omega^{\mathfrak a}, \omega^{\mathfrak b},\omega^{\mathfrak c}\in W^p$ satisfy \begin{equation}{\mathfrak a}<{\mathfrak c}\textrm{ and } {\mathfrak a}\leq {\mathfrak b}\leq{\mathfrak c}.\end{equation} \begin{Def}\label{7.1bis} A root $\gamma\in\triangle^+({\mathfrak c})$ is an {\bf increasing-mutation site} of $\omega^{\mathfrak b}\in W^p$ (in reference to $({\mathfrak a},{\mathfrak b},{\mathfrak c})$) if there exists a reduced form of $\omega^{\mathfrak c}$ as \begin{equation} \omega^{\mathfrak c}=\hat\omega\sigma_\gamma\omega^{\mathfrak b}. \end{equation} Let $W^p\ni\omega^{{\mathfrak b}'}=\sigma_\gamma\omega^{\mathfrak b}$. It follows that \begin{equation}\label{94} \omega^{{\mathfrak b}'}=\omega^{\mathfrak b}\sigma_{\alpha_s} \end{equation} for a unique $s\in Im(\pi_{{\mathfrak b}'})$. Such a site will henceforth be called an ${\mathfrak m}^+$ site. We will further say that $\gamma$ is a {\bf decreasing-mutation site}, or ${\mathfrak m}^-$ site (in reference to $({\mathfrak a},{\mathfrak b},{\mathfrak c})$) of $\omega^{{\mathfrak b}}\in W^p$ in case there exists a rewriting of $\omega^{{\mathfrak b}}$ as $\omega^{{\mathfrak b}}=\sigma_\gamma\omega^{{\mathfrak b}''}$ with ${\mathfrak a}\leq \omega^{{\mathfrak b}''}\in W^p$. Here, \begin{equation} \omega^{{\mathfrak b}}=\omega^{{\mathfrak b}''}\sigma_{\alpha_s} \end{equation} for a unique $s\in Im(\pi_{{\mathfrak b}})$. We view such sites as places where replacements are possible and will use the notation \begin{equation}\label{m+}{\mathfrak m}^{+}_{{\mathfrak a},{\mathfrak c}}:({\mathfrak a},{\mathfrak b},{\mathfrak c})\to ({\mathfrak a},{\mathfrak b}',{\mathfrak c}),\end{equation} and \begin{equation}{\mathfrak m}^-_{{\mathfrak a},{\mathfrak c}}:({\mathfrak a},{\mathfrak b},{\mathfrak c})\to ({\mathfrak a},{\mathfrak b}'',{\mathfrak c}),\end{equation} respectively, for the replacements while at the same time defining what we mean by replacements. Notice that ${\mathfrak a}={\mathfrak b}$ and ${\mathfrak b}'={\mathfrak c}$ are allowed in the first while ${\mathfrak b}={\mathfrak c}$ and ${\mathfrak b}''={\mathfrak a}$ are allowed in the second. Furthermore, $${\mathfrak m}_{{\mathfrak a},{\mathfrak c}}:({\mathfrak a},{\mathfrak b},{\mathfrak c})\to ({\mathfrak a},{\mathfrak b}_1,{\mathfrak c})$$ denotes the composition of any finite number of such maps ${\mathfrak m}^{\pm}_{{\mathfrak a},{\mathfrak c}}$ (in any order, subject to the limitations at any step stipulated above) We will further extend the meaning of ${\mathfrak m}_{{\mathfrak a},{\mathfrak c}}$ also to include the replacements $${\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})\to {\mathcal C}_q({\mathfrak a},{\mathfrak b}_1,{\mathfrak c}),$$ and even $${\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})\to {\mathcal Q}_q({\mathfrak a},{\mathfrak b}_1,{\mathfrak c}).$$At the seed level, we will refer to the replacements as {\bf Schubert mutations}. Similarly, we can define maps ${\mathfrak m}^{o,\pm}_{{\mathfrak a},{\mathfrak c}}$, and after that mutations as composites $${\mathfrak m}^{o}_{{\mathfrak a},{\mathfrak c}}:{\mathcal Q}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})\to {\mathcal Q}_q^o({\mathfrak a},{\mathfrak b}_1,{\mathfrak c}).$$ \end{Def} We need to define another kind of replacement: Consider\begin{equation}\label{maxim} {\mathfrak a}<{\mathfrak b}_1<{\mathfrak b}<{\mathfrak c}. \end{equation} \begin{Def}We say that $({\mathfrak a},{\mathfrak b},{\mathfrak c})$ is a {\bf d-splitting} of $({\mathfrak a},{\mathfrak c})$ if $${\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})={\mathcal C}_q({\mathfrak a},{\mathfrak c}).$$ In this case we will also say that $({\mathfrak a},{\mathfrak c})$ is a {\bf d-merger} of $({\mathfrak a},{\mathfrak b},{\mathfrak c})$. \end{Def} To make this more definitive, one might further assume that ${\mathfrak b}$ is maximal amongst those satisfying (\ref{maxim}), but we will not need to do this here. Similarly, \begin{Def}We say that $({\mathfrak a},{\mathfrak b},{\mathfrak c})$ is a {\bf u-splitting} of $({\mathfrak a},{\mathfrak c})$ if $${\mathcal C}_q^o({\mathfrak a},{\mathfrak b},{\mathfrak c})={\mathcal C}_q^o({\mathfrak a},{\mathfrak c}).$$ Similarly, we will in this case also say that $({\mathfrak a},{\mathfrak c})$ is a {\bf u-merger} of $({\mathfrak a},{\mathfrak b},{\mathfrak c})$. \end{Def} Our next definition combines the two preceding: \begin{Def}\label{def84}A Schubert creation replacement $$a^+_{{\mathfrak a},{\mathfrak c}}:({\mathfrak a},{\mathfrak c})\rightarrow ({\mathfrak a},{\mathfrak b}_1,{\mathfrak c})$$ consists in a d-splitting $$({\mathfrak a},{\mathfrak c})\rightarrow ({\mathfrak a},{\mathfrak b},{\mathfrak c})$$ followed by a replacement $m_{{\mathfrak a},{\mathfrak c}}$ applied to $({\mathfrak a},{\mathfrak b},{\mathfrak c})$. A Schubert annihilation replacement $$a^-_{{\mathfrak a},{\mathfrak c}}:({\mathfrak a},{\mathfrak b}_1,{\mathfrak c})\rightarrow ({\mathfrak a},{\mathfrak c})$$ is defined as the reverse process. Schubert creation/annihilation mutations $a^{o,\pm}_{{\mathfrak a},{\mathfrak c}}$ are defined analogously; $$a^{o,+}_{{\mathfrak a},{\mathfrak c}}: {\mathcal Q}_q^o({\mathfrak a},{\mathfrak c})\to {\mathcal Q}_q^o({\mathfrak a},{\mathfrak b}_1,{\mathfrak c}), $$and $$a^{o,-}_{{\mathfrak a},{\mathfrak c}}: {\mathcal Q}_q^o({\mathfrak a},{\mathfrak b}_1,{\mathfrak c})\to {\mathcal Q}_q^o({\mathfrak a},{\mathfrak c}). $$ We finally extend these Schubert creation/annihilation mutations into (we could do it more generally, but do not need to do so here) $${\mathcal Q}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)\rightarrow {\mathcal Q}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-2},\dots,{\mathfrak r}_{n\pm 1})$$ by inserting/removing an ${\mathfrak r}_x$ between ${\mathfrak r}_{\frac{n}2}$ and ${\mathfrak r}_{\frac{n}2+1}$ ($n$ even) or between ${\mathfrak r}_{\frac{n-1}2}$ and ${\mathfrak r}_{\frac{n+1}2}$ ($n$ odd). Similar maps are defined for the spaces ${\mathcal Q}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)$. \end{Def} In the sequel, we will encounter expressions of the form $\check B(u,v,s)$; \begin{equation}\label{99} \check B(u,v,s)=E_{u\sigma_s\Lambda_s,v\Lambda_s}^{-1}E_{ u\Lambda_s,v\sigma_s\Lambda_s}^{-1}\prod_{a_{ks<0}} E_{u\Lambda_k,v\Lambda_k}^{a_{ks}}\end{equation} where \begin{equation} E(u\Lambda_s,v\Lambda_s)\check B(u,v,s)=q^{-2(\Lambda_s,\alpha_s)}\check B(u,v,s)E(u\Lambda_s,v\Lambda_s), \end{equation} and where $\check B(u,v,s)$ commutes with all other elements in a given cluster. \begin{Def} We say that $\check B(u,v,s)$ implies the change $$E_{u\Lambda_s,v\Lambda_s}\to E_{u\sigma_s\Lambda_s,v\sigma_s\Lambda_s}.$$ \end{Def} We will only encounter such changes where the set with $E_{u\Lambda_s,v\Lambda_s}$ removed from the initial cluster, and $E_{u\sigma_s\Lambda_s,v\sigma_s\Lambda_s}$ added, again is a cluster. We further observe that a (column) vector with $-1$ at positions corresponding to $E_{u\sigma_s\Lambda_s,v\Lambda_s}$ and $E_{u\Lambda_s,v\sigma_s\Lambda_s}$ and $ {a_{ks}}$ at each position corresponding to a $E_{u\Lambda_k,v\Lambda_k}$ with $a_{ks}<0$ has the property that the symplectic form of the original cluster, when applied to it, returns a vector whose only non-zero entry is $-2(\Lambda_s,\alpha_s)$ at the position corresponding to $E_{u\Lambda_s,v\Lambda_s}$. Hence, this can be a column vector of the $B$ of a potential compatible pair. Even more can be ascertained: It can be seen that the last two lines of Theorem~\ref{toric} precisely states that with a $B$ matrix like that, the following holds: \begin{Prop}The change $E_{u\Lambda_s,v\Lambda_s}\to E_{u\sigma_s\Lambda_s,v\sigma_s\Lambda_s}$ implied by $\check B(u,v,s)$ is the result of a BFZ mutation. \end{Prop} \begin{Thm} The Schubert mutation $${\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})\rightarrow{\mathcal Q}_q({\mathfrak a},{\mathfrak b}',{\mathfrak c})$$ implied by a replacement ${\mathfrak m}^{+}_{{\mathfrak a},{\mathfrak c}}$ as in (\ref{m+}) is the result of series of BFZ mutations. \end{Thm} \proof The number $s$ is given by (\ref{94}) and remains fixed throughout. We do the replacement in a number of steps. We set ${\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})={\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(0)$ and perform changes \begin{eqnarray} &{\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})={\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(0)\rightarrow \\\nonumber& {\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(1)\rightarrow \dots\rightarrow {\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t_0)={\mathcal Q}_q({\mathfrak a},{\mathfrak b}',{\mathfrak c}). \end{eqnarray} We will below see that $t_0=s_{\mathfrak b}-s_{\mathfrak a}-1$. We set \begin{equation} \textrm{If } 0\leq t\leq t_o: {\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t)=({\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t),{\mathcal L}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t),{\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t)). \end{equation} The intermediate seeds ${\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t)$ with $0<t<t_0$ are not defined by strings $\tilde{\mathfrak a}\leq \tilde{\mathfrak b}\leq \tilde{\mathfrak c}$. At each $t$-level, only one column is replaced when passing from ${\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t)$ to ${\mathcal B}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})(t+1)$, and here (\ref{77}) is applied. Of course, the whole ${\mathcal B}$ matrix is given by (\ref{72}) and (\ref{75}) for a suitable seed. Specifically, using (\ref{77}) we introduce a family of expressions $\check B$ as in (\ref{99}) \begin{eqnarray}\label{b-conv-bar-t}{B}^{{\mathfrak a},{\mathfrak b}(t), {\mathfrak c}}_{m^+}(s,t)= E_{\omega(s_{\mathfrak a}+t+1)\Lambda_s,\omega^{{\mathfrak b}}\Lambda_s}^{-1} E_{\omega(s_{\mathfrak a}+t)\Lambda_s,\omega^{{\mathfrak b}'}\Lambda_s}^{-1} \prod E_{\omega(s,s_{\mathfrak a}+t+1)\Lambda_j,\omega^{{\mathfrak b}'}\Lambda_j} ^{-a_{js}}\\\nonumber =(E^u_{\mathfrak b}(s,s_{\mathfrak a}+t+1)E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t))^{-1}\prod E^u_{{\mathfrak b}}(j,\overline p(j,s,s_{\mathfrak a}+t+1))^{-a_{js}}, \end{eqnarray} implying the changes \begin{equation} E^u_{{\mathfrak b}}(s,s_{\mathfrak a}+t)\rightarrow E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t+1). \end{equation} If $\omega(s,s_{\mathfrak a}+t+1)=u_t\sigma_s$ and $v=\omega^{\mathfrak b}$ then this corresponds to \begin{equation} \left((u_t\sigma_s\Lambda_s,v\Lambda_s)(u_t\Lambda_s, v\sigma_s\Lambda_s)(u\Lambda_j,v\Lambda_j)^{a_{js}}\right)^{-1} \end{equation} Here are then in details how the changes are performed: \begin{eqnarray*}{\mathrm Step}(0):&&\\{\mathcal C}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})\ni E^d_{{\mathfrak a}}(s,s_{\mathfrak b}+1)&\rightarrow& E^u_{{\mathfrak b}'}(s,s_{\mathfrak a})\in {\mathcal C}_q({\mathfrak a},{\mathfrak b}(0),{\mathfrak c}) \ (renaming),\\ {B}_q^{{\mathfrak a},{\mathfrak b}, {\mathfrak c}}(s,s_{\mathfrak a})&\rightarrow&{B}^{{\mathfrak a},{\mathfrak b}(0), {\mathfrak c}}_{m^+}(s,0) \ (renaming), \\{\mathcal L}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})&\rightarrow& {\mathcal L}_q({\mathfrak a},{\mathfrak b}(0),{\mathfrak c}) \ (renaming), \\{\mathrm Step}(1): && (implied\ by \ {B}^{{\mathfrak a},{\mathfrak b}(0), {\mathfrak c}}_{m^+}(s,0) ), \\{\mathcal C}_q({\mathfrak a},{\mathfrak b}(0),{\mathfrak c}) \ni E^u_{{\mathfrak b}}(s,s_{\mathfrak a})&\rightarrow& E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+1) \in {\mathcal C}_q^d({\mathfrak a},{\mathfrak b}(1),{\mathfrak c}) , \\{B}_q^{{\mathfrak a},{\mathfrak b}, {\mathfrak c}}(s,s_{\mathfrak a}+1)& \rightarrow&{B}^{{\mathfrak a},{\mathfrak b}(1), {\mathfrak c}}_{m^+}(s,1) (by\ (\ref{77})), \\{\mathcal L}_q({\mathfrak a},{\mathfrak b}(0),{\mathfrak c})&\rightarrow& {\mathcal L}_q({\mathfrak a},{\mathfrak b}(1),{\mathfrak c}) \ (implied),\\{\mathrm Step}(2): && (implied\ by \ {B}^{{\mathfrak a},{\mathfrak b}(1), {\mathfrak c}}_{m^+}(s,1) ), \\ {\mathcal C}_q^d({\mathfrak a},{\mathfrak b}(1),{\mathfrak c}) \ni E^u_{{\mathfrak b}}(s,s_{\mathfrak a}+1)&\rightarrow& E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+2)\in {\mathcal C}_q^d({\mathfrak a},{\mathfrak b}(2),{\mathfrak c}),\\\vdots\ \\{\mathrm Step}(t+1): && (implied\ by \ {B}^{{\mathfrak a},{\mathfrak b}(t), {\mathfrak c}}_{m^+}(s,t) ),\\ {\mathcal C}_q^d({\mathfrak a},{\mathfrak b}(t),{\mathfrak c}) \ni E^u_{{\mathfrak b}}(s,s_{\mathfrak a}+t)&\rightarrow& E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t+1)\in {\mathcal C}_q^d({\mathfrak a},{\mathfrak b}(t+1),{\mathfrak c}) , \\{B}_q^{{\mathfrak a},{\mathfrak b}, {\mathfrak c}}(s,s_{\mathfrak a}+t)& \rightarrow&{B}^{{\mathfrak a},{\mathfrak b}(t), {\mathfrak c}}_{m^+}(s,t) (by\ (\ref{77})),\\{\mathcal L}_q({\mathfrak a},{\mathfrak b}(t),{\mathfrak c})&\rightarrow& {\mathcal L}_q({\mathfrak a},{\mathfrak b}(t+1),{\mathfrak c}) \ (implied). \end{eqnarray*} The last step is $t=s_{\mathfrak b}-s_{\mathfrak a}-1$. ${\mathfrak b}(0)={\mathfrak b}$, ${\mathfrak b}(s_{\mathfrak b}-s_{\mathfrak a}-1)={\mathfrak b}'$. It is easy to see that all intermediate sets indeed are seeds. What is missing now is to connect, via a change of basis transformation of the compatible pair, with the ``$E,F$'' matrices of \cite{bz}. Here we notice that both terms \begin{equation} (E^u_{\mathfrak b}(s,s_{\mathfrak a}+t+1)E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t))^{-1}(E^u_{\mathfrak b}(s,s_{\mathfrak a}+t))^{-1} \end{equation} and \begin{equation} \prod E^u_{{\mathfrak b}}(j,\overline p(j,s,s_{\mathfrak a}+t+1))^{-a_{js}}(E^u_{\mathfrak b}(s,s_{\mathfrak a}+t))^{-1} \end{equation} have the same $q$-commutators as $E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t+1)$. The two possibilities correspond to the two signs in formulas (3.2) and (3.3) in \cite{bz}. Indeed, the linear transformation \begin{equation}E(t):E^u_{\mathfrak b}(s,s_{\mathfrak a}+t)\rightarrow -E^u_{\mathfrak b}(s,s_{\mathfrak a}+t+1)-E^u_{{\mathfrak b}'}(s,s_{\mathfrak a}+t)-E^u_{\mathfrak b}(s,s_{\mathfrak a}+t) \end{equation} results in a change-of-basis on the level of forms: \begin{eqnarray}{\mathcal L}_q({\mathfrak a},{\mathfrak b}(t),{\mathfrak c})\rightarrow{\mathcal L}_q({\mathfrak a},{\mathfrak b}(t+1),{\mathfrak c})&=&E^T(t){\mathcal L}_q({\mathfrak a},{\mathfrak b}(t),{\mathfrak c})E(t),\\\nonumber {\mathcal B}_{m^+}^{{\mathfrak a},{\mathfrak b}(t),{\mathfrak c}}(s,t)\rightarrow{\mathcal B}_{m^+}^{{\mathfrak a},{\mathfrak b}(t+1),{\mathfrak c}}(s,t+1)&=&E(t){\mathcal B}_{m^+}^{{\mathfrak a},{\mathfrak b}(t),{\mathfrak c}}(s,t)F(t), \end{eqnarray} where $F(t)$ is a truncated part of $E(t)^T$ (the restriction to the mutable elements). With this, the proof is complete. \qed \begin{Thm} Any ${\mathcal Q}_q({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)$ can be obtained from ${\mathcal Q}_q({\mathfrak e},{\mathfrak p})$ as a sub-seed and any ${\mathcal Q}_q^o({\mathfrak r}_1,\dots, {\mathfrak r}_{n-1},{\mathfrak r}_n)$ can be obtained from ${\mathcal Q}_q^o({\mathfrak e},{\mathfrak p})$ as a sub-seed through a series of Schubert creation and annihilation mutations. These mutations are, apart from the trivial actions of renaming, splitting, merging, or simple restrictions, composites of BFZ-mutations. \end{Thm} \proof Apart from mergers and splittings (Definition~\ref{def84}), the mutations are composites of mutations of the form ${\mathcal Q}_q({\mathfrak a},{\mathfrak b},{\mathfrak c})\to {\mathcal Q}_q({\mathfrak a},{\mathfrak b}',{\mathfrak c})$. \qed \begin{Cor}\label{cor8.7}The algebras ${\mathcal A}_q^{d,{\mathfrak a},{\mathfrak c}}$ and ${\mathcal A}_q^{u,{\mathfrak a},{\mathfrak c}}$ are mutation equivalent and indeed are equal. We denote henceforth this algebra by ${\mathcal A}^{{\mathfrak a},{\mathfrak c}}$. This is the quadratic algebra generated by the elements $\beta_{c,d}$ with $c_{\mathfrak a}<d\leq c_{\mathfrak c}$. \end{Cor} We similarly denote the corresponding skew-field of fractions by ${\mathcal F}_q^{{\mathfrak a},{\mathfrak c}}$. \section{Prime} \begin{Def} \begin{equation}{\det}_{s}^{{\mathfrak a},{\mathfrak c}}:=E_{\omega^{\mathfrak a}\Lambda_s,\omega^{\mathfrak c}\Lambda_s}.\end{equation} \end{Def} \begin{Thm}\label{8.6} The 2 sided ideal $I({\det}_{s}^{{\mathfrak a},{\mathfrak c}})$ in ${\mathcal A}_q({\mathfrak a},{\mathfrak c})$ generated by the covariant and non-mutable element ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}$ is \underline{prime} for each $s$. \end{Thm} \proof Induction. The induction start is trivially satisfied. Let us then divide the induction step into two cases. First, let $Z_\gamma$ be an annihilation-mutation site of $\omega^{\mathfrak c}$ such that $\omega^{\mathfrak c}=\sigma_\gamma\omega^{{\mathfrak c}_1}=\omega^{{\mathfrak c}_1}\sigma_{\alpha_s}$ with $\omega^{{\mathfrak c}_1}\in W^p$. We have clearly ${\mathcal A}_q({\mathfrak a},{\mathfrak c})= {\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)\cup I({\det}_{s}^{{\mathfrak a},{\mathfrak c}})$. Furthermore, ${\mathcal A}_q({\mathfrak a},{\mathfrak c})\setminus {\mathcal A}_q({\mathfrak a},{\mathfrak c}_1) =I_\ell(Z_\gamma)$, where $I_\ell(Z_\gamma)$ denotes the left ideal generated by $Z_\gamma$. We might as well consider the right ideal, but not the 2-sided ideal since in general there will be terms ${\mathcal R}$ of lower order, c.f. Theorem~\ref{4.1}. It follows that \begin{equation}\label{Z}{\det}_{s}^{{\mathfrak a},{\mathfrak c}}=M_1Z_\gamma +M_2\end{equation} where $M_1,M_2\in {\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$ and $M_1\neq0$. Indeed, $M_1$ is a non-zero multiple of ${\det}_{s}^{{\mathfrak a},{\mathfrak c}_1}$. (If $s_{\mathfrak c}=1$ then $M_1=1$ and $M_2=0$.) We also record, partly for later use, that $Z_\gamma$ $q$-commutes with everything up to correction terms from ${\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$. Notice that we use Corollary~\ref{cor8.7}. Now consider an equation \begin{equation} {\det}_{s}^{{\mathfrak a},{\mathfrak c}}p_1=p_2p_3 \end{equation} with $p_1,p_2,p_3\in {\mathcal A}_q({\mathfrak a},{\mathfrak c})$. Use (\ref{Z}) to write for each $i=1,2,3$ \begin{equation} p_i=\sum_{k=0}^{n_i}({\det}_{s}^{{\mathfrak a},{\mathfrak c}})^kN_{i,k} \end{equation} where each $N_{i,k}\in {\mathbf L}_q({\mathfrak a},{\mathfrak c}_1)$ and assume that $N_{i,0}\neq0\textrm{ for }i=2,3$ Then $0\neq N_{0,2}N_{0,3}\in {\mathbf L}_q({\mathfrak a},{\mathfrak c}_1)$. At the same time, \begin{equation} N_{0,2}N_{0,3}=\sum_{k=1}^{n_i}({\det}_{s}^{{\mathfrak a},{\mathfrak c}})^k\tilde N_{i,k} \end{equation} for certain elements $\tilde N_{i,k} \in {\mathbf L}_q({\mathfrak a},{\mathfrak c}_1)$. Using the linear independence (\cite[Proposition~10.8]{bz}) we easily get a contradiction by looking at the leading term in ${\det}_{s}^{{\mathfrak a},{\mathfrak c}})$. Now in the general case, the $s$ in ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}$ is given and we may write ${\omega}^{\mathfrak c}={\omega}^{{\mathfrak c}_2}\sigma_{s}\tilde{\omega}$ where $\sigma_s$ does not occur in $\tilde{\omega}$. Let ${\omega}^{{\mathfrak c}_1}={\omega}^{{\mathfrak c}_2}\sigma_s$. It is clear that ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}={\det}_{s}^{{\mathfrak a},{\mathfrak c}_1}$ and by the previous, ${\det}_{s}^{{\mathfrak a},{\mathfrak c}_1}$ is prime in ${\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$. We have that ${\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$ is an algebra in its own right. Furthermore, \begin{equation}{\mathcal A}_q({\mathfrak a},{\mathfrak c})={\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)[Z_{\gamma_1},\dots,Z_{\gamma_n}], \end{equation} where the Lusztig elements $Z_{\gamma_1},\dots,Z_{\gamma_n}$ are bigger than the generators of ${\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$. In a PBW basis we can put them to the right. They even generate a quadratic algebra $\tilde{\mathcal A}_q$ in their own right! The equation we need to consider are of the form \begin{equation}p_1p_2={\det}_{s}^{{\mathfrak a},{\mathfrak c}_1}p_3 \end{equation} with $p_1,p_2,p_3\in {\mathcal A}_q({\mathfrak a},{\mathfrak c})$. The claim that at least one of $p_1,p_2$ contains a factor of ${\det}^{{\mathfrak r}_1}_{q,s}$ follows by easy induction on the $\tilde{\mathcal A}_q$ degree of $p_1p_2$, i.e. the sum of the $\tilde{\mathcal A}_q$ degrees of $p_1$ and $p_2$. \qed \section{Upper} Let $\omega^{\mathfrak a}, \omega^{\mathfrak c}\in W^p$ and ${\mathfrak a}<{\mathfrak c}$. \begin{Def} The cluster algebra ${\mathbf A}_q({\mathfrak a},{\mathfrak c})$ is the ${\mathbb Z}[q]$-algebra generated in the space ${\mathcal F}_q({\mathfrak a},{\mathfrak c})$ by the inverses of the non-mutable elements ${\mathcal N}_q({\mathfrak a},{\mathfrak c})$ together with the union of the sets of all variables obtainable from the initial seed ${\mathcal Q}_q({\mathfrak a},{\mathfrak c})$ by composites of quantum Schubert mutations. (Appropriately applied) \end{Def} Observe that we include ${\mathcal N}_q({\mathfrak a},{\mathfrak c})$ in the set of variables. \begin{Def} The upper cluster algebra ${\mathbf U}_q({\mathfrak a},{\mathfrak c})$ connected with the same pair $\omega^{\mathfrak a}, \omega^{\mathfrak c}\in W^p$ is the ${\mathbb Z}[q]$-algebra in ${\mathcal F}_q({\mathfrak a},{\mathfrak c})$ given as the intersection of all the Laurent algebras of the sets of variables obtainable from the initial seed ${\mathcal Q}_q({\mathfrak a},{\mathfrak c})$ by composites of quantum Schubert mutations. (Appropriately applied) \end{Def} \begin{Prop} $${\mathcal A}_q({\mathfrak a},{\mathfrak c})\subseteq {\mathbf A}_q({\mathfrak a},{\mathfrak c})\subset {\mathbf U}_q({\mathfrak a},{\mathfrak c}).$$ \end{Prop} \proof The first inclusion follows from \cite{leclerc}, the other is the quantum Laurent phenomenon. \qed \begin{Rem} Our terminology may seem a bit unfortunate since the notions of a cluster algebra and an upper cluster algebra already have been introduced by Berenstein and Zelevinsky in terms of all mutations. We only use quantum line mutations which form a proper subset of the set of all quantum mutations. However, it will be a corollary to what follows that the two notions in fact coincide, and for this reason we do not introduce some auxiliary notation. \end{Rem} \begin{Thm} $${\mathbf U}_q({\mathfrak a},{\mathfrak c})={\mathcal A}_q({\mathfrak a},{\mathfrak c})[({\det}_{s}^{{\mathfrak a},{\mathfrak c}})^{-1}; s\in Im(\pi_{{\mathfrak c}})].$$ \end{Thm} \proof This follows by induction on $\ell(\omega^{\mathfrak c})$ (with start at $\ell(\omega^{\mathfrak a})+1$) in the same way as in the proof of \cite[Theorem~8.5]{jz}, but for clarity we give the details: Let the notation and assumptions be as in the proof of Theorem~\ref{8.6}. First of all, the induction start is trivial since we there are looking at the generator of a Laurent quasi-polynomial algebra. Let then $u\in {\mathbf U}_q({\mathfrak a},{\mathfrak c})$. We will argue by contradiction, and just as in the proof of \cite[Theorem~8.5]{jz}, one readily sees that one may assume that $u\in {\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)[({\det}_{s}^{{\mathfrak a},{\mathfrak c}_1})^{-1}, {\det}_{s}^{{\mathfrak a},{\mathfrak c}}]$. Using (\ref{Z}) we may now write \begin{equation} \label{neg1} u=\left(\sum_{i=0}^K Z_\gamma^ip_i({\det}_{s}^{{\mathfrak a},{\mathfrak c}_1})^{k_i}\right)({\det}_{s}^{{\mathfrak a},{\mathfrak c}_1})^{-\rho}, \end{equation} with $p_i\in {\mathcal A}_q({\mathfrak a},{\mathfrak c}_1)$, $p_i\notin I({\det}_{s}^{{\mathfrak a},{\mathfrak c}_1})$, and $k_i\geq0$. Our assumption is that $\rho>0$. recall that the elements ${\det}_{s}^{{\mathfrak a},{\mathfrak c}_1}$ and ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}$ are covariant and define prime ideals in the appropriate algebras. Using the fact that ${\mathbf U}_q({\mathfrak a},{\mathfrak c})$ is an algebra containing ${\mathcal A}_q({\mathfrak a},{\mathfrak c})$, we can assume that the expression in the left bracket in (\ref{neg1}) is not in $I({\det}_{s}^{{\mathfrak a},{\mathfrak c}})$ and we may further assume that $p_i\neq0\Rightarrow k_i<\rho$. To wit, one can remove the factors of ${\det}_{s}^{{\mathfrak a},{\mathfrak c}}$, then remove the terms with $k_i\geq \rho$, then possibly repeat this process a number of times. Consider now the cluster ${\mathcal C}_q^{u}({\mathfrak a},{\mathfrak c})$. We know that $u$ can be written as a Laurent quasi-polynomial in the elements of ${\mathcal C}_q^{u}({\mathfrak a},{\mathfrak c})$. By factoring out, we can then write \begin{equation}\label{neg2} u=p\prod_{(c,d)\in{\mathbb U}^{u,{\mathfrak a},{\mathfrak c}}}(E^u_{\mathfrak c}(c,d))^{-\alpha_{c,d}}, \end{equation} where $p\in {\mathcal A}_q({\mathfrak a},{\mathfrak c})$, and $\alpha_{c,d}\geq0$. We will compare this to (\ref{neg1}). For the sake of this argument set $\tilde{\mathbb U}^{u,{\mathfrak a},{\mathfrak c}})=\{(c,d)\in {\mathbb U}^{u,{\mathfrak a},{\mathfrak c}})\mid \alpha_{c,d}>0\}$. Of course, ${\det}_s^{{\mathfrak a},{\mathfrak c}}\in {\mathcal C}^{u}({\mathfrak e},{\mathfrak r})$. ``Multiplying across'', we get from (\ref{neg1}) and (\ref{neg2}), absorbing possibly some terms into $p$: \begin{equation} (\sum_{i=0}^K Z^ip_i({\det}^{{\mathfrak a},{\mathfrak c}_1}_{s})^{k_i})\prod_{(c,d)\in\tilde{\mathbb U}^{u,{\mathfrak a},{\mathfrak c}}}(E^u_{\mathfrak c}(c,d))^{\alpha_{c,d}}=p({\det}^{{\mathfrak a},{\mathfrak c}_1}_{s})^{\rho}. \end{equation} Any factor of ${\det}^{{\mathfrak a},{\mathfrak c}}_{s}$ in $p$ will have to be canceled by a similar factor of $E^u_{\mathfrak c}(s,0)$ in the left-hand side, so we can assume that $p$ does not contain no factor of ${\det}^{{\mathfrak a},{\mathfrak c}}_{s}$. After that we can assume that $(s,0)\notin \tilde{\mathbb U}^{u,{\mathfrak a},{\mathfrak c}}$ since clearly ${\det}^{{\mathfrak a},{\mathfrak c}_1}_{s}\notin I({\det}^{{\mathfrak a},{\mathfrak c}}_{s})$. Using that $k_i<\rho$ it follows that there must be a factor of $({\det}^{{\mathfrak a},{\mathfrak c}_1}_{s})$ in $\prod_{(c,d)\in\tilde{\mathbb U}^{u,{\mathfrak a},{\mathfrak c}}}(E^u_{\mathfrak c}(c,d))^{\alpha_{c,d}}$. Here, as but noticed, $d=0$ is excluded. The other terms do not contain $Z_{s,1}$ but $({\det}^{{\mathfrak a},{\mathfrak c}_1}_{s})$ does. This is an obvious contradiction. \qed \section{The diagonal of a quantized minor} \begin{Def}Let ${\mathfrak a}<{\mathfrak b}$. The diagonal, ${\mathbb D}_{\omega^{\mathfrak a}(\Lambda_s),\omega^{\mathfrak b}(\Lambda_s)}$, of $E_{\omega^{\mathfrak a}(\Lambda_s),\omega^{\mathfrak b}(\Lambda_s)}$ is set to \begin{equation} {\mathbb D}_{\omega^{\mathfrak a}(\Lambda_s),\omega^{\mathfrak b}(\Lambda_s)}=q^{\alpha}Z_{s,s_{\mathfrak a}+1}\cdots Z_{s,s_{\mathfrak b}}, \end{equation} where \begin{equation} Z_{s,s_{\mathfrak b}}\cdots Z_{s,s_{\mathfrak a}+1}=q^{2\alpha}Z_{s,s_{\mathfrak a}+1}\cdots Z_{s,s_{\mathfrak b}} + {\mathcal R} \end{equation} where the terms ${\mathcal R}$ are of lower order \end{Def} \begin{Prop} $$E_{\omega^{\mathfrak a}(\Lambda_s),\omega^{\mathfrak b}(\Lambda_s)}={\mathbb D}_{\omega^{\mathfrak a}(\Lambda_s),\omega^{\mathfrak b}(\Lambda_s)}+{\mathcal R}$$ The terms in ${\mathcal R}$ are of lower order in our ordering induced by $\leq_L$. They can in theory be determined from the fact that the full polynomial belongs to the dual canonical basis. (\cite{bz},\cite{leclerc}). \end{Prop} \proof We prove this by induction on the length $s_{\mathfrak b}-s_{\mathfrak a}$ of any $s$-diagonal. When this length is $1$ we have at most a quasi-polynomial algebra and here the case is clear. Consider then a creation-mutation site where we go from length $r$ to $r+1$: Obviously, it is only the very last determinant we need to consider. Here we use the equation in Theorem~\ref{3.2} but reformulate it in terms of the elements $E_{\xi,\eta}$, cf. Theorem~\ref{toric}. Set $\omega^{{\mathfrak b}_1}=\omega^{{\mathfrak b}}\sigma_s$ and consider $E_{\omega^{\mathfrak a}(\Lambda_s),\omega^{{\mathfrak b}_1}(\Lambda_s)}$. Its weight is given as $$\omega^{{\mathfrak b}_1}(\Lambda_s)-\omega^{\mathfrak a}(\Lambda_s)=\beta_{s,s_{\mathfrak a}+1} \dots+\beta_{s,s_{\mathfrak b}+1}.$$ In the recast version of Theorem~\ref{3.2}, the terms on the left hand side are covered by the induction hypothesis. The second term on the right hand side contains no element of the form $Z_{s,s_{{\mathfrak b}_1}}$ and it follows that we have an equation \begin{equation}(Z_{s,s_{\mathfrak a}+2}\cdots Z_{s,s_{\mathfrak b}})E_{\omega^{\mathfrak a}(\Lambda_s),\omega^{{\mathfrak b}_1}(\Lambda_s)}=(Z_{s,s_{\mathfrak a}+2}\cdots Z_{s,s_{\mathfrak b}+1}) (Z_{s,s_{\mathfrak a}+1}\cdots Z_{s,s_{\mathfrak b}}) +{\mathcal R}. \end{equation} The claim follows easily from that. \qed Recall that in the associated quasi-polynomial algebra is the algebra with relations corresponding to the top terms, i.e., colloquially speaking, setting the lower order terms ${\mathcal R}$ equal to $0$. Let \begin{equation} {d}_{\omega^{\mathfrak r}_{s,t_1}(\Lambda_s),\omega^{\mathfrak r}_{s,t}(\Lambda_s)}=z_{s,t_1+1}\cdots z_{s,t}. \end{equation} The following shows the importance of the diagonals: \begin{Thm} \begin{eqnarray} d_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}d_{u_1\cdot\Lambda_{i_1}, v_1\cdot\Lambda_{i_1}}&=& q^G d_{u_1\cdot\Lambda_{i_1},v_1\cdot\Lambda_{i_1}} d_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}\Leftrightarrow\\ {\mathbb D}_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}{\mathbb D}_{u_1\cdot\Lambda_{i_1},v_1\cdot\Lambda_{i_1}}&=& q^G {\mathbb D}_{u_1\cdot\Lambda_{i_1},v_1\cdot\Lambda_{i_1}} {\mathbb D}_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}+ {\mathcal R} \end{eqnarray}In particular, if the two elements ${E}_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}{E}_{u_1\cdot\Lambda_{i_1}, v_1\cdot\Lambda_{i_1}}$ $q$-commute: \begin{equation} {E}_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}{E}_{u_1\cdot\Lambda_{i_1}, v_1\cdot\Lambda_{i_1}}= q^G {E}_{u_1\cdot\Lambda_{i_1},v_1\cdot\Lambda_{i_1}} {E}_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}} \end{equation}then $G$ can be computed in the associated quasi-polynomial algebra: \begin{equation} d_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}d_{u_1\cdot\Lambda_{i_1}, v_1\cdot\Lambda_{i_1}}= q^G D_{u_1\cdot\Lambda_{i_1},v_1\cdot\Lambda_{i_1}} d_{u\cdot\Lambda_{i_0},v\cdot\Lambda_{i_0}}.\end{equation} \end{Thm} \begin{Rem} One can also compute $G$ directly using the formulas in \cite{bz}. \end{Rem} \begin{Rem} The elements $E_{\xi,\eta}$ that we consider belong to the dual canonical basis. As such, they can in principle be determined from the highest order terms ${\mathbb D}_{\xi,\eta}$. \end{Rem} \section{Litterature} \end{document}

\begin{document} \title{On the Basis Property of the Root Functions of Some Class of Non-self-adjoint Sturm--Liouville Operators.} \author{Cemile Nur\\{\small Depart. of Math., Dogus University, Ac\i badem, Kadik\"{o}y, \ }\\{\small Istanbul, Turkey.}\ {\small e-mail: [email protected]} \and O. A. Veliev\\{\small Depart. of Math., Dogus University, Ac\i badem, Kadik\"{o}y, \ }\\{\small Istanbul, Turkey.}\ {\small e-mail: [email protected]}} \date{} \maketitle \begin{abstract} We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential $q$ such that the root functions of these operators do not form a Riesz basis. Key Words: Asymptotic formulas, Regular boundary conditions. Riesz basis. AMS Mathematics Subject Classification: 34L05, 34L20. \end{abstract} \section{Introduction and Preliminary Facts} Let $T_{1},T_{2},T_{3}$ and $T_{4}$ be the operators generated in $L_{2}[0,1]$ by the differential expression \begin{equation} l\left( y\right) =-y^{\prime\prime}+q(x)y \end{equation} and the following boundary conditions: \begin{equation} y_{0}^{\prime}+\beta y_{1}^{\prime}=0,\text{ }y_{0}-y_{1}=0, \end{equation} \begin{equation} y_{0}^{\prime}+\beta y_{1}^{\prime}=0,\text{ }y_{0}+y_{1}=0, \end{equation} \begin{equation} y_{0}^{\prime}-y_{1}^{\prime}=0,\text{ }y_{0}+\alpha y_{1}=0, \end{equation} and \begin{equation} y_{0}^{\prime}+y_{1}^{\prime}=0,\text{ }y_{0}+\alpha y_{1}=0 \end{equation} respectively, where $q(x)$ is a complex-valued summable function on $[0,1]$, $\beta\neq\pm1$ and $\alpha\neq\pm1.$ In conditions (2), (3), (4) and (5) if $\beta=1,$ $\beta=-1,$ $\alpha=1$ and $\alpha=-1$ respectively, then any $\lambda\in \mathbb{C} $ is an eigenvalue of infinite multiplicity. In (2) and (4) if $\beta=-1$ and $\alpha=-1$ then they are periodic boundary conditions; In (3) and (5) if $\beta=1$ and $\alpha=1$ then they are antiperiodic boundary conditions. These boundary conditions are regular but not strongly regular. Note that, if the boundary conditions are strongly regular, then the root functions form a Riesz basis (this result was proved independently in [6], [10] and [17]). In the case when an operator is regular but not strongly regular, the root functions generally do not form even usual basis. However, Shkalikov [20], [21] proved that they can be combined in pairs, so that the corresponding 2-dimensional subspaces form a Riesz basis of subspaces. In the regular but not strongly regular boundary conditions, periodic and antiperiodic boundary conditions are the ones more commonly studied. Therefore, let us briefly describe some historical developments related to the Riesz basis property of the root functions of the periodic and antiperiodic boundary value problems. First results were obtained by Kerimov and Mamedov [8]. They established that, if \[ q\in C^{4}[0,1],\ q(1)\neq q(0), \] then the root functions of the operator $L_{0}(q)$ form a Riesz basis in $L_{2}[0,1],$ where $L_{0}(q)$ denotes the operator generated by (1) and the periodic boundary conditions. The first result in terms of the Fourier coefficients of the potential $q$ was obtained by Dernek and Veliev [1]. They proved that if the conditions \begin{align} \lim_{n\rightarrow\infty}\frac{\ln\left\vert n\right\vert }{nq_{2n}} & =0,\text{ }\\ q_{2n} & \sim q_{-2n} \end{align} hold, then the root functions of $L_{0}(q)$ form a Riesz basis in $L_{2} [0,1]$, where $q_{n}=:(q,e^{i2\pi nx})$ is the Fourier coefficient of $q$ and everywhere, without loss of generality, it is assumed that $q_{0}=0.$ Here $(.,.)$ denotes the inner product in $L_{2}[0,1]$ and $a_{n}\sim b_{n}$ means that $a_{n}=O(b_{n})$ and $b_{n}=O(a_{n})$ as $\ n\rightarrow\infty.$ Makin [11] improved this result. Using another method he proved that the assertion on the Riesz basis property remains valid if condition (7) holds, but condition (6) is replaced by a less restrictive one: $q\in W_{1}^{s}[0,1],$ \[ q^{(k)}(0)=q^{(k)}(1),\quad\forall\,k=0,1,...,s-1 \] holds and $\mid q_{2n}\mid>cn^{-s-1}$ with some$\ \,c>0$ for sufficiently large $n,$ where $s$ is a nonnegative integer. Besides, some conditions which imply the absence of the Riesz basis property were presented in [11]. Shkalilov and Veliev obtained in [22] more general results which cover all results discussed above. The other interesting results about periodic and antiperiodic boundary conditions were obtained in [2-5, 7, 14-16, 24, 25]. The basis properties of regular but not strongly regular other some problems are studied in [9,12,13]. It was proved in [12] that the system of the root functions of the operator generated by (1) and the boundary conditions \begin{align*} y^{\prime}\left( 1\right) -\left( -1\right) ^{\sigma}y^{\prime}\left( 0\right) +\gamma y\left( 0\right) & =0\\ y\left( 1\right) -\left( -1\right) ^{\sigma}y\left( 0\right) & =0, \end{align*} forms an unconditional basis of the space $L_{2}[0,1]$, where $q\left( x\right) $ is an arbitrary complex-valued function from the class $L_{1}[0,1]$, $\gamma$ is an arbitrary nonzero complex constant and $\sigma=0,1$. Kerimov and Kaya proved [9] that the system of the root functions of the spectral problem \begin{align*} y^{\left( 4\right) }+p_{2}\left( x\right) y^{\prime\prime}+p_{1}\left( x\right) y^{\prime}+p_{0}\left( x\right) y & =\lambda y,\text{ }0<x<1,\\ y^{\left( s\right) }\left( 1\right) -\left( -1\right) ^{\sigma }y^{\left( s\right) }\left( 0\right) +\sum_{l=0}^{s-1}\alpha _{s,l}y^{\left( l\right) }\left( 0\right) & =0,\text{ }s=1,2,3,\\ y\left( 1\right) -\left( -1\right) ^{\sigma}y\left( 0\right) & =0, \end{align*} forms a basis in the space $L_{p}\left( 0,1\right) $, $1<p<\infty$, when $\alpha_{3,2}+\alpha_{1,0}\neq\alpha_{2,1}$, $p_{j}\left( x\right) \in W_{1}^{j}\left( 0,1\right) $, $j=1,2$, and $p_{0}\left( x\right) \in L_{1}\left( 0,1\right) $; moreover, this basis is unconditional for $p=2$, where $\lambda$ is a spectral parameter; $p_{j}\left( x\right) \in L_{1}\left( 0,1\right) $, $j=1,2,3$, are complex-valued functions; $\alpha_{s,l}$, $s=1,2,3$, $l=\overline{0,s-1}$ are arbitrary complex constants; and $\sigma=0,1$. It was shown in [19] that if \[ q\left( x\right) =q\left( 1-x\right) ,\text{ }\forall x\in\left[ 0,1\right] , \] then the spectrum of each of the problems $T_{1}$, and $T_{3}$, coincides with the spectrum of the periodic problem and the spectrum of each of the problems $T_{2},$ and $T_{4}$, coincides with the spectrum of the antiperiodic problem. In this paper we prove that if \begin{equation} \lim_{n\rightarrow\infty}\dfrac{\ln\left\vert n\right\vert }{ns_{2n}}=0, \end{equation} where $s_{k}=\left( q,\sin2\pi kx\right) ,$ then the large eigenvalues of the operators $T_{1}$ and $T_{3}$ are simple. Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (8) holds when $n$ is replaced by $n_{k},$ then the root functions of these operators do not form a Riesz basis. Similarly, if \begin{equation} \lim_{n\rightarrow\infty}\dfrac{\ln\left\vert n\right\vert }{ns_{2n+1}}=0, \end{equation} then the large eigenvalues of the operators $T_{2}$ and $T_{4}$ are simple and if there exists a sequence $\left\{ n_{k}\right\} $ such that (9) holds when $n$ is replaced by $n_{k},$ then the root functions of these operators do not form a Riesz basis. Moreover we obtain asymptotic formulas of arbitrary order for the eigenvalues and eigenfunctions of the operators $T_{1}$,$T_{2},T_{3}$ and $T_{4}$. \section{Main Results} We will focus only on the operator $T_{1}$. The investigations of the operators $T_{2},T_{3}$ and $T_{4}$ are similar. It is well-known that ( see formulas (47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{1}(q)$ consist of the sequences $\{\lambda_{n,1}\},\{\lambda_{n,2}\}$ satisfying \begin{equation} \lambda_{n,j}=(2n\pi)^{2}+O(n^{1/2}) \end{equation} for $j=1,2$. From this formula one can easily obtain the following inequality \begin{equation} \left\vert \lambda_{n,j}-(2\pi k)^{2}\right\vert =\left\vert 2(n-k)\pi \right\vert \left\vert 2(n+k)\pi\right\vert +O(n^{\frac{1}{2}})>n \end{equation} for $j=1,2;$ $k\neq n;$ $k=0,1,...;$ and $n\geq N,$ where we denote by $N$ a sufficiently large positive integer, that is, $N\gg1.$ It is easy to verify that if $q(x)=0$ then the eigenvalues of the operator $T_{1},$ denoted by $T_{1}(0),$ are $\lambda_{n}=\left( 2\pi n\right) ^{2}$ for $n=0,1,\ldots$ The eigenvalue $0$ is simple and the corresponding eigenfunction is $1.$ The eigenvalues $\lambda_{n}=\left( 2\pi n\right) ^{2}$ for $n=1,2,\ldots$ are double and the corresponding eigenfunctions and associated functions are \begin{equation} y_{n}\left( x\right) =\cos2\pi nx\text{ }\And\text{ }\phi_{n}\left( x\right) =\left( \frac{\beta}{1+\beta}-x\right) \frac{\sin2\pi nx}{4\pi n} \end{equation} respectively. Note that for any constant $c$, $\phi_{n}\left( x\right) +cy_{n}\left( x\right) $ is also an associated function. It can be shown that the adjoint operator $T_{1}^{\ast}(0)$ is associated with the boundary conditions: \begin{equation} y_{1}+\overline{\beta}y_{0}=0,\text{ }y_{1}^{\prime}-y_{0}^{\prime}=0. \end{equation} It is easy to see that, $0$ is a simple eigenvalue of $T_{1}^{\ast}(0)$ and the corresponding eigenfunction is $y_{0}^{\ast}\left( x\right) =x-\dfrac {1}{1+\overline{\beta}}$ . The other eigenvalues $\lambda_{n}^{\ast}=\left( 2\pi n\right) ^{2}$ for $n=1,2,\ldots$, are double and the corresponding eigenfunctions and associated functions are \begin{equation} y_{n}^{\ast}\left( x\right) =\sin2\pi nx\text{ }\And\text{ }\phi_{n}^{\ast }\left( x\right) =\left( x-\dfrac{1}{1+\overline{\beta}}\right) \frac {\cos2\pi nx}{4\pi n}\nonumber \end{equation} respectively. Let \begin{equation} \varphi_{n}\left( x\right) :=\frac{16\pi n\left( \beta+1\right) }{\beta -1}\phi_{n}\left( x\right) =\frac{4\left( \beta+1\right) }{\beta-1}\left( \dfrac{\beta}{1+\beta}-x\right) \sin2\pi nx \end{equation} and \begin{equation} \varphi_{n}^{\ast}\left( x\right) :=\frac{16\pi n\left( \overline{\beta }+1\right) }{\overline{\beta}-1}\phi_{n}^{\ast}\left( x\right) =\frac{4\left( \overline{\beta}+1\right) }{\overline{\beta}-1}\left( x-\dfrac{1}{1+\overline{\beta}}\right) \cos2\pi nx. \end{equation} The system of the root functions of $T_{1}^{\ast}(0)$ can be written as $\{f_{n}:n\in\mathbb{Z}\},$ where \begin{equation} f_{-n}=\sin2\pi nx,\text{ }\forall n>0\And\text{ }f_{n}=\varphi_{n}^{\ast }\left( x\right) ,\text{ }\forall n\geq0. \end{equation} One can easily verify that it forms a basis in $L_{2}[0,1]$ and the biorthogonal system $\{g_{n}:n\in\mathbb{Z}\}$ is the system of the root functions of $T_{1}(0),$ where \begin{equation} g_{-n}=\varphi_{n}\left( x\right) ,\forall n>0\text{ }\And\text{ }g_{n} =\cos2\pi nx,\forall n\geq0, \end{equation} since $\left( f_{n},g_{m}\right) =\delta_{n,m}.$ To obtain the asymptotic formulas for the eigenvalues $\lambda_{n,j}$ and the corresponding normalized eigenfunctions $\Psi_{n,j}(x)$ of $T_{1}(q)$ we use (11) and the well-known relations \begin{equation} (\lambda_{N,j}-(2\pi n)^{2})(\Psi_{N,j},\sin2\pi nx)=(q\Psi_{N,j},\sin2\pi nx) \end{equation} and \begin{equation} \left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi _{N,j},\varphi_{n}^{\ast}\right) -\gamma_{1}n\left( \Psi_{N,j},\sin2\pi nx\right) =\left( q\Psi_{N,j},\varphi_{n}^{\ast}\right) , \end{equation} where \[ \gamma_{1}=\frac{16\pi\left( \beta+1\right) }{\beta-1}, \] which can be obtained by multiplying both sides of the equality \[ -\left( \Psi_{N,j}\right) ^{\prime\prime}+q\left( x\right) \Psi _{N,j}=\lambda_{N,j}\Psi_{N,j} \] by $\sin2\pi nx$ and $\varphi_{n}^{\ast}$ respectively. It follows from (18) and (19) that \begin{equation} \left( \Psi_{N,j},\sin2\pi nx\right) =\frac{\left( q\left( x\right) \Psi_{N,j},\sin2\pi nx\right) }{\lambda_{N,j}-\left( 2\pi n\right) ^{2} };\text{ }N\neq n, \end{equation} \begin{equation} \left( \Psi_{N,j},\varphi_{n}^{\ast}\right) =\frac{\gamma_{1}n\left( q\left( x\right) \Psi_{N,j},\sin2\pi nx\right) }{\left( \lambda _{N,j}-\left( 2\pi n\right) ^{2}\right) ^{2}}+\frac{\left( q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right) }{\lambda_{N,j}-\left( 2\pi n\right) ^{2}};\text{ }N\neq n. \end{equation} Moreover, we use the following relations \begin{align} \left( \Psi_{N,j},\overline{q}\sin2\pi nx\right) & =\sum_{n_{1}=0} ^{\infty}[\left( q\varphi_{n_{1}},\sin2\pi nx\right) \left( \Psi_{N,j} ,\sin2\pi n_{1}x\right) +\\ & +\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) \left( \Psi_{N,j} ,\varphi_{n_{1}}^{\ast}\right) ],\nonumber \end{align} \begin{equation} \left( \Psi_{N,j},\overline{q}\varphi_{n}^{\ast}\right) =\sum_{n_{1} =0}^{\infty}\left[ \left( q\varphi_{n_{1}},\varphi_{n}^{\ast}\right) \left( \Psi_{N,j},\sin2\pi n_{1}x\right) +\left( q\cos2\pi n_{1} x,\varphi_{n}^{\ast}\right) \left( \Psi_{N,j},\varphi_{n_{1}}^{\ast}\right) \right] , \end{equation} \begin{align} \left\vert (q\Psi_{N,j},\sin2\pi nx)\right\vert & <4M,\\ \left\vert (q\Psi_{N,j},\varphi_{n}^{\ast})\right\vert & <4M, \end{align} for $N\gg1,$where $M=\sup\left\vert q_{n}\right\vert .$ These relations are obvious for $q\in L_{2}(0,1),$ since to obtain (22) and (23) we can use the decomposition of $\overline{q}\sin2\pi nx$ and $\overline{q}\varphi_{n}^{\ast }$ by basis (16). For $q\in L_{1}(0,1)$\ see Lemma 1 of [23]. To obtain the asymptotic formulas for the eigenvalues and eigenfunctions we iterate (18) and (19) by using (22), (23). First let us prove the following obvious asymptotic formulas for the eigenfunctions $\Psi_{n,j}$. The expansion of $\Psi_{n,j}$\ by basis (17) can be written in the form \begin{equation} \Psi_{n,j}=u_{n,j}\varphi_{n}\left( x\right) +v_{n,j}\cos2\pi nx+h_{n,j} \left( x\right) , \end{equation} where \begin{equation} u_{n,j}=\left( \Psi_{n,j},\sin2\pi nx\right) ,\text{ }v_{n,j}=\left( \Psi_{n,j},\varphi_{n}^{\ast}\right) , \end{equation} \[ h_{n,j}\left( x\right) =\sum_{\substack{k=0\\k\neq n}}^{\infty}\left[ \left( \Psi_{n,j},\sin2\pi kx\right) \varphi_{k}\left( x\right) +\left( \Psi_{n,j},\varphi_{k}^{\ast}\right) \cos2\pi kx\right] . \] Using (20), (21), (24) and (25) one can readily see that, there exists a constant $C$ such that \begin{equation} \sup\left\vert h_{n,j}\left( x\right) \right\vert \leq C\left( \sum_{k\neq n}\left( \frac{1}{\mid\lambda_{n,j}-\left( 2\pi k\right) ^{2}\mid}+\frac {n}{\left\vert \left( \lambda_{n,j}-\left( 2\pi k\right) ^{2}\right) ^{2}\right\vert }\right) \right) =O\left( \frac{\ln n}{n}\right) \end{equation} and by (26) we get \begin{equation} \Psi_{n,j}=u_{n,j}\varphi_{n}\left( x\right) +v_{n,j}\cos2\pi nx+O\left( \frac{\ln n}{n}\right) . \end{equation} Since $\Psi_{n,j}$\ is normalized, we have \[ 1=\left\Vert \Psi_{n,j}\right\Vert ^{2}=\left( \Psi_{n,j},\Psi_{n,j}\right) =\left\vert u_{n,j}\right\vert ^{2}\left\Vert \varphi_{n}\left( x\right) \right\Vert ^{2}+\left\vert v_{n,j}\right\vert ^{2}\left\Vert \cos2\pi nx\right\Vert ^{2}+ \] \[ +u_{n,j}\overline{v_{n,j}}\left( \varphi_{n}\left( x\right) ,\cos2\pi nx\right) +v_{n,j}\overline{u_{n,j}}\left( \cos2\pi nx,\varphi_{n}\left( x\right) \right) +O\left( \frac{\ln n}{n}\right) \] \[ =\left( \frac{8}{3}\dfrac{\left\vert \beta\right\vert ^{2}-\operatorname{Re} \beta+1}{\left\vert \beta-1\right\vert ^{2}}\right) \left\vert u_{n,j} \right\vert ^{2}+\frac{1}{2}\left\vert v_{n,j}\right\vert ^{2}+O\left( \frac{\ln n}{n}\right) , \] that is, \begin{equation} a\left\vert u_{n,j}\right\vert ^{2}+\frac{1}{2}\left\vert v_{n,j}\right\vert ^{2}=1+O\left( \frac{\ln n}{n}\right) , \end{equation} where \[ a=\frac{8}{3}\dfrac{\left\vert \beta\right\vert ^{2}-\operatorname{Re}\beta +1}{\left\vert \beta-1\right\vert ^{2}}. \] Note that $a\neq0$, since $\left\vert \beta\right\vert ^{2}+1>\left\vert \beta\right\vert .$ Now let us iterate (18). Using (22) in (18) we get \begin{gather*} \left( \lambda_{n,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi _{n,j},\sin2\pi nx\right) =\\ =\sum_{n_{1}=0}^{\infty}\left[ \left( q\varphi_{n_{1}},\sin2\pi nx\right) \left( \Psi_{n,j},\sin2\pi n_{1}x\right) +\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n_{1}}^{\ast}\left( x\right) \right) \right] . \end{gather*} Isolating the terms in the right-hand side of this equality containing the multiplicands $\left( \Psi_{n,j},\sin2\pi nx\right) $ and $\left( \Psi_{n,j},\varphi_{n}^{\ast}\left( x\right) \right) $ (i.e., case $n_{1}=n$ ), using\ (20) and (21) for the terms $\left( \Psi_{n,j},\sin2\pi n_{1}x\right) $ and \ $\left( \Psi_{n,j},\varphi_{n_{1}}^{\ast}\left( x\right) \right) $ respectively (in the case $n_{1}\neq n$) we obtain \begin{gather*} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n} ,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left( q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right) =\\ =\sum_{\substack{n_{1}=0\\n_{1}\neq n}}^{\infty}\left[ \left( q\varphi _{n_{1}},\sin2\pi nx\right) \left( \Psi_{n,j},\sin2\pi n_{1}x\right) +\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) \left( \Psi_{n,j} ,\varphi_{n_{1}}^{\ast}\left( x\right) \right) \right] \\ =\sum_{n_{1}}\left[ a_{1}\left( \lambda_{n,j}\right) \left( q\left( x\right) \Psi_{n,j},\sin2\pi n_{1}x\right) +b_{1}\left( \lambda _{n,j}\right) \left( q\left( x\right) \Psi_{n,j},\varphi_{n_{1}}^{\ast }\right) \right] . \end{gather*} where \begin{align*} a_{1}\left( \lambda_{n,j}\right) & =\frac{\left( q\varphi_{n_{1}} ,\sin2\pi nx\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}} +\frac{\gamma_{1}n_{1}\left( q\cos2\pi n_{1}x,\sin2\pi nx\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}\right) ^{2}},\\ b_{1}\left( \lambda_{n,j}\right) & =\frac{\left( q\cos2\pi n_{1} x,\sin2\pi nx\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}}. \end{align*} Using (22) and (23) for the terms $\left( q\Psi_{n,j},\sin2\pi n_{1}x\right) $ and $\left( q\Psi_{n,j},\varphi_{n_{1}}^{\ast}\right) $ of the last summation we obtain \begin{gather*} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n} ,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left( q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right) =\\ =\sum_{n_{1}}\left[ a_{1}\left( \lambda_{n,j}\right) \left( q\Psi _{n,j},\sin2\pi n_{1}x\right) +b_{1}\left( \lambda_{n,j}\right) \left( q\Psi_{n,j},\varphi_{n_{1}}^{\ast}\right) \right] =\\ =\sum_{n_{1}}a_{1}\left( \sum_{n_{2}=0}^{\infty}\left[ \left( q\varphi_{n_{2}},\sin2\pi n_{1}x\right) \left( \Psi_{n,j},\sin2\pi n_{2}x\right) +\left( q\cos2\pi n_{2}x,\sin2\pi n_{1}x\right) \left( \Psi_{n,j},\varphi_{n_{2}}^{\ast}\left( x\right) \right) \right] \right) +\\ +\sum_{n_{1}}b_{1}\left( \sum_{n_{2}=0}^{\infty}\left[ \left( q\varphi_{n_{2}},\varphi_{n_{1}}^{\ast}\right) \left( \Psi_{n,j},\sin2\pi n_{2}x\right) +\left( q\cos2\pi n_{2}x,\varphi_{n_{1}}^{\ast}\right) \left( \Psi_{n,j},\varphi_{n_{2}}^{\ast}\left( x\right) \right) \right] \right) . \end{gather*} Now isolating the terms for $n_{2}=n$ we get \begin{gather*} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n} ,\sin2\pi nx\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) -\left( q\cos2\pi nx,\sin2\pi nx\right) \left( \Psi_{n,j},\varphi_{n}^{\ast}\right) =\\ =\sum_{n_{1}}\left[ a_{1}\left( q\varphi_{n},\sin2\pi n_{1}x\right) +b_{1}\left( q\varphi_{n},\varphi_{n_{1}}^{\ast}\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) +\\ +\sum_{n_{1}}\left[ a_{1}\left( q\cos2\pi nx,\sin2\pi n_{1}x\right) +b_{1}\left( q\cos2\pi nx,\varphi_{n_{1}}^{\ast}\right) \right] \left( \Psi_{n,j},\varphi_{n}^{\ast}\left( x\right) \right) +\\ =\sum_{n_{1},n_{2}}\left( \left[ a_{1}\left( q\varphi_{n_{2}},\sin2\pi n_{1}x\right) +b_{1}\left( q\varphi_{n_{2}},\varphi_{n_{1}}^{\ast}\right) \right] \left( \Psi_{n,j},\sin2\pi n_{2}x\right) +\right) +\\ +\sum_{n_{1},n_{2}}\left[ a_{1}\left( q\cos2\pi n_{2}x,\sin2\pi n_{1}x\right) +b_{1}\left( q\cos2\pi n_{2}x,\varphi_{n_{1}}^{\ast}\right) \right] \left( \Psi_{n,j},\varphi_{n_{2}}^{\ast}\right) . \end{gather*} Here and further the summations are taken under the conditions $n_{i}\neq n$ and $n_{i}=0,1,...$ for $i=1,2,...$ Introduce the notations \begin{align*} C_{1} & =:a_{1},\text{ }M_{1}=:b_{1},\\ C_{2} & =:a_{1}a_{2}+b_{1}A_{2}=C_{1}a_{2}+M_{1}A_{2},\text{ }M_{2} =:a_{1}b_{2}+b_{1}B_{2}=C_{1}b_{2}+M_{1}B_{2},\\ C_{k+1} & =:C_{k}a_{k+1}+M_{k}A_{k+1},\text{ }M_{k+1}=:C_{k}b_{k+1} +M_{k}B_{k+1};\text{ }k=2,3,\ldots, \end{align*} where \begin{gather*} a_{k+1}=a_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\varphi _{n_{k+1}},\sin2\pi n_{k}x\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1} \right) ^{2}}+\dfrac{\gamma_{1}n_{k+1}\left( q\cos2\pi n_{k+1}x,\sin2\pi n_{k}x\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{k+1}\right) ^{2}\right) ^{2}},\\ b_{k+1}=b_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\cos2\pi n_{k+1}x,\sin2\pi n_{k}x\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1}\right) ^{2}},\\ A_{k+1}=A_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\varphi _{n_{k+1}},\varphi_{n_{k}}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1}\right) ^{2}}+\dfrac{\gamma_{1}n_{k+1}\left( q\cos2\pi n_{k+1} x,\varphi_{n_{k}}^{\ast}\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{k+1}\right) ^{2}\right) ^{2}},\\ B_{k+1}=B_{k+1}\left( \lambda_{n,j}\right) =\dfrac{\left( q\cos2\pi n_{k+1}x,\varphi_{n_{k}}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi n_{k+1}\right) ^{2}}. \end{gather*} Using these notations and repeating this iteration $k$ times we get \begin{gather} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-\left( q\varphi_{n} ,\sin2\pi nx\right) -\widetilde{A}_{k}\left( \lambda_{n,j}\right) \right] \left( \Psi_{n,j},\sin2\pi nx\right) =\nonumber\\ =\left[ \left( q\cos2\pi nx,\sin2\pi nx\right) +\widetilde{B}_{k}\left( \lambda_{n,j}\right) \right] \left( \Psi_{n,j},\varphi_{n}^{\ast}\left( x\right) \right) +R_{k}, \end{gather} where \begin{align*} \widetilde{A}_{k}\left( \lambda_{n,j}\right) & =\sum_{m=1}^{k}\alpha _{m}\left( \lambda_{n,j}\right) \text{, }\widetilde{B}_{k}\left( \lambda_{n,j}\right) =\sum_{m=1}^{k}\beta_{m}\left( \lambda_{n,j}\right) ,\\ \alpha_{k}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k}}\left[ C_{k}\left( q\varphi_{n},\sin2\pi n_{k}x\right) +M_{k}\left( q\varphi _{n},\varphi_{n_{k}}^{\ast}\right) \right] ,\\ \beta_{k}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k}}\left[ C_{k}\left( q\cos2\pi nx,\sin2\pi n_{k}x\right) +M_{k}\left( q\cos2\pi nx,\varphi_{n_{k}}^{\ast}\right) \right] ,\\ R_{k} & =\sum_{n_{1},\ldots,n_{k+1}}\left\{ C_{k+1}\left( q\Psi_{n,j} ,\sin2\pi n_{k+1}x\right) +M_{k+1}\left( q\Psi_{n,j},\varphi_{n_{k+1}} ^{\ast}\right) \right\} . \end{align*} It follows from (11), (24) and (25) that \begin{equation} \alpha_{k}\left( \lambda_{n,j}\right) =O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k}\right) ,\beta_{k}\left( \lambda_{n,j}\right) =O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k}\right) ,R_{k}=O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k+1}\right) . \end{equation} Therefore if we take limit in (31) for $k\rightarrow\infty$, we obtain \[ \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda _{n,j}\right) \right] u_{n,j}=\left[ P_{n}+B\left( \lambda_{n,j}\right) \right] v_{n,j}, \] where \begin{equation} P_{n}=\left( q\cos2\pi nx,\sin2\pi nx\right) ,\text{ }Q_{n}=\left( q\varphi_{n},\sin2\pi nx\right) , \end{equation} \begin{equation} A\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\alpha_{m}\left( \lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right) \text{, }B\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\beta_{m}\left( \lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right) . \end{equation} Thus iterating (18) we obtained (31). Now starting \ to iteration from (19) instead of (18) and using (23), (22) and arguing as in the previous iteration, we get \begin{equation} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A_{k}^{\prime }\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1} n+Q_{n}^{\ast}+B_{k}^{\prime}\left( \lambda_{n,j}\right) \right] u_{n,j}+R_{k}^{\prime}, \end{equation} where \begin{equation} P_{n}^{\ast}=\left( q\cos2\pi nx,\varphi_{n}^{\ast}\right) ,\text{ } Q_{n}^{\ast}=\left( q\varphi_{n},\varphi_{n}^{\ast}\right) , \end{equation} \begin{align*} A_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{m=1}^{k}\alpha _{m}^{\prime}\left( \lambda_{n,j}\right) \text{, }B_{k}^{\prime}\left( \lambda_{n,j}\right) =\sum_{m=1}^{k}\beta_{m}^{\prime}\left( \lambda _{n,j}\right) ,\\ \alpha_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots ,n_{k}}\left[ \widetilde{C}_{k}\left( q\cos2\pi nx,\sin2\pi n_{k}x\right) +\widetilde{M}_{k}\left( q\cos2\pi nx,\varphi_{n_{k}}^{\ast}\right) \right] ,\\ \beta_{k}^{\prime}\left( \lambda_{n,j}\right) & =\sum_{n_{1},\ldots,n_{k} }\left[ \widetilde{C}_{k}\left( q\varphi_{n},\sin2\pi n_{k}x\right) +\widetilde{M}_{k}\left( q\varphi_{n},\varphi_{n_{k}}^{\ast}\right) \right] ,\\ R_{k}^{\prime} & =\sum_{n_{1},\ldots,n_{k+1}}\left\{ \widetilde{C} _{k+1}\left( q\Psi_{n,j},\sin2\pi n_{k+1}x\right) +\widetilde{M} _{k+1}\left( q\Psi_{n,j},\varphi_{n_{k+1}}^{\ast}\right) \right\} , \end{align*} \[ \widetilde{C}_{k+1}=\widetilde{C}_{k}a_{k+1}+\widetilde{M}_{k}A_{k+1},\text{ }\widetilde{M}_{k+1}=\widetilde{C}_{k}b_{k+1}+\widetilde{M}_{k}B_{k+1};\text{ }k=0,1,2,\ldots, \] \begin{align*} \widetilde{C}_{1} & =A_{1}\left( \lambda_{n,j}\right) =\frac{\left( q\varphi_{n_{1}},\varphi_{n}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}}+\frac{\gamma_{1}n_{1}\left( q\cos2\pi n_{1}x,\varphi _{n}^{\ast}\right) }{\left( \lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}\right) ^{2}},\\ \widetilde{M}_{1} & =B_{1}\left( \lambda_{n,j}\right) =\frac{\left( q\cos2\pi n_{1}x,\varphi_{n}^{\ast}\right) }{\lambda_{n,j}-\left( 2\pi n_{1}\right) ^{2}}. \end{align*} Similar to (32) one can verify that \begin{equation} \alpha_{k}^{\prime}\left( \lambda_{n,j}\right) =O\left( \left( \frac {\ln\left\vert n\right\vert }{n}\right) ^{k}\right) ,\beta_{k}^{\prime }\left( \lambda_{n,j}\right) =O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k}\right) ,R_{k}^{\prime}=O\left( \left( \frac{\ln\left\vert n\right\vert }{n}\right) ^{k+1}\right) . \end{equation} If we take limit in (35) for $k\rightarrow\infty$, we obtain \[ \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime }\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1} n+Q_{n}^{\ast}+B^{\prime}\left( \lambda_{n,j}\right) \right] u_{n,j}, \] where \begin{equation} A^{\prime}\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\alpha_{m} ^{\prime}\left( \lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right) \text{, }B^{\prime}\left( \lambda_{n,j}\right) =\sum_{m=1}^{\infty}\beta_{m}^{\prime}\left( \lambda_{n,j}\right) =O\left( \frac{\ln\left\vert n\right\vert }{n}\right) . \end{equation} To get the main results of this paper we use the following system of equations, obtained above, with respect to $u_{n,j}$ and $v_{n,j}$ \begin{gather} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda _{n,j}\right) \right] u_{n,j}=\left[ P_{n}+B\left( \lambda_{n,j}\right) \right] v_{n,j},\\ \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime }\left( \lambda_{n,j}\right) \right] v_{n,j}=\left[ \gamma_{1} n+Q_{n}^{\ast}+B^{\prime}\left( \lambda_{n,j}\right) \right] u_{n,j}, \end{gather} where \begin{gather} Q_{n}=\left( q\varphi_{n},\sin2\pi nx\right) =\nonumber\\ =-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}\left( xq\left( x\right) ,\cos4\pi nx\right) -\frac{2\beta}{\beta-1}\left( q\left( x\right) ,\cos4\pi nx\right) \\ =-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+o\left( 1\right) , \end{gather} \begin{gather} P_{n}^{\ast}=\left( q\cos2\pi nx,\varphi_{n}^{\ast}\right) =\nonumber\\ =\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}\left( xq\left( x\right) ,\cos4\pi nx\right) -\frac{2}{\beta-1}\left( q\left( x\right) ,\cos4\pi nx\right) \\ =\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+o\left( 1\right) , \end{gather} \begin{equation} P_{n}=\left( q\cos2\pi nx,\sin2\pi nx\right) =\frac{1}{2}\left( q,\sin4\pi nx\right) =o\left( 1\right) , \end{equation} \begin{equation} Q_{n}^{\ast}=\left( q\varphi_{n},\varphi_{n}^{\ast}\right) =8\left( \frac{\beta_{1}+1}{\beta_{1}-1}\right) ^{2}\int_{0}^{1}q\left( x\right) \left( \dfrac{\beta_{1}}{1+\beta_{1}}-x\right) \left( x-\dfrac{1} {1+\beta_{1}}\right) \sin4\pi nxdx=o\left( 1\right) . \end{equation} Note that (39), (40) with (34), (38) give us \begin{gather} \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-Q_{n}+O\left( \dfrac {\ln\left\vert n\right\vert }{n}\right) \right] u_{n,j}=\left[ P_{n}+O\left( \dfrac{\ln\left\vert n\right\vert }{n}\right) \right] v_{n,j},\\ \left[ \lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}+O\left( \dfrac{\ln\left\vert n\right\vert }{n}\right) \right] v_{n,j}=\left[ \gamma_{1}n+Q_{n}^{\ast}+O\left( \dfrac{\ln\left\vert n\right\vert } {n}\right) \right] u_{n,j}. \end{gather} Introduce the notations \begin{align} c_{n} & =\left( q,\cos2\pi nx\right) \text{, }s_{n}=\left( q,\sin2\pi nx\right) \nonumber\\ c_{n,1} & =\left( xq,\cos2\pi nx\right) \text{, }s_{n,1}=\left( xq,\sin2\pi nx\right) \\ c_{n,2} & =\left( x^{2}q,\cos2\pi nx\right) \text{, }s_{n,2}=\left( x^{2}q,\sin2\pi nx\right) .\nonumber \end{align} In these notations we have \begin{equation} Q_{n}=-\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}c_{2n,1}-\frac{2\beta }{\beta-1}c_{2n} \end{equation} \begin{equation} P_{n}^{\ast}=\frac{2\left( \beta+1\right) }{\beta-1}\int_{0}^{1}xq\left( x\right) dx+\frac{2\left( \beta+1\right) }{\beta-1}c_{2n,1}-\frac{2} {\beta-1}c_{2n} \end{equation} \begin{equation} P_{n}=\frac{1}{2}s_{2n} \end{equation} \begin{equation} Q_{n}^{\ast}=-8\left( \frac{\beta+1}{\beta-1}\right) ^{2}s_{2n,2}+8\left( \frac{\beta+1}{\beta-1}\right) ^{2}s_{2n,1}-\frac{8\beta}{\left( \beta-1\right) ^{2}}s_{2n}. \end{equation} \begin{theorem} For $j=1,2$ the following statements hold: $(a)$ Any eigenfunction $\Psi_{n,j}$ of $T_{1}$ corresponding to the eigenvalue $\lambda_{n,j}$ defined in (10) satisfies \begin{equation} \Psi_{n,j}=\sqrt{2}\cos2\pi nx+O\left( n^{-1/2}\right) . \end{equation} Moreover there exists $N$ such that for all $n>N$ the geometric multiplicity of the eigenvalue $\lambda_{n,j}$ is $1$. $\left( b\right) $ A complex number $\lambda\in U(n)=:\{\lambda:\left\vert \lambda-\left( 2\pi n\right) ^{2}\right\vert \leq n\}$ is an eigenvalue of $T_{1}$ if and only if it is a root of the equation \begin{gather} \left[ \lambda-\left( 2\pi n\right) ^{2}-Q_{n}-A\left( \lambda\right) \right] \left[ \lambda-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime }\left( \lambda\right) \right] -\nonumber\\ -\left[ P_{n}+B\left( \lambda\right) \right] \left[ \gamma_{1} n+Q_{n}^{\ast}+B^{\prime}\left( \lambda\right) \right] =0. \end{gather} Moreover $\lambda\in U(n)$ is a double eigenvalue of $T_{1}$ if and only if \textit{it is a double root of} (55) . \end{theorem} \begin{proof} $\left( a\right) $ By (10) the left hand side of (48) is $O(n^{1/2}),$ which implies that $u_{n,j}=O(n^{-1/2}).$ Therefore from (29) we obtain (54). Now suppose that there are two linearly independent eigenfunctions corresponding to $\lambda_{n,j}$. Then there exists an eigenfunction satisfying \[ \Psi_{n,j}=\sqrt{2}\sin2\pi nx+o\left( 1\right) \] which contradicts (54). $(b)$ First we prove that the large eigenvalues $\lambda_{n,j}$ are the roots of the equation (55). It follows from (54), (27) and (15) that $v_{n,j}\neq0.$ If $u_{n,j}\neq0$ then multiplying the equations (39) and (40) side by side and then canceling $v_{n,j}u_{n,j}$ we obtain (55) . If $u_{n,j}=0$ then by (39) and (40) we have $P_{n}+B\left( \lambda_{n,j}\right) =0$ \ and $\lambda_{n,j}-\left( 2\pi n\right) ^{2}-P_{n}^{\ast}-A^{\prime}\left( \lambda_{n,j}\right) =0$ which mean that (55) holds. Thus in any case $\lambda_{n,j}$ is a root of (55). Now we prove that the roots of (55) lying in $U(n)$\ are the eigenvalues of $T_{1}.$ Let $F(\lambda)$ be the left-hand side of (55) which can be written as \begin{gather} F(\lambda)=(\lambda-\left( 2\pi n\right) ^{2})^{2}-\left( Q_{n}+A\left( \lambda\right) +P_{n}^{\ast}+A^{\prime}\left( \lambda\right) \right) \left( \lambda-\left( 2\pi n\right) ^{2}\right) +\\ +\left( Q_{n}+A\left( \lambda\right) \right) \left( P_{n}^{\ast }+A^{\prime}\left( \lambda\right) \right) -\left( P_{n}+B\left( \lambda\right) \right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\left( \lambda\right) \right) \nonumber \end{gather} and \[ G(\lambda)=(\lambda-\left( 2\pi n\right) ^{2})^{2}. \] One can easily verify that the inequality \begin{equation} \mid F(\lambda)-G(\lambda)\mid<\mid G(\lambda)\mid \end{equation} holds\ for all $\lambda$ from the boundary of $U(n).$ Since the function $G(\lambda)$ has two roots in the set $U(n),$ by the Rouche's theorem we obtain that $F(\lambda)$ has two roots in the same\ set.\ Thus\ $T_{1}$ has two eigenvalues (counting with multiplicities) lying in $U(n)$ that are the roots of (55). On the other hand, (55) has preciously two roots (counting with multiplicities) in $U(n).$ Therefore $\lambda\in U(n)$ is an eigenvalue of $T_{1}$ if and only if (55) holds. If \textit{ }$\lambda\in U(n)$ is a double eigenvalue of $T_{1}$ then it has no other eigenvalues\textit{ }in\textit{ }$U(n)$ and hence (55) has no other roots. This implies that $\lambda$ is a double root of (55). By the same way one can prove that if $\lambda$ is a double root of (55) then it is a double eigenvalue of $T_{1}.$ \end{proof} Let us consider (55) in detail. If we substitute $t=:\lambda-\left( 2\pi n\right) ^{2}$ then it becomes \begin{gather} t^{2}-\left( Q_{n}+A\left( \lambda\right) +P_{n}^{\ast}+A^{\prime}\left( \lambda\right) \right) t+\\ +\left( Q_{n}+A\left( \lambda\right) \right) \left( P_{n}^{\ast }+A^{\prime}\left( \lambda\right) \right) -\left( P_{n}+B\left( \lambda\right) \right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\left( \lambda\right) \right) =0.\nonumber \end{gather} The solutions of this equation are \[ t_{1,2}=\frac{\left( Q_{n}+P_{n}^{\ast}+A+A^{\prime}\right) \pm\sqrt {\Delta\left( \lambda\right) }}{2}, \] where \begin{equation} \Delta\left( \lambda\right) =\left( Q_{n}+P_{n}^{\ast}+A+A^{\prime}\right) ^{2}-4\left( Q_{n}+A\right) \left( P_{n}^{\ast}+A^{\prime}\right) +4\left( P_{n}+B\right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime}\right) \nonumber \end{equation} which can be written in the form \begin{equation} \Delta\left( \lambda\right) =\left( Q_{n}-P_{n}^{\ast}+A-A^{\prime}\right) ^{2}+4\left( P_{n}+B\right) \left( \gamma_{1}n+Q_{n}^{\ast}+B^{\prime }\right) . \end{equation} Clearly the eigenvalue $\lambda_{n,j}$\ is a root either of the equation \begin{equation} \lambda=\left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n} +P_{n}^{\ast}+A+A^{\prime}\right) -\sqrt{\Delta\left( \lambda\right) }\right] \end{equation} or of the equation \begin{equation} \lambda=\left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n} +P_{n}^{\ast}+A+A^{\prime}\right) +\sqrt{\Delta\left( \lambda\right) }\right] . \end{equation} Now let us examine $\Delta\left( \lambda_{n,j}\right) $ in detail. If (8) holds then one can readily see from (34), (38), (50), (51) and (59) that \begin{equation} \Delta\left( \lambda_{n,j}\right) =2\gamma_{1}ns_{2n}(1+o(1)). \end{equation} Taking into account the last three equality and (34), (38), (50), (51), we see that (60) and (61) have the form \begin{equation} \lambda=\left( 2\pi n\right) ^{2}-\frac{\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n} }(1+o(1)), \end{equation} \begin{equation} \lambda=\left( 2\pi n\right) ^{2}+\frac{\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n} }(1+o(1)). \end{equation} \begin{theorem} If (8) holds, then the large eigenvalues $\lambda_{n,j}$ are simple and satisfy the following asymptotic formulas \begin{equation} \lambda_{n,j}=\left( 2\pi n\right) ^{2}+\left( -1\right) ^{j}\frac {\sqrt{2\gamma_{1}}}{2}\sqrt{ns_{2n}}(1+o(1)). \end{equation} for $j=1,2.$ Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (8) holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{1}$ do not form a Riesz basis. \end{theorem} \begin{proof} To prove that the large eigenvalues $\lambda_{n,j}$ are simple let us show that one of the eigenvalues, say $\lambda_{n,1}$ satisfies (65) for $j=1$ and the other $\lambda_{n,2}$ satisfies (65) for $j=2.$ Let us prove that each of the equations (60) and (61) has a unique root in $U(n)$ by proving that \[ \left( 2\pi n\right) ^{2}+\frac{1}{2}\left[ \left( Q_{n}+P_{n}^{\ast }+A+A^{\prime}\right) \pm\sqrt{\Delta\left( \lambda\right) }\right] \] is a contraction mapping. For this we show that there exist positive real numbers $K_{1},K_{2},K_{3}$ such that \begin{equation} \mid A\left( \lambda\right) -A(\mu)\mid<K_{1}\mid\lambda-\mu\mid,\text{ }\mid A^{\prime}(\lambda)-A^{\prime}(\mu)\mid<K_{2}\mid\lambda-\mu\mid, \end{equation} \begin{equation} \left\vert \sqrt{\Delta\left( \lambda\right) }-\sqrt{\Delta\left( \mu\right) }\right\vert <K_{3}\mid\lambda-\mu\mid, \end{equation} where $K_{1}+K_{2}+K_{3}<1$. The proof of (66) is similar to the proof of (56) of the paper [26]. Now let us prove (67). By (62) and (8) we have \[ \left( \sqrt{\Delta\left( \lambda\right) }\right) ^{-1}=o(1). \] On the other hand arguing as in the proof of (56) of the paper [26] we get \[ \dfrac{d}{d\lambda}\Delta\left( \lambda\right) =O(1). \] Hence in any case we have \[ \frac{d}{d\lambda}\sqrt{\Delta\left( \lambda\right) }=\frac{\dfrac {d}{d\lambda}\Delta\left( \lambda\right) }{2\sqrt{\Delta\left( \lambda\right) }}=o(1). \] Thus by the fixed point theorem, each of the equations (60) and (61) has a unique root $\lambda_{1}$ and $\lambda_{2}$ respectively. Clearly by (63) and (64), we have $\lambda_{1}\neq\lambda_{2}$ which implies that the equation (55) has two simple root in $U\left( n\right) .$ Therefore by Theorem 1(b), $\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of $T_{1}$ lying in $U\left( n\right) ,$ that is, they are $\lambda_{n,1}$ and $\lambda_{n,2}$, which proves the simplicity of the large eigenvalues and the validity of (65). If there exists a sequence $\left\{ n_{k}\right\} $ such that (8) holds when $n$ is replaced by $n_{k}$, then by Theorem 1(a) \[ \left( \Psi_{n_{k},1},\Psi_{n_{k},2}\right) =1+O\left( n_{k}^{-1/2}\right) . \] Now it follows from the theorems of [20,21] (see also Lemma 3 of [24]) that the root functions of $T_{1}$ do not form a Riesz basis. \end{proof} Now let us consider the operators $T_{2}$, $T_{3}$ and $T_{4}.$ First we consider the operator $T_{3}$. It is well-known that ( see formulas (47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{3}(q)$ consist of the sequences $\{\lambda_{n,1,3}\},\{\lambda_{n,2,3}\}$ satisfying (10) when $\lambda_{n,j}$ is replaced by $\lambda_{n,j,3}.$ The eigenvalues, eigenfunctions and associated functions of $T_{3}$ are \begin{align*} \lambda_{n} & =\left( 2\pi n\right) ^{2};\text{ }n=0,1,2,\ldots\\ y_{0}\left( x\right) & =x-\dfrac{\alpha}{1+\alpha},\text{ }y_{n}\left( x\right) =\sin2\pi nx;\text{ }n=1,2,\ldots\\ \phi_{n}\left( x\right) & =\left( x-\dfrac{\alpha}{1+\alpha}\right) \frac{\cos2\pi nx}{4\pi n};\text{ }n=1,2,\ldots. \end{align*} respectively. The biorthogonal systems analogous to (16), (17) are \begin{equation} \left\{ \cos2\pi nx,\frac{4\left( 1+\overline{\alpha}\right) } {1-\overline{\alpha}}\left( \dfrac{1}{1+\overline{\alpha}}-x\right) \sin2\pi nx\right\} _{n=0}^{\infty} \end{equation} \begin{equation} \left\{ \sin2\pi nx,\frac{4\left( 1+\alpha\right) }{1-\alpha}\left( x-\dfrac{\alpha}{1+\alpha}\right) \cos2\pi nx\right\} _{n=0}^{\infty} \end{equation} respectively. Analogous formulas to (18) and (19) are \begin{equation} \left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi _{N,j},\cos2\pi nx\right) =\left( q\left( x\right) \Psi_{N,j},\cos2\pi nx\right) \end{equation} \begin{equation} \left( \lambda_{N,j}-\left( 2\pi n\right) ^{2}\right) \left( \Psi _{N,j},\varphi_{n}^{\ast}\right) -\gamma_{3}n\left( \Psi_{N,j},\cos2\pi nx\right) =\left( q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right) \end{equation} respectively, where \[ \gamma_{3}=\frac{16\pi\left( 1+\alpha\right) }{1-\alpha}. \] Instead of (16)-(19) using (68)-(71) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for $T_{3}.$ \begin{theorem} If (8) holds, then the large eigenvalues $\lambda_{n,j,3}$ are simple and satisfy the following asymptotic formulas \begin{equation} \lambda_{n,j,3}=\left( 2\pi n\right) ^{2}+\left( -1\right) ^{j}\frac {\sqrt{2\gamma_{3}}}{2}\sqrt{ns_{2n}}(1+o(1)). \end{equation} for $j=1,2.$ The eigenfunctions $\Psi_{n,j,3}$ corresponding to $\lambda _{n,j,3}$ obey \begin{equation} \Psi_{n,j,3}=\sqrt{2}\sin2\pi nx+O\left( n^{-1/2}\right) . \end{equation} Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (8) holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{3}$ do not form a Riesz basis. \end{theorem} Now let us consider the operator $T_{2}$. It is well-known that ( see formulas (47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{2}(q)$ consist of the sequences $\{\lambda_{n,1,2}\},\{\lambda_{n,2,2}\}$ satisfying \begin{equation} \lambda_{n,j,2}=(2n\pi+\pi)^{2}+O(n^{1/2}), \end{equation} for $j=1,2$. The eigenvalues, eigenfunctions and associated functions of $T_{2}$ are \begin{align*} \lambda_{n} & =\left( \pi+2\pi n\right) ^{2},\text{ }y_{n}\left( x\right) =\cos\left( 2n+1\right) \pi x,\\ \phi_{n}\left( x\right) & =\left( \frac{\beta}{\beta-1}-x\right) \frac{\sin\left( 2n+1\right) \pi x}{2\left( 2n+1\right) \pi} \end{align*} for $n=0,1,2,\ldots$respectively. The biorthogonal systems analogous to (16), (17) are \begin{equation} \left\{ \sin\left( 2n+1\right) \pi x,\frac{4\left( \overline{\beta }-1\right) }{\overline{\beta}+1}\left( x+\dfrac{1}{\overline{\beta} -1}\right) \cos\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty} \end{equation} \begin{equation} \left\{ \cos\left( 2n+1\right) \pi x,\frac{4\left( \beta-1\right) } {\beta+1}\left( \frac{\beta}{\beta-1}-x\right) \sin\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty} \end{equation} respectively. Analogous formulas to (18) and (19) are \begin{equation} \left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right) \left( \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right) =\left( q\left( x\right) \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right) \end{equation} \begin{equation} \left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right) \left( \Psi_{N,j},\varphi_{n}^{\ast}\right) -\left( 2n+1\right) \gamma _{2}\left( \Psi_{N,j},\sin\left( 2n+1\right) \pi x\right) =\left( q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right) \end{equation} respectively, where \[ \gamma_{2}=\frac{8\pi\left( \beta-1\right) }{\beta+1}. \] Instead of (16)-(19) using (75)-(78) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for $T_{2}.$ \begin{theorem} If (9) holds, then the large eigenvalues $\lambda_{n,j,2}$ are simple and satisfy the following asymptotic formulas \begin{equation} \lambda_{n,j,2}=\left( \left( 2n+1\right) \pi\right) ^{2}+\left( -1\right) ^{j}\frac{\sqrt{2\gamma_{2}}}{2}\sqrt{\left( 2n+1\right) s_{2n+1}}(1+o(1)). \end{equation} for $j=1,2.$ The eigenfunctions $\Psi_{n,j,2}$ corresponding to $\lambda _{n,j,2}$ obey \begin{equation} \Psi_{n,j,2}=\sqrt{2}\cos\left( 2n+1\right) \pi x+O\left( n^{-1/2}\right) . \end{equation} Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (9) holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{2}$ do not form a Riesz basis. \end{theorem} Lastly we consider the operator $T_{4}$. It is well-known that ( see formulas (47a), (47b)) in page 65 of [18] ) the eigenvalues of the operators $T_{4}(q)$ consist of the sequences $\{\lambda_{n,1,4}\},\{\lambda_{n,2,4}\}$ satisfying (74) when $\lambda_{n,j,2}$ is replaced by $\lambda_{n,j,4}.$ The eigenvalues, eigenfunctions and associated functions of $T_{4}$ are \begin{align*} \lambda_{n} & =\left( \pi+2\pi n\right) ^{2},\text{ }y_{n}\left( x\right) =\sin\left( 2n+1\right) \pi x,\\ \phi_{n}\left( x\right) & =\left( \frac{\alpha}{1-\alpha}+x\right) \frac{\cos\left( 2n+1\right) \pi x}{2\left( 2n+1\right) \pi} \end{align*} for $n=0,1,2,\ldots$respectively. The biorthogonal systems analogous to (16), (17) are \begin{equation} \left\{ \cos\left( 2n+1\right) \pi x,\frac{4\left( 1-\overline{\alpha }\right) }{1+\overline{\alpha}}\left( \dfrac{1}{1-\overline{\alpha} }-x\right) \sin\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty} \end{equation} \begin{equation} \left\{ \sin\left( 2n+1\right) \pi x,\frac{4\left( 1-\alpha\right) }{1+\alpha}\left( \dfrac{\alpha}{1-\alpha}+x\right) \cos\left( 2n+1\right) \pi x\right\} _{n=0}^{\infty} \end{equation} respectively. Analogous formulas to (18) and (19) are \begin{equation} \left( \lambda_{N,j}-\left( \pi+2\pi n\right) ^{2}\right) \left( \Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) =\left( q\left( x\right) \Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) , \end{equation} \begin{equation} \left( \lambda_{N,j}-\left( \left( 2n+1\right) \pi\right) ^{2}\right) \left( \Psi_{N,j},\varphi_{n}^{\ast}\right) -\left( 2n+1\right) \gamma _{4}\left( \Psi_{N,j},\cos\left( 2n+1\right) \pi x\right) =\left( q\left( x\right) \Psi_{N,j},\varphi_{n}^{\ast}\right) \end{equation} respectively, where \[ \gamma_{4}=\frac{8\pi\left( 1-\alpha\right) }{1+\alpha}. \] Instead of (16)-(19) using (81)-(84) and arguing as in the proofs of Theorem 1 and Theorem 2 we obtain the following results for $T_{4}.$ \begin{theorem} If (9) holds, then the large eigenvalues $\lambda_{n,j,4}$ are simple and satisfy the following asymptotic formulas \begin{equation} \lambda_{n,j,4}=\left( \left( 2n+1\right) \pi\right) ^{2}+\left( -1\right) ^{j}\frac{\sqrt{2\gamma_{4}}}{2}\sqrt{\left( 2n+1\right) s_{2n+1}}(1+o(1)). \end{equation} for $j=1,2.$ The eigenfunctions $\Psi_{n,j,4}$ corresponding to $\lambda _{n,j,4}$ obey \begin{equation} \Psi_{n,j,4}=\sqrt{2}\sin\left( 2n+1\right) \pi x+O\left( n^{-1/2}\right) . \end{equation} Moreover, if there exists a sequence $\left\{ n_{k}\right\} $ such that (9) holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{4}$ do not form a Riesz basis. \end{theorem} \begin{remark} Suppose that \begin{equation} \int_{0}^{1}xq\left( x\right) dx\neq0. \end{equation} If \begin{equation} \frac{1}{2}s_{2n}+B=o\left( \frac{1}{n}\right) , \end{equation} where $B$ is defined by (34), then arguing as in the proof of Theorem 2, we obtain that the large eigenvalues of the operator $T_{1}$ are simple. Moreover if there exists a sequence $\left\{ n_{k}\right\} $ such that (88) holds when $n$ is replaced by $n_{k},$ then the root functions of $T_{1}$ do not form a Riesz basis. The similar results can be obtained for the operators $T_{2},T_{3}$ and $T_{4}.$ \end{remark} \begin{remark} Using (31) and (35) and arguing as in the proof of Theorem 3 of [1] it can be obtained asymptotic formulas of arbitrary order for the eigenvalues and eigenfunctions of the operator $T_{1}.$ The similar formulas can be obtained for the operators $T_{2},T_{3}$ and $T_{4}.$ \end{remark} \end{document}

\begin{document} \title{Restoring broken entanglement by separable correlations} \begin{abstract} We consider two bosonic Gaussian channels whose thermal noise is strong enough to break bipartite entanglement. In this scenario, we show how the presence of separable correlations between the two channels is able to restore the broken entanglement. This reactivation occurs not only in a scheme of direct distribution, where a third party (Charlie) broadcasts entangled states to remote parties (Alice and Bob), but also in a configuration of indirect distribution which is based on entanglement swapping. In both schemes, the amount of entanglement remotely activated can be large enough to be distilled by one-way distillation protocols. \end{abstract} \pacs{03.65.Ud, 03.67.--a, 42.50.--p, 89.70.Cf} \author{Gaetana Spedalieri} \author{Stefano Pirandola} \email{[email protected]} \affiliation{Computer Science and York Centre for Quantum Technologies, University of York, York YO10 5GH, United Kingdom} \maketitle Entanglement is a fundamental physical resource in quantum information and computation. Once two parties, say Alice and Bob, share a suitable amount of entanglement, they can implement a variety of powerful protocols~\cite{NielsenBook,Mwilde}. In a scheme of direct distribution, there is a middle station (Charlie) possessing a bipartite system in an entangled state; one subsystem is sent to Alice and the other to Bob. Alternatively, in a scheme of indirect distribution, known as entanglement swapping, the distribution is mediated by a measurement process. Here Alice and Bob each has a bipartite system prepared in an entangled state. One subsystem is retained while the other is sent to Charlie. At his station, Charlie detects the two incoming subsystems by performing a suitable Bell measurement and communicates the classical outcome back to Alice and Bob. As a result of this process, the two subsystems retained by the remote parties are projected onto an entangled state. \begin{figure} \caption{Direct and indirect schemes for entanglement distribution in memoryless and correlated-noise environments. Charlie is the middle station transmitting to or receiving systems from Alice (left station) and Bob (right station). Ellipses represent entangled states, black-circles represent channels, and detectors are Bell measurements. (1) Direct distribution in a memoryless environment. If the individual channels are EB, then no distribution of entanglement is possile. We have $A|B^{\prime}$ and $A^{\prime}|B$ which implies $A^{\prime}|B^{\prime}$. (2) Entanglement swapping in a memoryless environment. We have that $a|A^{\prime}$ and $B^{\prime}|b$ implies $a|b$. (3) Direct distribution in a correlated environment. Despite $A|B^{\prime}$ and $A^{\prime}|B$ we have that $A^{\prime}-B^{\prime}$ is possible. Surprisingly, this can be realized by a separable environment. (4) Entanglement swapping in a correlated environment. Despite $a|A^{\prime}$ and $B^{\prime}|b$ we have that $a-b$ is possible. Surprisingly, this is realizable by a separable environment.} \label{scenario} \end{figure} In both configurations, entanglement distribution is possible as long as the action of the environment is not too strong. When decoherence is strong enough to destroy any input entanglement, the environment results into an entanglement breaking (EB) channel~\cite{EBchannels,HolevoEB}. By definition, a quantum channel $\mathcal{E}$\ is EB when its local action on one part of a bipartite state always results into a separable output state. In other words, given two systems, $A$ and $B$, in an arbitrary bipartite state $\rho_{AB}$, the output state $\rho_{AB^{\prime}}=(\mathcal{I}_{A}\otimes\mathcal{E} _{B})(\rho_{AB})$\ is always separable, where $\mathcal{I}_{A}$ is the identity channel applied to system $A$ and $\mathcal{E}_{B}$ is the EB channel applied to system $B$. Thus, if the input systems $A$ and $B$ were initially entangled (here denoted by the notation $A-B$), the output systems $A$ and $B^{\prime}$ are separable (here denoted by the notation $A|B^{\prime}$). The standard model of decoherence is assumed to be Markovian, where the travelling systems are subject to memoryless channels. For instance, consider the case of direct distribution depicted in the panel~(1) of Fig.~\ref{scenario}. In the standard Markovian description, the entangled state $\rho_{AB}$ of the input systems $A$ and $B$ is subject to a tensor product of channels $\mathcal{E}_{A}\otimes\mathcal{E}_{B}$. In this case, there is clearly no way to distribute entanglement if both $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ are EB channels. Suppose that Charlie tries to share entanglement with one of the remote parties by sending one of the two systems while keeping the other (one-system transmission). For instance, Charlie may keep system $A$ while transmitting system $B$ to Bob. The action of $\mathcal{I}_{A}\otimes\mathcal{E}_{B}$ destroys the initial entanglement, so that systems $A$ (kept) and $B^{\prime}$ (transmitted) are separable ($A|B^{\prime}$). Symmetrically, the action of $\mathcal{E}_{A}\otimes \mathcal{I}_{B}$ destroys the entanglement between system $A^{\prime}$ (transmitted) and system $B$ (kept), i.e., we have $A^{\prime}|B$. Then suppose that Charlie sends both his systems to Alice and Bob (two-system transmission). This strategy will also fail since the joint action of the two EB\ channels is given by the tensor product $\mathcal{E}_{A}\otimes \mathcal{E}_{B}=(\mathcal{E}_{A}\otimes\mathcal{I}_{B})(\mathcal{I}_{A} \otimes\mathcal{E}_{B})$. In other words, since we have one-system EB ($A|B^{\prime}$ and $A^{\prime}|B$) then we must have two-system EB ($A^{\prime}|B^{\prime}$). The previous reasoning can be extended to the case of indirect distribution as shown in panel (2) of Fig.~\ref{scenario} involving a Bell measurement by Charlie. Since the environment is memoryless ($\mathcal{E}_{A}\otimes \mathcal{E}_{B}$), we have that the absence of entanglement before the Bell measurement ($a|A^{\prime}$ and $B^{\prime}|b$) is a sufficient condition for the swapping protocol to fail, i.e., the remote systems $a$ and $b$ remains separable ($a|b$). Similarly to the previous case, if one-system transmission does not distribute entanglement, then two-system transmission cannot lead to entanglement generation via the swapping protocol. Here we discuss how the previous implications for direct and indirect distribution of entanglement are false in the presence of a correlated-noise environment: Two-system transmission can successfully distribute entanglement despite one-system transmission being subject to EB. In other words, by combining two EB\ channels into a joint suitably-correlated environment, we can reactivate the distribution of entanglement. We will show the physical conditions under which the environmental correlations are able to trigger the reactivation, therefore \textquotedblleft breaking entanglement-breaking\textquotedblright. The most remarkable finding is that we do not need to consider an entangled state for the environment: The injection of separable correlations from the environment is sufficient for the restoration. To better clarify these points, consider the schemes of direct and indirect distribution in the presence of a correlated-noise environment. In the scheme of direct distribution shown in panel~(3) of Fig.~\ref{scenario}, an input entangled state $\rho_{AB}$ is jointly transformed into an output state $\rho_{A^{\prime}B^{\prime}}=\mathcal{E}_{AB}(\rho_{AB})$. We assume that the dilation of the composite channel $\mathcal{E}_{AB}$ is realized by introducing a two-system environment, $E_{1}$ and $E_{2}$, in a bipartite state $\rho_{E_{1}E_{2}}$, which interacts with the incoming systems via two unitaries $U_{AE_{1}}$ (transforming $A$ and $E_{1}$) and $U_{BE_{2}}$ (transforming $B$ and $E_{2}$). In other words, the output state can be written in the form \begin{align} \rho_{A^{\prime}B^{\prime}} & =\mathrm{Tr}_{E_{1}E_{2}}[(U_{AE_{1}}\otimes U_{BE_{2}})\nonumber\\ & \times(\rho_{AB}\otimes\rho_{E_{1}E_{2}})(U_{AE_{1}}^{\dagger}\otimes U_{BE_{2}}^{\dagger})]. \label{eqDIL} \end{align} If the environmental state is not tensor product, i.e., $\rho_{E_{1}E_{2}} \neq\rho_{E_{1}}\otimes\rho_{E_{2}}$, then the composite channel cannot be decomposed into memoryless channels, i.e., $\mathcal{E}_{AB}\neq \mathcal{E}_{A}\otimes\mathcal{E}_{B}$. In any case, from the dilation given in Eq.~(\ref{eqDIL}), we can always define the reduced channels, $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$, acting on the individual systems. For instance, if only system $B$ is transmitted, then we have the evolved state \begin{align} \rho_{AB^{\prime}} & =(\mathcal{I}_{A}\otimes\mathcal{E}_{B})(\rho _{AB})\nonumber\\ & =\mathrm{Tr}_{E_{2}}[(I_{A}\otimes U_{BE_{2}})\rho_{AB}\otimes\rho_{E_{2} }(I_{A}\otimes U_{BE_{2}}^{\dagger})], \end{align} where $\rho_{E_{2}}=\mathrm{Tr}_{E_{1}}(\rho_{E_{1}E_{2}})$. A similar formula holds for the evolution of the other system $\rho_{A^{\prime}B}=(\mathcal{E} _{A}\otimes\mathcal{I}_{B})(\rho_{AB})$. Now, assuming that $\mathcal{I}_{A}\otimes\mathcal{E}_{B}$ and $\mathcal{E} _{A}\otimes\mathcal{I}_{B}$ are EB channels (so that $A|B^{\prime}$ and $B|A^{\prime}$), the composite channel $\mathcal{E}_{AB}$ can still preserve entanglement (so that $A^{\prime}-B^{\prime}$ is possible). In other words, we have a paradoxical situation where Charlie is not able to share entanglement with Alice or Bob, but still can distribute entanglement to them. This is clearly an effect of the injected correlations coming from the environmental state $\rho_{E_{1}E_{2}}$. As mentioned earlier, our main finding is that these correlations do not need to be strong: Entanglement distribution can be activated by separable correlations, i.e., by an environment which is in a separable state $\rho_{E_{1}E_{2}}$. This effect of reactivation can also be extended to entanglement distillation, which typically requires stronger conditions than entanglement distribution (demanded by the existence of effective distillation protocols). Despite the individual channels are EB, their combination into a separable environment enables Charlie to distribute distillable entanglement to Alice and Bob. This is easy to prove for an environment with finite memory, which can be decomposed as $\mathcal{E}_{AB}\otimes\mathcal{E}_{AB}\otimes\ldots$ In our investigation, we also consider the case of entanglement swapping in a correlated-noise environment as depicted in panel~(4) of Fig.~\ref{scenario}. Here Alice and Bob have two entangled states, $\rho_{aA}$\ and $\rho_{Bb}$, respectively. Systems $a$ and $b$ are retained, while systems $A$ and $B$ are transmitted to Charlie, therefore undergoing the joint quantum channel $\mathcal{E}_{AB}$. Before the Bell measurement, the global state is described by \begin{align} \rho_{aA^{\prime}B^{\prime}b} & =(\mathcal{I}_{a}\otimes\mathcal{E} _{AB}\otimes\mathcal{I}_{b})(\rho_{aA}\otimes\rho_{Bb})\nonumber\\ & =\mathrm{Tr}_{E_{1}E_{2}}[U(\rho_{aA}\otimes\rho_{E_{1}E_{2}}\otimes \rho_{Bb})U^{\dagger}]~, \end{align} where $U=I_{a}\otimes U_{AE_{1}}\otimes U_{E_{2}B}\otimes I_{b}$. As before, we consider the case where the reduced channels, $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$, are EB channels, so that no entanglement survives before the Bell measurement ($a|A^{\prime}$ and $B^{\prime}|b$). If the environment has no memory ($\rho_{E_{1}E_{2}}=\rho_{E_{1}}\otimes\rho_{E_{2}} $) there is no way to distribute entanglement to Alice and Bob ($a|b$). By contrast, if the environment has memory ($\rho_{E_{1}E_{2}}\neq\rho_{E_{1} }\otimes\rho_{E_{2}}$), then entanglement distribution is possible ($a-b$) and this distribution can be activated by a separable environmental state $\rho_{E_{1}E_{2}}$. Thus, we have the paradoxical situation where no bipartite entanglement survives at Charlie's station ($a|A^{\prime}$ and $B^{\prime}|b$), but still the swapping protocol is able to generate remote entanglement at Alice's and Bob's stations ($a-b$) thanks to the separable correlations injected by the environment. As before, these separable correlations can be strong enough to distribute distillable entanglement to the remote parties. The environmental reactivation of entanglement distribution can be proven~\cite{NJP} for quantum systems with Hilbert spaces of any dimension, both finite (discrete-variable systems) and infinite (continuous-variable systems~\cite{BraREV,BraREV2,RMP}). We remark that the phenomenon of reactivation in direct distribution is not surprising in specific lossless scenarios where the environment is \textquotedblleft twirling\textquotedblright, i.e., a classical mixture of operators of the type $U\otimes U$ or $U\otimes U^{\ast}$, with $U$ being a unitary~\cite{NJP}. In this case, it is easy to find a fixed point in the joint map of the environment, so that a state can be perfectly distributed, despite the fact that the local (single-system) channels may become entanglement breaking~\cite{NJP}. In discrete variables with Hilbert space dimensionality $d\geq2$, these fixed points are the multi-dimensional Werner states~\cite{Werner} (invariant under $U\otimes U$-twirling) and the multi-dimensional isotropic states~\cite{HOROs} (invariant under $U\otimes U^{\ast}$-twirling). Similarly, one can consider continuous-variable Werner states which are invariant under anti-correlated phase-space rotations (non-Gaussian twirlings)~\cite{NJP}. However, all these cases are artificial since they are associated with lossless environments. The phenomenon of reactivation becomes non-trivial in the presence of loss as typical for continuous variable systems in realistic Gaussian environments. In the configuration of indirect distribution, we can also find simple examples of reactivation with discrete variable systems (in particular, qubits) when the environment is lossless and $U\otimes U^{\ast}$-twirling. Suppose that $\rho_{aA}$ and $\rho_{bB}$ are Bell pairs, e.g., singlet states \begin{equation} \left\vert -\right\rangle =\frac{1}{\sqrt{2}}\left( \left\vert 0,1\right\rangle -\left\vert 1,0\right\rangle \right) ~. \end{equation} Then suppose that qubits $A$ and $B$ are subject to twirling, which means that $\rho_{AB}$ is transformed as \begin{equation} \rho_{A^{\prime}B^{\prime}}=\int dU~(U\otimes U^{\ast})~\rho_{AB}~(U\otimes U^{\ast})^{\dagger}~, \end{equation} where the integral is over the entire unitary group $\mathcal{U}(2)$ acting on the bi-dimensional Hilbert space and $dU$ is the Haar measure. Now the application of a Bell detection on the output qubits $A^{\prime}$ and $B^{\prime}$ has the effect to cancel the environmental noise. In fact, one can easily check that the output state of $a$ and $b$ will be projected onto a singlet state up to a Pauli operator, which is compensated via the communication of the Bell outcome. Again, the phenomenon becomes non-trivial when more realistic environments are taken into account, in particular, lossy environments as typical for continuous-variable systems. For this reason we discuss here the reactivation phenomenon using continuous-variable systems. In particular, we consider the bosonic modes of the electromagnetic field. The input modes are prepared in Gaussian states with Einstein-Podolsky-Rosen (EPR) correlations~\cite{RMP,EPR}, which are the most typical form of continuous variable entanglement. These modes are then assumed to evolve under the action of a lossy Gaussian environment. This type of environment is modelled by two beam splitters which mix the travelling modes, $A$ and $B$, with two environmental modes, $E_{1}$ and $E_{2}$, prepared in a bipartite Gaussian state $\rho_{E_{1}E_{2}}$ (separable or entangled). The reduced channels, $\mathcal{E}_{A}$ and $\mathcal{E}_{B} $,\ are two lossy channels whose transmissivities and thermal noises are such to make them EB channels. To achieve simple analytical results, in this manuscript we only consider the limit of large entanglement for the input states. The paper is structured as follows. In Sec.~\ref{SECgauss} we characterize the basic model of correlated Gaussian environment, which directly generalizes the standard model of thermal-loss environment. We identify the physical conditions under which the correlated Gaussian environment is separable or entangled. In Sec.~\ref{SECdirect}, we study the direct distribution of entanglement in the presence of the correlated Gaussian environment and assuming the condition of one-system EB. We provide the regimes of parameters under which remote entanglement is activated by the environmental correlations (in particular, separable correlations) and the stronger regimes where the generated remote entanglement is also distillable. This part is a review of results already known in the literature~\cite{NJP}. Then, in Sec.~\ref{SECindirect}, we generalize the theory of entanglement swapping to the correlated Gaussian environment. We consider swapping and distillation of entanglement, finding the regimes of parameters where these tasks are successful despite the EB condition. Finally, Sec.~\ref{SECconclusion} is for conclusion and discussion. \section{Correlated Gaussian environment\label{SECgauss}} We consider two beam splitters (with transmissivity $\tau$) which combine modes $A$ and $B$ with two environmental modes, $E_{1}$ and $E_{2}$, respectively. These ancillary modes are in a zero-mean Gaussian state $\rho_{E_{1}E_{2}}$ symmetric under $E_{1}$-$E_{2}$ permutation. In the memoryless model, the environmental state is tensor-product $\rho_{E_{1}E_{2} }=\rho\otimes\rho$, meaning that $E_{1}$ and $E_{2}$ are fully independent. In particular, $\rho$ is a thermal state with covariance matrix (CM) $\omega\mathbf{I}$, where the noise variance $\omega=2\bar{n}+1$ quantifies the mean number of thermal photons $\bar{n}$\ entering the beam splitter. Each interaction is then equivalent to a lossy channel with transmissivity $\tau$ and thermal noise $\omega$. \begin{figure} \caption{\textit{Left}. Correlated Gaussian environment, with losses $\tau$, thermal noise $\omega$ and correlations $\mathbf{G}$. The state of the environment $E_{1}$ and $E_{2}$ can be separable or entangled. \textit{Right}. Correlation plane $(g,g^{\prime})$ for the Gaussian environment, corresponding to thermal noise $\omega=2$. The black area identifies forbidden environments (correlations are too strong to be compatible with quantum mechanics). White area identifies physical environments, i.e., the subset of points which satisfy the bona-fide conditions of Eq.~(\ref{CMconstraints}). Whitin this area, the inner region labbeled by S identifies separable environments, while the two outer regions identify entangled environments. Figures adapted from Ref.~\cite{NJP} under a CC BY 3.0 licence (http://creativecommons.org/licenses/by/3.0/).} \label{ENVschemes} \end{figure} This Gaussian process can be generalized to include the presence of correlations between the environmental modes as depicted in the right panel of Fig.~\ref{ENVschemes}. The simplest extension of the model consists\ of taking the ancillary modes, $E_{1}$ and $E_{2}$, in a zero-mean Gaussian state $\rho_{E_{1}E_{2}}$ with CM given by the symmetric normal form \begin{equation} \mathbf{V}_{E_{1}E_{2}}(\omega,g,g^{\prime})=\left( \begin{array} [c]{cc} \omega\mathbf{I} & \mathbf{G}\\ \mathbf{G} & \omega\mathbf{I} \end{array} \right) ~, \label{EVE_cmAPP} \end{equation} where $\omega\geq1$ is the thermal noise variance associated with each ancilla, and the off-diagonal block \begin{equation} \mathbf{G=}\left( \begin{array} [c]{cc} g & \\ & g^{\prime} \end{array} \right) ~, \label{Gblock} \end{equation} accounts for the correlations between the ancillas. This type of environment can be separable or entangled (conditions for separability will be given afterwards). It is clear that, when we consider the two interactions $A-E_{1}$ and $B-E_{2}$ separately, the environmental correlations are washed away. In fact, by tracing out $E_{2}$, we are left with mode $E_{1}$ in a thermal state ($\mathbf{V}_{E_{1}}=\omega\mathbf{I}$) which is combined with mode $A$ via the beam-splitter. In other words, we have again a lossy channel with transmissivity $\tau$ and thermal noise $\omega$. The scenario is identical for the other mode $B$ when we trace out $E_{1}$. However, when we consider the joint action of the two environmental modes, the correlation block $\mathbf{G}$ comes into play and the global dynamics of the two travelling modes becomes completely different from the standard memoryless scenario. Before studying the system dynamics and the corresponding evolution of entanglement, we need to characterize the correlation block $\mathbf{G}$\ more precisely. In fact, the two correlation parameters, $g$ and $g^{\prime}$, cannot be completely arbitrary but must satisfy specific physical constraints. These parameters must vary within ranges which make the CM of Eq.~(\ref{EVE_cmAPP}) a bona-fide quantum CM. Given an arbitrary value of the thermal noise $\omega\geq1$, the correlation parameters must satisfy the following three bona-fide conditions~\cite{TwomodePRA,NJP} \begin{equation} |g|<\omega,~~~|g^{\prime}|<\omega,~~~\omega^{2}+gg^{\prime}-1\geq \omega\left\vert g+g^{\prime}\right\vert . \label{CMconstraints} \end{equation} \subsection{Separability properties} Once we have clarified the bona-fide conditions for the environment, the next step is to characterize its separability properties. For this aim, we compute the smallest partially-transposed symplectic (PTS) eigenvalue $\varepsilon$ associated with the CM\ $\mathbf{V}_{E_{1}E_{2}}$. For Gaussian states, this eigenvalue represents an entanglement monotone which is equivalent to the log-negativity~\cite{logNEG1,logNEG2,logNEG3} $\mathcal{E}=\max\left\{ 0,-\log\varepsilon\right\} $. After simple algebra, we get~\cite{NJP} \begin{equation} \varepsilon=\sqrt{\omega^{2}-gg^{\prime}-\omega|g-g^{\prime}|}~. \end{equation} Provided that the conditions of Eq.~(\ref{CMconstraints}) are satisfied, the separability condition $\varepsilon\geq1$ is equivalent to \begin{equation} \omega^{2}-gg^{\prime}-1\geq\omega|g-g^{\prime}|~. \label{sepCON} \end{equation} To visualize the structure of the environment, we provide a numerical example in Fig.~\ref{ENVschemes}. In the right panel of this figure, we consider the \textit{correlation plane} which is spanned by the two parameters $g$ and $g^{\prime}$. For a given value of the thermal noise $\omega$, we identify the subset of points which satisfy the bona-fide conditions of Eq.~(\ref{CMconstraints}). This subset corresponds to the white area in the figure. Within this area, we then characterize the regions which correspond to separable environments (area labelled by S) and entangled environments (areas labelled by E). \section{Direct distribution of entanglement in a correlated Gaussian environment\label{SECdirect}} Let us study the system dynamics and the entanglement propagation in the presence of a correlated Gaussian environment, reviewing some key results from the literature~\cite{NJP}. Suppose that Charlie has an entanglement source described by an EPR\ state $\rho_{AB}$ with CM \begin{equation} \mathbf{V}(\mu)=\left( \begin{array} [c]{cc} \mu\mathbf{I} & \mu^{\prime}\mathbf{Z}\\ \mu^{\prime}\mathbf{Z} & \mu\mathbf{I} \end{array} \right) ~, \label{CM_TMSV} \end{equation} where $\mu\geq1$, $\mu^{\prime}:=\sqrt{\mu^{2}-1}$, and $\mathbf{Z}$\ is the reflection matrix \begin{equation} \mathbf{Z}:=\left( \begin{array} [c]{cc} 1 & \\ & -1 \end{array} \right) ~. \label{ZetaMAT} \end{equation} We may consider the different scenarios depicted in the three panels of Fig.~\ref{twomodeEB}. Charlie may attempt to distribute entanglement to Alice and Bob as shown in Fig.~\ref{twomodeEB}(1), or he may try to share entanglement with one of the remote parties, as shown in Figs.~\ref{twomodeEB} (2) and~(3). \begin{figure} \caption{Scenarios for direct distribution of entanglement. (1) Charlie has two modes $A$ and $B$ prepared in an EPR state $\rho_{AB}$. In order to distribute entanglement to the remote parties, Charlie transmits the two modes through the correlated Gaussian environment characterized by transmissivity $\tau$, thermal noise $\omega$ and correlations $\mathbf{G}$. (2) Charlie aims to share entanglement with Alice. He then keeps mode $B$ while sending mode $A$ to Alice through the lossy channel $\mathcal{E}_{A}$. (3) \ Charlie aims to share entanglement with Bob. He then keeps mode $A$ while sending mode $B$ to Bob through the lossy channel $\mathcal{E}_{B}$.} \label{twomodeEB} \end{figure} Let us start considering the scenario where Charlie aims to share entanglement with one of the remote parties (one-mode transmission). In particular, suppose that Charlie wants to share entanglement with Bob (by symmetry the derivation is the same if we consider Alice). For sharing entanglement, Charlie keeps mode $A$ while sending mode $B$ to Bob as shown in Fig.~\ref{twomodeEB}(3). The action of the environment is therefore reduced to $\mathcal{I}_{A} \otimes\mathcal{E}_{B}$, where $\mathcal{E}_{B}$ is a lossy channel applied to mode $B$. It is easy to check~\cite{NJP} that the output state $\rho _{AB^{\prime}}$, shared by Charlie and Bob, is Gaussian with zero mean and CM \begin{equation} \mathbf{V}_{AB^{\prime}}=\left( \begin{array} [c]{cc} \mu\mathbf{I} & \mu^{\prime}\sqrt{\tau}\mathbf{Z}\\ \mu^{\prime}\sqrt{\tau}\mathbf{Z} & x\mathbf{I} \end{array} \right) , \label{ABpCM} \end{equation} where \begin{equation} x:=\tau\mu+(1-\tau)\omega~. \end{equation} Remarkably, we can compute closed analytical formulas in the limit of large $\mu$, i.e., large input entanglement. In this case, the entanglement of the output state $\rho_{AB^{\prime}}$ is quantified by the PTS\ eigenvalue \begin{equation} \varepsilon=\frac{1-\tau}{1+\tau}\omega~. \end{equation} The EB condition corresponds to the separability condition $\varepsilon\geq1$, which provides \begin{equation} \omega\geq\frac{1+\tau}{1-\tau}:=\omega_{\text{EB}}~, \label{EBcond} \end{equation} or equivalently $\bar{n}\geq\tau/(1-\tau)$. Despite the EB condition of Eq.~(\ref{EBcond}) regards an EPR\ input, it is valid for any input state. In other words, a lossy channel $\mathcal{E}_{B}$\ with transmissivity $\tau$ and thermal noise $\omega\geq\omega_{\text{EB}}$ destroys the entanglement of any input state $\rho_{AB}$. Indeed Eq.~(\ref{EBcond}) corresponds exactly to the well-known EB condition for lossy channels~\cite{HolevoEB}. The threshold condition $\omega=\omega_{\text{EB}}$ guarantees one-mode EB, i.e., the impossibility for Charlie to share entanglement with the remote party. Now the central question is the following: Suppose that Charlie cannot share any entanglement with the remote parties (one-mode EB), can Charlie still distribute entanglement to them? In other words, suppose that the correlated Gaussian environment has transmissivity $\tau$ and thermal noise $\omega=\omega_{\text{EB}}$, so that the lossy channels $\mathcal{E}_{A}$\ and $\mathcal{E}_{B}$ are EB. Is it still possible to use the joint channel $\mathcal{E}_{AB}$ to distribute entanglement to Alice and Bob? In the following, we explicitly reply to this question, discussing how entanglement can be distributed by a separable environment, with the distributed amount being large enough to be distilled by one-way distillation protocols~\cite{NJP}. Let us study the general evolution of the two modes $A$ and $B$ under the action of the environment as in Fig.~\ref{twomodeEB}(1). Since the input EPR\ state $\rho_{AB}$ is Gaussian and the environmental state $\rho _{E_{1}E_{2}}$ is Gaussian, the output state $\rho_{A^{\prime}B^{\prime}}$ is also Gaussian. This state has zero mean and CM given by~\cite{NJP} \begin{equation} \mathbf{V}_{A^{\prime}B^{\prime}}=\tau\mathbf{V}_{AB}+(1-\tau)\mathbf{V} _{E_{1}E_{2}}=\left( \begin{array} [c]{cc} x\mathbf{I} & \mathbf{H}\\ \mathbf{H} & x\mathbf{I} \end{array} \right) ~, \end{equation} where \begin{equation} \mathbf{H}:=\tau\mu^{\prime}\mathbf{Z}+(1-\tau)\mathbf{G}~. \end{equation} For large $\mu$, one can easily derive the symplectic spectrum of the output state \begin{equation} \nu_{\pm}=\sqrt{\left( 2\omega+g^{\prime}-g\pm|g+g^{\prime}|\right) (1-\tau)\tau\mu}~, \end{equation} and its smallest PTS\ eigenvalue~\cite{NJP} \begin{equation} \varepsilon=(1-\tau)\sqrt{(\omega-g)(\omega+g^{\prime})}~, \label{epsMAIN} \end{equation} quantifying the entanglement distributed to Alice and Bob. In the same limit, one can compute the coherent information~\cite{CohINFO,CohINFO2} $I(A\rangle B)$ between the two remote parties, which provides a lower bound to the number of entanglement bits per copy that can be distilled using one-way distillation protocols, i.e., protocols based on local operations and one-way classical communication. It is clear that one-way distillability implies two-way distillability, where both forward and backward communication is employed. After simple algebra, one achieves~\cite{NJP} \begin{equation} I(A\rangle B)=\log\frac{1}{e\varepsilon}~. \label{coheDIR} \end{equation} Thus, remote entanglement is distributed for $\varepsilon<1$ and is distillable for $\varepsilon<e^{-1}$. Now suppose that the environment has thermal noise $\omega=\omega_{\text{EB}}$ (one-mode EB). Then, we can write \begin{align} \varepsilon & =\sqrt{[1+\tau-(1-\tau)g][1+\tau+(1-\tau)g^{\prime} ]}\nonumber\\ & :=\varepsilon(\tau,g,g^{\prime}) \label{EBentEXP} \end{align} Answering the previous question corresponds to checking the existence of environmental parameters $\tau$, $g$ and $g^{\prime}$, for which\ $\varepsilon $ is sufficiently low: For a given value of the transmissivity $\tau$, we look for regions in the correlation plane $(g,g^{\prime})$ where $\varepsilon<1$ (remote entanglement is distributed) and possibly $\varepsilon<e^{-1}$ (remote entanglement is distillable). This is done in Fig.~\ref{dirTOT} for several numerical values of the transmissivity. \begin{figure}\label{dirTOT} \end{figure} In Fig.~\ref{dirTOT}, the environments identified by the gray activation area allow Charlie to distribute entanglement to Alice and Bob ($\varepsilon<1$), despite it is impossible for him to share entanglement with any of the remote parties. In other words, these environments are two-mode entanglement preserving (EP), despite being one-mode EB. Furthermore, one can identify sufficiently-correlated environments for which the entanglement distributed to the remote parties can also be distilled ($\varepsilon<e^{-1}$). The most remarkable feature in Fig.~\ref{dirTOT} is represented by the presence of separable environments in the activation area. In other words, there are separable environments which contain enough correlations to restore the distribution of entanglement to Alice and Bob. Furthermore, for sufficiently high transmissivities and correlations, these environments enable Charlie to distribute distillable entanglement. As we can note from Fig.~\ref{dirTOT}, the weight of separable environments in the activation area increases for increasing transmissivities, with the entangled environments almost disappearing for $\tau=0.9$. \section{Entanglement swapping in a correlated Gaussian environment\label{SECindirect}} In this section we consider the indirect distribution of entanglement, i.e., the protocol of entanglement swapping. We start with a brief review of this protocol in the ideal case of no noise. Then, we generalize its theory to the case of correlated-noise Gaussian environments, where we prove how entanglement swapping can be reactivated in the presence of one-mode EB. \subsection{Entanglement swapping in the absence of noise\label{SECSUBswap1}} Consider two remote parties, Alice and Bob, who possess two identical EPR states with CM given in Eq.~(\ref{CM_TMSV}). At Alice's station, the EPR\ state describes modes $a$ and $A$, while at Bob's station it describes modes $b$ and $B$. Alice and Bob keep modes $a$ and $b$, while sending modes $A$ and $B$ to Charlie, where a Bell measurement is performed. This means that the travelling modes $A$ and $B$ are combined in a balanced beam splitter whose output modes \textquotedblleft$-$\textquotedblright\ and \textquotedblleft$+$\textquotedblright\ are homodyned, with mode \textquotedblleft$-$\textquotedblright\ measured in the position quadrature and mode \textquotedblleft$+$\textquotedblright\ in the momentum quadrature. In other words, Charlie measures the two EPR quadratures $\hat{q}_{-} :=(\hat{q}_{A}-\hat{q}_{B})/\sqrt{2}$ and $\hat{p}_{+}:=(\hat{p}_{A}+\hat {p}_{B})/\sqrt{2}$. The Bell measurement provides two classical outcomes, $q_{-}$ and $p_{+}$, which can be compacted into a single complex variable $\gamma:=q_{-}+ip_{+}$. The classical variable $\gamma$ is finally communicated to Alice and Bob, with the result of projecting their remote modes $a$ and $b$ into a conditional state $\rho_{ab|\gamma}$ (see Fig.~\ref{swap}). \begin{figure} \caption{Entanglement swapping in the absence of noise. See text for explanation.} \label{swap} \end{figure} Since the input states are pure Gaussian and the Bell measurement is a Gaussian measurement which projects pure states into pure states, we have that the remote conditional state $\rho_{ab|\gamma}$ turns out to be a pure Gaussian state. This state has a measurement-dependent mean $\mathbf{x} =\mathbf{x}(\gamma)$ which Alice and Bob can always delete by conditional displacements. It is clear that these local unitaries do not alter the amount of entanglement in the state, as long as they are perfectly implemented. The conditional CM $\mathbf{V}_{ab|\gamma}$ can be computed using a simple input-output formula for Gaussian entanglement swapping~\cite{GaussSWAP}. We get \begin{equation} \mathbf{V}_{ab|\gamma}=\frac{1}{2\mu}\left( \begin{array} [c]{cc} (\mu^{2}+1)\mathbf{I} & (\mu^{2}-1)\mathbf{Z}\\ (\mu^{2}-1)\mathbf{Z} & (\mu^{2}+1)\mathbf{I} \end{array} \right) ~. \label{CMswapNLess} \end{equation} Its smallest PTS\ eigenvalue is equal to $\varepsilon=\mu^{-1}$, which means that remote entanglement is always generated for entangled inputs ($\mu>1$). Furthermore, remote entanglement is present in the form of EPR correlations since the two remote EPR quadratures $\hat{q}_{-}^{r}:=(\hat{q}_{a}-\hat {q}_{b})/\sqrt{2}$ and $\hat{p}_{+}^{r}:=(\hat{p}_{a}+\hat{p}_{b})/\sqrt{2}$ have variances \begin{equation} V(\hat{q}_{-}^{r})=V(\hat{p}_{+}^{r})=\mu^{-1}~. \end{equation} The simplest description of the entanglement swapping protocol can be given when we consider the limit for $\mu\rightarrow\infty$. In this case the initial states are ideal EPR states with quadratures perfectly correlated, i.e., $\hat{q}_{a}=\hat{q}_{A}$ and $\hat{p}_{a}=-\hat{p}_{A}$ for Alice, and $\hat{q}_{b}=\hat{q}_{B}$ and $\hat{p}_{b}=-\hat{p}_{B}$ for Bob. Then, the overall action of Charlie, i.e., the Bell measurement plus classical communication, corresponds to create a remote state with \begin{equation} \hat{q}_{b}=\hat{q}_{a}-\sqrt{2}q_{-},~\hat{p}_{b}=-\hat{p}_{a}-\sqrt{2} p_{+}~. \end{equation} The quadratures of the two remote modes are perfectly correlated, up to an erasable displacement. In other words, the ideal EPR\ correlations have been swapped from the initial states to the final conditional state $\rho _{ab|\gamma}$. \subsection{Entanglement swapping in the presence of correlated-noise\label{SECSUBswap2}} The theory of entanglement swapping can be extended to include the presence of loss and correlated noise. We consider our model of correlated Gaussian environment with transmission $\tau$, thermal noise $\omega$ and correlations $\mathbf{G}$. The modified scenario is depicted in Fig.~\ref{swapLOSS} .\begin{figure} \caption{Entanglement swapping in the presence of loss, thermal noise and environmental correlations (correlated Gaussian environment). The Bell detector has been simplified.} \label{swapLOSS} \end{figure} \subsubsection{Swapping of EPR\ correlations} For simplicity, we start by studying the evolution of the EPR correlations under ideal input conditions ($\mu\rightarrow+\infty$). After the classical communication of the outcome $\gamma$, the quadratures of the remote modes $a$ and $b$ satisfy the asymptotic relations \begin{align} \hat{q}_{b} & =\hat{q}_{a}-\sqrt{\frac{2}{\tau}}\left( q_{-}-\sqrt{1-\tau }\hat{\delta}_{q}\right) ,\label{eqrrr}\\ \hat{p}_{b} & =-\hat{p}_{a}-\sqrt{\frac{2}{\tau}}\left( p_{+}-\sqrt{1-\tau }\hat{\delta}_{p}\right) , \label{eqrrr2} \end{align} where $\hat{\delta}_{q}=(\hat{q}_{E_{1}}-\hat{q}_{E_{2}})/\sqrt{2}$ and $\hat{\delta}_{p}=(\hat{p}_{E_{1}}+\hat{p}_{E_{2}})/\sqrt{2}$ are noise variables introduced by the environment. Using previous Eqs.~(\ref{eqrrr}) and~(\ref{eqrrr2}), we construct the remote EPR\ quadratures $\hat{q}_{-}^{r}$\ and $\hat{p}_{+}^{r}$, and we compute the EPR variances \begin{equation} \boldsymbol{\Lambda}:=\left( \begin{array} [c]{cc} V(\hat{q}_{-}^{r}) & \\ & V(\hat{p}_{+}^{r}) \end{array} \right) \rightarrow\boldsymbol{\Lambda}_{\infty}=\frac{1-\tau}{\tau} (\omega\mathbf{I}-\mathbf{ZG})~, \label{lam1} \end{equation} where the limit is taken for $\mu\rightarrow+\infty$. Assuming the EB condition $\omega=\omega_{\text{EB}}$, we finally get \begin{equation} \boldsymbol{\Lambda}_{\infty,\text{EB}}=\frac{1}{\tau}\left[ (1+\tau )\mathbf{I}-(1-\tau)\mathbf{ZG}\right] . \label{lam2} \end{equation} In the case of a memoryless environment ($\mathbf{G=0}$) we see that $\boldsymbol{\Lambda}_{\infty,\text{EB}}=(1+\tau^{-1})\mathbf{I}\geq \mathbf{I}$, which means that the EPR\ correlations cannot be swapped to the remote systems. However, it is evident from Eq.~(\ref{lam2}) that there are choices for the correlation block $\mathbf{G}$\ such that the EPR\ condition $\boldsymbol{\Lambda}_{\infty,\text{EB}}<\mathbf{I}$ is satisfied. For instance, this happens when we consider $\mathbf{G=}g\mathbf{Z}$. In this case it is easy to check that $\boldsymbol{\Lambda}_{\infty,\text{EB}}<\mathbf{I}$ is satisfied for $\tau\geq1/4$ and $g>(1-\tau)^{-1}$. Under these conditions, EPR\ correlations are successfully swapped to the remote modes. In particular, for $\tau>1/2$ and $g>(1-\tau)^{-1}$ there are separable environments which do the job. \subsubsection{Swapping and distillation of entanglement} Here we discuss in detail how entanglement is distributed by the swapping protocol in the presence of a correlated Gaussian environment. In particular, suppose that Alice and Bob cannot share entanglement with Charlie because the environment is one-mode EB. Then, we aim to address the following questions: (i)~Is it still possible for Charlie to distribute entanglement to the remote parties thanks to the environmental correlations? (ii)~In particular, is the swapping successful when the environmental correlations are separable? (iii)~Finally, are Alice and Bob able to distill the swapped entanglement by means of one-way distillation protocols? Our previous discussion on EPR correlations clearly suggests that these questions have positive answers. Here we explicitly show this is indeed true for quantum entanglement by finding the typical regimes of parameters that the Gaussian environment must satisfy. In order to study the propagation of entanglement we first need to derive the CM\ $\mathbf{V}_{ab|\gamma}$\ of the conditional remote state $\rho _{ab|\gamma}$. As before, we have two identical EPR\ states at Alice's and Bob's stations with CM $\mathbf{V}(\mu)$ given in Eq.~(\ref{CM_TMSV}). The travelling modes $A$ and $B$ are sent to Charlie through a Gaussian environment with transmissivity $\tau$, thermal noise $\omega$ and correlations $\mathbf{G}$. After the Bell measurement and the classical communication of the result $\gamma$, the conditional remote state at Alice's and Bob's stations is Gaussian with CM~\cite{BellFORMULA} \begin{equation} \mathbf{V}_{ab|\gamma}=\left( \begin{array} [c]{cc} \mu\mathbf{I} & \\ & \mu\mathbf{I} \end{array} \right) -\frac{(\mu^{2}-1)\tau}{2}\left( \begin{array} [c]{cccc} \frac{1}{\theta} & & -\frac{1}{\theta} & \\ & \frac{1}{\theta^{\prime}} & & \frac{1}{\theta^{\prime}}\\ -\frac{1}{\theta} & & \frac{1}{\theta} & \\ & \frac{1}{\theta^{\prime}} & & \frac{1}{\theta^{\prime}} \end{array} \right) ~, \label{VabGamma} \end{equation} where \begin{equation} \theta=\tau\mu+(1-\tau)(\omega-g),~\theta^{\prime}=\tau\mu+(1-\tau )(\omega+g^{\prime})~. \label{thetas} \end{equation} From the CM of Eq.~(\ref{VabGamma}) we compute the smallest PTS eigenvalue $\varepsilon$ quantifying the remote entanglement at Alice's and Bob's stations. For large input entanglement $\mu\gg1$, we find a closed formula in terms of the environmental parameters, i.e., \begin{equation} \varepsilon=\frac{1-\tau}{\tau}\sqrt{(\omega-g)(\omega+g^{\prime} )}:=\varepsilon(\tau,\omega,g,g^{\prime})~, \label{SpectrumTOT} \end{equation} which is equal to Eq.~(\ref{epsMAIN}) up to a factor $\tau^{-1}$. As before, this eigenvalue not only determines the log-negativity but also the coherent information $I(a\rangle b)$\ associated with the remote state $\rho _{ab|\gamma}$. In fact, for large $\mu$, one can easily compute the asymptotic expression \begin{equation} I(a\rangle b)\rightarrow\log\frac{2}{e}\sqrt{\frac{\det\mathbf{V}_{b|\gamma} }{\det\mathbf{V}_{ab|\gamma}}}=\log\frac{1}{e\varepsilon}~, \label{IabCOHEswap} \end{equation} which is identical to the formula of Eq.~(\ref{coheDIR}) for the case of direct distribution. Thus, the PTS\ eigenvalue of Eq.~(\ref{SpectrumTOT}) contains all the information about the distribution and distillation of entanglement in the swapping scenario. For $\varepsilon<1$ entanglement is successfully distributed by the swapping protocol (log-negativity $\mathcal{E}>0$). Then, for the stronger condition $\varepsilon<e^{-1}$, the swapped entanglement can also be distilled into $I(a\rangle b)$ entanglement bits per copy by means of one-way protocols. Now, let us assume the condition of one-mode EB ($\omega=\omega_{\text{EB}}$) so that the bipartite states before measurement $\rho_{aA^{\prime}}$ and $\rho_{B^{\prime}b}$ are separable (see Fig.~\ref{swapLOSS}). We investigate the amount of entanglement generated in the remote modes $a$ and $b$ by computing the eigenvalue $\varepsilon(\tau,\omega_{\text{EB}},g,g^{\prime})$. In the standard memoryless case ($\mathbf{G}=\mathbf{0}$) we have $\varepsilon=1+\tau^{-1}$ which means that no entanglement can be swapped, as expected. To study the general case of correlated environment, we consider different numerical values of the transmissivity $\tau$, and we plot the $\varepsilon(\tau,\omega_{\text{EB}},g,g^{\prime})$ on the correlation plane. The results are shown in Fig.~\ref{total} and are similar to those achieved in Fig.~\ref{dirTOT} for direct distribution. \begin{figure}\label{total} \end{figure} In each panel of Fig.~\ref{total}, the physical values for the correlation parameters $(g,g^{\prime})$ are individuated by the non-black area. Remote entanglement is distributed ($\varepsilon<1$) for values of the correlation parameters belonging to the gray activation area. For $\tau\leq1/2$ (top two panels), we see that the activation area is confined within the region of entangled environments. The property that entangled environments are necessary for the reactivation of entanglement swapping at any $\tau\leq1/2$ is easy to prove. In fact, suppose that $\varepsilon<1$ holds. By using its formula in Eq.~(\ref{SpectrumTOT}) and the bona-fide conditions on the correlation parameters given in Eq.~(\ref{CMconstraints}), we can write $\varepsilon ^{2}<1$ as \begin{equation} \omega^{2}-gg^{\prime}+\omega(g^{\prime}-g)<\left( \frac{\tau}{1-\tau }\right) ^{2}~. \label{eqkk} \end{equation} Now, for $\tau\leq1/2$, we have $\tau^{2}(1-\tau)^{-2}\leq1$ and using this inequality in Eq.~(\ref{eqkk}), we derive \begin{equation} \omega^{2}-gg^{\prime}-1<\omega(g-g^{\prime})\leq\omega|g-g^{\prime}|~, \end{equation} which is the entanglement condition for the environment [i.e., the violation of Eq.~(\ref{sepCON})]. It is clear that the most interesting result holds for transmissivities $\tau>1/2$. In this regime, in fact, the distribution of remote entanglement can be activated by separable environments. As explicitly shown for $\tau=0.75$ and $0.9$, the activation area progressively invades the region of separable environments. In other words, separable correlations become more and more important for increasing transmissivities. Furthermore, for $\tau \gtrsim0.75$, separable environments are even able to activate the distribution of distillable entanglement ($\varepsilon<e^{-1}$). By comparing Fig.~\ref{dirTOT} and Fig.~\ref{total}, we see how entanglement is more easily generated and distilled by the direct protocol. This is a consequence of the extra factor $\tau^{-1}$ in Eq.~(\ref{SpectrumTOT}), whose influence becomes less important only at high transmissivities ($\tau\simeq1$). \section{Conclusion\label{SECconclusion}} In conclusion, we have investigated the distribution of entanglement in the presence of correlated-noise Gaussian environments, proving how the injection of separable correlations can recover from entanglement breaking. In order to derive simple analytical results we have considered here only the case of large entanglement for the input states. We have analyzed scenarios of direct distribution and indirect distribution, i.e., entanglement swapping. Surprisingly, the injection of the weaker separable correlations is sufficient to restore the entanglement distribution, as we have shown for wide regimes of parameters. Furthermore, the generated entanglement can be sufficient to be distilled by means of one-way protocols. The fact that separability can be exploited to recover from entanglement breaking is clearly a paradoxical behavior which poses fundamental questions on the intimate relations between local and nonlocal correlations. \end{document}

\begin{document} \title[Barycentric subdivisions of cubical complexes] {Face numbers of barycentric subdivisions of cubical complexes} \author{Christos~A.~Athanasiadis} \address{Department of Mathematics\\ National and Kapodistrian University of Athens\\ Panepistimioupolis\\ 15784 Athens, Greece} \email{[email protected]} \date{December 18, 2020} \thanks{ 2010 \textit{Mathematics Subject Classification.} Primary 05E45; \, Secondary 26C10, 52B12.} \thanks{ \textit{Key words and phrases}. Barycentric subdivision, cubical complex, $h$-polynomial, Eulerian polynomial, real-rootedness.} \begin{abstract} The $h$-polynomial of the barycentric subdivision of any $n$-dimensional cubical complex with nonnegative cubical $h$-vector is shown to have only real roots and to be interlaced by the Eulerian polynomial of type $B_n$. This result applies to barycentric subdivisions of shellable cubical complexes and, in particular, to barycentric subdivisions of cubical convex polytopes and answers affirmatively a question of Brenti, Mohammadi and Welker. \end{abstract} \maketitle \section{Introduction} \label{sec:intro} A fundamental problem in algebraic and geometric combinatorics is to characterize, or at least obtain significant information about, the face enumerating vectors of triangulations of various topological spaces, such as balls and spheres \cite{StaCCA}. Face enumerating vectors are often presented in the form of the $h$-polynomial (see Section~\ref{sec:enu} for definitions). Properties such as unimodality, log-concavity, $\gamma$-positivity and real-rootedness have been of primary interest \cite{Ath18, Bra15, Bre94b, Sta89}. One expects that the `nicer' the triangulation is combinatorially and the space being triangulated is topologically, the better the behavior of the $h$-polynomial is. Following this line of thought, Brenti and Welker~\cite{BW08} considered an important and well studied triangulation in mathematics, namely the barycentric subdivision. They studied the transformation of the $h$-polynomial of a simplicial complex $\Delta$ under barycentric subdivision and showed that the resulting $h$-polynomial has only real roots (a property with strong implications) for every simplicial complex $\Delta$ with nonnegative $h$-polynomial. They asked \cite[Question~3.10]{BW08} whether the $h$-polynomial of the barycentric subdivision of any convex polytope has only real roots, suspecting an affirmative answer (see \cite[p.~105]{MW17}). This question was raised again by Mohammadi and Welker~\cite[Question~35]{MW17} and, as is typically the case in face enumeration, it is far more interesting and more challenging for general polytopes and polyhedral complexes, than it is for simplicial polytopes and simplicial complexes. Somewhat surprisingly, no strong evidence has been provided in the literature that such a result may (or may not) hold beyond the simplicial setting. One should also note that barycentric subdivisions of boundary complexes of polytopes form a special class of flag triangulations of spheres and that the real-rootedness property fails for the $h$-polynomials of this more general class of triangulations in dimensions higher that four \cite{Ga05}. At present, it is unclear where the borderline between positive and negative results lies. Mohammadi and Welker (based on earlier discussions with Brenti) suggested the class of cubical polytopes as another good test case; see \cite[p.~105]{MW17}. Cubical complexes and polytopes are important and mysterious objects with highly nontrivial combinatorial properties (see, for instance, \cite{Ad96, BBC97, Jo93, JZ00}). They have been studied both for their own independent interest, and for the role they play in other areas of mathematics. Given the intricacy of their combinatorics, it comes as no surprise that the question of Brenti and Welker turns out to be more difficult for them than for simplicial complexes. The following theorem provides the first general positive result on this question, since \cite{BW08} appeared, and suggests that an affirmative answer should be expected at least for broad classes of nonsimplicial convex polytopes (or even more general cell complexes and posets). \begin{theorem} \label{thm:main} The $h$-polynomial of the barycentric subdivision of any shellable cubical complex has only real roots. In particular, barycentric subdivisions of cubical polytopes have this property. \end{theorem} The case of cubical polytopes was also studied recently by Hlavacex and Solus~\cite{HS20+}. Using the concept of shellability and the theory of interlacing polynomials, they gave an affirmative answer for cubical complexes which admit a special type of shelling and applied their result to certain families of cubical polytopes, such as cuboids, capped cubical polytopes and neighborly cubical polytopes. The proof of the aforementioned result of~\cite{BW08} applies a theorem of Br\"and\'en~\cite{Bra06} on the subdivision operator \cite[Section~3.3]{Bra15} to a formula for the $h$-polynomial of the barycentric subdivision of a simplicial complex (see Remark~\ref{rem:BW-formula}). The proof of Theorem~\ref{thm:main} is motivated by the proof of the result of~\cite{BW08}, given and extended to the setting of uniform triangulations of simplicial complexes in~\cite{Ath20+} (the latter was partially motivated by \cite[Example~8.1]{Bra15}). To explain further, we let $h(\Delta, x) = \sum_{i=0}^{n+1} h_i(\Delta) x^i$ denote the $h$-polynomial and ${\rm sd}(\Delta)$ denote the barycentric subdivision of an $n$-dimensional simplicial complex $\Delta$. As already shown in~\cite{BW08}, there exist polynomials with nonnegative coefficients $p_{n,k}(x)$ for $k \in \{0, 1,\dots,n+1\}$, which depend only on $n$ and $k$, such that \begin{equation} \label{eq:BW} h({\rm sd}(\Delta), x) \ = \ \sum_{k=0}^{n+1} h_k(\Delta) p_{n,k}(x) \end{equation} for every $n$-dimensional simplicial complex $\Delta$. For every $n \in {\mathbb N}$, the polynomials $p_{n,k}(x)$ can be shown \cite[Example~8.1]{Bra15} to have only real roots and to form an interlacing sequence. This implies that their nonnegative linear combination $h({\rm sd}(\Delta), x)$ also has only real (negative) roots and that it is interlaced by $p_{n,0}(x)$, which equals the classical $(n+1)$st Eulerian polynomial $A_{n+1}(x)$ \cite[Section~1.4]{StaEC1}. The interlacing condition implies that the roots of $h({\rm sd}(\Delta), x)$ are not arbitrary, but rather that they lie in certain intervals that depend only on the dimension $n$, formed by zero and the roots of $A_{n+1}(x)$. The polynomial $p_{n,k}(x)$ can be interpreted as the $h$-polynomial of the relative simplicial complex obtained from the barycentric subdivision of the $n$-dimensional simplex by removing all faces lying on $k$ facets of the simplex \cite[Section~5]{Ath20+} \cite[Section~4.2]{HS20+}. This paper presents a similar picture for cubical complexes. We define (see Definition~\ref{def:pBnk}) polynomials $p^B_{n,k}(x)$ for $k \in \{0, 1,\dots,n+1\}$ as the $h$-polynomials of relative simplicial complexes obtained from the barycentric subdivision of the $n$-dimensional cube by removing all faces lying on certain facets of the cube and prove (see Theorem~\ref{thm:h-trans}) that Equation~(\ref{eq:BW}) continues to hold when $\Delta$ is replaced by an $n$-dimensional cubical complex ${\mathcal L}$, $p_{n,k}(x)$ is replaced by $p^B_{n,k}(x)$ and the $h_k(\Delta)$ are replaced by the entries of the (normalized) cubical $h$-vector of ${\mathcal L}$, introduced and studied by Adin~\cite{Ad96}. We provide recurrences (see Proposition~\ref{prop:pBnk}) for the polynomials $p^B_{n,k}(x)$ which guarantee that they form an interlacing sequence for every $n \in {\mathbb N}$ and conclude that $h({\rm sd}({\mathcal L}), x)$ has only real (negative) roots and that it is interlaced by the $n$th Eulerian polynomial $B_n(x)$ of type $B$ for every $n$-dimensional cubical complex ${\mathcal L}$ with nonnegative cubical $h$-vector (see Corollary~\ref{cor:main}). This implies Theorem~\ref{thm:main}, since shellable cubical complexes were shown~\cite{Ad96} to have nonnegative cubical $h$-vector and boundary complexes of convex polytopes are shellable~\cite{BM71}. The main results of this paper apply to cubical regular cell complexes (equivalently, to cubical posets) and will be stated at this level of generality. What comes perhaps unexpectedly is the fact that the transformation of a cubical $h$-polynomial into a simplicial one can be so well behaved. Corollary~\ref{cor:main} has nontrivial applications to triangulations of simplicial complexes as well; see Remark~\ref{rem:simplicial}. \section{Face enumeration of simplicial and cubical complexes} \label{sec:enu} This section recalls some definitions and background on the face enumeration of simplicial and cubical complexes and their triangulations, and shellability. For more information and any undefined terminology, we recommend the books \cite{HiAC, StaCCA}. All cell complexes considered here are assumed to be finite. Throughout this paper, we set ${\mathbb N} = \{0, 1, 2,\dots\}$ and denote by $|S|$ the cardinality of a finite set $S$. \subsection{Simplicial complexes} \label{sec:simplicial} An $n$-dimensional \emph{relative simplicial complex} \cite[Section~III.7]{StaCCA} is a pair $(\Delta, \Gamma)$, denoted $\Delta / \Gamma$, where $\Delta$ is an (abstract) $n$-dimensional simplicial complex and $\Gamma$ is a subcomplex of $\Delta$. The \emph{$f$-polynomial} of $\Delta / \Gamma$ is defined as \[ f(\Delta / \Gamma, x) \ = \ \sum_{i=0}^{n+1} f_{i-1} (\Delta / \Gamma) x^i, \] where $f_j(\Delta / \Gamma)$ is the number of $j$-dimensional faces of $\Delta$ which do not belong to $\Gamma$. The \emph{$h$-polynomial} is defined as \begin{eqnarray*} \label{def:simpl-h} h(\Delta/\Gamma, x) & = & (1-x)^{n+1} f(\Delta/\Gamma, \frac{x}{1-x}) \ = \ \sum_{i=0}^{n+1} f_{i-1} (\Delta/\Gamma) \, x^i (1-x)^{n+1-i} \\ & = & \sum_{F \in \Delta/\Gamma} x^{|F|} (1-x)^{n+1-|F|} \ := \ \sum_{k=0}^{n+1} \, h_k (\Delta/\Gamma) x^k \nonumber \end{eqnarray*} \noindent and the sequence $h(\Delta/\Gamma) := (h_0(\Delta/\Gamma), h_1(\Delta/\Gamma),\dots,h_{n+1}(\Delta/\Gamma))$ is called the \emph{$h$-vector} of $\Delta/\Gamma$. Note that $f(\Delta / \Gamma, x)$ has only real roots if and only if so does $h(\Delta/\Gamma, x)$. When $\Gamma$ is empty, we get the corresponding invariants of $\Delta$ and drop $\Gamma$ from the notation. Thus, for example, $h(\Delta, x) = \sum_{k=0}^{n+1} h_k (\Delta) x^k$ is the (usual) $h$-polynomial of $\Delta$. Suppose now that $\Delta$ triangulates an $n$-dimensional ball. Then, the boundary complex $\partial \Delta$ is a triangulation of an $(n-1)$-dimensional sphere and the \emph{interior $h$-polynomial} of $\Delta$ is defined as $h^\circ(\Delta, x) = h(\Delta / \partial \Delta, x)$. The following statement is a special case of \cite[Lemma~6.2]{Sta87}. \begin{proposition} \label{prop:hsymmetry} {\rm (\cite{Sta87})} Let $\Delta$ be a triangulation of an $n$-dimensional ball. Let $\Gamma$ be a subcomplex of $\partial\Delta$ which is homeomorphic to an $(n-1)$-dimensional ball and $\bar{\Gamma}$ be the subcomplex of $\partial\Delta$ whose facets are those of $\partial\Delta$ which do not belong to $\Gamma$. Then, \[ h(\Delta / \bar{\Gamma}, x) \ = \ x^{n+1} h(\Delta / \Gamma, 1/x) . \] Moreover, $h^\circ(\Delta, x) = x^{n+1} h(\Delta, 1/x)$. \end{proposition} \subsection{Cubical complexes} \label{sec:cubical} A \emph{regular cell complex} \cite[Section 4.7]{OM} is a (finite) collection ${\mathcal L}$ of subspaces of a Hausdorff space $X$, called \emph{cells} or \emph{faces}, each homeomorphic to a closed unit ball in some finite-dimensional Euclidean space, such that: (a) $\varnothing \in {\mathcal L}$; (b) the relative interiors of the nonempty cells partition $X$; and (c) the boundary of any cell in ${\mathcal L}$ is a union of cells in ${\mathcal L}$. The \emph{boundary complex} of $\sigma \in {\mathcal L}$, denoted by $\partial \sigma$, is defined as the regular cell complex consisting of all faces of ${\mathcal L}$ properly contained in $\sigma$. A regular cell complex ${\mathcal L}$ is called \emph{cubical} if every nonempty face of ${\mathcal L}$ is combinatorially isomorphic to a cube. A convex polytope is called \emph{cubical} if so is its boundary complex. Given a cubical complex ${\mathcal L}$ of dimension $n$, we denote by $f_k({\mathcal L})$ the number of $k$-dimensional faces of ${\mathcal L}$. The cubical $h$-polynomial was introduced and studied by Adin~\cite{Ad96} as a (well behaved) analogue of the (simplicial) $h$-polynomial of a simplicial complex. Following \cite[Section~4]{EH00}, we define the (normalized) \emph{cubical $h$-polynomial} of ${\mathcal L}$ as \begin{equation} \label{def:cub-h} (1+x) h({\mathcal L}, x) \ = \ 1 \, + \, \sum_{k=0}^n \, f_k({\mathcal L}) \, x^{k+1} \left( \frac{1-x}{2} \right)^{n-k} + \ (-1)^n \, \widetilde{\chi} ({\mathcal L}) x^{n+2} , \end{equation} where $\widetilde{\chi} ({\mathcal L}) = -1 + \sum_{k=0}^n (-1)^k f_k({\mathcal L})$ is the reduced Euler characteristic of ${\mathcal L}$ (the only difference from Adin's definition is that all coefficients of $h({\mathcal L}, x)$ have been divided by $2^n$ and, therefore, are not necessarily integers). We note that $h({\mathcal L}, x)$ is indeed a polynomial in $x$ of degree at most $n+1$. The (normalized) \emph{cubical $h$-vector} of ${\mathcal L}$ is the sequence $h({\mathcal L}) = (h_0({\mathcal L}), h_1({\mathcal L}),\dots,h_{n+1} ({\mathcal L}))$, where $h({\mathcal L}, x) = \sum_{k=0}^{n+1} h_k({\mathcal L}) x^k$. Adin showed that $h({\mathcal L}, x)$ has nonnegative coefficients for every shellable cubical complex ${\mathcal L}$ \cite[Theorem~5~(iii)]{Ad96} (his result is stated for abstract cubical complexes with the intersection property, but the proof is valid without assuming the later). He asked whether the same holds whenever ${\mathcal L}$ is Cohen--Macaulay \cite[Question~1]{Ad96}. The coefficient $h_k({\mathcal L})$ is known to be nonnegative for every Cohen--Macaulay ${\mathcal L}$ for $k \in \{0, 1\}$, since $h_0({\mathcal L}) = 1$ and $h_1({\mathcal L}) = (f_0({\mathcal L}) - 2^n)/2^n$, for $k=n$ \cite[Corollary~1.2]{Ath12} and for $k = n+1$, since $h_{n+1}({\mathcal L}) = (-1)^n \widetilde{\chi} ({\mathcal L})$, and for every $k$ in the special case that ${\mathcal L}$ is the cubical barycentric subdivision of a Cohen--Macaulay simplicial complex \cite{Het96} (see also Remark~\ref{rem:simplicial}). \subsection{Barycentric subdivision and shellability} \label{sec:shell} The \emph{barycentric subdivision} of a regular cell complex ${\mathcal L}$ is denoted by ${\rm sd}({\mathcal L})$ and defined as the abstract simplicial complex whose faces are the chains $\sigma_0 \subset \sigma_1 \subset \cdots \subset \sigma_k$ of nonempty faces of ${\mathcal L}$. The natural restriction of ${\rm sd}({\mathcal L})$ to a nonempty face $\sigma \in {\mathcal L}$ is exactly the barycentric subdivision ${\rm sd}(\sigma)$ of (the complex of faces of) $\sigma$. Similarly, by the barycentric subdivision ${\rm sd}(Q)$ of a convex polytope $Q$ we mean that of the complex of faces of $Q$. Since ${\rm sd}(Q)$ is a cone over ${\rm sd}(\partial Q)$, we have $h({\rm sd}(Q), x) = h({\rm sd}(\partial Q), x)$. For the $n$-dimensional cube $Q$ we have $h({\rm sd}(Q), x) = B_n(x)$, where $B_n(x)$ is the Eulerian polynomial which counts signed permutations of $\{1, 2,\dots,n\}$ by the number of descents of type $B$; see, for instance, \cite[Chapter~11]{Pet15}. The following well known type $B$ analogue of Worpitzky's identity \cite[Equation~(13.3)]{Pet15} \begin{equation} \label{eq:Bn-gen} \frac{B_n(x)}{(1-x)^{n+1}} \ = \ \sum_{m \ge 0} \, (2m+1)^n x^m \end{equation} will make computations in the following section easier. A regular cell complex ${\mathcal L}$ is called \emph{pure} if all its facets (faces which are maximal with respect to inclusion) have the same dimension. Such a complex ${\mathcal L}$ is called \emph{shellable} if either it is zero-dimensional, or else there exists a linear ordering $\tau_1, \tau_2,\dots,\tau_m$ of its facets, called a \emph{shelling}, such that (a) $\partial \tau_1$ is shellable; and (b) for $2 \le j \le m$, the complex of faces of $\partial \tau_j$ which are contained in $\tau_1 \cup \tau_2 \cup \cdots \cup \tau_{j-1}$ is pure, of the same dimension as $\partial \tau_j$, and there exists a shelling of $\partial \tau_j$ for which the facets of $\partial \tau_j$ contained in $\tau_1 \cup \tau_2 \cup \cdots \cup \tau_{j-1}$ form an initial segment. A fundamental result of Bruggesser and Mani \cite{BM71} states that $\partial Q$ is shellable for every convex polytope $Q$. For the shellability of cubical complexes in particular, see \cite[Section~3]{EH00} \cite[Section~3]{HS20+}. \section{The $h$-vector transformation} \label{sec:h-trans} This section studies the transformation which maps the cubical $h$-vector of a cubical complex ${\mathcal L}$ to the (simplicial) $h$-vector of the barycentric subdivision ${\rm sd}({\mathcal L})$ and deduces Theorem~\ref{thm:main} from its properties. We begin with an important definition. \begin{definition} \label{def:pBnk} For $n \in {\mathbb N}$ and $k \in \{0, 1,\dots,n+1\}$ we denote by ${\mathcal C}_{n,k}$ the relative simplicial complex which is obtained from the barycentric subdivision of the $n$-dimensional cube by removing \begin{itemize} \item[$\bullet$] no face, if $k=0$, \item[$\bullet$] all faces which lie in one facet and $k-1$ pairs of antipodal facets of the cube (making a total of $2k-1$ facets), if $k \in \{1, 2,\dots,n\}$, \item[$\bullet$] all faces on the boundary of the cube, if $k=n+1$. \end{itemize} We define $p^B_{n,k}(x) = h({\mathcal C}_{n,k}, x)$ for $k \in \{0, n+1\}$, and $p^B_{n,k}(x) = 2 h({\mathcal C}_{n,k}, x)$ for $k \in \{1, 2,\dots,n\}$. \end{definition} The polynomials $p^B_{n,k}(x)$ are shown on Table~\ref{tab:pBnk} for $n \le 3$. For $n=4$, \[ p^B_{4,k}(x) \ = \ \begin{cases} 1 + 76x + 230x^2 + 76x^3 + x^4, & \text{if $k = 0$,} \\ 108x + 460x^2 + 196x^3 + 4x^4, & \text{if $k = 1$,} \\ 36x + 420x^2 + 300x^3 + 12x^4, & \text{if $k=2$,} \\ 12x + 300x^2 + 420x^3 + 36x^4, & \text{if $k=3$,} \\ 4x + 196x^2 + 460x^3 + 108x^4, & \text{if $k=4$,} \\ x + 76x^2 + 230x^3 + 76x^4 + x^5, & \text{if $k=5$}. \end{cases} \] \noindent Their significance stems from the following theorem. \begin{theorem} \label{thm:h-trans} For every $n$-dimensional cubical complex ${\mathcal L}$, \[ h({\rm sd}({\mathcal L}), x) \ = \ \sum_{k=0}^{n+1} h_k({\mathcal L}) p^B_{n,k}(x) . \] \end{theorem} {\scriptsize \begin{table}[hptb] \begin{center} \begin{tabular}{| l || l | l | l | l | l | l ||} \hline & $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ \\ \hline \hline $n=0$ & 1 & $x$ & & & \\ \hline $n=1$ & $1+x$ & $4x$ & $x+x^2$ & & \\ \hline $n=2$ & $1+6x+x^2$ & $12x+4x^2$ & $4x+12x^2$ & $x+6x^2+x^3$ & \\ \hline $n=3$ & $1+23x+23x^2+x^3$ & $36x+56x^2+4x^3$ & $12x+72x^2+12x^3$ & $4x+56x^2+36x^3$ & $x+23x^2+23x^3+x^4$ \\ \hline \end{tabular} \caption{The polynomials $p^B_{n,k}(x)$ for $n \le 3$.} \label{tab:pBnk} \end{center} \end{table}} The proof requires a few preliminary results. We first summarize some of the main properties of $p^B_{n,k}(x)$. \begin{proposition} \label{prop:pBnk} For every $n \in {\mathbb N}$: \begin{itemize} \itemsep=0pt \item[{\rm (a)}] The polynomial $p^B_{n,k}(x)$ has nonnegative coefficients for every $k \in \{0, 1,\dots,n+1\}$; its degree is equal to $n+1$, if $k = n+1$, and to $n$ otherwise. \item[{\rm (b)}] $p^B_{n,n+1-k}(x) = x^{n+1}p^B_{n,k}(1/x)$ for every $k \in \{0, 1,\dots,n+1\}$. \item[{\rm (c)}] $p^B_{n,0}(x) = B_n(x)$, $p^B_{n,n+1}(x) = xB_n(x)$ and $\sum_{k=0}^{n+1} p^B_{n,k}(x) = B_{n+1}(x)$. \item[{\rm (d)}] We have \[ p^B_{n+1,k+1}(x) \ = \ \begin{cases} 2 p^B_{n+1,0}(x) + 2 (x-1) p^B_{n,0}(x), & \text{if $k = 0$,} \\ p^B_{n+1,k}(x) + 2(x-1) p^B_{n,k}(x), & \text{if $1 \le k \le n$,} \\ (1/2) \cdot p^B_{n+1,n+1}(x) + (x-1) p^B_{n,n+1}(x), & \text{if $k=n+1$} . \end{cases} \] \item[{\rm (e)}] The recurrence \[ p^B_{n+1,k}(x) \ = \ \begin{cases} {\displaystyle \sum_{i=0}^{n+1} p^B_{n,i}(x)}, & \text{if $k = 0$}, \\ {\displaystyle 2x \sum_{i=0}^{k-1} p^B_{n,i}(x) \, + \, 2 \sum_{i=k}^{n+1} p^B_{n,i}(x)}, & \text{if $1 \le k \le n+1$}, \\ {\displaystyle x \sum_{i=0}^{n+1} p^B_{n,i}(x)}, & \text{if $k=n+2$} \end{cases} \] holds for $k \in \{0, 1,\dots,n+1\}$. \item[{\rm (f)}] We have \[ \frac{p^B_{n,k}(x)}{(1-x)^{n+1}} \ = \ \begin{cases} {\displaystyle \sum_{m \ge 0} \, (2m+1)^n x^m}, & \text{if $k = 0$,} \\ {\displaystyle \sum_{m \ge 0} \, (4m) (2m-1)^{k-1} (2m+1)^{n-k} x^m}, & \text{if $1 \le k \le n$,} \\ {\displaystyle \sum_{m \ge 1} \, (2m-1)^n x^m}, & \text{if $k=n+1$}. \end{cases} \] \end{itemize} \end{proposition} \begin{proof} We first note that, as discussed in Section~\ref{sec:shell}, $p^B_{n,0}(x) = h({\mathcal C}_{n,0},x) = B_n(x)$. Part (d) follows from Definition~\ref{def:pBnk} and the definition of the $h$-polynomial of a relative simplicial complex. Indeed, for $1 \le k \le n$, we have $f({\mathcal C}_{n+1,k+1}, x) = f({\mathcal C}_{n+1,k}, x) - 2 f({\mathcal C}_{n,k}, x)$. Hence, \begin{eqnarray*} h({\mathcal C}_{n+1,k+1}, x) & = & (1-x)^{n+2} f({\mathcal C}_{n+1,k+1}, \frac{x}{1-x}) \\ & & \\ & = & (1-x)^{n+2} f({\mathcal C}_{n+1,k}, \frac{x}{1-x}) \, - \, 2 (1-x) \cdot (1-x)^{n+1} f({\mathcal C}_{n,k}, \frac{x}{1-x}) \\ & & \\ & = & h({\mathcal C}_{n+1,k}, x) \, + \, 2(x-1) h({\mathcal C}_{n,k}, x) \end{eqnarray*} \noindent and \begin{eqnarray*} p^B_{n+1,k+1}(x) & = & 2 h({\mathcal C}_{n+1,k+1}, x) \ = \ 2 h({\mathcal C}_{n+1,k}, x) \, + \, 4 (x-1) h({\mathcal C}_{n,k}, x) \\ & = & p^B_{n+1,k}(x) \, + \, 2 (x-1) p^B_{n,k}(x). \end{eqnarray*} \noindent The same argument, similar to that in the proof of \cite[Corollary~5.6]{Ath20+}, works for $k \in \{0, n+1\}$. Part (f) follows from part (d) by straightforward induction on $k$ (for fixed $n$), where the base $k=0$ of the induction holds because of Equation~(\ref{eq:Bn-gen}). For part (c) we first note that $p^B_{n,n+1}(x) = h({\mathcal C}_{n,n+1},x) = h^\circ({\mathcal C}_{n,0}, x) = x^{n+1} h({\mathcal C}_{n,0}, 1/x) = x^{n+1} B_n(1/x) = xB_n(x)$. The identity for the sum of the $p^B_{n,k}(x)$ can be verified directly by summing that of part (f). For a more conceptual proof, one can use an obvious shelling of the boundary complex of the $(n+1)$-dimensional cube to write, as explained in \cite[Section~3]{HS20+}, the $h$-polynomial $B_{n+1}(x)$ of its barycentric subdivision as a sum of $h$-polynomials of relative simplicial complexes, each one combinatorially isomorphic to one of the ${\mathcal C}_{n,k}$. The details are left to the interested reader. Given (c), the recursion of part (e) follows easily by induction on $k$ from part (d) (this parallels the proof of \cite[Lemma~6.3]{Ath20+}). Part (b) is a consequence of Definition~\ref{def:pBnk} and Proposition~\ref{prop:hsymmetry}. Alternatively, it follows from part (f) and standard facts about rational generating functions; see \cite[Proposition~4.2.3]{StaEC1}. The nonnegativity of the coefficients of $p^B_{n,k}(x)$, claimed in part (a), follows from the recursion of part (e), as well as from general results \cite[Corollary~III.7.3]{StaCCA} on the nonnegativity of $h$-vectors of Cohen--Macaulay relative simplicial complexes. The statement about the degree of $p^B_{n,k}(x)$, claimed there, follows from either of parts (d), (e) or (f). \end{proof} We leave the problem to find a combinatorial interpretation of $p^B_{n,k}(x)$ open. Given part (c) of the proposition, one naturally expects that there is such an interpretation which refines one of the known combinatorial interpretations of $B_{n+1}(x)$. The following statement is a consequence of a more general result \cite[Proposition~7.6]{Sta92} of Stanley on subdivisions of CW-posets. To keep this paper self-contained, we include a proof. \begin{proposition} \label{prop:sdc-h} For every $n$-dimensional cubical complex ${\mathcal L}$, \[ h({\rm sd}({\mathcal L}), x) \ = \ (1-x)^{n+1} \, + \, x \, \sum_{k=0}^n f_k({\mathcal L}) (1-x)^{n-k} B_k(x) . \] \end{proposition} \begin{proof} Since every face of ${\rm sd}({\mathcal L})$ is an interior face of the restriction ${\rm sd}(\sigma)$ of ${\rm sd}({\mathcal L})$ to a unique face $\sigma \in {\mathcal L}$, we have \[ f({\rm sd}({\mathcal L}), x) \ = \ \sum_{\sigma \in {\mathcal L}} f^\circ ({\rm sd}(\sigma), x) \ = \ 1 \, + \sum_{\sigma \in {\mathcal L} {\smallsetminus} \{\varnothing\}} f^\circ({\rm sd}(\sigma), x) . \] Transforming $f$-polynomials to $h$-polynomials in this equation and recalling from Section~\ref{sec:enu} that $h^\circ({\rm sd}(\sigma), x) = x^{k+1} h({\rm sd}(\sigma), 1/x) = x^{k+1} B_k(1/x) = x B_k(x)$ for every nonempty $k$-dimensional face $\sigma \in {\mathcal L}$, we get \begin{eqnarray*} h({\rm sd}({\mathcal L}), x) & = & (1-x)^{n+1} f({\rm sd}({\mathcal L}), \frac{x}{1-x}) \\ & = & (1-x)^{n+1} \ + \sum_{\sigma \in {\mathcal L} {\smallsetminus} \{\varnothing\}} (1-x)^{n+1} f^\circ({\rm sd}(\sigma), \frac{x}{1-x}) \\ & = & (1-x)^{n+1} \ + \sum_{\sigma \in {\mathcal L} {\smallsetminus} \{\varnothing\}} (1-x)^{n-\dim(\sigma)} \, h^\circ({\rm sd}(\sigma), x) \\ & = & (1-x)^{n+1} \, + \ \sum_{k=0}^n f_k({\mathcal L}) (1-x)^{n-k} x B_k(x) \end{eqnarray*} and the proof follows. \end{proof} \noindent \emph{Proof of Theorem~\ref{thm:h-trans}}. Let us denote by $p({\mathcal L}, x)$ the right-hand side of the desired equality. Clearly, it suffices to show that $h({\rm sd}({\mathcal L}), x)/(1-x)^{n+1} = p({\mathcal L}, x)/(1-x)^{n+1}$. From Proposition~\ref{prop:sdc-h} and Equation~(\ref{eq:Bn-gen}) we deduce that \begin{eqnarray*} \frac{h({\rm sd}({\mathcal L}), x)}{(1-x)^{n+1}} & = & 1 \, + \, x \, \sum_{k=0}^n f_k({\mathcal L}) \, \frac{B_k(x)}{(1-x)^{k+1}} \ = \ 1 \, + \, \sum_{m \ge 0} \left( \, \sum_{k=0}^n f_k({\mathcal L}) (2m+1)^k \right) x^{m+1} \nonumber \\ & = & 1 \, + \, \sum_{m \ge 1} \left( \, \sum_{k=0}^n f_k({\mathcal L}) (2m-1)^k \right) x^m . \end{eqnarray*} Similarly, from part (f) of Proposition~\ref{prop:pBnk} we get \[ \frac{p({\mathcal L}, x)}{(1-x)^{n+1}} \ = \ \sum_{k=0}^{n+1} h_k({\mathcal L}) \, \frac{p^B_{n,k}(x)}{(1-x)^{n+1}} \ = \ 1 \, + \, \sum_{m \ge 1} a_{\mathcal L}(m) x^m, \] where \[ a_{\mathcal L}(y) \ := \ h_0({\mathcal L}) (2y+1)^n \, + \, \sum_{k=1}^n h_k({\mathcal L}) (4y) (2y-1)^{k-1} (2y+1)^{n-k} \, + \, h_{n+1}({\mathcal L}) (2y-1)^n. \] Thus, it remains to show that \begin{equation} \label{eq:final} \sum_{k=0}^n f_k({\mathcal L}) (2y-1)^k \ = \ a_{\mathcal L}(y) . \end{equation} We claim that this is, essentially, the defining Equation~(\ref{def:cub-h}) of the cubical $h$-polynomial of ${\mathcal L}$ in disguised form. Indeed, cancelling first the summand $1 + h_{n+1}({\mathcal L}) x^{n+2} = 1 + (-1)^n \widetilde{\chi} ({\mathcal L}) x^{n+2}$, and then a factor of $x$, from both sides of (\ref{def:cub-h}) gives that \[ \sum_{k=0}^n (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) x^k \ = \ \left( \frac{1-x}{2} \right)^n \, \sum_{k=0}^n f_k({\mathcal L}) \left( \frac{2x}{1-x} \right)^ k . \] Setting $2x/(1-x) = 2y-1$, so that $x = (2y-1)/(2y+1)$ and $(1-x)/2 = 1/(2y+1)$, the previous identity can be rewritten as \begin{equation} \label{def:cub-h2} \sum_{k=0}^n f_k({\mathcal L}) (2y-1)^k \ = \ \sum_{k=0}^n \, (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) (2y-1)^k (2y+1)^{n-k} . \end{equation} Since the right-hand side is readily equal to $a_{\mathcal L}(y)$, this proves Equation~(\ref{eq:final}) and the theorem as well. \qed To deduce Theorem~\ref{thm:main} from Theorem~\ref{thm:h-trans} and Proposition~\ref{prop:pBnk}, we need to recall a few definitions and facts from the theory of interlacing polynomials; for more information, see \cite[Section~8]{Bra15} and references therein. A polynomial $p(x) \in {\mathbb R}[x]$ is called \emph{real-rooted} if either it is the zero polynomial, or every complex root of $p(x)$ is real. Given two real-rooted polynomials $p(x), q(x) \in {\mathbb R}[x]$, we say that $p(x)$ \emph{interlaces} $q(x)$ if the roots $\{\alpha_i\}$ of $p(x)$ interlace (or alternate to the left of) the roots $\{\beta_j\}$ of $q(x)$, in the sense that they can be listed as \[ \cdots \le \beta_3 \le \alpha_2 \le \beta_2 \le \alpha_1 \le \beta_1 \le 0. \] A sequence $(p_0(x), p_1(x),\dots,p_n(x))$ of real-rooted polynomials is called \emph{interlacing} if $p_i(x)$ interlaces $p_j(x)$ for all $0 \le i < j \le n$. Assuming also that these polynomials have positive leading coefficients, every nonnegative linear combination of $p_0(x), p_1(x),\dots,p_n(x)$ is real-rooted and interlaced by $p_0(x)$. A standard way to produce interlacing sequences in combinatorics is the following. Suppose that $p_0(x), p_1(x),\dots,p_n(x)$ are real-rooted polynomials with nonnegative coefficients and set \[ q_k(x) \ = \ x \sum_{i=0}^{k-1} p_i(x) \, + \, \sum_{i=k}^n p_i(x) \] for $k \in \{0, 1,\dots,n+1\}$. Then, if the sequence $(p_0(x), p_1(x),\dots,p_n(x))$ is interlacing, so is $(q_0(x), q_1(x),\dots,q_{n+1}(x))$; see \cite[Corollary~8.7]{Bra15} for a more general statement. The following result is a stronger version of Theorem~\ref{thm:main}. \begin{corollary} \label{cor:main} The polynomial $h({\rm sd}({\mathcal L}), x)$ is real-rooted and interlaced by the Eulerian polynomial $B_n(x)$ for every $n$-dimensional cubical complex ${\mathcal L}$ with nonnegative cubical $h$-vector. In particular, $h({\rm sd}({\mathcal L}), x)$ and $h({\rm sd}(Q), x)$ are real-rooted and interlaced by $B_n(x)$ for every shellable, $n$-dimensional cubical complex ${\mathcal L}$ and every cubical polytope $Q$ of dimension $n+1$, respectively. \end{corollary} \begin{proof} By an application of the lemma on interlacing sequences just discussed, the recurrence of part (e) of Proposition~\ref{prop:pBnk} implies that $(p^B_{n,0}(x), p^B_{n,1}(x),\dots,p^B_{n,n+1}(x))$ is interlacing for every $n \in {\mathbb N}$ by induction on $n$. Therefore, being a nonnegative linear combination of the elements of the sequence by Theorem~\ref{thm:h-trans}, $h({\rm sd}({\mathcal L}), x)$ is real-rooted and interlaced by $p^B_{n,0}(x) = B_n(x)$ for every $n$-dimensional cubical complex ${\mathcal L}$ with nonnegative cubical $h$-vector. This proves the first statement. The second statement follows from the first since shellable cubical complexes are known to have nonnegative cubical $h$-vector \cite[Theorem~5~(iii)]{Ad96}, $h({\rm sd}(Q), x) = h({\rm sd}(\partial Q), x)$ for every convex polytope $Q$ and because boundary complexes of polytopes are shellable. \end{proof} \begin{remark} \label{rem:simplicial} \rm Let $\Delta$ be a simplicial complex with nonnegative $h$-vector and ${\mathcal L}$ be a cubical complex which is obtained from $\Delta$ by any operation which preserves nonnegativity of $h$-vectors. Corollary~\ref{cor:main} implies that $h({\rm sd}({\mathcal L}), x)$ is real-rooted. By a result of Hetyei~\cite{Het96}, such an operation is the cubical barycentric subdivision ${\mathcal L} = {\rm sd}_c(\Delta)$ (see \cite[p.~44]{Ath18}), also known as barycentric cover \cite[Section~2.3]{BBC97}, of $\Delta$. Then, ${\rm sd}({\mathcal L})$ becomes the interval triangulation of $\Delta$ \cite[Section~3.3]{MW17}. This argument shows that the interval triangulation of $\Delta$ has a real-rooted $h$-polynomial for every simplicial complex $\Delta$ with nonnegative $h$-vector and answers in the affirmative the question of \cite[Problem~33]{MW17}. Although there are other proofs of this fact in the literature (see \cite{Ath20+} and references therein), the approach via Corollary~\ref{cor:main} allows for more general results, e.g., by applying further cubical subdivisions of ${\rm sd}_c(\Delta)$ which preserve the nonnegativity of the cubical $h$-vector. \end{remark} \begin{remark} \label{rem:BW-formula} \rm Applying the reasoning of the proof of Proposition~\ref{prop:sdc-h} and of the first few lines of the proof of Theorem~\ref{thm:h-trans} to an $n$-dimensional simplicial complex $\Delta$ gives that \[ h({\rm sd}(\Delta), x) \ = \ (1-x)^{n+1} \, + \, x \, \sum_{k=0}^n f_k(\Delta) (1-x)^{n-k} A_{k+1}(x) \] and \[ \frac{h({\rm sd}(\Delta), x)}{(1-x)^{n+2}} \ = \ \sum_{m \ge 0} \left( \, \sum_{k=0}^{n+1} f_{k-1} (\Delta) m^k \right) x^m \ = \ \sum_{m \ge 0} \left( \, \sum_{k=0}^{n+1} h_k(\Delta) m^k (m+1)^{n+1-k} \right) x^m . \] This is the expression at which Brenti and Welker arrived \cite[Equation~(5)]{BW08} via a different route and which they used to show that $h({\rm sd}(\Delta), x)$ has only real roots, provided that $h_k(\Delta) \ge 0$ for all $k$. \end{remark} \begin{remark} \rm Replacing $2y-1$ by $x$ in (\ref{def:cub-h2}) shows that the equation \[ \sum_{k=0}^n f_k({\mathcal L}) x^k \ = \ \sum_{k=0}^n \, (h_k({\mathcal L}) + h_{k+1}({\mathcal L})) \, x^k (x+2)^{n-k} , \] together with the condition $h_0({\mathcal L}) = 1$, gives an equivalent way to define the normalized cubical $h$-vector of an $n$-dimensional cubical complex ${\mathcal L}$. \end{remark} \section{Closing remarks} \label{sec:rem} There is a large literature on the barycentric subdivision of simplicial complexes which relates to the work \cite{BW08}. Many of the questions addressed there make sense for cubical complexes. We only consider a couple of them here. \textbf{a}. Being real-rooted, $h({\rm sd}(\Delta), x)$ is unimodal for every $n$-dimensional simplicial complex $\Delta$ with nonnegative $h$-vector. Kubitzke and Nevo showed~\cite[Corolalry~4.7]{KN09} that the corresponding $h$-vector $(h_i({\rm sd}(\Delta))_{0 \le i \le n+1}$ has a peak at $i = (n+1)/2$, if $n$ is odd, and at $i = n/2$ or $i = n/2 + 1$, if $n$ is even. The analogous statement for cubical complexes follows from Theorem~\ref{thm:h-trans} since, as in the simplicial setting, the unimodal polynomial $p^B_{n,k}(x)$ has a peak at $i = (n+1)/2$, if $n$ is odd, at $i = n/2$ if $n$ is even and $k \le n/2$, and at $i = n/2 + 1$, if $n$ is even and $k \ge n/2 + 1$. The latter claim can be deduced from the recursion of part (e) of Proposition~\ref{prop:pBnk} by mimicking the argument given in the simplicial setting in~\cite[Section~2]{Mur10}. For general results on the unimodality of $h$-vectors of barycentric subdivisions of Cohen--Macaulay regular cell complexes, proven by algebraic methods, see Corollaries~1.2 and~5.12 in \cite{MY14}. \textbf{b}. The main result of~\cite{BS20} implies (see~\cite[Section~8]{Ath20+}) that $h({\rm sd}(\Delta), x)$ has a nonnegative real-rooted symmetric decomposition with respect to $n$ for every triangulation $\Delta$ of an $n$-dimensional ball. Does this hold if $\Delta$ is replaced by any cubical subdivision of the $n$-dimensional ball? Are these symmetric decompositions interlacing? Do the polynomials $p^B_{n,k}(x)$ have such properties? \textbf{c}. The subdivision operator (see \cite[Section~3.3]{Bra15}) has a natural generalization in the context of uniform triangulations of simplicial complexes \cite[Section~5]{Ath20+} which plays a role in that theory. It may be worth studying the cubical analogue of this operator further. \end{document}

\begin{document} \title[Nonunique asymptotic limit]{Finite-energy pseudoholomorphic planes with multiple asymptotic limits} \author[R.\ Siefring]{Richard Siefring} \address{Fakult\"at f\"ur Mathematik \\ Ruhr-Universit\"at Bochum \\ 44780 Bochum \\ Germany} \urladdr{\url{http://homepage.ruhr-uni-bochum.de/richard.siefring}} \email{\href{mailto:[email protected]}{[email protected]}} \date{September 30, 2016} \dedicatory{ Dedicated to Helmut Hofer on the occasion of his 60th birthday, and in warm remembrance of Kris Wysocki. } \begin{abstract} It's known from \cite{hwz:prop1, hwz:prop4, bourgeois} that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its nonremovable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $(M, \xi)$ with cooriented contact structure one can choose a contact form $\lambda$ with $\ker\lambda=\xi$ and a compatible complex structure $J$ on $\xi$ so that for the associated $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ there exist families of embedded finite-energy $\tilde J$-holomorphic cylinders and planes having embedded tori as limit sets. \end{abstract} \maketitle \tableofcontents \section{Introduction and statement of results} The study of punctured pseudoholomorphic curves in symplectizations of contact manifolds was introduced by Hofer in \cite{hofer93}. Specifically, considering a contact manifold $(M, \xi=\ker\lambda)$, Hofer introduced a class of $\mathbb{R}$-invariant almost complex structures and a notion of energy for a pseudoholomorphic map $\tilde u=(a, u):\mathbb{C}\to \mathbb{R}\times M$ and showed that if the energy of a pseudoholomorphic plane is finite, then there are sequences $s_{k}\to\infty$ so that the sequence of loops \[ t\in \mathbb{S}^{1}\approx \mathbb{R}/\mathbb{Z} \mapsto u(e^{2\pi(s_{k}+it)}) \] converge in $C^{\infty}(\mathbb{S}^{1}, M)$ to a periodic orbit $\gamma$ of the Reeb vector field of the contact form $\lambda$. In \cite[Theorem 1.2/1.3]{hwz:prop1}, Hofer, Wysocki, and Zehnder further show that if the periodic orbit $\gamma$ is nondegenerate, then the maps $u(s):\mathbb{S}^{1}\to M$ defined by $u(s)(t)=u(e^{s+it})$ satisfy \[ \lim_{s\to\infty}u(s)=\gamma \text{ in $C^{\infty}(\mathbb{S}^{1}, M)$} \] and in fact the convergence is exponential \cite[Theorem 1.4]{hwz:prop1}. There, immediately following the statement of Theorem 1.2, the authors mention that they expect this need not be the case in the event that the periodic orbit $\gamma$ is degenerate, but that they didn't know of an explicit example. To date no examples have appeared in the literature, and whether or not it is possible for a finite-energy plane to have multiple periodic orbits as asymptotic limits has remained an open question.\footnote{In fact, a claimed proof that no such examples exist has appeared in a recent (now-withdrawn) arXiv preprint.} We present some examples here. The examples we construct can be localized to any arbitrarily small neighborhood of a standard model of a transverse knot and since transverse knots exist in abundance in any contact manifold, we can prove the following. \begin{theorem}\label{t:main-theorem} Let $(M, \xi)$ be a contact manifold. Then there exists a contact form $\lambda$ on $M$ and a compatible complex structure $J$ on $\xi$ so that there exist finite-energy pseudoholomorphic planes and cylinders for the data $(\lambda, J)$ whose limit sets have image diffeomorphic to the $2$-torus. \end{theorem} We give a brief outline of what follows. In Section \ref{s:background} we begin by recalling some basic notions from contact geometry and pseudoholomorphic curves. Then, in Section \ref{s:prequant}, we explain a correspondence between gradient flow lines on exact symplectic manifolds and pseudoholomorphic cylinders in contact manifolds constructed as circle bundles over those symplectic manifolds. From this construction it is clear that one can construct pseudoholomorphic cylinders having more than one limit orbit by constructing gradient flow lines in a symplectic manifold having an alpha or omega limit set consisting of more than a single point. To this end, we construct in Section \ref{s:gradient} a function on the cylinder $\mathbb{R}\times S^{1}$ which will have the circle $\br{0}\times S^{1}$ as the omega limit set of any nontrivial gradient flow line with respect to any Riemannian metric and which can be chosen to be linear in the $\mathbb{R}$-variable and independent of the $S^{1}$-variable outside of any desired neighborhood of $\br{0}\times S^{1}$. Finally, in Section \ref{s:main-proof}, we apply the results of Sections \ref{s:prequant} and \ref{s:gradient} to construct finite-energy pseudoholomorphic cylinders and planes having tori as limit sets. We comment that while the construction of a pseudoholomorphic cylinder with tori as limit sets is a straightforward application of the results in Sections \ref{s:prequant} and \ref{s:gradient}, applying these results to construct a plane with multiple limit orbits is a bit trickier and requires finding a situation where these results can be applied to construct a cylinder with a removable singularity. We close this section with a remark about notation. In most of what follows we find it convenient to consider the circle as $\mathbb{R}/2\pi\mathbb{Z}$, although at some points --- specifically when considering domains of pseudoholomorphic cylinders or periodic orbits --- we will find it more convenient to consider the circle to be $\mathbb{R}/\mathbb{Z}$. To avoid ambiguity we will use the notations $S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ and $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$ to distinguish between the two. \end{ack} \section{Pseudoholomorphic curves in contact manifolds}\label{s:background} Here we recall some basic notions, primarily for the purpose of fixing notation. Let $M$ be an oriented $(2n+1)$-dimensional manifold. A $1$-form $\lambda$ is said to be a contact form on $M$ if \begin{equation}\label{e:contact-condition} \text{$\lambda\wedge d\lambda^{n}$ is a nowhere vanishing.} \end{equation} A contact form on $M$ determines a splitting \begin{equation}\label{e:splitting} TM=\mathbb{R}X_{\lambda}\oplus\xi \end{equation} where $\xi=\ker\lambda$ is a hyperplane distribution, called the \emph{contact structure}, and $X_{\lambda}$ is the \emph{Reeb vector field}, defined by \[ i_{X_{\lambda}}d\lambda=0 \qquad\text{ and }\qquad i_{X_{\lambda}}\lambda=1. \] We note that \eqref{e:contact-condition} implies that $d\lambda$ restricts to a nondegenerate form on $\xi$ and thus $(\xi, d\lambda)$ is a symplectic vector bundle over $M$. We recall that if $\lambda$ is a contact form on $M$ and $f:M\to\mathbb{R}$ is a smooth function, then $e^{f}\lambda$ is also a contact form since $d(e^{f}\lambda)=e^{f}(df\wedge\lambda+d\lambda)$ and hence \[ (e^{f}\lambda)\wedge d(e^{f}\lambda)^{n}=e^{(n+1)f}\lambda\wedge d\lambda. \] We note for later reference that a straightforward computation shows that the Reeb vector field for the contact form $e^{f}\lambda$ is related to the Reeb vector field for $\lambda$ by \begin{equation}\label{e:reeb-change-1} X_{e^{f}\lambda}=e^{-f}\bp{X_{\lambda}-X_{f}} \end{equation} where $X_{f}$ is the unique section of $\xi$ satisfying \begin{equation}\label{e:reeb-change-2} i_{X_{f}}d\lambda=-df+df(X_{\lambda})\lambda. \end{equation} That there is a unique section $X_{f}$ of $\xi$ satisfying \eqref{e:reeb-change-2} follows from nondegeneracy of $d\lambda$ on $\xi$ and the fact that both sides of \eqref{e:reeb-change-2} vanish on $X_{\lambda}$. Given a symplectic vector bundle $(E, \omega)$ over a given manifold $W$, a complex structure $J\in\operatorname{End}(E)$ is said to be \emph{compatible} with $\omega$ if the section of $E^{*}\otimes E^{*}$ defined by $g_{J}:=\omega(\cdot, J\cdot)$ is symmetric and positive definite on $E$. It is well known that the space of such $J$ is nonempty and contractible (see e.g.\ the discussion following Proposition 5 in Section 1.3 of \cite{hoferzehnder}). Given a contact manifold, $(M, \xi=\ker\lambda)$, we then define the set $\mathcal{J}(M, \xi)$ to be the set of complex structures on $\xi$ compatible with $d\lambda|_{\xi\times\xi}$. We observe that if a complex structure $J\in\operatorname{End}(\xi)$ is compatible with $d\lambda$, then it is also compatible with $d(e^{f}\lambda)$ since \[ d(e^{f}\lambda)-e^{f}d\lambda=e^{f}df\wedge\lambda \] which vanishes on $\xi\times\xi=\ker\lambda\times\ker\lambda$. Therefore, the set $\mathcal{J}(M, \xi)$ depends only on a choice of conformal symplectic structure on $\xi$, and not on the choice of a specific contact form inducing that structure. Given a manifold $M$ with contact form $\lambda$ and a compatible $J$, we can extend $J$ to an $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ by requiring \begin{equation}\label{e:R-invariant-J} \tilde J\partial_{a}=X_{\lambda} \qquad\text{ and }\qquad \tilde J|_{\pi_{M}^{*}\xi}=\pi_{M}^{*}J \end{equation} with $a$ the coordinate along $\mathbb{R}$ and $\pi_{M}:\mathbb{R}\times M\to M$ the coordinate projection. We consider quintuples $(\Sigma, j, \Gamma, a, u)$ where $(\Sigma, j)$ is a closed Riemann surface, $\Gamma\subset\Sigma$ is a finite set, called the set of \emph{punctures}, and $a:\Sigma\setminus\Gamma\to\mathbb{R}$ and $u:\Sigma\setminus\Gamma\to M$ are smooth maps. We say such a quintuple is \emph{pseudoholomorphic map for the data $(\lambda, J)$ on $M$} if $\tilde u=(a, u):\Sigma\setminus\Gamma\to\mathbb{R}\times M$ satisfies the equation \begin{equation}\label{e:j-hol} d\tilde u\circ j=\tilde J\circ \tilde u \end{equation} or, equivalently, if $u$ and $a$ satisfy \begin{equation}\label{e:j-hol-M} \begin{gathered} \pi_{\lambda}\circ du\circ j=J\circ \pi_{\lambda}\circ du \\ u^{*}\lambda\circ j=da \end{gathered} \end{equation} where $\pi_{\lambda}:TM\approx\mathbb{R}X_{\lambda}\oplus\xi\to\xi$ is the projection of $TM$ onto $\xi$ along $X_{\lambda}$. The \emph{Hofer energy} $E(u)$ of a pseudoholomorphic map $(\Sigma, j, \Gamma, a, u)$ is defined by \begin{equation}\label{e:hofer-energy-defn} E(u)=\sup_{\varphi\in\Xi}\int_{\Sigma\setminus\Gamma}\tilde u^{*}d(\varphi\lambda)= \sup_{\varphi\in\Xi}\int_{\Sigma\setminus\Gamma}d(\varphi(a) u^{*}\lambda) \end{equation} where $\Xi\subset C^{\infty}(\mathbb{R}, [0, 1])$ is the set of smooth functions $\varphi:\mathbb{R}\to[0, 1]$ with $\varphi'(t)\ge 0$ for all $s\in\mathbb{R}$, $\lim_{s\to-\infty}\varphi(s)=0$, and $\lim_{s\to\infty}\varphi(s)=1$. To each puncture in a pseudoholomorphic map we will a assign a quantity called the mass of the puncture. First, we will call a holomorphic embedding $\psi:[0, +\infty)\times \mathbb{S}^{1}\subset\mathbb{C}/i\mathbb{Z}\to\Sigma\setminus\Gamma$ a \emph{holomorphic cylindrical coordinate system} around $z_{0}\in\Gamma$ if $\lim_{s\to\infty}\psi(s, t)=z_{0}$. Given a holomorphic cylindrical coordinates $\psi$ around $z_{0}\in\Gamma$, we consider the family of loops $v(s)=(u\circ\psi)(s, \cdot):\mathbb{S}^{1}\to M$ and define the \emph{mass $m(z_{0})$ of the puncture $z_{0}$} by \begin{equation}\label{e:mass} m(z_{0})=\lim_{s\to\infty}\int_{\mathbb{S}^{1}}v(s)^{*}\lambda. \end{equation} The limit in this definition is well-defined as a result the compatibility of $J$ with $d\lambda$. Indeed, for $s_{1}>s_{0}$ we apply Stokes' theorem to compute \begin{align} \int_{\mathbb{S}^{1}}v(s_{1})^{*}\lambda-\int_{\mathbb{S}^{1}}v(s_{0})^{*}\lambda &=\int_{[s_{0}, s_{1}]\times \mathbb{S}^{1}}(u\circ\psi)^{*}d\lambda \label{e:mass-stokes} \\ &=\int_{[s_{0}, s_{1}]\times \mathbb{S}^{1}}d\lambda(u_{s}, u_{t})\,ds\wedge dt \notag \\ &=\int_{[s_{0}, s_{1}]\times \mathbb{S}^{1}}d\lambda(\pi_{\lambda}(u_{s}),\pi_{\lambda}(u_{t}))\,ds\wedge dt & i_{X_{\lambda}}d\lambda=0 \notag \\ &=\int_{[s_{0}, s_{1}]\times \mathbb{S}^{1}}d\lambda(\pi_{\lambda}(u_{s}),J\pi_{\lambda}(u_{s}))\,ds\wedge dt & \eqref{e:j-hol-M} \notag \end{align} and we observe the integrand in the final line above is nonnegative by compatibility of $J$ with $d\lambda$. Thus the integral in the definition \eqref{e:mass} of mass is an increasing function of $s$, which lets us conclude the limit is well-defined (although possibly infinite). It can, moreover, be shown that the mass is independent of the choice of holomorphic cylindrical coordinates near $z_{0}$. If is a straightforward exercise using \eqref{e:mass-stokes} and definition of Hofer energy to show that if a pseudoholomorphic map has finite Hofer energy, then all punctures have finite mass. Furthermore, punctures with mass $0$ can be shown to be removable, that is, one can find a pseudoholomorphic extension of the map $\tilde u$ over any puncture with mass $0$ (see \cite[pgs.\ 272-3]{hwz:prop2}). The behavior near punctures with nonzero mass is described by the following now well-known theorem of Hofer from \cite{hofer93}.\footnote{ Hofer only considers planes in \cite{hofer93} and proves the slightly weaker statement that there exists a sequence $s_{k}\to\infty$ so that corresponding loops $u\circ\psi(s_{k}, \cdot)$ converge to a periodic orbit, but the generalization of the proof to the result we state here is straightforward. In the survey \cite[Theorem 3.2]{hwz-survey}, the appropriate result is proven for a general pseudoholomorphic half-cylinder, albeit under a different notion of energy. The fact that this different notion of energy implies finite Hofer energy as defined by \eqref{e:hofer-energy-defn} is addressed in Theorem 5.1 of the same paper. } \begin{theorem} Let $M$ be a compact manifold equipped with a contact form $\lambda$ and compatible complex structure $J\in\mathcal{J}(M, \xi)$ on $\xi=\ker\lambda$. Let $(\Sigma, j, \Gamma, a, u)$ be a solution to \eqref{e:j-hol-M} and assume that $z_{0}\in\Gamma$ has mass $m(z_{0})=T\ne 0$. Then for every holomorphic cylindrical coordinate system $\psi:[0, \infty)\times \mathbb{S}^{1}\to\Sigma\setminus\Gamma$ around $z_{0}$, and every sequence $s_{k}\to\infty$ there exists a subsequence $s_{k_{j}}$ and a smooth map $\gamma:\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}\to M$ with $\dot\gamma=T\cdotX_{\lambda}\circ\gamma$ so that the sequence of loops $u\circ\psi(s_{k_{j}}, \cdot):\mathbb{S}^{1}\to M$ converge in $C^{\infty}(\mathbb{S}^{1}, M)$ to $\gamma$. \end{theorem} We will refer to the collection of periodic orbits obtained as limits of a given finite-energy pseudoholomorphic map as the \emph{limit set} of that map. As mentioned in the introduction, it can be shown under some suitable nondegeneracy assumptions that a puncture has a unique periodic orbit (up to reparametrization) in its limit set and that the convergence to that orbit is exponential \cite{hwz:prop1, hwz:prop4, HWZ:planes, bourgeois, mora}. In the absence of nondegeneracy however, it has until now remained an open question whether it's possible for a finite-energy pseudoholomorphic map to have more than one periodic orbit in the asymptotic limit set of a given puncture. \section{Prequantization spaces, gradient flows, and pseudoholomorphic cylinders}\label{s:prequant} In this section we explain a correspondence between gradient flows on symplectic manifolds and certain pseudoholomorphic cylinders in an associated prequantization space, that is, a contact manifold constructed as a principal $S^{1}$-bundle over the given symplectic manifold with the contact structure being given as the horizontal distribution determined by an appropriate connection on the bundle. For simplicity we focus on the case of trivial $S^{1}$-bundles over exact symplectic manifolds, since that is all we require for the proof of our main theorem, but we point out that the construction of pseudoholomorphic cylinders in a prequantization space from gradient flow lines in the base can be generalized to any prequantization space. Let $(W, \omega=d\beta)$ be an exact symplectic manifold and consider $S^{1}(\approx \mathbb{R}/2\pi\mathbb{Z})\times W$ equipped with the $1$-form \begin{equation}\label{e:prequant-form} \lambda=d\theta+\pi^{*}\beta. \end{equation} where $\pi:S^{1}\times W\to W$ is the canonical projection onto the second factor. The $1$-form $\lambda$ defined in this way is a contact form on $S^{1}\times W$ since \begin{align*} \lambda\wedge (d\lambda)^{n} &=(d\theta+\pi^{*}\beta)\wedge \pi^{*}\omega^{n} \\ &=d\theta\wedge\pi^{*}\omega^{n}>0. \end{align*} We will refer to a pair $(S^{1}\times W, d\theta+\pi^{*}\beta)$ consisting of a trivial $S^{1}$-bundle and a contact form arising in this way as a \emph{prequantization space} over the symplectic manifold $(W, \omega=d\beta)$. We observe that the Reeb vector field of the contact form \eqref{e:prequant-form} is given by $\partial_{\theta}$ and hence the splitting \eqref{e:splitting} induced on $TM$ by the contact form is given by \[ T(S^{1}\times W)\approx TS^{1}\oplus\xi. \] Thus $\xi$ is an $S^{1}$-invariant horizontal distribution of the bundle $S^{1}\times W\to W$ which gives us a one-to-one correspondence between the space $\Gamma(TW)$ of vector fields on $W$ and the space $\Gamma_{S^{1}}(\xi)$ of $S^{1}$-invariant sections of the contact structure $\xi$. This correspondence is given explicitly by the maps \begin{equation}\label{e:horizontal-lift} X\in T_{p}W\mapsto \tilde X:=-\beta(X)\,\partial_{\theta} +X \in \xi_{(\theta, p)}\subset T_{(\theta, p)}(S^{1}\times W) \end{equation} and \[ \tilde Y\in \xi_{(\theta, p)}\mapsto d\pi(\tilde Y)\in T_{p}W, \] where the plus sign in \eqref{e:horizontal-lift} is to be interpreted relative to the natural splitting \[ T_{(\theta, p)}(S^{1}\times W)\approx T_{\theta}S^{1}\oplus T_{p}W \] arising from the canonical projection onto the factors of the Cartesian product. The correspondence between vector fields on $W$ and $S^{1}$-invariant sections of $\xi$ generalizes to arbitrary tensor fields on $W$. In particular an endomorphism $A\in\operatorname{End}(TW)\approx TW^{*}\otimes TW$ of the form \[ A=\sum_{i}\alpha_{i}\otimes X_{i} \] lifts to an $S^{1}$-invariant endomorphism $\tilde A\in\operatorname{End}(\xi)\approx\xi^{*}\otimes\xi$ given by \[ \tilde A=\sum_{i}\pi^{*}\alpha_{i}\otimes \tilde X_{i}. \] Equivalently, we can define $\tilde A$ to be the unique section of $\operatorname{End}(\xi)$ satisfying \[ \tilde A\tilde X=\widetilde{AX} \] for every vector field $X$ on $TW$. We define $\mathcal{J}(W, \omega)$ to be the set of almost complex structures on $W$ compatible with the symplectic form $\omega$, that is, those $j\in\operatorname{End}(TW)$ which square to negative the identity and for which $g_{j}:=\omega(\cdot, j\cdot)$ is a Riemannian metric on $W$. According to the remarks of the previous paragraph, $j$ lifts to an $S^{1}$-invariant endomorphism $\tilde j$ of $\xi$ characterized by \[ \tilde j\tilde X=\widetilde{jX} \] for every vector field $X$ on $W$. From this equation together with the linearity of the map $X\mapsto\tilde X$ and the fact that \[ d\lambda=d(\pi^{*}\beta)=\pi^{*}(d\beta)=\pi^{*}\omega \] it follows that the $S^{1}$-invariant lift $\tilde j\in\operatorname{End}(\xi)$ of a compatible almost complex structure $j\in\mathcal{J}(W, \omega)$ on $W$ is an element of $\mathcal{J}(S^{1}\times W, \xi)$, i.e.\ a complex structure on $\xi$ compatible with $d\lambda$. Given a choice of compatible $j\in\mathcal{J}(W, \omega)$ we can associate two vector fields on $W$ to any smooth real-valued function $f$ on $W$: the Hamiltonian vector field $X_{f}$ and the gradient $\nabla f$ defined respectively by \[ i_{X_{f}}\omega=-df \qquad\text{ and }\qquad g_{j}(\nabla f, \cdot)=df. \] These vector fields are related by the equations \[ X_{f}=j\nabla f \qquad\text{ and }\qquad \nabla f=-jX_{f} \] since we can use the definition of $g_{j}$ and the antisymmetry of $\omega$ to compute \[ i_{j\nabla f}\omega=\omega(j\nabla f, \cdot) =-\omega(\cdot, j\nabla f) =-g_{j}(\cdot, \nabla f) =-df. \] From the observations of the previous paragraph, the respective $S^{1}$-invariant lifts of $\tilde j$, $\widetilde{\nabla f}$, and $\widetilde{X_{f}}$ of $j$, $\nabla f$ and $X_{f}$ satisfy \begin{equation}\label{e:ham-grad-lift} \widetilde{X_{f}}=\tilde j\widetilde{\nabla f} \qquad\text{ and }\qquad \widetilde{\nabla f}=-\tilde j\widetilde{X_{f}}. \end{equation} Continuing to let $f:W\to\mathbb{R}$ denote a smooth function on $W$, we can pull $f$ back to an $S^{1}$-invariant smooth function $\pi^{*}f$ on $S^{1}\times W$ and consider the contact form $\lambda_{f}$ defined by \[ \lambda_{f}=e^{\pi^{*}f}\lambda=e^{\pi^{*}f}(d\theta+\pi^{*}\beta). \] Since \begin{align*} i_{\tilde X_{f}}d\lambda &=i_{\tilde X_{f}}\pi^{*}\omega \\ &=\pi^{*}(i_{X_{f}}\omega) \\ &=-\pi^{*}df \\ &=-d(\pi^{*}f)+d(\pi^{*}f)(\partial_{\theta})\lambda \end{align*} it follows from \eqref{e:reeb-change-1}-\eqref{e:reeb-change-2} that \begin{equation}\label{e:reeb-deformed} X_{\lambda_{f}}=e^{-\pi^{*}f}(\partial_{\theta}-\tilde X_{f}). \end{equation} From this and \eqref{e:ham-grad-lift} we note at any point $p\in W$ where $f$ has a critical point, $X_{\lambda_{f}}(\theta, p)=e^{-f(p)}\partial_{\theta}$, and thus the fiber in $S^{1}\times W$ over $p$ is a periodic orbit of the Reeb vector field with period $2\pi e^{f(p)}$. We are now ready to state the main theorem of the section, which establishes a correspondence between gradient flows on a symplectic manifold $(W, \omega=d\beta)$ and pseudoholomorphic cylinders in the corresponding prequantization space $(S^{1}\times W, d\theta+\pi^{*}\beta)$. The idea of relating gradient flow lines of a Morse function to pseudoholomorphic cylinders in a contact manifold originates in \cite{bourgeois} (see also \cite{sft, behwz}). In the present context this relationship can be seen as a generalization to the contact setting of an idea of Floer from \cite{floer} (see also \cite{SZ, HS}). \begin{theorem}\label{t:gradient-flow-hol-cylinders} Let $(S^{1}\times W, \lambda=d\theta+\pi^{*}\beta)$ be a prequantization space over an exact symplectic manifold $(W, \omega)$, let $j\in\mathcal{J}(W, \omega=d\beta)$ be a compatible almost complex structure on $W$, and let $J=\tilde j\in\mathcal{J}(S^{1}\times W, \xi)$ be the corresponding $S^{1}$-invariant compatible complex structure on $\xi=\ker\lambda$. Given a smooth function $f:W\to\mathbb{R}$, consider smooth maps $\gamma:\mathbb{R}\to W$, $\theta:\mathbb{R}\to S^{1}$, and $a:\mathbb{R}\to\mathbb{R}$ satisfying the system of o.d.e.'s \begin{align} \dot\gamma(s)&=2\pi \nabla f(\gamma(s)) \label{e:ode-gamma} \\ \dot\theta(s)&=-2\pi \beta(\nabla f(\gamma(s))) \label{e:ode-theta}\\ \dot a(s)&=2\pi e^{f(\gamma(s))} \label{e:ode-a} \end{align} with $\nabla f$ denoting the gradient with respect to the metric $g_{j}=\omega(\cdot, j\cdot)$. Then the map $\tilde u=(a, u):\mathbb{R}\times\mathbb{S}^{1}(\approx\mathbb{R}/\mathbb{Z})\to \mathbb{R}\times S^{1}(\approx\mathbb{R}/2\pi\mathbb{Z})\times W$ defined by \[ \tilde u(s, t)=(a(s, t), u(s, t))=(a(s), \theta(s)+2\pi t, \gamma(s)) \] is a pseudoholomorphic cylinder for the data $(e^{\pi^{*}f}\lambda, J)$ with Hofer energy\footnote{ We remark that finiteness of the energy here does not immediately imply that the cylinders approach periodic orbits because $W$, being equipped with an exact symplectic form, is necessarily noncompact. Consider, for example, the symplectic manifold $(\mathbb{R}^{2}, dx\wedge dy=d(x\,dy))$ and the function $f(x, y)=\arctan x$. The pseudoholomorphic cylinders in the appropriate prequantization space covering gradient flow lines in the base have finite energy as a result of \eqref{e:energy-formula} since the function $f$ is bounded, but the cylinders do not approach periodic orbits since the function $f$ has no critical points. } \begin{equation}\label{e:energy-formula} E(u)=2\pi \lim_{s\to\infty}e^{f(\gamma(s))}\in [0, +\infty]. \end{equation} \end{theorem} \begin{proof} We compute using \eqref{e:horizontal-lift} \begin{align*} \tilde u_{s}(s, t) &=\dot a(s)\,\partial_{a}+\dot\theta(s)\,\partial_{\theta}+\dot\gamma(s) \\ &=2\pi e^{f(\gamma(s))}\,\partial_{a}-2\pi \beta(\nabla f(\gamma(s)))\,\partial_{\theta}+2\pi \nabla f(\gamma(s)) \\ &=2\pi \bp{e^{f(\gamma(s))}\,\partial_{a}+\widetilde{\nabla f}(u(s, t))} \intertext{and similarly using \eqref{e:reeb-deformed}} \tilde u_{t}(s, t) &=2\pi \partial_{\theta} \\ &=2\pi \bp{\partial_{\theta}-\widetilde{X_{f}}}(u(s, t))+2\pi \widetilde{X_{f}}(u(s, t)) \\ &=2\pi \bp{e^{f(\gamma(s))}X_{e^{f}\lambda}(u(s,t))+\widetilde{X_{f}}(u(s, t))}. \end{align*} It then follows from the definition \eqref{e:R-invariant-J} of the $\mathbb{R}$-invariant extension of $J$ to an almost complex structure on $\mathbb{R}\times S^{1}\times W$ and from \eqref{e:ham-grad-lift}, that $\tilde u$ satisfies the pseudoholomorphic map equation \eqref{e:j-hol}. It remains to compute the Hofer energy. To do that, we first compute \begin{align*} u^{*}\lambda &=\lambda(u_{s})\,ds+\lambda(u_{t})\,dt \\ &=\lambda(\widetilde{\nabla f})\,ds+\lambda(2\pi \partial_{\theta})\,dt \\ &=2\pi dt. \end{align*} We then consider a smooth, increasing function $\varphi:\mathbb{R}\to[0, 1]$ with $\lim_{s\to\infty}\varphi(s)=1$ and $\lim_{s\to-\infty}\varphi(s)=0$, and compute \begin{equation}\label{e:energy-computation} \begin{aligned} \int_{[s_{0}, s_{1}]\times\mathbb{S}^{1}}\tilde u^{*}d(\varphi e^{\pi^{*}f}\lambda) &=\int_{[s_{0}, s_{1}]\times\mathbb{S}^{1}}d(\varphi(a) e^{f(\gamma)}2\pi dt) \\ &=\bp{\int_{\br{s_{1}}\times\mathbb{S}^{1}}-\int_{\br{s_{0}}\times\mathbb{S}^{1}}}2\pi \varphi(a) e^{f(\gamma)}\,dt \\ &=2\pi \bp{\varphi(a(s_{1}))e^{f(\gamma(s_{1}))}-\varphi(a(s_{0}))e^{f(\gamma(s_{0}))}}. \end{aligned} \end{equation} From \eqref{e:ode-gamma} and \eqref{e:ode-a} we know that the function $e^{f\circ\gamma}$ is increasing and $a$ is strictly increasing with increasing derivative. Thus $\lim_{s\to\infty}a(s)=+\infty$ and we can conclude that \[ \lim_{s_{1}\to\infty}\varphi(a(s_{1}))e^{f(\gamma(s_{1}))} =\bp{\lim_{s_{1}\to\infty}\varphi(a(s_{1}))}\bp{\lim_{s_{1}\to\infty}e^{f(\gamma(s_{1}))}} =\varphi(+\infty)\lim_{s_{1}\to\infty}e^{f(\gamma(s_{1}))} =\lim_{s_{1}\to\infty}e^{f(\gamma(s_{1}))}. \] Again using that $e^{f\circ\gamma}$ is increasing we can know that $\lim_{s_{0}\to-\infty}e^{f(\gamma(s_{0}))}$ exists and is either positive or zero. If the case that this limit is positive, we know from \eqref{e:ode-a} that $\lim_{s_{0}\to-\infty}a(s_{0})=-\infty$ and hence that $\lim_{s_{0}\to-\infty}\varphi(a(s_{0}))=0$. In either case, we conclude that \[ \lim_{s_{0}\to-\infty}\varphi(a(s_{0}))e^{f(\gamma(s_{0}))} =0 \] because it's a product of increasing, positive functions, at least one of which limits to $0$ as $s_{0}\to-\infty$. Hence, taking limits in \eqref{e:energy-computation} above leads to \[ \int_{\mathbb{R}\times\mathbb{S}^{1}}\tilde u^{*}d(\varphi e^{f}\lambda)=2\pi \lim_{s\to\infty}e^{f(\gamma(s))} \] for any $\varphi\in\Xi$, which establishes $E(u)=2\pi \lim_{s\to\infty}e^{f(\gamma(s))}$ as claimed. \end{proof} \section{A gradient flow with a $1$-dimensional limit set}\label{s:gradient} In this section we construct a function so that the omega limit set of all of its nontrivial gradient flow lines is diffeomorphic to a circle. This function will be used in the next section in conjunction with Theorem \ref{t:gradient-flow-hol-cylinders} above to construct finite-energy cylinders and planes localized near a transverse knot which have tori as limit sets. The main theorem of this section is the following. \begin{theorem}\label{t:function-construction} For any $\delta>0$, there exists a smooth function $F_{\delta}:\mathbb{R}\times S^{1}(\approx\mathbb{R}/2\pi\mathbb{Z}) \to \mathbb{R}$ so that \begin{itemize} \item $F_{\delta}(s, t)=s$ for $s\le\delta$, \item $s\le F_{\delta}(s, t)< 0$ for $s\in(-\delta, 0)$, \item $F_{\delta}(s, t)=0$ for $s\ge 0$, \item $dF_{\delta}(s, t)\ne 0$ for $s<0$, \end{itemize} and so that for any choice of Riemannian metric on $\mathbb{R}\times S^{1}$, the solution to the initial value problem \[ \gamma(\tau)=\nabla F_{\delta}(\gamma(\tau)) \qquad \gamma(0)=(s_{0},t_{0}) \text{ with $s_{0}<0$} \] exists for all $\tau\ge 0$ and has the circle $\br{0}\times S^{1}$ as its omega limit set. \end{theorem} \begin{remark} The fact that there exist gradient flows with omega limit sets consisting of more than a single point has been known for some time and a qualitative description of a function like the one we construct below is given in \cite[pg.\ 261]{curry}. In \cite[Example 3, pgs.\ 13-14]{palis-demelo} a function in $\mathbb{R}^{2}$ is given for which it can be shown that there is at least one gradient flow line whose omega limit set is a circle. An interesting feature of the functions $F_{\delta}$ provided by our theorem is that every nontrivial flow line, independent of the metric, has the circle $\br{0}\times S^{1}$ as its omega limit set. Since the behavior of the functions $F_{\delta}$ is especially simple outside of a neighborhood of this limit set, this allows for a good deal of flexibility in constructing functions on a given Riemannian manifold whose gradients will have flow lines having a circle as an omega limit set. For example, it is a straightforward corollary of this theorem that one can construct smooth functions on any Riemannian $2$-manifold $(M^2, g)$ having any desired embedded circle as the limit set of some gradient flow line. Indeed, we can either identify a neighborhood of a given embedded circle with $(-\varepsilon, \varepsilon)\times \mathbb{R}/2\pi\mathbb{Z}$ or, in the nonorientable case, we can identify a double cover of a neighborhood of the circle with $(-\varepsilon, \varepsilon)\times\mathbb{R}/2\pi\mathbb{Z}$ with the nontrivial deck-transformation of the cover being given by the map $(s, t)\mapsto (-s, t+\pi)$. We then consider the function $G(s, t):=F_{\varepsilon/2}(s, t)+F_{\varepsilon/2}(-s, t+\pi)$ which is invariant under the action of the deck transformation in the nonoriented case, agrees with $-\abs{s}$ for $\abs{s}\in (\varepsilon/2, \varepsilon)$, and which will have $\br{0}\times S^{1}$ as the omega limit set of any gradient flow line starting in the neighborhood. The function can then be extended to a smooth function on the entire surface using an appropriate cutoff function. \end{remark} We will construct the function in the following paragraph and prove that it has the required properties in a series of lemmas. Throughout this section we will make no notational distinction between smooth functions with domain $\mathbb{R}\times S^{1}\approx\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}$ and functions on $\mathbb{R}^{2}$ which are $2\pi$-periodic in the second variable. We consider the function $G:\mathbb{R}\times \mathbb{R}/2\pi\mathbb{Z}=\br{(s, t)}\to\mathbb{R}$ defined by\footnote{ As will become clear from our proof, the $5/4$ in our example can be replaced with any constant strictly bigger than $1$ and strictly less than $\sqrt{2}$. } \[ G(s, t)= \begin{cases} e^{1/s}\bp{\sin(1/s+t)-5/4} & s<0 \\ 0 & s\ge 0 \end{cases} \] and we note that $G$ is smooth and that\footnote{ The left-most part of this inequality can be seen from the following argument. To show that $-\frac{9}{4}e^{1/s}-s$ is positive for all $s<0$, it suffices to show that $g(t)=\frac{9}{4}te^{t}+1$ is positive for all $t<0$. A straightforward argument using single-variable calculus then shows that $g(-1)=-\frac{9}{4}e^{-1}+1>0$ is the absolute minimum of the function $g$ on $\mathbb{R}$. } \begin{equation}\label{e:bounds-G} s< -\tfrac{9}{4} e^{1/s} \le G(s, t)\le -\tfrac{1}{4}e^{1/s} \text{ for all $s<0$.} \end{equation} For a given value $\delta>0$ we let $\eta:\mathbb{R}\to[0, 1]$ be a smooth cut-off function satisfying \[ \eta(t)= \begin{cases} 0 & s<-\delta \\ 1& s>-\delta/2 \end{cases} \] and $\eta'(t)\ge 0$ everywhere and define the function $F:\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{R}$ by \begin{equation}\label{e:F-definition} F(s, t)=(1-\eta(s))s+\eta(s)G(s, t). \end{equation} We observe that this definition with \eqref{e:bounds-G} implies that \begin{equation}\label{e:bounds-F} s \le F(s, t)\le -\tfrac{1}{4}e^{1/s} \text{ for all $s<0$} \end{equation} so the first three properties required of $F_{\delta}$ in the theorem are clearly satisfied. The fourth property, concerning the critical set, is then addressed by the following lemma. \begin{lemma}\label{l:F-crit-set} The set of critical points of the above defined function $F$ is $s\ge 0$. \end{lemma} \begin{proof} We first compute for $s<0$ \begin{equation}\label{e:G-s} \begin{aligned} G_{s}(s, t) &=-1/s^{2}e^{1/s}\bp{\sin(1/s+t)+\cos(1/s+t)-\tfrac{5}{4}} \\ &=-1/s^{2}e^{1/s}\bp{\sqrt{2}\sin(1/s+t+\tfrac{\pi}{4})-\tfrac{5}{4}} \\ \end{aligned} \end{equation} and \begin{equation}\label{e:G-t} G_{t}(s, t) =e^{1/s}\cos(1/s+t) \end{equation} and note then that \begin{align*} dG(s^{2}\,\partial_{s}+\partial_{t}) &=s^{2}G_{s}+G_{t} \\ &=-G \end{align*} which is everywhere positive for $s<0$. We then compute \begin{align*} dF(s, t) &=\eta'(s)(G(s, t)-s)\,ds+(1-\eta(s))\,ds+\eta(s)dG(s, t) \end{align*} and thus \[ dF(s^{2}\,\partial_{s}+\partial_{t}) =s^{2}\eta'(s)(G(s, t)-s)+(1-\eta(s))s^{2}+\eta(s)(-G), \] which we claim is always positive for $s<0$. Indeed the first term is always nonnegative since, as observed above in \eqref{e:bounds-G}, $G(s, t)\ge-\frac{9}{4}e^{1/s}>s$ for all $s<0$. Meanwhile the second two terms are the convex sum of positive quantities and thus always positive. We've thus found a vector field $v=s^{2}\,\partial_{s}+\partial_{t}$ for which $dF(v)>0$ for $s<0$ which shows that $F$ has no critical points for $s<0$, and hence the critical set of $F$ is $s\ge 0$ where $F$ vanishes identically. \end{proof} As an immediate corollary we are able to show that the $\mathbb{R}$-component of any nontrivial gradient flow line of $F$ converges to $0$ in forward time. \begin{lemma}\label{l:F-forward-time} For an arbitrary Riemannian metric $g$ on $\mathbb{R}\times S^{1}$ and a point $(s_{0}, t_{0})\in\mathbb{R}^{-}\times S^{1}$, the solution $\gamma(\tau)=(s(\tau), t(\tau))\in\mathbb{R}\times S^{1}$ to \begin{equation}\label{e:gradient-flow-ode} \gamma'(\tau)=\nabla^{g} F(\gamma(\tau)) \qquad \gamma(0)=(s_{0}, t_{0}) \end{equation} exists for all $\tau\ge 0$ and $\lim_{\tau\to\infty}s(\tau)=0$. \end{lemma} \begin{proof} Since $dF(s, t)=0$ for $s\ge 0$ and $F(s, t)=s$ agrees with $s$ for $s< -\delta$, we know that any solution to \eqref{e:gradient-flow-ode} stays bounded in a set of the form $[a, 0]\times S^{1}$ in forward time which implies that the solution exists for all $\tau\ge 0$. Given that the solution $\gamma(\tau)$ exists and is bounded in forward time, we know from general properties of gradient flows that $\lim_{\tau\to\infty}F(\gamma(\tau))$ exists and is equal to a critical value of $F$. Since we have just seen in Lemma \ref{l:F-crit-set}, that $0$ is the unique critical value of $F$, we conclude $\lim_{\tau\to\infty}F(\gamma(\tau))=0$. This with \eqref{e:bounds-F} implies that $\lim_{\tau\to\infty}s(\tau)=0$. \end{proof} The key step to proving the claim about the omega limit sets of flow lines of $F$ is the following lemma. \begin{lemma}\label{l:z-bounded} Let $\gamma(\tau)=(s(\tau), t(\tau))$ be a solution to \eqref{e:gradient-flow-ode}, and let $\tilde t:\mathbb{R}^{+}\to\mathbb{R}$ be a choice of lift of $t:\mathbb{R}^{+}\to S^{1}$. Then the function $z:\mathbb{R}^{+}\to\mathbb{R}$ defined by \[ z(\tau)=\frac{1}{s(\tau)}+\tilde t(\tau) \] is bounded. \end{lemma} \begin{proof} Let \[ \begin{bmatrix} A(s, t) & B(s, t) \\ B(s, t) & C(s, t) \end{bmatrix} \] be the matrix of the dual metric to $g$ with respect to the coordinate basis $\br{ds, dt}$ for $T^{*}(\mathbb{R}\times S^{1})$, and note positive definiteness tells us that $A(s, t)$ and $C(s, t)$ are positive for all $(s, t)\in\mathbb{R}\times S^{1}$. Furthermore, since $\gamma(\tau)$ remains in a compact region for all $\tau\ge 0$, we can conclude that the functions $A(\tau):=A(s(\tau), t(\tau))$, $B(\tau):=B(s(\tau), t(\tau))$, and $C(\tau):=C(s(\tau), t(\tau))$ are bounded and that $A(\tau)$ and $C(\tau)$ are bounded away from zero (or, equivalently that $A^{-1}(\tau)$ and $C^{-1}(\tau)$ are bounded). From Lemma \ref{l:F-forward-time} and the definition \eqref{e:F-definition} of $F$ it follows that $F(\gamma(\tau))=G(\gamma(\tau))$ for sufficiently large $\tau$. For such values of $\tau$ we use \eqref{e:G-s}-\eqref{e:G-t} with the boundedness of $A(\tau)$, $B(\tau)$, $C(\tau)$, and $A^{-1}(\tau)$ to compute \begin{align*} s'&=A(s, t) G_{s}(s, t)+B(s, t) G_{t}(s, t) \\ &=-s^{-2}e^{1/s}A(s, t)\bp{\sqrt{2}\sin(1/s+t+\tfrac{\pi}{4})-\tfrac{5}{4}+O(s^{2})} \intertext{and} \tilde t' &=t' \\ &=B(s, t)G_{s}(s, t)+C(s,t)G_{t}(s, t) \\ &=s^{-4}e^{1/s}A(s, t)O(s^{2}) \end{align*} with $O(s^{2})$ denoting, as usual, a function $h$ for which $s^{-2}h(s, t)$ remains bounded on a deleted neighborhood of $s=0$. We then have that \begin{align*} z' &=-s^{-2}s'+\tilde t' \\ &=s^{-4}e^{1/s}A(s, t)\bp{\sqrt{2}\sin(1/s+ t+\tfrac{\pi}{4})-\tfrac{5}{4}+O(s^{2})} \\ &=s^{-4}e^{1/s}A(s, t)\bp{\sqrt{2}\sin(z+\tfrac{\pi}{4})-\tfrac{5}{4}+O(s^{2})} \end{align*} and so, for sufficiently large values of $\tau$ (and thus sufficiently small values of $s(\tau)$), we'll have that \begin{equation}\label{e:z-inequality} \sqrt{2}\sin(z(\tau)+\tfrac{\pi}{4})-\tfrac{11}{8} \le \bp{s(\tau)^{-4}e^{1/s(\tau)}A(\tau)}^{-1}z'(\tau) \le \sqrt{2}\sin(z(\tau)+\tfrac{\pi}{4})-\tfrac{9}{8}. \end{equation} We claim this lets us conclude that $z(\tau)$ is bounded. Indeed, since $\frac{9}{8}\in(-\sqrt{2}, \sqrt{2})$, the solution set to the inequality \[ \sqrt{2}\sin(z+\tfrac{\pi}{4})-\tfrac{9}{8}<0 \] is a countable union of intervals which is invariant under translation by $2\pi\mathbb{Z}$. By \eqref{e:z-inequality}, $z(\tau)$ can't cross these intervals in the positive direction once $\tau$ is sufficiently large for \eqref{e:z-inequality} to hold. Similarly, since $\frac{11}{8}\in(-\sqrt{2}, \sqrt{2})$, $z(\tau)$ can't cross the intervals where \[ \sqrt{2}\sin(z+\tfrac{\pi}{4})-\tfrac{11}{8}>0 \] in the negative direction once $\tau$ is sufficiently large. We conclude that $z(\tau)$ is bounded for $\tau\in[0, \infty)$. \end{proof} We now complete the proof of the main theorem of the section \begin{proof}[Proof of Theorem \ref{t:function-construction}] By construction and Lemma \ref{l:F-crit-set}, $F$ satisfies all required properties, and it remains to show that the omega limit set of a solution to \eqref{e:gradient-flow-ode} is the circle $\br{0}\times S^{1}$. Let $\gamma(\tau)=(s(\tau), t(\tau))$ be a solution to \eqref{e:gradient-flow-ode} and let $\tilde t:\mathbb{R}^{+}\to\mathbb{R}$ be a lift of $t:\mathbb{R}^{+}\to S^{1}$. We have shown in Lemma \ref{l:z-bounded} above that the function $z=1/s+\tilde t$ is bounded on $[0, \infty)$. Since we know from Lemma \ref{l:F-forward-time} that $\lim_{\tau\to\infty}s(\tau)=0$ and since $s(\tau)<0$ for all $\tau\ge 0$, we can conclude that $\lim_{\tau\to\infty}\frac{1}{s(\tau)}=-\infty$. This in turn lets us conclude $\tilde t(\tau)$ approaches $+\infty$ as $\tau\to\infty$ or else $z$ would not be bounded. By continuity, the equation $\tilde t(\tau)=c$ has a solution $\tau_{c}$ for all $c\ge \tilde t(0)$. We then conclude that for any $\tau_{0}\in\mathbb{R}$ and any $t_{0}\in S^{1}$, there exists a $\tau_{t_{0}}>\tau_{0}$ so that $t(\tau_{t_{0}})=t_{0}$. This with the fact that $s(\tau)\to 0$ as $\tau\to\infty$ shows that $\br{0}\times S^{1}$ is the omega limit set of $\gamma$. \end{proof} \section{Finite-energy cylinders and planes with tori as limit sets}\label{s:main-proof} Here we prove our main theorem, Theorem \ref{t:main-theorem}, that is, we construct examples of finite-energy cylinders and finite-energy planes having tori as limit sets. The constructions take place in an arbitrarily small tubular neighborhood of a standard model of a transverse knot, so we begin by recalling some basic facts about transverse knots and explaining why this construction suffices to prove the main theorem. Let $(M^{2n+1}, \xi=\ker\lambda)$ be a contact manifold. An embedding $\gamma:S^{1}\to M$ is said to be a transverse knot if $\gamma$ is everywhere transverse to $\xi$ or equivalently, if $\lambda(\dot\gamma)$ is never zero. Transverse knots exist in abundance in any contact manifold. Indeed, by the well-known Darboux theorem for contact structures, there exists a contactomorphism --- that is a diffeomorphism preserving the contact structure --- between a neighborhood of any point in a contact manifold $(M^{2n+1}, \xi)$ and a neighborhood of $0$ in $\mathbb{R}^{2n+1}=\br{(z, x_{i}, y_{i})}$ equipped with the contact structure $\xi_{0}=\ker\lambda_{0}$ where $\lambda_{0}$ is the contact form \[ \lambda_{0}=dz+\alpha_{n} \] with \begin{equation}\label{e:alpha-defn} \alpha_{n}=\sum_{i=1}^{n}x_{i}\,dy_{i}-y_{i}\,dx_{i} \end{equation} (see e.g.\ \cite[Theorem 2.24]{geiges}). Since, for a given $k\in\mathbb{Z}\cap[1, n]$ and any constants $r>0$, $c_{i}$, $d_{i}\in\mathbb{R}$, circles of the form \[ \text{$x_{k}^{2}+y_{k}^{2}=r^{2}$, and $x_{i}=c_{i}$, $y_{i}=d_{i}$, for $i\ne k$} \] are easily seen to be transverse to the contact structure, we can conclude that transverse knots exist in every contact manifold and, indeed, that transverse knots exist in any neighborhood of a given point in a contact manifold. We next recall that one can use a Moser argument to prove a neighborhood theorem for transverse knots which tells us that there exists a contactomorphism between some neighborhood of any given transverse knot and a neighborhood of $S^{1}\times\br{0}$ in $S^{1}\times\mathbb{R}^{2n}=\br{(\theta, x_{i}, y_{i})}$ equipped with the contact structure $\xi_{0}=\ker \lambda_{0}$ where \begin{equation}\label{e:transverse-standard} \lambda_{0}=d\theta+\alpha_{n} \end{equation} with $\alpha_{n}$ as defined in \eqref{e:alpha-defn} above (see e.g.\ \cite[Theorem 2.32/Example 2.33]{geiges}). We will refer to $S^{1}\times\br{0}\subset (S^{1}\times\mathbb{R}^{2n}, \xi_{0})$ as the standard model of a transverse knot in $S^{1}\times\mathbb{R}^{2n}$. Given the facts recalled in the previous two paragraphs, it suffices for the proof of our main theorem to construct the desired finite-energy planes and cylinders in any given neighborhood of the standard model of a transverse knot in $S^{1}\times\mathbb{R}^{2n}$. Since $d\alpha_{n}=2\sum_{i=1}^{n}dx_{i}\wedge dy_{i}$ is a symplectic form on $\mathbb{R}^{2n}$, $S^{1}\times\mathbb{R}^{2n}$ equipped with the contact form \eqref{e:transverse-standard} has the structure of a prequantization space. We can thus apply Theorems \ref{t:gradient-flow-hol-cylinders} and \ref{t:function-construction} to construct a finite-energy cylinder having tori of periodic orbits as its limit sets. \begin{theorem} Let $r_{+}>r_{-}>0$. Then there exists a smooth function $F:S^{1}\times\mathbb{R}^{2n}$ and an almost complex structure $J\in\mathcal{J}(S^{1}\times\mathbb{R}^{2n}, \xi_{0})$ so that passing through every point $(\theta_{0}, p, z)\in S^{1}\times \mathbb{R}^{2(n-1)}\times \mathbb{R}^{2}$ with $\abs{z}\in (r_{-}, r_{+})$ is a finite-energy cylinder for the data $(e^{F}\lambda, J)$ with limit sets equal to the union of tori $S^{1}\times\br{p}\times\br{\abs{z}=r_{-}}\cup\br{\abs{z}=r_{+}} \in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$. \end{theorem} \begin{proof} With $F_{\delta}$ a function with the properties stated in Theorem \ref{t:function-construction}, we consider a function $G:\mathbb{R}\times S^{1}\to\mathbb{R}$ defined by \[ G(\rho, \phi)= \begin{cases} F_{1/4}(\rho, \phi) & \rho>-3/4 \\ -F_{1/4}(-\rho-1, \phi)-1 & \rho<-1/4 \end{cases} \] which defines a smooth function since $F_{1/4}(\rho, \phi)=\rho=-F_{1/4}(-\rho-1, \phi)-1$ for $\rho\in [-3/4, -1/4]$. For any initial condition $(\rho_{0}, \phi_{0})\in [-3/4, -1/4]\times S^{1}$ the forward gradient flow of $G$ for any metric agrees with that of $F_{1/4}$ and thus limits to $\br{0}\times S^{1}$. Similarly, for any initial condition $(\rho_{0}, \phi_{0})\in [-3/4, -1/4]\times S^{1}$ the backward gradient flow of $G$ for any metric agrees with that of $-F_{1/4}(-\rho-1, \phi)-1$ which is conjugated to the forward gradient flow of $F_{1/4}$ by reflection and translation and thus limits in backward time to $\br{-1}\times S^{1}$. We consider the diffeomorphism $p:\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}\to\mathbb{R}^{2}\setminus\br{0}$ defined by \[ p(\rho, \phi)=(r_{+}(r_{+}/r_{-})^{\rho}\cos \phi, r_{+}(r_{+}/r_{-})^{\rho}\sin \phi) \] which maps the circles $\br{-1}\times S^{1}$ and $\br{0}\times S^{1}$ to the circles $\abs{z}=r_{-}$ and $\abs{z}=r_{+}$ respectively. We then define a function $F:\mathbb{R}^{2n}\to\mathbb{R}$ by \[ F(x_{1}, y_{1}, \dots, x_{n}, y_{n})= \begin{cases} G(p^{-1}(x_{n}, y_{n})) & (x_{n}, y_{n})\ne 0 \\ -1 & (x_{n}, y_{n})= 0 \end{cases} \] which defines a smooth function since $G(\rho, \phi)=-F_{1/4}(-\rho-1, \phi)-1$ for $\rho\le -1$ and hence $G(p^{-1}(z))=-1$ for $\abs{z}<r_{-}$. We observe that for any metric on $\mathbb{R}^{2n}\approx\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$ for which the last $T\mathbb{R}^{2}$ is everywhere orthogonal to $T\mathbb{R}^{2(n-1)}$ we will have that $\nabla F=(0, \nabla G)\in \mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$ and thus the gradient flow of $F$ for initial points $(p, z)\in\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$ with $\abs{z}\in(r_{-}, r_{+})$ will have the circles $\abs{z}\in\br{r_{-}, r_{+}}$ as limit sets. Choosing then an almost complex structure $J\in\mathcal{J}(\mathbb{R}^{2n}, d\alpha_{n})$ on $\mathbb{R}^{2(n-1)}\times \mathbb{R}^{2}$ which preserves the two factors (for example the standard $J_{0}$ defined by $J_{0}\partial_{x_{i}}=\partial_{y_{i}}$), we know from Theorem \ref{t:gradient-flow-hol-cylinders} that gradient flow lines of $F$ on $\mathbb{R}^{2n}$ with respect to the metric $d\beta(\cdot, J\cdot)$ lift to finite-energy cylinders in $\mathbb{R}\times S^{1}\times\mathbb{R}^{2n}$ for the data $(e^{\pi^{*}F}\lambda, J)$. Since the nonconstant gradient flow lines for the function $F$ will have the circles $\abs{z}=r_{\pm}$ as limit sets, the corresponding finite-energy cylinders in $S^{1}\times\mathbb{R}^{2n}$ will have the tori $S^{1}\times\br{p}\times\br{\abs{z}=r_{\pm}}$ as limit sets. \end{proof} Using Theorem \ref{t:gradient-flow-hol-cylinders} to construct finite-energy a plane with a torus as a limit set is somewhat more subtle since the theorem only tells us how to construct a cylinder from a gradient flow line. To construct a plane we will use the theorem to construct a cylinder with a removable singularity. \begin{theorem}\label{t:plane-construction} Let $r_{0}>0$. Then there exists a smooth function $\tilde F:S^{1}\times\mathbb{R}^{2n}$ and an almost complex structure $J\in\mathcal{J}(S^{1}\times\mathbb{R}^{2n}, \xi_{0})$ so that passing through every point $(\theta_{0}, 0, z)\in S^{1}\times \mathbb{R}^{2(n-1)}\times \mathbb{R}^{2}$ with $\abs{z}<r_{0}$ is a finite-energy plane for the data $(e^{\tilde F}\lambda_{0}, J)$ with limit set equal to the embedded torus $S^{1}\times\br{0}\times\br{\abs{z}=r_{0}}\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$. \end{theorem} The strategy of the proof is to consider a set which is contactomorphic to the complement of the $x_{n}=y_{n}=0$ locus of a standard model of a transverse knot and show that this can be given the structure of a prequantization space with respect to the angular variable on $\br{(x_{n}, y_{n})}\setminus\br{0}$. We then use Theorems \ref{t:gradient-flow-hol-cylinders} and \ref{t:function-construction} to construct a pseudoholomorphic cylinder which has a removable puncture mapped to the the $x_{n}=y_{n}=0$ locus. We begin with a computational lemma. \begin{lemma}\label{l:plane-construction} Consider $W:=S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}=\br{(\theta, x_{i}, y_{i}, \rho)}$ equipped with the $1$-form \begin{equation}\label{e:beta-planes} \beta=e^{-2\rho}\bp{d\theta+\alpha_{n-1}} \end{equation} with $\alpha_{n-1}$ as defined in \eqref{e:alpha-defn}. Then: \begin{itemize} \item $d\beta$ is a symplectic form on $W$. \item Consider the corresponding prequantization space $(S^{1}\times W, \lambda:=d\phi+\pi^{*}\beta)$ over $W$. With $\lambda_{0}$ as defined in \eqref{e:transverse-standard}, the map \[ \Phi:(S^{1}\times W, \xi=\ker\lambda)\to (S^{1}\times \mathbb{R}^{2(n-1)}\times(\mathbb{R}^{2}\setminus\br{0}), \xi_{0}=\ker\lambda_{0}) \] defined by \begin{equation}\label{e:Phi-definition} \Phi(\phi, \theta, x_{i}, y_{i}, \rho)=(\theta, x_{i}, y_{i}, e^{\rho}\cos\phi, e^{\rho}\sin\phi) \end{equation} is a contactomorphism and, in particular, \begin{equation}\label{e:lambda-0-pullback} \Phi^{*}\lambda_{0}=e^{2\rho}\lambda. \end{equation} \item For any choice of $j_{0}\in\mathcal{J}(\mathbb{R}^{2(n-1)}, d\alpha_{n-1})$ the endomorphism $j_{1}\in\operatorname{End}(TW)$ defined by \begin{equation}\label{e:j1-definition} \begin{gathered} j_{1}(\theta, p, \rho)\partial_{\rho}=-e^{2\rho}\partial_{\theta}, \qquad j_{1}(\theta, p, \rho)\partial_{\theta}=e^{-2\rho}\partial_{\rho}, \text{ and} \\ j_{1}(\theta, p, \rho)v=j_{0}(p)v-\alpha_{n-1}(j_{0}(p)v)\,\partial_{\theta}+\alpha_{n-1}(v)e^{-2\rho}\,\partial_{\rho} \text{ for $v\in T\mathbb{R}^{2(n-1)}$}. \end{gathered} \end{equation} is an almost complex structure on $W$ compatible with $d\beta$, i.e.\ $j_{1}\in\mathcal{J}(W, d\beta)$, and the corresponding metric $g_{j_{1}}:=d\beta\circ(I\times j_{1})$ on $W$ is given by \begin{equation}\label{e:metric-plane} g_{j_{1}} = 2 \,d\rho\otimes d\rho +2e^{-4\rho}(d\theta+\alpha_{n-1})\otimes\bp{d\theta+\alpha_{n-1}} +e^{-2\rho}d\alpha_{n-1}\circ (I\times j_{0}). \end{equation} \item Let $\tilde j_{1}\in\mathcal{J}(S^{1}\times W, \xi)$ be the $S^{1}$-invariant complex structure on $\xi$ determined by $j_{1}$ as defined above, i.e.\ $\tilde j$ is the complex structure characterized by $\widetilde{j_{1}v}=\tilde j_{1}\tilde v$ with \begin{equation}\label{e:lift-planes-proof} \tilde v=-\beta(v)\partial_{\phi}+v=-e^{-2\rho}\bp{d\theta(v)+\alpha_{n-1}(v)}\partial_{\phi}+v \end{equation} the lift of $v$ to an $S^{1}$-invariant section of $\xi$ from \eqref{e:horizontal-lift}. Then $\Phi_{*}\tilde j_{1}=d\Phi\circ \tilde j_{1}\circ d\Phi^{-1}\in\mathcal{J}(S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}\setminus\br{0}, \xi_{0})$ admits a smooth extension to a compatible $J\in\mathcal{J}(S^{1}\times\mathbb{R}^{2n}, \xi_{0})$. \end{itemize} \end{lemma} Assuming for the moment the results of the lemma, we proceed with the proof of Theorem \ref{t:plane-construction}. \begin{proof}[Proof of Theorem \ref{t:plane-construction}] Given $r_{0}>0$ we define a smooth function $G:W\to\mathbb{R}$ by \[ G(\theta, p, \rho)=2(F_{1}(\rho-\log r_{0}, \theta)+\log r_{0}) \] with $F_{1}$ a function satisfying the properties given in Theorem \ref{t:function-construction} with $\delta=1$. We note that as a result of the definition and of Theorem \ref{t:function-construction}, $G(\theta, p, \rho)=2\rho$ for $\rho<\log r_{0}-1$ and $G(\theta, p, \rho)=2\log r_{0}$ for $\rho\ge\log r_{0}$. Moreover, since $\alpha_{n-1}=0$ along the $p=(x_{1}, y_{1}, \dots, x_{n-1}, y_{n-1})=0$ locus, we have that \begin{equation}\label{e:G-gradient} \nabla G(\theta, 0, \rho)=\frac{1}{2}\bp{G_{\rho}\partial_{\rho}+G_{\theta}e^{4\rho}\partial_{\theta}} =\partial_{\rho} F_{1}(\rho-\log r_{0}, \theta)\partial_{\rho}+\partial_{\theta} F_{1}(\rho-\log r_{0}, \theta)e^{4\rho}\partial_{\theta} \end{equation} where $\nabla G$ is the gradient with respect to the metric \eqref{e:metric-plane} on $W$. Therefore, for any initial point $w_{0}=(\theta_{0}, 0, \rho_{0})\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}$ with $\rho_{0}<\log r_{0}$, the solution $\gamma(s)$ to the equation \begin{equation}\label{e:ode-gamma-G} \dot\gamma(s)=2\pi\nabla G(\gamma(s)) \end{equation} stays within the embedded cylinder $S^{1}\times\br{0}\times\mathbb{R}\subset S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}$ and agrees with a gradient flow for the function $(\rho, \theta)\in \mathbb{R}\times S^{1}\mapsto 2(F_{1}(\rho-\log r_{0}, \theta)+\log r_{0})$ for an appropriate metric on the cylinder. Thus the flow exists in forward time and has the circle $S^{1}\times\br{0}\times\br{\log r_{0}}\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}$ as its omega limit set. Meanwhile, using the fact that $F_{1}(\rho-\log r_{0}, \theta)=\rho-\log r_{0}$ for $\rho<\log r_{0}-1$, we have from \eqref{e:G-gradient} that \[ \nabla G(\theta, 0, \rho)=\partial_{\rho} \qquad \text{ for $\rho<\log r_{0}-1$} \] and thus that the solution $\gamma$ to \eqref{e:ode-gamma-G} is given by $\gamma(s)=(\theta_{1}, 0, 2\pi s+s_{1})$ for sufficiently small $s$ with $\theta_{1}\in S^{1}$ and $s_{1}\in\mathbb{R}$ appropriate constants. Thus the flow exists indefinitely in backward time as well. Applying Theorem \ref{t:gradient-flow-hol-cylinders} we know that the map $\tilde u(s, t)=(a(s),\phi(s)+2\pi t, \gamma(s))\in\mathbb{R}\times S^{1}\times W$ where $a:\mathbb{R}\to\mathbb{R}$ and $\phi:\mathbb{R}\to S^{1}$ satisfy $\dot a(s)=2\pi e^{G(\gamma(s))}$ and $\dot\phi(s)=2\pi \beta(\nabla G(\gamma(s)))$ is a finite-energy cylinder for the data $(e^{\pi^{*}_{W}G}\lambda, \tilde j_{1})$ with energy $2\pi \lim_{s\to\infty}e^{G(\gamma(s))}=2\pi e^{2\log r_{0}}=2\pi r_{0}^{2}$ and the torus $S^{1}\times S^{1}\times \br{0}\times\br{\log r_{0}}$ as its limit set. Moreover, since $G(\theta, p, \rho)=2\rho$ and $\nabla G(\theta, p, \rho)=\partial_{\rho}$ for $\rho<\log r_{0}-1$, there exist constants $a_{1}\in\mathbb{R}$, $\theta_{1}\in S^{1}$, $s_{1}\in\mathbb{R}$, and $t_{1}\in S^{1}$ so that \begin{equation}\label{e:map-near-zero} \tilde u(s, t) =(\pi e^{4\pi s}+a_{1},t_{1}+2\pi t, \theta_{1},0, 2\pi s+s_{1}) \in\mathbb{R}\times S^{1}\times S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R} =\mathbb{R}\times S^{1} \times W. \end{equation} for sufficiently negative $s$. We next show that the map $\tilde v:=(a, \Phi\circ u):\mathbb{R}\times\mathbb{S}^{1}\to\mathbb{R}\times S^{1}\times\mathbb{R}^{2n}$, with $\Phi:(S^{1}\times W, \xi) \to (S^{1}\times \mathbb{R}^{2n}, \xi_{0})$ the contactomorphism defined in \eqref{e:Phi-definition}, has a removable singularity at $-\infty$ and thus extends to a pseudoholomorphic plane. We first note that \eqref{e:lambda-0-pullback} gives us \[ [\Phi^{-1}]^{*}\bp{e^{(\pi_{W}^{*}G)}\lambda}=e^{(\pi_{W}^{*}G-2\rho)\circ\Phi^{-1}}\lambda_{0} \] and, since $G(\theta, p, \rho)=2\rho$ for $\rho<\log r_{0}-1$, we'll have that $\tilde F:=(\pi_{W}^{*}G-2\rho)\circ\Phi^{-1}$ extends to a smooth function on $S^{1}\times\mathbb{R}^{2n}$ and thus $[\Phi^{-1}]^{*}\bp{e^{(\pi_{W}^{*}G)}\lambda}=e^{\tilde F}\lambda_{0}$ defines a contact form on $S^{1}\times\mathbb{R}^{2n}$. Since, by Lemma \ref{l:plane-construction}, the pushed-forward complex structure $\Phi_{*}\tilde j_{1}=d\Phi\circ j_{1}\circ d\Phi^{-1}$ has a smooth extension to a $J\in\mathcal{J}(S^{1}\times\mathbb{R}^{2n}, \xi_{0})$, it suffices to show that the map $\tilde v=(a, \Phi\circ u)$ has a smooth extension. To see this, we use the definition \eqref{e:Phi-definition} with \eqref{e:map-near-zero} to compute that \[ \tilde v=(a, \Phi\circ u)=(\pi e^{4\pi s}+a_{1}, \theta_{0},0, e^{2\pi s+s_{1}}\cos(t_{1}+2\pi t), e^{2\pi s+s_{1}}\sin(t_{1}+2\pi t))\in \mathbb{R}\times S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}^{2} \] for sufficiently negative $s$. Precomposing with the biholomorphic map $\psi:\mathbb{C}\setminus\br{0}\to\mathbb{R}\times S^{1}=\mathbb{C}/i\mathbb{Z}$ defined by \[ \psi(z)=(\log\abs{z}/2\pi, \arg{z}/2\pi) \] we find that \[ \tilde v(\psi(z))=(\pi\abs{z}^{2}+a_{1}, \theta_{0}, 0, e^{s_{1}+it_{1}}z) \in \mathbb{R}\times S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}(\approx \mathbb{C}) \] which clearly extends smoothly over $z=0$. We note moreover that the limit set $S^{1}\times S^{1}\times \br{0}\times\br{\log r_{0}}\subset S^{1}\times S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}$ gets mapped by $\Phi$ to the embedded torus $S^{1}\times\br{0}\times\br{\abs{z}= r_{0}}\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$, while the set of points $(\phi, \theta, 0, \rho)\in S^{1}\times S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}$ with $\rho<\log r_{0}$ gets mapped by $\Phi$ to the set $(\theta, 0, z)\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$ with $\abs{z}\in(0, r_{0})$. Thus, since we were able, by appropriate choice of initial point of the flow of $\nabla G$, to construct a pseudoholomorphic cylinder for the data $(e^{\pi_{W}G}\lambda, \tilde j_{1})$ through any point $(\phi, \theta, 0, \rho)\in S^{1}\times S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}$, we can construct a pseudoholomorphic plane for the data $(e^{\tilde F}\lambda_{0}, J)$ through any point $(\theta, 0, z)\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}$ with $\abs{z}<r_{0}$ as desired. This completes the proof. \end{proof} \begin{remark} If we choose an initial point in the proof of theorem to be a point $(\theta_{0}, p_{0}, \rho_{0})\in S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}$ with $p_{0}\ne 0$, one can still construct a finite-energy plane from the resulting flow line with a limit set consisting of more than a single orbit, although the limit set may be more complicated than a torus. Indeed, for a given choice of $j_{0}\in \mathcal{J}(\mathbb{R}^{2(n-1)}, d\alpha_{n-1})$, we let $g_{j_{0}}=d\alpha_{n-1}(\cdot, j_{0})$ denote the associated metric and observe that \[ \alpha_{n-1}(p)(v)=\frac{1}{2}d\alpha_{n-1}(p, v)=-\frac{1}{2}d\alpha_{n-1}(v, p)=\frac{1}{2}g_{j_{0}}(v, j_{0}p). \] Using this, one can compute the gradient of the function $G$ with respect to the metric \eqref{e:metric-plane} to be given by \begin{align*} \nabla G(\theta, p, \rho) &=2^{-1}\bp{ G_{\rho}\,\partial_{\rho}+e^{2\rho}G_{\theta}\bp{e^{2\rho}+\abs{p}_{j_{0}}}\,\partial_{\theta} - e^{2\rho}G_{\theta}j_{0}p } \\ &= \partial_{\rho} F_{1}(\rho-\log r_{0}, \theta)\,\partial_{\rho} +\partial_{\theta} F_{1}(\rho-\log r_{0}, \theta)e^{2\rho}\bp{e^{2\rho}+\abs{p}_{j_{0}}}\,\partial_{\theta} -\partial_{\theta} F_{1}(\rho-\log r_{0}, \theta) e^{2\rho} j_{0}p. \end{align*} with $\abs{\cdot}_{j_{0}}$ the norm with respect to the metric $g_{j_{0}}$. We note the the $\mathbb{R}^{2(n-1)}$-component of $\nabla G(\theta, p, \rho)$ is always orthogonal to $p$. If the almost complex structure $j_{0}$ is constant, we can thus conclude that $\abs{p}_{j_{0}}$ is constant along the flow. Thus, $\rho$- and $\theta$-components of the gradient flow for $G$ agree with a gradient flow for a shift of the function $F$ on $\mathbb{R}\times S^1$ for an appropriate metric (specifically the metric $g=d\rho\otimes d\rho+e^{-2\rho}\bp{e^{2\rho}+c^{2}}^{-1}d\theta\otimes d\theta$ with $c^{2}$ equal to the constant value of $\abs{p}_{j_{0}}$ along the flow line). From this one can argue that under the projection $S^{1}\times\mathbb{R}^{2(n-1)}\times\mathbb{R}^{2}\to S^{1}\times\br{0}\times\mathbb{R}^{2}$ the limit set of any plane obtained as a lift of a gradient flow of the function $G$ in our theorem will project to a torus. \end{remark} Finally, to complete the proof of Theorem \ref{t:plane-construction}, we give the proof of Lemma \ref{l:plane-construction} above. \begin{proof}[Proof of Lemma \ref{l:plane-construction}] We first show that $d\beta$ is a symplectic form. Computing, we have that \begin{equation}\label{e:d-beta} d\beta =e^{-2\rho}(-2\,d\rho\wedge d\theta-2\,d\rho\wedge\alpha_{n-1}+d\alpha_{n-1}) \end{equation} and hence \begin{align*} d\beta^{n} &=e^{-2n\rho}(-2\,d\rho\wedge d\theta-2\,d\rho\wedge\alpha_{n-1})\wedge(d\alpha_{n-1})^{n-1} \\ &=-2e^{-2n\rho}\,d\rho\wedge d\theta\wedge(d\alpha_{n-1})^{n-1} \end{align*} which is nowhere vanishing on $W=S^{1}\times \mathbb{R}^{2(n-1)}\times\mathbb{R}$. Hence $d\beta$ is a symplectic form on $W$ as claimed. Next, we show that the map $\Phi:(W, \xi=\ker\lambda) \to (S^{1}\times\mathbb{R}^{2(n-1)}\times(\mathbb{R}^{2}\setminus\br{0}), \xi_{0}=\ker\lambda_{0})$ defined in \eqref{e:Phi-definition} is a contactomorphism satisfying \eqref{e:lambda-0-pullback}. From the definition \eqref{e:Phi-definition} of the map, it's clear that $\Phi$ is a diffeomorphism and that \[ \Phi^{*}d\theta=d\theta \qquad \Phi^{*}dx_{i}=dx_{i} \qquad \Phi^{*}dy_{i}=dy_{i} \] for $i\in\mathbb{Z}\cap[1, n-1]$, while a straightforward computation shows that \begin{equation}\label{e:Phi-pullback} \Phi^{*}(x_{n}\,dy_{n}-y_{n}\,dx_{n})=e^{2\rho}\,d\phi \qquad \Phi^{*}(x_{n}\,dx_{n}+y_{n}\,dy_{n})=e^{2\rho}\,d\rho. \end{equation} Computing then gives \begin{align*} \Phi^{*}\lambda_{0} &=d\theta+\alpha_{n-1}+\Phi^{*}(x_{n}\,dy_{n}-y_{n}\,dx_{n}) \\ &=d\theta+\alpha_{n-1}+e^{2\rho}\,d\phi \\ &=e^{2\rho}\lambda \end{align*} which shows that $\Phi$ is a contactomorphism and establishes \eqref{e:lambda-0-pullback} as claimed. We next address the third point. The fact that $j_{1}^{2}\partial_{\rho}=-\partial_{\rho}$ and $j_{1}^{2}\partial_{\theta}=-\partial_{\theta}$ is immediate from the definition \eqref{e:j1-definition}. Meanwhile for $v\in T(\mathbb{R}^{2(n-1)})$, we use \eqref{e:j1-definition} twice with $j_{0}^{2}v=-v$ to compute \begin{align*} j_{1}^{2}v &=j_{1}\bp{j_{0}v-\alpha_{n-1}(j_{0}v)\,\partial_{\theta}+\alpha_{n-1}(v)e^{-2\rho}\,\partial_{\rho}} \\ &=j_{1}(j_{0}v)-\alpha_{n-1}(j_{0}v)j_{1}\partial_{\theta}+\alpha_{n-1}(v)e^{-2\rho}j_{1}\partial_{\rho} \\ &=j_{0}^{2}v-\alpha_{n-1}(j_{0}^{2}v)\,\partial_{\theta}+\alpha_{n-1}(j_{0}v)e^{-2\rho}\,\partial_{\rho} \\ &\hskip.25in -\alpha_{n-1}(j_{0}v)(e^{-2\rho}\,\partial_{\rho})+\alpha_{n-1}(v)e^{-2\rho}(-e^{2\rho}\partial_{\theta}) \\ &=-v \end{align*} which shows that $j_{1}$ is an almost complex structure on $W$. To check compatibility of $j_{1}$ with $d\beta$ we compute from \eqref{e:j1-definition} that \begin{align*} d\rho\circ j_{1} &=e^{-2\rho}\bp{d\theta+\alpha_{n-1}} \\ d\theta\circ j_{1} &=-e^{2\rho}\,d\rho-\alpha_{n-1}\circ j_{0}\circ d\pi_{\mathbb{R}^{2(n-1)}}. \\ dx_{i}\circ j_{1}&=dx_{i}\circ j_{0}\circ d\pi_{\mathbb{R}^{2(n-1)}} \\ dy_{i}\circ j_{1}&=dy_{i}\circ j_{0}\circ d\pi_{\mathbb{R}^{2(n-1)}} \end{align*} which with \eqref{e:d-beta} gives us \[ d\beta\circ(I\times j_{1}) = 2 \,d\rho\otimes d\rho +2e^{-4\rho}(d\theta+\alpha_{n-1})\otimes\bp{d\theta+\alpha_{n-1}} +e^{-2\rho}d\alpha_{n-1}\circ (I\times j_{0}). \] as claimed. By the assumption that $j_{0}$ is compatible with $d\alpha_{n-1}$, this is clearly symmetric and positive definite, and thus $j_{1}\in\mathcal{J}(W, d\beta)$ as claimed. Finally, we show that $\Phi_{*}\tilde j_{1}$ has a smooth extension to a compatible complex structure $J\in\mathcal{J}(S^{1}\times\mathbb{R}^{2n}, \xi_{0})$. The contact structure $\xi_{0}=\ker\lambda_{0}$ is spanned by the smooth sections \[ -\alpha_{n}(\partial_{x_{i}})\partial_{\theta}+\partial_{x_{i}} \qquad -\alpha_{n}(\partial_{y_{i}})\partial_{\theta}+\partial_{y_{i}} \] so it suffices to check that $\Phi_{*}\tilde j_{1}$ times each of these sections has a smooth continuation. We first observe that from the definition \eqref{e:Phi-definition} of $\Phi$ we immediately have \begin{equation}\label{e:Phi-pushforward-1} \Phi_{*}\partial_{\theta}=\partial_{\theta} \qquad \Phi_{*}\partial_{x_{i}}=\partial_{x_{i}} \qquad \Phi_{*}\partial_{y_{i}}=\partial_{y_{i}} \end{equation} for $i$ between $1$ and $n-1$, while \eqref{e:Phi-pullback} give us \begin{equation}\label{e:Phi-pushforward-2} \Phi_{*}\partial_{\rho}=x_{n}\,\partial_{x_{n}}+y_{n}\,\partial_{y_{n}} \qquad \Phi_{*}\partial_{\phi}=x_{n}\,\partial_{y_{n}}-y_{n}\,\partial_{x_{n}}. \end{equation} Thus, for $v\in T(\mathbb{R}^{2(n-1)})=\operatorname{span}\br{\partial_{x_{i}}, \partial_{y_{i}}}_{i=1}^{n-1}$, a straightforward computation using that $d\theta(v)=0$ along with \eqref{e:lift-planes-proof} and \eqref{e:Phi-pushforward-1} shows that \[ -\alpha_{n}(v)\partial_{\theta}+v=\Phi_{*}(-\alpha_{n-1}(v)\widetilde{\partial_{\theta}}+\tilde v). \] Computing further with this, the definition \eqref{e:j1-definition} of $j_{1}$, and $\tilde j_{1}\tilde v=\widetilde{j_{1}v}$ then shows that \begin{align*} (\Phi_{*}\tilde j_{1})(-\alpha_{n}(v)\partial_{\theta}+v) &=(\Phi_{*}\tilde j_{1})\Phi_{*}(-\alpha_{n-1}(v)\widetilde{\partial_{\theta}}+\tilde v) \\ &=\Phi_{*}( -\alpha_{n-1}(v)\widetilde{j_{1}\partial_{\theta}}+\widetilde{j_{1}v}) \\ &=\Phi_{*}(-e^{-2\rho}\alpha_{n-1}(v)\widetilde{\partial_{\rho}}+\widetilde{j_{0}v}-\alpha_{n-1}(j_{0}v)\widetilde{\partial_{\theta}}+\alpha_{n-1}(v)e^{-2\rho}\widetilde{\partial_{\rho}}) \\ &=\Phi_{*}(-\alpha_{n-1}(j_{0}v)\widetilde{\partial_{\theta}}+\widetilde{j_{0}v}) \\ &=-\alpha_{n-1}(j_{0}v)\partial_{\theta}+j_{0}v \end{align*} which clearly extends smoothly over the $x_{n}=y_{n}=0$ locus since there is no $x_{n}$- or $y_{n}$-dependence. Meanwhile, a straightforward computation using \eqref{e:Phi-definition} and \eqref{e:Phi-pushforward-2} shows that \[ \partial_{x_{n}}=\Phi_{*}\bp{e^{-\rho}\bp{\cos\phi\,\partial_{\rho}-\sin\phi\,\partial_{\phi}}} \quad\text{ and }\qquad \partial_{y_{n}}=\Phi_{*}\bp{e^{-\rho}\bp{\sin\phi\,\partial_{\rho}+\cos\phi\,\partial_{\phi}}} \] and using this with \eqref{e:Phi-definition}, \eqref{e:Phi-pushforward-1}, and \eqref{e:lift-planes-proof} shows that \[ -\alpha_{n}(\partial_{x_{n}})\partial_{\theta}+\partial_{x_{n}} =y_{n}\,\partial_{\theta}+\partial_{x_{n}} =\Phi_{*}(e^{\rho}\sin\phi\,\widetilde\partial_{\theta}+e^{-\rho}\cos\phi\,\widetilde\partial_{\rho}) \] and \[ -\alpha_{n}(\partial_{y_{n}})\partial_{\theta}+\partial_{y_{n}} =-x_{n}\,\partial_{\theta}+\partial_{y_{n}} =\Phi_{*}(-e^{\rho}\cos\phi\,\widetilde\partial_{\theta}+e^{-\rho}\sin\phi\,\widetilde\partial_{\rho}). \] Computing further using the definition \eqref{e:j1-definition} of $j_{1}$ with $\tilde j_{1}\tilde v=\widetilde{j_{1}v}$ shows that \[ \Phi_{*}\tilde j_{1}(-\alpha_{n}(\partial_{x_{n}})\partial_{\theta}+\partial_{x_{n}}) =-\alpha_{n}(\partial_{y_{n}})\partial_{\theta}+\partial_{y_{n}} =-x_{n}\,\partial_{\theta}+\partial_{y_{n}} \] and \[ \Phi_{*}\tilde j_{1}(-\alpha_{n}(\partial_{y_{n}})\partial_{\theta}+\partial_{y_{n}}) =\alpha_{n}(\partial_{x_{n}})\partial_{\theta}-\partial_{x_{n}} =-y_{n}\,\partial_{\theta}-\partial_{x_{n}} \] which also extend smoothly over $x_{n}=y_{n}=0$. This completes the proof. \end{proof} \end{document}

\begin{document} \title{The Submodular Santa Claus Problem \ in the Restricted Assignment Caseootnote{This research was supported by the Swiss National Science Foundation project 200021-184656 “Randomness in Problem Instances and Randomized Algorithms.”} \begin{abstract} The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem $n$ unsplittable resources have to be assigned to $m$ players, each with a monotone submodular utility function $f_i$. The goal is to maximize $\min_i f_i(S_i)$ where $S_1,\dotsc,S_m$ is a partition of the resources. The result by Goemans et al. implies a polynomial time $O(n^{1/2 +\varepsilon})$-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all $f_i$ are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources $\Gamma_i$ and the individual valuation functions are defined as $f_i(S) = f(S \cap \Gamma_i)$ for a global linear function $f$. This can also be interpreted as maximizing $\min_i f(S_i)$ with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant. Namely, if $f$ is a monotone submodular function, we can in polynomial time compute an $O(\log\log(n))$-approximate solution. \end{abstract} \pagebreak \section{Introduction} In the Santa Claus problem (sometimes referred to as Max-Min Fair Allocation) we are given a set of $n$ players $P$ and a set of $m$ indivisible resources $R$. In its full generality, each player $i\in P$ has a utility function $f_i:2^R\mapsto \mathbb{R}_{\ge 0}$, where $f_i(S)$ measures the happiness of player $i$ if he is assigned the resource set $S$. The goal is to find a partition of the resources that maximizes the happiness of the least happy player. Formally, we want to find a partition $\{S_i\}_{i\in P}$ of the resources that maximizes \begin{equation*} \min_{i\in P} f_i(S_i) . \end{equation*} Most of the recent literature on this problem focuses on cases where $f_i$ is a linear function for all players $i$. If we assume all valuation functions are linear, the best approximation algorithm known for this problem, designed by Chakrabarty, Chuzhoy, and Khanna~\cite{DBLP:conf/focs/ChakrabartyCK09}, has an approximation rate of $n^{\epsilon}$ and runs in time $n^{O(1/\epsilon)}$ for $\epsilon\in\Omega(\log\log(n)/\log(n))$. On the negative side, it is only known that computing a $(2 - \delta)$-approximation is NP-hard~\cite{LenstraSchmoysTardos}. Apart from this there has been significant attention on the so-called \emph{restricted assignment case}. Here the utility functions are defined by one linear function $f$ and a set of resources $\Gamma_i$ for each player $i$. Intuitively, player $i$ is interested in the resources $\Gamma_i$, whereas the other resources are worthless for him. The individual utility functions are then implicitly defined by $f_i(S)=f(S\cap \Gamma_i)$. In a seminal work Bansal and Srividenko~\cite{BansalSrividenko} provide a $O(\log \log (m)/\log \log \log (m))$-approximation algorithm for this case. This was improved by Feige~\cite{Feige} to an $O(1)$-approximation. Further progress on the constant or the running time was made since then, see e.g.~\cite{DBLP:journals/talg/AnnamalaiKS17, DBLP:conf/soda/DaviesRZ20, DBLP:conf/icalp/ChengM19, DBLP:conf/icalp/ChengM18, JANSEN2020106025, Asadpour_local_search, Polacek}. Let us now move to the non-linear case. Indeed, the problem becomes hopelessly difficult without any restrictions on the utility functions. Consider the following reduction from set packing. There are sets of resources $\{S_1,\dotsc,S_k\}$ and all utility functions are equal and defined by $f_i(S) = 1$ if $S_j \subseteq S$ for some $j$ and $f_i(S) = 0$ otherwise. Deciding whether there are $m$ disjoint sets in $S_1,\dotsc,S_k$ (a classical NP-hard problem) is equivalent to deciding whether the optimum of the Santa Claus problem is non-zero. In particular, obtaining any bounded approximation ratio for Santa Claus in this case is NP-hard. Two naturally arising properties of utility functions are monotonicity and submodularity, see for example the related submodular welfare problem~\cite{DBLP:journals/geb/LehmannLN06,DBLP:conf/stoc/Vondrak08} where the goal is to maximize $\sum_i f_i(S_i)$. A function $f$ is monotone, if $f(S) \le f(T)$ for all $S\subseteq T$. It is submodular, if $f(S\cup \{a\}) - f(S) \ge f(T\cup\{a\}) - f(T)$ for all $S\subseteq T$ and $a\notin T$. The latter is also known as the \emph{diminishing returns} property in economics. A standard assumption on monotone submodular functions (used throughout this work) is that the value on the empty set is zero, i.e., $f(\emptyset) = 0$. Goemans, Harvey, Iwata, and Mirrokni~\cite{goemans2009approximating} first considered the Santa Claus problem with monotone submodular utility functions as an application of their fundamental result on submodular functions. Together with the algorithm of~\cite{DBLP:conf/focs/ChakrabartyCK09} it implies an $O(n^{1/2+\epsilon})$-approximation in time $O(n^{1/\epsilon})$. In this paper we investigate the restricted assignment case with a monotone submodular utility function. That is, all utility functions are defined by $f_i(S)=f(S\cap \Gamma_i)$, where $f$ is a monotone submodular function and $\Gamma_i$ is a subset of resources for each players $i$. Before our work, the state-of-the-art for this problem was the $O(n^{1/2+\epsilon})$-approximation algorithm mentioned above, since none of the previous results for the restricted assignment case with a linear utility function apply when the utility function becomes monotone submodular. \subsection{Overview of results and techniques} Our main result is an approximation algorithm for the submodular Santa Claus problem in the restricted assignment case. \begin{theorem} \label{thm:main} There is a randomized polynomial time $O(\log \log (n))$-approximation algorithm for the restricted assignment case with a monotone submodular utility function. \end{theorem} Our way to this result is organised as follows. In Section~\ref{sec:reduction to hypergraph}, we first reduce our problem to a hypergraph matching problem (see next paragraph for a formal definition). We then solve this problem using Lovasz Local Lemma (LLL) in Section~\ref{sec:hypergraph problem}. In~\cite{BansalSrividenko} the authors also reduce to a hypergraph matching problem which they then solve using LLL, although both parts are substantially simpler. The higher generality of our utility functions is reflected in the more general hypergraph matching problem. Namely, our problem is precisely the weighted variant of the (unweighted) problem in~\cite{BansalSrividenko}. We will elaborate later in this section why the previous techniques do not easily extend to the weighted variant. \paragraph{The hypergraph matching problem.} After the reduction in Section \ref{sec:reduction to hypergraph} we arrive at the following problem. There is a hypergraph $\mathcal H = (P\cup R, \mathcal C)$ with hyperedges $\mathcal C$ over the vertices $P$ and $R$. We write $m = |P|$ and $n = |R|$. We will refer to hyperedges as configurations, the vertices in $P$ as players and $R$ as resources \footnote{We note that these do not have to be the same players and resources as in the Santa Claus problem we reduced from, but $n$ and $m$ do not increase.}. Moreover, a hypergraph is said to be regular if all vertices in $P$ and $R$ have the same degree, that is, they are contained in the same number of configurations. The hypergraph may contain multiple copies of the same configuration. Each configuration $C\in\mathcal C$ contains exactly one vertex in $P$, that is, $|C\cap P| = 1$. Additionally, for each configuration $C\in \mathcal{C}$ the resources $j\in C$ have weights $w_{j,C} \ge 0$. We emphasize that the same resource $j$ can be given different weights in two different configurations, that is, we may have $w_{j,C}\neq w_{j,C'}$ for two different configurations $C,C'$. We require to select for each player $i\in P$ one configuration $C$ that contains $i$. For each configuration $C$ that was selected we require to assign a subset of the resources in $C$ which has a total weight of at least $(1/\alpha) \cdot \sum_{j\in C} w_{j,C}$ to the player in $C$. A resource can only be assigned to one player. We call such a solution an $\alpha$-relaxed perfect matching. One seeks to minimize $\alpha$. We show that every regular hypergraph has an $\alpha$-relaxed perfect matching for some $\alpha=O(\log \log (n))$ assuming that $w_{j,C}\leq (1/\alpha) \cdot \sum_{j'\in C} w_{j',C}$ for all $j,C$, that is, all weights are small compared to the total weight of the configuration. Moreover, we can find such a matching in randomized polynomial time. In the reduction we use this result to round a certain LP relaxation and $\alpha$ essentially translates to the approximation rate. This result generalizes that of Bansal and Srividenko on hypergraph matching in the following way. They proved the same result for unit weights and uniform hyperedges, that is, $w_{j,C}=1$ for all $j,C$ and all hyperedges have the same number of resources\footnote{In fact they get a slightly better ratio of $\alpha = O(\log\log(m) / \log\log\log(m))$.}. In the next paragraph we briefly go over the techniques to prove our result for the hypergraph matching problem. \paragraph{Our techniques.} Already the extension from uniform to non-uniform hypergraphs (assuming unit weights) is highly non-trivial and captures the core difficulty of our result. Indeed, we show with a (perhaps surprising) reduction, that we can reduce our weighted hypergraph matching problem to the unweighted (but non-uniform) version by introducing some bounded dependencies between the choices of the different players. For sake of brevity we therefore focus in this section on the unweighted non-uniform variant, that is, we need to assign to each player a configuration $C$ and at least $|C| / \alpha$ resources in $C$. We show that for any regular hypergraph there exists such a matching for $\alpha = O(\log \log (n))$ assuming that all configurations contain at least $\alpha$ resources and we can find it in randomized polynomial time. Without the assumption of uniformity the problem becomes significantly more challenging. To see this, we lay out the techniques of Bansal and Srividenko that allowed them to solve the problem in the uniform case. We note that for $\alpha = O(\log(n))$ the statement is easy to prove: We select for each player $i$ one of the configurations containing $i$ uniformly at random. Then by standard concentration bounds each resource is contained in at most $O(\log(n))$ of the selected configurations with high probability. This implies that there is a fractional assignment of resources to configurations such that each of the selected configurations $C$ receives $\lfloor |C| / O(\log(n)) \rfloor$ of the resources in $C$. By integrality of the bipartite matching polytope, there is also an integral assignment with this property. To improve to $\alpha= O(\log \log (n))$ in the uniform case, Bansal and Srividenko proceed as follows. Let $k$ be the size of each configuration. First they reduce the degree of each player and resource to $O(\log(n))$ using the argument above, but taking $O(\log(n))$ configurations for each player. Then they sample uniformly at random $O(n \log(n) / k)$ resources and drop all others. This is sensible, because they manage to prove the (perhaps surprising) fact that an $\alpha$-relaxed perfect matching with respect to the smaller set of resources is still an $O(\alpha)$-relaxed perfect matching with respect to all resources with high probability (when assigning the dropped resources to the selected configurations appropriately). Indeed, the smaller instance is easier to solve: With high probability all configurations have size $O(\log(n))$ and this greatly reduces the dependencies between the bad events of the random experiment above (the event that a resource is contained in too many selected configurations). This allows them to apply Lov\'asz Local Lemma (LLL) in order to show that with positive probability the experiment succeeds for $\alpha = O(\log\log(n))$. It is not obvious how to extend this approach to non-uniform hypergraphs: Sampling a fixed fraction of the resources will either make the small configurations empty---which makes it impossible to retain guarantees for the original instance---or it leaves the big configurations big ---which fails to reduce the dependencies enough to apply LLL. Hence it requires new sophisticated ideas for non-uniform hypergraphs, which we describe next. Suppose we are able to find a set $\mathcal K\subseteq \mathcal C$ of configurations (one for each player) such that for each $K\in\mathcal K$ the sum of intersections $|K\cap K'|$ with smaller configurations $K'\in \mathcal K$ is very small, say at most $|K| / 2$. Then it is easy to derive a $2$-relaxed perfect matching: We iterate over all $K\in\mathcal K$ from large to small and reassign all resources to $K$ (possibly stealing them from the configuration that previously had them). In this process every configuration gets stolen at most $|K| / 2$ of its resources, in particular, it keeps the other half. However, it is non-trivial to obtain a property like the one mentioned above. If we take a random configuration for each player, the dependencies of the intersections are too complex. To avoid this we invoke an advanced variant of the sampling approach where we construct not only one set of resources, but a hierarchy of resource sets $R_0\supseteq \cdots \supseteq R_d$ by repeatedly dropping a fraction of resources from the previous set. We then formulate bad events based on the intersections of a configuration $C$ with smaller configurations $C'$, but we write it only considering a resource set $R_k$ of convenient granularity (chosen based on the size of $C'$). In this way we formulate a number of bad events using various sets $R_k$. This succeeds in reducing the dependencies enough to apply LLL. Unfortunately, even with this new way of defining bad events, the guarantee that for each $K\in\mathcal K$ the sum of intersections $|K\cap K'|$ with smaller configurations $K'\in \mathcal K$ is at most $|K| / 2$ is still too much to ask. We can only prove some weaker property which makes it more difficult to reconstruct a good solution from it. The reconstruction still starts from the biggest configurations and iterates to finish by including the smallest configurations but it requires a delicate induction where at each step, both the resource set expands and some new small configurations that were not considered before come into play. \paragraph{Additional implications of non-uniform hypergraph matchings to the Santa Claus problem.} We believe this hypergraph matching problem is interesting in its own right. Our last contribution is to show that finding good matchings in unweighted hypergraphs with fewer assumptions than ours would have important applications for the Santa Claus problem with linear utility functions. We recall that here, each player $i$ has its own utility function $f_i$ that can be any linear function. In this case, the best approximation algorithm is due to Chakrabarty, Chuzhoy, and Khanna~\cite{DBLP:conf/focs/ChakrabartyCK09} who gave a $O(n^{\epsilon})$-approximation running in time $O(n^{1/\epsilon})$. In particular, no sub-polynomial approximation running in polynomial time is known. Consider as before $\mathcal H = (P\cup R, \mathcal C)$ a non-uniform hypergraph with unit weights ($w_{j,C}=1$ for all $j,C$ such that $j\in C$). Finding the smallest $\alpha$ (or an approximation of it) such that there exists an $\alpha$-relaxed perfect matching in $\mathcal H$ is already a very non-trivial question to solve in polynomial time. We show, via a reduction, that a $c$-approximation for this problem would yield a $O((c\log^* (n))^2)$-approximation for the Santa Claus problem with arbitrary linear utility functions. In particular, any sub-polynomial approximation for this problem would significantly improve the state-of-the-art\footnote{We mention that our result on relaxed matchings in Section \ref{sec:hypergraph problem} does not imply an $O(\log \log (n))$-approximation for this problem since we make additional assumptions on the regularity of the hypergraph or the size of hyperedges.}. All the details of this last result can be found in Section \ref{sec:reduction santa claus}. \paragraph{A remark on local search techniques.} We focus here on an extension of the LLL technique of Bansal and Srividenko. However, another technique proved itself very successful for the Santa Claus problem in the restricted assignment case with a linear utility function. This is a local search technique discovered by Asadpour, Feige, and Saberi~\cite{Asadpour_local_search} who used it to give a non-constructive proof that the integrality gap of the configuration LP of Bansal and Srividenko is at most $4$. One can wonder if this technique could also be extended to the submodular case as we did with LLL. Unfortunately, this seems problematic as the local search arguments heavily rely on amortizing different volumes of configurations (i.e., the sum of their resources' weights or the number of resources in the unweighted case). Amortizing the volumes of configurations works well, if each configuration has the same volume, which is the case for the problem derived from linear valuation functions, but not the one derived from submodular functions. If the volumes differ then these amortization arguments break and the authors of this paper believe this is a fundamental problem for generalizing those arguments. \section{Reduction to hypergraph matching problem} \label{sec:reduction to hypergraph} In this section we give a reduction of the restricted submodular Santa Claus problem to the hypergraph matching problem. As a starting point we solve the configuration LP, a linear programming relaxation of our problem. The LP is constructed using a parameter $T$ which denotes the value of its solution. The goal is to find the maximal $T$ such that the LP is feasible. In the LP we have a variable $x_{i,C}$ for every player $i\in P$ and every configuration $C\in \mathcal{C}(i, T)$. The configurations $\mathcal{C}(i, T)$ are defined as the sets of resources $C \subseteq \Gamma_i$ such that $f(C) \ge T$. We require every player $i\in P$ to have at least one configuration and every resource $j \in R$ to be contained in at most one configuration. \begin{align*} \sum_{C\in \mathcal{C}(i, T)} & x_{i,C} \ge 1 \quad \text{ for all } i \in P \\ \sum_{i\in P}\sum_{C\in \mathcal{C}(i,T) : j \in C} & x_{i,C} \leq 1 \quad \text{ for all } j \in R \\ & x_{i,C} \geq 0 \quad \text{ for all } i\in P, C \in \mathcal{C}(i, T) \end{align*} Since this linear program has exponentially many variables, we cannot directly solve it in polynomial time. We will give a polynomial time constant approximation for it via its dual. This is similar to the linear variant in~\cite{BansalSrividenko}, but requires some more work. In their case they can reduce the problem to one where the separation problem of the dual can be solved in polynomial time. In our case even the separation problem can only be approximated. Nevertheless, this is sufficient to approximate the linear program in polynomial time. \begin{theorem}\label{thm:config-LP} The configuration LP of the restricted submodular Santa Claus problem can be approximated within a factor of $(1 - 1/e)/2$ in polynomial time. \end{theorem} We defer the proof of this theorem to Appendix~\ref{appendix_lp}. Given a solution $x^*$ of the configuration LP we want to arrive at the hypergraph matching problem from the introduction such that an $\alpha$-relaxed perfect matching of that problem corresponds to an $O(\alpha)$-approximate solution of the restricted submodular Santa Claus problem. Let $T^*$ denote the value of the solution $x^*$. We will define a resource $j\in R$ as \emph{fat} if \begin{equation*} f(\{j\}) \geq \frac{T^*}{100 \alpha} . \end{equation*} Resources that are not fat are called \emph{thin}. We call a configuration $C\in\mathcal{C}(i, T)$ thin, if it contains only thin resources and denote by $\mathcal{C}_t(i,T) \subseteq \mathcal{C}(i, T)$ the set of thin configurations. Intuitively in order to obtain an $O(\alpha)$-approximate solution, it suffices to give each player $i$ either one fat resource $j\in \Gamma_i$ or a thin configuration $C\in\mathcal{C}_t(i,T^*/O(\alpha))$. For our next step towards the hypergraph problem we use a technique borrowed from Bansal and Srividenko~\cite{BansalSrividenko}. This technique allows us to simplify the structure of the problem significantly using the solution of the configuration LP. Namely, one can find a partition of the players into clusters such that we only need to cover one player from each cluster with thin resources. All other players can then be covered by fat resources. Informally speaking, the following lemma is proved by sampling configurations randomly according to a distribution derived in a non-trivial way from the configuration LP. \begin{lemma}\label{lem:config-sample} Let $\ell \ge 12\log (n)$. Given a solution of value $T^*$ for the configuration LP in randomized polynomial time we can find a partition of the players into clusters $K_1\cup\cdots \cup K_k\cup Q = P$ and multisets of configurations $\mathcal{C}_h \subseteq \bigcup_{i\in K_h} \mathcal{C}_T(i, T^*/5)$, $h=1,\dotsc,k$, such that \begin{enumerate} \item $|\mathcal{C}_h| = \ell$ for all $h=1,\dotsc,k$ and \item Each small resource appears in at most $\ell$ configurations of $\bigcup_h \mathcal{C}_h$. \item given any $i_1\in K_1, i_2\in K_2,\dotsc,i_k\in K_k$ there is a matching of fat resources to players $P\setminus\{i_1,\dotsc,i_k\}$ such that each of these players $i$ gets a unique fat resource $j\in\Gamma_i$. \end{enumerate} \end{lemma} The role of the players $Q$ in the lemma above is that each one of them gets a fat resource for certain. The proof follows closely that in~\cite{BansalSrividenko}. For completeness we include it in Appendix~\ref{appendix_lp}. We are now ready to define the hypergraph matching instance. The vertices of our hypergraph are the clusters $K_1,\dotsc,K_k$ and the thin resources. Let $\mathcal{C}_1,\dotsc,\mathcal{C}_k$ be the multisets of configurations as in Lemma~\ref{lem:config-sample}. For each $K_h$ and $C\in\mathcal{C}_h$ there is a hyperedge containing $K_h$ and all resources in $C$. Let $\{j_1,\dotsc,j_\ell\} = C$ ordered arbitrarily, but consistently. Then we define the weights as normalized marginal gains of resources if they are taken in this order, that is, \begin{equation*} w_{j_i, C} = \frac{5}{T^*} f(\{j_i\} \mid \{j_1,\dotsc,j_{i-1}\}) = \frac{5}{T^*} (f(\{j_1,\dotsc,j_{i-1}, j_i\})-f(\{j_1,\dotsc,j_{i-1}\})). \end{equation*} This implies that $\sum_{j\in C} w_{j, C} \ge 5 f(C) / T^* \ge 1$ for each $C\in\mathcal{C}_h$, $h=1,\dotsc,k$. \begin{lemma} Given an $\alpha$-relaxed perfect matching to the instance as described by the reduction, one can find in polynomial time an $O(\alpha)$-approximation to the instance of restricted submodular Santa Claus. \end{lemma} \begin{proof} The $\alpha$-relaxed perfect matching implies that cluster $K_h$ gets some small resources $C'$ where $C'\subseteq C$ for some $C\in\mathcal{C}_h$ and $\sum_{j\in C'} w_{j, C} \ge 1/\alpha$. By submodularity we have that $f(C') \ge T^* / (5 \alpha)$. Therefore we can satisfy one player in each cluster using thin resources and by Lemma~\ref{lem:clusters} all others using fat resources. \end{proof} The proof above is the most critical place in the paper where we make use of the submodularity of the valuation function $f$. We note that since all resources considered are thin resources we have, by submodularity of $f$, the assumption that \begin{equation*} w_{j,C} \leq \frac{5}{T^*}f(\{j\}) \leq \frac{5}{T^*}\frac{T^*}{100\alpha} \leq \frac{5}{100\alpha} \sum_{j\in C} w_{j,C} \end{equation*} for all $j,C$ such that $j\in C$. This means that the weights are all small enough, as promised in introduction. From now on, we will assume that $\sum_{j\in C} w_{j,C}=1$ for all configurations $C$. This is w.l.o.g. since we can just rescale the weights inside each configuration. This does not hurt the property that all weights are small enough. \subsection{Reduction to unweighted hypergraph matching} Before proceeding to the solution of this hypergraph matching problem, we first give a reduction to an unweighted variant of the problem. We will then solve this unweighted variant in the next section. First, we note that we can assume that all the weights $w_{j,C}$ are powers of $2$ by standard rounding arguments. This only loses a constant factor in the approximation rate. Second, we can assume that inside each configuration $C$, each resource has a weight that is at least a $1/(2n)$. Formally, we can assume that \begin{equation*} \min_{j\in C}w_{j,C}\geq 1/(2n) \end{equation*} for all $C\in \mathcal{C}$. If this is not the case for some $C\in \mathcal{C}$, simply delete from $C$ all the resources that have a weight less than $1/(2n)$. By doing this, the total weight of $C$ is only decreased by a factor $1/2$ since it looses in total at most a weight of \begin{equation*} n\cdot \frac{1}{2n} = \frac{1}{2}. \end{equation*} (Recall that we rescaled the weights so that $\sum_{j\in C} w_{j,C}=1$). Hence after these two operations, an $\alpha$-relaxed perfect matching in the new hypergraph is still an $O(\alpha)$-relaxed perfect matching in the original hypergraph. From there we reduce to an unweighted variant of the matching problem. Note that each configuration contains resources of at most $\log (n)$ different possible weights (powers of $2$ from $1 /(2n)$ to $1 / \alpha$). We create the following new unweighted hypergraph $\mathcal H'=(P'\cup R,\mathcal C')$. The resource set $R$ remains unchanged. For each player $i\in P$, we create $\log (n)$ players, which later correspond each to a distinct weight. We will say that the players obtained from duplicating the original player form a \textit{group}. For every configuration $C$ containing player $i$ in the hypergraph $\mathcal H$, we add a set $\mathcal{S}_C=\{C_1,\ldots ,C_s, \ldots ,C_{\log(n)}\}$ of configurations in $\mathcal H'$. $C_s$ contains player $i_s$ and all resources that are given a weight $2^{-(s+1)}$ in $C$. In this new hypergraph, the resources are not weighted. Note that if the hypergraph $\mathcal H$ is regular then $\mathcal H'$ is regular as well. Additionally, for a group of player and a set of $\log(n)$ configurations (one for each player in the group), we say that this set of configurations is \textit{consistent} if all the configurations selected are obtained from the same configuration in the original hypergraph $\mathcal H$ (i.e. the selected configurations all belong to $\mathcal{S}_C$ for some $C$ in $\mathcal H$). Formally, we focus of the following problem. Given the regular hypergraph $\mathcal H'$, we want to select, for each group of $\log (n)$ players, a consistent set of configurations $C_1,\ldots, C_s, \ldots ,C_{\log(n)}$ and assign to each player $i_s$ a subset of the resources in the corresponding configuration $C_s$ so that $i_s$ is assigned at least $\left\lfloor |C_s|/\alpha\right\rfloor$ resources. No resource can be assigned to more than one player. We refer to this assignment as a consistent $\alpha$-relaxed perfect matching. Note that in the case where $|C_s|$ is small (e.g. of constant size) we are not required to assign any resource to player $i_s$. \begin{lemma} A consistent $\alpha$-relaxed matching in $\mathcal H'$ induces a $O(\alpha)$-relaxed matching in $\mathcal H$. \end{lemma} \begin{proof} Let us consider a group of $\log (n)$ players $i_1,\ldots , i_s, \ldots ,i_{\log (n)}$ in $\mathcal H'$ corresponding to a player $i$ in $\mathcal H$. These players are assigned a consistent set of configurations $C_1,\ldots , C_s, \ldots, C_{\log (n)}$ that correspond to a partition of a configuration in $\mathcal H$. Moreover, each player $i_s$ is assigned $\left\lfloor |C_s|/\alpha\right\rfloor$ resources from $C_s$. We have two cases. If $|C_s|\geq \alpha$ then we have that $i_s$ is assigned at least \begin{equation*} \left\lfloor |C_s|/\alpha\right\rfloor\geq |C_s|/(2\alpha) \end{equation*} resources from $C_s$. On the other hand, if $\left\lfloor |C_s|/\alpha\right\rfloor=0$ then the player $i_s$ might not be assigned anything. However, we claim that that the configurations $C_s$ of cardinality less than $\alpha$ can represent at most a $1/5$ fraction of the total weight of the configuration $C$ in the original weighted hypergraph. To see this note that the total weight they represent is upper bounded by \begin{equation*} \alpha \left(\sum_{k=\log(100\alpha/5)}^{\infty} \frac{1}{2^k}\right) = \alpha\left(\frac{5}{100\alpha}\sum_{k=0}^{\infty} \frac{1}{2^k}\right) \leq \frac{10}{100} = \frac{1}{10}\sum_{j\in C}w_{j,C}. \end{equation*} Hence, the consistent $\alpha$-relaxed matching in $\mathcal H'$ induces in a straightforward way a matching in $\mathcal H$ where every player gets at least a fraction $1/(2\alpha) \cdot (1-1/10) \geq 1/(3\alpha)$ of the total weight of the appropriate configuration. This means that the consistent $\alpha$-relaxed perfect matching in $\mathcal H'$ is indeed a $(3\alpha)$-relaxed perfect matching in $\mathcal H$. \end{proof} \section{Matchings in regular hypergraphs} \label{sec:hypergraph problem} In this section we solve the hypergraph matching problem we arrived to in the previous section. For convenience, we give a self contained definition of the problem before formulating and proving our result. \paragraph{Input:} We are given $\mathcal H = (P\cup R, \mathcal C)$ a hypergraph with hyperedges $\mathcal C$ over the vertices $P$ (players) and $R$ (resources) with $m = |P|$ and $n = |R|$. As in previous sections, we will refer to hyperedges as configurations. Each configuration $C\in\mathcal C$ contains exactly one vertex in $P$, that is, $|C\cap P| = 1$. The set of players is partitioned into groups of size at most $\log (n)$, we will use $A$ to denote a group. These groups are disjoint and contain all players. Finally there exists an integer $\ell$ such that for each group $A$ there are $\ell$ consistent sets of configurations. A consistent set of configurations for a group $A$ is a set of $|A|$ configurations such that all players in the group appear in exactly one of these configurations. We will denote by $\mathcal{S}_A$ such a set and for a player $i\in A$, we will denote by $\mathcal{S}_A^{(i)}$ the unique configuration in $\mathcal{S}_A$ containing $i$. Finally, no resource appears in more than $\ell$ configurations. We say that the hypergraph is regular (although some resources may appear in less than $\ell$ configurations). \paragraph{Output:} We wish to select a matching that covers all players in $P$. More precisely, for each group $A$ we want to select a consistent set of configurations (denoted by $\{\mathcal{S}_A^{(i)}\}_{i\in A}$). Then for each player $i\in A$, we wish to assign a subset of the resources in $\mathcal{S}_A^{(i)}$ to the player $i$ such that: \begin{enumerate} \item No resource is assigned to more than one player in total. \item For any group $A$ and any player $i\in A$, player $i$ is assigned at least \begin{equation*} \left\lfloor \frac{\mathcal{S}_A^{(i)}}{\alpha}\right\rfloor \end{equation*} resources from $\mathcal{S}_A^{(i)}$. \end{enumerate} We call this a consistent $\alpha$-relaxed perfect matching. Our goal in this section will be to prove the following theorem. \begin{theorem}\label{thm:unweighted_hypergraph} Let $\mathcal H=(P\cup R, \mathcal C)$ be a regular (non-uniform) hypergraph where the set of players is partitioned into groups of size at most $\log (n)$. Then we can, in randomized polynomial time, compute a consistent $\alpha$-relaxed perfect matching for $\alpha=O(\log \log (n))$. \end{theorem} We note that Theorem \ref{thm:unweighted_hypergraph} together with the reduction from the previous section will prove our main result (Theorem \ref{thm:main}) stated in introduction. \subsection{Overview and notations} To prove Theorem~\ref{thm:unweighted_hypergraph}, we introduce the following notations. Let $\ell \in \mathbb N$ be the regularity parameter as described in the problem input (i.e. each group has $\ell$ consistent sets and each resource appears in no more than $\ell$ configurations). As we proved in Lemma \ref{lem:config-sample} we can assume with standard sampling arguments that $\ell =300.000\log^{3}(n)$ at a constant loss. If this is not the case because we might want to solve the hypergraph matching problem by itself (i.e. not obtained by the reduction in Section \ref{sec:reduction to hypergraph}), the proof of Lemma \ref{lem:config-sample} can be repeated in a very similar way here. For a configuration $C$, its size will be defined as $|C\cap R|$ (i.e. its cardinality over the resource set). For each player $i$, we denote by $\mathcal{C}_i$ the set of configurations that contain $i$. We now group the configurations in $\mathcal{C}_i$ by size: We denote by $\mathcal{C}_{i}^{(0)}$ the configurations of size in $[0,\ell^{4})$ and for $k\ge 1$ we write $\mathcal{C}_{i}^{(k)}$ for the configurations of size in $[\ell^{k+3},\ell^{k+4})$. Moreover, define $\mathcal{C}^{(k)}=\bigcup_i \mathcal{C}_{i}^{(k)}$ and $\mathcal{C}^{(\ge k)} = \bigcup_{h\ge k} \mathcal{C}^{(h)}$. Let $d$ be the smallest number such that $\mathcal{C}^{(\ge d)}$ is empty. Note that $d\le \log(n) / \log(\ell)$. Now consider the following random process. \begin{experiment}\label{exp:sequence} We construct a nested sequence of resource sets $R=R_0 \supseteq R_1 \supseteq \ldots \supseteq R_d$ as follows. Each $R_k$ is obtained from $R_{k-1}$ by deleting every resource in $R_{k-1}$ independently with probability $(\ell-1) / \ell$. \end{experiment} In expectation only a $1/\ell$ fraction of resources in $R_{k-1}$ survives in $R_k$. Also notice that for $C \in \mathcal{C}^{(k)}$ we have that $\mathbb E[ |R_k \cap C| ] = \mathrm{poly}(\ell)$. The proof of Theorem~\ref{thm:unweighted_hypergraph} is organized as follows. In Section~\ref{sec:sequence}, we give some properties of the resource sets constructed by Random Experiment~\ref{exp:sequence} that hold with high probability. Then in Section~\ref{sec:LLL}, we show that we can find a single consistent set of configurations for each group of players such that for each configuration selected, its intersection with smaller selected configurations is bounded if we restrict the resource set to an appropriate $R_k$. Restricting the resource set is important to bound the dependencies of bad events in order to apply Lovasz Local Lemma. Finally in Section~\ref{sec:reconstruction}, we demonstrate how these configurations allows us to reconstruct a consistent $\alpha$-relaxed perfect matching for an appropriate assignment of resources to configurations. \subsection{Properties of resource sets}\label{sec:sequence} In this subsection, we give a precise statement of the key properties that we need from Random Experiment~\ref{exp:sequence}. The first two lemmas have a straight-forward proof. The last one is a generalization of an argument used by Bansal and Srividenko \cite{BansalSrividenko}. Since the proof is more technical and tedious, we also defer it to Appendix~\ref{appendix_sequence} along with the proof of the first two statements. We start with the first property which bounds the size of the configurations when restricted to some $R_k$. This property is useful to reduce the dependencies while applying LLL later. \begin{lemma} \label{lma-size} Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. For any $k\geq 0$ and any $C\in\mathcal{C}^{(\geq k)}$ we have \begin{equation*} \frac{1}{2} \ell^{-k}|C| \le |R_k \cap C| \le \frac{3}{2} \ell^{-k}|C| \end{equation*} with probability at least $1-1/n^{10}$. \end{lemma} The next property expresses that for any configuration the sum of intersections with configurations of a particular size does not deviate much from its expectation. In particular, for any configuration $C$, the sum of it's intersections with other configurations is at most $|C|\ell$ as each resource is in atmost $\ell$ configurations. By the lemma stated below, we recover this up to a multiplicative constant factor when we consider the appropriately weighted sum of the intersection of $C$ with other configurations $C'$ of smaller sizes where each configuration $C' \in \mathcal{C}^{(k)}$ is restricted to the resource set $R_k$. \begin{lemma} \label{lma-overlap-representative} Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. For any $k\geq 0$ and any $C\in\mathcal{C}^{(\geq k)}$ we have \begin{equation*} \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C\cap R_k| \leq \frac{10}{\ell^{k}} \left(|C|+\sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C| \right) \end{equation*} with probability at least $1-1/n^{10}$. \end{lemma} We now define the notion of \emph{good} solutions which is helpful in stating our last property. Let $\mathcal{F}$ be a set of configurations, $\alpha:\mathcal{F} \rightarrow \mathbb N$, $\gamma \in\mathbb N$, and $R'\subseteq R$. We say that an assignment of $R'$ to $\mathcal{F}$ is $(\alpha,\gamma)$-good if every configuration $C\in \mathcal{F}$ receives at least $\alpha(C)$ resources of $C\cap R'$ and if no resource in $R'$ is assigned more than $\gamma$ times in total. Below we obtain that given a $(\alpha,\gamma)$-good solution with respect to resource set $R_{k+1}$, one can construct an almost $(\ell \cdot \alpha,\gamma)$-good solution with respect to the bigger resource set $R_{k}$. Informally, starting from a good solution with respect to the final resource set and iteratively applying this lemma would give us a good solution with respect to our complete set of resources. \begin{lemma} \label{lma-good-solution} Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. Fix $k\geq 0$. Conditioned on the event that the bounds in Lemma~\ref{lma-size} hold for $k$, then with probability at least $1 - 1/n^{10}$ the following holds for all $\mathcal{F}\subseteq \mathcal{C}^{(\geq k+1)}$, $\alpha:\mathcal{F} \rightarrow \mathbb N$, and $\gamma \in\mathbb N$ such that $\ell^3/1000\leq \alpha(C) \leq n $ for all $C\in\mathcal{F}$ and $\gamma \in \{1,\dotsc,\ell\}$: If there is a $(\alpha,\gamma)$-good assignment of $R_{k+1}$ to $\mathcal{F}$, then there is a $(\alpha',\gamma)$-good assignment of $R_k$ to $\mathcal{F}$ where \begin{equation*} \alpha'(C) \ge \ell \left(1-\frac{1}{\log (n)} \right) \alpha(C) \end{equation*} for all $C\in\mathcal{F}$. Moreover, this assignment can be found in polynomial time. \end{lemma} Given the lemmata above, by a simple union bound one gets that all the properties of resource sets hold. \subsection{Selection of configurations}\label{sec:LLL} In this subsection, we give a random process that selects one consistent set of configurations for each group of players such that the intersection of the selected configurations with smaller configurations is bounded when considered on appropriate sets $R_k$. We will denote $\mathcal{S}_A$ the selected consistent set for group $A$ and for ease of notation we will denote $K_i=\mathcal{S}_A^{(i)}$ the selected configuration for player $i\in A$. For any integer $k$, we write $\mathcal K^{(k)}_i = \{K_i\}$ if $K_i\in\mathcal C^{(k)}_i$ and $\mathcal K^{(k)}_i = \emptyset$ otherwise. As for the configuration set, we will also denote $\mathcal K^{(k)}=\bigcup_{i}\mathcal K^{(k)}_i$ and $\mathcal K= \bigcup_{k}\mathcal K^{(k)}$. The following lemma describes what are the properties we want to have while selecting the configurations. For better clarity we also recall what the properties of the sets $R_0,\dotsc,R_d$ that we need are. These hold with high probability by the lemmata of the previous section. \begin{lemma}\label{lma:main-LLL} Let $R =R_0\supseteq\dotsc\supseteqR_d$ be sets of fewer and fewer resources. Assume that for each $k$ and $C\in \mathcal C_i^{(k)}$ we have \begin{equation*} 1/2 \cdot \ell^{k - h} \le |C\cap R_h| \le 3/2 \cdot \ell^{- h} |C| < 3/2 \cdot \ell^{k - h + 4} \end{equation*} for all $h=0,\dotsc,k$. Then there exists a selection of one consistent set $\mathcal{S}_A$ for each group $A$ such for all $k=0,\dotsc, d$, $C\in \mathcal C^{(k)}$ and $j=0,\dotsc,k$ then we have \begin{equation*} \sum_{j\leq h\le k} \sum_{K\in\mathcal K^{(h)}} \ell^{h} |K \cap C \cap R_h| \le \frac{1}{\ell} \sum_{j\leq h\le k} \sum_{C'\in\mathcal C^{(h)}} \ell^{h} |C' \cap C \cap R_h| + 1000 \frac{d + \ell}{\ell}\log(\ell) |C| . \end{equation*} Moreover, this selection of consistent sets can be found in polynomial time. \end{lemma} Before we prove this lemma, we give an intuition of the statement. Consider the sets $R_1,\dotsc,R_d$ constructed as in Random Experiment~\ref{exp:sequence}. Then for $C'\in\mathcal C^{(h)}$ we have $\mathbb{E}[\ell^h |C'\cap C\cap R_h|] = |C'\cap C|$. Hence \begin{equation*} \sum_{h\le k} \sum_{K\in\mathcal K^{(h)}} |K \cap C| = \mathbb{E}[\sum_{h\le k} \sum_{K\in\mathcal K^{(h)}} \ell^h |K \cap C \cap R_h|] \end{equation*} Similarly for the right-hand side we have \begin{multline*} \mathbb{E}[\frac{1}{\ell} \sum_{j \le h\le k} \sum_{C'\in\mathcal C^{(h)}} \ell^h |C' \cap C \cap R_h| + O(\frac{d + \ell}{\ell}\log(\ell) |C|)] \\ = \frac{1}{\ell}\underbrace{\sum_{j\le h\le k} \sum_{C'\in\mathcal C^{(h)}} |C' \cap C|}_{\le \ell |C|} + O\left(\frac{d + \ell}{\ell}\log(\ell) |C|\right) = O\left(\frac{d + \ell}{\ell}\log(\ell) |C|\right) . \end{multline*} Hence the lemma says that each resource in $C$ is roughly covered $O((d + \ell)/\ell \cdot \log(\ell))$ times by smaller configurations. We now proceed to prove the lemma by performing the following random experiment and by Lovasz Local Lemma show that there is a positive probability of success. \begin{experiment} For each group $A$, select one consistent set $\mathcal{S}_A$ uniformly at random. Then for each player $i \in A$ set $K_i=\mathcal \mathcal{S}_A^{(i)}$. \end{experiment} For all $h=0,\dotsc,d$ and $i\in P$ we define the random variable \begin{equation*} X^{(h)}_{i,C} = \sum_{K\in\mathcal K^{(h)}_i} |K \cap C \cap R_h| \le \min\{3/2 \cdot \ell^4, |C\cap R_h|\} . \end{equation*} Let $X^{(h)}_C = \sum_{i=1}^m X^{(h)}_{i, C}$. Then \begin{equation*} \mathbb{E}[X^{(h)}_C] \le \frac{1}{\ell} \sum_{C'\in\mathcal C^{(h)}} |C'\cap C\cap R_h| \le |C\cap R_h| . \end{equation*} We define a set of bad events. As we will show later, if none of them occur, the properties from the premise hold. For each $k$, $C\in\mathcal C^{(k)}$, and $h\le k$ let $B_C^{(h)}$ be the event that \begin{equation*} X_C^{(h)} \ge \begin{cases} \mathbb{E}[X_C^{(h)}] + 63 |C\cap R_h| \log(\ell) &\text{ if $k - 5 \le h \le k$}, \\ \mathbb{E}[X_C^{(h)}] + 135 |C\cap R_h| \log(\ell) \cdot \ell^{-1} &\text{ if $h \le k - 6$}. \end{cases} \end{equation*} There is an intuitive reason as to why we define these two different bad events. In the case $h\leq k-6$, we are counting how many times $C$ is intersected by configurations that are much smaller than $C$. Hence the size of this intersection can be written as a sum of independent random variables of value at most $O(\ell^4)$ which is much smaller than the total size of the configuration $|C\cap R_h|$. Since the random variables are in a much smaller range, Chernoff bounds give much better concentration guarantees and we can afford a very small deviation from the expectation. In the other case, we do not have this property hence we need a bigger deviation to maintain a sufficiently low probability of failure. However, this does not hurt the statement of Lemma~\ref{lma:main-LLL} since we sum this bigger deviation only a constant number of times. With this intuition in mind, we claim the following. \begin{claim} For each $k$, $C\in\mathcal C^{(k)}$, and $h\le k$ we have \begin{equation*} \mathbb{P}[B_C^{(h)}] \le \exp\left(- 2\frac{|C \cap R_h|}{\ell^9} - 18\log(\ell)\right) . \end{equation*} \end{claim} \begin{proof} Consider first the case that $h \ge k - 5$. By a Chernoff bound (see Proposition~\ref{cor:chernoff}) with \begin{equation*} \delta = 63\frac{|C\cap R_h| \log(\ell)}{\mathbb{E}[X_C^{(h)}]} \ge 1 \end{equation*} we get \begin{equation*} \mathbb{P}[B_C^{(h)}] \le \exp\bigg(-\frac{\delta \mathbb{E}[X^{(h)}_C]}{3 |C\cap R_h|}\bigg) \le \exp(-21\log(\ell))) \le \exp\bigg(-2 \underbrace{\frac{|C\cap R_h|}{\ell^{9}}}_{\le 3/2} - 18\log(\ell)\bigg). \end{equation*} Now consider $h \le k - 6$. We apply again a Chernoff bound with \begin{equation*} \delta = 135\frac{|C\capR_h| \log(\ell)}{\ell \mathbb{E}[X_C^{(h)}]} \ge \frac{1}{\ell} . \end{equation*} This implies \begin{multline*} \mathbb P[B_C^{(h)}] \le \exp\left(-\frac{\min\{\delta,\delta^2\} \mathbb{E}[X^{(h)}_C]}{3 \cdot 3/2 \cdot \ell^4}\right) \le \exp\left(-30\frac{|C\capR_h| \log(\ell)}{\ell^6} \right) \\ \le \exp\left(-2 \frac{|C\cap R_h|}{\ell^9} - 18\log(\ell)\right) . \qedhere \end{multline*} \end{proof} \begin{proposition}[Lovasz Local Lemma (LLL)]\label{prop:LLL} Let $B_1, \dotsc, B_t$ be bad events, and let $G = (\{B_1,\dotsc,B_t\}, E)$ be a dependency graph for them, in which for every $i$, event $B_i$ is mutually independent of all events $B_j$ for which $(B_i, B_j)\notin E$. Let $x_i$ for $1\le i \le t$ be such that $0 < x(B_i) < 1$ and $\mathbb{P}[B_i]\le x(B_i) \prod_{(B_i,B_j)\in E} (1-x(B_j))$. Then with positive probability no event $B_i$ holds. \end{proposition} Let $k\in\{0,\dotsc,d\}$, $C\in\mathcal C^{(k)}$ and $h\le k$. For event $B_C^{(h)}$ we set \begin{equation*} x(B_C^{(h)}) = \exp(-|C\capR_h| / \ell^9 - 18\log(\ell)) . \end{equation*} We now analyze the dependencies of $B_C^{(h)}$. The event depends only on random variables $\mathcal{S}_A$ for groups $A$ that contain at least one player $i$ that has a configuration in $\mathcal C^{(h)}_i$ which overlaps with $C\cap R_h$. The number of such configurations (in particular, of such groups) is at most $\ell |C\cap R_h|$ since the hypergraph is regular. In each of these groups, we count at most $\log (n)$ players, each having $\ell$ configurations hence in total at most $\ell\cdot \log (n)$ configurations. Each configuration $C'\in \mathcal{C}^{(h')}$ can only influence those events $B^{(h')}_{C''}$ where $C' \cap C'' \cap R_{h'} \neq \emptyset$. Since $|C'\cap R_{h'}|\leq 3/2\cdot \ell^4$ and since each resource appears in at most $\ell$ configurations, we see that each configuration can influence at most $3/2 \cdot \ell^5$ events. Putting everything together, we see that the bad event $B_C^{(h)}$ is independent of all but at most \begin{equation*} (\ell |C\cap R_h|) \cdot (\ell\cdot \log (n)) \cdot (3/2 \cdot \ell^5) = 3/2\cdot \ell^7 \cdot \log (n) |C\cap R_h| \leq |C\cap R_h|\ell^8 \end{equation*} other bad events. We can now verify the condition for Proposition~\ref{prop:LLL} by calculating \begin{align*} x(B_C^{(h)}) & \prod_{(B_C^{(h)}, B_{C'}^{(h')})\in E} (1 - x(B_{C'}^{(h')})) \\ &\ge \exp(-|C\capR_h|/\ell^9 - 18\log(\ell)) \cdot (1 - \ell^{-18})^{|C\capR_h|\ell^8} \\ &\ge \exp(-|C\capR_h|/\ell^9 - 18\log(\ell)) \cdot \exp(- |C\capR_h| / \ell^9) \\ &\ge \exp(-2|C\capR_h|/\ell^9 - 18\log(\ell)) \ge \mathbb{P}[B^{(h)}_C] . \end{align*} By LLL we have that with positive probability none of the bad events happen. Let $k\in\{0,\dotsc,d\}$ and $C\in\mathcal C^{(k)}$. Then for $k - 5 \le h \le k$ we have \begin{equation*} \ell^{h} X^{(h)}_C \le \ell^{h} \mathbb{E}[X_C^{(h)}] + 63 \ell^{h} |C\capR_h|\log(\ell) \le \ell^{h} \mathbb{E}[X_C^{(h)}] + 95 |C|\log(\ell) . \end{equation*} Moreover, for $h\le k-6$ it holds that \begin{equation*} \ell^{h} X^{(h)}_C \le \ell^{h} \mathbb{E}[X_C^{(h)}] + 135 \ell^{h-1} |C\capR_h|\log(\ell) \le \ell^{h} \mathbb{E}[X_C^{(h)}] + 203 |C|\log(\ell) \cdot \ell^{-1} . \end{equation*} We conclude that, for any $0\leq j\leq k$, \begin{align*} \sum_{j\leq h\le k} \sum_{K\in\mathcal K^{(h)}} \ell^{h} |K \cap C \cap R_h| &\le \sum_{j\leq h\le k} \ell^{h} \mathbb{E}[X^{(h)}_{C}] + 1000 \frac{(k-j + 1) + \ell}{\ell} |C| \log(\ell) \\ &\le \frac{1}{\ell} \sum_{j\leq h\le k} \ell^{h} \sum_{C'\in\mathcal C^{(h)}} |C'\cap C \cap R_h| + 1000 \frac{d + \ell}{\ell} |C| \log(\ell) . \end{align*} This proves Lemma~\ref{lma:main-LLL}. \begin{remark}{\rm Since there are at most $\mathrm{poly}(n,m,\ell)$ bad events and each bad event $B$ has $\frac{x(B)}{1-x(B)}\le1/2$ (because $x(B)\le \ell^{-18}$), the constructive variant of LLL by Moser and Tardos~\cite{moser2010constructive} can be applied to find a selection of configurations such that no bad events occur in randomized polynomial time.} \end{remark} \subsection{Assignment of resources to configurations}\label{sec:reconstruction} In this subsection, we show how all the previously established properties allow us to find, in polynomial time, a good assignment of resources to the configurations $\mathcal{K}$ chosen as in the previous subsection. We will denote as in the previous subsection $\mathcal{K}_i^{(k)}=\{K_i\}$ if $K_i\in \mathcal{C}_i^{(k)}$ and $\mathcal{K}_i^{(k)}=\emptyset$ otherwise. We also define $\mathcal{K}^{(k)}=\bigcup_{i}\mathcal{K}_i^{(k)}$ and $\mathcal{K}^{(\geq k)}=\bigcup_{h\geq k}\mathcal{K}^{(k)}$. Finally we define the parameter \begin{equation*} \gamma = 100.000 \frac{d+\ell}{\ell}\log(\ell) , \end{equation*} which will define how many times each resource can be assigned to configurations in an intermediate solution. Note that $d\le\log(n)/\log(\ell)$. By our choice of $\ell=300.000\log^3(n)$, we have that $\gamma \leq 310.000 \log \log (n)$. Lemma~\ref{lma:main-LLL} implies the following bound. For sake of brevity, the proof is deferred to Appendix~\ref{appendix_reconstruct}. \begin{claim} \label{cla:reconstruct} For any $k\geq 0$, any $0\leq j\leq k$, and any $C\in \mathcal{K}^{(k)}$ \begin{equation*} \sum_{j\leq h\leq k}\sum_{K\in \mathcal{K}^{(h)}} \ell^{h}|K\cap C \cap R_h| \leq 2000\frac{d+\ell}{\ell}\log (\ell) |C| \end{equation*} \end{claim} The main technical part of this section is the following lemma that is proved by induction. \begin{lemma} \label{lem:reconstruct} For any $j\geq 0$, there exists an assignment of resources of $R_j$ to configurations in $\mathcal{K}^{(\geq j)}$ such that no resource is taken more than $\gamma$ times and each configuration $C\in \mathcal{K}^{(k)}$ ($k\geq j$) receives at least \begin{equation*} \left(1-\frac{1}{\log (n)} \right)^{2(k-j)}\ell^{k-j} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-j}|K\cap C \cap R_h| \end{equation*} resources from $R_k$. \end{lemma} Before proceeding to the proof, we first give intuition of why this is what we want to prove. Note that the term $\ell^{k-j}|C\cap R_k|$ is roughly equal to $\ell^{-j}|C|$ by the properties of the resource sets (precisely Lemma \ref{lma-size}). The second term \begin{equation*} \sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-j}|K\cap C \cap R_h| \end{equation*} can be shown to be \begin{equation*} O\left(\ell^{-j}\frac{d+\ell}{\ell}\log (\ell) |C| \right)= O (\ell^{-j}\log \log (n) |C|) \end{equation*} by Claim \ref{cla:reconstruct}. Hence by choosing $\gamma$ to be $\Theta (\log \log (n))$ we get that the bound in Lemma \ref{lem:reconstruct} will be $\Theta (\ell^{-j}|C|)$. At the end of the induction, we have $j=0$ which indeed implies that we have an assignment in which configurations receive \begin{equation*} \Theta (\ell^{-0}|C|)=\Theta(|C|) \end{equation*} resources and such that each resource is assigned to at most $O (\log \log (n))$ configurations. \begin{proof} We start from the biggest configurations and then iteratively reconstruct a good solution for smaller and smaller configurations. Recall $d$ is the smallest integer such that $\mathcal{K}^{(\geq d)}$ is empty. Our base case for these configurations in $\mathcal{K}^{(\geq d)}$ is vacuously satisfied. Now assume that we have a solution at level $j$, i.e. an assignment of resources to configurations in $\mathcal{K}^{(\geq j)}$ such that no resource is taken more than $\gamma$ times and each configuration $C\in \mathcal{K}^{(k)}$ such that $k\geq j$ receives at least \begin{equation*} \left(1-\frac{1}{\log (n)} \right)^{2(k-j)}\ell^{k-j} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-j}|K\cap C \cap R_h| \end{equation*} resources from $R_j$. We show that this implies a solution at level $j-1$ in the following way. First by Lemma~\ref{lma-good-solution}, this implies an assignment of resources of $R_{j-1}$ to configurations in $\mathcal{K}^{(\geq j)}$ such that each $C\in \mathcal{K}^{(k)}$ receives at least \begin{align*} &\left(1-\frac{1}{\log (n)} \right)\ell \left(\ell^{k-j} \left(1-\frac{1}{\log (n)} \right)^{2(k-j)} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-j}|K\cap C \cap R_h|\right)\\ &=\left(1-\frac{1}{\log (n)} \right)^{2(k-(j-1))-1} \ell^{k-(j-1)} |C\cap R_k|-\frac{3}{\gamma}\left(1-\frac{1}{\log (n)} \right)\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-(j-1)}|K\cap C \cap R_h|\\ &\geq \left(1-\frac{1}{\log (n)} \right)^{2(k-(j-1))-1}\ell^{k-(j-1)} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-(j-1)}|K\cap C \cap R_h| \end{align*} resources and no resource of $R_{j-1}$ is taken more than $\gamma$ times. Note that we can apply Lemma \ref{lma-good-solution} since we have by Claim \ref{cla:reconstruct} and Lemma \ref{lma-size} \begin{align*} &\left(1-\frac{1}{\log (n)} \right)^{2(k-j)}\ell^{k-j} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-j}|K\cap C \cap R_h| \\ &\geq \frac{\ell^{k-j}}{e^2}|C\cap R_k| - \frac{3}{\gamma}2000\ell^{-j}\frac{d+\ell}{\ell}\log(\ell)|C|\\ &\geq \ell^{-j}|C|\left(\frac{1}{2e^2}-\frac{6000}{\gamma}\frac{d+\ell}{\ell}\log(\ell)\right)\\ &\geq \frac{\ell^{-j}|C|}{3e^2}>\frac{\ell^3}{1000} \end{align*} Now consider configurations in $\mathcal{K}^{(j-1)}$ and proceed for them as follows. Give to each $C\in\mathcal{K}^{(j-1)}$ all the resources in $C\cap R_{j-1}$ except all the resources that appear in more than $\gamma$ configurations in $\mathcal{K}^{(j-1)}$. Since each deleted resource is counted at least $\gamma$ times in the sum $\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C\cap R_{j-1}|$, we have that each configuration $C$ in $\mathcal{K}^{(j-1)}$ receives at least \begin{equation*} |C\cap R_{j-1}|-\frac{1}{\gamma}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C\cap R_{j-1}| \end{equation*} resources and no resource is taken more than $\gamma$ times by configurations in $\mathcal{K}^{(j-1)}$. Notice that now every resource is taken no more than $\gamma$ times by configurations in $\mathcal{K}^{(\geq j)}$ and no more than $\gamma$ times by configurations in $\mathcal{K}^{(j-1)}$ which in total can sum up to $2\gamma$ times. Therefore to finish the proof consider an resource $i\in R_{j-1}$. This resource is taken $b_i$ times by configurations in $\mathcal{K}^{(\geq j)}$ and $a_i$ times by configurations in $\mathcal{K}^{(j-1)}$. If $a_i+b_i \leq \gamma$, nothing needs to be done. Otherwise, denote by $O$ the set of problematic resources (i.e. resources $i$ such that $a_i+b_i>\gamma$). For every $i\in O$, select uniformly at random $a_i+b_i-\gamma$ configurations in $\mathcal{K}^{(\geq j)}$ that currently contain resource $i$ and delete the resource from these configurations. When this happens, each configuration in $C\in \mathcal{K}^{(\geq j)}$ that contains $i$ has a probability of $(a_i+b_i-\gamma)/b_i$ to be selected to loose this resource. Hence the expected number of resources that $C$ looses with such a process is \begin{equation*} \mu = \sum_{i\in O\cap C} \frac{a_i+b_i-\gamma}{b_i} \end{equation*} It is not difficult to prove the following claim. However, for better clarity we defer its proof to appendix \ref{appendix_reconstruct}. \begin{claim} For any $C\in \mathcal{K}^{(\geq j)}$, \label{cla:reconstruct_mu} \begin{equation*} \frac{1}{\gamma^2}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\leq \mu \leq \frac{2}{\gamma} \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O| \end{equation*} \end{claim} Assume then that $\mu \leq \frac{|C\cap R_k|}{10^{12}\log^3 (n)}$. Note that $C$ cannot loose more than $\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|$ resources in any case. Therefore, by assumption on $\mu$, and since \begin{equation*} \mu\geq \frac{1}{\gamma^2}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\ , \end{equation*} we have that \begin{equation*} \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\leq \frac{\gamma^2}{10^{12}\log^3 (n)} |C\cap R_k|\leq \frac{10^{11} \log^2 \log (n)}{10^{12}\log^3 (n)}|C\cap R_k|\leq \frac{1}{\log (n)}|C\cap R_k|\ . \end{equation*} Therefore $C$ looses at most $|C\cap R_k|/\log (n)$ resources. Otherwise we have that \begin{equation*} \mu > \frac{|C\cap R_k|}{10^{12}\log^2 (n)} \geq \frac{\ell^3}{10^{12} \log^3 (n)} \geq 200\log(n) \end{equation*} by Lemma~\ref{lma-size}. Hence noting $X$ the number of deleted resources in $C$ we have that \begin{equation*} \mathbb P\left(X\geq \frac{3}{2}\mu \right) \leq \exp\left(-\frac{\mu}{12} \right)\leq \frac{1}{n^{10}}. \end{equation*} With high probability no configuration looses more than \begin{equation*} \frac{3}{2}\mu \leq \frac{3}{\gamma}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\leq \frac{3}{\gamma}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}| \end{equation*} resources. Hence each configuration $C\in \mathcal{K}^{(\geq j)}$ ends with at least \begin{align*} &\left(1-\frac{1}{\log (n)} \right)^{2(k-(j-1))-1}\ell^{k-(j-1)} |C\cap R_k|-\frac{3}{\gamma}\sum_{j\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-(j-1)}|K\cap C \cap R_h|\\ &-\frac{1}{\log (n)}\left(1-\frac{1}{\log (n)} \right)^{2(k-(j-1))-1}\ell^{k-(j-1)} |C\cap R_k| - \frac{3}{\gamma}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}|\\ &\geq \left(1-\frac{1}{\log (n)} \right)^{2(k-(j-1))}\ell^{k-(j-1)} |C\cap R_k|-\frac{3}{\gamma}\sum_{j-1\leq h\leq k} \sum_{K\in \mathcal{K}^{(h)}} \ell^{h-(j-1)}|K\cap C \cap R_h| \end{align*} resources which concludes the proof. \end{proof} \begin{corollary} \label{reconstruct_corollary} There exists an assignment of resources $R$ to $\mathcal{K}$ such that each configuration $C\in \mathcal{K}$ receives at least $\left\lfloor |C|/(100\gamma) \right\rfloor$ resources. Moreover, this assignment can be found in polynomial time. \end{corollary} \begin{proof} Lemma \ref{lem:reconstruct} for $k=0$ and Claim \ref{cla:reconstruct} together imply that we can assign at least \begin{equation*} \frac{|C|}{2e^2}-\frac{6000}{100.000}|C|\geq \frac{|C|}{100} \end{equation*} resources to every $C\in \mathcal{K}$ such that no resource in $R$ is assigned more than $\gamma$ times. In particular, we can fractionally assign at least $|C| / (100\gamma)$ resources to each $C\in \mathcal{K}$ such that no resource is assigned more than once. By integrality of the bipartite matching polytope, the corollary follows. \end{proof} \section{Further connections between hypergraph matching and Santa Claus} \label{sec:reduction santa claus} In Section~\ref{sec:hypergraph problem} we essentially prove that every regular (non-uniform) hypergraph has an $\alpha$-relaxed perfect matching for some $\alpha=O(\log \log (n))$, assuming that all hyperedges contain at least $\alpha$ resources. This means that we give a sufficient condition for a hypergraph to have a good relaxed matching. A natural optimization problem that arises from this is the following: Given any unweighted hypergraph, which is not necessarily regular nor all hyperedges necessarily contain many resources, what is the minimum $\alpha$ such that there exists an $\alpha$-relaxed perfect matching in this hypergraph? In this section, we investigate the relationship between this problem and the Santa Claus problem with linear utility functions. Formally, the two problems considered are precisely the following. \paragraph{Matching in general hypergraphs.} Consider a (non-uniform) hypergraph $\mathcal H=(P\cup R, \mathcal{C})$ with unit weights, that is, $w_{j,C}=1$ for all $j,C$ such that $j\in C$. The problem is to find the minimum $\alpha$ such that $\mathcal H$ has an $\alpha$-relaxed perfect matching (and output such a matching). \paragraph{The Santa Claus with linear utility functions.} In this case, each player $i$ has an arbitrary linear utility function $f_i$. We note that there is no relationship assumed between the utility functions of different players. The goal is to assign resources to players to maximize the minimum utility among players. As mentioned in introduction, the best approximation algorithm for this problem is an $O(n^\epsilon)$-approximation running in time $O(n^{1/\epsilon})$. We show by a straightforward reduction that a $c$-approximation for the Santa Claus problem immediately implies a $c$-approximation for the matching problem. Interestingly, there is also a close connection in the opposite direction. \begin{theorem}\label{thm:reduction} A $c$-approximation algorithm to the hypergraph matching problem in general hypergraphs yields an $O((c\log^* (n))^2)$-approximation algorithm to the Santa Claus problem. \end{theorem} We mention that we implicitly refer to polynomial time algorithms even when not specified. All the proofs of this section are deferred to Appendix \ref{appendix:reduction}. We also mention that Theorem~\ref{thm:reduction} implies that any sub-polynomial approximation to the matching problem would be a significant improvement of the state-of-the-art for Santa Claus with arbitrary linear utility functions. \paragraph{Remark.} Since hypergraphs considered here might be non-regular and some hyperedges might contain very few resources, our result in Section \ref{sec:hypergraph problem} does not imply any approximation for the optimization problem considered here. Our reduction in this section makes a crucial use of small hyperedges containing only one resource. This shows that handling the small hyperedges is one of the core difficulties in this case. \begin{equation*} \min_{i\in P} f(S_i\cap R_i) \end{equation*} is maximized. A very natural extension of this problem is to consider $f$ as non necessarily linear. For instance, $f$ can be submodular. We show that our result also implies a $O(\log \log (|R|))$-approximation when $f$ is submodular. \section{Conclusion} We investigated the submodular Santa Claus in the restricted assignment case and gave a $O(\log \log (n))$-approximation for this problem. This represents a significant generalization of the results for the linear case. The submodularity of the utility function introduced new obstacles compared to the linear case. These difficulties are captured by the fact that we need to solve a new matching problem in non-uniform hypergraphs that generalizes the case of uniform hypergraphs which has been already studied in the context of the restricted Santa Claus problem with a linear utility function. Under the assumption that the hypergraph is regular and all edges are sufficiently large, we proved that there is always a $\alpha$-relaxed perfect matching for $\alpha = O(\log \log (n))$. This result generalizes the work of Bansal and Srividenko~\cite{BansalSrividenko}. It remains an intriguing question whether one can get $\alpha = O(1)$ as it is possible in the uniform case. One idea (similar to Feige's proof in the uniform case~\cite{Feige}) would be to view our proof as a sparsification theorem and to apply it several times. Given a set of hyperedges such that every player has $\ell$ hyperedges and every resource appears in no more than $\ell$ hyperedges, one would like to select $\textrm{polylog}(\ell)$ hyperedges for each player such that all resources appear in no more than $\textrm{polylog}(\ell)$ of the selected hyperedges. It is not difficult to see than our proof actually achieves this when $\ell=\textrm{polylog}(n)$. However, repeating this after the first step seems to require new ideas since our bound on the number of times each resource is taken is $\Omega \left(\frac{d+\ell}{\ell}\log(\ell) \right)$ where $\ell$ is the current sparsity and $d$ the number of configuration sizes. For the first step, we conveniently have that $d=O(\log (n))=O(\ell)$ but after the first sparsification, it may not be true. We also provided a reduction from the Santa Claus with arbitrary linear utility functions to the hypergraph matching problem in general hypergraphs. This shows that finding the smallest $\alpha$ such that a hypergraph has an $\alpha$-relaxed perfect matching (or approximating it) is a very non-trivial problem (even within a sub-polynomial factor). Another interesting question is to improve the $O(\log^*(n))^2$ factor in the reduction to a constant. \appendix \section{Concentration bounds} \begin{proposition}[Chernoff bounds (see e.g.~\cite{mitzenmacher2017probability})] \label{chernoff} Let $X=\sum_i X_i$ be a sum of independent random variables such that each $X_i$ can take values in a range $[0,1]$. Define $\mu=\mathbb E(X)$. We then have the following bounds \begin{equation*} \mathbb P \left(X\geq (1+\delta)\mathbb E(X) \right) \leq \exp\left(-\frac{\min\{\delta,\delta^2\} \mu}{3} \right) \end{equation*} for any $\delta>0$. \begin{equation*} \mathbb P \left(X\leq (1-\delta)\mathbb E(X) \right) \leq \exp\left(-\frac{\delta^2 \mu}{2} \right) \end{equation*} for any $0<\delta<1$. \end{proposition} The following proposition follows immediately from Proposition \ref{chernoff} by apply it with $X'=X/a$. \begin{proposition} \label{cor:chernoff} Let $X=\sum_i X_i$ be a sum of independent random variables such that each $X_i$ can take values in a range $[0,a]$ for some $a>0$. Define $\mu=\mathbb E(X)$. We then have the following bounds \begin{equation*} \mathbb P \left(X\geq (1+\delta)\mathbb E(X) \right) \leq \exp\left(-\frac{\min\{\delta,\delta^2\} \mu}{3a} \right) \end{equation*} for any $\delta>0$. \begin{equation*} \mathbb P \left(X\leq (1-\delta)\mathbb E(X) \right) \leq \exp\left(-\frac{\delta^2 \mu}{2a} \right) \end{equation*} for any $0<\delta<1$. \end{proposition} in regular hypergraphs with sufficiently large hyperedges, we can assume that $\ell =300.000\log^{3}(n)$ at a constant loss: If $\ell$ is smaller than $300.000\log^{3}(n)$, then we simply duplicate all hyperedges an appropriate number of times. If $\ell$ is larger, we select for each player $300.000\log^{3}(n)$ configurations uniformly at random from his configurations. The expected number of times a resource appears in a configuration with this process is at most $300.000\log^{3}(n)$. Hence, the probability that a resource appears more than $600.000\log^{3}(n)$ times is at most $\exp \left(- 1/3 \cdot 100.000\log^{3}(n)\right)\leq 1/n^{10}$ by a standard Chernoff bound (see Proposition~\ref{cor:chernoff}). Hence with high probability this event does not happen for any resource. We now have that each player has $300.000\log^{3}(n)$ configurations and each resource does not appear in more than $600.000\log^{3}(n)$ configurations. Taking for each configuration $C$ only $\lfloor |C| / 2 \rfloor$ resources we can reduce the latter bound to $300.000\log^3(n)$ as well: The previous argument gives a half-integral matching of resources to configurations satisfying the mentioned guarantee. Then by integrality of the bipartite matching polytope there is also an integral one. \section{Omitted proofs from Section~\ref{sec:reduction to hypergraph}}\label{appendix_lp} \subsection{Solving the configuration LP} The goal of this section is to prove Theorem~\ref{thm:config-LP}. We consider the dual of the configuration LP (after adding an artificial minimization direction $\min 0^T x$). \begin{align*} \max \sum_{i\in P} y_i &- \sum_{j\in R} z_j \\ \sum_{j\in C} z_j &\ge y_i \quad \text{ for all } i\in P, C\in\mathcal{C}(i, T) \\ y_j, z_i &\ge 0 \end{align*} Observe that the optimum of the dual is either $0$ obtained by $y_i = 0$ and $z_j = 0$ for all $i,j$ or it is unbounded: If it has any solution with $\sum_{i\in P} y_i - \sum_{j\in R} z_j > 0$, the variables can be scaled by an arbitrary common factor to obtain any objective value. If it is unbounded, this can therefore be certified by providing a feasible solution $y, z$ with \begin{equation} \sum_{i\in P} y_i - \sum_{j\in R} z_j \ge 1 \tag{$*$}. \end{equation} We approximate the dual in the variant with constraint $(*)$ instead of a maximization direction using the ellipsoid method. The separation problem of the dual is as follows. Given $z_j$, $y_i$ find a player $i$ and set $C$ with $g(C\cap \Gamma_i) \ge T$ such that $\sum_{j\in C} z_j < y_i$. To this end, consider the related problem of maximizing a monotone submodular function subject to knapsack constraints. In this problem we are given a monotone submodular function $g$ over a ground set $E$ and the goal is to maximize $g(E')$ over all $E'\subseteq E$ with $\sum_{j\in E'} a_j \le b$. Here $a_j \ge 0$ is a weight associated with $j\in E$ and $b$ is a capacity. For this problem Srividenko gave a polynomial time $(1 - 1/e)$-approximation algorithm~\cite{DBLP:journals/orl/Sviridenko04}. It is not hard to see that this can be used to give a constant approximation for the variation where strict inequality is required in the knapsack constraint: Assume w.l.o.g.\ that $0 < a_j < b$ for all $j$. Then run Srivideko's algorithm to find a set $E'$ with $\sum_{j\in E'} a_j \le b$. Notice that $g(E')$ is at least $(1 - 1/e)\mathrm{OPT}$, also when $\mathrm{OPT}$ is the optimal value with respect to strict inequality. If $E'$ contains only one element then equality in the knapsack constraint cannot hold and we are done. Otherwise, split $E'$ into two arbitrary non-empty parts $E''$ and $E'''$. It follows that $\sum_{j\in E''} a_j < b$ and $\sum_{j\in E'''} a_j < b$. Moreover, either $g(E'') \ge g(E') / 2$ or $g(E'') \ge g(E') / 2$. Hence, this method yields a $c$-approximation for $c = (1 - 1/e)/2$. We now demonstrate how to use this to find a $c$-approximation to the configuration LP. Let $\mathrm{OPT}$ be the optimum of the configuration LP. It suffices to solve the problem of finding for a given $T$ either a solution of value $c T$ or deciding that $T > \mathrm{OPT}$. This can then be embedded into a standard dual approximation framework. We run the ellipsoid method on the dual of the configuration LP with objective value $c T$ and constraint $(*)$. This means we have to solve the separation problem. Let $z, y$ be the variables at some state. We first check whether $(*)$ is satisfied, that is $\sum_{i\in P} y_i - \sum_{j\in R} z_j \ge 1$. If not, we return this inequality as a separating hyperplane. Hence, assume $(*)$ is satisfied and our goal is to find a violated constraint of the form $\sum_{j\in C} z_j < y_i$ for some $i\in P$ and $C\in \mathcal{C}(i, T)$. For each player $i$ we maximize $f$ over all $S\subseteq \Gamma_i$ with $\sum_{j\in S} z_j < y_i$. We use the variant of Srividenko's algorithm described above to obtain a $c$-approximation for each player. If for one player $i$ the resulting set $S$ satisfies $f(S) \ge c T$, then we have found a separating hyperplane to provide to the ellipsoid method. Otherwise, we know that $f(S) < T$ for all players $i$ and $S\subseteq\Gamma_i$ with $\sum_{j\in S} z_j < y_i$. In other words, for all players $i$ and all $C\in\mathcal{C}(i,T)$ it holds that $\sum_{j\in C} z_j \ge y_i$, i.e., $z, y$ is feasible for objective value $T$ and hence $\mathrm{OPT} > T$. If the ellipsoid method terminates without concluding that $\mathrm{OPT} > T$, we can derive a feasible primal solution with objective value $cT$: The configurations constructed for separating hyperplanes suffice to prove that the dual is bounded. These configurations can only be polynomially many by the polynomial running time of the ellipsoid method. Hence, when restricting the primal to these configurations it must remain feasible. To obtain the primal solution we now only need to solve a polynomial size linear program. This concludes the proof of Theorem~\ref{thm:config-LP}. \subsection{Clusters} This section is devoted to prove Lemma~\ref{lem:config-sample}. The arguments are similar to those used in~\cite{BansalSrividenko}. \begin{lemma} \label{lem:clusters} Let $x^*$ be a solution to the configuration LP of value $T^*$. Then $x^*$ can be transformed into some $x'_{i,C} \ge 0$ for $i\in P$, $C\in\mathcal{C}_t(i, T^*)$ which satisfies the following. There is a partition of the players into clusters $K_1\cup\cdots \cup K_k \cup Q = P$ that satisfy the following. \begin{enumerate} \item any thin resource $j$ is fractionally assigned at most once, that is, \begin{equation*} \sum_{i\in P} \sum_{C\in \mathcal{C}_t(i,T^*):j\in C}x'_{i, C} \le 1 \end{equation*} We say that the \textit{congestion} on item $j$ is at most 1. \item every cluster $K_j$ gets at least $1/2$ thin configurations in $x'$, that is, \begin{equation*} \sum_{i\in K_j} \sum_{C\in \mathcal{C}_t(i,T^*)} x'_{i, C} \ge 1/2 ; \end{equation*} \item given any $i_1\in K_1, i_2\in K_2,\dotsc,i_k\in K_k$ there is a matching of fat resources to players $P\setminus\{i_1,\dotsc,i_k\}$ such that each of these players $i$ gets a unique fat resource $j\in\Gamma_i$. \end{enumerate} \end{lemma} The role of the set of players $Q$ in the lemma above is that each of them gets one fat resource for certain. \begin{proof} We first transform the solution $x^*$ as follows. For every configuration $C$ (for player $i$) that contains at least one fat resource and such that $x^*_{i,C}>0$, we select arbitrarily one of these fat resources $j$ and we set $x^*_{i,\{j\}}=x^*_{i,C}$ and then we set $x^*_{i,C}=0$. It is clear that this does not increase the congestion on resources and now every configuration that has non-zero value is either a thin configuration or a singleton containing one fat resource. Therefore we can consider the bipartite graph $G$ formed between the players and the fat resources where there is an edge between player $i$ and fat resource $j$ if the corresponding configuration $C=\{j\}$ is of non zero value (i.e. $x^*_{i,C}>0$). The value of such an edge will be exactly the value $x^*_{i,C}$. We now make G acyclic by doing the following operation until there exists no cycle anymore. Pick any cycle (which must have even length since the graph is bipartite) and increase the coordinate of $x^*$ corresponding to every other edge in the cycle by a small constant. Decrease the value corresponding to the remaining edges of the cycle by the same constant. This ensures that fat resources are still (fractionally) taken at most once and that the players still have one unit of configurations fractionally assigned to them. We continue this until one of the edge value becomes 0 or 1. If an edge becomes 0, delete that edge and if it becomes 1, assign the corresponding resource to the corresponding player forever. Then delete the player and the resource from the graph and add the player to the cluster $Q$. By construction, every added player to $Q$ is assigned a unique fat resource. Notice that when we stop, each remaining player still has at least 1 unit of configurations assigned to him and every fat resource is still (fractionally) taken at most once. Hence we get a new assignment vector where the assignments of fat resources to players form a forest. We also note that the congestion on thin resources did not increase during this process (it actually only decreased either when we replace fat configurations by a singleton and when players are put into the set $Q$ and deleted from the instance). We show below how to get the clusters for any tree in the forest. \begin{enumerate} \item If the tree consists of a single player, then it trivially forms its own cluster. By feasibility of the original solution $x^*$, condition 2 of the lemma holds. \item If there is a fat resource that has degree 1, assign it to its player, add the player to $Q$ and delete both the player and resource. Continue this until every resource has a degree of at least 2. This step adds players to cluster $Q$. By construction, every added player is assigned a unique fat resource. \item While there is a resource of degree at least 3, we perform the following operation. Root the tree containing such a resource at an arbitrary player. Consider a resource $j$ of degree at least 3 such that the subtree rooted at this resource contains only resources of degree 2. Because this resource must have at least 2 children in the tree $i_1,i_2,\ldots$ (which are players) and because \begin{equation*} \sum_{i\in P} \sum_{C : j\in C} x^*_{i,C}\leq 1, \end{equation*} it must be that one of the children (say $i_1$) satisfies $x^*_{i_1,\{j\}}\leq 1/2$. We then delete the edge $(j,i_1)$ in the tree and set $x^*_{i_1,\{j\}}$ to 0. \item Every resource now has degree exactly 2. We form a cluster for each tree in the forest. The cluster will contain the players and fat resources in the tree. We note that in every tree, only the player at the root lost at most $1/2$ unit of a fat resource by the previous step in the construction. By the degree property of resources and because the graph contains no cycle, it must be that in each cluster $K$ we have $|R(K)|=|P(K)|-1$ where $|R(K)|$ is the number of resources in the cluster and $|P(K)|$ the number of players. Because each resource is assigned at most once, and because only one player in the cluster lost at most $1/2$ unit of a fat resource, it must be that the cumulative amount of thin configurations assigned to players in $K$ is at least \begin{equation*} |P(K)|-|R(K)|-1/2=1/2. \end{equation*} This gives the second property of the lemma. For the third property, notice that for any choice of player $i\in K$, we can root the tree corresponding to the cluster $K$ at the player $i$ and assign all the fat resources in $K$ to their only child in the tree (they all have degree 2). This gives the third property of the lemma. As each of these steps individually maintained maintained a congestion of at most $1$ on every thin resource, we indeed get a new solution $x'$ and the associated clusters with the required properties. \end{enumerate} \end{proof} Lemma~\ref{lem:clusters} implies that for each cluster we need to cover only one player with a thin configuration. Then the remaining players can be covered with fat resources. We will now replace $x'$ by a solution $x''$ which takes slightly worse configurations $\mathcal{C}_t(i, T^*/5)$, but satisfies (2) in Lemma~\ref{lem:clusters} with $2$ instead of $1/2$. This can be achieved by splitting each configuration $C\in \mathcal{C}_t(i, T^*)$ in $4$ disjoint parts $C_1, C_2, C_3, C_4\in \mathcal{C}_t(i, T^* / 5)$. Let $C_1 \subseteq C$ with $f(C_1) \ge T^* / 5$ minimal in the sense that $f(C_1 \setminus \{j\}) < T^* / 5$ for all $j\in C_1$. Let $j_1 \in C_1$. By submodularity and because $j_1$ is thin it holds that \begin{equation*} f(C \setminus C_1) \ge f(C) - f(C_1 \setminus \{j_1\}) - f(\{j_1\}) \ge 4T^* / 5 - T^*/100 . \end{equation*} Hence, in the same way we can select $C_2\subseteq C\setminus C_1$, $C_3\subseteq C\setminus (C_1 \cup C_2)$ and $C_4\subseteq C\setminus (C_1 \cup C_2 \cup C_3)$. We now augment $x'$ to $x''$ by initializing $x''$ with $0$ and then for each $i$ and $C\in\mathcal{C}(i,T^*)$ increasing $x''_{i, C_1}$, $x''_{i, C_2}$, $x''_{i, C_3}$, and $x''_{i, C_4}$ by $x'_{i, C}$. Here $C_1, C_2, C_3, C_4 \in \mathcal{C}(i, T^* / 5)$ are the configurations derived from $C$ by splitting it as described above. Finally, we sample for each cluster some $\ell \geq 12\log(n)$ many configurations with the distribution of $x''$ to obtain the statement of Lemma~\ref{lem:config-sample} which we restate for convenience. \begin{customthm}{\ref{lem:config-sample}} (restated) Let $\ell \ge 12\log (n)$. Given a solution of value $T^*$ for the configuration LP in randomized polynomial time we can find a partition of the players into clusters $K_1\cup\cdots \cup K_k\cup Q = P$ and multisets of configurations $\mathcal{C}_h \subseteq \bigcup_{i\in K_h} \mathcal{C}_T(i, T^*/5)$, $h=1,\dotsc,k$, such that \begin{enumerate} \item $|\mathcal{C}_h| = \ell$ for all $h=1,\dotsc,k$ and \item Each small resource appears in at most $\ell$ configurations of $\bigcup_h \mathcal{C}_h$. \item given any $i_1\in K_1, i_2\in K_2,\dotsc,i_k\in K_k$ there is a matching of fat resources to players $P\setminus\{i_1,\dotsc,i_k\}$ such that each of these players $i$ gets a unique fat resource $j\in\Gamma_i$. \end{enumerate} \end{customthm} \begin{proof} We start with the clusters obtained with Lemma~\ref{lem:clusters} and the solution $x''$ described above. Recall that \begin{equation*} \sum_{i\in K_h} \sum_{C\in C_t(i,T^*/5)} x''_{i,C} \geq 2 \end{equation*} for each cluster $K_h$. We assume w.l.o.g.\ that equality holds by reducing some variables $x''_{i, C}$. Clearly then each resource is still contained in at most one configuration in total. For each cluster $K_h$, we sample a configuration that contains a player in this cluster according to the probability distribution given by the values $\{x''_{i,C}/2 \}_{i\in K_h, C\in \mathcal{C}_t (i,T^*/5)}$. By the assumption of equality stated above this indeed defines a probability distribution. We repeat this process $\ell$ times. We first note that for one iteration, each resource is in expectation contained in \begin{equation*} \sum_{i\in P} \sum_{C\in \mathcal{C}(i,T^*/5):j\in C} x''_{i,C}/2 \leq 1/2 \end{equation*} selected configurations. Hence in expectation all the resource are contained in $\ell/2$ selected configurations after $\ell$ iterations. By a standard Chernoff bound (see Proposition \ref{chernoff}), we have that with probability at most \begin{equation*} \exp \left( -\ell/6\right)\leq 1/n^2 \end{equation*} a resource is contained in more than $\ell$ configurations. By a union bound, it holds that all resources are contained in at most $\ell$ selected configurations with high probability. \end{proof} \section{Omitted proofs from Section~\ref{sec:sequence}}\label{appendix_sequence} \begin{customthm}{\ref{lma-size}}(restated) Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. For any $k\geq 0$ and any $C\in\mathcal{C}^{(\geq k)}$ we have \begin{equation*} \frac{1}{2} \ell^{-k}|C| \le |R_k \cap C| \le \frac{3}{2} \ell^{-k}|C| \end{equation*} with probability at least $1-1/n^{10}$. \end{customthm} \begin{proof} The lemma trivially holds for $k=0$. For $k>0$, by assumption $C\in\mathcal{C}^{(\geq k)}$ hence $|C|\geq \ell^{k+3}$. Since each resource of $R=R_0$ survives in $R_k$ with probability $\ell^{-k}$ we clearly have that in expectation \begin{equation*} \mathbb E(|R_k\cap C|) = \ell^{-k}|C| \end{equation*} Hence the random variable $X=|R_k\cap C|$ is a sum of independent variables of value either $0$ or $1$ and such that $\mathbb E (X)\geq \ell^3$. By a standard Chernoff bound (see Proposition \ref{cor:chernoff}), we get \begin{equation*} \mathbb P\left(X\notin \left[\frac{\mathbb{E}(X)}{2}, \frac{3\mathbb{E}(X)}{2}\right]\right) \leq 2 \exp \left(-\frac{\mathbb E(X)}{12} \right) \leq 2 \exp \left(-\frac{300.000\log^3 (n)}{12} \right) \leq \frac{1}{n^{10}} \end{equation*} since by assumption $\ell \geq 300.000\log^3 (n)$. \end{proof} \begin{customthm}{\ref{lma-overlap-representative}}(restated) Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. For any $k\geq 0$ and any $C\in\mathcal{C}^{(\geq k)}$ we have \begin{equation*} \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C\cap R_k| \leq \frac{10}{\ell^{k}} \left(|C|+\sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C| \right) \end{equation*} with probability at least $1-1/n^{10}$. \end{customthm} \begin{proof} The expected value of the random variable $X=\sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C\cap R_k|$ is \begin{equation*} \mathbb E(X) = \frac{1}{\ell^k} \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C|. \end{equation*} Since each resource is in at most $\ell$ configurations, $X$ is a sum of independent random variables that take value in a range $[0,\ell]$. Then by a standard Chernoff bound (see Proposition \ref{cor:chernoff}), we get \begin{equation*} \mathbb P\left(X\ge 10 \left(\frac{|C|}{\ell^k} + \mathbb E(X)\right) \right) \leq \exp\left(-\frac{3|C|}{\ell^{k+1}}\right) \leq \frac{1}{n^{10}} , \end{equation*} since by assumption, $|C|\geq \ell^{k+3}$ and $\ell \geq 300.000\log ^3(n)$. \begin{comment} \begin{align*} \mathbb P\left(X\ge 10\cdot \mathbb E(X)\right) &\leq \exp\left(-\frac{3\mathbb E(X)}{\log (n)}\right) \\ &\leq \exp\left(-\frac{3\ell^{-k}|C|}{\log^6 (n)}\right) \\ &\leq \exp\left(-\frac{3\ell}{\log^6 (n)}\right) \\ &= o\left(\frac{1}{n^{10}}\right) \end{align*} Otherwise if $1\leq \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C|\leq\frac{|C|}{\log^5 (n)}$, then by choosing $\delta = \frac{10|C|}{l^k \mathbb E(X)}$ we first note that $\delta \geq 10\log^5 (n)>1$ since $\mathbb E(X) = \frac{1}{\ell^k}\cdot \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C| \leq \frac{|C|}{\ell^k\log^5 (n)}$. By plugging in a Chernoff bound (Proposition \ref{cor:chernoff}) we have that \begin{align*} \mathbb P\left(X\ge (1+\delta)\mathbb E(X)\right) &\leq \exp\left(-\frac{\delta\mathbb E(X)}{3}\right) \\ &\leq \exp\left(-\frac{10|C|}{l^k}\right)\\ &\leq \exp\left(-10\ell\right)\\ &=o\left(\frac{1}{n^{10}} \right) \end{align*} Hence we have \begin{equation*} \mathbb P\left(X\ge \mathbb E(X)+\frac{10|C|}{\ell^k}\right)=o\left(\frac{1}{n^{10}} \right) \end{equation*} The last case when $\mathbb E(X)=0$ is trivial since in this case there is no randomness and we always get \begin{equation*} \sum_{C'\in \mathcal{C}^{(k)}} |C'\cap C\cap R_k|=0 \end{equation*} \end{comment} \end{proof} We finish by the proof of the last property. As mentioned in the main body of the paper, this statement is a generalization of some ideas that already appeared in \cite{BansalSrividenko}. However, in \cite{BansalSrividenko}, the situation is simpler since they need to sample down the resource set only once (i.e. there are only two sets $R_1\subseteq R$ and not a full hierarchy of resource sets $R_d\subseteq R_{d-1}\subseteq \cdots \subseteq R_1 \subseteq R$). Given the resource set $R_1$, they want to select configurations and give to each selected configuration $K$ all of its resource set $|K\cap R_1|$ so that no resource is assigned too many times. In our case the situation is also more complex than that since at every step the selected configurations receive only a fraction of their current resource set. Nevertheless, we extend the ideas of Bansal and Srividenko to our more general setting. We recall the main statement before proceeding to its proof. \begin{customthm}{\ref{lma-good-solution}}(restated) Consider Random Experiment~\ref{exp:sequence} with $\ell\geq 300.000\log^{3} (n)$. Fix $k\geq 0$. Conditioned on the event that the bounds in Lemma~\ref{lma-size} hold for $k$, then with probability at least $1 - 1/n^{10}$ the following holds for all $\mathcal{F}\subseteq \mathcal{C}^{(\geq k+1)}$, $\alpha:\mathcal{F} \rightarrow \mathbb N$, and $\gamma \in\mathbb N$ such that $\ell^3/1000\leq \alpha(C) \leq n $ for all $C\in\mathcal{F}$ and $\gamma \in \{1,\dotsc,\ell\}$: If there is a $(\alpha,\gamma)$-good assignment of $R_{k+1}$ to $\mathcal{F}$, then there is a $(\alpha',\gamma)$-good assignment of $R_k$ to $\mathcal{F}$ where \begin{equation}\label{property:3} \alpha'(C) \ge \ell \left(1-\frac{1}{\log (n)} \right) \alpha(C) \end{equation} for all $C\in\mathcal{F}$. Moreover, this assignment can be found in polynomial time. \end{customthm} \begin{comment} \begin{lemma}(Property 3) For any $k\geq 0$, any family of configurations $\mathcal{F}\subseteq \mathcal{C}^{(\geq k+1)}$, any integers $\alpha_1,\alpha_2,\ldots ,\alpha_{|\mathcal{F}|}$ and $\gamma$ such that $n\geq \alpha_j\geq \ell^2/1000$ for all $j$ and $\gamma=O(\log (n))$, if there is a $(\alpha_1,\ldots ,\alpha_{|\mathcal{F}|},\gamma)$-good assignment of $R_{k+1}$ to $\mathcal{F}$, then there is a $(\alpha'_1,\ldots ,\alpha'_{|\mathcal{F}|},\gamma)$-good assignment of $R_k$ to $\mathcal{F}$ with $\alpha'_j=\ell \left(1-\frac{1}{\log (n)} \right) \alpha_j$. Moreover, this assignment can be found in polynomial time. \end{lemma} \end{comment} We first provide the definitions of a flow network that allows us to state a clean condition whether a good assignment of resources exists or not. We then provide the high probability statements that imply the lemma. For any subset of configurations $\mathcal F \subseteq \mathcal{C}^{(\geq k+1)}$, resource set $R_k$, $\alpha:\mathcal{F} \rightarrow \mathbb N$, and any integer $\gamma$, consider the following directed network (denoted by $\mathcal N (\mathcal F, R_k, \alpha,\gamma)$). Create a vertex for each configuration in $\mathcal F$ as well as a vertex for each resource. Add a source $s$ and sink $t$. Then add a directed arc from $s$ to the vertex $C\in\mathcal F$ with capacity $\alpha(C)$. For every pair of a configuration $C$ and a resource $i$ such that $i\in C$ add a directed arc from $C$ to $i$ with capacity $1$. Finally, add a directed arc from every resource to the sink of capacity $\gamma$. See Figure \ref{fig:network_flow} for an illustration. \begin{figure} \caption{The directed network and an $s$-$t$ cut} \label{fig:network_flow} \end{figure} We denote by \begin{equation*} \textrm{maxflow}\left(\mathcal N (\mathcal F, R_k, \alpha,\gamma)\right) \end{equation*} the value of the maximum $s$-$t$ flow in $\mathcal N (\mathcal F, R_k, \alpha,\gamma)$. Before delving into the technical lemmas, we provide a brief road map for the proof. First, we argue that for any subset of configurations, in the two networks induced on this subset and the consecutive resource sets (which are $R_k$ and $R_{k+1}$), the value of the maximum flow differs by approximately a factor $\ell$ (this is Lemma \ref{lem:flow_conservation} stated below). Then by a union bound over all possible subsets of configurations, we say that the above argument consecutively holds with good probability. This helps us conclude that a good assignment of the resource set $R_{k+1}$ implies that there is a good assignment of the resource set $R_k$. Notice that if one does not have the above argument with respect to all subsets of configurations at once, it is not necessary that a good assignment of resources must exist. In particular, we need Lemma \ref{lem:flow_black_box} to show that if on \textit{all} subsets of configurations the maximum flow is multiplied by \textit{approximately} $\ell$ when we expand the resource set from $R_{k+1}$ to $R_k$, then an $(\alpha,\gamma)$-good assignment of $R_{k+1}$ implies an $(\alpha',\gamma)$-good assignment of $R_k$, where $\alpha'$ is almost equal to $\ell \alpha$. \input{black-box-lemma} \begin{lemma} \label{lem:flow_conservation} Let $\mathcal F\subseteq \mathcal C^{\geq (k+1)}$, $\alpha:\mathcal{F} \rightarrow \mathbb N$ such that $\ell^3/1000 \leq \alpha(C) \leq n$ for all $C\in\mathcal{F}$, and $1 \le \gamma \le \ell$. Denote by $\mathcal N$ the network $\mathcal N (\mathcal F, R_k, \ell \cdot \alpha,\gamma)$ and by $\Tilde{\mathcal{N}}$ the network $\mathcal N (\mathcal F, R_{k+1},\alpha,\gamma)$. Then \begin{equation*} \mathrm{maxflow}\left(\mathcal{N}\right)\geq \frac{\ell}{1+0.5/\log (n)} \mathrm{maxflow}\left(\Tilde{\mathcal{N}}\right) \end{equation*} with probability at least $1-1/(n\ell)^{20|\mathcal F|}$. \end{lemma} \begin{proof} We use the max-flow min-cut theorem that asserts that the value of the maximum flow in a network is equal to the value of the minimum $s$-$t$ cut in the network. Consider a minimum cut $S$ of network $\mathcal N$ with $s\in S$ and $t\notin S$. Denote by $c(S)$ the value of the cut. We will argue that with high probability this cut induces a cut of value at most $c(S) / \ell \cdot (1+0.5/\log(n))$ in the network $\Tilde{\mathcal N}$. This directly implies the lemma. Denote by $\mathcal C'$ the set of configurations of $\mathcal{F}$ that are in $S$, i.e., on the source side of the cut, and $\mathcal C''=\mathcal{F}\setminus \mathcal C'$. Similarly consider $R'$ the set of resources in the $s$ side of the cut and $R''= R_k\setminus R'$. With a similar notation, we denote $\Tilde R' = R'\cap R_{k+1}$ the set of resources of $R'$ surviving in $R_{k+1}$; and $\Tilde R'' = R''\cap R_{k+1}$. Finally, denote by $\Tilde S$ the cut in $\Tilde{\mathcal N}$ obtained by removing resources of $R'$ that do not survive in $R_{k+1}$ from $S$, i.e., $\Tilde S = \{s\}\cup \mathcal C' \cup R'$. The value of the cut $S$ of $\mathcal N$ is \begin{equation*} c(S) = \sum_{C\in \mathcal C''} \ell \cdot \alpha(C) + e(\mathcal C',R'')+ \gamma |R'| \end{equation*} where $e(X,Y)$ denotes the number of edges from $X$ to $Y$. The value of the cut $\Tilde S$ in $\Tilde{\mathcal N}$ is \begin{equation*} c( \Tilde S) = \sum_{C\in \mathcal C''} \alpha(C) + e(\mathcal C',\Tilde R'')+ \gamma |\Tilde R'| \end{equation*} We claim the following properties. \begin{claim} \label{cla:size_configurations} For every $C\in \mathcal F$, the outdegree of the vertex corresponding to $C$ in $\mathcal N$ is at least $\ell^4/2$. \end{claim} Since $C\in \mathcal{C}^{(\geq k+1)}$ and by Lemma \ref{lma-size}, we clearly have that $|C\cap R_k|\geq \ell^4/2$. \begin{claim} \label{cla:size_cut} It holds that \begin{equation*} c(S)\geq \frac{|\mathcal{F}| \ell^3}{1000} . \end{equation*} \end{claim} We have by assumption on $\alpha(C)$ \begin{multline*} c(S) = \sum_{C\in \mathcal C''} \ell \cdot \alpha(C) + e(\mathcal C',R'')+ \gamma |R'| \geq \sum_{C\in \mathcal C''} \frac{\ell^3}{1000} + e(\mathcal C',R'')+ \gamma |R'|\\ \geq \frac{|\mathcal C''|\ell^3}{1000} + e(\mathcal C',R'')+ \gamma |R'| \end{multline*} Now consider the case where $e(\mathcal C',R'')\leq |\mathcal C'|\ell^3 / 1000$. Since each vertex in $\mathcal C'$ has outdegree at least $\ell^4/2$ in the network $\mathcal N$ (by Claim~\ref{cla:size_configurations}) it must be that $e(\mathcal C',R')\geq |\mathcal C'|\ell^4 / 2 - |\mathcal C'|\ell^3 / 1000 > |\mathcal C'|\ell^4 / 3$. Using that each vertex in $R'$ has indegree at most $\ell$ (each resource is in at most $\ell$ configurations), this implies $|R'|\geq |\mathcal C'|\ell^3 / 3$. Since $\gamma \geq 1$ we have in all cases that $e(\mathcal C',R'')+ \gamma |R'|\geq |\mathcal C'|\ell^3 / 1000$. Hence \begin{equation*} c(S) \geq \frac{|\mathcal C''|\ell^3}{1000} + \frac{|\mathcal C'|\ell^3}{1000} = \frac{|\mathcal{F}| \ell^3}{1000} . \end{equation*} This proves Claim~\ref{cla:size_cut}. We can now finish the proof of the lemma. Denote by $X$ the value of the random variable $e(\mathcal C',\Tilde{R''})+ \gamma |\Tilde{R'}|$. We have that \begin{equation*} \mathbb E[X] = \frac{1}{\ell}(e(\mathcal C',R'')+ \gamma |R'|). \end{equation*} Moreover, $X$ can be written as a sum of independent variables in the range $[0, \ell]$ since each vertex is in at most $\ell$ configurations and $\gamma \le \ell$ by assumption. By a Chernoff bound (see Proposition \ref{cor:chernoff}) with \begin{equation*} \delta = \frac{0.5 c(S)}{\log(n) \cdot (c(S)-\sum_{C\in \mathcal C''} \alpha(C))} \geq \frac{0.5}{\log(n)} \end{equation*} we have that \begin{multline*} \mathbb P\left(X\geq \mathbb E(X)+\frac{0.5 c(S)}{\ell\log(n)}\right) \leq \exp\left(-\frac{\min\{\delta,\delta^2\}\mathbb E(X)}{3\ell} \right) \\ \leq \exp\left(-\frac{c(S)}{12\ell^2\log^2 (n)} \right) \leq \exp\left(-\frac{|\mathcal F|\ell^3}{12.000\ell^2\log^2 (n)} \right) \leq \frac{1}{(n\ell)^{20|\mathcal{F}|}} , \end{multline*} where the third inequality comes from Claim~\ref{cla:size_cut} and the last one from the assumption that $\ell\geq 300.000\log^{3}(n)$. Hence with probability at least $1-1/(n\ell)^{20|\mathcal{F}|}$, we have that \begin{equation*} c( \Tilde S) = \sum_{C\in \mathcal C''} \alpha(C) + e(\mathcal C',\Tilde R'')+ \gamma |\Tilde R'| \leq \frac{1}{\ell}c(S)+\frac{0.5}{\ell \log (n)}c(S) .\qedhere \end{equation*} \end{proof} We are now ready to prove Lemma~\ref{lma-good-solution}. Note that Lemma \ref{lem:flow_conservation} holds with probability at least $1-1/(n\ell)^{20|\mathcal{F}|}$. Given the resource set $R_k$ and a cardinality $s = |\mathcal{F}|$ there are $O((n\ell)^{2s})$ ways of defining a network satisfying the conditions from Lemma~\ref{lem:flow_conservation} ($(m\ell)^s\le (n\ell)^s$ choices of $\mathcal{F}$, $n^{s}$ choices for $\alpha$ and $\ell$ choices for $\gamma$). By a union bound, we can assume that the properties of Lemma~\ref{lem:flow_conservation} hold for every possible network with probability at least $1 - 1/n^{10}$. Assume now there is a $(\alpha,\gamma)$-good assignment of $R_{k+1}$ to some family $\mathcal{F}$. Then by Lemma~\ref{lem:flow_black_box} the $\mathrm{maxflow}(\mathcal{N}(\mathcal{F}',R_{k+1}, \alpha,\gamma))$ is exactly $\sum_{C\in \mathcal{F}'}\alpha(C)$ for any $\mathcal{F}'\subseteq \mathcal{F}$. By Lemma~\ref{lem:flow_conservation}, this implies that $\mathrm{maxflow}(\mathcal{N}(\mathcal{F}',R_{k}, \ell \cdot \alpha,\gamma))$ is at least $\ell/(1+0.5/\log(n)) \sum_{C\in \mathcal{F}'}\alpha(C)$. By Lemma \ref{lem:flow_black_box}, this implies a $(\alpha',\gamma)$-good assignment from $R_k$ to $\mathcal{F}$, where \begin{equation*} \alpha'(C) = \lfloor\ell/(1+0.5/\log(n))\rfloor \alpha(C) \ge \ell / (1 + 1/\log(n)) \alpha(C) \geq \ell(1 - 1/\log(n)) \alpha(C). \end{equation*} \section{Omitted proofs from Section~\ref{sec:reconstruction}}\label{appendix_reconstruct} \begin{customcla}{\ref{cla:reconstruct}}(restated) For any $k\geq 0$, any $0\leq j\leq k$, and any $C\in \mathcal{K}^{(k)}$ \begin{equation*} \sum_{j\leq h\leq k}\sum_{K\in \mathcal{K}^{(h)}} \ell^{h}|K\cap C \cap R_h| \leq 2000\frac{d+\ell}{\ell}\log (\ell) |C|. \end{equation*} \end{customcla} \begin{proof}[Proof of Claim \ref{cla:reconstruct}] By Lemma~\ref{lma:main-LLL} we have that \begin{equation*} \sum_{j\leq h\leq k}\sum_{K\in \mathcal{K}^{(h)}} \ell^{h}|K\cap C \cap R_h| \leq \frac{1}{\ell} \sum_{j\leq h\leq k}\sum_{C'\in \mathcal{C}^{(h)}} \ell^{h}|C'\cap C \cap R_h| + 1000\frac{d+\ell}{\ell}\log (\ell) |C|. \end{equation*} Furthermore, by Lemma \ref{lma-overlap-representative}, we get \begin{equation*} \sum_{C'\in \mathcal{C}^{(h)}} \ell^{h}|C'\cap C \cap R_h| \leq \ell^{h}\frac{10}{\ell^h}\left(|C|+\sum_{C'\in \mathcal{C}^{(h)}} |C'\cap C| \right). \end{equation*} Finally note that each resource appears in at most $\ell$ configurations, hence \begin{equation*} \sum_{j\leq h\leq k}\sum_{C'\in \mathcal{C}^{(h)}} |C'\cap C| \leq \ell |C|. \end{equation*} Putting everything together we conclude \begin{align*} \sum_{j\leq h\leq k}\sum_{K\in \mathcal{K}^{(h)}} \ell^{h}|K\cap C \cap R_h| &\leq \frac{1}{\ell} \sum_{j\leq h\leq k}\sum_{C'\in \mathcal{C}^{(h)}} \ell^{h}|C'\cap C \cap R_h| + 1000\frac{d+\ell}{\ell}\log (\ell) |C| \\ &\leq \frac{1}{\ell} \sum_{j\leq h\leq k}10\left( |C|+\sum_{C'\in \mathcal{C}^{(h)}}|C'\cap C|\right) + 1000\frac{d+\ell}{\ell}\log (\ell) |C|\\ &\leq \frac{k-j}{\ell}10|C|+10|C|+1000\frac{d+\ell}{\ell}\log (\ell) |C|\\ &\leq 20|C|+1000\frac{d+\ell}{\ell}\log (\ell) |C|\\ &\leq 2000\frac{d+\ell}{\ell}\log (\ell) |C|.\qedhere \end{align*} \end{proof} \begin{customcla}{\ref{cla:reconstruct_mu}}(restated) For any $C\in \mathcal{K}^{(\geq j)}$, \begin{equation*} \frac{1}{\gamma^2}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\leq \mu \leq \frac{2}{\gamma} \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|. \end{equation*} \end{customcla} \begin{proof}[Proof of Claim \ref{cla:reconstruct_mu}] Note that we can write \begin{equation*} \mu = \sum_{i\in O\cap C} \frac{a_i+b_i-\gamma}{b_i} \leq \max_{i\in O\cap C}\left\lbrace \frac{a_i+b_i-\gamma}{a_ib_i} \right\rbrace \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|. \end{equation*} The reason for this is that each resource $i$ accounts for an expected loss of $(a_i+b_i-\gamma)/b_i$ while it is counted $a_i$ times in the sum \begin{equation*} \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|. \end{equation*} Similarly, \begin{equation*} \mu = \sum_{i\in O\cap C} \frac{a_i+b_i-\gamma}{b_i} \geq \min_{i\in O\cap C}\left\lbrace \frac{a_i+b_i-\gamma}{a_ib_i} \right\rbrace \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|. \end{equation*} Note that by assumption we have that $a_i+b_i>\gamma$. This implies that either $a_i$ or $b_i$ is greater than $\gamma/2$. Assume w.l.o.g. that $a_i\geq \gamma/2$. Since by assumption $a_i\leq \gamma$ we have that \begin{equation*} \frac{a_i+b_i-\gamma}{a_ib_i}\leq \frac{b_i}{a_ib_i} =\frac{1}{a_i} \leq \frac{2}{\gamma}. \end{equation*} In the same manner, since $a_i+b_i>\gamma$ and that $a_i,b_i\leq \gamma$, we can write \begin{equation*} \frac{a_i+b_i-\gamma}{a_ib_i}\geq \frac{1}{a_ib_i} \geq \frac{1}{\gamma^2}. \end{equation*} We therefore get the following bounds \begin{equation*} \frac{1}{\gamma^2}\sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|\leq \mu \leq \frac{2}{\gamma} \sum_{K\in \mathcal{K}^{(j-1)}}|K\cap C \cap R_{j-1}\cap O|, \end{equation*} which is what we wanted to prove. \end{proof} \section{Omitted proofs from Section~\ref{sec:reduction santa claus}}\label{appendix:reduction} \subsection{From matchings to Santa Claus} The idea in this reduction is to replace each player by a set of players, one for each of the $t$ configuration containing him. These players will share together $t-1$ large new resources, but to satisfy all, one of them has to get other resources, which are the original resources in the corresponding configuration. \begin{description} \item[Players.] For every vertex $v \in P$, and every hyperedge $C \in \mathcal{C}$ that $v$ belongs to, we create a player $p_{v,C}$ in the Santa Claus instance. \item[Resources.] For every vertex $u \in R$, create a resource $r_{u}$ in the Santa Claus instance. For any vertex $v \in P$ such that it belongs to $t$ edges in $\mathcal{C}$, create $t-1$ resources $r_{v,1}, r_{v,2}, \ldots, r_{v,t-1} $. \item[Values.] For any resource $r_{u}$ for some $u \in R$ and any player $p_{v,C}$ for some $C \in \mathcal{C}$, the resource has a value $\frac{1}{|C|-1}$ if $u \in C$, otherwise it has value $0$. Any resource $r_{v,i}$ for some $v \in P$ and $i \in \mathbb N$, has value $1$ for any player $p_{v,C}$ for some $C \in \mathcal{C}$ and $0$ to all other players. \end{description} It is easy to see that given an $\alpha$-relaxed matching in the original instance, one can construct an $\alpha$-approximate solution for the Santa Claus instance. For the other direction, notice that for each $v \in P$, there exists a player $p_{v,C}$ for some $C \in \mathcal{C}$, such that it gets resources only of the type $r_{u}$. One can simply assign the resource $u \in R$ to the player $v$ for any resource $r_{u}$ assigned to $p_{v,C}$. \subsection{From Santa Claus to matchings} This subsection is devoted to the proof of Theorem \ref{thm:reduction}. \begin{proof} We write $(\log)^k(n) = \underbrace{\log \cdots \log}_{\times k}(n)$ and $(\log)^0(n) = n$. \paragraph*{Construction.} We describe how to construct a hypergraph matching instance from a Santa Claus instance in four steps by reducing to the following more and more special cases. \paragraph{(1) Geometric grouping.} In this step, given arbitrary $v_{ij}$, we reduce it to an instance such that $\mathrm{OPT} = 1$ and for each $i, j$ we have $v_{ij} = 2^{-k}$ for some integer $k$ and $1/(2n) < v_{ij} \le 1$. This step follows easily from guessing $\mathrm{OPT}$, rounding down the sizes, and omitting all small elements in a solution. \paragraph{(2) Reduction to O(log*(n)) size ranges.} Next, we reduce to an instance such that for each player $i$ there is some $k \le \log^*(2n)$ such that for each resource $j$, $v_{ij}\in\{0, 1\}$ or $1/(\log)^k(2n) < v_{ij} \le 1/(\log)^{k+1}(2n)$. We explain this step below. Each player and resource is copied to the new instance. However, we will also add auxiliary players and resources. Let $i$ be a player. In the optimal solution there is some $0 \le k \le \log^*(2n)$ such that the values of all resources $j$ with $1/(\log)^k(2n) < v_{ij} \le 1/(\log)^{k+1}(2n)$ assigned to player $i$ sum up to at least $1/\log^*(2n)$. Hence, we create $\log^*(2n)$ auxiliary players which correspond to each $k$ and each of which share an resource with the original player that has value $1$ for both. The original player needs to get one of these resources, which means one of the auxiliary players needs to get a significant value from the resources with $1/(\log)^k(2n) < v_{ij} \le 1/(\log)^{k+1}(2n)$. This reduction loses a factor of at most $\log^*(2n)$. Hence, $\mathrm{OPT} \geq 1/\log^*(2n)$. \begin{comment} \item[(3) Reduction to 3 sizes.] We further reduce to an instance such that for each player $i$ there is some value $v_i$ such that for each resource $j$, $v_{ij}\in\{0, v_i, 1\}$. Let $i$ be some player who has only resources of value $v_{ij}\in\{0,1\}$ or $1/(\log)^k(2n) < v_{ij} \le 1/(\log)^{k+1}(2n)$. There are at most $\log((\log)^k(2n)) \leq (\log)^{k+1}(2n)$ distinct values of the latter kind. The idea is to assign bundles of resources of value $1/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ to the player $i$. For each distinct value, we create sufficiently many (say, $2n$) auxiliary players. These auxiliary players each share a new resource with $i$, which has value $1$ for this player and value $1/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ for $i$. If $i$ takes such an resource, the auxiliary player should collect a value of $0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ of resources of his particular value. Hence, we set the values for these resources for this player to $v_{ij} / \left(0.5/\left(\log^*(2n)(\log)^{k+1}(2n) \right) \right)$. We lose a factor of $O(\log^*(2n))$ in this step as there are at least $\lceil (0.5(\log)^{k+1}(2n))/\log^*(2n) \rceil$ bundles of size at least $0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ for player $i$. Now rescale the instance appropriately to get $\mathrm{OPT}=1$. \end{comment} \paragraph{(3) Reduction to 3 sizes.} We further reduce to an instance such that for each player $i$ there is some value $v_i$ such that for each resource $j$, $v_{ij}\in\{0, v_i, 1\}$. Let $i$ be some player who has only resources of value $v_{ij}\in\{0,1\}$ or $1/(\log)^k(2n) < v_{ij} \le 1/(\log)^{k+1}(2n)$ for some integer $k$. There are at most $\log((\log)^k(2n)) \leq (\log)^{k+1}(2n)$ distinct values of the latter kind. The idea is to assign bundles of resources of value $0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ to the player $i$. Fix a resource value $s$ such that $1/(\log)^k(2n) <s\le 1/(\log)^{k+1}(2n)$. We denote by $R_s$ the set of resources $j$ such that $v_{ij}=s$. We define the integer \begin{equation*} b=\left\lceil \frac{0.5}{s\log^*(2n)(\log)^{k+1}(2n)}\right\rceil \end{equation*} which is the number of resources of value $s$ that are needed to make a bundle of total value at least $0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$. We remark that if $s>0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ we have $b=1$. However, since $s\leq 1/(\log)^{k+1}(2n)$, the value of a bundle never exceeds $1/(\log)^{k+1}(2n)$ in the instance of step (2). Then we create \begin{equation*} \left\lfloor |R_s|/b\right\rfloor \end{equation*} auxiliary players $i_1,i_2,\ldots $ and auxiliary resources $j_1,j_2,\ldots$ (note that we create 0 player and resource if $|R_s|<b$). Each auxiliary player $i_\ell$ shares resource $j_\ell$ with player $i$. This resource has value $2/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ for player $i$ and value $1$ for player $i_\ell$. Then for all resources $j\in R_s$, we set $v_{ij}=0$ and \begin{equation*} v_{i_\ell j}=\frac{1}{(\log^*(2n))^2b} \end{equation*} for any auxiliary player $i_\ell$ that was created. We see that we are now in the case where for each player $i$, there exists some $v_i$ such that $v_{ij}\in \{0,v_i,1\}$ for all resources $j$. We claim the following. \begin{claim} \label{cla:reduction_3_OPT} In the instance created at step (3), we have that $\mathrm{OPT}\geq 1/(\log^*(2n))^2$. \end{claim} \begin{proof} To see this, take an assignment of resources to player that gives $1/\log^*(2n)$ value to every player in the instance obtained at the end of step (2). Define $R_i$ to be the set of resources assigned to player $i$ in this solution. Either $R_i$ contains a resource of value $1$ or only resources that are in a range $(1/(\log)^k(2n),1/(\log)^{k+1}(2n)]$ for some integer $k$. In the first case, nothing needs to be done as the resource $j$ of value $1$ assigned to $i$ still satisfies $v_{ij}=1$ in the new instance. Hence we assign $j$ to $i$ and all auxiliary players created for player $i$ get their auxiliary resource of value 1. In the second case, fix a resource value $s$. Let $R_{i,s}$ be the set of resources assigned to $i$ for which $v_{ij}=s$ and $b$ defined as before. We select $\left\lfloor |R_{i,s}|/b \right\rfloor$ auxiliary players to receive $b$ resources from $R_{i,s}$ and player $i$ takes the corresponding auxiliary resources. The remaining auxiliary players of the corresponding value take their auxiliary resource. Doing this, we ensure that all auxiliary players receive either a value of 1 (by taking the auxiliary resource) or $1/(\log^*(2n))^2$ by taking resources assigned to $i$ in the instance of step (2). Moreover, we claim that $i$ receives a total value of at least $1/(\log^*(2n))^2$. To see this, we have $3$ cases depending on the value of $b$ and $\left\lfloor |R_{i,s}|/b \right\rfloor$. \begin{itemize} \item If $b=1$, then $\left\lfloor |R_{i,s}|/b \right\rfloor=|R_{i,s}|$. We note that the value of a bundle of $b$ resources of size $s$ never exceeds $1/(\log)^{k+1}(2n)$ in instance (2). Since each auxiliary resource represents a value of $2/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ to player $i$ in instance (3), it must be that player $i$ receives in instance (3) at least a $2/\log^*(2n)$ fraction of the value he would receive in instance (2). \item If $b>1$ and $\left\lfloor |R_{i,s}|/b \right\rfloor>0$. Then we have that $\left\lfloor |R_{i,s}|/b \right\rfloor\geq |R_{i,s}|/(2b)$. Since in this case we have $s<0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ it must be that each bundle of $b$ resources of size $s$ represents a total value of at most $1/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$. Since the value of auxiliary resources is twice this value and because $\left\lfloor |R_{i,s}|/b \right\rfloor\geq |R_{i,s}|/(2b)$ it must be that in this case player $i$ receives in instance (3) at least the same value he would receive in instance (2). \item If $\left\lfloor |R_{i,s}|/b \right\rfloor=0$, then player $i$ receives $0$ value from resources of this value. However, when we combine all the values $s$ for which $\left\lfloor |R_{i,s}|/b \right\rfloor=0$, it represents to player $i$ in instance (2) a total value of at most \begin{equation*} 0.5/\left( \log^*(2n)(\log)^{k+1}(2n) \right)\cdot (\log)^{k+1}(2n) = 0.5/\log^*(2n) \end{equation*} since there are at most $(\log)^{k+1}(2n)$ different resource values. \end{itemize} Putting everything together, we see that in the first two cases, player $i$ receives at least a $2/\log^*(2n)$ fraction of the value he would receive in instance (2) and that he looses at total value of at most $0.5/\log^*(2n)$ in the third case. Since in instance (2) we have that $\mathrm{OPT}\geq 1/\log^*(2n)$ we see that in instance (3) player $i$ receives a value at least \begin{equation*} (2/\log^*(2n))\cdot (1/\log^*(2n)-0.5/\log^*(2n))\geq 1/(\log^*(2n))^2. \end{equation*} \end{proof} Finally, we also claim that it is easy to reconstruct an approximate solution to the instance obtained at step (1) from an approximate solution to the instance at step (3). \begin{claim}\label{cla:reduction_3_to_1} A $c$-approximate solution to the instance obtained at step (3) induces a $O((c\log^*(2n))^2)$-approximate solution to the instance obtained at step (1). \end{claim} \begin{proof} To see this, note that a $c$-approximate solution must give at least $1/(c(\log^*(2n))^2)$ value to every player since $\mathrm{OPT}\geq 1/(\log^*(2n))^2$ (by Claim \ref{cla:reduction_3_OPT}). This means that each player $i$ either takes a resource of value 1 which has also value 1 for him in the instance at step (1) or he must take at total value of $1/(c(\log^*(2n))^2)$ in auxiliary resources and the corresponding auxiliary players must take bundles of resources that represent a value of at least \begin{equation*} 0.5/\left( c\log^*(2n)(\log)^{k+1}(2n)\right) \end{equation*} for player $i$ in the instance at step (1). We simply assign all the resources appearing in these bundles to the player $i$ in the instance of step (1). Since the value of an auxiliary resource for player $i$ is $2/\left( \log^*(2n)(\log)^{k+1}(2n) \right)$ it must be that player $i$ takes at least \begin{equation*} \frac{1/(c(\log^*(2n))^2)}{2/\left( \log^*(2n)(\log)^{k+1}(2n) \right)} = \frac{(\log)^{k+1}(2n)}{2c\log^*(2n)} \end{equation*} auxiliary resources. Since each auxiliary resource brings a value of \begin{equation*} 0.5/\left( c\log^*(2n)(\log)^{k+1}(2n)\right) \end{equation*} to player $i$ (in the instance at step (1)) then player $i$ receives in total a value of at least \begin{equation*} \frac{1}{(2c\log^*(2n))^2} \end{equation*} in the instance of step (1). \end{proof} Before the last step, we rescale the instance appropriately to get $\mathrm{OPT}=1$ (we keep the property that each player $i$ has 3 distinct sizes 0,1 and $v_i$). \paragraph{(4) Reduction to hypergraph matching.} For each player create a vertex in $P$ and for each resource create a vertex in $R$. For each player add one hyperedge for each resource he values at $1$ (containing $i$ and this resource). Moreover, for every player $i$, add $1/v_i$ \textit{new} vertices to $P$ and the same number of \textit{new} resources to $R$. Pair these $1/v_i$ new vertices in $P$ and $R$ together (one from $R$ and one from $P$) and for each pair add a hyperedge containing these two vertices in the pair. Add another hyperedge for $i$ containing $i$ and all corresponding $1/v_i$ new vertices in $R$. Finally, for each new vertex in $P$ and each resource that $i$ values at $v_i$, add a hyperedge containing them. See Figure \ref{fig:reduction} for an illustration: New resources and players are marked as squares and hyperedges containing only 2 vertices are marked as simple edges. \begin{figure} \caption{An example of the reduction to hypergraph matching for player $i$ with $v_i=1/2$.} \label{fig:reduction} \end{figure} We claim that there exists a $1$-relaxed perfect matching in this instance. Since $\mathrm{OPT}=1$ there is an assignment of resources to players such that every player gets a value $1$. If player $i$ takes one resource of value $1$, give to player $i$ the corresponding hyperedge and the resource in it in the hypergraph. All the new players get the new resource they are paired to. If player $i$ takes $1/v_i$ resources of value $v_i$, give to player $i$ in the hypergraph all the $1/v_i$ new resources contained in the new hyperedge. Then we give to each new player the hyperedge (and the resource in it) corresponding to a resource that is assigned to $i$ in instance from step (3). This is indeed a $1$-relaxed perfect matching. \paragraph*{Correctness.} In the reduction we arrive at step (3) for which we prove that a $c$-approximate solution can be used to easily reconstruct a $O((c\log^*(2n))^2)$-approximate solution to the original instance (in Claim \ref{cla:reduction_3_to_1}). It remains to show that a $c$-relaxed perfect matching in the instance (4) induces a $c$-approximate solution to step (3). To see this, note that a $c$-relaxed perfect matching in the instance (4) either gives to player $i$ the resource in one hyperedge corresponding to a resource of value $1$ to player $i$ in instance (3). In that case we assign this resource to player $i$ in instance (3). Or it gives at least $1/(cv_i)$ new resources to player $i$. In this case, it must be that each new player paired to one of these resources takes one resource of value $v_i$ in instance (3). We give these resources to $i$ in instance (3). In this case $i$ receives a total value of $v_i/(cv_i)=1/c$ which ends the proof. We finish by remarking that the size of our construction is indeed polynomial in the size of the original instance. This is clear for step (1). In step (2), only $O(\log^* (n))$ new players and items are created for each player in the original instance. In step (3), for each player $i$ and each resource size $v_{ij}$, at most a polynomial number of resources and players are created. As for the last step, $O(1/v_i)$ new resources and players are created for each player $i$ which is also polynomial since $v_i=\Omega (1/n)$. The number of hyperedges in the hypergraph is also clearly polynomial in the number of vertices in our construction. \end{proof} \begin{comment} \paragraph*{Correctness.} Steps (1) and (2) are easy and we omit it. For step (3), notice that player $i$ might only be able to get a value of $O(1/(\log^*(2n))^2)$. Hence, we lose at most a factor of $O(\log^*(2n))$. It is not hard to see that one can reconstruct a solution to the instance produced by step ($2$) given a solution to instance produced by step ($3$) and vice-versa. For the (4) step, notice that if the player $i$ of value $v_i$ in the Santa Claus instance takes a resource of value 1, then the new vertices of $P$ and $R$ can form a matching and the vertex $i$ can take the same resource. On the other hand, if there exists a 1-relaxed matching in the hypergraph matching instance, we argue as follows. If the vertex $i$ takes an edge with single vertex, the player $i$ is given the corresponding resource of value 1. Otherwise, vertex $i$ must have taken the edge with $1/v_i$ new vertices corresponding to $i$ and consequently $1/v_i$ new vertices in $P$ must have taken edges with vertices corresponding to resources of value $v_i$ each. We simply assign the resources corresponding to these vertices to the player $i$. Hence, any $c$-approximate solution to the hypergraph matching problem yields a $O((c\log^*(n))^2)$ solution to the Santa Claus problem. We lose a factor of $O(1)$ in step (1), $O(\log^*(n))$ factors each in steps (2) and (3) due to the reduction and factors of $O(c)$ each in steps (3) and (4) while reconstructing the solution. We also note that the size of the newly constructed instance is at most polynomially larger than the size of the original instance. This is clear for step (1). In step (2), only $O(\log^* (n))$ new players and items are created for each player in the original instance. In step (3), for each player $i$ and each resource size $v_{ij}$, at most a polynomial number of resources and players are created. As for the last step, $O(1/v_i)$ new resources and players are created for each player $i$ which is also polynomial since $v_i=\Omega (1/n)$. The number of hyperedges in the hypergraph is also clearly polynomial in the number of vertices in our construction. \end{comment} \end{document}

\begin{document} \title{ extbf{More on limited packings in graphsootnote{Supported by NSFC No.11531011.} \begin{abstract} A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain the tight Nordhaus-Gaddum-type result of this parameter for general $k$. At last, we investigate the relationship among the open packing number, the packing number and $2$-limited packing number of trees. \noindent\textbf{Keywords:} $k$-limited packing, opening packing, Nordhaus-Gaddum-type result \noindent\textbf{AMS subject classification 2010:} 05C69, 05C70 \end{abstract} \section{Introduction} All graphs in this paper are undirected, simple and nontrivial. We follow \cite{BM} for graph theoretical notation and terminology not described here. Let $G$ be a graph, we use $V(G), E(G), diam(G), \Delta(G)$ and $\delta(G)$ to denote the vertex set, edge set, diameter, maximum degree, and minimum degree of $G$, respectively. Take a vertex $v\in V(G)$, the \emph{open neighbourhood} of $v$ is defined as the set of all vertices adjacent to $v$ in $G$, the set $N[v] = \{v\}\cup N(v)$ is called \emph {the closed neighbourhood} of $v$ in $G$. A set $D$ of vertices in a graph $G$ is called a \emph{dominating set} if each vertex in $V(G)\setminus D$ has at least one neighbour in $D$. The \emph{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality of a dominating set in $G$. The theory of dominating sets, introduced formally by Ore \cite{O} and Berge \cite{Berge}, has been the subject of many recent papers due to its practical and theoretical interest. For more information on domination topics we refer to the books \cite{HHS, HHS1}. A domination set $D$ of a graph $G$ is called a \emph{total dominating set} if $G[D]$ has no isolated vertex, and the minimum cardinality of a total dominating set in $G$ is called the \emph{total domination number} of $G$, denoted by $\gamma_t(G)$. Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi \cite{CDH}, and has been well studied (see, for example, \cite{FH,FHMP,HY,Y}). On the other side, the \emph{open packing} of a graph $G$ is a set $S$ of vertices in $G$ such that for each vertex $v$ of $G$, $|N(v)\cap S|\leq 1$. The \emph{open packing number} of a graph $G$, denoted by $\rho^{0}(G)$, is the maximum cardinality among all open packings in $G$. The open packing of a graph has been studied in \cite{H,HSE}. The well-known \emph{packing} (\emph{$2$-packing}) of a graph $G$ is a set $B$ of vertices in $G$ such that $|N[v]\cap B|\leq 1$ for each vertex $v$ of $G$. The \emph{packing number} $\rho(G)$ of a graph $G$ is the maximum cardinality of a packing in $G$. The packing of a graph has been well studied in the literature \cite{B,C,MM,TV}. Dominating sets and packings of graphs are two good models for many utility location problems in operations research. But the corresponding problems have a very different nature: the former is a minimization problem (dominating sets) to satisfy some reliability requirements, the latter is a maximization problem not to break some (security) constraints. Consider the following scenarios: Network security: A set of sensors is to be deployed to covertly monitor a facility. Too many sensors close to any given location in the facility can be detected. Where should the sensors be placed so that the total number of sensors deployed is maximized? Market Saturation: A fast food franchise is moving into a new city. Market analysis shows that each outlet draws customers from both its immediate city block and from nearby city blocks. However it is also known that a given city block cannot support too many outlets nearby. Where should the outlets be placed? Codes: Information is to be transmitted between two interested parties. This data is first represented by bit strings (codewords) of length $n$. It is desirable to be able to use as many of these $2^n$ strings as possible. However, if a single bit of a codeword is altered during transmission, we should still be able to recover the piece of data correctly by employing a ``nearest neighbour" decoding algorithm. How many code words can be used as a function of $n$? A graph model of these scenarios might maximize the size of a vertex subset subject to the constraint that no vertex in the graph is near too many of the selected vertices. Motivated by the packing of graphs, Gallant et al. relaxed the constraints and introduced the concept of the \emph{$k$-limited packing} in graphs in \cite{GGHR}. A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, $|N[v]\cap B|\leq k$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing set in $G$. It is clear that $L_1(G) = \rho(G)$. The problem of finding a $1$-limited packing of maximum size for a graph is shown to be NP-complete in \cite{HS0}. In \cite{DLN}, it is shown that the problem of finding a maximum size $k$-limited packing is NP-complete even in split or bipartite graphs. For more results on $k$-limited packings of graphs, we refer to \cite{BBG,GZ,GGHR,LN}. The remainder of this paper will be organized as follows. In Section $2$, we give the technical preliminaries, including notations and relevant known results on open packings and $k$-limited packings of graphs. In Section $3$, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. Based on them, we obtain the tight Nordhaus-Gaddum-type result of this parameter for general $k$. In Section $4$, we focus on the $2$-limited packing number of graphs, including trees and graphs with diameter two. And we get the better upper bound of the $2$-limited packing number of graphs with large diameter. In Section $5$, we investigate the relationship among the open packing number, the $1$-limited packing number and the $2$-limited packing number of trees. \section{Preliminaries} The notation we use is mostly standard. For $B \subseteq V(G)$, let $\overline{B}=V(G)\backslash B$, and $G[B]$ denote the subgraph of $G$ induced by $B$. Given $t$ graphs $G_1,\ldots, G_t$, the \emph{union} of $G_1,\ldots,G_t$, denoted by $G_1\cup \cdots \cup G_t$, is the graph with vertex set $V(G_1)\cup \cdots \cup V(G_t)$ and edge set $E(G_1)\cup \cdots \cup E(G_t)$. In particular, let $tG$ denote the vertex-disjoint \emph{union} of $G_1,\ldots G_t$ for $G_1= \cdots =G_t=G$. We next state some relevant known results on $k$-limited packings of graphs, which will be needed later. \begin{lem}{\upshape\cite{HS}}\label{lem15} Let $G$ be a graph of order at least $3$. Then $\rho^{0}(G) = 1$ if and only if $diam(G)\leq 2$ and every edge of $G$ lies on a triangle. \end{lem} \begin{lem}{\upshape\cite{HS}}\label{lem17} If $G$ is a graph of diameter $2$, then $\rho^{0}(G)\leq 2$. \end{lem} \noindent\textbf{Remark 1.} It is clear that graphs with diameter $1$, which are exactly complete graphs, have opening packing number at most $2$. Thus, if $G$ is a graph of diameter at most $2$, then $\rho^{0}(G)\leq 2$. \begin{lem}{\upshape\cite{R}}\label{lem33} If $T$ is any tree of order at least $2$, then $\rho^{0}(T)=\gamma_t(T)$. \end{lem} \begin{lem}{\upshape\cite{MS}}\label{lem16} For any graph $G$, $L_1(G) = 1$ if and only if $diam(G)\leq2$. \end{lem} \begin{lem}{\upshape\cite{BBG}}\label{lem31} For any graph $G$ of order $n$, $L_1(G)\geq \frac{n}{\Delta(G)^2+1}.$ \end{lem} \begin{lem}{\upshape\cite{MM}}\label{lem34} For any tree T, $L_1(T)=\gamma(T).$ \end{lem} \begin{lem}{\upshape\cite{MS}}\label{lem7} For any connected graph $G$ and integer $k\in \{1,2\}$, $$ L_k(G)\geq \lceil \frac{k+kdiam(G)}{3}\rceil .$$ \end{lem} \begin{lem}{\upshape\cite{S}} \label{lem2} Let $G$ be a graph of order $n$. Then $L_2(G)+L_2(\overline{G})\leq n+2$, and this bound is tight. \end{lem} Since a $k$-limited packing of a graph is also a $(k+1)$-limited packing, we immediately obtain the following inequalities: $L_1(G)\leq L_2(G)\leq\cdots\leq L_{k}(G)\leq L_{k+1}(G)\leq\cdots$. Furthermore, the authors obtained the stronger result in \cite{MSH}. \begin{lem}{\upshape\cite{MSH}}\label{lem5} Let $G$ be a connected graph of order $n$ and $k\leq \Delta(G)$. Then $L_{k+1}(G)\geq L_k(G)+1$. Moreover, $L_{k}(G)\geq L_1(G)+k-1$, and this bound is tight. \end{lem} \noindent\textbf{Remark 2.} Based on the proof of Lemma \ref{lem5} in \cite{MSH}, the condition of the connectivity of $G$ in Lemma \ref{lem5} can be deleted. \begin{lem}{\upshape\cite{BBG}}\label{lem10} For any graph $G$ of order $n$, $L_k(G)\leq \frac{kn}{\delta(G)+1}.$ \end{lem} In the sequel, let $P_n$, $C_n$, $K_n$, and $K_{s,t}$ denote the path of order $n$, cycle of order $n$, complete graph of order $n$, and complete bipartite graph of order $s+t$, respectively. It is clear that $L_k(P_n)=L_k(C_n)=n$ for $k\geq 3$. \begin{lem}{\upshape \cite{GGHR}}\label{lem3} Let $m, n, k\in \mathbb{N}$. Then $(i)$ $L_k(P_n)=\lceil \frac{kn}{3}\rceil $ for $k=1,2$, $(ii)$ $L_k(C_n)=\lfloor\frac{kn}{3}\rfloor$ for $k=1,2$ and $n\geq 3$, $(iii)$ $L_k(K_{n})=min\{k,n\},$ $(iv)$ $L_k(K_{m,n})= \begin{cases} 1& \text{if $k=1$},\\ min\{k-1,m\}+min\{k-1,n\}& \text{if $k > 1$}. \end{cases}$ \end{lem} \begin{lem}{\upshape\cite{GGHR}}\label{lem4} If $G$ is a graph, then $L_k(G)\leq k\gamma(G)$. Furthermore, the equality holds if and only if for any maximum $k$-limited packing $B$ in $G$ and any minimum dominating set $D$ in $G$ both the following hold: $(i)$ For any $b\in B$ we have $\mid N[b]\cap D\mid=1$, $(ii)$ For any $d\in D$ we have $\mid N[d]\cap B\mid=k$. \end{lem} It is worth mentioning that we generalize the results of Lemma \ref{lem16}, Lemma \ref{lem7} and lemma \ref{lem2} to general $k$-limited packing parameter of graphs, and characterize all the trees $T$ satisfying $L_{2}(T)= L_1(T)+1$ in Lemma \ref{lem5} later, which are parts of our job. \section{$k$-limited packing} In this section we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain the tight Nordhaus-Gaddum-type result for this parameter. It is clear to obtain the following result. \begin{pro}\label{prop1} If $G$ is a graph of order $n$ with $n\leq k$, then $L_k(G)=n$. \end{pro} \noindent\textbf{Remark 3.} Actually, the above condition that $n\leq k$ can be weakened to $\Delta(G)+1 \leq k$. So, we only need to consider the $k$-limited packing number for graphs $G$ with $\Delta(G)\geq k$. \begin{pro}\label{prop2} If $G$ is a graph of order $k+1$, then \begin{equation*} L_k(G)= \left\{ \begin{array}{ll} k & \hbox{ if $\Delta(G)=k$,}\\ k+1 & \hbox{ otherwise.} \\ \end{array} \right. \end{equation*} \end{pro} \begin{proof} Let $G$ be a graph of order $k+1$. Then $k \leq L_k(G)\leq k+1$. Let $\Delta(G)=k$. Assume to the contrary that $L_k(G)= k+1$. It is obtained that $V(G)$ is the unique maximum $k$-limited packing of $G$. Let $v_0$ be a vertex with maximum degree $k$ in $G$. Then $|N[v_0]\cap V(G)|=k+1$, which is a contradiction. Thus, $L_k(G)= k$. It remains to show the other case. Let $\Delta(G)\leq k-1$. Obviously, $V(G)$ is a $k$-limited packing of $G$, it follows that $L_k(G)=k+1$. \end{proof} For a given graph $G$ of order less than $k+2$, we can determine its $k$-limited packing number by Proposition \ref{prop1} and Proposition \ref{prop2}. So we are concerned with graphs of order at least $k+2$ in the following. \begin{pro}\label{prop3} If $G$ is a graph of order at least $k+2$, then $L_k(G)\geq k$. \end{pro} The following result is a generalization of Lemma \ref{lem16}. \begin{thm}\label{th1} Let $G$ be a graph of order $n$. Then $L_k(G)=k$ if and only if one of the following conditions holds: $(i)$ $n=k$, $(ii)$ $\Delta(G)=k$, where $n=k+1$, $(iii)$ for each $(k+1)$-subset $X$ of $V(G)$, $G[X]$ has maximum degree $k$ or the $k+1$ vertices of $X$ have a common neighbour, where $n\geq k+2$. \end{thm} \begin{proof} The statement holds for $n\leq k+1$ by Proposition \ref{prop1} and Proposition \ref{prop2}, thus we may assume that $n\geq k+2$ in the following. Notice that $k \leq L_k(G)\leq n$ for $n\geq k+2$. Let $G$ be a graph of order $n$ such that for each $(k+1)$-subset $X$ of $V(G)$, $G[X]$ has maximum degree $k$ or the $k+1$ vertices of $X$ have a common neighbour. Assume that $G$ has a $k$-limited packing $B$ with at least $k+1$ vertices. Let $X$ be a $(k+1)$-subset of $B$. Obviously, $X$ is also a $(k+1)$-subset of $V(G)$. If $G[X]$ has a vertex $v_0$ with degree $k$, then $|N[v_0]\cap B|\geq k+1$, which is a contradiction. If the $k+1$ vertices of $X$ have a common neighbour $a$, then $|N[a]\cap B|\geq k+1$, which is also a contradiction. Thus, $L_k(G)= k$. It remains to show the converse. Let $G$ be a graph of order $n$ such that $L_k(G)= k$. Assume that there exists a $(k+1)$-subset $X_0$ of $V(G)$ such that $G[X_0]$ has maximum degree at most $k-1$ and each vertex outside $X_0$ is adjacent to at most $k$ vertices in $X_0$. It follows that $X_0$ is a $k$-limited packing of $G$, which implies that $L_k(G)\geq k+1$, a contradiction. Therefore, if $G$ is a graph of order at least $k+2$ with $L_k(G)= k$, then for each $(k+1)$-subset $X$ of $V(G)$, $G[X]$ has maximum degree $k$ or the $k+1$ vertices of $X$ have a common neighbour. \end{proof} The following result is an immediate and obvious corollary of the above theorem. \begin{cor}\label{coro2} Let $G$ be a graph of order at least $k+1$ such that $L_k(G)=k$. Then $diam(G)\leq 2$. \end{cor} Next, we present a lower bound of the $k$-limited packing number of a graph in terms of it diameter for $k\geq 3$, which is a generalization of Lemma \ref{lem7}. \begin{thm}\label{th5} Let $G$ be a connected graph and $ \Delta(G)\geq k\geq 3$. Then $L_k(G)\geq diam(G)+k-2$. Moreover, the lower bound is tight. \end{thm} \begin{proof} Let $P=v_1v_2\cdots v_{diam(G)+1}$ be a path of length $diam(G)$ in $G$. Obviously, for each vertex $v_i$ on $P$, $|N[v_i]\cap V(P)|\leq 3$. We claim that $V(P)$ is a $3$-limited packing in $G$. Assume to the contrary that there exists a vertex $u$ outside $P$ such that $|N(u)\cap V(P)|\geq 4$. Let $N(u)\cap V(P)=\{v_{i_1},\ldots,v_{i_d}\}$ with $i_1\leq \cdots \leq i_d$ and $d\geq 4$. Then $P'=v_1\cdots v_{i_1}uv_{i_d}\cdots v_{diam(G)+1}$ is a path between $v_1$ and $v_{diam(G)+1}$, whose length is less than $diam(G)$, a contradiction. Thus, $L_3(G)\geq |V(P)|=diam(G)+1$. And by Lemma \ref{lem5}, we have $L_k(G)\geq L_3(G)+k-3\geq diam(G)+1+k-3=diam(G)+k-2.$ Corollary \ref{coro2} shows that non-complete graphs $G$ of order at least $k+1$ with $L_k(G)=k$ are ones satisfying $L_k(G)=diam(G)+k-2$. \end{proof} Recall that \emph{the girth} of a graph $G$ is the length of a shortest cycle in $G$, denoted by $g(G)$. \begin{thm}\label{th7} If $G$ is a graph with girth $g(G)$, then $L_1(G)\geq \lfloor\frac{g(G)}{3}\rfloor$. Moreover, the lower bound is tight. \end{thm} \begin{proof} Let $G$ be a graph and $C$ be a cycle of length $g(G)$ in $G$. The statement is evidently true for $g(G)\leq 4$. Thus, we only need to consider the case when $g(G)\geq 5$. Let $B$ be a maximum $1$-limited packing of $C.$ Then $|B|=L_1(C)=\lfloor\frac{g(G)}{3}\rfloor$ by Lemma \ref{lem3}. Next we will show that $B$ is also a $1$-limited packing of $G$, which implies that $L_1(G)\geq \lfloor\frac{g(G)}{3}\rfloor$. It is sufficient to show that each vertex $v$ outside $C$ has at most one neighbour on $C$. Assume to the contrary that there is a vertex $v_0$ outside $C$ that is adjacent to two vertices, say $x,$ $y$, on $C$. Let $P$ be the shortest path between $x$ and $y$ on $C$. If $|V(P)|\leq 3$, then $x$, $v_0$ and $y$ are on either a $C_3$ or $C_4$ in $G$, contradicting $g(G)\geq 5.$ Now we may assume that $|V(P)|\geq 4$. Let $C'$ be the cycle obtained from $C$ replacing $P$ by the path $xv_0y$. Then the length of $C'$ is less than $g(G)$, which is a contradiction. Thus, each vertex outside $C$ has at most one neighbour on $C$. Furthermore, cycles are graphs $G$ with $L_1(G)= \lfloor\frac{g(G)}{3}\rfloor$ by Lemma \ref{lem3}. The proof is complete. \end{proof} Next, we give the lower bound of the $k$-limited packing number of a graph with respect to its girth for general $k\geq 2$. \begin{thm}\label{th2} If $G$ is a graph with girth $g(G)$, then $L_2(G)\geq \lfloor\frac{2g(G)}{3}\rfloor$ and $L_k(G)\geq g(G)+k-3$ for $ \Delta(G)\geq k \geq 3$. Moreover, the lower bounds are tight. \end{thm} \begin{proof} The statement trivially holds for $g(G)\leq 3$. Thus, we may assume that $g(G)\geq 4$ in the following. Let $C$ be a cycle of length $g(G)$ in $G$. We first present the following claim. \textbf{Claim 1.} Each vertex outside $C$ has at most two neighbours on $C$. \noindent\textbf{Proof of Claim 1:} Suppose to the contrary that there is a vertex $v$ outside $C$ such that $v$ is adjacent to three vertices, say $x,$ $y,$ $z$, on $C$. Let $P$ be the shortest path containing $x,$ $y,$ $z$ on $C$ such that the end vertices of $P$ are contained in $\{x,y,z\}.$ Without loss of generality, assume that $x$, $y$ are the end vertices of $P.$ Obviously, $|V(P)|\geq 3.$ Suppose that $|V(P)|=3$. It follows that $z$ is adjacent to both $x$ and $y$. Therefore, $G[\{x,z,v\}]$ is exactly $C_3$, which contradicts to that $g(G)\geq 4.$ Suppose that $|V(P)|\geq 4$. Let $C'$ be the cycle obtained from $C$ replacing $P$ by the path $xvy$. Then the length of $C'$ is less than $g(G)$, which is a contradiction. Thus, each vertex outside $C$ has at most two neighbours on $C$. Let $B$ be a maximum $2$-limited packing of $C.$ Then $|B|=L_2(C)=\lfloor\frac{2g(G)}{3}\rfloor$ by Lemma \ref{lem3}. By Claim $1$, we obtain that $B$ is also a $2$-limited packing of $G$. Thus, $L_2(G)\geq|B| =\lfloor\frac{2g(G)}{3}\rfloor$. Moreover, cycles are graphs $G$ with $L_2(G)= \lfloor\frac{2g(G)}{3}\rfloor$ by Lemma \ref{lem3}. Observe that $V(C)$ is a maximum $3$-limited packing of $C$. And by Claim $1$, it is known that $V(C)$ is also a $3$-limited packing of $G$, which implies that $L_3(G)\geq g(G)$. It follows from Lemma \ref{lem5} and Remark $2$ that $L_k(G)\geq L_3(G)+k-3\geq g(G)+k-3$ for $k\geq 3$. Furthermore, graphs $G$ of order at least $k+1$ with triangles, satisfying $L_k(G)=k$, have the property that $L_k(G)=g(G)+k-3$. \end{proof} Next, we turn to study the upper bound of the $k$-limited packing number of a graph. \begin{thm}\label{th3} If $G$ is a graph of order $n$, then $L_k(G)\leq n+k-1-\Delta(G)$. \end{thm} \begin{proof} Let $v_0$ be a vertex of maximum degree $\Delta(G)$ in $G$. If $k\geq \Delta(G)+1$, then it is clear that $V(G)$ is a $k$-limited packing of $G$, and hence $L_k(G)=n\leq n+k-1-\Delta(G)$. Thus, we may assume that $k< \Delta(G)+1$ in the following. Let $B$ be a maximum $k$-limited packing of $G$. Since $|N[v_0]\cap B|\leq k$, it follows that there exist at least $\Delta(G)+1-k$ vertices in $N[v_0]\setminus B$, which means that $|\overline{B}|\geq \Delta(G)+1-k$. Thus, $L_k(G)=|B|=n-|\overline{B}|\leq n-(\Delta(G)+1-k)=n+k-1-\Delta(G)$. \end{proof} We define the graph class $\mathcal{G}$ consisting of all graphs $G$ constructed as follows. Let $G$ be a graph of order $n$ such that $V(G)=A_0\cup B_0$ has the following properties: $(i)$ $|A_0\cap B_0|=2$, $(ii)$ $G[A_0]$ has a spanning star, and each component of $G[B_0]$ is $K_1$ or $K_2$, $(iii)$ for each vertex $v\in \overline{B_0}$, $|N(v)\cap B_0|\leq 2$. The following result shows that $\mathcal{G}$ is the set of all graphs $G$ of order $n$ with $L_2(G)= n+1-\Delta(G)$. \begin{cor}\label{cor14} If $G$ is a graph of order $n$, then $L_2(G)\leq n+1-\Delta(G)$. Moreover, $L_2(G)= n+1-\Delta(G)$ if and only if $G \in \mathcal{G}$. \end{cor} \begin{proof} We first restate the proof for Theorem \ref{th3}. Let $B$ be a maximum $2$-limited packing in $G$. Obviously, each component of $G[B]$ is $K_1$ or $K_2$, and $|N[v]\cap B|\leq 2$ for each vertex $v$ of $G$. Let $v_0$ be a vertex of maximum degree $\Delta(G)$. Since $|N[v_0]\cap B|\leq 2$, it follows that there exist at least $\Delta(G)-1$ vertices in $N[v_0]\setminus B$. Thus, $L_2(G)=|B|=n-|\overline{B}|\leq n-(\Delta(G)-1)=n+1-\Delta(G)$. Let $G$ be a graph of order $n$ such that $L_2(G)=n+1-\Delta(G)$. It is easily obtained that $G$ has the following properties: $(P1)$ $|N[v_0]\cap B|=2$, $(P2)$ $V(G)\setminus N[v_0]\subset B$. By the above argument, we have $G \in \mathcal{G}$ with $N[v_0]=A_0$ and $B=B_0$. It remains to show the converse. Suppose that $G \in \mathcal{G}$. It is sufficient to show that $L_2(G)\geq n+1-\Delta(G)$. Let $A_0\cap B_0=\{v_p,v_q\}$ and $|A_0|=t+1$, where $v_0$ is a vertex of degree $t$ in $G[A_0]$. Observe that $d(v_0)\geq t$. Furthermore, we obtain the following claim. \textbf{Claim 1.} $\Delta(G)=t$. \noindent\textbf{Proof of Claim 1:} Since each vertex in $B_0$ has degree at most $1$ in $G[B_0]$, it follows that each of $v_p,v_q$ is adjacent to at most one vertex in $B_0$. On the other hand, each of $v_p,v_q$ is adjacent to at most $t-1$ vertices in $A_0\setminus\{v_p,v_q\}$. Thus, $d(v_p)\leq t$ and $d(v_q)\leq t$. For each vertex $v$ in $A_0\setminus\{v_p,v_q\}$, $v$ is adjacent to at most $t-2$ vertices in $A_0\setminus\{v,v_p,v_q\}$ and at most two vertices in $B_0$, thus $d(v)\leq t$ for each vertex $v$ in $A_0\setminus\{v_p,v_q\}$. For each vertex $u$ in $B_0\setminus\{v_p,v_q\}$, $u$ is adjacent to at most $t-1$ vertices in $A_0\setminus\{v_p,v_q\}$ and at most one vertex in $B_0$, hence $d(u)\leq t$ for each vertex $u$ in $B_0\setminus\{v_p,v_q\}$. Thus $\Delta(G)\leq t$. But $d(v_0)\geq t$, which means that $\Delta(G)=t$. Notice that $B_0$ is a $2$-limited packing of $G$ with $|B_0|=n-|A_0|+2=n+1-\Delta(G)$, then $L_2(G)\geq n+1-\Delta(G)$. We complete the proof. \end{proof} \begin{cor}\label{coro1} Let $G$ be a $d$-regular graph of order $n$ such that $L_k(G)= n+k-1-d$, where $k\leq d$. Then $d\geq \frac{n}{2}$. \end{cor} \begin{proof} If $d=n-1$, then $G$ is a complete graph with $L_k(G)=k$ for $n\geq k+1\geq 2$, and the result follows from $d=n-1\geq \frac{n}{2}$. Thus, we may assume that $d\leq n-2$. Suppose that $L_k(G)= n+k-1-d$. Let $B$ be a maximum $k$-limited packing of $G$ with $|B|=n+k-1-d$, and $v$ be a vertex of $G$. Since $|N[v]\cap B|\leq k$, it follows that $|N[v]\cap \overline{B}|\geq d+1-k$. Assume that $|N[v]\cap \overline{B}|> d+1-k$. Then $|B|<n-(d+1-k)=n+ k-1-d$, a contradiction. Thus, there exist exactly $d+1-k$ vertices, say $v_1, \ldots, v_{d+1-k}$, in $N[v]\cap \overline{B}$, furthermore, $\overline{B}=\{v_1, \ldots, v_{d+1-k}\}$. Let $U=V(G)\setminus N[v]$. Since $d\leq n-2$, it follows that $|U|> 0$. Observe that $U\subseteq B$. Consider a vertex $u_i$ in $U$, there exist at most $k-1$ neighbours in $B$, therefore $u_i$ is adjacent to at least $d-(k-1)$ vertices in $\overline{B}$. But $|\overline{B}|=d+1-k$, it follows that for each vertex $u_i$ in $U$, $u_i$ is adjacent to all the vertices in $\overline{B}$. That is, each vertex $v_i$ in $\overline{B}$ is adjacent to all the $n-d-1$ vertices in $U$. Note that $d(v_i)=d$ and $v_i$ has at least one neighbour in $N[v]$, it follows that $n-d-1+1\leq d$. Thus, we obtain that $d\geq \frac{n}{2}$. \end{proof} To end this section, we present the tight Nordhaus-Gaddum-type result for $k$-limited packing numbers of graphs $G$ and $\overline{G}$ for $k\geq1$. We first establish the tight Nordhaus-Gaddum-type lower bound for this parameter, and characterize all the graphs obtaining this lower bound. \begin{pro}\label{prop6} If $G$ is a graph of order at least $k$, then $L_k(G)+L_k(\overline{G})\geq 2k$. Moreover, $L_k(G)+L_k(\overline{G})=2k$ if and only if $G$ has one of the following properties: $(i)$ $G$ has exactly $k$ vertices, $(ii)$ for each $(k+1)$-subset $X$ of $V(G)$, $G[X]$ has maximum degree $k$ and there is a vertex outside $X$ such that it is not adjacent to any vertex of $X$, or there is a vertex outside $X$ such that it is adjacent to all the vertices of $X$ and $G[X]$ has an isolated vertex, or there are one vertex outside $X$ such that it is adjacent to all the vertices of $X$ and another vertex outside $X$ such that it is not adjacent to any vertex of $X$. \end{pro} \begin{proof} Since it is impossible that $\Delta(G)=\Delta(\overline {G})=k$ for $|V(G)|=k+1$, it follows from Proposition \ref{prop2} that $L_k(G)+L_k(\overline{G})>2k$ for $|V(G)|=k+1$. And observe that it is also impossible that for some $(k+1)$-subset $X$ of $V(G)$, both $G[X]$ and $\overline{G}[X]$ has maximum degree $k$. Thus, the result follows from Theorem \ref{th1}. \end{proof} The tight Nordhaus-Gaddum-type upper bounds for $k$-limited packing numbers of graphs $G$ and $\overline{G}$ in the following theorem are a generalization of Lemma \ref{lem2}. \begin{thm}\label{th10} Let $G$ be a graph of order $n$. Then \begin{equation*} L_k(G)+L_k(\overline{G})\leq \left\{ \begin{array}{ll} 2n & \hbox{ if $k\geq \max\{\Delta(G),\Delta(\overline {G})\}+1$,}\\ n+2k-2 & \hbox{ if $ k \leq \min\{\Delta(G),\Delta(\overline {G})\}$, }\\ 2n-1 & \hbox{ otherwise. } \\ \end{array} \right. \end{equation*} Moreover, the upper bounds are tight. \end{thm} \begin{proof} Suppose that $k\geq \max\{\Delta(G),\Delta(\overline {G})\}+1$. It is clear that $L_k(G)+L_k(\overline{G})=n+n=2n$. Suppose that $\max\{\Delta(G),\Delta(\overline {G})\}+1> k \geq \min\{\Delta(G),\Delta(\overline {G})\}+1$. Without loss of generality, we assume that $\Delta(\overline {G})+1> k\geq \Delta(G)+1$. It follows that $L_k(G)=n$ and $L_k(\overline{G})<n$. Therefore, $L_k(G)+L_k(\overline{G})\leq 2n-1$. To show that the upper bound is tight. Let $G$ be a graph of order $k+1$ with $\Delta(G)<k$ such that $G$ has isolated vertices. By Proposition \ref{prop2}, $L_k(G)+L_k(\overline{G})=2n-1$. It remains to consider the case when $ k \leq \min\{\Delta(G),\Delta(\overline {G})\}$. By Theorem \ref{th3}, we have $L_k(G)\leq n+k-1-\Delta(G)$ and $L_k(\overline{G})\leq n+k-1-\Delta(\overline{G})$. Thus, \begin{eqnarray*} L_k(G)+L_k(\overline{G}) &\leq& (n+k-1-\Delta(G))+ (n+k-1-\Delta(\overline{G}))\\ &=& 2n+2k-2-(\Delta(G)+\Delta(\overline{G}))\\ &\leq& 2n+2k-2-(\Delta(G)+\delta(\overline{G}))\\ &=& 2n+2k-2-(n-1)\\ &=& n+2k-1. \end{eqnarray*} Next, we claim that it is impossible that $L_k(G)+L_k(\overline{G})= n+2k-1$. Assume to the contrary that $L_k(G)+L_k(\overline{G})= n+2k-1$. It follows that both $G$ and $\overline{G}$ are regular graphs with $ L_k(G)= n+k-1-\Delta(G)$ and $ L_k(\overline{G})= n+k-1-\Delta(\overline{G})$. Notice that $G$ is a $\Delta(G)$-regular graph and $\overline{G}$ is a $\Delta(\overline{G})$-regular graph, then $\Delta(G)+\Delta(\overline{G})=n-1$. Since $ L_k(G)= n+k-1-\Delta(G)$ and $ L_k(\overline{G})= n+k-1-\Delta(\overline{G})$, it follows from Corollary \ref{coro1} that $\Delta(G)\geq \frac{n}{2}$ and $\Delta(\overline{G})\geq \frac{n}{2}$, which implies that $\Delta(G)+\Delta(\overline{G})> n-1$, which is a contradiction. Thus, $L_k(G)+L_k(\overline{G})\leq n+2k-2$. The following examples show that the upper bound is best possible. Let $G=K_n-e$, where $e$ is an edge of $K_n$ and $n\geq 3$. Then $L_1(G)= 1$ by Theorem \ref{th1}. On the other side, $\overline{G}=K_2\cup (n-2)K_1$, then $L_1(\overline{G})= n-1$. It is obtained that $\min\{\Delta(G),\Delta(\overline {G})\}\geq 1$ and $L_1(G)+L_1(\overline{G})= n+2k-2=n$. \end{proof} \section{ $2$-limited packing} In \cite{GGHR}, the authors bounded the $2$-limited packing number for a graph in terms of its order. \begin{lem}{\upshape\cite{GGHR}}\label{lem26} If $G$ is a connected graph with $|V(G)|\geq3$, then $L_2(G)\leq\frac{4}{5}|V(G)|$. \end{lem} Furthermore, they imposed constraints on the minimum degree of $G$, and obtained the following result. \begin{lem}{\upshape\cite{GGHR}}\label{le1} If $G$ is a connected graph, and $\delta(G)\geq k$, then $L_k(G)\leq\frac{k}{k+1}|V(G)|$. \end{lem} By Lemma \ref{le1}, we have $L_2(G)\leq\frac{2}{3}|V(G)|$ for graphs with $\delta(G)\geq 2$. It is known that trees are graphs with minimum degree $1$. We find a class of trees $T$ with $2$-limited packing number at most $\frac{2}{3}|V(T)|$. The minimum degree of a graph $G$ taken over all non-leaf vertices is denoted by $\delta'(G)$. \begin{thm}\label{thm30} If $T$ is a tree with $\delta'(T)\geq 4$, then $L_2(T)\leq\frac{2}{3}|V(T)|$. \end{thm} \begin{proof} Since $\delta'(T)\geq 4$, it follows that $|V(T)|\geq 5$. Let $B$ be a maximum $2$-limited packing of $T.$ By induction on the order of $T$. If $|V(T)|=5$, then $T=K_{1,4}$, and hence $L_2(T)=2\leq \frac{2}{3}|V(T)|$ by Lemma \ref{lem3}. Let $T$ be a tree of order at least $6$. It is known that $T$ can be regarded as a rooted tree. Take a leaf vertex $v$ of $T$, which is the lowest level in the rooted tree $T$. Let $v_0$ be the unique neighbour of $v$ in $T$, and $L_0$ be the set of leaf vertices in $N(v_0)$. Since $v_0$ is adjacent to at least three leaf vertices, we have $|L_0|\geq 3$. Let $T_0$ be the subtree obtained from $T$ by deleting all the vertices of $L_0$. By the inductive hypothesis, $L_2(T_0)\leq \frac{2|V(T_0)|}{3}\leq \frac{2(|V(T)|-3)}{3}=\frac{2}{3}|V(T)|-2.$ Since $|N[v_0]\cap B|\leq 2$, it follows that $|L_0\cap B|\leq 2$. Hence, $L_2(T)\leq L_2(T_0)+2=\frac{2}{3}|V(T)|$. \end{proof} It is shown that both the opening packing number and the $1$-limited packing number of a graph with diameter at most $2$ are small in Remark $1$ and Lemma \ref{lem16}. These results naturally lead to the following problem: can the $2$-limited packing number of a graph $G$ be bounded by a constant for $diam(G)\leq 2$? It is known that the graph with order $n$ and diameter $1$, which is exactly $K_n$, has $2$-limited packing number $2$ by Lemma \ref{lem3}. Thus, we only need to investigate the $2$-limited packing number of graphs with diameter $2$. Theorem \ref{th6} answers the above question. \begin{thm}\label{th6} For any positive integer $a$ with $a\geq 2$, there exists a graph $G$ with $diam(G)=2$ such that $L_2(G)=a.$ \end{thm} \begin{proof} We construct a graph $G$ with $diam(G)=2$ such that $L_2(G)=a$ for $a\geq 2$ as follows. First, suppose that $X=\{x_1,x_2,\ldots, x_a\}$ and $Y=\{y_1,y_2,\ldots, y_{\frac{a(a-1)}{2}}\}$ with $X\cap Y=\emptyset$. Let $G$ be a graph with $V(G)=X\cup Y$ such that $G[X]$ consists of $a$ isolated vertices, $G[Y]$ is a clique and each pair of distinct vertices in $X$ has a unique common neighbour in $Y$. Obviously, it is true that $diam(G)=2$. Now we need to show that $L_2(G)=a$. Notice that $|V(G)|=a+\frac{a(a-1)}{2}$ and $\Delta(G)=\frac{a(a-1)}{2}+1$, then $L_2(G)\leq |V(G)|+1-\Delta(G)=a$ by Corollary \ref{cor14}. Observe that $X$ is a $2$-limited packing of $G$, thus $L_2(G)= a$. \end{proof} But we can find graphs $G$ with $diam(G)=2$ such that $L_2(G)$ is small. First, we give some auxiliary lemmas. \begin{lem}{\upshape\cite{JP}}\label{lem20} Every planar graph of diameter $2$ has domination number at most $2$ except for the graph $F$ of Fig. $1$ which has domination number $3$. \end{lem} \begin{figure} \caption{A counterexample $F$ of Lemma \ref{lem20}} \end{figure} \begin{lem}\label{lem21} If $G$ is a graph with order $n$ and $\Delta(G)=n-1$, then $L_2(G)=2$. \end{lem} \begin{proof} By Lemma \ref{lem7}, we have $L_2(G)\geq \lceil \frac{2diam(G)+2}{3}\rceil \geq \lceil\frac{4}{3}\rceil=2$. On the other hand, $L_2(G)\leq n-\Delta(G)+1=2$ by Corollary \ref{cor14}. Thus, $L_2(G)=2$. \end{proof} \begin{lem}\label{lem22} Let $G$ be a graph with diameter $2$. Then $(i)$ if $G$ has a cut vertex, then $L_2(G)=2$, $(ii)$ if $G$ is a planar graph, then $L_2(G)\leq 4$. \end{lem} \begin{proof} Firstly we prove part $(i)$. Let $v_0$ be a cut vertex of $G$. We claim that for any vertex $u\in V(G)\setminus \{v_0\}$, $d(u,v_0)=1$. Suppose that there exists a vertex $u_0\in V(G)\setminus \{v_0\}$ such that $d(u_0,v_0)=2$. Let $w_0$ be another vertex of $G$ such that $w_0$ and $u_0$ are contained in different components of $G-v_0$. Then $d(u_0,w_0)\geq 3$, which contradicts to $diam(G)=2$. Thus, for any vertex $u\in V(G)\setminus \{v_0\}$, $d(u,v_0)=1$. It is obtained that $d(v_0)=|V(G)|-1$, hence $L_2(G)=2$ by Lemma \ref{lem21}. Next we prove part $(ii)$. Let $G$ be a planar graph with diameter $2$. Then it follows from Lemma \ref{lem4} and Lemma \ref{lem20} that $L_2(G)\leq 2\gamma(G)\leq 4$ except for the graph $F$ of Fig. $1$. It remains to verify the graph $F$ in Fig. $1$. Let $B$ be a maximum $2$-limited packing of $F$. Observe that $|\{u_1,\ldots,u_4\}\cap B|\leq 2$ and $|\{v_0,\ldots,v_4\}\cap B|\leq 2$, it follows that $L_2(F)\leq 4$. \end{proof} Next, we get the better upper bound of the $2$-limited packing number of graphs with large diameter. \begin{thm}\label{th9} If $G$ is a connected graph of order $n$, then $L_2(G)\leq n+1-\Delta(G)-\lfloor \frac{diam(G)-4}{3} \rfloor$. \end{thm} \begin{proof} If $diam(G)\leq 2$, then $L_2(G)\leq |V(G)|+1-\Delta(G)\leq n+1-\Delta(G) -\lfloor \frac{diam(G)-4}{3} \rfloor $ by Corollary \ref{cor14}. Thus, we may assume that $diam(G)\geq 3$ in the following. Let $B$ be a maximum $2$-limited packing in $G$, and $u$ be a vertex of degree $\Delta(G)$. Then $|N[u]\cap B|\leq 2$. Let $P$ be a path of length $diam(G)$ between $x$ and $y$ in $G$. We claim that $|V(P)\cap N[u]|\leq3$, otherwise using the same argument in proof of Theorem \ref{th5}, we can find a path between $x$ and $y$, whose length is less than $diam(G)$, a contradiction. It is also obtained that $|\{x,y\}\cap N[u]|\leq 1$, otherwise $d(x,y)\leq 2$, which contradicts to $diam(G)\geq 3$. Without loss of generality, assume that $x\in V(G)\setminus N[u]$. \textbf{Case 1.} $V(P)\cap N[u]=\emptyset$. By Lemma \ref{lem3}, $P$ has at most $\lceil\frac{2|V(P)|}{3}\rceil $ vertices in $B$. Then $|V(P)\cap \overline {B}|\geq \lfloor\frac{|V(P)|}{3}\rfloor$. On the other hand, $|N[u]\cap \overline {B}|\geq \Delta(G)+1-2=\Delta(G)-1$. Thus, \begin{eqnarray} | \overline {B}| &\geq & \Delta(G)-1+\lfloor\frac{|V(P)|}{3}\rfloor \nonumber\\ &=& \Delta(G)-1+\lfloor \frac{diam(G)+1}{3} \rfloor\nonumber. \end{eqnarray} \textbf{Case 2.} $V(P)\cap N[u]\neq\emptyset$. Let $P_x$, $P_y$ be the paths obtained from $P$ by deleting all the vertices in $N[u]$ such that $P_x$ and $P_y$ contain $x$, $y$, respectively. Let $H=P_x\cup P_y$. It is worth mentioning that $V(P_y)= \emptyset$ if $y \in N[u]$. Observe that $|V(H)|=|V(P_x)|+|V(P_y)|\geq |V(P)|-3 = diam(G)-2$. Since $H$ has at most $\lceil\frac{2|V(P_x)|}{3}\rceil+\lceil\frac{2|V(P_y)|}{3} \rceil$ vertices in $B$ by Lemma \ref{lem3}, it follows that $|V(H)\cap \overline{B}|\geq \lfloor\frac{|V(P_x)|}{3}\rfloor+\lfloor\frac{|V(P_y)|}{3} \rfloor$. Since $|N[u]\cap \overline {B}|\geq \Delta(G)+1-2=\Delta(G)-1$ and $V(H)\cap N[u]=\emptyset$, we have \begin{eqnarray} | \overline{B}| &\geq & \Delta(G)-1+\lfloor\frac{|V(P_x)|}{3}\rfloor+\lfloor\frac{|V(P_y)|}{3} \rfloor \nonumber\\ &\geq& \Delta(G)-1+\lfloor \frac{|V(H)|-2}{3} \rfloor\nonumber\\ &\geq& \Delta(G)-1+\lfloor \frac{diam(G)-4}{3} \rfloor \end{eqnarray} Combining Case $1$ and Case $2$, we have $| \overline{B}|\geq \Delta(G)-1+\lfloor \frac{diam(G)-4}{3} \rfloor$. Hence, $L_2(G)=|B|\leq n+1-\Delta(G)-\lfloor \frac{diam(G)-4}{3} \rfloor$. \noindent\textbf{Remark 4.} The upper bound in Theorem \ref{th9} is better than that in Corollary \ref{cor14} for $diam(G)\geq 7$. \end{proof} \section{ Comparing $L_2(T)$ with $L_1(T)$ and $\rho^{0}(T)$} In this section, we study the relationship among the $2$-limited packing number, the $1$-limited packing number and the open packing number of trees. \begin{lem}{\upshape\cite{HHS}}\label{lem27} For any graph $G$, $L_1(G)\leq \rho^{0}(G)\leq 2L_1(G)$. \end{lem} Similarly, we consider the relationship between the $2$-limited packing number and the $1$-limited packing number of graphs. \begin{pro}\label{prop4} For any graph $G$ with edges, $L_1(G)+1\leq L_2(G) \leq \frac{2(\Delta(G)^2+1)}{\delta(G)+1}L_1(G).$ \end{pro} \begin{proof} The Lower bound is evidently true for $\Delta(G)\geq 1$ by Lemma \ref{lem5} and Remark $2$. It remains to verify the upper bound. Combining Lemma \ref{lem31} and Lemma \ref{lem10}, we have \begin{eqnarray*} L_1(G) &\geq& \frac{|V(G)|}{\Delta(G)^2+1}\\ &=& \frac{2|V(G)|}{\delta(G)+1}\frac{\delta(G)+1}{2(\Delta(G)^2+1)}\\ &\geq& L_2(G)\frac{\delta(G)+1}{2(\Delta(G)^2+1)}. \end{eqnarray*} That is, $L_2(G) \leq \frac{2(\Delta(G)^2+1)}{\delta(G)+1}L_1(G).$ \end{proof} With respect to trees, the above result can be further improved. Recall that a star is a tree with diameter at most $2$. Define a $t$-spider to be a tree obtained from a star by subdividing $t$ of its edges once. \begin{thm}\label{th11} For any tree $T$, $L_1(T)+1\leq L_2(T)\leq 2L_1(T)$. Moreover, $L_1(T)+1= L_2(T)$ if and only if $T$ is a $t$-spider with $0\leq t<\Delta(T)$. \end{thm} \begin{proof} The lower bound holds from $\Delta(T)\geq 1$ by Lemma \ref{lem5}, and the upper bound follows from $L_2(T)\leq 2\gamma (T)=2L_1(T)$ by Lemma \ref{lem34} and Lemma \ref{lem4}. Next, we show that $L_1(T)+1= L_2(T)$ if and only if $T$ is a $t$-spider with $0\leq t<\Delta(T)$. Let $T$ be a $t$-spider with $V(T)=\{r,v_1,\ldots,v_t, w_1,\ldots,w_t, u_1,\ldots,u_s\}$ and $E(T)=\{rv_1,\ldots,rv_t\}\cup \{ru_1,\ldots,ru_s\} \cup \{v_1w_1,\ldots,v_tw_t\}$, where $s\geq1$ and $t\geq 0$. If $t=0$, then $T$ is a star and the results follows from $L_2(T)=2$ and $L_1(T)=1$. Now we assume that $t\geq1$. Notice that $|V(T)|=1+2t+s$ and $\Delta(T)=t+s$. By Corollary \ref{cor14}, we have $L_2(T)\leq |V(T)|+1-\Delta(T)=t+2$. Observe that $\{ v_1,u_1, w_1,\ldots,w_t\}$ is a $2$-limited packing of $T$, it follows that $L_2(T)= t+2$. On the other side, it follows from Theorem \ref{th3} that $L_1(T)\leq |V(T)|-\Delta(T)=t+1$. And it is clear that $\{w_1,\ldots,w_t, u_1\}$ is a $1$-limited packing of $T$, so $L_1(T)= t+1$. Thus, $L_1(T)+1= L_2(T)$. It remains to show the converse. Let $T$ be a tree with $L_1(T)+1= L_2(T)$. We first give the following claim. \textbf{Claim 1.} $diam(T)\leq 4$. \noindent\textbf{Proof of Claim 1:} Assume to the contrary that there is a path $Q=v_1\cdots v_6$ of order $6$ in $T$. Let $B_1$ be a maximum $1$-limited packing of $T$. By Lemma \ref{lem3}, it is obtained that $|V(Q)\cap B_1|\leq 2$. To have a contradiction, we aim to find a $2$-limited packing that contains $|B_1|+2$ vertices in $T$. Suppose that $V(Q)\cap B_1=\emptyset$. We claim that $B_2=B_1\cup\{v_1,v_6\}$ is a $2$-limited packing of $T$. It is clear that $|N[v_i]\cap B_2|\leq 2$ for each vertex $v_i$ on $Q$. Consider each vertex $u$ outside $Q$. First, we know $|N[u]\cap B_1|\leq 1$ for each vertex $u$ outside $Q$. Since $T$ is a tree and has no cycle, it follows that each vertex $u$ outside $Q$ has at most one neighbour on $Q$, which means $|N[u]\cap \{v_1,v_6\}|\leq 1$. Thus, $|N[u]\cap B_2|\leq 2$ for each vertex $u$ outside $Q$. As a result, we obtain that $B_2=B_1\cup\{v_1,v_6\}$ is a $2$-limited packing of $T$. Suppose that $V(Q)\cap B_1=\{v_i\}$ for some $1\leq i\leq 6$. If $i=1$, then $B_1\cup\{v_2,v_6\}$ is a $2$-limited packing of $T$. It is worth mentioning that $v_2$ is not adjacent to any vertex in $B_1\setminus\{v_1\}$, otherwise $|N[v_2]\cap B_1|\geq2$, a contradiction. If $2\leq i\leq5$, then $B_1\cup\{v_1,v_6\}$ is a $2$-limited packing of $T$. If $i=6$, then $B_1\cup\{v_1,v_5\}$ is a $2$-limited packing of $T$. Suppose that $V(Q)\cap B_1=\{v_i,v_j\}$ for some $1\leq i\neq j\leq 6$. If $(i,j)\in\{(1,4),(3,6)\}$, then $ B_1\cup\{v_2,v_5\}$ is a $2$-limited packing of $T$. If $(i,j)\in\{(1,5),(1,6),(2,5),(2,6)\}$, then $(B_1\setminus \{v_i,v_j\})\cup\{v_1,v_2,v_5,v_6\}$ is a $2$-limited packing of $T$. By the above argument, we have $L_2(T)\geq L_1(T)+2$, which is a contradiction. Thus, it is obtained that $diam(T)\leq4$. Suppose that $diam(T)\leq4$. Let $F$ be a tree with diameter $4$ and a unique vertex $f_0$ of maximum degree $3$ such that $f_0$ is adjacent to two leaf vertices. Let $B_1$ be a maximum $1$-limited packing of $T$. It is clear that $|V(F)\cap B_1|\leq 2$. We claim that $T$ has no $F$ as a subtree. Suppose to the contrary that $F$ is a subtree of $T$. By the similar argument when $T$ contains a path $P_6$, we always find a $2$-limited packing with $L_1(T)+2$ vertices in $T$ as depicted in Fig. 2, which is a contradiction. Thus, $T$ has no $F$ as a subtree, which implies that $T$ is a $t$-spider with $0\leq t\leq \Delta(T)$. Notice that if $\Delta(T)=1$, then it is clear that $t=0$. For $\Delta(T)\geq2$, we claim that $t< \Delta(T)$. Assume to the contrary that $t=\Delta(T)\geq 2$. Let $r$ be a vertex of maximum degree $t$ with $N(r)=\{v_1,\ldots,v_t\}$ and $N(v_i)=\{r,w_i\}$ for $1\leq i\leq t$, where $w_1,\ldots,w_t$ are $t$ leaf vertices of $T$. Observe that $\{v_1, v_2, w_1,\ldots,w_t\}$ is $2$-limited packing of $T$, it follows that $L_2(T)\geq t+2$. On the other hand, $\{r, v_i, w_i\}$ has at most one vertex in a $1$-limited packing of $T$ for each $1\leq i\leq t$, it follows that $L_1(T)\leq t$. Thus, $L_2(T)\geq L_1(T)+2$, which is a contradiction. As a result, $T$ is a $t$-spider with $0\leq t< \Delta(T)$. \begin{figure} \caption{The three cases can arise on the number of $|V(F)\cap B_1|$. In each case, the blue points correspond to the vertices in $B_1$, the black points correspond to the vertices outside $B_1$, and the circled black points correspond to the vertices outside $B_1$ that will be added into $B_2$.} \end{figure} \end{proof} \noindent\textbf{Remark 5.} By the proof of Theorem \ref{th11}, we know that $L_2(T)= 2L_1(T)$ if and only if $L_2(T)= 2\gamma (T)$. And all the trees $T$ with $L_2(T)= 2\gamma (T)$ are characterized in \cite{GGHR}. \vskip 0.3cm Similarly, we compare the $2$-limited packing number with the open packing number of trees. We first define a class of trees, which is needed in the following theorem. Let $\mathcal{T}$ be the set of trees $T$ whose vertex set can be partitioned into two disjoint subsets $S_0$ and $R_0$, satisfying the following properties: $(i)$ $T[S_0]=aK_2$, and each copy of $K_2$ has at least one vertex with degree $1$ in $T$, where $a$ is an positive integer, $(ii)$ for each $r\in R_0$, $|N(r)\cap S_0|=1$. \begin{thm}\label{th12} For any tree $T$, $\rho^{0}(T)\leq L_2(T)\leq2\rho^{0}(T)$. Moreover, $\rho^{0}(T)= L_2(T)$ if and only if $T\in \mathcal{T}$. \end{thm} \begin{proof} For the lower bound, we give the stronger result that $ L_2(G)\geq \rho^{0}(G)$ for any graph $G$. Let $S$ be an open packing of $G$ with $|S|=\rho^{0}(G)$. Then $|N(v)\cap S|\leq 1$ for each vertex $v$ of $G$. Obviously, $|N[v]\cap S|\leq 2$ for each vertex $v$ of $G$. It is obtained that $S$ is a $2$-limited packing of $G$, therefore $\rho^{0}(G)\leq L_2(G)$. On the other hand, since $\gamma(T)\leq \gamma_t(T)$, it follows that $L_2(T)\leq2\gamma(T)\leq 2\gamma_t(T)=2\rho^{0}(T)$ by Lemma \ref{lem33} and Lemma \ref{lem4}. Next, we show that $\rho^{0}(T)= L_2(T)$ if and only if $T\in \mathcal{T}$. Let $T$ be a tree in $\mathcal{T}$. If $T$ has only two vertices, then $T=K_2$, and hence the result trivially holds. Now we assume that $|V(T)|\geq 3$. Observe that $S_0$ is an open packing of $T$, if follows that $\rho^{0}(T)\geq 2a$. By the definition of $T$, it is obtained that $V(T)$ can be partitioned into $V_1\cup \cdots\cup V_a$ such that $G[V_i]$ is a star for each $1\leq i\leq a$. Notice that $V_i$ has at most two vertices in a $2$-limited packing of $T$ for each $1\leq i\leq a$, then $L_2(T)\leq 2a$. Since $\rho^{0}(T)\leq L_2(T)$, it follows that $\rho^{0}(T)= L_2(T)$ for each tree $T$ in $\mathcal{T}$. Conversely, suppose that $\rho^0(T)= L_2(T)$. Let $S$ be a maximum open packing of $T$. It is known that each component of $T[S]$ is $K_1$ or $K_2$. To show $T\in \mathcal{T}$, we give the following claims. \textbf{Claim 1.} $T[S]=tK_2$. \noindent\textbf{Proof of Claim 1:} Suppose to the contrary that there is at least one isolated vertex, say $v$, in $T[S]$. Since $T$ is connected, it follows that $v$ has a neighbour, say $r$, in $\overline {S}$. Let $B=S\cup\{r\}$. Next we show that $B$ is a $2$-limited packing of $T$. Since $r$ is not adjacent to any vertex in $S\setminus\{v\}$, it follows that for each vertex $v$ in $B$, $|N[v]\cap B|\leq 2$. On the other hand, for each vertex $u$ in $\overline{B}$, we have $|N[u]\cap S|\leq 1$, and hence $|N[u]\cap B|\leq 2$. Thus, $B$ is a $2$-limited packing of $T$, and $L_2(T)\geq |B|=|S|+1$, which is a contradiction. Thus, there is no isolated vertex in $T[S]$, so $T[S]=tK_2$. \textbf{Claim 2.} For each $r\in \overline{S}$, $|N(r)\cap S|=1$. \noindent\textbf{Proof of Claim 2:} By the definition of the open packing, we know $|N(r)\cap S|\leq 1$ for any vertex $r\in \overline{S}$. To show this claim, it remains to prove $|N(r)\cap S|\geq 1$ for any vertex $r\in \overline{S}$. Assume that there is a vertex $r_0\in \overline{S}$ such that $N(r_0)\cap S=\emptyset$. Then $S\cup \{r_0\}$ is a $2$-limited packing of $T$, so $L_2(T)\geq \rho^{0}(T)+1$, a contradiction. Hence, we have $|N(r)\cap S|=1$ for each $r\in \overline{S}$. \textbf{Claim 3.} Each component of $T[S]$ has at least one vertex with degree $1$ in $T$. \noindent\textbf{Proof of Claim 3:} Suppose that $T[S]$ has one component $K_2=v_1v_2$, where $d(v_i)\geq2$ for $i=1,2$. It is obtained that there is a path $P=uv_1v_2w$ in $T$, where $u,w\in \overline{S}$. Notice that each vertex on $P$ has the property that its neighbours outside $P$ are not contained in $S$, otherwise there is a vertex on $P$ such that it has at least two neighbours in $S$, which is a contradiction. It is obtained that $(S\setminus \{v_2\})\cup\{u,w\}$ is a $2$-limited packing of $T$, which means $L_2(T)\geq \rho^0(T)+1$, which is a contradiction. Thus, each component of $T[S]$ has at least one vertex with degree $1$ in $T$. By the above claims, we get that if $\rho^0(T)= L_2(T)$, then $T\in \mathcal{T}$ with $S=S_0$, this completes the proof. \end{proof} Graphs with $\rho^{0}(G)=1$, graphs with $L_k(G)=k$ for $k=1,2$ are characterized in Lemma \ref{lem15} and Theorem \ref{th1}, respectively. So we assume that $a\geq 2$ in the following theorem. \begin{thm}\label{th8} For each pair of integers $a$ and $b$ with $a\geq 2$ and $a+1\leq b\leq 2a$, there exists a tree $T$ such that $\rho^{0}(T)=L_1(T)=a$ and $L_2(T)=b$. \end{thm} \begin{proof} Suppose $a$ and $b$ are two positive integers with $a+1\leq b\leq 2a$. Let $b=a+r$ with $1\leq r\leq a$. To construct a tree $T$ with $\rho^{0}(T)=L_1(T)=a$ and $L_2(T)=a+r$ for $a\geq2$ and $1\leq r\leq a$, we distinguish the following two cases. \textbf{Case 1.} $a=r$. Suppose that $Q_i=x_iy_iz_i$ is a path of order $3$ for $1\leq i\leq a$. Let $T$ be the tree obtained from $Q_1\cup\cdots \cup Q_a$ by adding the edge $y_iy_{i+1}$ for $1 \leq i\leq a-1$. First, we show that $L_2(T)=a$. Since each $V(Q_i)$ has at most two vertices in a $2$-limited packing of $T$ for $1\leq i\leq a$, we have $L_2(T)\leq 2a$. It is observed that $\{x_1, \cdots, x_a, z_1,\cdots,z_a\}$ is a $2$-limited packing of $T$, thus $L_2(T)=2a$. Next, we show that $\rho^{0}(T)=L_1(T)=a$. Note that $L_1(T)\leq \rho^{0}(T)$ by Lemma \ref{lem27}, then it is sufficient to show that $L_1(T)\geq a$ and $\rho^{0}(T)\leq a$. Obviously, $\{x_1, \cdots, x_a\}$ is a $1$-limited packing of $T$, thus $L_1(T)\geq a$. It remains to show that $\rho^{0}(T)\leq a$. It is clear that $\{y_1, \cdots, y_a\}$ is a total dominating set of $T$, which implies that $\gamma_t(T)\leq a$. By Lemma \ref{lem33}, we have $\rho^{0}(T)=\gamma_t(T)\leq a$. \textbf{Case 2.} $1\leq r\leq a-1$. Consider a star $A=K_{1,a}$ with $V(A)=\{v_0,v_1,\ldots,v_a\}$ and $d(v_0)=a$. Let $T$ be the tree obtained from $A$ by adding two pendent edges $v_iw_i$ and $v_iw_i^{'}$ to each $v_i$ of $A$ for $1\leq i\leq r-1$, and one pendent edge $v_iw_i$ at each $v_i$ of $A$ for $r\leq i\leq a-1$. Fig. $3$ gives an example for $a=8$, $b=12$. \begin{figure} \caption{A graph with $\rho^{0}(T)=L_1(T)=8$ and $L_2(T)=12$} \end{figure} To obtain that $L_1(T)=\rho^{0}(T)=a$, it suffices to prove that $L_1(T)\geq a$ and $\rho^{0}(T)\leq a$ by Lemma \ref{lem27}. Obviously, $\{w_1,\ldots,w_{a-1}, v_a\}$ is a $1$-limited packing of $T$, so $L_1(T)\geq a$. On the other hand, let $S$ be a maximum open packing of $T$ with $|S|=\rho^{0}(T)$. It only need to show $|S|\leq a$. Suppose that $v_0\in S$. It follows that $\{v_i: 1\leq i\leq a\}$ has at most one vertex in $S$, and $\{w_i,w_j': 1\leq i\leq a-1, 1\leq j\leq r-1\}$ has no vertex in $S$. It is obtained that $S=\{v_0, v_i\}$ for some $i$ with $1\leq i\leq a$, and hence $|S|=2\leq a$. Suppose that $v_0 \notin S$. If $v_a\in S$, then $\{v_i: 1\leq i\leq a-1\}$ has no vertex in $S$ and $\{w_i,w_i'\}$ has at most one vertex in $S$ for each $1\leq i\leq a-1$, and hence $|S|\leq a$. If $v_a\notin S$, then both $\{v_i: 1\leq i\leq a-1\}$ and $\{w_i,w_i'\}$ for each $1\leq i\leq a-1$ have at most one vertex in $S$, so $|S|\leq a$. It remains to show that $L_2(T)= a+r$ with $1\leq r\leq a-1$. Note that $T$ has $2a+r-1$ vertices and $\Delta(T)=a$. By Corollary \ref{cor14}, we have $L_2(T)\leq |V(T)|+1-\Delta(T)= a+r$. Observe that $\{v_a,v_{a-1}\}\cup\{w_i,w_j': 1\leq i\leq a-1, 1\leq j\leq r-1\}$ is a $2$-limited packing of $T$, it follows that $L_2(T)= a+r$. \end{proof} \end{document}

\begin{document} \begin{center} {\bf Combinatorial Sums $\sum_{k\equiv r(\mbox{mod } m)}{n\choose k}a^k$ and Lucas Quotients (II)} \vskip 20pt {\bf Jiangshuai Yang}\\ {\smallit Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China}\\ {\tt [email protected]}\\ \vskip 10pt {\bf Yingpu Deng}\\ {\smallit Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People's Republic of China}\\ {\tt [email protected]}\\ \end{center} \vskip 30pt \centerline{\bf Abstract} \noindent In \cite{dy}, we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod }m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed in terms of some recurrent sequences with orders not exceeding $\varphi{(m)}$ and give some new congruences. \pagestyle{myheadings} \thispagestyle{empty} \baselineskip=12.875pt \vskip 30pt \section{Introduction} \noindent Let $p$ be an odd prime, using the formula for the sum $$\sum_{k\equiv r(\mbox{mod }8)}{n\choose k},$$ Sun \cite{s1995} proved that \[\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{1}{k\cdot2^k}\equiv\sum\limits_{k=1} ^{[\frac{3p}{4}]}\frac{(-1)^{k-1}}{k}\pmod p.\] Later, Shan and E.T.H.Wang \cite{sw} gave a simple proof of the above congruence. In \cite{sun5}, Sun proved five similar congruences by using the formulas for Fibonacci quotient and Pell quotient. In \cite{s2002}, Sun showed that the sum $$\sum_{k\equiv r(\mbox{mod }m)}{n\choose k},$$ where $n,m$ and $r$ are integers with $m,n>0$, can be expressed in terms of some recurrent sequences with orders not exceeding $\varphi{(m)}/2$, and obtained the following congruence \[\sum_{k=1}^{\frac{p-1}{2}}\frac{3^k}{k}\equiv\sum_{k=1}^{\left[\frac{p}{6}\right]}\frac{(-1)^k}{k} \pmod p. \] In \cite{dy}, we studied more general sum \begin{equation}\label{generalsum} \sum_{k\equiv r(\mbox{mod }m)}{n\choose k}a^k, \end{equation} and obtained congruences for Lucas quotients of two infinite families of Lucas sequences. See (\cite{dy} Theorems 4.10 and 5.4). In this paper, we continue studying the sum. We show that it can be expressed in terms of some recurrent sequences with orders not exceeding $\varphi{(m)}$, and obtain some new congruences. For $x\in\mathbb{R}$, we use $[x]$ to denote the integral part of $x$ i.e., the largest integer $\leq x$. For odd prime $p$ and integer $b$, let $\left(\frac bp\right)$ denote the Legendre symbol and $q_p(b)$ denote the Fermat quotient $(b^{p-1}-1)/p$ if $p\nmid b$. When $c,d\in\mathbb{Z}$, as usual $(c,d)$ stands for the greatest common divisor of $c$ and $d$. For any positive integer $m$, let $\zeta_m=e^{\frac{2\pi i}{m}}$ be the primitive $m$-th root of unity and let $\varphi({m})$, $\mu{(m)}$ denote the Euler totient function and M$\ddot{\textup{o}}$bius function respectively. Throughout this paper, we fix $a\neq 0,\pm1$. \section{Main Results} \begin{definition}\label{defsum} {\rm Let $n,m,r$ be integers with $n>0$ and $m>0$. We define $$\left[\begin{array}{c}n \\ r\\\end{array}\right] _{m}(a):=\sum_{\substack{k=0\\k\equiv r({\mbox{mod }}m)}}^n\binom nk a^k,$$ where ${n\choose k}$ is the binomial coefficient with the convention ${n\choose k}=0$ for $k<0$ or $k>n$.} \end{definition} \noindent Then we have the following theorem. \begin{theorem}\label{Maintheorem} Let $m,n\in\mathbb{Z}^+$, and $k\in\mathbb{Z}$. Write $$W_{n}(k,m)=\sum_{\substack{l=1\\(l,m)=1}}^m\zeta_m^{-kl}(1+a\zeta_m^l)^n,$$ and $$A_{m}(x)=\prod_{\substack{l=1\\(l,m)=1}}^m(x-1-a\zeta_m^l)=\sum\limits_{s=0}^{\varphi(m)}b_sx^s.$$ Then $$A_{m}(x)\in \mathbb{Z}[x] \quad and \quad\sum\limits_{s=0}^{\varphi(m)}b_sW_{n+s}(k,m)=0.$$ Moreover, for any $r\in\mathbb{Z}$ we have \begin{equation*} \left[\begin{array}{c}n \\r\\\end{array}\right] _{m}(a)=\frac1m\sum\limits_{d\mid m }W_{n}(r,d). \end{equation*} \end{theorem} \begin{proof} It is easy to see that the coefficients of $A_{m}(x+1)$ are symmetric polynomials in those primitive $m$-th roots of unity with integer coefficients. Sicne \[\Phi_m(x)=\prod_{\substack{l=1\\(l,m)=1}}^m(x-\zeta_m^l)\in\mathbb{Z}[x],\] $A_{m}(x+1)\in \mathbb{Z}[x]$ by Fundamental Theorem on Symmetric Polynomials. Therefore $A_{m}(x)\in \mathbb{Z}[x].$ For any positive integer $n$, we clearly have \begin{align*} \sum\limits_{s=0}^{\varphi(m)}b_sW_{n+s}(k,m) &=\sum\limits_{s=0}^{\varphi(m)}b_s\sum_{\substack{l=1\\(l,m)=1}}^m\zeta_m^{-kl}(1+a\zeta_m^l)^{n+s}\\ &=\sum_{\substack{l=1\\(l,m)=1}}^m\zeta_m^{-kl}(1+a\zeta_m^l)^{n}\sum\limits_{s=0}^{\varphi(m)}b_s(1+a\zeta_m^l)^{ s}\\ &=\sum_{\substack{l=1\\(l,m)=1}}^m\zeta_m^{-kl}(1+a\zeta_m^l)^{n}A_{m}(1+a\zeta_m^l)\\ &=0. \end{align*} Let $r\in\mathbb{Z}$, then we have \begin{align*} \left[\begin{array}{c}n \\ r\\\end{array}\right] _{m}(a) &=\sum\limits_{k=0}^n\binom nka^k\cdot\frac1m\sum\limits_{l=1}^m\zeta_m^{(k-r)l}\\ &=\frac1m\sum\limits_{l=1}^m\zeta_m^{-rl}(1+a\zeta_m^l)^n\\ &=\frac1m\sum\limits_{d\mid m}\sum_{\substack{b=1\\(b,d)=1}}^d\zeta_d^{-rb}(1+a\zeta_d^b)^n\\ &=\frac1m\sum\limits_{d\mid m}W_{n}(r,d). \end{align*} This ends the proof. \end{proof} Note that the theorem is a generalization of Theorem 1 of \cite{s2002}. \begin{remark}\label{Wremark} The last result shows that $\left[\begin{array}{c}n \\ r\\\end{array}\right] _{m}(a)$ can be expressed in terms of some linearly recurrent sequences with orders not exceeding $\varphi{(m)}.$ \end{remark} Now we list $A_{m}(x)$ for $1\leq m\leq6$: \begin{align*} &A_{1}(x)=x-1-a, A_{2}(x)=x-1+a,\\ &A_{3}(x)=x^2-(2-a)x+a^2-a+1, A_{4}(x)=x^2-2x+a^2+1,\\ &A_{5}(x)=x^4-(4-a)x^3+(a^2-3a+6)x^2+(a^3-2a^2+3a+4)+a^4-a^3+a^2-a+1,\\ &A_{6}(x)=x^2-(a+2)x+a^2+a+1. \end{align*} \begin{lemma}\textup{(\cite{s2002})}\label{Molemma} Let $m,c$ be integers with $m>0$. Then we have \begin{equation*} \sum\limits_{d\mid m}\mu(\frac md)d\delta_{d\mid c}=\varphi(m)\frac{\mu(m/(c,m))}{\varphi(m/(c,m))}, \end{equation*} where \[\delta_{d\mid c}=\begin{cases}1,&\mbox{ if }d\mid c \mbox{ holds};\\0,&\mbox{otherwise}.\end{cases} \] \end{lemma} \begin{proof} We can find that both sides are multiplicative with respect to $m$, thus we only need to prove it when $m$ is a prime power. For any prime $p$ and positive integer $k$, we have \begin{align*} \sum\limits_{d\mid p^k}\mu(\frac {p^k}d)d\delta_{d\mid c} &=\sum\limits_{s=0}^k\mu(p^{k-s})p^s\delta_{p^s\mid c}\\ &=p^k\delta_{p^k\mid c}-p^{k-1}\delta_{p^{k-1}\mid c}\\ &=\begin{cases}p^k-p^{k-1}&\textup{if}\; p^k\mid c,\\ -p^{k-1}&\textup{if}\;p^{k-1}\parallel c,\\ 0&\textup{if} \;p^{k-1}\nmid c.\end{cases}\\ &=\varphi(p^k)\frac{\mu(p^k/(c,p^k))}{\varphi(p^k/(c,p^k))}. \end{align*} This concludes the proof. \end{proof} \begin{theorem}\label{Wtheorem} Let $m,n\in\mathbb{Z}^+ ,r\in\mathbb{Z}$. Then \begin{equation*} W_{n}(r,m)=\varphi(m)\sum\limits_{k=0}^n\frac{\mu(m/(k-r,m))}{\varphi(m/(k-r,m))}\binom nka^k. \end{equation*} \end{theorem} \begin{proof} By Theorem \ref{Maintheorem}, Lemma \ref{Molemma} and M$\ddot{\textup{o}}$bius Inversion Theorem, we have \begin{align*} W_{n}(r,m) &=\sum\limits_{d\mid m}\mu(\frac md)d\left[\begin{array}{c}n \\r\\\end{array}\right]_d(a)\\ &=\sum\limits_{d\mid m}\mu(\frac md)d\sum\limits_{k=0}^n\binom nka^k\delta_{d\mid k-r} \\ &= \sum\limits_{k=0}^n\binom nka^k\sum\limits_{d\mid m}\mu(\frac md)d\delta_{d\mid k-r}\\ &=\varphi(m)\sum\limits_{k=0}^n\frac{\mu(m/(k-r,m))}{\varphi(m/(k-r,m))}\binom nka^k. \end{align*} \end{proof} \begin{corollary}\label{Wcorollary} Let $m,n$ be two relatively prime positive integers. Then we have \begin{equation*} W_{n}(0,m)-\varphi(m)-\mu(m)a^n=\varphi(m)n\sum\limits_{k=1}^{n-1}\frac{\mu(m/(m,k))}{\varphi(m/(m,k))}\binom{n-1}{k-1}\frac{a^k}{k} \end{equation*} and \begin{equation*} W_{n}(n,m)-\varphi(m)a^n-\mu(m)=\varphi(m)n\sum\limits_{k=1}^{n-1}\binom{n-1}{k-1}\frac{\mu(m/(m,k))}{\varphi(m/(m,k))}\frac{a^{n-k}}{k}. \end{equation*} \end{corollary} \begin{proof} Since $\binom nk=\frac nk\binom{n-1}{k-1}$ for $1\leq k\leq n$, we can derive the results by setting $r=0,n$ respectively in Theorem \ref{Wtheorem}, \end{proof} \begin{corollary}\label{Wpcorlllary} Let $m\in\mathbb{Z}^+$ and $p$ be an odd prime not dividing $am$. Then we have \begin{equation*} \frac{W_{p}(0,m)- \varphi(m)-\mu(m)a^p}{p}\equiv-\varphi(m)\sum\limits_{k=1}^{p-1}\frac{\mu(m/(m,k))}{\varphi(m/(m,k))}\cdot\frac{(-a)^k}{k}\pmod p, \end{equation*} and \begin{equation*} \frac{W_{p}(p,m)-\varphi(m)a^p-\mu(m)}{p}\equiv\varphi(m)\sum\limits_{k=1}^{p-1}\frac{\mu(m/(m,k))}{\varphi(m/(m,k))}\cdot\frac{1}{k(-a)^{k-1}}\pmod p. \end{equation*} \end{corollary} \begin{proof} Since $\binom{p-1}{k}=(-1)^k$ for $0\leq k\leq p-1$, the results follow from Corollary \ref{Wcorollary}. \end{proof} \section{Some New Congruences} In this section, we give some new congruences by using the results of \cite{dy}. \begin{lemma}\label{3uvlemma} Let $p\nmid 3a(2-a)(a^3+1)$ be and odd prime, and $\{u_n\}_{n\geq0},\{v_n\}_{n\geq0}$ be the Lucas sequences defined as $$u_0=0,\;u_1=1,\;u_{n+1}=(2-a)u_n-(a^2-a+1)u_{n-1}\;\textup{for}\;n\geq1;$$ $$v_0=2,\;v_1=(2-a),\;v_{n+1}=(2-a)v_n-(a^2-a+1)v_{n-1}\;\textup{for}\;n\geq1.$$ Then we have: \\ \begin{description} \item[(1)] \[ \frac{u_p-\left(\frac{-3}{p}\right)}{p}\equiv\sum\limits_{k=1}^{\frac{p-1}2}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2} +\left(\frac{-3}{p}\right)\left(q_p(a)-q_p(2)+\frac12q_p(3)\right)\pmod p;\] \item[(2)] \[ \quad\frac{v_{p}-(2-a)}{p}\equiv (2-a)\left[-\frac12\sum\limits_{k=1}^{\frac{p-1}2}\frac{(-3)^k}{k}\cdot\left(\frac{a}{2-a}\right)^{2k}-q_p(2)+q_p(2-a)\right]\pmod p.\] \end{description} \end{lemma} \begin{proof} By Lemmas 2.1 and 2.2 of \cite{dy}, we have $u_p=\frac1{a\sqrt{-3}} \left[\left(\frac{2-a}{2}+\frac a2\sqrt{-3}\right)^p- \left(\frac{2-a}{2}-\frac a2\sqrt{-3}\right)^p\right]$ , $v_p= \left(\frac{2-a}{2}+\frac a2\sqrt{-3}\right)^p+ \left(\frac{2-a}{2}-\frac a2\sqrt{-3}\right)^p$, and $u_p\equiv\left(\frac{-3}{p}\right)\pmod p$, $v_p=(2-a)u_p-2(a^2-a+1)u_{p-1} =2u_{p+1}-(2-a)u_p\equiv(2-a)\pmod p$. Then \begin{align*} 2^{p-1}u_p &=\sum_{\substack{k=0\\k\; odd}}^{p}\binom pk(2-a)^{p-k}(a\sqrt{-3})^{k-1}\\ &=a^{p-1}(-3)^{\frac{p-1}{2}}+\sum\limits_{k=1}^{\frac{p-1}{2}}\binom p{2k-1}(2-a)^{p-2k+1}a^{2k-2}(-3)^{k-1}\\ &=a^{p-1}(-3)^{\frac{p-1}{2}}+p\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\binom {p-1}{2k-2}(2-a)^{p-2k+1}a^{2k-2}\\ &\equiv a^{p-1}(-3)^{\frac{p-1}{2}}+ p\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2}\pmod {p^2},\\ \end{align*} and \begin{align*} 2^{p-1}v_p &=\sum_{\substack{k=0\\k\; even}}^{p}\binom pk(2-a)^{p-k}(a\sqrt{-3})^k\\ &=(2-a)^p+\sum\limits_{k=1}^{\frac{p-1}{2}}\binom p{2k}(2-a)^{p-2k}a^{2k}(-3)^k\\ &=(2-a)^p+p\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^k}{2k}\binom {p-1}{2k-1}(2-a)^{p-2k}a^{2k}\\ &\equiv(2-a)^p-\frac{2-a}2 p\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^k}{k}\cdot\left(\frac{a}{2-a}\right)^{2k}\pmod {p^2}.\\ \end{align*} Hence (1) and (2) follow from Lemma 2.6(1) of \cite{dy}. \end{proof} \begin{corollary}\label{30corollary} Let $p\nmid 3a(2-a)(a^3+1)$ be and odd prime. Then we have \[\sum\limits_{k=1}^{[\frac{p}{3}]}\frac{(-a)^{3k}}{k}\equiv(2-a)\left[\frac12\sum\limits_{k=1}^{\frac{p-1}2}\frac{(-3)^k}{k}\cdot\left(\frac{a}{2-a}\right)^{2k} +q_p(2)-q_p(2-a)\right]-(a+1)q_p(a+1)\pmod p.\] \end{corollary} \begin{proof} The result follows from Lemma 4.9 of \cite{dy} and Lemma \ref{3uvlemma}(2). \end{proof} \begin{theorem}\label{31theorem} Let $p\nmid 3a(a-1)(2-a)(a^3+1)$ be an odd prime, and $\{u_n\}_{n\geq0}$ be the Lucas sequence defined as $$u_0=0,\;u_1=1,\;u_{n+1}=(2-a)u_n-(a^2-a+1)u_{n-1}\;\textup{for}\;n\geq1.$$ \begin{description} \item[(1)] If $p\equiv1\pmod 3$, we have \[\frac{u_{p-1}}{p}\equiv-\frac{2}{a(a-1)}\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1}+\frac{a+1}{3a(a-1)}\left(q_p(a^2-a+1)-2q_p(a+1)\right)\pmod p\] and \begin{align*} \sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1} &\equiv\frac{a(a-1)}{a-2}\sum\limits_{k-1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2}\\ &+\frac{a(a-1)}{a-2} [q_p(a)-q_p(2)+\frac12q_p(3)]\\ &-\frac{1}{3}(a+1)q_p(a+1)- \frac{ a^2-a+1}{3(a-2)}q_p(a^2-a+1) \pmod p. \end{align*} \item[(2)] If $p\equiv2\pmod 3$, we have \[\frac{u_{p+1}}{p}\equiv\frac{2(a^2-a+1)}{a(a-1)}\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2}-\frac{a^3+1}{3a(a-1)}\left(q_p(a^2-a+1)-2q_p(a+1)\right)\pmod p\] and \begin{align*} \sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2} &\equiv-\frac{a(a-1)}{a-2}\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2}\\ &\quad+\frac{a(a-1)}{a-2} [q_p(a)-q_p(2)+\frac12q_p(3)]\\ &\quad-\frac{1}{3}(a+1)q_p(a+1)-\frac{ a^2-a+1}{3(a-2)}q_p(a^2-a+1) \pmod p. \end{align*} \end{description} \end{theorem} \begin{proof} Since$(2-a)^2-4(a^2-a+1)=-3a^2$, we have $p\mid u_{p-\left(\frac{-3}{p}\right)}$ by Lemma 2.2 of \cite{dy}. Let $\{v_n\}_{n\geq0}$ be the Lucas sequence define as Lemma \ref{3uvlemma}. (1) By Lemma 2.1 and Theorem 4.1 of \cite{dy}, we have $-(a+1)u_p+(a^2-a+1)u_{p-1}=3\left[\begin{array}{c}p \\ 2\\\end{array}\right] _{3}(a)-(1+a)^p$ and $v_{p-1}=2u_p-(2-a)u_{p-1}$. Thus by Lemma 2.4 of \cite{dy}, we have \begin{align*} 3a(a-1)u_{p-1} &=6\left[\begin{array}{c}p \\ 2\\\end{array}\right] _{3}(a)-2(1+a)^p+(a+1)v_{p-1}\\ &\equiv-6p\sum\limits_{k-1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1}+(a+1)\left[(v_p-2)-2((a+1)^{p-1}-1)\right]\pmod{p^2} \end{align*} and \begin{align*} 3a(a-1)(u_{p}-1) &=3(2-a)\left[\begin{array}{c}p \\ 2\\\end{array}\right] _{3}(a)-(2-a)(1+a)^p+(a^2-a+1)v_{p-1}-3a(a-1)\\ &\equiv-3(2-a)p\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1}-(2-a)(1+a)((a+1)^{p-1}-1)\\ &\quad+(a^2-a+1)(v_{p-1}-2) \pmod{p^2}. \end{align*} Thence by Lemma 2.7 of \cite{dy} and Lemma \ref{3uvlemma}(1), \[\frac{u_{p-1}}{p}\equiv-\frac{2}{a(a-1)}\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1}+\frac{a+1}{3a(a-1)}\left(q_p(a^2-a+1)-2q_p(a+1)\right)\pmod p,\] and \begin{align*} \sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-a)^{3k-1}}{3k-1} &\equiv\frac{a(a-1)}{a-2}\sum\limits_{k-1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2}\\ &+\frac{a(a-1)}{a-2} [q_p(a)-q_p(2)+\frac12q_p(3)]\\ &-\frac{1}{3}(a+1)q_p(a+1)- \frac{ a^2-a+1}{3(a-2)}q_p(a^2-a+1) \pmod p. \end{align*} (2) By Lemma 2.1 and Theorem 4.1 of \cite{dy}, we have $-u_{p+1}+(a+1)u_{p}=3\left[\begin{array}{c}p \\ 1\\\end{array}\right] _{3}(a)-(1+a)^p$ and $v_{p+1}=(2-a)u_{p+1}-2(a^2-a+1)u_p$. Thus by Lemmas 2.4 of \cite{dy}, we have \begin{align*} -3a(a-1) u_{p+1} &=6(a^2-a+1)\left[\begin{array}{c}p \\ 1\\\end{array}\right] _{3}(a)-2(a^2-a+1)(1+a)^p+(a+1)v_{p+1}\\ &\equiv-6(a^2-a+1)p\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2}-2(a^2-a+1)(a+1)\left[(a+1)^{p-1}-1\right]\\ &\quad+(a+1)\left[v_{p+1}-2(a^2-a+1)\right]\pmod{p^2} \end{align*} and \begin{align*} -3a(a-1)(u_{p}+1) &=3(2-a)\left[\begin{array}{c}p \\ 1\\\end{array}\right] _{3}(a)- (2-a)(1+a)^p+ v_{p+1}-3a(a-1)\\ &\equiv-3 (2-a )p\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2}- (2-a)(a+1)\left[(a+1)^{p-1}-1\right]\\ &\quad +v_{p+1}-2(a^2-a+1) \pmod{p^2}. \end{align*} Thence by Lemma 2.7 of \cite{dy} and Lemma \ref{3uvlemma}(1), \[\frac{u_{p+1}}{p}\equiv\frac{2(a^2-a+1)}{a(a-1)}\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2}-\frac{a^3+1}{3a(a-1)}\left(q_p(a^2-a+1)-2q_p(a+1)\right)\pmod p,\] and \begin{align*} \sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-a)^{3k-2}}{3k-2} &\equiv-\frac{a(a-1)}{a-2}\sum\limits_{k=1}^{\frac{p-1}{2}}\frac{(-3)^{k-1}}{2k-1}\cdot\left(\frac{a}{2-a}\right)^{2k-2}\\ &+\frac{a(a-1)}{a-2} [q_p(a)-q_p(2)+\frac12q_p(3)]\\ &-\frac{1}{3}(a+1)q_p(a+1)-\frac{ a^2-a+1}{3(a-2)}q_p(a^2-a+1) \pmod p. \end{align*} \end{proof} Set $a=-2$ in Corollary \ref{30corollary} and Theorem \ref{31theorem}, we have the following two corollaries. \begin{corollary} Let $p\neq3,7$ be and odd prime. Then we have \begin{description} \item[(1)] \[\sum\limits_{k=1}^{[\frac p3]}\frac{8^k}{k}\equiv\sum\limits_{k=1}^{\frac {p-1}{2}}\frac{2}{k}\cdot\left(-\frac34\right)^k-4q_p(2)\pmod p.\] \item[(2)] If $p\equiv1 \pmod 3$, \[\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{8^k}{3k-1}\equiv4\sum\limits_{k=1}^{\frac {p-1}{2}}\frac{1}{2k-1}\cdot\left(-\frac34\right)^k-\frac 32q_p(3)+\frac 76q_p(7)\pmod p.\] If $p\equiv2 \pmod 3$, \[\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{8^k}{3k-2}\equiv-8\sum\limits_{k=1}^{\frac {p-1}{2}}\frac{1}{2k-1}\cdot\left(-\frac34\right)^k-3q_p(2)+\frac 73q_p(7)\pmod p.\] \end{description} \end{corollary} \begin{corollary}\label{-2corollary} Let $p\neq3,7$ be and odd prime, and $\{u_n\}_{n\geq0}$ be the Lucas sequence defined as $$u_0=0,\;u_1=1,\;u_{n+1}=4u_n-7u_{n-1}\;\textup{for}\;n\geq1.$$ Then, if $p\equiv1\pmod 3$, \[\frac{u_{p-1}}p\equiv-\frac{1}{6}\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{8^k}{3k-1}-\frac{1}{18} q_p(7)\pmod p,\] if $p\equiv2\pmod 3$, \[\frac{u_{p+1}}p\equiv\frac{7}{12}\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{8^k}{3k-2}+\frac{7}{18} q_p(7)\pmod p.\] \end{corollary} The following theorem can reduce the summation terms occurring in the expression of Lucas quotients in Corollary 4.11 of \cite{dy} and Corollary 3.5. \begin{theorem} Let $p\neq3,7$ be an odd prime, and $\left\{u_n\right\}_{n\geq0}$ be the Lucas sequence defined as Corollary 3.5. Then if $p\equiv1\pmod3$, \begin{align*} \frac{u_{p-1}}p &\equiv\frac16\sum\limits_{k=1}^{\frac{p-1}{6}}\frac{64^k}{k}+\frac13q_p(7)+\frac12q_p(3)\\ &\equiv-\frac{1}{3}\sum\limits_{k=1}^{\frac{p-1}{6}}\frac{64^k}{6k-1}-\frac{1}{18} q_p(7)+\frac 16q_p(3) \pmod p, \end{align*} if $p\equiv2\pmod3$, \begin{align*} \frac{u_{p+1}}p &\equiv-\frac76\sum\limits_{k=1}^{\frac{p-5}{6}}\frac{64^k}{k}-\frac73q_p(7)-\frac72q_(3)\\ &\equiv\frac{7}{6}\sum\limits_{k=1}^{\frac{p+1}{6}}\frac{64^k}{6k-2}+\frac{7}{18} q_p(7)+\frac 76q_p(3)\pmod p. \end{align*} \end{theorem} \begin{proof} By Lemma 2.4 and Theorem 4.5 of \cite{dy}, if $p\equiv1\pmod 3$, \[3^{p-1}-(-3)^{\frac{p-1}{2}}=\left[\begin{array}{c}p \\2 \\\end{array}\right] _3(2)\equiv-p\sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-2)^{3k-1}}{3k-1}\pmod{p^2},\] if $p\equiv2\pmod 3$, \[3^{p-1}+(-3)^{\frac{p-1}{2}}=\left[\begin{array}{c}p \\1 \\\end{array}\right] _3(2)\equiv-p\sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-2)^{3k-2}}{3k-2}\pmod{p^2}.\] Thus by the Lemma 2.6 of \cite{dy}, we have \[ \sum\limits_{k=1}^{\frac{p-1}{3}}\frac{(-8)^{k}}{3k-1}\equiv q_p(3)\pmod p \;\textup{if} \;p\equiv1 \pmod 3,\] and \[ \sum\limits_{k=1}^{\frac{p+1}{3}}\frac{(-8)^{k}}{3k-2}\equiv-2q_p(3) \pmod p \;\textup{if} \;p\equiv2 \pmod 3.\] Hence the results follow from Corollaries 4.7 and 4.11 of \cite{dy} and Corollary \ref{-2corollary}. \end{proof} \section{A Specific Lucas Sequence} \noindent Let $A,B\in\mathbb{Z}$. The Lucas sequences $u_n=u_n(A,B)(n\in\mathbb{N})$ and $v_n=v_n(A,B)(n\in\mathbb{N})$ are defined by \[u_0=1,\;u_1=1,\; u_{n+1}=Bu_n-Au_{n-1}(n\geq1);\] \[v_0=2,\;v_1=B,\; v_{n+1}=Bv_n-Av_{n-1}(n\geq1).\] Next we give some properties of the Lucas sequences with $A=5$ and $B=2$. We need some lemmas. Let $D=B^2-4A.$ \begin{lemma}\label{ulucasmod} Let $p$ be an odd prime not dividing $DA$. \begin{description} \item[(1)] If $p\equiv1\pmod 4$, then $p\mid u_{\frac{p-1}{4}}$ if and only if $v_{\frac{p-1}{2}}\equiv2A^{\frac{p-1}{4}}\pmod p$ and $p\mid v_{\frac{p-1}{4}}$ if and only if $v_{\frac{p-1}{2}}\equiv-2A^{\frac{p-1}{4}}\pmod p$. \item[(2)] If $p\equiv3\pmod 4$, then $p\mid u_{\frac{p+1}{4}}$ if and only if $v_{\frac{p+1}{2}}\equiv2A^{\frac{p+1}{4}}\pmod p$ and $p\mid v_{\frac{p+1}{4}}$ if and only if $v_{\frac{p+1}{2}}\equiv-2A^{\frac{p+1}{4}}\pmod p$. \end{description} \end{lemma} \begin{proof} (1) and (2) follow from the fact that $v_{2n}=v_n^2-2A^n=Du_n^2+2A^n.$ \end{proof} \begin{lemma}\textup{(\cite{sun4})}\label{uvlucasmod} Let $p$ be an odd prime and $A'$ be an integer such that $4A'\equiv B^2-4A\pmod p $. Let $u_n'=u_n(A',B),\;v_n'=v_n(A',B)$. Then we have \[ u_{\frac{p+1}{2}}\equiv\frac12\left(\frac2p\right)v'_{\frac{p-1}{2}}\pmod p,\;u_{\frac{p-1}{2}}\equiv-\left(\frac2p\right)u'_{\frac{p-1}{2}}\pmod p, \] \[v_{\frac{p+1}{2}}\equiv\left(\frac2p\right)v'_{\frac{p+1}{2}}\pmod p,\;v_{\frac{p-1}{2}}\equiv2\left(\frac2p\right)u'_{\frac{p+1}{2}}\pmod p. \] \end{lemma} \begin{remark} \begin{description} \item[(1)] Let $S_n=u_n(1,4),\;T_n=v_n(1,4)$. For any prime $p>3$, by the facts that $u'_n=u_n(3,4)=\frac{1}{2}(3^n-1)$ and $v_n'=v_n(3,4)=3^n+1$, we have \begin{align*} &S_{\frac{p+1}{2}}\equiv\frac12\left(\frac{2}{p}\right)\left[\left(\frac{3}{p}\right)+1\right]\pmod p,\quad S_{\frac{p-1}{2}}\equiv-\frac12\left(\frac{2}{p}\right)\left[\left(\frac{3}{p}\right)-1\right]\pmod p,\\ &T_{\frac{p+1}{2}}\equiv \left(\frac{2}{p}\right)\left[3\left(\frac{3}{p}\right)+1\right]\pmod p,\quad T_{\frac{p-1}{2}}\equiv\left(\frac{2}{p}\right)\left[3\left(\frac{3}{p}\right)-1\right]\pmod p. \end{align*} Thus by Lemma \ref{ulucasmod}, $p\mid S_{[\frac{p+1}{4}]}$ iff $p\equiv1,19\pmod {24}$ and $p\mid T_{[\frac{p+1}{4}]}$ iff $p\equiv7,13\pmod {24}$. Sun \cite{s2002} got these by studying the sum (\ref{generalsum}) for $a=1$ and $m=12$; \item[(2)] Let $P_n=u_n(-1,2),\;Q_n=v_n(-1,2)$ and $u_n'=u_n(2,2),\;v_n=v_n'(2,2).$ For any odd prime $p$, by the facts that $u_{4n}'=0,u_{4n+1}'=(-4)^n,u_{4n+2}'=u_{4n+3}'=2(-4)^n$ and $v_{4n}'=v_{4n+1}'=2(-4)^n,u_{4n+2}'=0,u_{4n+3}'=(-4)^{(n+1)}$, we have \[P_{\frac{p-\left(\frac2p\right)}{2}}\equiv\begin{cases}0\pmod p,& \textup{if}\,p\equiv1\pmod 4,\\(-1)^{[\frac{p+5}{8}]}2^{\frac{p-3}{4}}\pmod p,& \textup{if}\;p\equiv3\pmod 4,\end{cases} \] \[Q_{\frac{p-\left(\frac2p\right)}{2}}\equiv\begin{cases}(-1)^{[\frac{p}{8}]}2^{\frac{p+3}{4}}\pmod p,& \textup{if}\;p\equiv1\pmod 4,\\0\pmod p,& \textup{if}\;p\equiv3\pmod 4,\end{cases}\] and \[P_{\frac{p+\left(\frac2p\right)}{2}}\equiv(-1)^{[\frac{p+1}{8}]}2^{[\frac{p }{4}]}\pmod p,\quad Q_{\frac{p+\left(\frac2p\right)}{2}}\equiv(-1)^{[\frac{p+5}{8}]}2^{[\frac{p+5}{4}]}\pmod p.\] \end{description} Sun got \cite{sun2} these by studying the sum (\ref{generalsum}) for $a=1$ and $m=8$. \end{remark} \begin{lemma}\label{uv1lucasmod} Let $p\nmid B$ be an odd prime and $A'$ be an integer such that $A'\equiv \frac{A}{B^2}\pmod p $. Let $u_n'=u_n(A',1),\;v_n'=v_n(A',1).$ Then we have \[u_{\frac{p+1}{2}}\equiv\left(\frac Bp\right)u'_{\frac{p+1}{2}}\pmod p,\quad u_{\frac{p-1}{2}}\equiv\frac1B\left(\frac Bp\right)u_{\frac{p-1}{2}}'\pmod p,\] \[v_{\frac{p+1}{2}}\equiv B\left(\frac Bp\right)v'_{\frac{p+1}{2}} \pmod p,\quad v_{\frac{p-1}{2}}\equiv \left(\frac Bp\right)v_{\frac{p-1}{2}}'\pmod p.\] \end{lemma} \begin{proof} By Lemma 2.1 of \cite{dy} and $D'=1-4A'\equiv\frac{D}{B^2}\pmod p,$ we have \begin{align*} u_{n} &=2\sum_{\substack{k=0\\k\;odd}}^{n}\binom{n}{k}\left(\frac{B}2\right)^{n-k}\left(\frac{D}2\right)^{\frac{k-1}{2}}\\ &=2B^{n-1}\sum_{\substack{k=0\\k\;odd}}^{n}\binom{n}{k}\left(\frac{1}2\right)^{\frac{p-1}{2}-k}\left(\frac{D}{2B^2}\right)^{\frac{k-1}{2}}\\ &\equiv2B^{n-1}\sum_{\substack{k=0\\k\;odd}}^{n}\binom{n}{k}\left(\frac{1}2\right)^{n-k}\left(\frac{D'}{2}\right)^{\frac{k-1}{2}}\\ &=B^{n-1}u'_n\pmod p, \end{align*} and \begin{align*} v_{n} &=2\sum_{\substack{k=0\\k\;even}}^{n}\binom{n}{k}\left(\frac{B}2\right)^{n-k}\left(\frac{D}2\right)^{\frac{k}{2}}\\ &=2B^{n}\sum_{\substack{k=0\\k\;even}}^{n}\binom{n}{k}\left(\frac{1}2\right)^{\frac{p-1}{2}-k}\left(\frac{D}{2B^2}\right)^{\frac{k}{2}}\\ &\equiv2B^{n}\sum_{\substack{k=0\\k\;even}}^{n}\binom{n}{k}\left(\frac{1}2\right)^{n-k}\left(\frac{D'}{2}\right)^{\frac{k}{2}}\\ &=B^{n}v'_n\pmod p, \end{align*} Thus \begin{align*} &u_{\frac{p+1}{2}}\equiv B^{\frac{p-1}{2}}u'_{\frac{p+1}{2}}\equiv\left(\frac Bp\right)u'_{\frac{p+1}{2}}\pmod p,\\ & u_{\frac{p-1}{2}}\equiv B^{\frac{p-3}{2}}u'_{\frac{p-1}{2}}\equiv\frac1B\left(\frac Bp\right)u_{\frac{p-1}{2}}'\pmod p, \\ &v_{\frac{p+1}{2}}\equiv B^{\frac{p+1}{2}}v'_{\frac{p+1}{2}}\equiv B\left(\frac Bp\right)v'_{\frac{p+1}{2}} \pmod p,\\ & v_{\frac{p-1}{2}}\equiv B^{\frac{p-1}{2}}v'_{\frac{p-1}{2}}\equiv \left(\frac Bp\right)v_{\frac{p-1}{2}}'\pmod p. \end{align*} \end{proof} \begin{theorem} \label{52lucas} Let $p\neq5$ be an odd prime and $\{U_n\}_{n\geq0}$ and $\{V_n\}_{n\geq0}$ be the Lucas sequences defined as $$U_0=0,U_1=1,U_{n+1} =2U_{n}-5U_{n-1} \;\textup{for}\;n\geq1;$$ $$V_0=2,V_1=2,V_{n+1}=2V_{n}-5V_{n-1}\;\textup{for}\;n\geq1.$$ \begin{description} \item[(1)] If $p\equiv\pm1\pmod 5$, we have \begin{align*} &U_{\frac{p+\left(\frac{-1}{p}\right)}{2}}\equiv\left(\frac{-1}{p}\right)(-1)^{[\frac{p+5}{10}]}5^{[\frac{p}{4}]}\pmod p,\\ &U_{\frac{p-\left(\frac{-1}{p}\right)}{2}}\equiv0\pmod p,\\ &V_{\frac{p+\left(\frac{-1}{p}\right)}{2}}\equiv2(-1)^{[\frac{p+5}{10}]}5^{[\frac{p}{4}]} \pmod p,\\ &V_{\frac{p-\left(\frac{-1}{p}\right)}{2}}\equiv2(-1)^{[\frac{p+5}{10}]}5^{[\frac{p+1}{4}]}\pmod p. \end{align*} \item[(2)] If $p\equiv\pm2\pmod 5$, we have \begin{align*} &U_{\frac{p+\left(\frac{-1}{p}\right)}{2}}\equiv\frac12\left(\frac{-1}{p}\right)(-1)^{[\frac{p+5}{10}]}5^{[\frac{p}{4}]}\pmod p,\\ &U_{\frac{p-\left(\frac{-1}{p}\right)}{2}}\equiv\frac12\left(\frac{-1}{p}\right)(-1)^{[\frac{p+5}{10}]}5^{[\frac{p+1}{4}]}\pmod p,\\ &V_{\frac{p+\left(\frac{-1}{p}\right)}{2}}\equiv4(-1)^{[\frac{p-5}{10}]}5^{[\frac{p}{4}]} \pmod p,\\ &V_{\frac{p-\left(\frac{-1}{p}\right)}{2}}\equiv0\pmod p. \end{align*} \end{description} \end{theorem} \begin{proof} Let $F_n=u_n(-1,1)$ and $L_n=v_n(-1,1)$ be Fibonacci sequence and its companion. Then by Lemmas \ref{uvlucasmod} and \ref{uv1lucasmod}, we have \[U_{\frac{p+1}{2}}\equiv\frac12L_{\frac{p-1}{2}}\pmod p, \quad U_{\frac{p-1}{2}}\equiv-\frac12F_{\frac{p-1}{2}}\pmod p,\] \[V_{\frac{p+1}{2}}\equiv2L_{\frac{p+1}{2}}\pmod p,\quad V_{\frac{p-1}{2}}\equiv2F_{\frac{p+1}{2}}\pmod p.\] Thus by Corollaries 1 and 2 of \cite{ss}, we can derive the results. \end{proof} \begin{remark} In \cite{dy}, we gave some congruences for the Lucas quotient $ U_{p-\left(\frac{-1}p\right)}/p$ by studying the sum (\ref{generalsum}) for $a=-2$ and $m=4$. \end{remark} \begin{corollary} Let $p\neq5$ be an odd prime, $\{U_n\}_{n\geq0}$ and $\{V_n\}_{n\geq0}$ be Lucas sequences defined as above. \begin{description} \item[(1)] If $p\equiv1\pmod4$, then $p\mid U_{\frac{p-1}{4}}$ if and only if $p\equiv1\pmod {20}$ and $p\mid V_{\frac{p-1}{4}}$ if and only if $p\equiv9\pmod {20}$. \item[(2)] If $p\equiv3\pmod4$, then $p\mid U_{\frac{p+1}{4}}$ if and only if $p\equiv19\pmod {20}$ and $p\mid V_{\frac{p+1}{4}}$ if and only if $p\equiv11\pmod {20}$. \end{description} \end{corollary} \begin{proof} (1) and (2) follow from Lemma \ref{ulucasmod} and Theorem \ref{52lucas}. \end{proof} \noindent \textbf{Acknowledgments}\quad The work of this paper was supported by the NNSF of China (Grant No. 11471314), and the National Center for Mathematics and Interdisciplinary Sciences, CAS. \end{document}

\begin{document} \preprint{\includegraphics[scale=0.4]{iqus_logo.png} $\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ Preprint number: IQuS@UW-21-054} \hspace{0.1cm} \title{Quantum Imaginary Time Propagation algorithm for preparing thermal states} \newcommand{\iqusfil}{InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington, Seattle, Washington 98195, USA} \author{Francesco~Turro \orcidlink{https://orcid.org/0000-0002-1107-2873}} \affiliation{\iqusfil} \begin{abstract} Calculations at finite temperatures are fundamental in different scientific fields, from nuclear physics to condensed matter. Evolution in imaginary time is a prominent classical technique for preparing thermal states of quantum systems. We propose a new quantum algorithm that prepares thermal states based on the quantum imaginary time propagation method, using a diluted operator with ancilla qubits to overcome the non-unitarity nature of the imaginary time operator. The presented method is the first that allows us to obtain the correct thermal density matrix on a general quantum processor for a generic Hamiltonian. We prove its reliability in the actual quantum hardware computing thermal properties for two and three neutron systems. \end{abstract} \maketitle Calculations at finite temperature are essential for understanding quantum systems across scientific fields. In particular, the thermodynamic properties of nuclear matter play a crucial role in heavy-ion collisions, astrophysics, and general nuclear applications. Some examples are nuclear reactions in the evolution of matter in the early universe and inside the core of stars~\cite{RevModPhys_83_195,annurev-astro-081811-125543,annurev-nucl-020620-063734}, supernova explosions and the phase diagram of QCD~\cite{de2010simulating,shuryak2017strongly}. The recent detection of gravitational waves~\cite{abbott2017gw170817} can provide constraints on the equation of state of nuclear matter at high densities, used, for example, to describe the composition of Neutron Stars~\cite{burgio2021neutron,lattimer2021neutron,haensel2007neutron}. According to statistical mechanics, the idealization of the density matrix that describes the quantum system with a thermal bath at temperature $T$, and the thermal expectation value of an observable $O$, are given by \begin{equation} \rho = e^{-\beta H}\,, \qquad \langle O \rangle = \frac{\Tr{\rho O}}{Z_0} \,, \label{eq:thermal_states} \end{equation} where $H$ is the Hamiltonian of the system, $\beta=\frac{1}{k_B T}$, $k_B$ the Boltzmann constant, and $Z_0=\Tr{\rho}$ is the partition function. The imaginary time propagation (ITP) method is a popular classical algorithm to prepare a thermal state of a quantum system. This algorithm was originally designed to prepare the ground state of quantum systems, where one dissipates an arbitrary quantum state to reach the ground state using the ITP operator, $e^{-\tau\, H}$, where $\tau$ is the imaginary time. Well-known classical techniques for computing thermodynamic properties are Quantum Monte Carlo methods and their improvements, like Auxiliary Field Monte Carlo \cite{baeurle2002field}, Continuous-time Quantum Monte Carlo \cite{gull2011continuous} and Path Integral Monte Carlo \cite{barker1979quantum}. It is a notorious problem that the required classical computational resources grow exponentially with the number of particles. Moreover, many nuclear systems are mainly composed of fermions. Hence, the fermion sign problem emerges, slowing down the progress in studying complex systems~\cite{troyer2005}. Following Feynman's idea on the efficiency in simulating quantum systems using quantum hardware~\cite{feynman}, it is desired to develop quantum algorithms that efficiently compute the thermodynamic properties of quantum systems. Different quantum algorithms that implement imaginary time methods and thermal state preparation have been proposed~\cite{motta2020determining,mcardle2019variational,warren2022adaptive,holmes2022quantum,sagastizabal2021variational,consiglio2023variational,turro2022imaginary}. Refs.~\cite{motta2020determining,mcardle2019variational} presents a hybrid quantum-classical algorithm, called Quantum Imaginary Time Evolution (QITE), based on variational ansatz. Applications of the QITE method for thermal states can be found in Ref.~\cite{motta2020determining,davoudi2022toward,sun2021quantum,getelina2023adaptive,wang2023critical,leadbeater2023nonunitary}. Instead, Ref.~\cite{turro2022imaginary} illustrates a quantum algorithm for implementing the imaginary time operator using ancilla qubits with a unitary operator. This letter presents the first quantum algorithm (to the best of our knowledge) that allows us to obtain the correct density matrix on quantum hardware. This is done by implementing a modified version of the QITP operator of Ref.~\cite{turro2022imaginary}. Moreover, the proposed quantum algorithm is independent of the initial variational ansatz and of the set of classical variables that one has using QITE. In particular, this algorithm can be implemented in studying phase transition, because QITE may become prohibitively expensive due to the large correlation length. In this letter, we start from the work of Ref.~\cite{turro2022imaginary}, presenting a quantum algorithm that prepares the thermal state implementing the imaginary time propagation. We also upgrade the QITP algorithm of Ref.~\cite{turro2022imaginary}, improving success probability. We implemented the proposed algorithm in computing the partition function $Z_0$ for spin systems of two and three neutrons in the IBM~\cite{ibm} and Quantinuum H$1$-$1$ quantum hardware~\cite{quantinuum}. We also evaluate the thermal expectation value of some observables. The obtained results are compatible with the analytical behavior. \textit{Quantum algorithm. } Our algorithm starts with the initialization of $n_s$ qubits, where we map our physical system, in the so-called maximally mixed state whose density matrix is given by $\mathbb{1}/ 2^{n_s} $ ($\mathbb{1}$ indicates the $2^{n_s}\times 2^{n_s}$ identity matrix). The first proposal was discussed in Ref.~\cite{white2009minimally}, using $n_s$ ancilla qubits (details and our implementation can also be found in App.~A). For small quantum systems, $2-4$ qubits, the ancilla cost is not expensive, but this can be a limitation for bigger systems. However, this initialization is not strictly necessary, and improvements can be implemented, like minimally entangled typical thermal states \cite{white2009minimally,stoudenmire2010minimally} and Canonical Thermal Pure Quantum State~\cite{sugiura2013canonical,sugiura2012thermal}. After the initialization of the state in the quantum processor, we should implement the imaginary time operator. We start by summarizing the basic steps of the algorithm in QITP~\cite{turro2022imaginary}. Being the imaginary time operator non-unitary, one could work in a diluted Hilbert space to employ a unitary form. Explicitly, we add an ancilla qubit in the $\ket{0}$ state. Then, we implement the following operator \begin{equation} QITP_{gs} = \begin{pmatrix} \frac{e^{-\tau (H-E_T)}}{\sqrt{1+e^{-2\,\tau (H-E_T)}}} & \frac{1}{\sqrt{1+e^{-2\,\tau (H-E_T)}}}\\ \frac{-1}{\sqrt{1+e^{-2\,\tau (H-E_T)}}} & \frac{e^{-\tau (H-E_T)}}{\sqrt{1+e^{-2\,\tau (H-E_T)}}}\\ \end{pmatrix} \,, \end{equation} where $E_T$ indicates the so-called trial energy, an algorithm parameter that should be tuned (see Ref.~\cite{turro2022imaginary} for more details). Then, after the action of the $QITP_{gs}$ operator and measuring the ancilla qubit in $\ket{0}$, the system state is closer to the ground state than the initial one due to the action of $\frac{e^{-\tau (H-E_T)}}{\sqrt{1+e^{-2\,\tau (H-E_T)}}}$ operator. We have to modify the form of the $QITP_{gs}$ operator such that, after measuring the ancilla qubit $\ket{0}$, we get the correct form of thermal state $e^{-\beta H}$. A straightforward modification is described by \begin{equation} QITP_{th} = \begin{pmatrix} \sqrt{p}\,e^{-\tau (H-E_T)} & \sqrt{ 1- p\,e^{-2\tau (H-E_T)}}\\ -\sqrt{ 1-p\, e^{-2\tau (H-E_T)}} &\sqrt{p}\,e^{-\tau (H-E_T)}\\ \end{pmatrix} \,,\label{eq:thermal_QITP_op} \end{equation} where $0<p\le 1$ is a free parameter, representing the success probability in measuring the ancilla qubit in the $\ket{0}$ state in the limit of $\tau \xrightarrow[]{}0 $. For $E_T\le E_0$ ($E_0$ represents the ground energy), we can also set $p=1$, which removes the exponential decay of the success probability. Details for implementing this ITP operator can be found in App.~B. This form requires the least ancilla qubit number (only one) for implementing $QITP_{th}$. Blocking encoding and qubitization~\cite{low2019hamiltonian,low2017optimal,tang2023cs} can be explored to compile the imaginary time operator $e^{-\beta H}$ using more than one ancilla qubit. After implementing the operator in Eq.~\eqref{eq:thermal_QITP_op} and measuring the ancilla qubit in the $\ket{0}$ state, we find that the physical qubits are in the correct thermal state (a demonstration can be found in App.~C). The steps of the proposed algorithm to prepare thermal states in quantum processors are as follows: \begin{enumerate} \item Start with all the qubits in the $\ket{0}$. \item Implement the gates of the dashed square in Fig.~\ref{fig:qc_classical} (Its action gives us the physical system qubits in $\mathbb{1}/2^{n_s}$ state). \item Employ the the $QITP_{th}$ operator of Eq.~\eqref{eq:thermal_QITP_op} with a additional ancilla qubit, setting $\tau=\frac{\beta}{2}$. \item Measure the ancilla qubit in $\ket{0}$. \end{enumerate} In the worst case scenario, the required number of qubits is $2\, n_s+N_\beta$, where $n_s$ qubits are needed to map the system, $n_s$ ancilla qubits to prepare the maximally mixed state, and $N_\beta$ additional qubits to apply the $QITP_{th}$ operator. Moreover, using more ancilla qubits, one can apply the Trotter decomposition to simplify the compilation of the $QITP_{th}$. The success probability $P_s$ of the proposed algorithm (equal to the probability to measure all the ancilla qubit in $\ket{0}$) for a perfect noiseless quantum computer is given by \begin{equation} P_s=\frac{1}{2^{n_s}} \Tr\left[ p^{N_\beta}\,e^{-\beta (H-E_T)}\right] =\frac{p^{N_\beta}}{2^{n_s}} Z(\beta) \,. \end{equation} Setting $p=1$, the success probability is proportional to the partition function, $Z(\beta)=\Tr\left[ e^{-\beta (H-E_T)}\right]$. However, in the worst case scenario, for $\beta \xrightarrow[]{}+\infty$, the success probability decays as $\frac{1}{2^{n_s}}$. This is mostly related to the fact that the state of the quantum system is mostly in the ground state. A solution to this decay could be the application of the amplitude amplification method~\cite{brassard2002quantum} to enhance the success probability. \begin{figure} \caption{Quantum circuits for preparing the thermal state in quantum processors. The dashed square shows the gates for initializing the system qubits on the maximally entangled mixed state.} \label{fig:qc_classical} \end{figure} \begin{figure} \caption{Feynman diagrams of the Leading Order of Chiral Effective Field Theory} \label{fig:feynman} \end{figure} \textit{Results} As a first test, we prepare the thermal state of a spin system of two neutrons fixed in their position. The considered interaction is the spin-dependent part of the leading order of chiral effective field theory~\cite{Chiral_review1,Chiral_review2,tews2016quantum}, using the parameters of Ref.~\cite{holland2020optimal}. The Feynman diagrams of such interaction is shown in Fig.~\ref{fig:feynman}. This system can be mapped to two qubits. We implement the thermal $QITP_{th}$ quantum algorithm for different values of $\beta$ using five qubits (two for the system, two for preparing the maximally mixed state and one for implementing $QITP_{th}$). The employed quantum circuits in this work are built with the \texttt{Qiskit} package~\cite{qiskit} and compiled with the \texttt{pytkey} package~\cite{pytket_paper} for the Quantinuum machine. Additionally, we also compute the expectation value of the $\sigma_z$ for the first neutron using the \texttt{H1-1} hardware (adding an extra qubit). Our procedure is described in App.~D. Panel (a) of Fig.~\ref{fig:2n_partition_function} shows our results for the partition function, $Z_0=\Tr{e^{-\beta (H-E_T)}}$. In our tests, we set $E_T$ equal to the ground energy $E_0$ and $p$ in Eq.~\eqref{eq:thermal_QITP_op} equal to $0.8$ to diagnose the algorithm in the non-optimal situation. The dashed line represents the analytical values of the partition function (obtained by classically computing the thermal density matrix and tracing it). The magenta circles and orange squares represent the results obtained from \texttt{H1-1} Quantinuum using both 200 shots. The quantum circuits for the squares compute the partition function and the expectation value, instead, for the circles only the partition function. In the same panel (a), the results from the IBM \texttt{ibmq\textunderscore manila} (green diamonds) and \texttt{ibmq\textunderscore quito} (pink triangles) are shown as well. While we do not implement any error mitigation methods on the Quantinuuum results, on the IBM hardware we employ the randomizing compiling technique~\cite{wallman2016noise,hashim2021randomized}, using 8 randomized quantum circuits with 64000 total shots (8$\times$8000). Our results are compatible (within two sigma) with the analytical partition function values. Panel (b) of Fig.~\ref{fig:2n_partition_function} presents the results for the expectation value of $\sigma_z$ (orange squares) using the same obtained probabilities of the panel (a) with different analysis (see App.~D). The dashed line represents the analytical curve. Also, in this case, the obtained results are compatible with the analytical values. However, with the increase of $\beta$, the error bars get larger due to statistical error because the probability of measuring the ancilla qubits in $\ket{0}$ decreases. A good solution would be to run the quantum circuits with more shots, reducing the statistical errors. Nevertheless, these big error bars are mostly caused by how we compile the hermitian operator. Hence, different strategies, like decomposing in the sum of Pauli matrices, can reduce them. A discussion about tests of these two different compiling methods can be found at the end of App.~D. \begin{figure}\label{fig:2n_partition_function} \end{figure} The next test is preparing thermal states for a spin system of three neutrons. The used Hamiltonian is given by the sum of three two-body terms, ignoring a three-body potential. We start by evaluating the partition function as a function of $\beta$ implementing a single $\beta$ step. Additionally, we add an extra ancilla qubit to evaluate the expectation value of the Hamiltonian $H$. The final quantum circuits use seven qubits for preparing the thermal state and an additional one to evaluate the expectation value of the Hamiltonian. Panel (a) and panel (b) of Fig.~\ref{fig:3n_single_trotter} present the results for the partition function and the expectation value of the Hamiltonian, respectively. The orange squares, brown circles, and violet diamonds indicate the Quantinuum, the IBM \texttt{ibmq\textunderscore nairobi} and \texttt{ibmq\textunderscore olso} results, respectively. The dashed line represents the analytical curves. Even though employing randomizing compiling, we observe the IBM results are noisier than the Quantinuum ones. Indeed, this can be easily explained by the required implementation of swap gates to correctly compile the $QITP_{th}$ operator due to the non-all-to-all connectivity on IBM hardware. The addition of swap gates increases the depth of quantum circuits and the contribution of noise. Nevertheless, our results are still compatible with the analytical values at two sigma. One can observe the same increasing behavior of the error bars for the expectation value. Therefore, we can reduce the error bar with a more number of shots. \begin{figure}\label{fig:3n_single_trotter} \end{figure} \begin{figure}\label{fig:3n_trotterdecomposition} \end{figure} We also test the proposed quantum algorithm implementing the Trotter decomposition for preparing the thermal states of the three neutron spin systems. Specifically, we split $\beta$ in smaller steps and the full thermal propagators into a product of the two-body propagators. Consequently, each layer of the quantum circuit employs a single two-body contribution. Panel (a) of Fig.~\ref{fig:3n_trotterdecomposition} shows the results obtained from the \texttt{H1-1} Quantinuum processor with orange squares. The dashed line represents the analytical curve applying the full Hamiltonian, and the black stars indicate the analytical values applying the Trotter decomposition. Also, we obtain full compatibility with the analytical values in this case. In the runs, we duplicate the number of shots from $200$ to $400$. In the same figure, we also report the total number of used qubits and implemented CNOT gates for the different $\beta$ values in the lower table. We also compute the expectation value of $\sigma_z$ of the first neutron as a function of $\beta$ with the same implemented quantum circuit. as shown in panel (b) of Fig.~\ref{fig:3n_trotterdecomposition}. We observe the same behavior for error bars of Fig.~\ref{fig:2n_partition_function}. A solution may be to increase the number of shots to reduce the error bars. \textit{Conclusions } This work has presented a quantum algorithm that prepares the quantum thermal states in quantum processors for a generic Hamiltonian. This algorithm is based on the quantum imaginary time propagation method using ancilla qubits to get a unitary version of the imaginary time operator. Upgrading of the quantum imaginary time propagation in Ref.~\cite{turro2022imaginary} has been reported, solving the decaying of the success probability. We have discussed the proposed algorithm's validity and reliability in the present quantum hardware. Indeed, we have reported our tests in computing thermal expectation values for simple spin nuclear systems. These simulations have been implemented in different quantum processor hardware, ion-trap (Quantinuum) and superconducting devices (IBM quantum processors). The obtained results are compatible with the analytical values proving the reliability of the proposed quantum algorithm, also in the presence of deep quantum circuits, especially using the \texttt{H1-1} Quantinuum processor. This algorithm and the algorithm of Ref.~\cite{motta2020determining} provide a good starting point in preparing thermal states using imaginary time methods on quantum processors. With the leveraging of quantum processors, these algorithms can be applied to solve today's hard problems in nuclear physics, thermalization problems in Quantum Chromodynamics, quantum chemistry, condensed matter applications, and other fields. \textit{Acknowledgments} We thank Francesco Pederiva, Alessandro Roggero and the whole IQuS group for useful discussions. In particular, in the IQuS group, we are grateful to Marc Illa Subi\~{n}\`{a}, Anthony Ciavarella, and Martin Savage for the support and correction of the text. This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, InQubator for Quantum Simulation (IQuS) (\url{https://iqus.uw.edu}) under Award Number DOE (NP) Award DE-SC0020970 via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. We acknowledge the use of Quantinnum and IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. A discussion session inspired this work at the "Next-Generation Computing for Low-Energy Nuclear Physics: from Machine Learning to Quantum Computing" IQuS workshop in August 2022. The obtained results are shown in App.~E. The reported data are obtained from simulations run in February 2023. \appendix \section{Appendix A: Preparation of the maximally mixed state}\label{app:Classicalstate_proof} This section describes a method to obtain the maximally mixed state. We start adding ancilla qubits for each system qubit. All the $2\,n_s$ qubits are initialized in $\ket{0}$ state. Applying the Hadamard gate and CNOT gate, as shown in Fig.~\ref{fig:qc_classical}, and measuring the ancilla qubits, we will obtain the maximally mixed state $\rho=\frac{\mathbb{1}}{2^{n_s}}$. We will prove this procedure for a single qubit system. After the action of the H and CNOT gates to the state $\ket{00}$, we get \begin{equation} \rho = \text{CNOT}\,H\ket{00}\bra{00}H\,\text{CNOT}= \begin{pmatrix} \frac{1}{2} & 0 & 0& \frac{1}{2} \\ 0 & 0 & 0&0\\ 0 & 0 & 0&0\\ \frac{1}{2} & 0 & 0& \frac{1}{2} \\ \end{pmatrix}\,. \end{equation} Now, we eliminate the ancilla qubit, for example, measuring it (without interest in the outcome). In mathematical terms, this corresponds to the partial trace. Hence, by doing it, the system state becomes \begin{equation} \rho^1_{cl}=\Tr_{ancilla}\left[\rho \right]= \begin{pmatrix} \frac{1}{2} & 0\\ 0& \frac{1}{2}\\ \end{pmatrix}\,. \end{equation} Iterating this method for $n_s$ qubits, we obtain our desired $2^n$-maximally mixed state. Indeed, we have \begin{equation} \rho_{MME}= \otimes_{n_s} \left(\rho^1_{cl}\right)_{n_s}= \otimes_{n_s} \left(\frac{\mathbb{1_{2\times2}}}{2} \right)_{n_s}= \frac{1}{2^{n_s}} \mathbb{1}_{2^{n_s}\times2^{n_s}} \end{equation} \section{Appendix B: Implementation of $QITP_{th}$ operator}\label{app:implementation_qitp} This section will discuss how the $QITP_{th}$ operator of Eq.~\eqref{eq:thermal_QITP_op} could be compiled in quantum circuits. We start by diagonalizing the Hamiltonian $H$, \begin{equation} U H U^\dagger = E\,, \label{eq:Hdiagonalization} \end{equation} where $U$ is the eigenstate matrix and $E$ represent the eigenvalue diagonal matrix. Therefore, applying Eq.~\eqref{eq:Hdiagonalization} , the $QITP_{th}$ operator can be rewritten in the following form \begin{widetext} \begin{equation} QITP_{th}\,=\,U\, \begin{pmatrix} \sqrt{p} e^{-\beta (E-E_T)}&\sqrt{1-p\,e^{-2\beta (H-E_T)}}\\ -\sqrt{1-p\,e^{-2\beta (H-E_T)}}&\sqrt{p} e^{-\beta (E-E_T)}\\ \end{pmatrix} \,U^\dagger\,, \end{equation} \end{widetext} where we assume $E_T\le E_0$. The central matrix of the right side of the equation, the matrix with $E$, contains all the physical information. The matrix $U$ is only a change of the computational basis. The central part of $QITP_{th}$ is described by the Cosine Sine decomposition matrix. Hence, this can be decomposed as a product of controlled $R_y$ rotations, where the different angle $\theta_i$ are given by $\theta_i=\arccos\left(\sqrt{p} e^{-\beta(E_i-E_T)}\right)$. Moreover, employing the Grey Code \cite{mottonen2005decompositions,barenco1995elementary,tang2023cs}, this operator can be compiled using $2^{n_s}$ CNOT and $2^{n_s}-1$ $R_y$ gates. Fig.~\ref{fig:Greycode} shows an example of a two-qubit system. \begin{figure} \caption{Example in compiling the $QITP_{th}$ matrix for two qubit system. The upper qubits represent the system ones, and the lower one indicates the ancilla qubit.} \label{fig:Greycode} \end{figure} \section{Appendix C: Demonstration of thermal preparation} This appendix proves that we obtain the correct thermal density matrix described by Eq.~\eqref{eq:thermal_states} by implementing all these steps of the proposed quantum algorithm. We assume that we have got the maximally mixed state $\rho_{MME}=\frac{1}{2^{n_s}} \mathbb{1}$. Applying the $QITP_{th}$ operator and measuring the ancilla in the $\ket{0}$ state we have \begin{widetext} \begin{equation} \begin{split} \rho&=P_0\,QITP_{th}\left(\frac{\beta}{2}\right)\,\begin{pmatrix} \rho_{MME} & 0\\ 0&0 \end{pmatrix} \,\left(QITP_{th}\left(\frac{\beta}{2}\right)\right)^\dagger \,P_0\\ &= P_0 \begin{pmatrix} e^{-\frac{\beta}{2} (H-E_T)} \rho_{MME} e^{-\frac{\beta}{2} (H-E_T)} & e^{-\frac{\beta}{2} (H-E_T)} \rho_{MME} \sqrt{1-e^{-\beta (H-E_T)}} \\ \sqrt{1-e^{-\beta H}} \rho_{MME} e^{-\frac{\beta}{2} (H-E_T)} &\sqrt{1-e^{-\beta (H-E_T)}} \rho_{MME} \sqrt{1-e^{-\beta H}} \end{pmatrix} P_0\\ &= \begin{pmatrix} \frac{1}{2^n} e^{-\frac{\beta}{2} (H-E_T)} \rho_{MME} e^{-\frac{\beta}{2} (H-E_T)} &0\\ 0&0\\ \end{pmatrix} = \frac{1}{2^n} e^{-\beta (H-E_T)} \otimes \ket{0}\bra{0}\,, \end{split} \label{eq:rho_th} \end{equation} \end{widetext} where $P_0$ indicates the projector to the $\ket{0}$ state of the ancilla. We have proved that the density matrix is proportional to the thermal density matrix. \section{Appendix D: Compilation of an observable } \label{app:compi_observ} This appendix presents how we can compile and evaluate the expectation value of an observable $O$ implementing a single quantum circuit. Usually, the observable is hermitian but not a unitary operator. Therefore, we have to transform it into one. Using the same spirit of the compilation of the $QITP_{th}$ operator, we define the operator $A$ \begin{equation} A= \sqrt{\frac{O-\lambda^O_0}{\left|O-\lambda^O_0\right|}}\,, \label{eq:A_definitation} \end{equation} where $\lambda_0$ is the lowest eigenvalue of $O$ and $|O-\lambda^O_0|= \max_{\lambda_i} |\lambda_i-\lambda_0|$. With this transformation, we shrink the spectrum of $O$ to be between 0 and 1 (the unitary condition requires this), keeping the hermiticity of $A$. We follow the same procedure of the $QITP_{th}$ operator, we add an extra ancilla qubit in $\ket{0}$ state and we define the following unitary operator $U_O$ as \begin{equation} U_O=\begin{pmatrix} A & \sqrt{1-A^2}\\ -\sqrt{1-A^2} & A\\ \end{pmatrix}\,. \end{equation} This new operator is unitary and can be used to evaluate the thermal expectation value of $O$. Indeed, the expectation value of $\langle O \rangle$ is given by \begin{equation} \langle O \rangle = \langle A^2 \rangle |O-\lambda^O_0| + \lambda_0\,, \label{eq:thermal_O_as_function_A} \end{equation} where $|O-\lambda^O_0|$ and $\lambda_0$ are defined in Eq.~\eqref{eq:A_definitation}. To demonstrate the validity of this compiling method, we start by assuming we prepare the system qubits in the maximally mixed state, $\rho_{MME}=\frac{\mathbb{1}}{2^n}$. We add two ancilla qubits, one for the $QITP_{th}$ operator and one for the $U_0$ operator. Applying first the $QITP_{th}$ operator and using the result of Eq.~\eqref{eq:rho_th}, we get: \begin{widetext} \begin{equation} \rho_1=\begin{pmatrix} e^{-\frac{\beta}{2} H} \rho_{MME} e^{-\frac{\beta}{2} H} & e^{-\frac{\beta}{2} H} \rho_{MME} \sqrt{1-e^{-\beta H}}& 0 &0\\ \sqrt{1-e^{-\beta H}} \rho_{MME} e^{-\frac{\beta}{2} H} &\sqrt{1-e^{-\beta H}} \rho_{MME} \sqrt{1-e^{-\beta H}} & 0 &0\\ 0 &0 &0 &0\\ 0 &0 &0 &0\\ \end{pmatrix}\,. \end{equation} \end{widetext} Then, using $\rho_{MME}=\frac{\mathbb{1}}{2^{n_s}}$ and $A^\dagger=A$, the action of the operator $U_O$ give us \newcommand\scalemath[2]{\scalebox{#1}{\mbox{\ensuremath{\displaystyle #2}}}} \begin{widetext} \begin{equation} \scalemath{0.75}{ \rho_2=\frac{1}{2^{n_s}} \begin{pmatrix} A e^{-\beta H} A & -A e^{-\frac{\beta}{2} H} \sqrt{1-e^{-\beta H}} A & -A e^{-\beta H} \sqrt{1-A^2}& A e^{-\frac{\beta}{2} H} \sqrt{1-e^{-\beta H}} \sqrt{1-A^2}\\ -A \sqrt{1-e^{-\beta H}} e^{-\frac{\beta}{2} H} A& A \sqrt{1-e^{-\beta H}} \sqrt{1-e^{-\beta H}} A & A \sqrt{1-e^{-\beta H}} e^{-\frac{\beta}{2} H} \sqrt{1-A^2} & -A \sqrt{1-e^{-H \beta}} \sqrt{1-e^{-H \beta}} \sqrt{1-A^2} \\ -\sqrt{1-A^2} e^{-\beta H} A & -\sqrt{1-A^2} e^{-\frac{\beta}{2} H} \sqrt{1-e^{\beta H}} A & \sqrt{1-A^2} e^{-\beta H} \sqrt{1-A^2} & -\sqrt{1-A^2} e^{-\frac{\beta}{2} H} \sqrt{1-e^{-H \beta}} \sqrt{1-A^2} \\ \sqrt{1-A^2} \sqrt{1-e^{-H \beta}} e^{-\frac{\beta}{2} H} A & -\sqrt{1-A^2} \sqrt{1-e^{-H \beta}} \sqrt{1-e^{-H \beta}} A & -\sqrt{1-A^2} \sqrt{1-e^{-H t}} e^{-\frac{\beta}{2} H} \sqrt{1-A^2}& \sqrt{1-A^2} \sqrt{1-e^{-H \beta}} \sqrt{1-e^{-H \beta}} \sqrt{1-A^2}\\ \end{pmatrix} }\label{eq:dm_expec_A}\,. \end{equation} \end{widetext} We observe that the probability of measuring both ancilla qubits in $\ket{0}$, indicated with $P_{00}$, is equal to the numerator part for the thermal expectation value of $A^2$, $\Tr{A^2\,e^{-\beta H}}$. Moreover, the partition function, $Z_0$, is obtained by the sum of $P_{00}$ and the probability of measuring the ancilla qubit for $U_O$ in $\ket{1}$ and QITP ancilla in $\ket{0}$, indicated with $P_{10}$ (the third diagonal element in Eq.~\eqref{eq:dm_expec_A}). Hence, we have \begin{widetext} \begin{equation} \begin{split} P_{00} &= \Tr{A e^{-\frac{\beta}{2} H} \rho_{MME} e^{-\frac{\beta}{2} H} A^\dagger}= \Tr{ A^2 e^{-\beta H} }\\ P_{00}+P_{10} &= \Tr{A e^{-\frac{\beta}{2} H} \rho_{MME} e^{-\frac{\beta}{2} H} A} + \Tr{\sqrt{1-A^2} e^{-\frac{\beta}{2} H} \rho_{MME} e^{-\frac{\beta}{2} H} \sqrt{1-A^2} }\\ & = \Tr{e^{-\frac{\beta}{2} H} \rho_{MME} e^{-\frac{\beta}{2} H}}= \Tr{e^{-\beta H}}=Z_0 \,.\label{eq:thermal_expec} \end{split} \end{equation} \end{widetext} where, in the last line, we used the unitary condition of $U_O$, $\Tr{A \rho A + \sqrt{1-A^2} \rho \sqrt{1-A^2}}=\Tr{\rho}$. Using Eq.~\eqref{eq:thermal_expec}, we can compute the thermal expectation of $\langle A^2 \rangle$ using \begin{equation} \langle A^2 \rangle\,=\,\frac{1}{Z_0} \Tr{ A^2 e^{-\beta H} }\,=\, \frac{P_{00}}{P_{00}+P_{01}} \,. \end{equation} The expectation value of $O$ is recovered via Eq.~\eqref{eq:thermal_O_as_function_A}. \begin{figure*} \caption{Obtained results for uncertainties of different thermal expectations expanding the observable in Pauli operators (dashed lines) and using the presented method (solid lines) as a function of $\beta$ for different positions of the two neutrons. In (a) and (c) panels, we use 200 shots; in panels (b) and (d), 2000. } \label{fig:uncert_pauli_vs_ancilla} \end{figure*} \subsection{Test with the Pauli expansion} As reported in the main text, the experimental error bars of thermal expectation values of observable become bigger with the increase of $\beta$. Hence, at the end of this section, we also report our tests about the difference between implementing the proposed algorithm and expanding the observable in the sum of Pauli operators. In particular, we focus on computing the uncertainties of the expectation values of different observables for the spin system of two neutrons by implementing the two different compiling methods. We evaluate the uncertainties for a single qubit operator, $\sigma_z$, and a two-qubit operator, the Hamiltonian $H$. They are evaluated as a function of $\beta$ for different nuclear Hamiltonians, where we change the position of the two neutrons. The simulations are implemented on the noiseless IBM emulator. Panels (a) and (b) of Fig.~\ref{fig:uncert_pauli_vs_ancilla} show the uncertainties for a single Pauli matrix ($\sigma_z$), panels (c) and (d) of Fig.~\ref{fig:uncert_pauli_vs_ancilla} for the Hamiltonian (a generic two-qubit operator). Solid lines indicate the results with the ancilla method, and dashed lines with the Pauli expansion. In panels (a) and (c), we use 200 shots, in panels (b) and (d), 2000 (for the Pauli expansion, we use 200 or 2000 for each quantum circuit). We observe that the uncertainties for $\sigma_z$ are better for the Pauli expansion. Nevertheless, for a generic single qubit operator, the final uncertainties for the Pauli expansion would generally be twice bigger due to the contribution of the four generators ($\{\mathbb{1},\sigma_x,\sigma_y,\sigma_z\}$). For the two-qubit operator, we notice that the uncertainties of the proposed method (with an extra ancilla) are similar to or smaller than the results obtained from the Pauli expansion. Despite the low uncertainties of the Pauli expansion for a single qubit, we have used the presented compiling method to compute the expectation value because we have saved more computational credits. \section{Appendix E: Obtained Data} The obtained data are shown in the following tables. Tab.~\ref{tab:chi2} reports the reduced chi-square for each simulation. Tab.~\ref{tab:2n_singlestep} for the two neutrons, Tab.~\ref{tab:3n_singlestep} for the three neutrons with a single step, Tab.~\ref{tab:3n_trotter} for the three neutrons implementing the Trotter decomposition. \begin{table*}[h] \centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{$\frac{\chi^2}{N}$}& \multicolumn{4}{|c|}{2 neutrons} &\multicolumn{3}{|c|}{3 neutrons} & Trotter 3 neutrons \\ & \texttt{H1-1} (only $Z_0$)& \texttt{H1-1} (both $Z_0$ and $\sigma_z$)& \texttt{ibmq\textunderscore quito}&\texttt{ibmq\textunderscore manila}& \texttt{H1-1}&\texttt{ibmq\textunderscore olso}&\texttt{ibmq\textunderscore naroibi} & \texttt{H1-1} \\ \hline $Z_0$ & 0.93&0.44&2.2 &1.42&0.62 &0.72 &2.06&0.77\\ $Ob$ & &0.34&&&0.17&&&0.44\\ \hline \end{tabular} \caption{Reduced $\chi^2$ for each simulations} \label{tab:chi2} \end{table*} \begin{table*}[h] \centering \begin{tabular}{|c|cccc|cc|} \hline \multirow{2}{*}{$\beta$} & \multicolumn{4}{|c|}{$Z_0$} &\multicolumn{2}{c|}{$\langle \sigma_z^1\rangle$ } \\ & \texttt{H1-1} &\texttt{ibm\textunderscore manila} &\texttt{ibmq\textunderscore quito} & analytical & \texttt{H1-1} & analytical \\ \hline 0.025 & 2.4(2) & &2.7& & &\\ 0.075 & 1.50(16) & &1.6& & &\\ 0.125 & 1.35(16) & &1.2& & &\\ 0.175 & 1.12(15) & &1.1& & &\\ \hline 0.00 & 3.9(2) & & & 4.0& 0.01(11) &0.00\\ 0.05 & 2.1(2) & & & 2.0& 0.00(17) &0.01\\ 0.10 & 1.40(17) & & & 1.4& 0.1(2) &0.11\\ 0.15 & 1.38(17) & & & 1.1& 0.2(2) &0.20\\ 0.20 & 1.12(16) & & & 1.1& 0.1(3) &0.27\\ \hline 0.00 & & 3.3(5) & 3.1(4) & 4.0& &\\ 0.025 & & 2.9(3) & 2.9(2) & 2.7& &\\ 0.05 & & 2.1(2) & 2.1(2) & 2.0& &\\ 0.075 & & 1.69(14) & 1.73(13) & 1.60& &\\ 0.10 & & 1.60(13) & 1.55(13) & 1.37& &\\ 0.125 & & 1.39(11) & 1.50(12) & 1.23& &\\ \hline \end{tabular} \caption{Results of the two neutron spin system} \label{tab:2n_singlestep} \end{table*} \begin{table*}[h] \centering \begin{tabular}{|c|cccc|cc|} \hline \multirow{2}{*}{$\beta$} & \multicolumn{4}{|c|}{$Z_0$} &\multicolumn{2}{c|}{$\langle H \rangle$ } \\ & \texttt{H1-1} &\texttt{ibmq\textunderscore olso} & \texttt{ibmq\textunderscore nairobi} & analytical & \texttt{H1-1} & analytical \\ \hline 0.005 & 6.3(5) & & & 6.7& 0.0(4) &-2.4\\ 0.015 & 5.4(4) & & & 4.9& -6(4) &-7.2\\ 0.025 & 3.5(4) & & & 3.8& -9(5) &-11.7\\ 0.045 & 2.7(3) & & & 2.5& -17(5) &-19.3\\ 0.065 & 1.8(3) & & & 1.9& -21(5) &-24.7\\ \hline 0.00 & & 6.8(9) & 7.3(6) & 8.0& &\\ 0.01 & & 5.4(6) & 5.8(5) & 5.7& &\\ 0.02 & & 4.7(8) & 4.9(7) & 4.3& &\\ 0.03 & & 4.2(6) & 4.3(4) & 3.4& &\\ 0.04 & & 3.4(4) & 3.9(4) & 2.7& &\\ 0.05 & & 3.2(4) & 3.6(4) & 2.3& &\\ \hline \end{tabular} \caption{Results of the three neutron spin system without applying the Trotter decomposition} \label{tab:3n_singlestep} \end{table*} \begin{table*}[h] \centering \begin{tabular}{|c|cc|cc|} \hline \multirow{2}{*}{$\beta$} &\multicolumn{2}{|c|}{$Z_0$} &\multicolumn{2}{|c|}{$\langle \sigma_z^1\rangle$ }\\ &\texttt{H1-1} & analytical &\texttt{H1-1} & analytical\\ \hline 0.000& 8.0(3)&8.0&0.03(6)&0.00\\ 0.005& 5.6(3)&5.9&-0.15(10)&-0.03\\ 0.01& 5.0(4)&4.5&-0.05(13)&-0.06\\ 0.015& 3.1(4)&3.6&0.0(2)&-0.08\\ 0.02& 2.8(4)&2.9&0.0(3)&-0.10\\ \hline \end{tabular} \caption{Results of the three neutron spin system without using the Trotter decomposition} \label{tab:3n_trotter} \end{table*} \end{document}

\begin{document} \title{Lateral interatomic dispersion forces} \author{Pablo Barcellona} \email{[email protected]} \affiliation{Physikalisches Institut, Albert-Ludwigs-Universit\"at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany} \author{Robert Bennett} \affiliation{Physikalisches Institut, Albert-Ludwigs-Universit\"at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany} \affiliation{School of Physics \& Astronomy, University of Glasgow, Glasgow, G12 8QQ, United Kingdom} \author{Stefan Yoshi Buhmann} \affiliation{Physikalisches Institut, Albert-Ludwigs-Universit\"at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany} \date{\today} \begin{abstract} Van der Waals forces between atoms and molecules are universally assumed to act along the line separating them. Inspired by recent works on effects which can propel atoms parallel to a macroscopic surface via the Casimir--Polder force, we predict a lateral van der Waals force between two atoms, one of which is in an excited state with non-zero angular momentum and the other is isotropic and in its ground state. The resulting force acts in the same way as a planetary gear, in contrast to the rack-and-pinion motion predicted in works on the lateral Casimir--Polder force in the analogous case, for which the force predicted here is the microscopic origin. We illustrate the effect by predicting the trajectories of an excited caesium in the vicinity of ground-state rubidium, finding behaviour qualitatively different to that if lateral forces are ignored. \end{abstract} \maketitle Descriptions of macroscopic phenomena are often informed and improved by understanding the underlying microscopic processes. Examples are found throughout condensed matter physics, for instance the BCS theory of superconductivity \cite{Bardeen1957} or the Lifshitz theory of Casimir forces \cite{E.M.Lifshitz1956}. The latter explains Casimir's original result \cite{Casimir1948} for the attraction between two perfectly conducting parallel plates in terms of correlations between the fluctuating charge distributions of their elementary atomic constituents. This is part of a broad class of phenomena known as dispersion interactions (c.f. \cite{Buhmann2012BothBooks}), the most familiar being the Van der Waals force between two neutral atoms. Closely related to this is the Casimir--Polder force \new{that a} neutral atom feels in proximity to a material body. In recent years, lateral Casimir (surface--surface) and Casimir--Polder (atom--surface) \footnote{In this work we refer to atom-atom forces at all distances as van der Waals forces (in contrast to some authors who refer to the long-distance atom-atom interaction as a Casimir-Polder force), and all atom-surface forces as Casimir-Polder forces. This should not be taken as denial of the fact that Casimir and Polder derived atom-atom \emph{and} atom surface forces at all distances in their seminal paper \cite{Casimir1948a}.} forces have received attention due to their potential to realise contactless force transmission \cite{Ashourvan2007,Nasiri2012}, as well as novel types of sensors and clocks \cite{Miri2008}. All of these works rely on corrugated surfaces \cite{Messina2009,Dalvit2008,Chen2002,Rodrigues2006,Emig2003,Dobrich2008}, gratings \cite{Lambrecht2008,Contreras-Reyes2010,Bender2014,Buhmann2016}, or gyrotopic response \cite{Polevoi1985}. A number of more recent works have discussed the intriguing possibility of engineering modes propagating along a flat, featureless planar interface \cite{Rodriguez-Fortuno2013,LeKien2016,Mueller2013,Xi2013,Lin2013,Neugebauer2014,Manjavacas2017} or nanofiber \cite{Petersen2014} in such a way that an atom or second object placed nearby will feel a force dragging it along the surface. In this Letter we will reveal the microscopic origins of this latter force. The resonant Casimir--Polder (CP) force on an atom can be expressed in terms of the dyadic Green's tensor $\overbar{\tens{G}}\left(\mathbf{r},\mathbf{r}',\omega \right)$ describing propagation of electromagnetic waves of frequency $\omega$ from point $\B{r}'$ to $\B{r}$ subject to boundary conditions imposed by material geometry. For a two-level atom at position $\B{r}_\text{A}$ with time-dependent excited-state occupancy $p(t)$ \new{it} is given by \cite{Scheel2015,OudeWeernink2018} \begin{align} \textbf{F}^\text{res}(\textbf{r}_\text{A},t) &= 2\mu _0 p(t) \omega_\text{A}^2 \notag \\ &\quad \times \text{Re}\Big[ \nabla \textbf{d}_{10}^\text{A} \cdot \overbar{\tens{G}}\left( \textbf{r},\textbf{r}_\text{A},\omega_\text{A} \right) \cdot \textbf{d}_{01}^\text{A} \Big]_{\textbf{r} = \textbf{r}_\text{A}}, \label{CasPol} \end{align} where $\omega_\text{A}$ is the transition frequency and $ \textbf{d}_{01}^\text{A}=\textbf{d}_{10}^{\text{A}*} $ is the (complex) transition dipole moment from the upper to lower level, and $\mu_0$ is the permeability of free space. \new{There is also a non-resonant force originating in the contribution from photons with frequencies different to the atomic transition, but as shown in the Supplementary Material \footnote{See Supplementary Material [url] for detailed derivations of the forces, emission rates and emission spectra.} the contribution of this for the parameters we will choose is negligible compared to the resonant terms.} Most derivations of Casimir--Polder forces proceed by finding the position-dependent energy shift of the atomic levels, then taking a spatial derivative to find the force. If the atom has a complex polarisability (and corresponding complex dipole moment) then the Casimir--Polder force is not conservative, meaning that it cannot be derived as the gradient of an energy shift. We seek a microscopic version of the non-conservative force given by Eq.~\eqref{CasPol}, which was derived from the Lorentz force law. From a microscopic point of view, a macroscopic medium is a collection of a large number of atoms --- the imposition of macroscopic boundary conditions is simply a neat and powerful way of summarising their collective behaviour. We thus begin by replacing the material body found in accounts of the lateral Casimir--Polder force with a collection of neutral atoms. This is done by taking the dilute-gas limit (\new{in which the polarisability volume of each atom is much smaller than the cube of the mean interatomic spacing}) in a similar manner to that done by Lifshitz \cite{E.M.Lifshitz1956} via a Born-expansion of the dyadic Green's tensor (see, for example, \cite{Purcell1973,Buhmann2006,Sherkunov2007}) \begin{align}\label{BornExpansion} &\overbar{\tens{G}}\left(\mathbf{r},\mathbf{r}',\omega \right) = {\tens{G}}\left( \mathbf{r},\mathbf{r}',\omega \right)\notag \\ &\!\!+ \mu _0 \omega ^2\!\!\int \! \text{d}^3 r'' \!\rho \left( \mathbf{r}'' \right){\tens{G}}\left( \mathbf{r},\mathbf{r}'',\omega \right) \cdot \bm{\alpha}_\text{B} \left( \omega \right) \cdot {\tens{G}}\left( \mathbf{r}'',\mathbf{r}',\omega \right) \new{+ \ldots} \end{align} where $\rho(\B{r})$ is the number density of a collection of arbitrarily-placed atoms with identical polarisibilities $\bm{\alpha}_\text{B}\left( \omega \right)$, and ${\tens{G}}\left( \mathbf{r},\mathbf{r}',\omega \right)$ is the known Green's tensor of the background environment which could for example be unbounded vacuum, but need not be. Using the Born-expanded Green's tensor \eqref{BornExpansion} \new{with a delta-distributed number density} in the expression \eqref{CasPol} for the \new{resonant} force, one finds that $\textbf{F}^\text{res}(\textbf{r}_\text{A},t)=\bar{\textbf{F}}^\text{res}(\textbf{r}_\text{A},t) + \int d^3 r' \rho (\B{r}') \B{F}^\text{res}(\B{r}_\text{A},\B{r}',t)$, where $\bar{\textbf{F}}^\text{res}(\textbf{r}_\text{A},t)$ is the force felt between atom A and the background bodies alone, and; \begin{align}\label{ForceA} \textbf{F}^\text{res}(\textbf{r}_\text{A},\B{r}',t)= & 2\mu _0^2p(t)\omega_\text{A}^4 \text{Re} \big[\nabla\textbf{d}_{10}^\text{A} \cdot {\tens{G}}\left( \textbf{r},\textbf{r}',\omega_\text{A} \right)\notag \\ & \cdot \bm{\alpha}_\text{B}(\omega_\text{A})\cdot {\tens{G}}\left( \textbf{r}',\textbf{r}_\text{A},\omega_\text{A} \right) \cdot \textbf{d}_{01}^\text{A} \big] _{\textbf{r} = \textbf{r}_\text{A}}. \end{align} This is an atom-atom (van der Waals) force felt by atom A due to the presence of a (non-identical) atom B at $\B{r}'=\B{r}_\text{B}$ with dynamic polarisability tensor $\bm{\alpha}_\text{B}(\omega)$, valid as long the atoms are far enough apart that there is no appreciable wave-function overlap. Equation \eqref{ForceA} is made up of both the interaction of atom A with its own field as reflected by atom B, and the interaction with the quantised electromagnetic vacuum field. For most naturally-arising situations, the atomic dipoles can be considered to be randomly oriented, leaving an average force which pulls the particles linearly together (or, in some rare cases, pushes them apart). The situation changes drastically if one of the atoms has a complex dipole moment, corresponding to an atomic transition with different magnetic quantum numbers --- loosely thought of as a continuous rotation. As we will show, the resulting force causes atom A to orbit atom B. Extending the analogy of the lateral Casimir--Polder force with a rack and pinion to our situation, the interaction considered here could be considered as an atom-scale, contactless version of planetary gearing as illustrated in Fig.~\ref{Mechanical}. \begin{figure} \caption{Mechanical analogies to the lateral Casimir--Polder force studied in previous works, and the lateral interatomic force discussed here. Dashed (green) arrows represent forces, while solid arrows (black, white) represent motion. In all cases the entity on the right (blue) is considered as being fixed in space. } \label{Mechanical} \end{figure} We will illustrate this by taking atom A to be caesium undergoing a D2 transition from the highest hyperfine state $\ket{6^2\text{P}_{3/2}, F=5, M_\text{F} = 5} \equiv \ket{1}$ to the hyperfine ground state $\ket{6^2\text{S}_{1/2}, F=4, M_\text{F} = 4}\equiv \ket{0}$, and atom B to be rubidium in its ground state (\new{$5^2\text{S}_{1/2}$}, polarizability $\bm{\alpha}_\text{B} = \alpha_\text{B} \text{diag}(1,1,1)$, where $\alpha_\text{B} = 4\pi \varepsilon_0 \times 293\mathrm{\AA}^3$ at the caesium D2 wavelength of 852nm \cite{Sansonetti2005,SteckData}). The magnitude of the transition dipole moment between these two caesium levels is ${d}_\text{A} \equiv |\B{d}^\text{A}_{10}| = 2.68 \times 10^{-29}$Cm \cite{Scheel2015,SteckData}, while its components in the lab frame depend on the character of the light which excites the transition. Assuming a right-circularly polarised laser beam propagates along the $y$ direction of a cartesian co-ordinate system, the transition dipole moment can be written as: \begin{equation}\label{DipoleMoment} \B{d}^\text{A}_{10} = \frac{d_\text{A}}{\sqrt{2}} (\mathrm{i},0,1) \; . \end{equation} We assume that \new{the atoms are in free space, with} atom B at the origin and atom A in the $xz$ plane at position $z = r_\text{A} \cos\theta_\text{A}$, $x = r_\text{A} \sin \theta_\text{A}$. The two lateral components of the \new{resonant} force are in the $\theta$ and $y$ directions, and are found by inserting the free-space Green's tensor $\tens{G}^{(0)}$ into \eqref{ForceA}. As shown in, e.g., Ref.~\cite{Buhmann2012BothBooks}, this is given explicitly by \begin{equation} \tens{G}^{(0)}\left( \mathbf{r},\mathbf{r}',\omega \right) = \left( \tens{I} + \frac{c^2}{\omega ^2}\nabla \nabla \right)\frac{e^{\mathrm{i}\omega \left| \mathbf{r} - \mathbf{r}' \right|/c}}{4\pi \left| \mathbf{r} - \mathbf{r}' \right|} \label{G0Eq} \end{equation} where $c$ is the speed of light. Using cylindrical coordinates $\mathbf{r}=\left(r\sin \theta,y,r\cos \theta \right)$, $\nabla f= \frac{\partial f}{\partial r} \hat{\bm{r}} + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\bm{\theta}} + \frac{\partial f}{\partial y} \hat{\bm{y}} $ we find that the $y$ component of the force \new{$F^\text{res}_y = \B{F}^\text{res}\cdot \hat{\B{y}}$} vanishes; \begin{equation}\label{lateralfy} F^\text{res}_y(r_\text{A},t)=0 \end{equation} and the $\theta$ component \new{$F^\text{res}_\theta = \B{F}^\text{res}\cdot \hat{\bm{\theta}}$} is: \begin{equation}F^\text{res}_\theta(r_\text{A},t) = - \frac{p(t)}{40\pi ^2\varepsilon _0^2c^5r_\text{A}^2}d_{\text{A}}^2\alpha_\text{B}(\omega _{\text{A}})\omega _{\text{A}}^5 g \left( \frac{\omega_{\text{A}} r_\text{A}}{c} \right) \label{lateralf} \end{equation} where $\varepsilon_0$ is the permittivity of free space and we have defined \begin{align} g(\eta) \equiv\frac{5}{2\eta^5}[ 6\eta\left( \eta^2 - 3 \right) &\cos (2\eta) \notag \\ &+ \left( 9 - 15\eta^2 + \eta^4 \right)\sin (2\eta) ]. \end{align} \new{The lateral force shown in Eq.~\eqref{lateralf} is our main result, but as a point of comparison we also report the normal force} $F^\text{res}_r = \B{F}^\text{res}\cdot \hat{\bm{r}}$: \begin{align} F^\text{res}_r (r_{\text{A}},t) =&- \frac{15 p \left(t \right)}{16\pi ^2\varepsilon _0^2 r_\text{A}^7 }d_{\text{A}}^2 \alpha_{\text{B}}(\omega _{\text{A}}) h \left( \frac{\omega_{\text{A}} r_{\text{A}}}{c} \right), \label{Fr} \end{align} where: \begin{align}\label{hOfEta} h \left(\eta \right) =&\frac{1}{15}\Big[ 3\left(5-8 \eta^2 + \eta^4 \right)\cos (2\eta)\notag \\ & +\eta \left( 30 - 10\eta^2 + \eta^4 \right)\sin (2\eta) \Big]. \end{align} \new{Similar normal forces between (non-rotating) excited and ground state atoms are well-studied, having been considered by the authors of Refs~\cite{Power1995,Milonni2015,Donaire2015c,Donaire2016a,Jentschura2017a,Jentschura2017,Barcellona2016} with particular emphasis on the oscillating distance dependence, but the lateral force \eqref{lateralf} predicted here has not previously been discussed.} The van der Waals interaction in the near field \new{(non-retarded)} limit $\omega_\text{A} r_\text{A}/c \ll 1$, is given by Eqs.~(\ref{lateralf}) and (\ref{Fr}), where $\lim _{\eta \to 0}g\left( \eta \right) = 1, \lim _{\eta \to 0}h\left( \eta \right) = 1$, \new{while the far-field (retarded) limit is found from Eqs.~(\ref{lateralf}) and (\ref{Fr}) by taking $\omega_\text{A} r_\text{A}/c \gg 1$}. It is interesting to note that the forces are independent of $\theta_\text{A}$ which also results from symmetry considerations. Formulae (\ref{lateralf}) and (\ref{Fr}) account for retardation effects via the function $g$ in the limit $\omega_{\text{A}} r_\text{A}/c \gg 1$, which arises because of the finite velocity of light. In the retarded regime the time taken for the photon to reach the second atom and reflect back to the first atom become comparable with the time scale of the dipole fluctuations themselves. In this case the orientation of the dipole at the time of emission may differ from its orientation at the time of absorption of the reflected photon, reducing the attractive force as compared to the ideal case of parallel alignment. Our next step is to recognise that the excited-state interatomic force can be understood as a recoil force originating from the exchange of excitations with the environment, \new{for which we present an alternative derivation of Eq.~\eqref{ForceA} [and thereby Eqs \eqref{lateralfy} and \eqref{lateralf}], based on emission spectra instead of forces \cite{Sherkunov2009}.} \new{To do this we begin by calculating the spontaneous decay rate for atom A in the excited state $\left| 1\right\rangle $ in the presence of a second atom B. As shown explicitly in the supplementary material, in free space this is given by;} \begin{align}\label{GammaRShifted}\Gamma(\textbf{r}_\text{A},\textbf{r}_\text{B})= \frac{ 2\mu _0^2}{\hbar} \omega _\text{A}^4 \text{Im} \Big[ \textbf{d}_{10}^\text{A} \cdot \tens{G}^{(0)}\left( \textbf{r}_\text{A},\textbf{r}_\text{B},\omega _\text{A} \right)\notag \\ \cdot \bm{\alpha}_\text{B} (\omega_\text{A})\cdot \tens{G}^{(0)}\left( \textbf{r}_\text{B},\textbf{r}_\text{A},\omega _\text{A} \right) \cdot \textbf{d}_{01}^\text{A} \Big] \, . \end{align} \new{We can define a momentum-space emission rate density $\gamma$ as \begin{equation} \Gamma(\mathbf{r}_\text{A},\mathbf{r}_\text{B})= \int \text{d}^3k \gamma \left(\mathbf{k}; \mathbf{r}_\text{A}, \mathbf{r}_\text{B}\right), \end{equation} which is the rate at which light with wavevector $\mathbf{k}$ is emitted, if the atom $\text{A}$ is in the excited state. Since the free-space Green's tensor can be Fourier transformed $\tens{G}^{(0)} \left( \textbf{r},\textbf{r}',\omega\right)=\left(2\pi\right)^{-3}\int \text{d}^3k \text{e}^{\text{i} \mathbf{k} \cdot \left(\mathbf{r}-\mathbf{r}'\right)} \tens{G}^{(0)} \left( \mathbf{k},\omega\right)$ the rate density reads: \begin{align}\label{GammaRDef} \gamma(\mathbf{k}; \textbf{r}_\text{A},\textbf{r}_\text{B})= & \frac{ 2\mu _0^2}{\left(2\pi\right)^3\hbar} \omega _\text{A}^4 \text{Im} \Big[\text{e}^{\text{i} \mathbf{k} \cdot \left(\mathbf{r}_\text{A}-\mathbf{r}_\text{B}\right)} \textbf{d}_{10}^\text{A} \cdot \tens{G}^{(0)}\left( \mathbf{k},\omega _\text{A} \right) \notag \\ &\cdot {\bm{\alpha}}_\text{B} (\omega_\text{A})\cdot \tens{G}^{(0)}\left( \textbf{r}_\text{B},\textbf{r}_\text{A},\omega _\text{A} \right) \cdot \textbf{d}_{01}^\text{A} \Big]. \end{align}} \new{Explicit evaluation of the rate density in our particular setup (see supplemental material) reveals that $\gamma(-\mathbf{k}; \textbf{r}_\text{A},\textbf{r}_\text{B}) \ne \gamma(\mathbf{k}; \textbf{r}_\text{A},\textbf{r}_\text{B})$, showing that the net recoil force is, as expected, not zero.} \new{This can be explained by noting that the momentum-space recoil force density is given by $-\gamma \hbar \mathbf{k}$ (the minus signs accounting for the fact that we are considering recoils), so that the total resonant force on atom A is given by \begin{equation}\label{FResFromGamma} \mathbf{F}^{\text{res}} \left(\mathbf{r}_\text{A},\mathbf{r}_\text{B},t\right)=-p\left(t\right) \int \text{d}^3k \hbar \mathbf{k} \gamma \left(\mathbf{k}; \mathbf{r}_\text{A}, \mathbf{r}_\text{B}\right). \end{equation} Since $\nabla \text{e}^{\text{i} \mathbf{k}\cdot \mathbf{r}}=\text{i} \mathbf{k} \text{e}^{\text{i} \mathbf{k}\cdot \mathbf{r}} $ we immediately find the recoil force Eq.~\eqref{ForceA}, which leads to the lateral forces \eqref{lateralfy} and \eqref{lateralf}. } We are now left with a remarkable conclusion. The asymmetry that atom B represents in the environment of atom A causes the latter to preferentially release its excitation in a direction perpendicular to the line joining them, propelling A around B like a planetary gear. When combined with the oscillatory nature of the \new{resonant} force that atom B exerts on atom A, we also find that the sign of this torque can be varied by changing the distance between the atoms, as shown in Fig.~\ref{Plot1D}, \begin{figure} \caption{Lateral [solid, Eq.~\eqref{lateralf}] and normal [dashed, Eq.~\eqref{Fr}] \new{resonant} forces on a caesium atom (D2 transition) due to the presence of a rubidium atom at the origin. The numbered dots are those used later for trajectory simulations. Each chosen distance is comfortably larger than the atomic radii ($\sim10$\AA), consistent with our assumption of independent polarisibilities. } \label{Plot1D} \end{figure} where we also plot the corresponding normal \new{resonant} force \eqref{Fr}. Having seen that a lateral interatomic dispersion force is possible, we now turn our attention to its magnitude and prospects for experimental observation. In the absence of external driving, the atomic population (and therefore the recoil force) decays on average like $e^{-\Gamma t}$, meaning that the torque quickly becomes unobservably small. In order to combat this, we \new{introduce a coherent driving, for which it us useful to go} into the vacuum picture where the interaction of an atom with a coherent field can be considered as being made up of a classical driving field plus the vacuum field \cite{Pegg1980,Dutra1994,Fuchs2018c}. We consider atom A to be continuously driven by a circularly-polarised classical laser field propagating in the positive $y$-direction: \begin{equation}\label{DrivingLaser} \mathbf{E}_{\text{L}}\left( t \right) = E_0\B{e}_{\text{R}} e^{ - \text{i} \omega _{\text{L}}t}/2 + \text{c.c.} \end{equation} where $E_0$ is the field's amplitude, $\omega_{\text{L}}$ its frequency and $\B{e}_{\text{R}}= \left(-\text{i},\; 0, \; 1 \right)/\sqrt 2$. The effect of the driving laser is accounted for by the real Rabi frequency ${ \hbar \Omega = \mathbf{d}_{10}^{\text{A}} \cdot \B{e}_{\text{R}} E_0= d_\text{A} E_0 }$. Solving the optical Bloch equations \new{for the interaction of the laser field with atom A in the absence of atom B} in the long time limit ($t\gg \Gamma^{-1}$), the expectation value of the dipole moment operator of atom A is then given by \begin{align}\label{DipoleExpct} \left\langle \mathbf{d}^\text{A}(t) \right\rangle &= \frac{\sqrt{2}\Omega \Delta }{2\Delta ^2 + \Omega^2} d_\text{A}\big( \sin \left( \omega _\text{L}t \right),0, -\cos \left( \omega _\text{L}t \right) \big) \end{align} where $\Delta = \omega_\text{L}-\omega_\text{A}$ is the detuning of the laser field from the atomic resonance, and we have also assumed $\Gamma \ll |\Delta|$. In the absence of atom B, atom A simply rotates in the $x-z$ plane with the same frequency as the laser, which is not surprising. The presence of atom B breaks the symmetry of the electromagnetic environment experienced by atom A. To quantify this effect we use Eq.~\eqref{ForceA} with an excited state population given by (\new{see, for example, \cite{Mollow1969}}); \begin{equation}\label{Prob} p(t) =\frac{ \Omega^2}{4\Delta ^2 + 2\Omega^2} \end{equation} In the strong interaction limit $\Omega \gg |\Delta|$, the effect of the resultant force in is shown in Fig.~\ref{StreamPlot}, \begin{figure} \caption{Simulated trajectories for a caesium atoms starting at rest for the four points shown in Fig.~\ref{Plot1D}. Shown in the background is the potential energy function found by integrating the normal \new{resonant} force in the radial direction.} \label{StreamPlot} \end{figure} where we place atoms initially at rest on the $x$ axis at the positions indicated by the dots in Fig.~\ref{Plot1D} and compute their trajectories. \new{The illuminating light should be set up in such a way that it has a constant amplitude over the trajectory of atom A, while affecting atom B as little as possible. This could be achieved, for example, by tailoring atom B's level structure, or through the use of structured light.} It is seen that under such continuous laser driving the lateral force causes atom A to be ejected after slightly more than half an orbit of the fixed, isotropic atom B. In Fig.~\ref{VelocityPlot} \begin{figure} \caption{Velocities gained along the four trajectories simulated in Fig.~\ref{StreamPlot}. } \label{VelocityPlot} \end{figure} we plot the velocity gained as a function of time, finding $12-15\mu$m/s for the parameters chosen here. To reach these velocities takes a relatively long time (on the order of a second) since the force is so weak. However, there are several routes to combat this by enhancement of the interaction. One might expect that use of Rydberg atoms with their large dipole moments (quadratic in the principle quantum number $n$), however the energy difference of adjacent states scaled as $n^{-3}$ meaning that the force derived here is strongly suppressed for such systems. Finally, we note that the interaction could be enhanced by placing the pair of atoms in a cavity, in much the same was as the spontaneous decay rate of a quantum emitter can be enhanced through the Purcell factor \cite{Purcell1946}. To conclude, we have demonstrated the existence of a lateral Van der Waals force on an excited, circularly polarised atom due to the placement of an isotropic, ground state atom nearby. We have outlined how the effect might be experimentally accessed by selectively pumping the atom to a Zeeman sub-level. Control of the lateral force direction and magnitude can be experimentally implemented by changing the handedness of the illuminating light and the distance between the two atoms. Our work is the first demonstration of the most elementary lateral force that can act on a circularly polarised emitter, without the influence of a surface. Nevertheless, our expression of the force in terms of the dyadic Green's tensor means that additional macroscopic objects can be introduced without fundamental changes to the method, opening up the effect detailed here to Purcell-type enhancement. In the longer term, the force could find applications in optomechanics as a new actuation method, as well as in any of the numerous fields in which Van der Waals forces play a pivotal role. \acknowledgments{\new{The authors thank Gabriel Dufour for valuable feedback on the manuscript, and the Deutsche Forschungsgemeinschaft for financial support (grant BU 1803/3-1476)}.} \begin{thebibliography}{49} \makeatletter \providecommand \@ifxundefined [1]{ \@ifx{#1\undefined} } \providecommand \@ifnum [1]{ \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \@ifx [1]{ \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \natexlab [1]{#1} \providecommand \enquote [1]{``#1''} \providecommand \bibnamefont [1]{#1} \providecommand \bibfnamefont [1]{#1} \providecommand \citenamefont [1]{#1} \providecommand \href@noop [0]{\@secondoftwo} \providecommand \href [0]{\begingroup \@sanitize@url \@href} \providecommand \@href[1]{\@@startlink{#1}\@@href} \providecommand \@@href[1]{\endgroup#1\@@endlink} \providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax} \providecommand \@@startlink[1]{} \providecommand \@@endlink[0]{} \providecommand \url [0]{\begingroup\@sanitize@url \@url } \providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }} \providecommand \urlprefix [0]{URL } \providecommand \Eprint [0]{\href } \providecommand \doibase [0]{http://dx.doi.org/} \providecommand \selectlanguage [0]{\@gobble} \providecommand \bibinfo [0]{\@secondoftwo} \providecommand \bibfield [0]{\@secondoftwo} \providecommand \translation [1]{[#1]} \providecommand \BibitemOpen [0]{} \providecommand \bibitemStop [0]{} \providecommand \bibitemNoStop [0]{.\EOS\space} \providecommand \EOS [0]{\spacefactor3000\relax} \providecommand \BibitemShut [1]{\csname bibitem#1\endcsname} \let\auto@bib@innerbib\@empty \bibitem [{\citenamefont {Bardeen}\ \emph {et~al.}(1957)\citenamefont {Bardeen}, \citenamefont {Cooper},\ and\ \citenamefont {Schrieffer}}]{Bardeen1957} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Bardeen}}, \bibinfo {author} {\bibfnamefont {L.~N.}\ \bibnamefont {Cooper}}, \ and\ \bibinfo {author} {\bibfnamefont {J.~R.}\ \bibnamefont {Schrieffer}},\ }\href {\doibase 10.1103/PhysRev.106.162} {\bibfield {journal} {\bibinfo {journal} {Physical Review}\ }\textbf {\bibinfo {volume} {106}},\ \bibinfo {pages} {162} (\bibinfo {year} {1957})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Lifshitz}(1956)}]{E.M.Lifshitz1956} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Lifshitz}},\ }\href {http://www.jetp.ac.ru/cgi-bin/e/index/e/2/1/p73?a=list} {\bibfield {journal} {\bibinfo {journal} {Journal of Experimental and Theoretical Physics}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {73} (\bibinfo {year} {1956})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Casimir}(1948)}]{Casimir1948} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {H.~B.~G.}\ \bibnamefont {Casimir}},\ }\href {\doibase citeulike-article-id:8810715} {\bibfield {journal} {\bibinfo {journal} {Proc. K. Ned. Akad.}\ }\textbf {\bibinfo {volume} {360}},\ \bibinfo {pages} {793} (\bibinfo {year} {1948})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Buhmann}(2012)}]{Buhmann2012BothBooks} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~Y.}\ \bibnamefont {Buhmann}},\ }\href@noop {} {\emph {\bibinfo {title} {{Dispersion Forces}}}},\ \bibinfo {series} {Springer Tracts in Modern Physics}, Vol.\ \bibinfo {volume} {247}\ (\bibinfo {publisher} {Springer},\ \bibinfo {address} {Berlin, Heidelberg},\ \bibinfo {year} {2012})\BibitemShut {NoStop} \bibitem [{Note1()}]{Note1} \BibitemOpen \bibinfo {note} {In this work we refer to atom-atom forces at all distances as van der Waals forces (in contrast to some authors who refer to the long-distance atom-atom interaction as a Casimir-Polder force), and all atom-surface forces as Casimir-Polder forces. This should not be taken as denial of the fact that Casimir and Polder derived atom-atom \emph {and} atom surface forces at all distances in their seminal paper \cite {Casimir1948a}.}\BibitemShut {Stop} \bibitem [{\citenamefont {Ashourvan}\ \emph {et~al.}(2007)\citenamefont {Ashourvan}, \citenamefont {Miri},\ and\ \citenamefont {Golestanian}}]{Ashourvan2007} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Ashourvan}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Miri}}, \ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Golestanian}},\ }\href {\doibase 10.1103/PhysRevLett.98.140801} {\bibfield {journal} {\bibinfo {journal} {Physical Review Letters}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo {pages} {140801} (\bibinfo {year} {2007})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Nasiri}\ \emph {et~al.}(2012)\citenamefont {Nasiri}, \citenamefont {Miri},\ and\ \citenamefont {Golestanian}}]{Nasiri2012} \BibitemOpen \bibfield {author} 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\begin{document} \title{Towards Automatic Stress Analysis using Scaled Boundary Finite Element Method with Quadtree Mesh of High-order Elements} \author{Hou~Man\corref{cor1}} \ead{[email protected]} \cortext[cor1]{Corresponding author. Tel.: +612 93855030} \author{Chongmin~Song, Sundararajan~Natarajan, Ean~Tat~Ooi, Carolin~Birk\corref{}} \address{School of Civil and Environmental Engineering} \address{The University of New South Wales, Sydney, NSW 2052, Australia} \begin{abstract} This paper presents a technique for stress and fracture analysis by using the scaled boundary finite element method (SBFEM) with quadtree mesh of high-order elements. The cells of the quadtree mesh are modelled as scaled boundary polygons that can have any number of edges, be of any high orders and represent the stress singularity around a crack tip accurately without asymptotic enrichment or other special techniques. Owing to these features, a simple and automatic meshing algorithm is devised. No special treatment is required for the hanging nodes and no displacement incompatibility occurs. Curved boundaries and cracks are modelled without excessive local refinement. Five numerical examples are presented to demonstrate the simplicity and applicability of the proposed technique.\end{abstract} \begin{keyword} scaled boundary finite-element method; quadtree mesh; high order elements; polygon elements \end{keyword} \maketitle \section{Introduction} Finite Element Analysis (FEA) is the most widely used analysing tool in Computer Aided Engineering (CAE). One key factor to achieve an accurate FEA is the layout of the finite element mesh, including both mesh density and element shape \citep{Yerry1983a}. Regions containing complex boundaries, rapid transitions between geometric features or singularities require finer discretisation \citep{Cheng1996,Greaves1999}. This leads to the development of adaptive meshing techniques that assure the solution accuracy without sacrificing the computational efficiency \citep{Tabarraei2005a,Lo2010}. The construction of a high quality mesh, in general, takes the most of the analysis time \citep{Hughes2005}. The recent rapid development of the isogeometric analysis \citep{Hughes2005,Nguyen-Thanh2011a,Simpson2013}, which suppressed the meshing process, has emphasised the significance of mesh automation in engineering design and analysis. Quadtree in FEA is a kind of hierarchical tree-based techniques for adaptive meshing of a 2D geometry \citep{Greaves1999}. It discretises the geometry into a number of square cells of different size. The process is illustrated in Fig.\,\ref{fig:qtreerep1} using a circular domain. The geometry is first covered with a single square cell, also known as the root cell of the quadtree (Fig.\,\ref{fig:qtreerep1}a). As shown in Fig.\,\ref{fig:qtreerep1}b, the root cell is subdivided into 4 equal-sized square cells and each of the cells is recursively subdivided to refine the mesh until certain termination criteria are reached. In this example, a cell is subdivided to better represent the boundary of the circle and the subdivision stops when the predefined maximum number of division is reached. The final mesh is obtained after deleting all the cells outside the domain (Fig.\,\ref{fig:qtreerep1}c). The cell information is stored in a tree-type data structure, in which the root cell is at the highest level. It is common practice to limit the maximum difference of the division levels between two adjacent cells to one \citep{Yerry1983a,GVS2001}. This is referred to the $2:1$ rule and the resulting mesh is called a balanced \citep{GVS2001} or restricted quadtree mesh \citep{Tabarraei2005a}. A balanced quadtree mesh not only ensures there is no large size difference between adjacent cells, but also reduces the types of quadtree cells in a mesh to the 6 shown in Fig.\,\ref{fig:qtreecell}. Owing to its simplicity and large degree of flexibility, the quadtree mesh is also recognised in large-scale flood/tsunami simulations \citep{Liang2008,Popinet2011} and image processing \citep{Morvan2007}. \begin{figure} \caption{Generation of quadtree mesh on a circular domain. (a) Cover the domain with a square root cell (b) Subdivide the square cells (c) Select the cells based on the domain boundary} \label{fig:qtreerep1} \end{figure} \begin{figure} \caption{6 main types of master quadtree cells with $2:1$ rule enforced} \label{fig:qtreecell} \end{figure} It is, however, not straightforward to integrate a quadtree mesh in a FEA. The two major issues are illustrated by Fig.\,\ref{fig:qtreerep}, which shows the quadtree mesh of the top-right quadrant of the circular domain in Fig.\,\ref{fig:qtreerep1}. \begin{enumerate} \item \emph{Hanging nodes} Middle nodes, shown as solid dots in Fig.\,\ref{fig:qtreerep}, exist at the common edges between the adjacent cells with different division levels. When conventional quadrilateral finite elements are used, a middle node is connected to the two smaller elements (lower level) but not to the larger element (higher level). This leads to incompatible displacement along the edges and the middle nodes are called the hanging nodes \citep{Greaves1999}. \item \emph{Fitting of curved boundary} Quadtree cells are composed of horizontal and vertical lines only. As shown in Fig.\,\ref{fig:qtreerep}, the quadtree cells intersected with the curved boundary have to be further divided into smaller ones to improve the fitting of the boundary. Generally, the mesh has to be refined in the area surrounding the boundary. Despite this, the boundary may still not be smooth (Fig.\,\ref{fig:qtreerep1}c) and may result in unrealistically high stresses. Additional procedure is required to conform the mesh to the boundary. \end{enumerate} There exist a number of different approaches to ensure displacement compatibility when hanging nodes are present \citep{Ebeida2010,Legrain2011,Tabarraei2008a,Ainsworth2007}. Three typical approaches among all are briefly discussed here. The first one is to subdivide the higher level quadtree cells next to a hanging node into smaller triangular elements \citep{Yerry1983a,Bern1994,Alyavuz2009} as shown in Fig\,\ref{fig:qtreerep}. Additional nodes may be added to improve the mesh quality and/or reduce the number of element types. These techniques lead to a final mesh that only contains conforming triangular elements. A similar approach was adpoted by \citet{Ebeida2010}, in which the quadtree mesh was subdivided into a conforming mesh dominated by quadrilateral elements. The second approach introduces special conforming shape functions \citep{Gupta1978} to ensure the displacement compatibility. An early work by Gupta \citep{Gupta1978} reported the development of a transition element that had additional node along its side. A conforming set of shape functions was derived based on the shape functions of the bilinear quadrilateral elements. Owing to its simplicity and applicability, Gupta's work was further extended by \citet{Mcdill1987} and \citet{Lo2010} to hexahedral elements. Fries et al. \citep{Fries2011} investigated two approaches to handle the hanging nodes within the framework of the extended finite element method (XFEM). They were different in whether the enriched degrees-of-freedom (DOFs) were assigned to the hanging node. A similar work was reported by Legrain et al. \citep{Legrain2011}, in which the selected DOFs are enriched and properly constrained to ensure the continuity of the field. \begin{figure} \caption{Quadtree mesh of the top-right quadrant of a circular domain. Demonstration of subdivision (dashed lines) is given in two quadtree cells with hanging nodes on their sides.} \label{fig:qtreerep} \end{figure} The third approach is to model the quadtree cells as \emph{n-}sided polygon elements by treating hanging nodes as vertices of the polygon. This approach generally requires a set of polygonal basis function. Special techniques are usually required to integrate the resulting equations over arbitrary polygon domain \citep{Natarajan2009}. This development was initiated by \citet{wachspress1975rational} who showed the use of rational basis functions for elements with arbitrary number of sides. Tabarraei and Sukumar \citep{Tabarraei2005a,Tabarraei2007} in their work adapted their polygon element \citep{Sukumar2004} to quadtree mesh. The set of polygonal basis functions was derived using Laplace interpolant. By using an affine map on the reference polygon, the conforming shape functions of a quadtree cell with the same number of vertices (including the hanging nodes) were obtained. They also reported a fast technique for computing the global stiffness matrix, making use of the quadtree structure by defining parent elements \citep{Tabarraei2007}. In this way, the elemental stiffness matrix has to be computed only 15 times (4-node cell not included) for a balanced quadtree mesh (when $2:1$ rule is enforced). Further development of their work with XFEM was also reported in \citep{Tabarraei2008a}. As mentioned in a recent paper \citep{Sukumar2013}, the development of high-order polygon elements received relatively less attention. Mibradt and Pick \citep{Milbradt2008} devised high order basis functions for polygons based on the natural element coordinates. Those basis functions, however, are not complete polynomials. Rand et al. \citep{Rand2013} developed a quadratic serendipity element for arbitrary convex polygons based on generalised barycentric coordinates. The potential of using their approach for higher order serendipity elements on convex polygons was also reported. Based on the same approach, Sukumar \citep{Sukumar2013} recently developed the quadratic serendipity shape functions that were applicable for convex and nonconvex polygons and were complete quadratic polynomials. The shape functions were obtained through solving an optimisation problem, which was derived from the maximum-entropy principle. Besides dealing with the hanging nodes in a quadtree mesh, fitting complex boundaries is another challenging part in the mesh generation. \citet{Yerry1983a} proposed trimming the quadtree cells, intersected with the boundary, into polygons before a further subdivision process into triangles or quadrilaterals. Alternatively those cells were first subdivided and some of the vertices were repositioned based on their projections onto the boundary \citep{Greaves1999,Alyavuz2009}. In \citet{Ebeida2010} and \citet{Liang2010}, after the subdivision, a buffer zone was introduced between the boundary and the internal quadtree cells. A compatible mesh was then constructed to fill up this zone. All these techniques require an additional optimisation step to ensure the final mesh quality. Within the framework of XFEM, quadtree cells intersected with the boundary were not modified in pre-processing stage \citep{Fries2011}. However, when constructing the stiffness matrix, the domain boundary is still required to identify the portion of the cell within the domain for numerical integration. In the integration process, that portion of the cell within the domain is either subdivided into geometric sub-cells \citep{Dreau2010} or treated as a polygon \citep{Natarajan2010}. The scaled boundary finite element method (SBFEM) provides an attractive alternate technique to construct polygon elements (scaled boundary polygon) \citep{Ooi2012a,Ooi2013} (Fig.\,\ref{fig:sbfempolygon}). It is a semi-analytical procedure developed by Song and Wolf to solve boundary value problems \citep{song1997scaled}. The only requirement for a scaled boundary polygon is that its entire boundary is visible from the \emph{scaling centre} \citep{song1997scaled}. Only the edges of the polygon are discretised into line elements. The number of line elements on an edge can be as many as required. Any type of displacement-based line elements, including high-order spectral elements\textcolor{black}{{} \citep{vu2006use}}, can be used. The domain of the scaled boundary polygon is constructed by scaling from its scaling centre to its boundary, and the solution within the polygon is expressed semi-analytically \citep{Ooi2012a,Ooi2013}. A salient feature of the scaled boundary polygons is that stress singularities occurring at crack and notch tips, formed by one or several materials, can be accurately modelled without resorting to asymptotic enrichment and local mesh refinement. Its high accuracy and flexibility in mesh generation lead to simple remeshing procedures when modelling crack propagation \citep{yang2006,Ooi2009,Ooi2010a,Ooi2013}. \begin{figure} \caption{ Scaled boundary representation of a polygon} \label{fig:sbfempolygon} \end{figure} \begin{figure} \caption{Scaled boundary representation of quadtree cells} \label{fig:sbfequadtreerep} \end{figure} This paper presents a technique for the stress and fracture analysis by integrating the scaled boundary finite element method (SBFEM) with quadtree mesh of high-order elements. This integrated technique possesses the following features: \begin{enumerate} \item Hanging nodes are treated without cell subdivision. Each quadtree cell is modelled as a scaled boundary polygon as shown in Fig.\,\ref{fig:sbfequadtreerep}. The edges of a quadtree cell can be divided into more than one line element to ensure displacement compatibility with the adjacent smaller cells. Hanging-nodes are thus treated the same as other nodes. Owing to the SBFE formulations \citep{Ooi2012a}, no additional procedure is required to compute the shape functions for the quadtree cells. High-order elements can also be used within each quadtree cell directly. \item The entire quadtree meshing process is simple and automatic. The boundary of the problem domain is defined using signed distance functions \citep{Talischi2012}. Only seed points \citep{Greaves1999} are required to be predefined to control the mesh density. Owing to the ability of the SBFEM in constructing polygon elements of, practically, arbitrary shape and order, the quadtree cells trimmed by curved boundaries are simply treated as a non-square scaled boundary polygon. High-order elements can be used to fit curved boundaries closely. The resulting mesh conforms to the boundary without excessive mesh refinement (see Fig.\,\ref{fig:sbfequadtreerep}). \item No local mesh refinement or asymptotic enrichment is required for a quadtree cell containing a crack tip to accurately model the stress singularity. \end{enumerate} The present paper is organised as follows. The summary of the SBFEM and its application to quadtree cells are first presented in the next section. It is followed by the developed algorithm of quadtree mesh generation in Section\,\ref{sec:Quadtree-mesh-generation}. Five examples are given in Section\,\ref{sec:Numerical-examples} with detailed discussion on accuracy and convergence. Finally, conclusions of the present work are stated in Section\,\ref{sec:Conclusion}. \section{Scaled boundary finite element method on quadtree cells\label{sec:Scaled-boundary-finite}} This section summarises the scaled boundary finite element method for 2D stress and fracture analysis. Only the key equations that are related to its use with a quadtree mesh are listed. A detailed derivation of the method based on a virtual work approach is given in \citet{deeks2002virtual}. \subsection{Element formulation} The SBFEM can be formulated on quadtree cells by treating each cell as a polygon with arbitrary number of sides (Fig.~\ref{fig:sbfequadtreerep}). In each cell, a local coordinate system $(\xi,\eta)$ is defined at a point called the scaling centre from which the entire boundary is visible. $\xi$ is the radial coordinate with $\xi=0$ at the scaling centre and $\xi=1$ at the cell boundary. The edges of each cell are discretised using one-dimensional finite elements with a local coordinate $\eta$ having an interval of $-1\leq\eta\leq1$. It is noted that the hanging nodes appearing in the quadtree structure do not require any special treatment in the SBFEM formulation. They are simply used as end nodes of the 1D elements. The coordinate transformation between the Cartesian $(x,y)$ and the local $(\xi,\eta)$ coordinate systems are given by the scaled boundary transformation equations \citep{song1997scaled}: \begin{align} \mathbf{x}(\xi,\eta)= & \xi\mathbf{N}(\eta)\mathbf{x}_{\mathrm{b}}\label{eq:coordtrans} \end{align} where $\mathbf{x}(\xi,\eta)=[x(\xi,\eta)\; y(\xi,\eta)]^{\mathrm{T}}$ is the Cartesian coordinates of a point in the cell, $\mathbf{N}(\eta)$ is the shape function matrix and $\mathbf{x}_{\mathrm{b}}=[\begin{array}{ccccc} x_{1} & y_{1} & \ldots & x_{n} & y_{n}\end{array}]^{\mathrm{T}}$ is the vector of nodal coordinates of a cell with $n$ nodes. The displacement field in each cell $\mathbf{u}(\xi,\eta)$ is interpolated as \begin{align} \mathbf{u}(\xi,\eta)= & \mathbf{N}(\eta)\mathbf{u}(\xi)\label{eq:dispfield} \end{align} \noindent where $\mathbf{u}(\xi)$ are radial displacement functions and are obtained by solving the scaled boundary finite element equation in displacement \citep{song1997scaled}: \begin{align} \mathbf{E}_{0}\xi^{2}\mathbf{u}(\xi)_{,\xi\xi}+(\mathbf{E}_{0}-\mathbf{E}_{1}+\mathbf{E}_{1}^{\mathrm{T}})\xi\mathbf{u}(\xi)_{,\xi}-\mathbf{E}_{2}\mathbf{u}(\xi)= & 0\label{eq:sbfedispeqn} \end{align} \noindent with coefficient matrices \begin{align} \mathbf{E}_{0}= & \int_{-1}^{+1}\mathrm{\mathbf{B}}_{1}^{\mathrm{T}}(\eta)\mathrm{\mathbf{D\mathrm{\mathbf{B}}_{\mathrm{1}}}(\eta)}|\mathbf{J}(\eta)|d\eta\label{eq:e0}\\ \mathbf{E}_{1}= & \int_{-1}^{+1}\mathrm{\mathbf{B}}_{2}^{\mathrm{T}}(\eta)\mathrm{\mathbf{D\mathrm{\mathbf{B}}_{\mathrm{1}}}(\eta)}|\mathbf{J}(\eta)|d\eta\label{eq:e1}\\ \mathbf{E}_{2}= & \int_{-1}^{+1}\mathrm{\mathbf{B}}_{2}^{\mathrm{T}}(\eta)\mathrm{\mathbf{D\mathrm{\mathbf{B}}_{\mathrm{2}}}(\eta)}|\mathbf{J}(\eta)|d\eta\label{eq:e2} \end{align} \noindent where $\mathbf{D}$ is the material constitutive matrix, $\mathbf{B}_{1}(\eta)$ and $\mathbf{B}_{2}(\eta)$ are the SBFEM strain-displacement matrices and $|\mathbf{J}(\eta)|$ is the Jacobian on the boundary required for coordinate transformation. Eq.\,\eqref{eq:sbfedispeqn} is solved by introducing the variable $\mathbf{X}(\xi)$ \citep{wolf2003scaled} \begin{align} \mathbf{X}(\xi)= & [\begin{array}{cc} \mathbf{u}(\xi) & \quad\mathbf{q}(\xi)\end{array}]^{\mathrm{T}}\label{eq:Xksi} \end{align} \noindent where \begin{align} \mathbf{q}(\xi)= & \mathbf{E}_{0}\xi\mathbf{u}(\xi)_{,\xi}+\mathbf{E}_{1}^{\mathrm{T}}\mathbf{u}(\xi)\label{eq:qksi} \end{align} \noindent so that Eq.\,\eqref{eq:sbfedispeqn} is transformed into a first order ordinary differential equation with twice the number of unknowns: \begin{align} \xi\mathbf{X}(\xi)_{,\xi}= & -\mathbf{Z}\mathbf{X}(\xi)\label{eq:firstord} \end{align} \noindent with the Hamiltonian matrix $\mathbf{Z}$ \citep{wolf2003scaled} \begin{align} \mathbf{Z}= & \left[\begin{array}{cc} \mathbf{E}_{0}^{-1}\mathbf{E}_{1}^{\mathrm{T}} & -\mathbf{E}_{0}^{-1}\\ \mathbf{E}_{1}\mathbf{E}_{0}^{-1}\mathbf{E}_{1}^{\mathrm{T}}-\mathbf{E}_{2} & -\mathbf{E}_{1}\mathbf{E}_{0}^{-1} \end{array}\right]\label{eq:hamilton} \end{align} An eigenvalue decomposition of the $\mathbf{Z}$ results in \begin{align} \mathbf{Z}\left[\begin{array}{cc} \boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}} & \boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(p)}}\\ \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(n)}} & \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(p)}} \end{array}\right]= & \left[\begin{array}{cc} \boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}} & \boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(p)}}\\ \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(n)}} & \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(p)}} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{\Lambda}^{\mathrm{(n)}} & 0\\ 0 & \boldsymbol{\Lambda}^{\mathrm{(p)}} \end{array}\right]\label{eq:eigendecomp} \end{align} \noindent where $\boldsymbol{\Lambda}^{(\mathrm{n})}$ and $\boldsymbol{\Lambda}^{(\mathrm{p})}$ are the eigenvalue matrices with real parts satisfying $\mathrm{Re}(\lambda(\boldsymbol{\Lambda}^{(\mathrm{n})})<0$ and $\mathrm{Re}(\lambda(\boldsymbol{\Lambda}^{(\mathrm{p})})>0$, respectively. $\boldsymbol{\Phi}_{\mathrm{u}}^{(\mathrm{n})}$ and $\boldsymbol{\Phi}_{\mathrm{q}}^{(\mathrm{n})}$ are the corresponding eigenvectors of $\boldsymbol{\Lambda}^{(\mathrm{n})}$ whereas $\boldsymbol{\Phi}_{\mathrm{u}}^{(\mathrm{p})}$ and $\boldsymbol{\Phi}_{\mathrm{q}}^{(\mathrm{p})}$ are the eigenvectors corresponding to $\boldsymbol{\Lambda}^{(\mathrm{p})}$. For bounded domains such as those considered in this paper, only the eigenvalues satisfying $\mathrm{Re}(\lambda(\boldsymbol{\Lambda}^{(\mathrm{n})})<0$ lead to finite displacements at the scaling centre. Using Eq.\,\eqref{eq:eigendecomp} and Eq.\,\eqref{eq:firstord}, the solutions for $\mathbf{u}(\xi)$ and $\mathbf{q}(\xi)$ are \begin{align} \mathbf{u}(\xi)= & \boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\xi^{-\boldsymbol{\Lambda}^{\mathrm{(n)}}}\mathbf{c}^{\mathrm{(n)}}\label{eq:uksisol}\\ \mathbf{q}(\xi)= & \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(n)}}\xi^{-\boldsymbol{\Lambda}^{\mathrm{(n)}}}\mathbf{c}^{\mathrm{(n)}}\label{eq:qksisol} \end{align} The integration constants $\mathbf{c}^{(\mathrm{n})}$ in Eq.\,\eqref{eq:uksisol} and Eq.\,\eqref{eq:qksisol} are obtained from the nodal displacements at the cell boundary $\mathbf{u}_{\mathrm{b}}=\mathbf{u}(\xi=1)$ as \begin{align} \mathbf{c}^{(\mathrm{n})}= & \left(\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\right)^{-1}\mathbf{u}_{\mathrm{b}}\label{eq:intcons} \end{align} \noindent The stiffness matrix of each quadtree cell is formulated as \citep{wolf2003scaled} \begin{align} \mathbf{K}= & \boldsymbol{\Phi}_{\mathrm{q}}^{\mathrm{(n)}}\left(\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\right)^{-1}\label{eq:stf} \end{align} \noindent Substituting Eq.\,\eqref{eq:uksisol} into Eq.\,\eqref{eq:dispfield}, the displacement field in a cell is \begin{align} \mathbf{u}(\xi,\eta)= & \mathbf{N}(\eta)\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\xi^{-\boldsymbol{\Lambda}^{\mathrm{(n)}}}\mathbf{c}^{\mathrm{(n)}}\label{eq:dispfieldsol} \end{align} \noindent Using the Hooke's law and the strain-displacement relationship, the stress at a point in a cell is \citep{wolf2003scaled} \begin{align} \boldsymbol{\sigma}(\xi,\eta)= & \boldsymbol{\Psi}_{\sigma}(\eta)\xi^{-\boldsymbol{\Lambda}^{\mathrm{(n)}}-\mathbf{I}}\mathbf{c}^{\mathrm{(n)}}\label{eq:stresfield} \end{align} \noindent where $\boldsymbol{\Psi}_{\sigma}(\eta)=\left[\begin{array}{ccc} \boldsymbol{\Psi}_{\sigma_{xx}}(\eta) & \boldsymbol{\Psi}_{\sigma_{yy}}(\eta) & \boldsymbol{\Psi}_{\tau_{xy}}(\eta)\end{array}\right]^{\mathrm{T}}$ is the stress mode \begin{align} \boldsymbol{\Psi}_{\sigma}(\eta)= & \mathbf{D}\left(-\mathbf{B}_{1}(\eta)\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\boldsymbol{\Lambda}^{\mathrm{(n)}}+\mathbf{B}_{2}(\eta)\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(n)}}\right)\label{eq:stresmod} \end{align} \subsection{Evaluation of stress intensity factors} Fig.\,\ref{fig:crackrep} shows how a crack is modelled with a quadtree cell. The crack tip is chosen as the scaling centre. The crack surfaces are not discretised. The line elements discretising the cell boundary do not form a closed loop. \begin{figure} \caption{Modelling of a crack with the scaled boundary finite element method.} \label{fig:crackrep} \end{figure} When a crack is modelled by the SBFEM, two eigenvalues, $\lambda_{i}$, $i=1,\,2$ satisfying $-1<\mathrm{Re}(\lambda_{i})\leq0$ appear in $\boldsymbol{\Lambda}^{(\mathrm{n})}$. From Eq.\,\ref{eq:stresfield}, it can be discovered that these eigenvalues lead to a stress singularity as $\xi\rightarrow0$. Using the two modes corresponding to these two eigenvalues, the singular stresses are expressed as \begin{align} \boldsymbol{\sigma}(\xi,\eta)= & \boldsymbol{\Psi}_{\sigma}^{\mathrm{(s)}}(\eta)\xi^{-\boldsymbol{\Lambda}^{\mathrm{(s)}}-\mathbf{I}}\mathbf{c}^{\mathrm{(s)}}\label{eq:singstrefield} \end{align} \noindent where \begin{align} \boldsymbol{\Lambda}^{(\mathrm{s})}= & \left[\begin{array}{cc} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right]\label{eq:singeigenval} \end{align} \noindent and $\mathbf{c}^{(\mathrm{s})}$ are the integration constants corresponding to $\boldsymbol{\Lambda}^{(\mathrm{s})}$. The singular singular stress modes $\boldsymbol{\Psi}_{\sigma}^{(\mathrm{s})}(\eta)=\left[\begin{array}{ccc} \boldsymbol{\Psi}_{\sigma_{xx}}^{(\mathrm{s})}(\eta) & \boldsymbol{\Psi}_{\sigma_{yy}}^{(\mathrm{s})}(\eta) & \boldsymbol{\Psi}_{\tau_{xy}}^{(\mathrm{s})}(\eta)\end{array}\right]^{\mathrm{T}}$ is written as \begin{align} \boldsymbol{\Psi}_{\sigma}^{\mathrm{(s)}}(\eta)= & \mathbf{D}\left(-\mathbf{B}_{1}(\eta)\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(s)}}\boldsymbol{\Lambda}^{\mathrm{(s)}}+\mathbf{B}_{2}(\eta)\boldsymbol{\Phi}_{\mathrm{u}}^{\mathrm{(s)}}\right)\label{eq:singstremode} \end{align} \noindent where $\boldsymbol{\Phi}_{\mathrm{u}}^{(\mathrm{s})}$ are the modal displacements in $\boldsymbol{\Phi}_{\mathrm{u}}^{(\mathrm{n})}$ corresponding to $\boldsymbol{\Lambda}^{(\mathrm{s})}$. The stress intensity factors can be computed directly from their definitions. For a crack that is aligned with the Cartesian coordinate system as shown in Fig.\,\ref{fig:crackrep}, the stress intensity factors are defined as \begin{align} \left\{ \begin{array}{c} K_{\mathrm{I}}\\ K_{\mathrm{II}} \end{array}\right\} = & \lim_{r\rightarrow0}\left\{ \begin{array}{c} \sqrt{2\pi r}\left.\sigma_{yy}\right|_{\theta=0}\\ \sqrt{2\pi r}\left.\tau_{xy}\right|_{\theta=0} \end{array}\right\} \label{eq:sifdef} \end{align} \noindent Substituting the stress components in Eq.\,\eqref{eq:singstrefield} into Eq.\,\eqref{eq:sifdef} and using the relation $\xi=r/L_{\mathrm{A}}$ ($L_{\mathrm{A}}$ is the distance from the scaling centre to the boundary along the direction of the crack, see Fig.\,\ref{fig:crackrep}) at $\theta=0$ leads to \begin{align} \left\{ \begin{array}{c} K_{\mathrm{I}}\\ K_{\mathrm{II}} \end{array}\right\} = & \sqrt{2\pi L_{\mathrm{A}}}\left\{ \begin{array}{c} \boldsymbol{\Psi}_{\sigma_{yy}}^{\mathrm{(s)}}(\eta(\theta=0))\mathbf{c}^{\mathrm{(s)}}\\ \boldsymbol{\Psi}_{\tau_{xy}}^{\mathrm{(s)}}(\eta(\theta=0))\mathbf{c}^{\mathrm{(s)}} \end{array}\right\} \label{eq:sifsbfe} \end{align} \section{Quadtree mesh generation\label{sec:Quadtree-mesh-generation}} This section presents the developed algorithm for quadtree mesh generation. Fig.\,\ref{flowchart} shows the flow chart of the overall process. The entire generation process is automatic with minimal number of inputs required from the user, which include \begin{itemize} \item Maximum allowed number of seed points in a cell $(s_{max})$, \item Seed points on each boundary $(s_{b})$ and region of interest $(s_{roi})$, \item Maximum difference between the division levels of adjacent cells $(d_{max})$, which is equal to 1 for a balanced quadtree mesh. \end{itemize} \begin{figure} \caption{Flow chart of the quadtree mesh generation.} \label{flowchart} \end{figure} This section is organised based on Fig\,\ref{flowchart}. It first presents defining geometry using signed distance function, and assigning seed points on the boundary and the regions of interest. Detailed explanations of the meshing steps, which include generating the initial quadtree grid, trimming the boundary quadtree cells into polygons and merging cells surrounding a crack tip, are then followed. To facilitate the description of the meshing steps, Fig.\,\ref{qtreedes0} shows a square plate with a circular hole and two local refinement features to be used as an example throughout this section. An efficient computation of the global stiffness matrix, by taking advantage on the quadtree mesh, is described at the end of this section. \begin{figure} \caption{Example to illustrate the quadtree mesh generation process: a square plate with a circular hole. An additional circle and an inclined line (dashed lines) are included to control local mesh density.} \label{qtreedes0} \end{figure} \subsection{Define geometry using signed distance function} The geometry is defined by using the signed distance function \citep{Persson2004}. It provides all the essential information of a geometry and can be operated with simple Boolean operations to build up more complex geometries \citep{Talischi2012}. The signed distance function of a point $\mathbf{x}\in\mathbb{R}^{2}$ associated with a domain $\Omega$, which is a subset of $\mathbb{R}^{2}$, is given as \begin{equation} d_{\Omega}(\mathbf{x})=s_{\Omega}(\mathbf{x})\min_{\mathbf{y\in\partial\Omega}}\left\Vert \mathbf{x}-\mathbf{y}\right\Vert , \end{equation} where $\partial\Omega$ represents the boundary of the domain and $\left\Vert \mathbf{x}-\mathbf{y}\right\Vert $ is the \emph{Euclidean norm} in $\mathbb{R}^{2}$ with $\mathbf{y}\in\partial\Omega$. The sign function $s_{\Omega}(\mathbf{x})$ is equal $-1$ when $\mathbf{x}$ lies inside the domain and is equal 1 otherwise. This definition of the signed distance function is visualised in Fig.\,\ref{dispfunc}. A number of distance functions in MATLAB for simple geometries are given in \citet{Talischi2012}, including their Boolean operations. \begin{figure} \caption{Signed distance function of the points inside the domain ($\mathbf{x}_{1}$), on the boundary ($\mathbf{x}_{2}$) and outside the domain ($\mathbf{x}_{3}$ and $\mathbf{x}_{4}$)} \label{dispfunc} \end{figure} For each boundary and region of interest, a set of pre-defined seed points \citep{Greaves1999} is introduced to control the quadtree mesh density. There require four sets of predefined seed points for the example in Fig.\,\ref{qtreedes0}. Two sets are for the square and the circular hole representing the actual domain boundary. The number of seed points directly controls the local density of the quadtree cells and the quality of fitting the boundary. This is further discussed in Section\,\ref{sub:Polygon-boundary-cells}. The other two sets are for the large circle and inclined line controlling local mesh density only. \subsection{Initialise quadtree grid} The meshing process starts with covering the problem domain with a single square cell (the root cell). The dimension of the root cell is based on the larger one between the maximum vertical and maximum horizontal dimension of the geometry. The developed algorithm will check the number of seed points in the cell. If the number is larger than the predefined maximum allowed number, the cell will be divided into 4 equal-sized cells. This generation process is applied recursively until all the cells have seed points no more than the predefined value. For each recursive loop, the maximum difference between the division levels of adjacent cells $(d_{max})$ is enforced. For cells that have division level difference with the adjacent cells larger than $d_{max}$, the higher level cell is subdivided into 4 equal-sized cells. Fig.\,\ref{qtreedes1} shows the initial quadtree grid of the example in Fig.\,\ref{qtreedes0}. \begin{figure} \caption{Initial quadtree grid of the example in Fig.~\,\ref{qtreedes0}. Vertices with solid square markers are on the boundary, with square box markers are inside the domain, and without any markers are outside the domain.} \label{qtreedes1} \end{figure} \subsection{Trim boundary cells into polygons\label{sub:Polygon-boundary-cells}} The initial quadtree grid shown in Fig.\,\ref{qtreedes1} does not conform to the boundary. Those cells that have edges intersected with the boundary need to be identified and trimmed. By using the signed distance function, the locations of the vertices (inside the domain, on the domain boundary or outside the domain as shown in Fig.\,\ref{qtreedes1}) are identified based on the sign and value of the function. For edges containing two vertices with opposite signs, they are identified as the edges intersected with the boundary. For each of those edges, the intersection point with the boundary is computed. Some quadtree cells could have vertices very close to the boundary in comparison with the lengths of their edges. After trimming, poorly shaped polygon cells with some edges much shorter than the others could be generated and may adversely affect the mesh quality. To avoid this situation, the vertices that are within a threshold distance away from the boundary are identified and then moved to their closest points on the boundary. In the present work, $1/10$ of the length of the cell edge (based on the smallest cell attaching to the vertex) is used as the threshold value. The edges connecting to these vertices will no longer be cut by the boundary. The trade-off of this process is the presence of additional non-square cells that lead to additional computation of the stiffness matrix. This is discussed in Section\,\ref{sub:An-efficient-assembly}. \begin{figure} \caption{Model curved boundary by quadtree refinement or using high-order elements. Nodes are represented with small circles along the cell edges.} \label{qtreedes2} \end{figure} At the end of the trimming process, the edges of a cell cut by the boundary are updated with the intersection points and the enclosed segment of boundary is added to the cell. This will result in polygon cells. After trimming the quadtree in Fig.\,\ref{qtreedes1}, the polygon cells around the hole of the example problem is shown in Fig.\,\ref{qtreedes2}. It is clear from Fig.\,\ref{qtreedes2} that the circular boundary is not represented accurately if a single linear element is used on the edge of the cell. In order to represent the curved boundary more accurately, two alternates are available in the developed algorithm. The first is to reduce the element size ($h$-refinement). This is achieved by increasing the number of seed points on the curved boundary. Fig.\,\ref{qtreedes3} shows the initial quadtree layout of the example problem after increasing the seed points around the hole by 4 times. It can be seen by comparing Fig.\,\ref{qtreedes1} with Fig.~\ref{qtreedes3} that the refinement is limited to a small region around the hole. The refined quadtree (Fig\,\ref{qtreedes2}) demonstrates the improvement of capturing the circular boundary. \begin{figure} \caption{Quadtree mesh after refinement} \label{qtreedes3} \end{figure} The second option to improve the modelling of curved boundaries is to utilise high-order elements ($p$-refinement). Fig.\,\ref{qtreedes2} shows the example problem with each line segment on the circular boundary modelled with a 4th order element. With this approach, curved boundaries can be captured more accurately using fewer elements. Both options to improve the modelling of the boundaries can be applied simultaneously without conflicts. The numerical accuracy of both approaches is discussed through numerical examples given in Section\,\ref{sec:Numerical-examples}. \subsection{Merge cells surrounding a crack tip} Owing to the capability of the SBFEM for fracture analysis \citep{Song2002}, the domain containing a crack tip is modelled with a single cell. In the stress solution, the variation along the radial direction, including the stress singularity, is given analytically and the variation along the circumference of the cell is represented numerically by the line elements on boundary. To obtain accurate results, sufficient nodes have to present on the boundary of the cell to cover the angular variation of the solution . In the developed algorithm, the size of a cell containing a crack tip is controlled, as shown in Fig.\,\ref{qtreedes8} with an inclined crack, by a predefined set of seed points on a circle. \begin{figure} \caption{Quadtree mesh for a crack problem before and after merging cells. The two crack tips are marked with a cross. The two circles are to control the size of quadtree cells covering the crack tips.} \label{qtreedes8} \end{figure} For problems with cracks, only one additional step is required after the initial mesh is generated. The cells surrounding the crack tip are refined to the same division level and then merged into a single cell as shown in Fig.\,\ref{qtreedes8}. This step avoids having a crack tip too close to the edges of the cell, which could affect the mesh quality and the solution accuracy \citep{Ooi2010a}. After the cells are merged, the intersection point between the edge of the resulting cell and the crack is computed to define the two crack mouth points. The other cells on the crack path are split by the crack into two cells. The splitting process is similar to the trimming of cells by the boundary, but two vertices are created at every intersection point between the cell edge and the crack to split the original cells. \subsection{An efficient construction of the global stiffness matrix\label{sub:An-efficient-assembly}} The global stiffness matrix is simply the assembly of the stiffness matrices of each master quadtree and polygon cell. When the $2:1$ rule is enforced to the mesh, only 6 main types of master quadtree cells are present as given in Fig.\,\ref{fig:qtreecell}. By rotating the geometry of the master cells orthogonally, the maximum number of types of these master quadtree cells are 24. For isotropic homogeneous materials, rotation does not have effect on 4-node or 8-node cells and only two 2 rotations are required for the first type of 6-node cell (the top one in Fig.\,\ref{fig:qtreecell}). The maximum number of master quadtree cells that require stiffness matrix calculation reduces to 16 (only 15 in \citet{Tabarraei2007} as 4-node cell is excluded). After the mesh generation, the algorithm will check which master cells out of the 16 appear in the mesh. Their stiffness matrices are then computed and stored. During the stiffness assembling process, the stiffness matrix of each regular quadtree cell is directly extracted from those computed stiffness matrices. For the polygon cells and those irregular quadtree cells (with their vertices moved to fit the boundaries), individual stiffness matrix calculation is required. This approach clearly improves the computational efficiency of constructing the global stiffness matrix, especially for large scale problems that contain a significant number of cells. With the use of high-order elements in the quadtree mesh, this assembling approach becomes even more economical. \section{Numerical examples\label{sec:Numerical-examples}} This section presents five numerical examples to highlight the capability and the performance of the proposed technique. In the first example, an infinite plate with a circular hole is modelled and the results are compared with the analytical solution. The proposed technique is then used to analyse a square plate with multiple holes to highlight the automatic meshing capability in handling transition between geometric features. In the third example, a square plate with a central hole and multiple cracks is studied to demonstrate the performance of the proposed technique in handling complicated geometries with singularities. Thereafter, a square plate with two cracks cross each other is analysed. It is aimed to emphasise the automation and simplicity of the mesh generation in the proposed technique. In the first four examples, the same material properties, with Young's modulus $E=100$ and Poisson's ratio $v=0.3$, are used. The final example is a cracked nuclear reactor under internal pressure. It is aimed to show the simplicity of the present technique in modelling practical non-regular structures. The computation time reported in this section is based on a desktop PC with Intel(R) Core(TM) i7 3.40GHz CPU and 16GB of memory. The proposed technique is implemented in MATLAB and the computation time is extracted in interactive mode of MATLAB. \subsection{Infinite plate with a circular hole under uniaxial tension} \subsubsection{Modelling using exact boundary condition} An infinite plate containing a circular hole with radius $a$ at its centre is considered in this example. The plate is subject to a uniaxial tensile load as shown in Fig.\,\ref{openhole}. The analytical solution of the stresses in polar coordinates $(r,\theta)$ is given by \citep{Sukumar2001}: \begin{align} \sigma_{11}(r,\theta) & =1-\frac{a^{2}}{r^{2}}\left(\frac{3}{2}\cos2\theta+\cos4\theta\right)+\frac{3a^{4}}{2r^{4}}\cos4\theta\nonumber \\ \sigma_{22}(r,\theta) & =-\frac{a^{2}}{r^{2}}\left(\frac{1}{2}\cos2\theta-\cos4\theta\right)-\frac{3a^{4}}{2r^{4}}\cos4\theta\label{eq:exohstr}\\ \sigma_{12}(r,\theta) & =-\frac{a^{2}}{r^{2}}\left(\frac{1}{2}\sin2\theta+\sin4\theta\right)+\frac{3a^{4}}{2r^{4}}\sin4\theta\nonumber \end{align} The displacement solutions are: \begin{align} u_{1}(r,\theta) & =\frac{a}{8\mu}\left[\frac{r}{a}(\kappa+1)\cos\theta+\frac{2a}{r}\left((1+\kappa)\cos\theta+\cos3\theta\right)-\frac{2a^{3}}{r^{3}}\cos3\theta\right]\nonumber \\ u_{2}(r,\theta) & =\frac{a}{8\mu}\left[\frac{r}{a}(\kappa-3)\sin\theta+\frac{2a}{r}\left((1-\kappa)\sin\theta+\sin3\theta\right)-\frac{2a^{3}}{r^{3}}\sin3\theta\right],\label{eq:exohdisp} \end{align} where $\mu$ is the shear modulus and $\kappa=\frac{3-v}{1+v}$ is the Kolosov constant for plane stress condition. The problem is solved by analysing a finite dimension of the plate with a dimension of $L\times L$ (see Fig.\,\ref{openhole}). Analytical traction (Eq.\,\ref{eq:exohstr}) is applied at the four edges of this finite plate. \begin{figure} \caption{Infinite plate with a circular hole under uniaxial tension} \label{openhole} \end{figure} \begin{figure} \caption{Mesh of a finite square plate with a circular hole ($L/a=10$)} \label{ohm1} \label{ohm2} \label{openholemesh} \end{figure} Fig.\,\ref{ohm1} shows the quadtree mesh of the plate for $L/a=10$. Each edge on a quadtree cell is discretised with 1st order line elements. The $2:1$ rule is enforced. Based on the proposed technique, the curved boundary is handled as shown in Fig.\,\ref{ohm2} with polygon cells. Convergence study is conducted based on the $h-$refinement. Three different element orders $(p=1,2,4)$ are investigated. Fig.\,\ref{openholecon} shows the present results of the relative error in the displacement norm $\left\Vert {\rm {\rm \mathbf{u}}-{\rm \mathbf{u}}}^{h}\right\Vert _{L^{2}(\Omega)}$, with ${\rm \mathbf{u}}$ the analytical solution given in Eq.\,\eqref{eq:exohdisp} and ${\rm \mathbf{u}}^{h}$ the solution computed by the proposed technique. The results show that all three types of elements have monotonic convergence. For higher order elements, more accurate results with similar number of DOF are obtained and the convergence rate is also faster. \begin{figure} \caption{Convergence results of the infinite plate with a circular hole, where $p$ is the element order and $m$ is the slope of the fitted line } \label{openholecon} \end{figure} There are 37 out of 100 cells calculated for the stiffness matrices. Among those 37 cells, 9 are regular quadtree cells and 28 are polygon cells surrounding the hole. For the remaining cells, their stiffness matrices are simply extracted from those 9 regular quadtree cells. To further demonstrate the accuracy of the proposed technique, $\sigma_{\theta}/\sigma$ along $A-B$ (see Fig.\,\ref{openhole}) is plotted in Fig.\,\ref{openholesigmatheta} using the mesh given in Fig.\,\ref{ohm1} with 4th order elements. It can be seen that the results of the proposed technique agree well with the analytical solution, which has $\sigma_{\theta}/\sigma=3$ at $A$ ($\theta=90^{\circ},r=a$). For points away from $A$, $\sigma_{\theta}/\sigma$ approaches 1. \begin{figure} \caption{Thin square plate with a single circular hole under uniaxial tension} \label{openholesigmatheta} \end{figure} \subsubsection{Approximation of infinite plate by varying $L/a$ ratio } The same infinite plate can be approximated by increasing the $L/a$ ratio. The application of quadtree mesh facilitates such a study. Only the left and right sides of the plate are subjected to uniaxial in-plane tension stress $\sigma$. The element order used in this study is $p=4$. The same mesh given in Fig.\,\ref{ohm1} is used for $L/a=10$. The adaptive capability of quadtree mesh leads to the same mesh pattern for all $L/a$ ratios. Fig.\,\ref{ohm3} shows the cells around the hole for $L/a=640$ and it is exactly the same as the one shown in Fig.\,\ref{ohm2}. For $L/a=640$, although there are 316 cells in total, only 37 cells are calculated for the stiffness matrices, which is the same as the previous study. The results of $\sigma_{\theta}/\sigma$ at $A$ with varying $L/a$ ratio are given in Table\,\ref{openholesigmathetavsratio}. It is seen that the analytical solution ($\sigma_{\theta}/\sigma=3$) is quickly approached when increasing the $L/a$ ratio. \begin{figure} \caption{Cell pattern around the hole for $L/a=640$} \label{ohm3} \end{figure} \begin{table} \caption{Normalised stress ($\sigma_{\theta}/\sigma$) at $A$ of the thin square plate with a circular hole} \centering{} \begin{tabular}{ccc} \hline $L/a$ ratio & No. of Nodes & $\sigma_{\theta}/\sigma$ at $A$\tabularnewline \hline 10 & 860 & 3.3591\tabularnewline 40 & 1428 & 3.0204\tabularnewline 160 & 1996 & 3.0049\tabularnewline 640 & 2564 & 2.9991\tabularnewline \hline \end{tabular}\label{openholesigmathetavsratio} \end{table} \subsection{Square plate with multiple holes} A unit square plate with 9 randomly distributed holes of different sizes, shown in Fig.\,\ref{multiholes}, is analysed. This example highlights the automation and flexibility of the proposed technique in handling the mesh transition between features with various dimensions. The ability of capturing curved boundaries accurately using high-order elements is also demonstrated. The displacements at the bottom edge of the plate are fully constrained and a uniform tension $P=1$ is applied at the top edge of the plate. A set of consistent units are chosen. \begin{figure} \caption{Thin square plate multiple holes under uniaxial tension} \label{multiholes} \end{figure} To validate the results, the same problem is solved using the commercial FEA software ANSYS V14.5. The plate is discretised using 8-node quadrilateral elements (PLANE183). In order to demonstrate the automation and performance of the proposed technique, similar user inputs are given to ANSYS to generate a mesh for comparison. In ANSYS, the square plate is divided into 4 equal-sized quadrants such that a centre key point is created for result comparison. The 9 holes are introduced to the plate by subtracting their areas from the square plate. The mesh constructed in ANSYS is unstructured (paving). A single variable $(N)$ is used to control the mesh density and is used for mesh refinement. For each hole, the number of element around them is equal to $4N$. And for all the straight lines, the size of the elements is set to be $1/3N$, which gives each outer edge approximately $3N$ elements. Note that the mesh used in ANSYS is far from optimal and structured mesh should be used for better performance. However, the main objective using paving mesh and only controlling the boundary element divisions is to show how the proposed technique and ANSYS perform when minimal number of controlling variables are used for the meshing. For the proposed technique, seed points with $s_{b}=(4N\times s_{max})$, where $s_{max}$ is the maximum allowed number of seed points in a cell, are set on the circular holes to generate a mesh with similar number of boundary division as the one in ANSYS. Fig.\,\ref{multiholesmesh} shows the ANSYS mesh (1358 elements) and the quadtree mesh (1169 cells). Both meshes can effectively handle the mesh transition between the holes. As commented earlier, while the mesh in ANSYS can be further improved by designing a structured layout, it would also require additional time and investigation effort. The amount of additional effort depends on the complexity of the geometry and user experience. And for the proposed technique, the resulting mesh is always in a structured manner (see Fig.\,\ref{mhm2}) without additional effort. The time of generating the quadtree mesh is around 3s using the computer with details outlined in the beginning of this section. \begin{figure} \caption{Mesh of a square plate with 9 circular holes} \label{mhm1} \label{mhm2} \label{multiholesmesh} \end{figure} For the stiffness calculation, there are 364 out of 1169 cells calculated. Among those 364 cells, 12 are master quadtree cells and 352 are polygon cells surrounding the holes. The stiffness matrices for all the other cells are simply extracted from the calculated master quadtree cells. The total time from constructing the stiffness to obtaining the displacement solutions is less than 3.2s when using 5th order elements. Table\,\ref{multiholestab1} shows the convergence of the displacement components at the centre point $A$ with increasing element order using mesh in Fig.\,\ref{mhm2}. In order to highlight the convergence performance, Fig.\,\ref{multiholescon} shows the relative error of the present results of the displacement vector sum at point $A$. The error is calculated based on the converged ANSYS results, which converged to the first 6 significant digits. Also shown in the same figure are the relative errors of the ANSYS results and another set of results of the proposed technique. They are both generated through a series of $h-$refinement with the use of 2nd order elements. It is observed that the present results with $p-$refinement converge with the fastest rate. And for the $h-$refinement, the present results are basically converging at the same rate as those of ANSYS with slightly better accuracy. The present results demonstrate that for the same accuracy, much less number of DOFs is required when using high-order elements, and curved boundaries are also accurately modelled with minimal number of cells. \begin{table} \caption{Centre displacement results of a square plate with 9 circular holes} \centering{} \begin{tabular}{cccc} \hline Elem. Order & No. of Nodes & $u_{x}\times10^{4}$ at $A$ & $u_{y}\times10^{3}$ at $A$\tabularnewline \hline 2 & 4379 & 4.98298 & 6.67236\tabularnewline 3 & 7157 & 4.98350 & 6.67359\tabularnewline 4 & 9935 & 4.98241 & 6.67374\tabularnewline 5 & 12713 & 4.98197 & 6.67379\tabularnewline \hline \end{tabular}\label{multiholestab1} \end{table} \begin{figure} \caption{Convergence results of the centre displacement vector sum for the square plate with multiple holes, where $p$ is the element order and $m$ is the slope of the fitted line } \label{multiholescon} \end{figure} In order to further demonstrate the overall consistency of the present results, Fig.\,\ref{multiholessy} shows the contour plots of $\sigma_{y}$, from both ANSYS and the proposed technique. Good agreement is observed from the contour plots. \begin{figure} \caption{$\sigma_{y}$ of the square plate with random pattern of holes} \label{mhsya} \label{mhsy} \label{multiholessy} \end{figure} \subsection{Square plate with multiple cracks emanating from a hole} A square plate of length $L$ with a centre hole of radius of $r$ given in Fig.~\ref{holemulticracks} is considered. $n$ cracks with crack length $a$ emanate from the hole. This example aims to show the simplicity and effectiveness of the proposed technique to solve problems with singularities. \begin{figure} \caption{Thin square plate with cracks emanating from a hole under bi-axial tension} \label{holemulticracks} \end{figure} In this example, to approach the assumption of an infinite plate, $r/L=0.01$ is considered. A parametric study is performed considering $n=2,4,8$ cracks surrounding the hole with various $s=\frac{a}{a+r}$ ratio. The element order used in this study is $p=4$, which is capable to model the circular boundary accurately as shown in the first example. The present results are compared with the reference solution of the stress intensity factor given in \citet{tada2000stress}. Fig.\,\ref{holemulticracksmesh} shows the mesh around the central hole, with 4 and 8 cracks around the edge and $s=0.6$. Based on the proposed technique, no refinement is required around the crack tips. This facilitates the study with multiple cracks and results in less computational effort when comparing to the conventional FEM. \begin{figure} \caption{Mesh of the square plate with cracks emanating from a hole. The black dots represent the crack tips.} \label{hmcm1} \label{hmcm2} \label{holemulticracksmesh} \end{figure} Fig.\,\pageref{holemulticracksres} shows the stress intensity factor ($F_{I}=K_{I}/(\sigma\sqrt{\pi a})$) computed from the proposed technique for different value of $s$. Excellent agreement with the reference solution \citep{tada2000stress} is observed. This demonstrates the accuracy of the proposed technique in dealing with stress singularities as well as the feasibility in handling geometry with complicated features. \begin{figure} \caption{Stress intensity of the square plate with cracks emanating from a hole} \label{holemulticracksres} \end{figure} \subsection{Square plate with two cracks cross each other } A square plate of length $L$ with two cracks cross each other is considered. The dimensions of the plate and the cracks as well as the boundary conditions are shown in Fig.~\ref{crosscrack}. This example highlights the automatic mesh generation of the proposed technique and the capacity to handle problems with complicated crack configuration. \begin{figure} \caption{Thin square plate with two cracks cross each other} \label{crosscrack} \end{figure} Fig.\,\ref{crosscrackmesh} shows the quadtree mesh of the proposed technique. The mesh only requires defining seed points on the domain boundary, along the cracks and around the crack tips to control the quadtree mesh density. The mesh generation is fully automatic without the requirement of dividing area regions. The resulting mesh contains a total of 216 cells. During the construction of the stiffness matrix, only 48 cells are computed, which contains 9 master quatree cells and 39 polygon cells. For the same problem, it would require a few more steps to generate a mesh in FEA. These include defining crack tip regions that directly affect the solution accuracy, and designing proper refinement strategy that directly affects the convergence performance. For example in ANSYS, a command \emph{``}KSCON'' needs to be issued to each crack tip in order to generate two circular layers of elements (1 layer singular elements) around the tip. The radius of the two circular layers of elements is solely based on user experience and \emph{trial-and-error}. Shape warning on the elements would occur if the settings of that command are not consistent with the global mesh. Moreover, automatic $h-$refinement is not applicable when ``KSCON'' is activated. It would, therefore, require multiple steps to conduct convergence study with $h-$refinement, such as reducing the radius of the circular layers of element around the crack tips and increasing elements in circumferential direction around the crack tips. \begin{figure} \caption{Mesh of a square plate with two cracks cross each other (216 cells)} \label{crosscrackmesh} \end{figure} Table\,\ref{crosscracktab} shows the crack opening displacements ($op1$ and $op2$ in Fig.\,\ref{crosscrack}) with increasing element order. Similar to previous examples, the present results converge rapidly with minimal number of nodes increased. The results between using the 2nd order elements and using the 5th order elements are different with less than $0.04\%$. \begin{table*} \caption{Crack opening displacement: $u_{op1}$ for the opening $op1$ and $u_{op2}$ for the opening $op2$} \noindent \centering{} \begin{tabular}{cccc} \hline Elem. Order & No. of Nodes & $u_{op1}\times10^{-3}$ & \multicolumn{1}{c}{$u_{op2}\times10^{-3}$}\tabularnewline \hline 2 & 818 & 5.1300 & 6.9710\tabularnewline 3 & 1333 & 5.1274 & 6.9727\tabularnewline 4 & 1848 & 5.1279 & 6.9726\tabularnewline 5 & 2363 & 5.1280 & 6.9725\tabularnewline \hline \end{tabular}\label{crosscracktab} \end{table*} \subsection{Cracked nuclear reactor under internal pressure} In this final example, a nuclear reactor under internal pressure \citep{Simpson2013} is analysed. Due to symmetry, only a quadrant of the reactor is modelled. The geometry, material properties, loading and dimension are shown in Fig.\,\ref{nreact}. Also shown in the figure are the two cracks introduced on the outer boundary. This example shows the flexibility of the proposed technique and the developed meshing algorithm to model more practical structures. \begin{figure} \caption{Cracked nuclear reactor under internal pressure} \label{nreact} \end{figure} Fig.\,\ref{nreactmesh} shows the quadtree mesh used in this example, which contains a total of 160 cells. Using the proposed technique only requires seed points to be defined at the boundaries to control the quadtree mesh density. No additional requirement for the cells containing the crack tips is necessary. The time spent on generating the quadtree mesh is less than 0.8s using the same computer with details outlined in the beginning of this section. The calculation of stiffness matrix involves computing 44 out of the 160 cells, in which 12 are master quadtree cells and 32 are polygon cells. The total time from constructing the stiffness matrix to obtaining the displacement solution is less than 0.7s when using 5th order elements. \begin{figure} \caption{Mesh for the quadrant of the reactor with 2 cracks (160 cells)} \label{nreactmesh} \end{figure} A convergence study is conducted by increasing the order of the element without changing the quadtree layout in Fig.\,\ref{nreactmesh}. Table\,\ref{nreacttab1} shows the two crack opening displacements at points $A$ and $B$ (Fig.\,\ref{nreact}). The present results converge quickly with the element order increased. The difference between using the 2nd order elements and using the 5th order elements is less than 0.1\%. This further highlights the advantage of using high-order elements that can model curved boundary more accurately with minimal number of cells. \begin{table*} \caption{Crack opening displacement: $u_{A}$ for opening at$(25,40)$, $u_{B}$ for opening at $(45,75)$} \noindent \centering{} \begin{tabular}{cccc} \hline Elem. Order & No. of Nodes & $u_{A}\times10^{2}$ & \multicolumn{1}{c}{$u_{B}\times10^{2}$}\tabularnewline \hline 2 & 635 & 7.14372 & 2.60404\tabularnewline 3 & 1031 & 7.15066 & 2.60336\tabularnewline 4 & 1427 & 7.15077 & 2.60334\tabularnewline 5 & 1823 & 7.15077 & 2.60332\tabularnewline \hline \end{tabular}\label{nreacttab1} \end{table*} \section{Conclusion\label{sec:Conclusion}} This paper has presented a numerical technique to automate stress and fracture analysis using the SBFEM and quadtree mesh of high-order elements. Owing to the nature of the SBFEM, the proposed technique has no specific requirement, such as deriving conforming shape functions or sub-triangulation, to handle quadtree cells with hanging nodes. High-order elements are used within each quadtree cell directly. The quadtree mesh generation is fully automatic and involves minimal number of user inputs and operation steps. Boundaries are modelled with scaled boundary polygons and this allows the proposed technique to conform the boundary without excessive mesh refinement. The meshing algorithm is also applicable for problems with singularities. The use of quadtree mesh leads to an efficient approach to compute the global stiffness matrix. This facilitates the analysis that requires a significant number of cells using high-order elements. Five numerical examples are presented to highlight the functionality and performance of the proposed technique. The present results show excellent agreement with analytical solutions and those computed by the FEM. \section*{Reference} \section*{\textmd{\normalsize{ }}} \end{document}

\begin{document} \title{Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves} \tableofcontents \section*{Abstract} The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory. \section{Introduction} The Trefftz Discontinuous Galerkin (TDG) method proposed in \cite{git09} is a mesh based method for approximating solutions of the Helmholtz equation. This method generalizes the Ultra Weak Variational Formulation (UWVF)~of the same problem~\cite{cessenat_phd,despres} by allowing different weighting strategies on penalty terms in the TDG method. Error analysis \cite{buf07,git09,HMP11,hmp13,hmp15} and computational experience \cite{hut03} show that the method can be an efficient way of approximating solutions of the Helmholtz equation. It has also become clear that the method works best in an $hp$-mode (see \cite{hmp15}) where large elements are used away from boundaries in the computational domain, together with larger numbers of plane waves. However because of the use of simple Trefftz functions (usually plane waves element by element), it has to be assumed that the coefficients in the governing partial differential equation are piecewise constant. Of course smoothly varying coefficient functions could be first approximated by a piecewise constant function and then the resulting perturbed Helmholtz equation could be solved by TDG or UWVF. But this would require small elements and hence defeat some of the potential advantages of using large elements in a Trefftz based scheme. To circumvent the difficulty with smoothly varying coefficients, we propose to use approximate solutions of the underlying partial differential equation constructed element by element where the coefficients are variable. In this work we use the Generalized Plane Waves (GPW) of \cite{IGD2013,IG2015} as a basis for the TDG type scheme. Note that smoothly varying coefficients arise in the simulation of electromagnetic wave propagation in tokamaks where the permittivity is spatially variable and may even become negative. Indeed the original design of GPWs in \cite{IGD2013,IG2015} was motivated precisely by this application. To describe the setting for applying GPWs in more detail let us consider the following model problem from \cite{hmp13}. Given a bounded Lipschitz polyhedron $\Omega$ that is star shaped with respect to the origin and a larger Lipschitz polyhedron $\Omega_R$ containing $\overline{\Omega_D}$, we define the computational domain to be the annulus $\Omega=\Omega_R\setminus\overline{\Omega_D}$. The two boundaries of $\Omega$ are $\Gamma_D=\partial\Omega_D$ and $\Gamma_R=\partial\Omega_R$ and we use a normal $\mathbf{n}$ that is outwards from $\Omega$. Because we shall use some regularity results from \cite{hmp13} we need to assume that $\Omega_R$ is star-like with respect to a ball of radius $\gamma_Rd_\Omega$ centered at the origin where $\gamma_R>0$ and $d_\Omega$ is the diameter of $\Omega$. Suppose we are give a wave number $\kappa>0$. In addition given a strictly positive, piecewise smooth and bounded real function $\epsilon\in L^{\infty}(\Omega)$ and another function $g_R\in L^2(\Gamma_R)$, we want to approximate the solution $u$ of \begin{eqnarray} \Delta u+\kappa^2 \epsilon u&=&0\mbox{ in }\Omega,\label{helm}\\ u&=&0\mbox{ on }\Gamma_D,\label{dirich}\\ {\partial_n u}+{\rm{}i}\kappa u&=&{\rm{}i}\kappa g\mbox{ on }\Gamma_R.\label{imp} \end{eqnarray} As pointed out in \cite{hmp13} this is a model problem for scattering (for example of an $s$-polarized electromagnetic wave from a perfect conductor embedded in a dielectric in two dimensions). The impedance boundary condition is then a simple radiation boundary condition. As we shall detail shortly, if we assume that the function $\epsilon$ is piecewise analytic on each element, we can approximate it by a power series. With this in hand we shall give details of a recursive algorithm for generating the coefficients of basis functions on each element that satisfy (\ref{helm}) to high accuracy. These are constructed so that if the coefficient $\epsilon$ is constant on an element, the resulting basis function is just a plane wave. Using these generalized plane waves we can prove convergence of a modified TDG scheme. The resulting discrete problem obtains high order convergence for smooth solutions as is the case for the standard TDG or UWVF. We only consider $h$-convergence in this paper. As we shall see, the main disadvantage of the use of GPWs in the TDG method is the need to integrate over elements in the grid. We would prefer to use them in a generalized UWVF avoiding this integration. However, although numerical experiments are encouraging \cite{LM_thesis} we do not have a theoretical justification of this approach. The paper proceeds as follows. In the next section we briefly outline our modification of the basic TDG method. Then in Section~\ref{GPW} we show how to construct GPWs and then obtain two new error estimates for these functions that will underlie our error analysis. We also show that piecewise linear functions can be approximated element by element using the GPW functions. In Section~\ref{EE} we derive error estimates for the new TDG scheme with GPW basis functions. Finally in Section~\ref{NT} we give some basic numerical tests of the new algorithm. \section{The Plane Wave Discontinuous Galerkin Method} Even with a variable coefficient $\epsilon$, the choice of domain $\Omega$ and the conditions on $\epsilon$ guarantee that if $g_R\in H^s(\Gamma_R)$ for some sufficiently small $s>0$ (depending on the interior angles of $\Gamma_R$) then $u\in H^{3/2+s}(\Omega)$~\cite[Theorem 2.3]{hmp13}. This regularity will allow us to develop consistent fluxes for $u$ and it's derivatives. Unfortunately, because $\epsilon$ is variable, the dependence on $\kappa$ of the continuity constants for estimates in this paper is not easy to track. Therefore we note now that constants in the analysis will depend in an unspecified way on $\kappa$. As is usual for DG schemes, we start with a mesh and continue to define the method using definitions from \cite{hmp13}. Suppose we cover $\Omega$ by a finite element mesh ${\cal T}_h$ of regular triangular elements $K$ of maximum diameter $h$ (in fact more general domains can easily be allowed). The diameter of an element $K$ is denoted $h_K$. In addition following \cite{hmp13} we assume \begin{enumerate} \item {\em Local quasi-uniformity}: there exists a constant $\tau\geq 1$ independent of $h$ such that \[ \tau^{-1}\leq \frac{h_{K_1}}{h_{K_2}}\leq \tau\mbox{ for all triangles }K_1,K_2\mbox{ meeting at any edge}. \] \item {\em Quasi-uniformity close to $\Gamma_R$}: For all triangles $K$ touching $\Gamma_R$ there is a constant $\tau_R$ independent of $h$ such that \[ \frac{h}{h_K}\leq \tau_R. \] \end{enumerate} Let $\mathbf{n}_K$ denote the unit outward normal to element $K$. Let $K$ and $K'$ denote two elements in ${\cal T}_h$ meeting at an edge $e$ then, on $e$, we make the standard definitions of the average value and jump of functions across $f$: \begin{eqnarray*} \mv{u}=\frac{u|_{K}+u|_{K'}}{2},&\quad&\mv{\mathbf{\sigma}}=\frac{\mathbf{\sigma}|_K+\mathbf{\sigma}|_{K'}}{2},\\ \jmp{u}=u|_K\mathbf{n}_K+u|_{K'}\mathbf{n}_{K'},&\quad& \jmp{\mathbf{\sigma}}=\mathbf{\sigma}|_{K}\cdot\mathbf{n}_K+\mathbf{\sigma}|_{K'}\cdot\mathbf{n}_{K'}. \end{eqnarray*} We denote by ${\cal E}_h$ the set of all edges in the mesh. Then let \begin{itemize} \item ${\cal E}_I$ denote the set of all edges in the mesh interior to $\Omega$, \item ${\cal E}_D$ is the set of all boundary edges on $\Gamma_D$, \item ${\cal E}_R$ is the set of all boundary edges on $\Gamma_R$. \end{itemize} We also need three positive penalty parameters that are functions of position on the skeleton of the mesh: $\alpha$, $\beta$ and $\delta$. At this point these are simply assumed to be positive functions of position on ${\cal E}$ and will be given in more detail shortly. Using the above defined jumps and average values, we are lead to consider the following standard sesquilinear form for TDG \cite{hmp13,MelenkEsterhazy12,kapita14}: \begin{eqnarray} A_h(u,v)&=& \int_{\Omega}\left(\nabla_hu\cdot\nabla_h\overline{v} -\kappa^2\epsilon u\,\overline{v}\right)\,dA-\int_{{\cal E}_I}\left(\avg{\nabla_h u}\cdot\jmp{\overline{v}} +\jmp{ u}\cdot\avg{\nabla_h\overline{v}}\right)\,ds\nonumber\\&& -\frac{1}{i\kappa}\int_{{\cal E}_I}\beta\jmp{\nabla_h u}\jmp{\nabla_h\overline{v}}\,ds +{i\kappa}\int_{{\cal E}_I}\alpha\jmp{ u}\cdot\jmp{\overline{v}}\,ds-\int_{{\cal E}_R}\delta u {\partial_n \overline{v}}\,ds\nonumber\\&& -\int_{{\cal E}_R}\delta{\partial_n u}\overline{v}\,ds -\frac{1}{i\kappa}\int_{{\cal E}_R}\delta{\partial_n u}{\partial_n \overline{v}}\,ds+i\kappa\int_{{\cal E}_D}\alpha u\overline{v}\,ds\nonumber\\&& +i\kappa\int_{{\cal E}_R}(1-\delta)u\overline{v}\,ds -\int_{{\cal E}_D} \left({\partial_n u}\overline{v}+u{\partial_n \overline{v}}\right)\,ds.\label{Ahalt} \end{eqnarray} Here $\nabla_h$ is the piecewise defined gradient and $\partial_n u =\nabla_h u\cdot n$ element by element. In addition the right had side is given by \[ F(v)=-\frac{1}{{\rm{}i}\kappa}\int_{{\cal E}_R}\delta g\,{\partial_n \overline{v}}\,ds+\int_{{\cal E}_R} (1-\delta )g\overline{v}\,ds \] By virtue of the regularity of the solution of (\ref{helm})-(\ref{imp}) noted above, it satisfies \[ A_h(u,v)=F(v)\] for all sufficiently smooth test functions $v$ (for example piecewise $H^2$ is sufficient). Now suppose we wish to discretize the problem. Let $V_h\subset \Pi_{K\in T_h}H^2(K)$ be a finite dimensional space. If $V_h$ is chosen to consist of piecewise smooth solutions of (\ref{helm}), we have the standard TDG and seek an approximate $u_h\in V_h$ that satisfies \[ A_h(u_h,v)=F(v)\quad\mbox{ for all }v\in V_h. \] For piecewise constant media, the space $V_h$ can be chosen in many ways. One choice uses Bessel functions (good for conditioning but bad for computational speed because of the need for quadrature), another, more standard choice, uses plane waves (typically worse conditioned but easier to use since integrals can be computed in closed form)~\cite{HMP13I}. However if $\epsilon(x,y)$ is non-constant on an element we cannot use simple solutions of the Helmholtz equation. In this paper we assume that $\epsilon(x,y)$ is a smooth function on each element (but may be discontinuous between elements). Then, as we shall shortly describe, the space $V_h$ can be constructed using ``Generalized Plane Waves'' (GPWs) that approximately satisfy the Helmholtz equation. However we have been unable to prove convergence for the standard TDG method in this case, and instead add a stabilizing term to the sesquilinear form so define \[ B_h(u,v)=A_h(u,v)+\frac{i}{\kappa^2}\int_\Omega\gamma (\Delta u+\kappa^2\epsilon u)\overline{(\Delta v+\kappa^2\epsilon v)}\,dA. \] Here $\gamma>0$ is a new penalty parameter that is a piecewise constant function of position on the mesh. We note that the new term vanishes on elements with constant material coefficients allowing plane waves to be used and their the method reduces to the standard TDG. Now we seek $u_h\in V_h$ such that \begin{equation}\label{Bhelm} B_h(u_h,v)=F(v)\quad\forall v\in V_h. \end{equation} In the next section we describe how to construct GPWs element by element and hence complete the specification of the method. \section{Generalized Plane Waves}\label{GPW} In this section we focus specifically on GPWs. Firstly we describe the design process, including an explicit algorithm to build a local set of GPWs on a given element of the mesh $\mathcal T_h$. Secondly we turn to interpolation of a solution of \eqref{helm} and prove error estimates. We provide various interpolation properties of such a set of local GPWs on a given element of the mesh, and derive a global interpolation property on the whole domain $\Omega$ by piecewise GPWs. Thirdly we prove a result on approximation of the space of bi-variate polynomials of degree 1 by GPWs. This result will be useful for the error analysis. \subsection{Design and interpolation properties} GPWs have been introduced in \cite{LM_thesis,IGD2013}. They generalize the use of classical plane waves, as exact solutions of an equation with piecewise constant coefficients, to the case of variable coefficients. The GPWs are not exact solutions of (\ref{helm}) but approximately solve the equation element by element. Their design process is based on a Taylor expansion and ensures that the homogeneous equation is locally satisfied up to a given order on each element $K$ of the mesh. On a given element $K$, consider the centroid $(x_{K},y_{K})$. A GPW on $K$ is a function $\varphi = e^{P}$ where \begin{equation}\label{eq:polnotation} P(x,y)=\sum_{i=0}^{{\rm{}d}_P} \sum_{j=0}^{{\rm{}d}_P-i} \lu{i}{j} \left(x-x_K\right)^i \left(y-y_K\right)^j, \end{equation} ${\rm{}d}_P$ being the total degree of the polynomial $P$. A GPW is designed to be an approximate solution of the Helmholtz equation: the polynomial coefficients $\left\{ \lu{i}{j},0\leq i+j\leq {\rm d}_P \right\}$ are computed from the Taylor expansion of the variable coefficient $\epsilon$ in order for the function $\varphi = e^{P}$ to satisfy \begin{equation}\label{eq:-lapl+al} [ \Delta + \kappa^{2}\epsilon ] e^{P(x,y)} = O \left(\| (x,y)-(x_K,y_K) \|^q\right). \end{equation} The parameter $q$ is the order of approximation of the equation. Canceling all the terms of order less than $q$ in the Taylor expansion \eqref{eq:-lapl+al} is equivalent to a non linear system of $q(q+1)/2$ non linear equations. The unknowns of this system are the $({\rm{}d}_P+1)({\rm{}d}_P+2)/2$ coefficients of $P$. Setting simultaneously ${\rm{}d}_P = q+1$ and giving the values of the $2q+3$ coefficients $\left\{ \lu{i}{j}, i\in \{0,1\}, 0\leq j \leq q+1-i \right\}$ leads to a unique solution of the non linear system. This solution is explicitly expressed as \begin{equation}\label{eq:IF} \begin{array}{l} \forall (i,j) \text{ s.t. } 0\leq i+j \leq q-1, \\ \displaystyle \lu{i+2}{j} = \frac1{(i+2)(i+1)}\Bigg(-\kappa^2\frac{\partial_x^i \partial_y^j \epsilon\left(x_K,y_K\right)}{i!j!} - (j+2)(j+1) \lu{i}{j+2} \\ \displaystyle \phantom{\frac{\partial_x^i\beta\left(\mathbf{g}_K\right) \partial_y^j \beta\left(\mathbf{g}_K\right)}{i!j!} = } -\sum_{k=0}^{i} \sum_{l=0}^{j} (i-k+1)(k+1) \lu{i-k+1}{j-l} \lu{k+1}{l} \\ \displaystyle \phantom{\frac{\partial_x^i\beta\left(\mathbf{g}_K\right) \partial_y^j \beta\left(\mathbf{g}_K\right)}{i!j!} = } -\sum_{k=0}^{j} \sum_{l=0}^{i} (j-k+1)(k+1) \lu{i-l}{j-k+1} \lu{l}{k+1}\Bigg), \end{array} \end{equation} where $\partial_x=\partial/\partial_x$ and $\partial_y=\partial/\partial y$. More precisely, as defined in \cite{IG2015}, a GPW at $(x_K,y_K)$ corresponds to the following normalization : \begin{itemize} \item[$\bullet$] $\lambda_{0,0}=0$, \item[$\bullet$] $(\lambda_{1,0},\lambda_{0,1})=N (\cos \theta,\sin\theta)$, for some $N\in\mathbb C$ and $\theta\in\mathbb R$, \item[$\bullet$] $ \lu{i}{j}=0$ for $i\in\{0,1\}$ and $1<i+j\leq q+1$. \end{itemize} A local set of linearly independent GPWs is then obtained for a given value of $N$ by considering $p$ equi-spaced directions $\theta_{l} = 2\pi(l-1)/p$ for $1\leq l \leq p$. The interpolation properties of this set of functions are the main topic of \cite{IG2015}. The main result of that paper provides a sufficient condition on the parameters p and q to achieve a high order interpolation of a smooth solution of \eqref{helm} by GPWs, as well as a high order interpolation of its gradient. We will denote by $GPW_\kappa^{p,q}(K)$ the space spanned by the $p$ GPWs corresponding to $\theta_{l} = 2\pi(l-1)/p$ for $1\leq l \leq p$ and $N=\sqrt{ -\kappa^2 \epsilon(x_K,y_K)}$, or $N=\imath \kappa\sqrt{ \epsilon(x_K,y_K)}$. Let ${\mathcal C}^k(S)$ denote the set of functions with $k$ continuous derivatives on a set $S$. As a reminder, with the present notation, the interpolation result reads: \begin{theorem}\label{th:u-ua} Consider $K\in \mathcal T_h$ together with $n\in\mathbb N$ such that $n>0$. Assume that $q\geq n+1$, $p=2n+1$ and $\mathbf{g}_K=(x_K,y_K)\in K$ is the centroid of $K$. Finally suppose the coefficient $\epsilon \in\mathcal C^{q-1}(K)$. Consider a solution $u$ of scalar wave equation \eqref{helm}, satisfying $u\in \mathcal C^{n+1}$. Then there is a function $u_a\in GPW_\kappa^{p,q}(K)$ implicitly depending on $\epsilon$ and its derivatives, and a constant $C(\kappa,K,n)$, implicitly depending on $\epsilon$ and its derivatives as well, such that: for all $\mathbf{m}\in K$ \e{\label{eq:gradumua} \left\{ \begin{array}{l} \left| u\left(\mathbf{m}\right)-u_a\left(\mathbf{m}\right)\right| \leq C(\kappa,K,n) \left|\mathbf{m}-\mathbf{g}_K\right|^{n+1} \left\| u \right\|_{\mathcal C^{n+1}(K)} ,\\%\phantom \nabla \left\| \nabla u\left(\mathbf{m}\right)-\nabla u_a\left(\mathbf{m}\right)\right\| \leq C(\kappa,K,n) \left|\mathbf{m}-\mathbf{g}_K\right|^{n} \left\| u \right\|_{\mathcal C^{n+1}(K)}. \end{array} \right. } \end{theorem} The following interpolation property of higher order derivatives stems directly from the proof of the previous theorem. \begin{theorem}\label{th:u-ua2} Consider $K\in \mathcal T_h$ together with $n\in\mathbb N$ such that $n>0$. Assume that $q\geq n+1$, $p=2n+1$ and $\mathbf{g}_K=(x_K,y_K)\in K$ is the centroid of $K$. Finally suppose the coefficient $\epsilon \in\mathcal C^{q-1}(K)$. Consider a solution $u$ of scalar wave equation \eqref{helm}, satisfying $u\in \mathcal C^{n+1}$. Then the function $u_a\in GPW_\kappa^{p,q}(K)$ and the constant $C(\kappa,K,n)$ provided by Theorem \ref{th:u-ua} also satisfies : for all $\mathbf{m}\in K$ and all $j$ such that $0\leq j \leq k$ \e{\label{eq:derumua} \left| \partial_x^{j}\partial_y^{k-j}u\left(\mathbf{m}\right)-\partial_x^{j}\partial_y^{k-j}u_a\left(\mathbf{m}\right)\right| \leq C(\kappa,K,n)\frac{(n+1)!}{(n+1-k)!} \left|\mathbf{m}-\mathbf{g}_K\right|^{n+1-k} \left\| u \right\|_{\mathcal C^{n+1}(K)} , } where $k\leq n$. Moreover there is a constant $\mathfrak C(\kappa,K,n)$ such that for all $\mathbf{m}\in K$ \e{\label{eq:opua} \left| [\Delta +\kappa^2\epsilon]u_a\left(\mathbf{m}\right)\right| \leq \mathfrak C(\kappa,K,n) \left|\mathbf{m}-\mathbf{g}_K\right|^{n+1} \left\| u \right\|_{\mathcal C^{n+1}(K)} . } \end{theorem} \begin{proof} The interpolation property of GPWs for any derivative of $u$ directly stems from the Taylor expansion of $u-u_{a}$, exactly as for the gradient and this proves \eqref{eq:derumua}. The design of GPWs directly yields that for all $l$ such that $1\leq l \leq p$ the corresponding GPW satisfies $$ \left| [\Delta +\kappa^2\epsilon]\varphi_{l} \right| \leq C_{l} \left|\mathbf{m}-\mathbf{g}_K\right|^{q }. $$ Moreover $\displaystyle u_a= \sum_{l=1}^{2n+1} \mathsf X_l \varphi_l$ and it was already noticed in \cite{IG2015} that $|\mathsf X_l|\leq C(\kappa,K,n)\|u\|_{\mathcal C^{n+1}}$. As a result $$ \left| [\Delta +\kappa^2\epsilon]u_a\left(\mathbf{m}\right)\right| =\left| \sum_{l=1}^{2n+1} \mathsf X_l [\Delta +\kappa^2\epsilon]\varphi_l \right| \leq C(\kappa,K,n)\|u\|_{\mathcal C^{n+1}}\left|\mathbf{m}-\mathbf{g}_K\right|^{q } \sum_{l=1}^{2n+1} C_{l} $$ and so \eqref{eq:opua} holds $\displaystyle \mathfrak C (\kappa,K,n) =C(\kappa,K,n) \sum_{l=1}^{2n+1} C_{l} $. \end{proof} The last step to build, from the local functions spaces $GPW_\kappa^{p,q}(K)$, a set of GPWs on the whole domain $\Omega$: the GPWs space $V_h$ is naturally defined as $ \prod_{K\in T_h}GPW_\kappa^{p,q}(K)$. Note that $p$ and $q$ can vary from element to element. As a result, we have the following estimate for $[\Delta +\kappa^2\epsilon] (u-v_h)$, where $u$ is a smooth solution of Equation \eqref{helm}: \begin{lemma}\label{helm-est} Suppose that $u$ is a solution of scalar wave equation \eqref{helm} which belongs to $\mathcal C^{n+1}(\Omega)$. Then the function $v_h\in V_h=\prod_{K\in T_h}GPW_\kappa^{p,q}(K)$, provided element by element by Theorem \ref{th:u-ua}, satisfies : there exists a constant $C$ independent of $h$ such that \[ \Vert [\Delta +\kappa^2\epsilon] (u-v_h) \Vert_{L^2(\Omega)} \leq C\mbox{area}(\Omega)^{1/2}\left(\max_{K\in T_h} h_K\right)^q\Vert u\Vert_{{\mathcal C}^q(\Omega)} \] where $h_K$ is the radius of $K$. \end{lemma} \subsection{Approximation of linear polynomials} The result \cite[Lemma 3.10]{git09} addresses the approximation of bi-variate polynomials of degree 1 by classical plane waves, and here we are interested in the approximation of bi-variate polynomials of degree 1 by GPWs. This result is needed to apply the $h$-based analysis of \cite{git09} or \cite{kapita14}. \begin{lemma}\label{linfun} Consider $\hat K\in [0,1]^2$ the reference element. Suppose $n\in\mathbb N$ is such that $n\geq 2$. For $p=2n+1$ and $q\geq n+1$ there is a constant $C$ independent of $\kappa$ (but not of $p$) such that $$ \inf_{v\in GPW_\kappa^{p,q}\left(\hat K\right)} \|f-v\|_{0,\hat K} \leq C\kappa^2|\epsilon(x_{\hat K},y_{\hat K})|\|f\|_{0,\hat K},\ \forall f\in \mathcal P_1(\mathbb R^2). $$ \end{lemma} \begin{remark}The proof strongly relies on the fact that the GPW space is designed with the normalization $(\lambda_{1,0}^k,\lambda_{0,1}^k) = \imath \kappa\sqrt{ \epsilon(x_{\hat K},y_{\hat K})}(\cos\theta_k,\sin\theta_k)$, for equi-spaced angles $\theta_k$, for $1\leq k \leq p$.\end{remark} \begin{proof} For the sake of clarity we define $\tilde \kappa = \kappa\sqrt{ \epsilon(x_{\hat K},y_{\hat K})}$. Consider $$ b_j :=(i\tilde \kappa)^{-[j/2]}\sum_{k=1}^p \alpha_k^{(j)} \varphi_k $$ where the $\varphi_k$s are the GPWs, $\alpha_k^{(j)} = (\mathsf{M}_p)^{-1}$, $p=2m+1$ and $\mathsf{M}_p\in\mathbb R^{p,p}$ is defined for $1\leq k,l \leq p$ by $$ (\mathsf M_p)_{kl} :=\left\{ \begin{array}{l} 1\text{ for }l = 1 \\\displaystyle \cos\left(\frac{l}{2}\theta_k\right)\text{ for }l \text{ even} \\\displaystyle \sin\left(\frac{l-1}{2}\theta_k\right)\text{ for }l \geq 3 \text{ odd} \end{array} \right. $$ We know that $$ \sum_{k=1}^p \alpha_k^{(j)} \varphi_k = \sum_{n=0}^\infty\frac{1}{n!}\sum_{k=1}^p \alpha_k^{(j)} (P_k(x,y))^n , $$ and, if $\hat{\mathbf x} = (\hat x,\hat y)=(x-x_{\hat K},y-y_{\hat K})$, then $$ \begin{array}{rl} (P_k(x,y))^n &\displaystyle= \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} +\tilde \kappa^2 f_{k,q}({\mathbf{x}}) \right)^n \text{ see Lemma \ref{lem:Pk}} \\&\displaystyle= \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^n +\tilde \kappa^n\sum_{j=0}^{n-1}\begin{pmatrix} n\\ j \end{pmatrix} \imath^j\tilde \kappa^{n-j}\left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^j f_{k,q}({\mathbf{x}})^{n-j} \\&\displaystyle= \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^n +\tilde \kappa^{n+1}g_{k,q}({\mathbf{x}}), \end{array} $$ the function $g_{k,q}$, defined as $g_{k,q}({\mathbf{x}}) = \sum_{j=0}^{n-1}\begin{pmatrix} n\\ j \end{pmatrix} \imath^j\tilde \kappa^{n-1-j}\left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^j f_{k,q}({\mathbf{x}})^{n-j}$, being uniformly bounded for $\mathbf x =(x,y)\in \hat K$ as $\tilde \kappa\rightarrow 0$. We assume that $(\lambda_{1,0}^k,\lambda_{0,1}^k) = \imath \tilde \kappa(\cos\theta_k,\sin\theta_k)$. Define $$ K_{ 0 }^n(\mathbf{x}) = \frac{1}{2\pi}\int_{-\pi}^{\pi} \left(\begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\\hat y \end{pmatrix}\right)^n d\theta \quad \text{ and }\forall 1\leq j\leq n: $$ $$ K_{2j }^n(\mathbf{x}) = \frac{1}{\pi}\int_{-\pi}^{\pi} \left(\begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix}\right)^n \cos(j\theta)d\theta \text{, } K_{2j+1}^n (\mathbf{x})= \frac{1}{\pi}\int_{-\pi}^{\pi} \left(\begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix}\right)^n \sin(j\theta)d\theta. $$ The leading order term of $(P_k(x,y))^n$ as $\tilde \kappa\rightarrow 0$ is $\left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^n$. As shown in \cite{git09} , it can be written $ \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right)^n =(\imath\tilde\kappa)^n\sum_{l=1}^{2n+1}K_l^n(\mathbf{x})\mathsf{M}_{lk} $ so that $$ \begin{array}{l}\displaystyle \sum_{n=0}^\infty\frac{1}{n!}\sum_{k=1}^p \alpha_k^{(j)} \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right) ^n \\\displaystyle = \sum_{n=0}^\infty\frac{(\imath\tilde \kappa)^n}{n!} \sum_{l=1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}\\\displaystyle = \sum_{n=0}^m\frac{(\imath\tilde \kappa)^n}{n!} \sum_{l=1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}+ \sum_{n=m+1}^\infty\frac{(\imath\tilde \kappa)^n}{n!} \sum_{l=1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}, \end{array} $$ and since $K_j^n=0$ if $[j/2]>n$ and $\sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}=\delta_{jl}$ for $1\leq l,j\leq p$, it yields $$ \begin{array}{rl}\displaystyle \sum_{n=0}^\infty\frac{1}{n!}\sum_{k=1}^p \alpha_k^{(j)} \left( \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} \right) ^n &\displaystyle = \sum_{n=[j/2]}^m\frac{(\imath\tilde \kappa)^n}{n!}\left(K_j^n(\mathbf{x})+ \sum_{l=p+1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}\right)\\&\displaystyle+ \sum_{n=m+1}^\infty\frac{(\imath\tilde \kappa)^n}{n!} \left(K_j^n(\mathbf{x})+ \sum_{l=p+1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}\right)\\ &\displaystyle = \sum_{n=[j/2]}^m\frac{(\imath\tilde \kappa)^n}{n!}K_j^n(\mathbf{x})+ \tilde \kappa^{m+1}R_j(\tilde \kappa,\mathbf{x}) \end{array} $$ since $l/2\geq m+1\Rightarrow [K_l^n = 0$ for $n\leq m]$, and where the remainder function $R_j$ defined by $$ R_j(\kappa,\mathbf{x}) = \frac{1}{\tilde \kappa^{m+1}} \sum_{n=m+1}^\infty\frac{(\imath\tilde \kappa)^n}{n!} \left(K_j^n(\mathbf{x})+ \sum_{l=p+1}^{2n+1}K_l^n(\mathbf{x}) \sum_{k=1}^p \alpha_k^{(j)}\mathsf{M}_{lk}\right) $$ is uniformly bounded on $\hat K$. So $$ \sum_{k=1}^p \alpha_k^{(j)} \varphi_k = \sum_{n=[j/2]}^m \left( \frac{(\imath\tilde \kappa)^n}{n!}K_j^n(\mathbf{x}) +\tilde \kappa^{n+1}g_{k,q}(\mathbf{x}) \right) +\tilde \kappa^{m+1}R_j(\tilde \kappa,\mathbf{x}) $$ Since $b_j(\mathbf{x}) =(i\tilde \kappa)^{-[j/2]}\sum_{k=1}^p \alpha_k^{(j)} \varphi_k $ it clearly shows that $$ \lim_{\tilde \kappa\rightarrow 0} b_j(\mathbf{x}) = \frac{1}{{[j/2]}!}K_j^{[j/2]}(\mathbf{x}). $$ As a consequence, the definition of $K_j^{[j/2]}$ combined with $((\lambda_{1,0}^k,\lambda_{0,1}^k) = \imath \tilde \kappa(\cos\theta_k,\sin\theta_k))$ lead to $$ b_1(\mathbf{x}) = 1 + O(\tilde \kappa^2),\ b_2(\mathbf{x}) = x-x_K + O(\tilde \kappa^2),\ b_3(\mathbf{x}) = y-y_K + O(\tilde \kappa^2). $$ \end{proof} To complete the proof of Lemma \ref{linfun}, we need to prove the following result. \begin{lemma}\label{lem:Pk} Suppose $n\in\mathbb N$ is such that $n\geq 2$. For $p=2n+1$ and $q\geq n+1$, consider the basis of $p$ functions $\varphi_k\in GPW_\kappa^{p,q}\left(\hat K\right)$ approximating \eqref{helm} at the point $\mathbf g_{\hat K}=(x_{\hat K},y_{\hat K})$ at order $q$, and, for all $k$ such that $1\leq k\leq p$, the corresponding polynomials $\displaystyle P_k(x,y)=\sum_{0\leq i+j\leq q+1} \lambda_{i,j}^k (x-x_{\hat K})^i (y-y_{\hat K})^j$ satisfying $\varphi_k=\exp P_k$. These polynomials satisfy $$ P_k(x,y)= \begin{pmatrix} \lambda_{1,0}^k\\ \lambda_{0,1}^k \end{pmatrix} \cdot \begin{pmatrix} \hat x \\ \hat y \end{pmatrix} +\kappa^2 \epsilon(x_K,y_K) f_{k,q}({\mathbf{x}}) $$ where $\mathbf x = (x,y)$, $(\hat x,\hat y)=(x-x_{\hat K},y-y_{\hat K})$ and the remainder function $f_{k,q}$ is uniformly bounded on $\hat K$. \end{lemma} \begin{proof} The normalization $(\lambda_{1,0}^k,\lambda_{0,1}^k) = \imath \kappa\sqrt{ \epsilon(x_K,y_K)}(\cos\theta_k,\sin\theta_k)$ implies that $\lu{2}{0}^k=0$ so that the induction formula \eqref{eq:IF} reads: \begin{equation}\left\{ \begin{array}{cl} \lu{2}{j}^k &\displaystyle = -\frac{1}{2}\frac{\kappa^2\partial_y^j\epsilon(x_K,y_K)}{j!} \\ \lu{3}{j}^k &\displaystyle = -\frac{1}{6}\left( \frac{\kappa^2\partial_x\partial_y^j\epsilon(x_K,y_K)}{j!} + 4\lu{2}{j}^k\lambda_{1,0}^k\right)\\ \lu{i+2}{j}^k &\displaystyle = -\frac{1}{(i+2)(i+1)}\Bigg( \frac{\kappa^2\partial_x^i\partial_y^j\epsilon(x_K,y_K)}{i!j!} + (j+2)(j+1)\lu{i}{j+2}^k\lambda_{1,0}^k \qquad \forall i>1,\\ &\displaystyle \phantom = + \sum_{k=1}^{i-1}\sum_{l=0}^{j} (i-k+1)(k+1)\lu{i-k+1}{j-l}^k\lu{k+1}{l}^k\\ &\displaystyle \phantom = + \sum_{l=1}^{i-1}\sum_{k=0}^{j} (j-k+1)(k+1)\lu{i-l}{j-k+1}^k\lu{l}{k+1}^k\Bigg) \end{array}\right. \end{equation} This clearly completes the proof by induction. \end{proof} More precisely, the result needed in the following result that corresponds to Lemma 3.12 from \cite{git09}. We state it here to specify the GPWs parameters, and provide no more than a sketch of the proof since it relies on Lemma \ref{linfun} but not specifically on the basis function set. \begin{corollary} \label{lincor} Suppose that $n\in\mathbb N$ is such that $n\geq 2$. For $p=2n+1$ and $q\geq n+1$, suppose $w_h^c$ is a linear function on a triangle $K$ then there is a GPW function $w_h\in GPW_\kappa^{p,q}(K)$ such that \[ \Vert w_h^c-w_h\Vert_{L^2(K)}\leq Ch_K^2\Vert w_h^c\Vert_{L^2(K)}. \] \end{corollary} \begin{proof} By translation and dilation by $1/h_K$ we can map an element $K$ to an element $\tilde{K}\subset \hat{K}= (0,1)^2$. Let $\hat{w}^c$ denote the transformed polynomial and note that $\hat{w}_h^c\in \mathcal P_1(\mathbb R^2)$. Let $\hat P$ the $L^2(\hat K)$-projection onto the plane wave space $GPW_{\hat{\kappa}}^{p,q}(\hat{K})$ where $\hat{\kappa} = h_K \kappa$. Applying Lemma \ref{linfun}, we get { \begin{equation}\label{eq:est} \| (I-\hat P )\hat{w}_h^c\|_{0,\hat K} \leq C\hat{\kappa}^2|\epsilon(x_{\hat K},y_{\hat K})| \| \hat{w}_h^c \|_{0,\hat K}. \end{equation}} The conclusion then follows by transforming back to $K$ using the fact that the transformation from a triangle $K$ to the reference triangle $\tilde K$ changes the frequency into $\hat \kappa = h_K \kappa$ { and the bound $\| \hat{w}_h^c \|_{0,\hat K}\leq C \| \hat{w}_h^c \|_{0,\tilde K} $ with $C$ independent of $\tilde{K}$ which holds because $ \hat{w}_h^c$ is a linear polynomial and the mesh is regular.} \end{proof} \section{Error Estimates}\label{EE} In this section we start by establishing a straightforward error estimate using the coercivity and boundedness of the sesquilinear form $B_h(\cdot,\cdot)$, and then prove convergence in the global $L^2$ norm. We define the obvious modification of the DG norm from \cite{hmp13} by \begin{eqnarray*} \Vert u\Vert_{DG}^2&=& \frac{1}{\kappa}\Vert \beta^{1/2}\jmp{\nabla_h u}\Vert_{L^2({\cal E}_I)}^2+\kappa\Vert \alpha^{1/2}\jmp{ u}\Vert_{L^2({\cal E}_I)}^2+\frac{1}{\kappa}\Vert \delta^{1/2}\partial_nu\Vert_{L^2({\cal E}_R)}^2\\&&\quad+\kappa\Vert (1-\delta)^{1/2}u\Vert_{L^2({\cal E}_R)}^2+\kappa\Vert \alpha^{1/2}u\Vert_{L^2({\cal E}_D)}^2+\frac{1}{\kappa^2}\Vert \gamma^{1/2}(\Delta_h u+\kappa^2\epsilon u)\Vert_{L^2(\Omega)}^2 \end{eqnarray*} where $\Delta_h$ is the Laplacian defined piecewise element by element. Obviously this is a semi-norm, but due to the new term is also a norm even on functions that do not exactly satisfy the Helmholtz equation. \begin{lemma} The semi-norm $\Vert \cdot\Vert_{DG}$ is a norm. \end{lemma} \begin{proof} Suppose $\Vert u\Vert_{DG}=0$ then $u$ satisfies $\Delta u+\kappa^2\epsilon u=0$ element-wise, and the normal derivatives and function values have no jump across interior edges. So $\Delta u+\kappa^2\epsilon u=0$ in $\Omega$. In addition the Cauchy data vanishes and so $u=0$ in $\Omega$. Hence $\Vert \cdot\Vert_{DG}$ is a norm. \end{proof} We also need a new DG+ norm: \begin{eqnarray*} \Vert u\Vert_{DG+}^2&=&\Vert u\Vert_{DG}^2+\kappa\Vert \beta^{-1/2}\avg{u}\Vert_{L^2({\cal E}_I)}^2+\frac{1}{\kappa}\Vert \alpha^{-1/2}\avg{\nabla_h u}\Vert_{L^2({\cal E}_I)}^2+ \kappa\Vert\delta^{-1/2} u\Vert^2_{L^2({\cal E}_R)}\\&&\quad+\frac{1}{\kappa}\Vert \alpha^{-1}\partial_n u\Vert^2_{L^2({\cal E}_D)}+\kappa^2\Vert\gamma^{-1/2} u\Vert_{L^2(\Omega)}. \end{eqnarray*} The following estimates hold: \begin{lemma} Under the assumption that $\alpha>0$, $\beta>0$, $1>\delta>0$ and $\gamma>0$ in the generalized TDG, and provided $u$ is such that $\Vert u\Vert_{DG}$ is finite, \[ \Im(B_h(u,u))\geq \Vert u\Vert_{DG}^2. \] Provided $\Vert u\Vert_{DG+}$ and $\Vert v\Vert_{DG}$ are finite, there exists a constant $C$ independent of $\kappa$, $u$ and $v$ such that \[ |B_h(u,v)|\leq C \Vert u\Vert_{DG+}\Vert v\Vert_{DG}. \] \end{lemma} \begin{proof} The coercivity estimate follows from in the usual way by considering $\Im (B_h(u,u))$ and using the assumption that $\epsilon$ is real \cite{HMP11}. To obtain the desired continuity, we integrate the term $\nabla_h u\cdot\nabla_h v$ term in the definition of $A_h(\cdot,\cdot)$ by parts to obtain, for any $u,v\in V_h$, \begin{eqnarray} A_h(u,v)&=&-\int_\Omega u(\overline{\Delta_h v+\kappa^2\epsilon v})\,dA+\int_{{\cal E}_I} \avg{u}\jmp{\nabla_h \overline{v}}\,ds-\int_{{\cal E}_I}\avg{\nabla_h u}\jmp{\overline{v}}\,ds\nonumber \\&& +\int_{{\cal E}_R}(1-\delta) u{\partial_n \overline{v}}\,ds-\frac{1}{{\rm{}i}\kappa}\int_{{\cal E}_I}\beta\jmp{\nabla_h u}\jmp{\nabla_h \overline{v}}\,ds\nonumber\\ &&+{\rm{}i}\kappa\int_{{\cal E}_I}\alpha\jmp{u}\jmp{\overline{v}}\,ds-\frac{1}{{\rm{}i}\kappa}\int_{{\cal E}_R}\delta{\partial_n u}{\partial_n \overline{v}} \,ds\nonumber\\ &&+{\rm{}i}\kappa\int_{{\cal E}_R}(1-\delta)u\overline{v}\,ds-\int_{{\cal E}_R}\delta {\partial_n u}\,\overline{v}\,ds\label{UWVF}\\ &&-\int_{{\cal E}_D}{\partial_n u}\overline{v}+i\kappa\int_{{\cal E}_D}\alpha u\overline{v}. \nonumber \end{eqnarray} The result now follows from the definition of $B_h(u,v)$ and the Cauchy-Schwarz inequality. \end{proof} The following result is now a standard consequence of the above estimates~\cite{MelenkEsterhazy12,hmp13}: \begin{lemma} There is a unique solution $u_h\in V_h$ that satisfies (\ref{Bhelm}), and the following error estimate holds with constant $C$ independent of $\kappa$, $u$, and $u_h$: \begin{equation} \Vert u-u_h\Vert_{DG}\leq C \Vert u-v\Vert_{DG+}\quad \mbox{ for all }v\in V_h. \label{ceatype} \end{equation} \end{lemma} To obtain an order estimate, we need to make specific choices of the parameters $\alpha$, $\beta$, $\delta$ and $\gamma$. There are several choices in the literature depending on the precise setting of the problem (see for example \cite{buf07,git09,hmp13,hmp15}). In this paper we shall make the classical UWVF choice \cite{buf07}: \begin{equation} \alpha=\beta=\gamma=\delta=1/2,\label{p_uwvf} \end{equation} so that we can use results from~\cite{kapita14}. Then we choose for $\gamma$ \[ \gamma=\gamma_0 h_K^r \] where $\gamma_0$ is constant and $ r\geq 0$. We shall examine the role of $r$ later. Using the estimates from Section~\ref{GPW} we can then prove the following error estimate \begin{theorem}\label{th:u-uh} Suppose $n\in\mathbb N$ is such that $n\geq 2$ and consider $p=2n+1$ and $q\geq n+1$. Suppose $V_h$ is formed from $q$th order GPWs element by element using $p$ directions. Then the solution $u_h\in V_h$ of (\ref{Bhelm}) exists for all $h>0$ independent of $\kappa$ and it satisfies the following estimate with constant $C$ independent of $\kappa$, $u$, and $u_h$: \[ \Vert u-u_h\Vert_{DG}\leq C (h^{n-1/2}+h^{q+r/2}). \] Here $C$ depends on the $\Vert u\Vert_{{\cal C}^{\max(n+1,q)}(\Omega)}$ norm of $u$. \end{theorem} \begin{remark} Since we need $q\geq n+1$ in the GPW theory, we see that the choice $q=n+1$ guarantees that the approximation of the Helmholtz equation is high enough order. \end{remark} \begin{proof} We pick $v\in V_h$ in equation (\ref{ceatype}) element by element to be the approximation by GPWs denoted by $u_a$ in Theorem \ref{th:u-ua2}. We now need to estimate each term in $\Vert u-v\Vert_{DG+}$. Using Lemma~\ref{helm-est} the new term \begin{equation}\label{Luest} \Vert \gamma^{1/2}(\Delta_h (u-v)+\kappa^2\epsilon (u-v)\Vert_{L^2(\Omega)} = \Vert \gamma^{1/2}(\Delta_h v+\kappa^2\epsilon v)\Vert_{L^2(\Omega)} \leq Ch^{q+r/2} \Vert u\Vert_{{\cal C}^q}. \end{equation} The remaining terms can be estimated in using Theorem \ref{th:u-ua2}. For example \begin{eqnarray*} \Vert \alpha^{-1/2}\avg{\nabla_h (u-v)}\Vert^2_{L^2({\cal E}_I)}&\leq &C \sum_K\Vert \alpha^{-1/2}\nabla_h (u-v)\Vert_{L^2(\partial K)}^2\\ &\leq&C\sum_K \max_{e\in\partial K}\alpha^{-1}(e)\Vert \nabla_h (u-v)\Vert_{L^2(\partial K)}^2\\ &\leq&C\sum_K \max_{e\in\partial K}\alpha^{-1}(e)\left[h_K^{-1}\Vert \nabla_h (u-v)\Vert_{L^2(K)}^2+h_K\Vert\nabla\nabla (u-v)\Vert_{L^2(K)}^2\right] \end{eqnarray*} where we have used the standard trace estimate on $\partial K$. Using Theorem \ref{th:u-ua2} with $k=2$ and Theorem \ref{th:u-ua} we obtain \begin{eqnarray*} \Vert \alpha^{-1/2}\avg{\nabla_h (u-v)}\Vert^2_{L^2({\cal E}_I)}&\leq &C\sum_K \max_{e\in\partial K}\alpha^{-1}(e)h^{2n-1}_K h_K^2\Vert u\Vert^2_{{\cal C}^{n+1}(K)}\\&\leq& C\max_{e\in {\cal E}_I}\alpha^{-1}(e)h^{2n-1}\Vert u\Vert_{{\cal C}^{n+1}(\Omega)}^2. \end{eqnarray*} Of course under our assumptions $\alpha=1/2$. The remaining terms are estimated in the same way. \end{proof} We now use the standard duality approach to prove an $L^2(\Omega)$ norm estimate on the error. \begin{theorem}\label{th:L2cv} Suppose we choose $r=3$ in the penalty parameter $\gamma$, $p=2n+1$, $n\geq 2$ and $q=n+1$. Then there exists a constant $C$ depending on $\kappa$ but independent of $h$ such that \[ \Vert u-u_h\Vert_{L^2(\Omega)}\leq C h^{s} \Vert u-u_h\Vert_{DG} \] for some $s$ with $0<s<1/2$ depending on $\Omega$ (given in \cite[Theorem 2.3]{hmp13}). \end{theorem} Under best possible conditions we then have the following convergence estimate: \begin{corollary}\label{uuhcor} Suppose $u$ is a smooth solution of the Helmholtz equation, that $r=3$, $p=2n+1$, $n\geq 2$ and $q= n+1$ then \[ \Vert u-u_h\Vert_{L^2(\Omega)}\leq C h^{n+s-1/2} \] \end{corollary} \begin{remark} Since $s\leq 1/2$ the maximum rate of convergence predicted for the method assuming a smooth solution and best regularity is $O(h^n)$. \end{remark} \begin{proof} Define the dual variable $z\in H^1(\Omega)$ to satisfy \begin{eqnarray*} \Delta z+\kappa^2\epsilon z&=& u-u_h\mbox{ in }\Omega,\\ {\partial_n z}-i\kappa z&=&0\mbox{ on }\Gamma_R,\\ z&=&0\mbox{ on }\Gamma_D. \end{eqnarray*} Under the assumptions on the domain, it is easy to see that $z\in H^{3/2+s}(\Omega)$, $s>1/2$, \cite{hmp13} is sufficiently regular that \[ A_h(\xi,z)=\int_\Omega \xi\overline{(u-u_h)}\,dA \] for all test function $\xi$ that are $H^2$ piecewise smooth. This follows from (\ref{UWVF}). Hence, by the definition of $B_h$ \[ B_h(\xi,z)=\frac{1}{\kappa^2}\int_\Omega \gamma (\Delta_h\xi+\kappa^2\epsilon \xi)\overline{(u-u_h)}\,dA+\int_\Omega \xi\overline{(u-u_h)}\,dA \] so choosing $\xi=u-u_h$ and letting $z_h\in V_h$ be arbitrary \[ \Vert u-u_h\Vert_{L^2(\Omega)}^2=B(u-u_h,z-z_h)-\frac{1}{\kappa^2}\int_\Omega \gamma (\Delta_h(u-u_h)+\kappa^2\epsilon (u-u_h))\overline{(u-u_h)}\,dA. \] The second term on the right hand side can be estimated using the Cauchy-Schwarz and arithmetic-geometric mean inequality to give \begin{eqnarray*} \left|\int_\Omega \gamma (\Delta_h(u-u_h)+\kappa^2\epsilon (u-u_h))\overline{(u-u_h)}\,dA\right| &\leq &\frac{1}{k}\Vert u-u_h\Vert_{DG}\Vert \gamma^{1/2} (u-u_h)\Vert_{L^2(\Omega)}\\&\leq& \frac{\gamma_{max}}{2\kappa^2}\Vert u-u_h\Vert_{DG}^2+\frac{1}{2}\Vert u-u_h\Vert_{L^2(\Omega)}^2, \end{eqnarray*} where $\gamma_{max}=\max_{x\in\Omega}\gamma=O(h^r)$. To estimate $B_h(u-u_h,z-z_h)$ we integrate the grad-grad term in $A_h(u,v)$ by parts onto $u$ to obtain \begin{eqnarray} A_h(u,v)&=& -\int_{\Omega}(\Delta_h u+\kappa^2\epsilon u)\overline v\,dA+\int_{{\cal E}_I}\left(\jmp{\nabla_h u}\avg{\overline{v}} -\jmp{ u}\cdot\avg{\nabla_h\overline{v}}\right)\,ds-\frac{1}{i\kappa}\int_{{\cal E}_I}\beta\jmp{\nabla_h u}\jmp{\nabla_h\overline{v}}\,ds\nonumber\\&& +{i\kappa}\int_{{\cal E}_I}\alpha\jmp{ u}\cdot\jmp{\overline{v}}\,ds-\int_{{\cal E}_R}\frac{\delta}{i\kappa} (i\kappa u-{\partial_n u}){\partial_n \overline{v}}\,ds\nonumber\\&& +\int_{{\cal E}_R}(1-\delta)({\partial_n u}-i\kappa u)\overline{v}\,ds+\int_{{\cal E}_D}u(i\kappa\alpha \overline{v}-\partial_n\overline{v})\,ds. \label{Ahap}\end{eqnarray} Using this in the definition of $B_h(u,v)$ shows that \[ |B_h(u,v)|\leq C\Vert u\Vert_{DG}\Vert v\Vert_{DG+} \] where $C$ is independent of $u$ and $v$ so that we have the estimate \begin{equation} \Vert u-u_h\Vert_{L^2(\Omega)}^2\leq C \Vert u-u_h\Vert_{DG}\Vert z-z_h\Vert_{DG+} +\frac{\gamma_{max}}{2\kappa^2}\Vert u-u_h\Vert_{DG}^2. \label{eqest} \end{equation} It is now necessary to choose $z_h$. Following the proof of \cite[Theorem 5.6]{kapita14}, let $z_h^c$ denote the continuous piecewise linear finite element interpolant of $z$. We choose $z_h\in V_h$ to be the GPW approximation of $z_h^c$ constructed in Lemma~\ref{lincor}). Of course \[ \Vert z-z_h\Vert_{DG+}\leq \Vert z-z^c_h\Vert_{DG+}+\Vert z_h^c-z_h\Vert_{DG+} \] and it remains to estimate each term. Estimates from the proof of \cite[Theorem 5.6]{kapita14} show that on each interior edge in the mesh \begin{eqnarray*} \Vert\alpha^{-1/2}\avg{\nabla_h(z-z^c_{h})}\Vert_{L^2(e)} &\leq& C\sum_{j=1}^2h_{K_j}^{s}\vert z\vert_{H^{3/2+s}(K_j)}\\ \Vert\beta^{-1/2}\avg{z-z^c_{h}}\Vert_{L^2(e)}&\leq&C\sum_{j=1}^2h^{1+s}_{K_j}\vert z\vert_{H^{3/2+s}(K_j)}, \end{eqnarray*} with corresponding entries results for the jumps in the above quantities and for boundary terms. In addition \begin{eqnarray*} \Vert \gamma^{1/2}(\Delta_h (z-z_h^c)+\kappa^2\epsilon (z-z_h^c)\Vert_{L^2(\Omega)} &=& \Vert \gamma^{1/2}(u-u_h-(\Delta_h z_h^c+\kappa^2\epsilon z_h^c))\Vert_{L^2(\Omega)}\\ &\leq& Ch^{r/2}\Vert u-u_h\Vert_{L^2(\Omega)}+ \Vert \gamma^{1/2}(\Delta_h z_h^c+\kappa^2\epsilon z_h^c)\Vert_{L^2(\Omega)}. \end{eqnarray*} On an element $K$ we can use the regularity of the mesh to establish local inverse estimates and prove: \begin{eqnarray*} && \Vert \gamma^{1/2}\Delta_h z_h^c\Vert_{L^2(K)}\leq Ch_K^{r/2-1}\Vert z_h^c\Vert_{H^1(K)}\\ & \leq &Ch_K^{r/2-1}(\Vert z_h^c-z\Vert_{H^1(K)}+\Vert z\Vert_{H^1(K)})\leq Ch_K^{r/2-1}\Vert z\Vert_{H^{3/2+s}(K)}. \end{eqnarray*} Proceeding similarly for the lower order term, we conclude that provided $r/2>1$ we have \[ \Vert \gamma^{1/2}(\Delta_h (z-z_h^c)+\kappa^2\epsilon (z-z_h^c))\Vert_{L^2(\Omega)}\leq Ch^{r/2-1}\Vert z\Vert_{H^{3/2+s}(\Omega)}. \] In addition we must estimate \begin{eqnarray*} \Vert \gamma^{-1/2}(z-z_h^c)\Vert^2_{L^2(\Omega)}&=&\sum_K\int_K\gamma^{-1}(z-z_h^c)^2\,dA\leq C\sum_Kh_K^{3+2s-r}\Vert z\Vert_{H^{3/2+s}(K)}^2\\ &\leq& Ch^{3+2s-r}\Vert z\Vert_{H^{3/2+s}(\Omega)}^2. \end{eqnarray*} Taken together, if $3+2s\geq r\geq 2$ we have \[ \Vert z-z^c_h\Vert_{DG+}\leq C(h^{r/2-1}+h^{3/2+s-r/2})\Vert z\Vert_{H^{3/2+s}(\Omega)} \] A good choice is then $r=3$ since in that case $r/2-1\geq s$ and using the a priori estimate for $z$ from \cite[Theorem 2.3]{hmp13} \[ \Vert z-z^c_h\Vert_{DG+}\leq Ch^s\Vert u-u_h\Vert_{L^2(\Omega)}. \] It now remains to estimate $\Vert z_h^c-z_h\Vert_{DG+}$. As we have seen there are two troublesome terms: $\Vert \gamma^{-1/2}(z_h^c-z_h)\Vert^2_{L^2(\Omega)}$ and $\Vert \gamma^{1/2}(\Delta_h (z^c_h-z_h)+\kappa^2\epsilon (z_h^c-z_h))\Vert_{L^2(\Omega)}$ with the remaining terms following using Lemma \ref{linfun} as in \cite{kapita14}. Using first a local inverse estimate, then Lemma \ref{linfun}, \[ \Vert \gamma^{1/2}(\Delta_h (z^c_h-z_h)+\kappa^2\epsilon (z_h^c-z_h))\Vert_{L^2(K)}\leq Ch_K^{r/2-2}\Vert z_h^c-z_h\Vert_{L^2(K)} \leq Ch_K^{r/2}\Vert z_h^c\Vert_{L^2(K)} \] so that, squaring and adding, and using the a priori estimate for $z$ from \cite[Theorem 2.3]{hmp13} \begin{eqnarray*} \Vert \gamma^{1/2}(\Delta_h (z^c_h-z_h)+\kappa^2\epsilon (z_h^c-z_h))\Vert_{L^2(\Omega)}&\leq &Ch_K^{r/2}\Vert z_h^c\Vert_{L^2(\Omega)}\\ &\leq& Ch_K^{r/2}(\Vert z-z_h^c\Vert_{L^2(K)}+\Vert z\Vert_{L^2(K)})\\&\leq& Ch^{r/2}\Vert u-u_h\Vert_{L^2(\Omega)}. \end{eqnarray*} To estimate the global $L^2$ term, again using Lemma \ref{linfun}, \[ \Vert \gamma^{-1/2}(z_h^c-z_h)\Vert^2_{L^2(K)}\leq Ch_K^{-r/2} \Vert (z_h^c-z_h)\Vert^2_{L^2(K)} \leq Ch_K^{2-r/2} \Vert z_h^c\Vert^2_{L^2(K)} \] Adding over all elements and using the a priori estimate for $z$ from \cite[Theorem 2.3]{hmp13} \[ \Vert \gamma^{-1/2}(z_h^c-z_h)\Vert^2_{L^2(\Omega)}\leq Ch^{2-r/2} \Vert u-u_h\Vert^2_{L^2(K)} \] We have thus proved that when $r=3$ and noting $0<s<1/2$ we have \[ \Vert z-z_h\Vert\leq Ch^{s}\Vert u-u_h\Vert_{L^2(\Omega)}. \] so we conclude the desired result using this result in (\ref{eqest}).\end{proof} \section{Numerical Tests}\label{NT} We now test the GPW based RDG method on two test problems with a known solution: Airy waves (linear variation in $\epsilon$) and Weber Waves (quadratic variation in $\epsilon$). {In Figs.~\ref{ghqs}, \ref{ghcomb}, \ref{Wcomb} we plot the relative $L^2(\Omega)$ error in the computed solution against a parameter labeled $C/h$. This is computed using the total number of degrees of freedom $N_{\rm{}dof}$ and the number of directions per element $p=2n+1$ as $\sqrt{N_{\rm{}dof}/p}$. We choose this parameter since, in our theorems, convergence is in terms of mesh size rather than total number of degrees of freedom.} \subsection{Airy Waves.}\label{Airy} The simplest example of a spatially dependent refractive index is $\epsilon(x,y)=-y$ on the domain $[-1,1]\times[-1,1]$. We can then choose Dirichlet boundary data (for our theory we need an impedance boundary condition, but the same result holds in the case provided $\kappa$ is not an eigenvalue of the domain) such that the exact solution is \[ u(x,y)=Ai(\kappa^{2/3}y) \] where $Ai(r)$ is the Airy function as shown in the left panel of Fig.~\ref{FL1}. \begin{figure} \caption{Left: Exact Airy function solution. Right: Initial mesh.} \label{FL1} \end{figure} This solution is oscillatory for $y<0$ and exponentially decaying for $y>0$. In all the experiments, we make the choice \[ \alpha=\beta=\delta=1/2, \] The initial mesh for the experiments is shown Fig.~\ref{FL1} right panel. \subsubsection{The case $\gamma=h^3$} { Starting with the mesh in Fig.~\ref{FL1} and using uniform refinement, we have computed the error in approximating the Airy function solution when $\gamma=h^3$ and $\kappa=15$. The order of approximation of the Helmholtz equation $q$ is set to 1, 3, 4, and 5 and the corresponding results are respectively marked with diamonds, circles, crosses and squares. Our goal is to demonstrate that an appropriate choice of $n$ and $q$ can produce high order convergence. Indeed our theory predicts that we should choose $q=n+1$, $n\geq 2$ and expect $O(h^n)$ convergence in the $L^2(\Omega)$ norm since the Airy function solution is smooth and the domain is convex (see Corollary~\ref{uuhcor}). Results are shown in Fig.~\ref{ghqs} and Fig.~\ref{ghcomb}. {Fig.~\ref{ghqs} (left panel) demonstrates the need for GPWs in order to obtain high order convergence. When $q=1$, the GPWs are plane waves and we see no obvious convergence when $n=1$ (three plane waves per element), but at most third order convergence for $n>1$. This suggests that one approach using an $h$-refinement strategy is to use simple plane waves with $n=3$ to obtain third order convergence under mesh refinement (it appears that $n=3$ offers a useful improvement in accuracy over $n=2$ even if the order of convergence is the same). To obtain fourth or higher order convergence we need true GPWs with $q>1$. } The case $n=1$ is also interesting. Regardless of $q$ we do not see obvious convergence when $n=1$, whereas for a constant medium the plane wave basis with $n=1$ gives $O(h^2)$ convergence \cite{cessenat_phd}. The variable refractive index seems to require $n>1$. {This is not unreasonable since when $n=1$ the plane waves do not approximate linear polynomials well, and hence may not converge for a solution corresponding to smoothly varying coefficients. When $n>1$, piecewise linears are well approximated by plane waves and so we expect (and see) convergence in this case ~\cite{git09,kapita14}.} {For $n=2$ regardless of the choice $q=1,\cdots,5$ we see $O(h^{3})$ convergence, and for $n=3$ we get $O(h^4)$ convergence provided $q>1$, while if $q=1$ we get $O(h^2)$ convergence. Finally for $n=4$ we only have $O(h^{3})$ convergence when $q=1$, but $O(h^5)$ convergence for $q>1$ (with some deterioration on the finest mesh when $q=3$ or $q=5$). This deterioration may be due to the usual conditioning problem experience by plane wave type methods since when $n=4$ the condition number of the system is roughly $10^{20}$ which may impact convergence. The last cases $n=3,4$ confirms that $q$ must increase as $n$ increases although apparently more slowly than we predict. In addition, with an adequate choice of $q$ we appear to see $O(h^{n+1})$ convergence for $n>1$. This is the same order as has been found experimentally using $2n+1$ plane waves when $\epsilon$ is constant!~\cite{cessenat_phd}. An optimal error analysis in that case is also illusive~\cite{buf07}.} To try to clarify the best relationship between $q$ and $n$ we focus on the cases $q=n-1$, $q=n$ and $q=n+1$ in the left panel of Fig.~\ref{ghcomb}. Again the case $n=1$ fails to converge regardless of $q$, but otherwise the most reliable convergence is seen when $n=q$. \begin{figure} \caption{$L^2(\Omega)$ norm convergence when $\gamma=h^3$ (left panel), $\gamma=h$ (middle panel) and $\gamma=0$ (right panel). The dotted lines in each figure show the order of convergence. } \label{ghqs} \end{figure} \begin{figure} \caption{$L^2(\Omega)$ norm convergence when $\gamma=h^3$ (left panel), $\gamma=h$ (middle panel) and $\gamma=0$ (right panel). The dotted lines are reference lines showing $O(h^2)$, $O(h^{3})$, $O(h^{4})$, and $O(h^5)$ convergence. } \label{ghcomb} \end{figure} \subsubsection{The case $\gamma=h$.} In this section we describe numerical results obtained by increasing the parameter $\gamma$ from $h^3$ to $\gamma =h$, and we compare convergence rates obtained for different combinations of the order of approximation of the equation by the basis functions, $q$, and the number of basis functions per element, $p=2n+1$. This choice for $\gamma$ violates the hypothesis of Theorem \ref{th:L2cv}, but should result in greater stability. Figure \ref{ghqs} (middle panel) displays series of results for several choices of $q$, while $n$ varies from 1 to 4. The results are broadly similar to those in the left panel of the same figure, although the case $n=4$ shows a slowing of convergence on fine meshes for $q=3$ and $q=4$. In this case it appears that $q=n+1$ is indeed a good choice. Figure \ref{ghcomb} (middle panel) displays series of results for $n=q$, $n=q-1$, and $n=q+1$,. These convergence studies emphasize the fact that the three choices $n=q$, $n=q-1$, and $n=q+1$ seem to result in approximately the same rate of convergence, suggesting that $q=n-1$ would be the best choice for a fixed value of $n$. The best rates of convergence obtained are 3 for $n=2$, 4 for $n=3$, and 5 for $n=4$. Although the convergence rates are similar to those when $\gamma=h^3$, the accuracy attained on a given mesh is slightly worse. This suggests that choosing $\gamma$ larger than $O(h^3)$ is not useful (other tests, not shown, with $\gamma=1$ and $\gamma=10^3$ show similar results but even worse error at a particular mesh). \subsubsection{The case $\gamma=0$.} Our theoretical analysis requires that $\gamma>0$ even to obtain convergence in the DG norm but this term requires integration over the interior of all the elements (unlike the standard PWDG or UWVF) and we would prefer to drop it. In addition we saw that $\gamma=O(h^3)$ gives better results than $\gamma=O(h)$ so we want to test if an even smaller penalty is better. In Fig~\ref{ghqs} and \ref{ghcomb} (right panels) we show results when $\gamma=0$. Overall the results are similar to previous results. Provided $q$ is chosen large enough, we can obtain $O(h^{n+1})$ convergence. In fact the mesh now seems slightly more stable! } \subsection{Weber waves} { In this section we approximate what we term Weber waves. These are solutions of the following problem \[ \Delta u + \kappa^2\left(\frac{x_2^2}{4}-\frac{a}{\kappa}\right)u=0 \] in the domain $\Omega=[-1,1]^2$ subject to \[ u(x_2,y_2)=P_o(\sqrt{\kappa}x_2,a)\mbox{ on }\partial \Omega \] where $w(x_2)=P_o(x_2,a)$ is the odd solution of Weber's differential equation \[ \frac{d^2w}{dx_2^2}+\left(\frac{x_2^2}{4}-a\right)=0 \] defined in \cite{ban04} and implemented in \cite{ban_matlab}. We choose $a=5$ and $\kappa=50$ which gives the solution in Fig.~\ref{para_exact}. For this choice of $a$, $\kappa$ and domain, the solution is evanescent for $|x_2|<\sqrt(2/5)$ and oscillatory otherwise. So this example again tests how well GPWs can approximate both traveling and evanescent solutions. \begin{figure} \caption{Exact solution for Weber's equation with $a=5$} \label{para_exact} \end{figure} \begin{figure} \caption{Analogues of Fig.~\ref{ghqs} (left) and Fig.~\ref{ghcomb} (right) for the Weber wave example. } \label{Wcomb} \end{figure} Results are shown in Fig.~\ref{Wcomb}. Broadly the same picture emerges for the Weber example as for the Airy example. We see $O(h^{n+1})$ convergence (this is not completely clear when $n=4$) provided $q$ is large enough. \section{Conclusion} We have provided a modification to the TDG approach that allows the approximation of solutions of the Helmholtz equation in which the refractive index is piecewise smooth using Generalized Plane Waves. The resulting numerical scheme maintains one advantage of TDG: the number of degrees of freedom per element increases linearly with the order of approximation of the method. But the method looses one advantage of pure TDG: there is now a need to perform numerical integration element by element. This is required because we introduce a new stabilization term, and also because the GPW basis functions are not exact solutions of the adjoint problem. Theory suggests a choice of parameters that balances polynomial degree with the number of GPWs in the basis element by element. This is examined in detail using Airy's equation to provide an exact solution, and substantiated further by using Weber's example. In the Airy case we have also studied if our new stabilization term is necessary: the numerical results in this one simple case suggest that it can be ignored, but much more testing (for example with less smooth solutions with curved wavefronts) and theoretical backup would be needed to confirm this. Our testing also suggests that our predicted choice of polynomial degree $q=n+1$ may be excessive. In summary, we have achieved a first theoretical convergence result for GPWs in a TDG setting. Our numerical investigations suggest that the theory is not optimal so far, but do show examples where GPWs can provide accurate solutions to wave propagation problems in which the coefficients are smooth functions of position.} \end{document}

\begin{document} \title[Similarity and commutators of matrices over PIDs]{Similarity and commutators of matrices over principal ideal rings} \author{Alexander Stasinski} \begin{abstract} We prove that if $R$ is a principal ideal ring and $A\in\M_{n}(R)$ is a matrix with trace zero, then $A$ is a commutator, that is, $A=XY-YX$ for some $X,Y\in\M_{n}(R)$. This generalises the corresponding result over fields due to Albert and Muckenhoupt, as well as that over $\Z$ due to Laffey and Reams, and as a by-product we obtain new simplified proofs of these results. We also establish a normal form for similarity classes of matrices over PIDs, generalising a result of Laffey and Reams. This normal form is a main ingredient in the proof of the result on commutators. \end{abstract} \address{Department of Mathematical Sciences, Durham University, South Rd, Durham, DH1 3LE, UK} \email{[email protected]} \maketitle \section{Introduction} Let $R$ denote an arbitrary ring. If a matrix $A\in\M_{n}(R)$ is a commutator, that is, if $A=[X,Y]=XY-YX$ for some $X,Y\in\M_{n}(R)$, then $A$ must have trace zero. The problem of when the converse holds goes back at least to Shoda \cite{Shoda} who showed in 1937 that if $K$ is a field of characteristic zero, then every $A\in\M_{n}(K)$ with trace zero is a commutator. Shoda's argument fails in positive characteristic, but Albert and Muckenhoupt \cite{Albert-Muckenhoupt} found another argument valid for all fields. The first result for rings which are not fields was obtained by Lissner \cite{Lissner} who proved that if $R$ is a principal ideal domain (PID) then every $A\in\M_{2}(R)$ with trace zero is a commutator. A motivation for Lissner's work was the relation with a special case of Serre's problem on projective modules over polynomial rings, nowadays known as the Quillen-Suslin theorem (see \cite[Sections~1-2]{Lissner}). Lissner's result on commutators in $\M_{2}(R)$ for $R$ a PID was rediscovered by Vaserstein \cite{Vaserstein/87} and Rosset and Rosset \cite{Rosset}, respectively. Vaserstein also formulated the problem of whether every $A\in\M_{n}(\Z)$ with trace zero is a commutator for $n\geq3$ (see \cite[Section~5]{Vaserstein/87}). A significant breakthrough was made by Laffey and Reams \cite{Laffey-Reams} who settled Vaserstein's problem in the affirmative. However, their proofs involve steps which are special to the ring of integers $\Z$ and do not generalise to other rings in any straightforward way. The most crucial step of this kind is an appeal to Dirichlet's theorem on primes in arithmetic progressions. The analogue of Dirichlet's theorem, although true in the ring $\F_{q}[x]$, fails for other Euclidean domains such as $\C[x]$ or discrete valuation rings. Nevertheless, in \cite{Laffey-notes} Laffey asked whether any matrix with trace zero over a Euclidean domain is a commutator. Until now this appears to have remained an open problem even for $n=3$, except for the cases where $R$ is a field or $\Z$. In the present paper we answer Laffey's question by proving that if $R$ is any PID and $A\in\M_{n}(R)$ is a matrix with trace zero, then $A$ is a commutator. This is achieved by extending the methods of Laffey and Reams and in particular removing the need for Dirichlet's theorem. Another of our main results is a certain (non-unique) normal form for similarity classes of matrices over PIDs, itself a generalisation of a result proved in \cite{Laffey-Reams} over $\Z$. The normal form, while interesting in its own right and potentially for other applications, is also a key ingredient in the proof of the main result on commutators. We now describe the contents of the paper in more detail. In Section~\ref{sec:Regular-elements} we define regular elements in $\M_{n}(R)$ for an arbitrary ring $R$ and state some of their basic properties. Regular elements play a central role in the problem of writing matrices as commutators because of the criterion of Laffey and Reams, treated in Section~\ref{sec:LF-criterion}. The criterion says that if $R$ is a PID and $A,X\in\M_{n}(R)$ with $X$ regular mod every maximal ideal of $R$, then a necessary and sufficient condition for $A$ to be a commutator is that $\Tr(X^{r}A)=0$ for $r=0,1,\dots,n-1$. This was proved in \cite{Laffey-Reams} for $R=\Z$, but the proof goes through for any PID with only a minor modification. In Section~\ref{sec:Comm-fields} we apply the Laffey-Reams criterion for fields to give a short proof of the theorem of Albert and Muckenhoupt mentioned above. We actually prove a stronger and apparently new result, namely that in the commutator one of the matrices may be taken to be regular (see Proposition~\ref{sec:Comm-fields}). Section~\ref{sec:Similarity} is concerned with similarity of matrices over PIDs, that is, matrices up to conjugation by invertible elements. Our first main result is Theorem~\ref{thm:LF-normalform} stating that every non-scalar element in $\M_{n}(R)$ is similar to one in a special form. This result was established by Laffey and Reams over $\Z$. However, a crucial step in their proof uses the fact that $2$ is a prime element in $\Z$, and the analogue of this does not hold in an arbitrary PID. To overcome this, our proof involves an argument based on the surjectivity of the map $\SL_{n}(R)\rightarrow\SL_{n}(R/I)$ for an ideal $I$, which in a certain sense lets us avoid any finite set of primes, in particular those of index $2$ in $R$ (see Lemma~\ref{lem:b12-avoidsprimes}). This argument is evident especially in the proof of Proposition~\ref{prop:3x3-normalform}. Apart from this, our proof uses the methods of \cite{Laffey-Reams}, although we give a different argument, avoiding case by case considerations, and have made Lemma~\ref{lem:row-column} explicit. Our second main result is Theorem~\ref{thm:Main} whose proof occupies Section~\ref{sec:Proof-Main}, and follows the lines of \cite[Section~4]{Laffey-Reams}. There are two new key ideas in our proof. First, there is again an argument which at a certain step allows us to avoid finitely many primes, including those of index $2$ in $R$. This step in the proof is the choice of $q$ and uses a special case of Lemma~\ref{lem:GCD}\,\ref{enu:GCD-lemma abx}. Secondly, we apply Lemma~\ref{lem:Centr-product} to obtain a set of generators of the centraliser of a certain matrix modulo a product of distinct primes; see (\ref{eq:Centr-span-severalprimes}). It is this set of generators together with our choice of $q$ and an appropriate choice of $t$ in (\ref{eq:at+y}) which allows us to avoid Dirichlet's theorem. It is interesting to note that the proofs of our main results, Theorems~\ref{thm:LF-normalform} and \ref{thm:Main}, despite being rather different, both involve the technique of avoiding finitely many primes, in particular those of index $2$ in $R$. Our proof of Theorem~\ref{thm:Main} also simplifies parts of the proof of Laffey and Reams over $\Z$ since we avoid some of the case by case considerations present in the latter. By a theorem of Hungerford, Theorem~\ref{thm:Main}, once established, easily extends to any principal ideal ring (not necessarily an integral domain); see Corollary~\ref{cor:Coroll-Main}. The final Section~\ref{sec:Further-directions} discusses the possibility of generalising Theorem~\ref{thm:Main} to other classes of rings such as Dedekind domains, and mentions some known counter-examples. We end this introduction by mentioning some recent work on matrix commutators. In \cite{Mesyan} Mesyan proves that if $R$ is a ring (not necessarily commutative) and $A\in\M_{n}(R)$ has trace zero, then $A$ is a sum of two commutators. This result was proved for commutative rings in earlier unpublished work of Rosset. In \cite{Lam-Khurana-Gen-comm} Khurana and Lam study {}``generalised commutators'', that is, elements of the form $XYZ-ZYX$, where $X,Y,Z\in\M_{n}(R)$. They establish in particular that if $R$ is a PID, then every element in $\M_{n}(R)$, $n\geq2$, is a generalised commutator. Although these results may seem closely related to the commutator problem studied in the present paper, the proofs are in fact very different. \subsection*{Notation and terminology} We use $\N$ to denote the natural numbers $\{1,2,\dots\}$. Throughout the paper a ring will always mean a commutative ring with identity. In Sections~\ref{sec:LF-criterion}-\ref{sec:Proof-Main} $R$ will be a PID, unless stated otherwise. Let $R$ be a ring. We denote the set of maximal ideals of $R$ by $\Specm R$ and the ring of $n\times n$ matrices over $R$ by $\M_{n}(R)$. For $A,B\in\M_{n}(R)$ we call $[A,B]=AB-BA$ the \emph{commutator} of $A$ and $B$. Let $A\in\M_{n}(R)$. A matrix $B\in\M_{n}(R)$ is said to be \emph{similar} to $A$ if there exists a $g\in\GL_{n}(R)$ such that $gAg^{-1}=B$. The transpose of $A$ is denoted by $A^{T}$ and the trace of $A$ by $\Tr(A)$. We write $C_{\M_{n}(R)}(A)$ for the centraliser of $A$ in $\M_{n}(R)$, that is, \[ C_{\M_{n}(R)}(A)=\{B\in\M_{n}(R)\mid[A,B]=0\}. \] Let $f(x)=a_{0}+a_{1}x+\dots+x^{n}\in R[x]$ be the characteristic polynomial of $A$. We will refer to the \emph{companion matrix} associated to $A$ (or to $f$) as the matrix $C\in\M_{n}(R)$ such that \[ C=(c_{ij})=\begin{cases} c_{i,i+1}=1 & \text{for }1\leq i\leq n-1,\\ c_{ni}=-a_{i-1} & \text{for }1\leq i\leq n,\\ c_{ij}=0 & \text{otherwise}. \end{cases} \] The identity matrix in $\M_{n}(R)$ is denoted by $1$ or sometimes $1_{n}$. For $u,v\in\N$ we write $E_{uv}$ for the matrix units, that is, $E_{uv}=(e_{ij})$ with $e_{uv}=1$ and $e_{ij}=0$ otherwise. The size of the matrices $E_{uv}$ is suppressed in the notation and will be determined by the context. \section{\label{sec:Regular-elements}Regular elements} Let $\bfG$ be a reductive algebraic group over a field $K$ with algebraic closure $\overline{K}$. An element $x\in G=\bfG(\overline{K})$ is called \emph{regular} if $\dim C_{G}(x)$ is minimal, and it is known that this minimal dimension equals the rank $\rk G$ (see \cite{Steinberg-regular} and \cite[Section~14]{dignemichel}). Similarly, if $\mfg$ is the Lie algebra of $\bfG$ an element $X\in\mfg(\overline{K})$ is called \emph{regular} if $\dim C_{G}(X)=\rk G$, where $G$ acts on $\mfg$ via the adjoint action. In the case $\bfG=\GL_{n}$ there are several equivalent characterisations of regular elements in $\mfg(K)=\M_{n}(K)$. More precisely, the following is well-known: \begin{prop} \label{prop:Reg-fields}Let $K$ be a field and $X\in\M_{n}(K)$. Then the following is equivalent \begin{enumerate} \item \label{enu:reg-fields 1}$X$ is regular, \item \label{enu:reg-fields 5-1}There exists a vector $v\in K^{n}$ such that $\{v,Xv,\dots,X^{n-1}v\}$ is a basis for $K^{n}$ over $K$, \item \label{enu:reg-fields 5}The set $\{1,X,\dots,X^{n-1}\}$ is linearly independent over $K$, \item \label{enu:reg-fields 1-1}$X$ is similar to its companion matrix $C$ as well as to $C^{T}$, \item \label{enu:reg-fields 2}$C_{\M_{n}(K)}(X)=K[X]$. \end{enumerate} \end{prop} Regular elements of $\M_{n}(K)$ are sometimes called \emph{non-derogatory }or\emph{ cyclic.} For matrices over arbitrary rings we make the following definition. \begin{defn} Let $R$ be a ring. A matrix $X\in\M_{n}(R)$ is called \emph{regular} if there exists a vector $v\in R^{n}$ such that $\{v,Xv,\dots,X^{n-1}v\}$ is a basis for $R^{n}$ over $R$.\end{defn} \begin{prop} \label{prop:Reg-rings}Let $R$ be a ring and $X\in\M_{n}(R)$. Then the following is equivalent \begin{enumerate} \item \label{enu:reg-rings 1}$X$ is regular, \item \label{enu:reg-rings 2-1}$X$ is similar to its companion matrix $C$ as well as to $C^{T}$, \item \label{enu:reg-rings 3}$C_{\M_{n}(R)}(X)=R[X]$. \end{enumerate} \end{prop} The proof of Proposition \ref{prop:Reg-rings} is the same as in the classical case of matrices over fields. In the following we will use the properties of regular elements expressed in Propositions~\ref{prop:Reg-fields} and \ref{prop:Reg-rings} without explicit reference. \begin{comment} Even though Condition~\ref{enu:reg-fields 2} makes sense over any ring, it does not lead to a good notion of regular element in general. For example, if $R=k[s,t]/(s^{2},st,t^{2})$ and $X=\begin{pmatrix}s & t\\ 0 & 0 \end{pmatrix}$ we have $C_{\GL_{2}(R)}(X)=ZK$, where $Z$ is the centre of $\GL_{2}(R)$ and $K$ is the kernel of the reduction map $\GL_{2}(R)\to\GL_{2}(k)$. one can regard $\GL_{2}(R)$ as a linear algebraic group over $k$ via the Greenberg functor, so we have the classical notion of regular elements in $\GL_{2}(R)$. With this notion $X$ is not regular since $\dim C_{\GL_{2}(R)}(X)=10$ while the centraliser of for example the element $\begin{pmatrix}1 & 0\\ 0 & 0 \end{pmatrix}$ has dimension $6$. On the other hand, $C_{\GL_{2}(R)}(X)$ is abelian since $K$ is. However, it has been shown by Hill \cite[Theorem~3.6]{Hill_regular} that when $R$ is a local Artinian principal ideal ring this type of anomaly cannot occur and \ref{enu:reg-fields 2} is equivalent to \ref{enu:reg-fields 4}. \begin{rem} Over an arbitrary ring $R$, we may not have a uniquely defined minimal polynomial of a matrix over $R$ (cf.~\cite[Ex.~7.30]{Brown-Matrices}). In case $R$ is an integrally closed domain the minimal polynomial is uniquely defined and divides the characteristic polynomial (see \cite{Frisch}), but even in this case the analogue of part \ref{enu:reg-fields 3} of Proposition~\ref{prop:Reg-fields} for $X\in\M_{n}(R)$ does not imply that $X$ is regular, and furthermore the analogue of part \ref{enu:reg-fields 5} does not imply that $X$ is regular. For a simple counter-example in both of these cases, take for instance $X=\left(\begin{smallmatrix}0 & 2\\ 0 & 0 \end{smallmatrix}\right)\in\M_{2}(\Z)$.\end{rem} \end{comment} If $\phi:R\to S$ is a homomorphism of rings we also use $\phi$ to denote the induced homomorphism $\M_{n}(R)\rightarrow\M_{n}(S)$. \begin{lem} \label{lem:Reg-extnscalars}Let $\phi:R\to S$ be a homomorphism of rings. If $X\in\M_{n}(R)$ is regular, then $\phi(X)$ is regular. \end{lem} \begin{proof} Suppose that $X$ is regular. By definition there exists a vector $v\in R^{n}$ such that $\{v,Xv,\dots,X^{n-1}v\}$ is an $R$-basis for $R^{n}$. Then $\{v\otimes1,Xv\otimes1,\dots,X^{n-1}v\otimes1\}$ is an $S$-basis for $R^{n}\otimes_{R}S$ (cf.~\cite[XVI, Proposition~2.3]{Lang-Algebra}). Let $\phi(v)\in S^{n}$ be the image of $v$ under component-wise application of $\phi$. Under the isomorphism $R^{n}\otimes_{R}S\rightarrow S^{n}$, the elements $X^{i}v\otimes1$ are sent to $\phi(X)^{i}\phi(v)$, so $\{\phi(v),\phi(X)\phi(v),\dots,\phi(X)^{n-1}\phi(v)\}$ is a basis for $S^{n}$. Thus $\phi(X)$ is regular. \end{proof} Let $R$ be a ring and $X\in\M_{n}(R)$. If $\mfp$ is an ideal of $R$ we use $X_{\mfp}$ to denote the image of $X$ under the canonical map $\pi:\M_{n}(R)\to\M_{n}(R/\mfp$), that is, $X_{\mfp}=\pi(X)$. For a general ring $R$ an element in $\M_{n}(R)$ which is regular modulo every maximal ideal may not be regular. However, if $R$ is a local ring, the situation is favourable: \begin{lem} \label{lem:Reg-locring}Assume that $R$ is a local ring with maximal ideal $\mfm$. Then $X\in\M_{n}(R)$ is regular if and only if $X_{\mfm}\in\M_{n}(R/\mfm)$ is regular.\end{lem} \begin{proof} If $X$ is regular, then $X_{\mfm}$ is regular by Lemma~\ref{lem:Reg-extnscalars}. Conversely, suppose that $X_{\mfm}$ is regular and choose $v\in(R/\mfm)^{n}$ such that $(R/\mfm)^{n}=(R/\mfm)[X_{\mfm}]v$. Let $\hat{v}\in R^{n}$ be a lift of $v$. Then $R^{n}=R[X]\hat{v}+\mfm M$ for some submodule $M$ of $R^{n}$, and Nakayama's lemma yields $R^{n}=R[X]\hat{v}$ , so $X$ is regular.\end{proof} \begin{prop} \label{prop:Reg-mod-m}Let $R$ be an integral domain with field of fractions $F$, and let $X\in\M_{n}(R)$. If $X_{\mfm}$ is regular for some maximal ideal $\mfm$ of $R$, then $X$ is regular as an element of $\M_{n}(F)$. \end{prop} \begin{proof} Suppose that $X_{\mfm}$ is regular for some maximal ideal $\mfm$ of $R$. Let $R_{\mfm}$ be the localisation of $R$ at $\mfm$, and let $j:R\to R_{\mfm}$ be the canonical homomorphism. Since the diagram \[ \xymatrix{R\ar[d]\ar[r]^{j} & R_{\mfm}\ar[d]\\ R/\mfm\ar[r]\sp-{\cong} & R_{\mfm}/\mfm } \] commutes, Lemma~\ref{lem:Reg-locring} implies that $j(X)$ is regular. If $\sum_{i=0}^{n-1}r_{i}X^{i}=0$ for some $r_{i}\in R$, then $\sum_{i=0}^{n-1}j(r_{i})j(X)^{i}=0$. But since $j(X)$ is regular, we must have $j(r_{i})=0$ for all $i=0,\dots,n-1$. Since $R$ is an integral domain $j$ is injective, so $r_{i}=0$ for $i=0,\dots,n-1$. Now, if $\sum_{i=0}^{n-1}s_{i}X^{i}=0$ for some $s_{i}\in F$, then clearing denominators shows that $s_{i}=0$ for all $i=0,\dots,n-1$. Hence, by Proposition~\ref{prop:Reg-fields}~\ref{enu:reg-fields 5} the matrix $X$ is regular as an element of $\M_{n}(F)$. \end{proof} The following result has appeared in \cite[Proposition~6]{Vaserstein-Wheland}. \begin{lem} \label{lem:reg-triang}Let $R$ be an arbitrary ring and $A=(a_{ij})\in\M_{n}(R)$ a matrix such that $a_{i,i+1}=1$ for all $1\leq i\leq n$ and $a_{ij}=0$ for all $j\geq i+2$. Then $A$ is regular. \end{lem} \begin{proof} Let $\{e_{1}=(1,0,\dots,0)^{T},e_{2}=(0,1,0,\dots,0)^{T},\dots,e_{n}=(0,\dots,0,1)^{T}\}$ be the standard basis for $R^{n}$. Then the matrix \[ B=(e_{1},Ae_{1},\dots,A^{n-1}e_{1}) \] is upper triangular with $1$s on the diagonal, so $B\in\SL_{n}(R)$. Now for $1\leq i\leq n-1$ we have \[ B^{-1}ABe_{i}=B^{-1}A^{i}e_{1}=e_{i+1} \] (since $Be_{i+1}=A^{i}e_{1}$ ). Thus $B^{-1}AB$ is a companion matrix, and so $A$ is regular. \end{proof} \section{\label{sec:LF-criterion}\foreignlanguage{british}{The criterion of Laffey and Reams }} \selectlanguage{british} Throughout this section $R$ is a PID and $F$ its field of fractions. In Theorem~\ref{prop:Criterion} we give a criterion for a matrix in $\M_{n}(R)$ to be a commutator discovered by Laffey and Reams \cite[Section~3]{Laffey-Reams}. This criterion plays an important role in our proof of the main theorem. Laffey and Reams proved the criterion for matrices over fields and over $\Z$, and we only need minor modifications of their proofs, together with Proposition~\ref{prop:Reg-mod-m}, to prove it over arbitrary PIDs. The following result is from \cite[Section~3]{Laffey-Reams}. We reproduce the proof here for completeness. \begin{prop} \label{prop:LF-criterion-fields}Let $K$ be a field and $X\in\M_{n}(K)$ be regular. Let $A\in\M_{n}(K)$. Then $A=[X,Y]$ for some $Y\in\M_{n}(K)$ if and only if \foreignlanguage{english}{$\Tr(X^{r}A)=0$ for all $r=0,\dots,n-1$.}\end{prop} \selectlanguage{english} \begin{proof} Since $\{1,X,\dots,X^{n-1}\}$ is linearly independent over $K$ the subspace \[ V=\{A\in\M_{n}(K)\mid\Tr(X^{r}A)=0\text{ for }0,1,\dots,n-1\} \] has dimension $n^{2}-n$. The kernel of the linear map $\M_{n}(R)\rightarrow\M_{n}(R)$, $Y\mapsto[X,Y]$ is equal to the centraliser $C_{\M_{n}(K)}(X)$, which has dimension $n$ since $X$ is regular. Thus the image $[X,\M_{n}(K)]$ of the map $Y\mapsto[X,Y]$ has dimension $n^{2}-n$. But if $A\in[X,\M_{n}(K)]$ there exists a $Y\in\M_{n}(K)$ such that for every $r=0,1,\dots,n-1$ we have \[ \Tr(X^{r}A)=\Tr(X^{r}(XY-YX))=\Tr(X^{r+1}Y)-\Tr(X^{r}YX)=0. \] Thus $A\in V$ and so $[X,\M_{n}(K)]\subseteq V$. Since $\dim V=\dim[X,\M_{n}(K)]$ we conclude that $V=[X,\M_{n}(K)]$. \end{proof} \selectlanguage{british} \begin{prop} \label{prop:LF-XYM}Let $X\in\M_{n}(R)$ be such that $X_{\mfp}$ is regular for every maximal ideal $\mfp$ in $R$. Suppose that $M\in\M_{n}(F)$ is such that $[X,M]\in\M_{n}(R)$. Then there exists an $Y\in\M_{n}(R)$ such that $[X,M]=[X,Y]$.\end{prop} \selectlanguage{english} \begin{proof} There exists an element $m\in R$ such that $mY\in\M_{n}(R)$, and we have $[X,mY]=m[X,Y]$. Assume that $d\in R$ is chosen so that it has the minimal number of irreducible factors with respect to the property that $[X,C]=d[X,Y]$ for some $C\in\M_{n}(R)$. If $d$ is a unit we are done, so assume that $p$ is an irreducible factor of $d$. Then $[X,C]\in p\M_{n}(R)$, so $X_{(p)}$ commutes with $C_{(p)}$. But since $X_{(p)}$ is regular, we have $C_{(p)}=f(X_{(p)})$, for some polynomial $f(T)\in R[T]$. Hence $C-f(X)=pD$ for some $D\in\M_{n}(R)$. But this implies that $[X,C]=[X,pD]=p[X,D]$ and thus $(dp^{-1})[X,Y]=[X,D]$, giving a contradiction to our choice of $d$. Hence $d$ is a unit and so $[X,Y]=[X,M]$ with $M=d^{-1}C\in\M_{n}(R)$.\end{proof} \begin{prop} \label{prop:Criterion}Let $A\in\M_{n}(R)$ and let $X\in\M_{n}(R)$ be such that $X_{\mfp}$ is regular for every maximal ideal $\mfp$ in $R$. Then $A=[X,Y]$ for some $Y\in\M_{n}(R)$ if and only if $\Tr(X^{r}A)=0$ for $r=0,\dots,n-1$.\end{prop} \begin{proof} Clearly the condition $\Tr(X^{r}A)=0$ for all $r\geq0$ is necessary for $A$ to be of the form $[X,Y]$ with $Y\in\M_{n}(R)$. Conversely, suppose that $\Tr(X^{r}A)=0$ for $r=0,1,\dots,n-1$. By \foreignlanguage{british}{Proposition~\ref{prop:Reg-mod-m}} $X$ is regular \foreignlanguage{british}{as an element in $\M_{n}(F)$ so Proposition~\ref{prop:LF-criterion-fields} implies that $A=[X,M]$ for some $M\in\M_{n}(F)$. But now the result follows from Proposition~\ref{prop:LF-XYM}.} \end{proof} \section{\label{sec:Comm-fields}Commutators over fields} Let $K$ be a field. Using the criterion of Laffey and Reams over fields (Proposition~\ref{prop:LF-criterion-fields}) we give a swift proof of the theorem of Albert and Muckenhoupt \cite{Albert-Muckenhoupt} that every matrix with trace zero in $\M_{n}(K)$ is a commutator. Note that if $R$ is any ring and $A,X,Y\in\M_{n}(R)$ are such that $A=[X,Y]$, then for every $g\in\GL_{n}(R)$ we have $gAg^{-1}=[gXg^{-1},gYg^{-1}]$. Thus $A$ is a commutator if and only if any matrix similar to $A$ is. Let $n\in\N$ with $n\geq2$ and $k=\lfloor n/2\rfloor$. The following matrices were considered by Laffey and Reams \cite[Section~4]{Laffey-Reams} who also established the properties stated below. \[ P_{n}=(p_{ij})=\begin{cases} p_{ii}=1 & \text{for }i=2,4,\dots,2k,\\ p_{i,i-2}=1 & \text{for }i=3,4,\dots n,\\ p_{ij}=0 & \text{otherwise}. \end{cases} \] Depending on the context we will consider $P_{n}$ as an element of $\M_{n}(R)$ where $R$ is a ring. For any $m\in\N$ and $a\in R$ we will use $J_{m}(a)$ to denote the $m\times m$ Jordan block with eigenvalue $a$ and $1$s on the subdiagonal. Over any $R$ the matrix $P_{n}$ is similar to $J_{k}(1)\oplus J_{n-k}(0)$ (cf.~\cite[p.~681]{Laffey-Reams}), and thus it is regular by Lemma~\ref{lem:reg-triang}. For any $A=(a_{ij})\in\M_{n}(R)$ let $c(A)=\sum_{i=1}^{k}a_{2i,2i}$ and $d(A)=\sum_{i=1}^{n-1}a_{i,i+1}$. Suppose now that $R$ is a PID and that $a_{ij}=0$ for $j\geq i+2$. Observe that for any $r\in\N$, $P_{n}^{r}$ has the same diagonal as $P_{n}$ and the $(i,j)$ entry of $P_{n}^{r}$ is $0$ if $i\neq j$ and $i<j+2$. Thus \begin{equation} \Tr(P_{n}^{r}A)=c(A),\,\text{ for }r\in\N.\label{eq:Tr_c(A)} \end{equation} \begin{prop} \label{prop:Main-fields}Let $K$ be a field and let $A\in\M_{n}(K)$ be a matrix with trace zero. Then $A=[X,Y]$ for some $X,Y\in\M_{n}(K)$, where $X$ is regular. More precisely, if $A$ is non-scalar $X$ can be chosen to be conjugate to $P_{n}$, while if $A$ is scalar we can take $X=J_{n}(0)$.\end{prop} \begin{proof} Assume first that $A$ is non-scalar. It then follows from the rational normal form that $A$ is similar to a matrix $B=(b_{ij})$ with $b_{12}=1$ and $b_{ij}=0$ for $j\geq i+2$, so we have $A=gBg^{-1}$ for some $g\in\GL_{n}(K)$. Define $z\in\SL_{n}(K)$ as \[ z=1+c(B)E_{21}. \] Then the $(i,j)$ entry of $z^{-1}Bz$ is $0$ for $j\geq i+2$ and $c(z^{-1}Bz)=0$, so by (\ref{eq:Tr_c(A)}) we have $\Tr(P_{n}^{r}z^{-1}Bz)=0$ for $r=0,\dots,n-1$. By Proposition~\ref{prop:LF-criterion-fields} it follows that $B=[zP_{n}z^{-1},Y]$ for some $Y\in\M_{n}(K)$, and thus $A=[gzP_{n}(gz)^{-1},gYg^{-1}]$. Assume on the other hand that $A$ is a scalar. Then $\Tr(J_{n}(0)^{r}A)=0$ for $r=0,\dots,n-1$, and Proposition~\ref{prop:LF-criterion-fields} implies that $A=[J_{n}(0),Y]$, for some $Y\in\M_{n}(K)$. \end{proof} \section{\label{sec:Similarity}Matrix similarity over a PID} In this section we extend the results of \cite[Section~2]{Laffey-Reams} on similarity of matrices over $\Z$ to matrices over an arbitrary PID $R$. \begin{lem} \label{lem:b12-avoidsprimes}Let $A\in\M_{n}(R)$ be non-scalar, and let $S$ be a finite set of maximal ideals of $R$ such that $A_{\mfp}\in\M_{n}(R/\mfp)$ is non-scalar for every $\mfp\in S$. Then $A$ is similar to a matrix $B=(b_{ij})\in\M_{n}(R)$ such that $b_{12}\notin\mfp$ for all $\mfp\in S$.\end{lem} \begin{proof} It is well known that for any PID $R$ and any non-zero ideal $\mfa$ of $R$ the natural map \begin{equation} \SL_{n}(R)\longrightarrow\SL_{n}(R/\mfa)\label{eq:strongapprox} \end{equation} is surjective. This follows for example from the fact that $R/\mfa$ is the product of local rings and that over local rings $\SL_{n}$ is generated by elementary matrices (see~\cite[2.2.2~and~2.2.6]{Rosenberg_K-theory}). Moreover, if we take $\mfa=\prod_{\mfp\in S}\mfp$ the Chinese remainder theorem implies that we have an isomorphism \begin{equation} \SL_{n}(R/\mfa)\longiso\prod_{\mfp\in S}\SL_{n}(R/\mfp).\label{eq:Chineserem} \end{equation} Let $\mfp\in S$. Since $A_{\mfp}$ is non-scalar and $R/\mfp$ is a field the rational canonical form for matrices in $\M_{n}(R/\mfp)$ implies that there exists a $g_{\mfp}\in\GL_{n}(R/\mfp)$ such that $g_{\mfp}A_{\mfp}g_{\mfp}^{-1}$ is a matrix whose $(1,2)$ entry is non-zero. Since $\GL_{n}(R/\mfp)=T(R/\mfp)\SL_{n}(R/\mfp)$, where $T(R/\mfp)$ is the diagonal subgroup of $\GL_{n}(R/\mfp)$, we may take $g_{\mfp}$ to be in $\SL_{n}(R/\mfp)$. Suppose that $g_{\mfp}$ is chosen in this way for every $\mfp\in S$. By the surjectivity of the maps (\ref{eq:strongapprox}) and (\ref{eq:Chineserem}), there exists a $g\in\SL_{n}(R)$ such that the image of $g$ in $\SL_{n}(R/\mfp)$ is $g_{\mfp}$ for all $\mfp\in S$. Let $B=(b_{ij})=gAg^{-1}$. Then $B$ is a matrix such that $b_{12}$ is non-zero modulo every $\mfp\in S$. \end{proof} The following lemma will be used repeatedly in the proof of Proposition~\ref{prop:3x3-normalform} and Theorem~\ref{thm:LF-normalform}. It can informally be described as saying that if the off-diagonal entries in a row (column) of a matrix $A\in\M_{n}(R)$ with $n\geq3$ have a greatest common divisor $d$, then $A$ is similar to a matrix in which the corresponding row (column) has off-diagonal entries $d,0,\dots,0$. \begin{lem} \label{lem:row-column}Let $A=(a_{ij})\in\M_{n}(R)$, $n\geq3$. Let $1\leq u\leq n$ and $1\leq v\leq n$ be fixed. Let $r\in R$ be a generator of the ideal $(a_{uj}\mid1\leq j\leq n,\, u\neq j)$, and let $c\in R$ be a generator of the ideal $(a_{iv}\mid1\leq i\leq n,\, i\neq v)$. Then $A$ is similar to a matrix $B=(b_{ij})$ such that if $u=1$ we have $b_{u2}=r$ and $b_{uj}=0$ for all $3\leq j\leq n$, and if $u\geq2$ we have $b_{u1}=r$ and $b_{uj}=0$ for all $1\leq j\leq n$ such that $j\notin\{1,u\}$. Moreover, $A$ is similar to a matrix $C=(c_{ij})$ such that if $v=1$ we have $c_{2v}=r$ and $c_{iv}=0$ for all $3\leq i\leq n$, and if $v\geq2$ we have $c_{1v}=c$ and $c_{iv}=0$ for all $1\leq i\leq n$ such that $i\notin\{1,v\}$.\end{lem} \begin{proof} The proof follows the lines of \cite[Ch.~III, Section~2]{Newman}. For $1\leq i<j\leq n$ and $\left(\begin{smallmatrix}x & y\\ z & w \end{smallmatrix}\right)\in\SL_{2}(R)$, let \begin{align*} M_{ij} & =M_{ij}(x,y,z,w)\\ & =1_{n}+(x-1)E_{ii}+yE_{ij}+zE_{ji}+(w-1)E_{jj}\in\SL_{n}(R). \end{align*} Note that $M_{ij}^{-1}=M_{ij}(w,-y,-z,x)$. Let $3\leq j\leq n$. Direct computation shows that the first row in $B_{1}\coloneqq M_{2j}^{-1}AM_{2j}$ is \begin{align*} (a_{11},a_{12}x+a_{13}z,a_{12}y+a_{13}w,a_{14},\dots,a_{1n}) & \quad\text{if }j=3,\\ (a_{11},a_{12}x+a_{1j}z,a_{13},\dots,a_{1,j-1},a_{12}y+a_{1j}w,a_{1,j+1},\dots,a_{1n}) & \quad\text{if }j>3. \end{align*} Now let $3\leq j\leq n$ be the smallest integer such that $a_{1j}\neq0$ (if no such $j$ exists the assertion of the lemma holds trivially for $A$ and $u=1$). Let $d\in R$ be a generator of $(a_{12},a_{1j})$ and set \[ y=a_{1j}d^{-1},\quad w=-a_{12}d^{-1}. \] Then $(y,w)=(1)$ and hence $x,z\in R$ may be determined so that $xw-yz=1$. Thus $a_{12}x+a_{1j}z=-d$. With these values of $x,y,z,w$ all the entries of $A_{1}$ in positions $(1,3),\dots,(1,j)$ are zero, and the $(1,2)$ entry generates the ideal ($a_{12},a_{1j})$. Repeating the process, let $j<k\leq n$ be the smallest integer such that $a_{1k}\neq0$. Then $B_{2}\coloneqq M_{2k}^{-1}B_{1}M_{2k}$ has all its entries $(1,3),\dots,(1,k)$ zero and its $(1,2)$ entry generates the ideal $(a_{12},a_{1j},a_{1k})$. Proceeding in this way, we obtain a matrix $B=(b_{ij})$ similar to $A$ such that $b_{12}$ is a generator of $(a_{1j}\mid2\leq j\leq n)$ and $b_{1j}=0$ for $3\leq j\leq n$ (the generator $b_{12}$ can be replaced by any other generator of $(a_{1j}\mid2\leq j\leq n)$ by a diagonal similarity transformation of $B$). This shows the existence of $B$ for $u=1$. For $u\geq2$, observe that if we let $W_{u}=(w_{ij}^{(u)})\in\GL_{n}(R)$ be any permutation matrix such that $w_{1u}^{(u)}=w_{u1}=1$, then \[ A'=(a_{ij}')=W_{u}AW_{u}^{-1} \] is a matrix such that $a_{11}'=a_{uu}$ and $\{a_{1j}'\mid2\leq j\leq n\}=\{a_{uj}\mid1\leq j\leq n,u\neq j\}$. Informally, the off-diagonal entries in the $u$-th row of $A$ are the same as the off-diagonal entries in the first row of $A'$, up to a permutation. Thus the existence of $B$ for $u\geq2$ follows from the argument for $u=1$ above. For the existence of $C$ for $v=1$, let $3\leq i\leq n$ and $C_{1}\coloneqq M_{2i}^{-1}AM_{2i}$. Direct computation shows that the first column in $C_{1}$ is \begin{align*} (a_{11},a_{21}x+a_{31}y,a_{21}z+a_{31}w,a_{41},\dots,a_{n1})^{T} & \quad\text{if }i=3,\\ (a_{11},a_{21}x+a_{i1}y,a_{31},\dots,a_{i-1,1},a_{21}z+a_{i1}w,a_{i+1,1},\dots,a_{n1})^{T} & \quad\text{if }i>3. \end{align*} Now let $3\leq i\leq n$ be the smallest integer such that $a_{i1}\neq0$ (if no such $i$ exists the assertion of the lemma holds trivially for $A$ and $v=1$). Let $e\in R$ be a generator of $(a_{21},a_{i1})$ and set \[ z=a_{i1}e^{-1},\quad w=-a_{21}d^{-1}. \] Then $(z,w)=(1)$ and hence $x,y\in R$ may be determined so that $xw-yz=1$. Thus $a_{21}x+a_{i1}y=-e$. With these values of $x,y,z,w$ all the entries of $C_{1}$ in positions $(3,1),\dots,(i,1)$ are zero, and the $(2,1)$ entry generates the ideal ($a_{21},a_{i1})$. Repeating the process in analogy with the above argument, we obtain a matrix $C$ satisfying the assertion of the lemma for $v=1$. For $v\geq2$ we may use the matrix $W_{v}$ as above to reduce to the case where $v=1$.\end{proof} \begin{prop} \label{prop:3x3-normalform}Let $A\in\M_{3}(R)$ be non-scalar. Then $A$ is similar to a matrix $B=(b_{ij})\in\M_{3}(R)$ such that $b_{12}\mid b_{ij}$ for all $i\neq j$ and $b_{12}\mid(b_{ii}-b_{jj})$ for all $1\leq i,j\leq3$.\end{prop} \begin{proof} Write $A=aI+bA'$, where $a,b\in R$, $b\neq0$ and where, if $A'=(a_{ij}')$, we have $(a_{ii}'-a_{jj}',a_{ij}'\mid i\neq j,1\leq i,j\leq3)=(1)$. Note that $A_{\mfp}'$ is non-scalar for every maximal ideal $\mfp$ of $R$ and that the proposition will follow for $A$ if we can show it for $A'$, that is, if we can show that $A'$ is similar to a matrix whose $(1,2)$ entry is a unit. Without loss of generality we may therefore assume that $A=A'$ so that $A$ satisfies \[ (a_{ii}-a_{jj},a_{ij}\mid i\neq j,1\leq i,j\leq3)=(1). \] Note that any matrix similar to $A$ will also satisfy this. Let \[ S\coloneqq\{\mfp\in\Specm R\mid|R/\mfp|=2\}. \] Note that $S$ is a finite set since in any PID (or any Dedekind domain) there are only finitely many maximal ideals of any given finite index. Since $A_{\mfp}$ is not scalar for any maximal ideal $\mfp$ of $R$, Lemma~\ref{lem:b12-avoidsprimes} implies that $A$ is similar to a matrix $B=(b_{ij})$ such that $b_{12}\notin\mfp$ for all $\mfp\in S$. Among all such matrices choose one for which the number of distinct primes which divide $b_{12}$ is least possible, and subject to this, for which the number of not necessarily distinct prime factors is minimal. By Lemma~\ref{lem:row-column} applied to the first row in $B$, we see that there exists a matrix $B'$ similar to $B$ whose $(1,3)$ entry is zero and whose $(1,2)$ entry, being equal to a generator of $(b_{12},b_{13})$, has no more distinct prime factors than $b_{12}$. Hence we may assume that $B$ has been replaced by $B'$ so that $b_{13}=0$. We thus have the following condition on $B$: \MyQuote{The matrix $B=(b_{ij})$ is similar to $A$, $b_{12}\notin\mfp$ for all $\mfp\in S$, $b_{13}=0$, the entry $b_{12}$ has the smallest number of distinct prime factors among all the matrices similar to $A$ and among all matrices with these properties $B$ is such that $b_{12}$ has the minimal number of not necessarily distinct prime factors.}Note first that by Lemma~\ref{lem:row-column} applied to the second column in $B$, there exists a matrix similar to $B$ whose $(1,2)$ entry is a generator of $(b_{12},b_{32})$. Thus, by $(*)$ we must have $b_{12}\mid b_{32}$, so $b_{32}=b_{12}a$ for some $a\in R$. Let \[ B_{1}=(b_{ij}^{(1)})=(1-E_{31}a)B(1-E_{31}a)^{-1}. \] Then $b_{12}^{(1)}=b_{12}$ and $b{}_{13}^{(1)}=b_{32}^{(1)}=0$ so that \[ B_{1}=\begin{pmatrix}b_{11}^{(1)} & b_{12} & 0\\ b_{21}^{(1)} & b_{22}^{(1)} & b_{23}^{(1)}\\ b_{31}^{(1)} & 0 & b_{33}^{(1)} \end{pmatrix}. \] In particular, $B'$ satisfies $(*)$. \begin{claim} \label{Claim I}The entry $b_{12}$ divides both $b_{33}^{(1)}-b_{11}^{(1)}$ and $b_{31}^{(1)}$. \end{claim} Let $y\in R$. The first row of the matrix $(1+E_{13}y)B_{1}(1+E_{13}y)^{-1}$ is \[ (b_{11}^{(1)}+yb_{31}^{(1)},\, b_{12},\, y(b_{33}^{(1)}-b_{11}^{(1)}-yb_{31}^{(1)})). \] Thus, by $(*)$ and Lemma~\ref{lem:row-column} applied to the first row in $(1+E_{13}y)B_{1}(1+E_{13}y)^{-1}$ we conclude that $b_{12}$ divides $y(b_{33}^{(1)}-b_{11}^{(1)}-yb_{31}^{(1)})$ for any $y\in R$. Let \[ (b_{12})=\mfp_{1}^{e_{1}}\cdots\mfp_{\nu}^{e_{\nu}} \] be the factorisation of $(b_{12})$, where $\nu\in\N$, $e_{i}\in\N$ and the ideals $\mfp_{i}\in\Specm R$ are distinct for $1\leq i\leq\nu$. By $(*)$ and the definition of $S$ we know that $|R/\mfp_{i}|\geq3$ for any $1\leq i\leq\nu$. Hence there exist elements $y_{i},y_{i}'\in R/\mfp_{i}$ such that \begin{equation} y_{i}\neq0,\quad y_{i}'\neq0,\quad y_{i}\neq y_{i}',\quad\text{for }i=1,\dots,\nu.\label{eq:yi-yi'} \end{equation} By the Chinese remainder theorem we have \[ R/(b_{12})\cong\prod_{i=1}^{\nu}R/\mfp_{i}^{e_{i}}. \] Let $\lambda=(y_{1},\dots,y_{\nu}),\lambda'=(y_{1}',\dots,y_{\nu}')\in\prod_{i=1}^{\nu}R/\mfp_{i}^{e_{i}}$. Then $\lambda$ and $\lambda'$ can be considered as elements in $R/(b_{12})$ and because of (\ref{eq:yi-yi'}) each of $\lambda,\lambda'$ and $\lambda-\lambda'$ is a unit in $R/(b_{12})$. In particular, each of $\lambda,\lambda'$ and $\lambda-\lambda'$ is coprime to $b_{12}$. We know from the above that $b_{12}$ divides $y(b_{33}^{(1)}-b_{11}^{(1)}-yb_{31}^{(1)})$ for any $y\in R$. In particular, choosing $y=\lambda,\lambda',\lambda-\lambda'$, respectively, we obtain $b_{31}^{(1)}(\lambda-\lambda')\in(b_{12})$, hence $b_{31}^{(1)}\in(b_{12})$ and $b_{33}^{(1)}-b_{11}^{(1)}\in(b_{12})$. This proves the claim. By Claim~\ref{Claim I} there exist elements $\alpha,\beta\in R$ such that \[ b_{33}^{(1)}-b_{11}^{(1)}=\alpha b_{12}\quad\text{and}\quad b_{31}^{(1)}=\beta b_{12}. \] Let \[ B_{2}=(b_{ij}^{(2)})=(1+E_{21}(-\alpha+\beta))(1+E_{31})B_{1}(1+E_{31})^{-1}(1+E_{21}(-\alpha+\beta))^{-1}. \] Then $b_{12}^{(2)}=b_{32}^{(2)}=b_{12}$ and $b_{13}^{(2)}=b_{31}^{(2)}=0$ so that \[ B_{2}=\begin{pmatrix}b_{11}^{(2)} & b_{12} & 0\\ b_{21}^{(2)} & b_{22}^{(2)} & b_{23}^{(2)}\\ 0 & b_{12} & b_{33}^{(2)} \end{pmatrix}. \] Moreover, let \[ B'_{2}=(1-E_{31})B_{2}(1-E_{31})^{-1}=\begin{pmatrix}b_{11}^{(2)} & b_{12} & 0\\ b_{23}^{(2)}+b_{21}^{(2)} & b_{22}^{(2)} & b_{23}^{(2)}\\ b_{33}^{(2)}-b_{11}^{(2)} & 0 & b_{33}^{(2)} \end{pmatrix} \] and \[ B''_{2}=(1-E_{33})B_{2}(1-E_{33})^{-1}=\begin{pmatrix}b_{33}^{(2)} & b_{12} & 0\\ b_{23}^{(2)}+b_{21}^{(2)} & b_{22}^{(2)} & b_{21}^{(2)}\\ b_{11}^{(2)}-b_{33}^{(2)} & 0 & b_{11}^{(2)} \end{pmatrix}. \] We will now show that $B_{2}$ has the property that $b_{12}\mid b_{ij}^{(2)}$ for all $i\neq j$ and $b_{12}\mid(b_{ii}^{(2)}-b_{jj}^{(2)})$ for all $1\leq i,j\leq3$. This follows from the following fact applied to the matrices $B'_{2}$ and $B''_{2}$. \begin{claim} \label{Claim II}Suppose that $C=(c_{ij})\in\M_{n}(R)$ satisfies $(*)$ and that $c_{32}=0$. Then $c_{12}\mid c_{ij}$ for all $i,j$ such that $(i,j)\neq(2,1)$ and $i\neq j$, and $c_{12}\mid(c_{ii}-c_{jj})$ for all $1\leq i,j\leq3$. \end{claim} To prove the claim, let $x\in R$ and \[ X=(x_{ij})=(1+E_{32}x)C(1+E_{32}x)^{-1}. \] Then \begin{align*} x_{12} & =c_{12},\\ x_{32} & =x(c_{22}-c_{33}-xc_{23}), \end{align*} and by Lemma~\ref{lem:row-column} applied to the second column in $X$ we conclude that $c_{12}$ divides $x(c_{22}-c_{33}-xc_{23})$ for any $x\in R$. By $(*)$ and the same argument as in the proof of Claim~\ref{Claim I} we obtain \[ c_{12}\mid(c_{22}-c_{33})\quad\text{and}\quad c_{12}\mid c_{23}. \] Next, for $y\in R$ let \[ Y=(y_{ij})=(1+E_{13}y)C(1+E_{13}y)^{-1}. \] Then \begin{align*} y_{12} & =c_{12},\\ y_{13} & =y(c_{33}-c_{11}-yc_{31}), \end{align*} and by Lemma~\ref{lem:row-column} applied to the first row in $Y$ and the same argument as for the matrix $X$ (that is, using $(*)$ and the same argument as in the proof of Claim~\ref{Claim I}) we obtain \[ c_{12}\mid(c_{33}-c_{11})\quad\text{and}\quad c_{12}\mid c_{31}, \] whence also $c_{12}\mid(c{}_{22}-c_{11})$. This proves Claim~\ref{Claim II} for $C$. Applying Claim~\ref{Claim II} to the matrices $B'_{2}$ and $B''_{2}$, respectively, we conclude that $B_{2}$ has the property that $b_{12}\mid b_{ij}^{(2)}$ for all $i\neq j$ and $b_{12}\mid(b_{ii}^{(2)}-b_{jj}^{(2)})$ for all $1\leq i,j\leq3$. Since $B_{2}$ is similar to $B$ (and $B$ is similar to $A$), we have \[ (b_{ii}-b_{jj},b_{ij}\mid i\neq j,1\leq i,j\leq3)=(1), \] so $b_{12}$ must be a unit. This proves the proposition. \end{proof} We now use Proposition~\ref{prop:3x3-normalform} to prove the corresponding result for matrices in $\M_{n}(R)$ for all $n\geq3$. More precisely, we have \begin{thm} \label{thm:LF-normalform}Let $A\in\M_{n}(R)$ with $n\geq3$, be non-scalar. Then $A$ is similar to a matrix $B=(b_{ij})\in\M_{n}(R)$ such that $b_{12}\mid b_{ij}$ for all $i\neq j$ and $b_{12}\mid(b_{ii}-b_{jj})$ for all $1\leq i,j\leq n$. Moreover, $B$ may be chosen with $b_{ij}=0$ for all $i,j$ such that $j\geq i+2$ and $1\leq i\leq n-2$.\end{thm} \begin{proof} As in the proof of Proposition~\ref{prop:3x3-normalform}, we may assume that \[ (a_{ii}-a_{jj},a_{ij}\mid i\neq j,1\leq i,j\leq n)=(1), \] and choose a matrix $B$ satisfying the following condition \MyQuote{The matrix $B=(b_{ij})$ is similar to $A$, $(b_{12},2)=(1)$, $b_{1j}=0$ for $j\geq 3$, the entry $b_{12}$ has the smallest number of distinct prime factors among all the matrices similar to $A$ and among all matrices with these properties $B$ is such that $b_{12}$ has the minimal number of not necessarily distinct prime factors.}If for some $i,j$ the entry $b_{12}$ does not divide $b_{ii}-b_{jj}$, then $b_{12}$ does not divide $b_{11}-b_{vv}$ for some $v$. If $v\geq4$ let $W_{v}=(w_{ij}^{(v)})\in\GL_{n}(R)$ be any permutation matrix such that $w_{11}^{(v)}=w_{22}^{(v)}=1$, $w_{v3}^{(v)}=1$ and $w_{3v}^{(v)}=1$. Then $W_{v}BW_{v}^{-1}$ has $(1,2)$ entry equal to $b_{12}$ and $(3,3)$ entry equal to $b_{vv}$, so we may assume that $b_{12}$ does not divide $b_{11}-b_{22}$ or $b_{11}-b_{33}$. Consider the submatrix \[ B_{0}=(b_{ij})_{1\leq i,j\leq3} \] of $B$ and note that any similarity $B_{0}\mapsto g^{-1}B_{0}g$ for $g\in\GL_{3}(R)$ may be achieved by $B\mapsto(g\oplus I_{n-3})B(g\oplus I_{n-3})^{-1}$. By the minimality property of $b_{12}$ expressed in $(*)$ and the argument in the proof of Proposition~\ref{prop:3x3-normalform} applied to $B_{0}$ we conclude that $b_{12}$ divides both $b_{11}-b_{22}$ and $b_{11}-b_{33}$, which is a contradiction. Thus \[ b_{12}\mid(b_{ii}-b_{jj})\text{ for all }1\leq i,j\leq n\quad\text{and}\quad b_{12}\mid b_{ij}\text{ for all }i\neq j,\,1\leq i,j\leq3. \] Similarly, for any $4\leq v\leq n$ the matrix $W_{v}BW_{v}^{-1}$ has $(3,1)$ entry equal to $b_{v1}$, so by $(*)$ and the argument in the proof of Proposition~\ref{prop:3x3-normalform} applied to $B_{0}$ we conclude that $b_{12}\mid b_{v1}$. Hence \[ b_{12}\mid b_{v1}\text{ for all }4\leq v\leq n. \] Furthermore, by $(*)$ and Lemma~\ref{lem:row-column} applied to the second column in $B$, we see that \[ b_{12}\mid b_{i2}\text{ for all }i\neq2. \] Let $1\leq u,v\leq n$ be such that $u\geq3$ and $v\neq u$. For $x\in R$ let \[ X_{u}=(x_{ij}^{(u)})=(1+E_{u2})B(1+E_{u2})^{-1}, \] so that $x_{v2}^{(u)}=b_{v2}-b_{vu}$ and in particular $x_{12}^{(u)}=b_{12}$. By $(*)$ and Lemma~\ref{lem:row-column} applied to the second column in $X_{u}$ we see that $b_{12}\mid x_{v2}^{(u)}$ and since $b_{12}\mid b_{v2}$ we conclude that $b_{12}\mid b_{vu}$. Hence \[ b_{12}\mid b_{vu}\text{ for all }u\geq3,\, v\neq u. \] We have thus shown that $B$ has the property that $b_{12}\mid b_{ij}$ for all $i\neq j$ and $b_{12}\mid(b_{ii}-b_{jj})$ for all $1\leq i,j\leq n$. For the second statement we follow \cite[III,~2]{Newman}. Conjugating $B$ by $1_{2}\oplus M_{3j}\in\GL_{n}(R)$ for a suitable $M_{3j}\in\GL_{n-2}(R)$ (cf.~the proof of Lemma~\ref{lem:row-column}), we can replace $B$ by a matrix $B_{1}$ in which the first row equals that of $B$ and whose $(2,j)$ entries are zero whenever $j\geq4$. Conjugating $B_{1}$ by $1_{3}\oplus M_{4j}\in\GL_{n}(R)$ for a suitable $M_{4j}\in\GL_{n-3}(R)$, we can replace $B_{1}$ by a matrix $B_{2}$ in which the first two rows equal those of $B_{1}$ and whose $(3,j)$ entries are zero whenever $j\geq5$. Proceeding inductively in this way, we obtain a matrix $C=(c_{ij})$ similar to $B$ such that $c_{12}=b_{12}$ and $c_{ij}=0$ for $i,j$ such that $j\geq i+2$ and $1\leq i\leq n-2$. But since $B\equiv b_{11}1_{n}\bmod{(b_{12})}$ we also have $C\equiv b_{11}1_{n}\bmod{(b_{12})}$, so $C$ has the desired form. \end{proof} Using Theorem~\ref{thm:LF-normalform} it is now easy to prove the following result. The following proof is entirely analogous to that of Laffey and Reams for $R=\Z$. \begin{prop} Let $A\in\M_{n}(R)$, $n\geq3$ have trace zero, and suppose that for every $\mfp\in\Specm R$ and every $a\in R/\mfp$, $a\neq0$ we have $A_{\mfp}\neq a1_{n}$. Then $A$ is similar to a matrix $B=(b_{ij})\in\M_{n}(R)$ where $b_{ii}=0$ for all $1\leq i\leq n$.\end{prop} \begin{proof} If $A_{\mfp}=0$ for some $\mfp$, we can write $A=mA'$, where $m\in R$ and $A'$ is such that for every $\mfp\in\Specm R$ and every $a\in R/\mfp$ we have $A_{\mfp}'\neq a1_{n}$. Since $A'$ must be non-scalar Theorem~\ref{thm:LF-normalform} implies that $A'$ is similar to a matrix $A''=(a_{ij}'')$ such that $a_{12}''\mid a_{ij}''$ for all $i\neq j$ and $a_{12}''\mid(a_{ii}''-a_{jj}'')$ for all $1\leq i,j\leq n$. Since $A''$ satisfies $A_{\mfp}''\neq a1_{n}$ for any $\mfp\in\Specm R$ and $a\in R/\mfp$, the entry $a_{12}''$ must be a unit. We may therefore assume without loss of generality that $A=A''$, so that in particular $a_{12}$ is a unit. We now prove that $A$ is similar to a matrix with zero diagonal by induction on $n$. If $n=2$, the matrix \[ (1+E_{21}a_{11}a_{12}^{-1})A(1+E_{21}a_{11}a_{12}^{-1})^{-1} \] has zero diagonal. If $n>2$, conjugating $A$ by a matrix of the form $1+\alpha E_{n1}$, $\alpha\in R$, we may assume that $a_{n2}=1$, and then conjugating $A$ by a matrix of the form $1+\beta E_{21}$, $\beta\in R$, we may further assume that $a_{11}=0$. Thus we may assume that $A$ is of the form \[ \begin{pmatrix}0 & x\\ y^{T} & A_{1} \end{pmatrix}, \] where $x,y\in R^{n-1}$, $A_{1}=(a_{ij}^{1})\in\M_{n-1}(R)$ with $a_{n-1,1}^{1}=1$ and $\Tr(A_{1})=0$. By Theorem~\ref{thm:LF-normalform} $A_{1}$ is similar to a matrix $A_{2}=(a_{ij}^{2})$ such that $a_{12}^{2}\mid a_{ij}^{2}$ for all $i\neq j$ and $a_{12}^{2}\mid(a_{ii}^{2}-a_{jj}^{2})$ for all $1\leq i,j\leq n$. Since $(A_{2})_{\mfp}\neq a1_{n}$ for all $\mfp\in\Specm R$ and $a\in R/\mfp$, the entry $a_{12}^{2}$ must be a unit. So by induction there exists a $Q\in\GL_{n-1}(R)$ such that $QA_{1}Q^{-1}=B_{1}$ is a matrix with zeros on the diagonal. But then \[ B=(1_{1}\oplus Q)A(1_{1}\oplus Q)^{-1} \] has the desired form. \end{proof} A matrix in $\M_{n}(R)$ satisfying the conditions on the matrix $B$ in Theorem~\ref{thm:LF-normalform} will be said to be in \emph{Laffey-Reams form}. \section{\label{sec:Proof-Main}Proof of the main result} In this section we give a proof of our main theorem on commutators, Theorem~\ref{thm:Main}. We first prove a couple of lemmas used in the proof. \begin{lem} \label{lem:GCD}Let $R$ be a PID. Then the following holds: \begin{enumerate} \item \label{enu:GCD-lemma abx}Let $a,b\in R$ be such that $(a,b)=(1)$, and let $S$ be a finite set of maximal ideals of $R$. Then there exists an $x\in R$ such that for all $\mfp\in S$ we have $a+bx\notin\mfp$. \item \label{enu:GCD-lemma pqt}Let $\alpha,\beta\in R$ be such that $(\alpha,\beta)=(1)$. Suppose that $\mfp$ is a maximal ideal of $R$ such that $|R/\mfp|\geq3$. Then for every finite set $S$ of maximal ideals of $R$ such that $\mfp\notin S$ there exists a $t\in R$ such that $t\notin\mfp$, $t\in\mfq$ for all $\mfq\in S\setminus\{\mfp\}$ and $\alpha t+\beta\notin\mfp$. \item \label{enu:GCD-lemma abc}Let $a,b,c\in R$ be such that $(a,b,c)=(1)$, $(a,b)\neq(1)$ and $(a,c)\neq(1)$. Then there exists an $x\in R$ such that $(a+cx,b-ax)=(1)$. \end{enumerate} \end{lem} \begin{proof} To prove \ref{enu:GCD-lemma abx}, take $x$ to be a generator of the product \[ \prod_{\substack{\mfp\in S\\ a\notin\mfp } }\mfp \] and let $x=1$ if there is no $\mfp\in S$ such that $a\notin\mfp$. Let $\mfp\in S$ be such that $a\in\mfp$. If $a+bx\in\mfp$, then $ $$bx\in\mfp$ and since $(a,b)=(1)$ we have $x\in\mfp$, which contradicts the definition of $x$. On the other hand, let $\mfp\in S$ be such that $a\notin\mfp$. If $a+bx\in\mfp$, then by the definition of $x$ we have $bx\in\mfp$, so $a\in\mfp$, which is a contradiction. Thus in either case, $a+bx\notin\mfp$. Next, we prove \ref{enu:GCD-lemma pqt}. Since $|R/\mfp|\geq3$ there exist two elements $r_{1},r_{2}\in R\setminus\mfp$ such that $r_{1}-r_{2}\notin\mfp$. Let $s\in R$ be such that \[ (s)=\prod_{\mfq\in S\setminus\{\mfp\}}\mfq. \] Then for $i=1,2$ we have $r_{i}s\notin\mfp$ and $r_{i}s\in\mfq$ for all $\mfq\in S\setminus\{\mfp\}$. Furthermore, if $\alpha r_{i}s+\beta\in\mfp$ for $i=1,2$, then $\alpha\in\mfp$ and $\beta\in\mfp$, contradicting the hypothesis $(\alpha,\beta)=(1)$. Thus we may assume that $\alpha r_{1}s+\beta\notin\mfp$, and $t=r_{1}s$ yields the desired element. We now prove \ref{enu:GCD-lemma abc}. We first show that $a+cx$ and $b-ax$ are relatively prime as elements of $R[x]$, that is, that none of them is a multiple of the other. Indeed, if $a+cx=m(b-ax)$ for some $m\in R$, then $a=mb$ and $c=-ma$ so $(1)=(a,b,c)=(mb,b,-m^{2}b)=(b)$, which is impossible since $(a,b)\neq(1)$. Similarly, if $n(a+cx)=b-ax$ for some $n\in R$, then $b=na$ and $a=-nc$ so $(1)=(a,b,c)=(-nc,-n^{2}c,c)=(c)$, which is impossible. Let $K$ be the field of fractions of $R$. Since $a+cx$ and $b-ax$ are relatively prime as elements of $R[x]$, they are relatively prime as element of $K[x]$. Thus there exists $f_{0},g_{0}\in K[x]$ such that $(a+cx)f_{0}+(b-ax)g_{0}=1$, and so there exists some $f,g\in R[x]$ such that \begin{equation} (a+cx)f+(b-ax)g=D\in R\setminus\{0\}.\label{eq:fgD} \end{equation} Let $S$ be the set of maximal ideals dividing $(D)$. By \ref{enu:GCD-lemma abx} we can choose $x\in R$ such that for all $\mfp\in S$ we have $a+cx\notin\mfp$. Now if $\mfp$ is a maximal ideal of $R$ such that $a+cx\in\mfp$ and $b-ax\in\mfp$, then $D\in\mfp$, by (\ref{eq:fgD}), and so $a+cx\notin\mfp$; a contradiction. Thus there is no $\mfp\in\Specm R$ such that $a+cx\in\mfp$ and $b-ax\in\mfp$, that is, $a+cx$ and $b-ax$ are relatively prime. \end{proof} The following result is the Chinese remainder theorem for centralisers of matrices over quotients of $R$. It will be used at a crucial step in our proof of Theorem~\ref{thm:Main}. \begin{lem} \label{lem:Centr-product}Let $X\in\M_{n}(R)$ and let $\mfp_{1},\dots,\mfp_{\nu}$, $\nu\in\N$ be maximal ideals in $R$. Then the map \begin{align*} C_{\M_{n}(R/(\mfp_{1}\cdots\mfp_{\nu}))}(X_{(\mfp_{1}\cdots\mfp_{\nu})}) & \longrightarrow\prod_{i=1}^{\nu}C_{\M_{n}(R/\mfp_{i})}(X_{\mfp_{i}})\\ g & \longmapsto(g_{\mfp_{1}},\dots,g_{\mfp_{\nu}}), \end{align*} is an isomorphism. \end{lem} \begin{proof} Let $\mathcal{C}=C_{\M_{n}(R/(\mfp_{1}\cdots\mfp_{\nu}))}(X_{(\mfp_{1}\cdots\mfp_{\nu})})$. Then $\mathcal{C}$ is a module over $R$. By the Chinese remainder theorem we have an isomorphism $R/(\mfp_{1}\cdots\mfp_{\nu})\rightarrow\prod_{i=1}^{\nu}R/\mfp_{i}$ given by $a\mapsto(a_{\mfp_{1}},\dots,a_{\mfp_{\nu}})$, and tensoring this by $\mathcal{C}$ yields \begin{align*} \mathcal{C} & \cong R/(\mfp_{1}\cdots\mfp_{\nu})\otimes_{R}\mathcal{C}\cong\big(\prod_{i=1}^{\nu}R/\mfp_{i}\big)\otimes_{R}\mathcal{C}\cong\prod_{i=1}^{\nu}(R/\mfp_{i}\otimes_{R}\mathcal{C})\\ & \cong\prod_{i=1}^{\nu}C_{\M_{n}(R/\mfp_{i})}(X_{\mfp_{i}}). \end{align*} Tracking the maps shows that the effect of the above isomorphisms on elements is given by \[ g\longmapsto1\otimes g\longmapsto(1_{\mfp_{1}},\dots,1_{\mfp_{\nu}})\otimes g\longmapsto(1_{\mfp_{1}}\otimes g,\dots,1_{\mfp_{\nu}}\otimes g)\longmapsto(g_{\mfp_{1}},\dots,g_{\mfp_{\nu}}). \] \end{proof} We now give the proof of our main theorem. Note that our proof in the case $n=2$ is different from the case $n\geq3$, and that for $n=2$, while our argument is not the shortest possible, yields the stronger result that any $A\in\M_{2}(R)$ with trace zero can be written as $A=[X,Y]$ for some $X,Y\in\M_{2}(R)$ and $X$ regular. \begin{thm} \label{thm:Main}Let $R$ be a PID and let $A\in\M_{n}(R)$ be a matrix with trace zero. Then $A=[X,Y]$ for some $X,Y\in\M_{n}(R)$. \end{thm} \begin{proof} For $n=1$ the result is trivial. First assume that $n=2$. By taking out a suitable factor we may assume that the matrix \[ A=\begin{pmatrix}a & b\\ c & -a \end{pmatrix} \] satisfies $(a,b,c)=(1)$. Let $X=\begin{pmatrix}0 & 1\\ x_{1} & x_{2} \end{pmatrix}\in\M_{2}(R)$. By Lemma~\ref{lem:reg-triang} the matrix $X$ is regular so it is regular mod $\mfp$ for every maximal ideal $\mfp$ of $R$. Furthermore, \[ \Tr(XA)=bx_{1}-ax_{2}+c, \] so if $(a,b)=(1)$ we can find $x_{1}$ and $x_{2}$ such that $\Tr(XA)=0$, and Proposition~\ref{prop:Criterion} implies that $A=[X,Y]$, for some $Y\in\M_{2}(R)$. Similarly, the transpose $X^{T}$ of $X$ is also regular, and \[ \Tr(X^{T}A)=cx_{1}-ax_{2}+b, \] so if $(a,c)=(1)$ we can find $x_{1}$ and $x_{2}$ such that $\Tr(X^{T}A)=0$, and so $A=[X^{T},Y]$, for some $ $$Y\in\M_{2}(R)$. Hence, in case $(a,b)=(1)$ or $(a,c)=(1)$ we are done. Assume therefore that $(a,b)\neq(1)$ and $(a,c)\neq(1)$. If we let $T=1+xE_{12}\in\M_{2}(R)$ for some $x\in R$, we have \[ TAT^{-1}=\begin{pmatrix}a+cx & b-ax-x(a+cx)\\ c & -a-cx \end{pmatrix}. \] Now $a+cx$ and $b-ax-x(a+cx)$ are relatively prime if and only if $a+cx$ and $b-ax$ are relatively prime. By Lemma~\ref{lem:GCD}\,\ref{enu:GCD-lemma abc} we can choose $x\in R$ such that $(a'+c'x,b'+a''x)=(1)$, and hence such that the $(1,1)$ and $(1,2)$ entries in $TAT^{-1}$ are relatively prime. As we have already seen, this means that we can find $x_{1}$ and $x_{2}$ such that $\Tr(XTAT^{-1})=0$, so Proposition~\ref{prop:Criterion} yields $A=[T^{-1}XT,Y]$ for some $Y\in\M_{2}(R)$. Assume now that $n\geq3$. If $A$ is a scalar matrix we obviously have $\Tr(J_{n}(0)^{r}A)=0$ for all $r\geq0$, so Proposition~\ref{prop:Criterion} yields the desired conclusion. We may therefore henceforth assume that $A$ is non-scalar. Write $A=(a_{ij})$ for $1\leq i,j,\leq n$. By Theorem~\ref{thm:LF-normalform} we may assume that $A$ is in Laffey-Reams form. If $d\in R$ is such that $(a_{ij},a_{ii}-a_{jj}\mid i\neq j,1\leq i,j\leq n)=(d)$, we can write $A=dA'$ where $A'=(a'_{ij})\in\M_{n}(R)$ is in Laffey-Reams form and $(a_{11}',a_{12}')=(1)$. It thus suffices to assume that $A=A'$ so that $(a_{11},a_{12})=(1)$, $a_{12}\mid a_{ij}$ for $i\neq j$, $a_{12}\mid(a_{ii}-a_{jj})$ for $1\leq i,j\leq n$, and $a_{ij}=0$ for $j\geq i+2$. Let $k=\lfloor n/2\rfloor$. For $x,y,q\in R$ define the matrix $X=(x_{ij})\in\M_{n}(R)$ by \[ (x_{ij})=\begin{cases} x_{ii}=-y & \text{for }i=2,4,\dots,2k,\\ x_{21}=x,\\ x_{31}=q,\\ x_{j,j-1}=1 & \text{for }j=3,4,\dots,n,\\ x_{ij}=0 & \text{otherwise}. \end{cases} \] \begin{comment} For example, for $n=5$ we have \[ X=\begin{pmatrix}0 & 0 & 0 & 0 & 0\\ x & -y & 0 & 0 & 0\\ q & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & -y & 0\\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}. \] \end{comment} Recall that for any $B=(b_{ij})\in\M_{n}(R)$ we write $c(B)=\sum_{i=1}^{k}b_{2i,2i}$.We have \[ \Tr(XA)=xa_{12}+a_{23}+\dots+a_{n-1,n}-yc(A). \] We claim that $\Tr(XA)=0$ implies that $\Tr(X^{r}A)=0$ for all $r\geq0$. To see this, observe that the matrix $X^{2}+yX$ is lower triangular and its $(i,j)$ entry is $0$ if $j\geq i-1$. Since $\Tr(E_{ij}A)=0$ if $j<i-1$ (since $a_{ij}=0$ for $j\geq i+2$), it follows that $\Tr((X^{2}+yX)A)=0$, so if $\Tr(XA)=0$ we get $\Tr(X^{2}A)=0$. More generally, using the fact that $X$ is lower triangular, we have $\Tr((X^{r}+yX^{r-1})A)=0$, and working inductively we get $\Tr(X^{r}A)=0$ for all $r\geq0$. Assume for the moment that $a_{12}\mid c(A)$ and let $M=1-c(A)a_{12}^{-1}E_{21}\in\M_{n}(R)$. Then \[ c(MAM^{-1})=0, \] so Proposition~\ref{prop:Criterion} together with (\ref{eq:Tr_c(A)}) and the fact that $P_{n}$ is regular imply $MAM^{-1}=[P_{n},Y]$, for some $Y\in\M_{n}(R)$. Thus in this case $A=[M^{-1}P_{n}M,M^{-1}YM]$, so we may henceforth assume that \begin{equation} a_{12}\nmid c(A).\label{eq:a12-notdiv-c(A)} \end{equation} We now show that there exist elements $x,y\in R$ with $(x,y)=(1)$ and such that $\Tr(XA)=0$. To this end, consider the equation \[ xa_{12}+a_{23}+\dots+a_{n-1,n}=yc(A),\qquad x,y\in R. \] Since $a_{12}$ divides $a_{23},\dots,a_{n-1,n}$, this may be written \begin{equation} a_{12}(x+l)=yc(A),\label{eq:Diophant-xy} \end{equation} for some $l\in R$. Let $d\in R$ be a generator of $(a_{12},c(A))$. Then (\ref{eq:Diophant-xy}) is equivalent to \begin{align*} x & =hc(A)d^{-1}-l\\ y & =ha_{12}d^{-1}, \end{align*} for any $h\in R$. Choose $h$ to be a generator of the product of all maximal ideals $\mfp$ of $R$ such that $a_{12}d^{-1}\in\mfp$ and $l\notin\mfp$ (and let $h=1$ if no such $\mfp$ exist). Suppose that $(x,y)\in(p)$ for some prime element $p\in R$. Then $y\in(p)$ and so $a_{12}d^{-1}\in(p)$ or $h\in(p)$. If $a_{12}d^{-1}\in(p)$ and $l\not\in(p)$, then $h\in(p)$, so $x\notin(p)$. If $a_{12}d^{-1}\in(p)$ and $l\in(p)$, then $h\notin(p)$ and since $(a_{12}d^{-1},c(A)d^{-1})=(1)$ we have $x\notin(p)$. Furthermore, if $h\in(p)$ then $l\notin(p)$ so $x\notin(p)$. Thus $(x,y)=(1)$. If $y$ is a unit then $a_{12}d^{-1}$ must be a unit, and so $a_{12}\mid c(A)$, contradicting (\ref{eq:a12-notdiv-c(A)}). Thus $y$ is not a unit, and so $x^{2}a_{12}\notin(ya_{12})$. Since $a_{12}$ divides each of $a_{11}-a_{22}$, $a_{21}$, $a_{31}$ and $a_{32}$, we have $xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\in(ya_{12})$. Thus, we must have \begin{equation} x^{2}a_{12}+xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\neq0.\label{eq:not-zero} \end{equation} From now on let $x$ and $y$ be as above, so that $(x,y)=(1)$ and $\Tr(XA)=0$. Next, we specify the entry $q$ in $X$. Let $S_{0}$ be the set of maximal ideals $\mfp$ of $R$ such that $x^{2}a_{12}+xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\in\mfp$, and let \[ S=S_{0}\cup\{\mfp\in\Specm R\mid|R/\mfp|=2\}. \] Note that $S$ is a finite set because of (\ref{eq:not-zero}) together with the fact that for any PID $R'$ (or any Dedekind domain), there are only finitely many $\mfp\in\Specm R'$ such that $|R'/\mfp|=2$. By Lemma~\ref{lem:GCD}\,\ref{enu:GCD-lemma abx} (with $r=1$) we can thus choose $q\in R$ such that \[ x+qy\notin\mfp,\quad\text{for all }\mfp\in S. \] Assume from now on that $q$ has been chosen in this way. Let $V$ be the set of maximal ideals of $R$ such that $x+qy\in\mfp$, that is, \[ V=\{\mfp\in\Specm R\mid x+qy\in\mfp\}. \] By the choice of $q$ we thus have in particular that \begin{equation} \mfp\in V\Longrightarrow x^{2}a_{12}+xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\notin\mfp.\label{eq:pinV-polynotinp} \end{equation} Note that for every $\mfp\in V$ we have $y\notin\mfp$ since $(x,y)=(1)$. Note also that $S\cap V=\varnothing$. We claim that $X_{\mfp}\in\M_{n}(R/\mfp)$ is regular for every maximal ideal $\mfp$ not in $V$. To show this, let $\mfp\in(\Specm R)\setminus V$ and let \[ M=\begin{pmatrix}x+qy & 0\\ q & 1 \end{pmatrix}\oplus1_{n-2}\in\M_{n}(R). \] Since $x+qy\not\in\mfp$ the image $M_{\mfp}\in\M_{n}(R/\mfp)$ of $M$ is invertible and, letting $y_{\mfp}$ denote the image of $y$ in $R/\mfp$, we have \[ M_{\mfp}X_{\mfp}M_{\mfp}^{-1}=(m_{ij})=\begin{cases} m_{ii}=-y_{\mfp} & \text{for }i=2,4,\dots,2k,\\ m_{j,j-1}=1 & \text{for }j=2,3,\dots,n,\\ m_{ij}=0 & \text{otherwise}. \end{cases} \] It follows from Lemma~\ref{lem:reg-triang} that $M_{\mfp}X_{\mfp}M_{\mfp}^{-1}$ is regular, and thus $X_{\mfp}$ is regular. By our choice of $q$ we have $\mfp\notin V$ if $\mfp\in S$, so $X_{\mfp}$ is regular for any $\mfp\in S$, and $S$ is non-empty. By Proposition~\ref{prop:Reg-mod-m} we have that $X$ is regular as an element in $\M_{n}(F)$, where $F$ is the field of fractions of $R$. By our choice of $x$ and $y$ we have $\Tr(X^{r}A)=0$ for $r=0,1,\dots,n-1$, so Proposition~\ref{prop:LF-criterion-fields} implies that we can write $A=[X,Q]$, for some $Q\in\M_{n}(F)$. Clearing denominators in $Q$ we find that there exists a non-zero element $m_{0}\in R$ such that $m_{0}A\in[X,\M_{n}(R)]$. We now highlight a step which we will refer to in the following: \MyQuote{Let $m\in R$ be such that it has the minimal number of (not necessarily distinct) prime factors among all $m'\in R$ such that $m'A\in[X,\M_n(R)]$, and let $Q\in\M_n(R)$ be such that $mA=[X,Q]$.} We show that the only maximal ideals containing $m$ are those in $V$. Suppose that $\mfp=(p)\in(\Specm R)\setminus V$ and that $m\in\mfp$. Then $0=[X_{\mfp},Q{}_{\mfp}]$, and since $X_{\mfp}$ is regular there exists a polynomial $f\in R[T]$ such that $Q=f(X)+pQ'$ for some $Q'\in\M_{n}(R)$, so $mA=[X,f(X)+pQ']=[X,pQ']$ and thus $mp^{-1}A=[X,Q']$, which contradicts $(*)$. Thus, if $m\in\mfp$ for some $\mfp\in\Specm R$, then we must have $\mfp\in V$. Let $\mfp_{1},\mfp_{2},\dots,\mfp_{\nu}$, $\nu\in\N$ be the elements of $V$ such that $m\in\mfp_{i}$. For each $\mfp_{i}$, choose a generator $p_{i}\in R$, so that $\mfp_{i}=(p_{i})$, for $i=1,\dots,\nu$. We then have \[ (m)=(p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{\nu}^{e_{\nu}}), \] for some $e_{i}\in\N$, $1\leq i\leq\nu$. The strategy is now to show that $X$ can be replaced by a matrix $X_{1}$ which is regular mod $\mfp$ for every $\mfp\in V$. Let \[ N=1+qE_{21}\in\M_{n}(R). \] For ease of calculation we will consider the matrices \[ A_{0}=NAN^{-1},\quad X_{0}=NXN^{-1},\quad Q_{0}=NQN^{-1}. \] Let $\mfp\in V$ be any of the ideals $\mfp_{1},\mfp_{2},\dots,\mfp_{\nu}$. We have \begin{equation} (X_{0})_{\mfp}=\begin{pmatrix}0 & 0\\ 0 & W_{\mfp} \end{pmatrix}=(0)\oplus W_{\mfp},\label{eq:Wp} \end{equation} where $W_{\mfp}\in\M_{n-1}(R/\mfp)$ is regular. We wish to determine the dimension of the centraliser \[ C(\mfp):=C_{\M_{n}(R/\mfp)}((X_{0})_{\mfp}). \] Since $(x,y)=(1)$, we have $y_{\mfp}\neq0$, so the Jordan form of $(X_{0})_{\mfp}$ is \[ J_{k}(-y_{\mfp})\oplus J_{n-k-1}(0)\oplus J_{1}(0), \] where $k=\lfloor n/2\rfloor$, as before. We have an isomorphism of $R/\mfp$-vector spaces \[ C(\mfp)\cong C_{\M_{k}(R/\mfp)}(J_{k}(-y_{\mfp}))\oplus C_{\M_{n-k}(R/\mfp)}(J_{k}(0)\oplus J_{1}(0)). \] Since $\dim C_{\M_{k}(R/\mfp)}(J_{k}(-y_{\mfp}))=k$ it remains to determine the dimension of $C_{\M_{n-k}(R/\mfp)}(J_{n-k-1}(0)\oplus J_{1}(0))$. A matrix \[ H=\begin{pmatrix}H_{11} & H_{12}\\ H_{21} & H_{22} \end{pmatrix}\in\M_{n-k}(R/\mfp), \] where $H_{11}$ is a $(n-k-1)\times(n-k-1)$ block, $H_{22}$ is a $1\times1$ block, and the other blocks are of compatible sizes, commutes with $J_{n-k-1}(0)\oplus J_{1}(0)$ if and only if \[ H_{11}J_{n-k-1}(0)=J_{n-k-1}(0)H_{11},\quad H_{12}\in\begin{pmatrix}R/\mfp\\ 0 \end{pmatrix},\quad H_{21}\in(0,R/\mfp). \] Hence $\dim C_{\M_{n-k}(R/\mfp)}(J_{n-k-1}(0)\oplus J_{1}(0))=n-k-1+1+1+1$, and so \[ \dim C(\mfp)=n+2, \] that is, $(X_{0})_{\mfp}$ is subregular (cf.~\cite{Springer-Steinberg}). Next, we need the dimension of $(R/\mfp)[(X_{0})_{\mfp}]$ (the algebra of polynomials in $(X_{0})_{\mfp}$ over the field $R/\mfp$). Since $(R/\mfp)[(X_{0})_{\mfp}]\cong(0)\oplus(R/\mfp)[W_{\mfp}]$ and $W_{\mfp}$ is regular, we have $\dim(R/\mfp)[(X_{0})_{\mfp}]=n-1$. We now find a basis for $C(\mfp)$. We know that $(R/\mfp)[(X_{0})_{\mfp}]$ is an $(n-1)$-dimensional subspace of $C(\mfp)$. Moreover, direct verification shows that $E_{11}$ and $E_{12}+y_{\mfp}E_{13}$ are in $C(\mfp)$. Let $\kappa=n+1-2\lfloor(n+1)/2\rfloor$, that is, $\kappa$ is $0$ if $n$ is odd and $1$ if $n$ is even. Then we also have \[ E_{n1}+\kappa y_{\mfp}E_{n-1,1}\in C(\mfp). \] Since $(X_{0})_{\mfp}$ is lower triangular and the first column of $(X_{0})_{\mfp}^{i}$ is $0$ for all $i\in\N$, the intersection of $(R/\mfp)[(X_{0})_{\mfp}]$ with the $R/\mfp$-span $\langle E_{11},E_{12}+y_{\mfp}E_{13},E_{n1}+\kappa y_{\mfp}E_{n-1,1}\rangle$ is $0$. Since $\{E_{11},E_{12}+y_{\mfp}E_{13},E_{n1}+\kappa y_{\mfp}E_{n-1,1}\}$ is linearly independent, $\dim(R/\mfp)[(X_{0})_{\mfp}]=n-1$ and $\dim C(\mfp)=n+2$, we must have \begin{equation} C(\mfp)=\langle(R/\mfp)[(X_{0})_{\mfp}],E_{11},E_{12}+y_{\mfp}E_{13},E_{n1}+\kappa y_{\mfp}E_{n-1,1}\rangle.\label{eq:Centr-span} \end{equation} We observe that the matrix $E_{n1}+\kappa yE_{n-1,1}\in\M_{n}(R)$, whose image in $\M_{n}(R/\mfp)$ is $E_{n1}+\kappa y_{\mfp}E_{n-1,1}$, satisfies \begin{equation} E_{n1}+\kappa yE_{n-1,1}\in C_{\M_{n}(R)}(X_{0}).\label{eq:E+kyE-inC(X)} \end{equation} Let $\mfa=\prod_{i=1}^{\nu}\mfp_{i}$, so that $\mfa=(p_{1}\cdots p_{\nu})$. By (\ref{eq:Centr-span}) and Lemma~\ref{lem:Centr-product} we have \begin{equation} C_{\M_{n}(R/\mfa)}(X_{\mfa})=\langle(R/\mfa)[(X_{0})_{\mfa}],E_{11},E_{12}+y_{\mfa}E_{13},E_{n1}+\kappa y_{\mfa}E_{n-1,1}\rangle.\label{eq:Centr-span-severalprimes} \end{equation} Since $[X_{0},Q_{0}]=mA_{0}$ we have $([X_{0},Q_{0}])_{\mfa}=0$, that is, $(Q_{0})_{\mfa}\in C_{\M_{n}(R/\mfa)}((X_{0})_{\mfa})$. Hence, by (\ref{eq:Centr-span-severalprimes}) \[ Q_{0}=f(X_{0})+\alpha E_{11}+\beta(E_{12}+yE_{13})+\gamma(E_{n1}+\kappa yE_{n-1,1})+p_{1}\cdots p_{\nu}D, \] for some $\alpha,\beta,\gamma\in R$, $f(T)\in R[T]$ and $D\in\M_{n}(R)$. Using (\ref{eq:E+kyE-inC(X)}) we get \begin{align} [X_{0},Q_{0}] & =[X_{0},f(X_{0})+\alpha E_{11}+\beta(E_{12}+yE_{13})\label{eq:X0-Q0}\\ & \quad+\gamma(E_{n1}+\kappa yE_{n-1,1})+p_{1}\cdots p_{\nu}D]\nonumber \\ & =[X_{0},\alpha E_{11}+\beta(E_{12}+yE_{13})+p_{1}\cdots p_{\nu}D]\nonumber \\ & =[X_{0},Q_{1}],\nonumber \end{align} where \[ Q_{1}:=\alpha E_{11}+\beta(E_{12}+yE_{13})+p_{1}\cdots p_{\nu}D. \] Let $i\in\N$ be such that $1\leq i\leq\nu$. If $(\alpha,\beta)\subseteq\mfp_{i}$ then $[X_{0},Q_{1}]\in p_{i}[X_{0},\M_{n}(R)]$ and so $mp_{i}^{-1}A\in[X,\M_{n}(R)]$, contradicting $(*)$. Thus either $\alpha\notin\mfp_{i}$ or $\beta\notin\mfp_{i}$. We show that the case where $\alpha\in\mfp_{i}$ and $\beta\notin\mfp_{i}$ cannot arise. Since $mA_{0}=[X_{0},Q_{0}]=[X_{0},Q_{1}]$, we have $m\cdot\Tr(Q_{1}A_{0})=0$, whence $\Tr(Q_{1}A_{0})=0$. Together with $\alpha\in\mfp_{i}$ and $\beta\notin\mfp_{i}$ this implies that \[ \Tr((E_{12}+yE_{13})A_{0})\in\mfp_{i}. \] Recalling that $A_{0}=NAN^{-1}$ we thus get \[ -q^{2}a_{12}+q(a_{11}-a_{22})+a_{21}+ya_{31}-qya_{32}\in\mfp_{i} \] and, after multiplying by $y^{2}$, \[ -q^{2}y^{2}a_{12}+qy^{2}(a_{11}-a_{22})+y^{2}(a_{21}+ya_{31}-qya_{32})\in\mfp_{i}. \] Since $\mfp_{i}\in V$ we have $qy\in-x+\mfp_{i}$ and so \[ x^{2}a_{12}+xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\in\mfp_{i}. \] But by our choice of $q$ we have \[ x^{2}a_{12}+xy(a_{11}-a_{22})-y^{2}(a_{21}+ya_{31}+xa_{32})\notin\mfp, \] for all $\mfp\in V$, which together with (\ref{eq:pinV-polynotinp}) yields a contradiction. Therefore we cannot have $\alpha\in\mfp_{i}$ and $\beta\notin\mfp_{i}$, so we must have $\alpha\notin\mfp_{i}$. We have thus shown that \[ \alpha\notin\mfp_{i},\quad\text{for all }i=1,\dots,\nu. \] By Lemma~\ref{lem:GCD}\,\ref{enu:GCD-lemma pqt} and our choice of $S$ there exists a $t\in R$ such that \begin{equation} t\notin\mfp_{i}\quad\text{and}\quad\alpha t+y\notin\mfp_{i},\quad\text{for all }i=1,\dots,\nu.\label{eq:at+y} \end{equation} Define the matrix \begin{align*} X_{1} & =X_{0}+tQ_{1}. \end{align*} Let $\mfp$ be any of the ideals $\mfp_{1},\mfp_{2},\dots,\mfp_{\nu}$. Let $\alpha_{\mfp},\beta_{\mfp},t_{\mfp}$ denote the images of $\alpha$, $\beta$ and $t$ in $R/\mfp$, respectively. As before, let $y_{\mfp}$ denote the image of $y$ in $R/\mfp$. If we let \[ L_{\mfp}=\begin{pmatrix}1 & \beta_{\mfp}\alpha_{\mfp}^{-1} & y_{\mfp}\beta_{\mfp}\alpha_{\mfp}^{-1}\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\oplus1_{n-3}\in\M_{n}(R/\mfp), \] then direct verification shows that $L_{\mfp}(X_{1})_{\mfp}L_{\mfp}^{-1}=\alpha_{\mfp}t_{\mfp}E_{11}\oplus W_{\mfp}$, where $W_{\mfp}$ is the matrix in (\ref{eq:Wp}). Since $W_{\mfp}$ is regular and neither of its eigenvalues $0$ or $-y_{\mfp}$ equals $\alpha_{\mfp}t_{\mfp}$ by (\ref{eq:at+y}), the matrix $\alpha_{\mfp}t_{\mfp}E_{11}\oplus W_{\mfp}$, and hence $(X_{1})_{\mfp}\in\M_{n}(R/\mfp)$, is regular. We thus see that $(X_{1})_{\mfp_{i}}$ is regular for all $i=1,\dots,\nu$. By (\ref{eq:X0-Q0}) we have \[ mA_{0}=[X_{0},Q_{0}]=[X_{0},Q_{1}]=[X_{1},Q_{1}], \] and since $(X_{1})_{\mfp_{i}}$ is regular and $m\in\mfp_{i}$ for all $i=1,\dots,\nu$, we get $Q_{1}=g_{i}(X_{1})+p_{i}Q{}_{1}^{(i)}$, for some $g_{i}(T)\in R[T]$ and $Q{}_{1}^{(i)}\in\M_{n}(R)$. Thus \[ mA_{0}=[X_{1},g_{i}(X_{1})+p_{i}Q{}_{1}^{(i)}]=p_{i}[X_{1},Q{}_{1}^{(i)}], \] and so $mp_{i}^{-1}A_{0}=[X_{1},Q{}_{1}^{(i)}]$. Repeating the argument if necessary, we obtain $mp_{i}^{-e_{i}}A_{0}\in[X_{1},\M_{n}(R)]$. Running through each $i=1,\dots,\nu$ we obtain $A_{0}=[X_{1},Y]$ for some $Y\in\M_{n}(R)$, and hence $A=[N^{-1}X_{1}N,NYN^{-1}]$. \end{proof} By a theorem of Hungerford \cite{Hungerford} every principal ideal ring (PIR) is a finite product of rings, each of which is a homomorphic image of a PID. Together with Theorem~\ref{thm:Main} this immediately implies the following: \begin{cor} \label{cor:Coroll-Main}Let $R$ be a PIR (not necessarily an integral domain) and let $A\in\M_{n}(R)$, $n\geq2$, be a matrix with trace zero. Then $A=[X,Y]$ for some $X,Y\in\M_{n}(R)$. \end{cor} We end this section by proving a strengthened version of Theorem~\ref{thm:Main} for $n=3$. \begin{prop} \label{prop:n3regX}Let $R$ be a PID and let $A\in\M_{3}(R)$ be a matrix with trace zero. Then $A=[X,Y]$ for some $X,Y\in\M_{3}(R)$ such that $X_{\mfp}$ is regular for all $\mfp\in\Specm R$. \end{prop} \begin{proof} As in the proof of Theorem~\ref{thm:Main} we may assume that $A$ is in Laffey-Reams form. Define the matrix \[ X=\begin{pmatrix}0 & 0 & 0\\ x & -y & 0\\ q & z & 0 \end{pmatrix}\in\M_{3}(R). \] The same argument as in the proof of Theorem~\ref{thm:Main} shows that $\Tr(XA)=0$ implies that $\Tr(X^{r}A)=0$ for all $r\geq0$. Let $a_{23}'\in R$ be such that $a_{23}=a_{12}a_{23}'$, and let $d\in R$ be a generator of $(a_{12},c(A))$. The condition $\Tr(XA)=0$ is then equivalent to \begin{align*} x & =hc(A)d^{-1}-a_{23}'z\\ y & =ha_{12}d^{-1}, \end{align*} for any $h\in R$. We claim that the system of equations \begin{equation} \begin{cases} x=hc(A)d^{-1}-a_{23}'z\\ y=ha_{12}d^{-1}\\ xz+qy=1 \end{cases}\label{eq:n3-system} \end{equation} has a solution in $x,y,q,z,h\in R$. Indeed, substituting the first two equations in the last, we get \[ -a_{23}'z^{2}+h(c(A)d^{-1}z+qa_{12}d^{-1})=1, \] and since $(c(A)d^{-1},a_{12}d^{-1})=(1)$ we can choose $z$ and $q$ in $R$ such that $ $$c(A)d^{-1}z+qa_{12}d^{-1}=1$, and it then remains to take \[ h=1+a_{23}'z^{2}. \] Suppose now that $x,y,q,z,h\in R$ is a solution of (\ref{eq:n3-system}), and let $\mfp\in\Specm R$. We show that $X_{\mfp}$ is regular. The characteristic polynomial of $X$ is \[ \lambda^{2}(\lambda+y)\in R[\lambda]. \] We have \[ X^{2}=\begin{pmatrix}0 & 0 & 0\\ -xy & y^{2} & 0\\ xz & -yz & 0 \end{pmatrix}. \] Thus, if $y\notin\mfp$ then $(X_{\mfp})^{2}\neq0$, and if $y\in\mfp$, then we must have $xz\not\in\mfp$, so $(X_{\mfp})^{2}\neq0$ also in this case. Furthermore, since $xz+qy=1$ we have \[ X(X+y)=E_{31}\neq0. \] Thus the minimal polynomial of $X_{\mfp}$ must equal the characteristic polynomial, so $X_{\mfp}$ is regular. Since we have $\Tr(X^{r}A)=0$ for all $r\geq0$, Proposition~(\ref{prop:Criterion}) implies that $A=[X,Y]$, for some $Y\in\M_{3}(R)$. \end{proof} We remark that while the matrix $X$ in the proof of the above proposition is regular modulo every $\mfp\in\Specm R$, it is not necessarily regular. Moreover, while for $n=4$ we can find an analogous matrix \[ X=\begin{pmatrix}0 & 0 & 0 & 0\\ x & -y & 0 & 0\\ q & z & 0 & 0\\ 0 & 0 & 1 & -y \end{pmatrix} \] such that $\Tr(AX)=0$ and $xz+yq=1$, in this case the matrix $X_{\mfp}$ may fail to be regular for some $\mfp\in\Spec R$. \section{\label{sec:Further-directions}Further directions} If $R$ is a field or if $R$ is a PID and $n=2$, we have shown that every $A\in\M_{n}(R)$ with trace zero can be written $A=[X,Y]$ where $X,Y\in\M_{n}(R)$ and $X$ is regular. Our proof of Theorem~\ref{thm:Main} shows that for any PID $R$, $n\geq2$ and every $A\in\M_{n}(R)$ with trace zero we have $A=[X,Y]$ for some $X,Y\in\M_{n}(R)$ where $X_{\mfp}$ is regular for all but finitely many maximal ideals $\mfp$ of $R$. Moreover, Proposition~\ref{prop:n3regX} says that when $n=3$ the matrix $X$ can be chosen such that $X_{\mfp}$ is regular for all maximal ideals $\mfp$. \begin{problem*} For $n\geq4$ and $A=[X,Y]$, is it always possible to choose $X$ such that $X_{\mfp}$ is regular for all maximal ideals $\mfp$? \end{problem*} This problem is interesting insofar as a proof, if possible, would be likely to yield a substantially simplified proof of Theorem~\ref{thm:Main}. It is natural to ask for generalisations of Theorem~\ref{thm:Main} to rings other than PIRs. We first mention some counter-examples. It was shown by Lissner \cite{Lissner} that the analogue of Theorem~\ref{thm:Main} fails when $n=2$ and $R=k[x,y,z]$, where $k$ is a field, and more generally that for $R=k[x_{1},\dots,x_{2n-1}]$ there exist matrices in $\M_{n}(R)$ with trace zero which are not commutators (see \cite[Theorem~5.4]{Lissner}). Rosset and Rosset \cite[Lemma~1.1]{Rosset} gave a sufficient criterion for a $2\times2$ trace zero matrix over any commutative ring not to be a commutator. They showed however, that a Noetherian integral domain cannot satisfy their criterion unless it has dimension at least $3$. This means that their criterion is not an obstruction to a $2\times2$ trace zero matrix over a one or two-dimensional Noetherian domain being a commutator. Still, if $R$ is the two-dimensional domain $\R[x,y,z]/(x^{2}+y^{2}+z^{2}-1)$ it can be shown that there exists a matrix in $\M_{2}(R)$ with trace zero which is not a commutator (this example goes back to Kaplansky; see \cite[Section~4, Example~1]{Swan/62}, \cite[p.~532]{Lissner-OPrings} or \cite[Section~3]{Rosset}). A ring $R$ is called an \emph{OP-ring} if for every $n\geq1$ every vector in $\bigwedge^{n-1}R^{n}$ is decomposable, that is, of the form $v_{1}\wedge\dots\wedge v_{n-1}$ for some $v_{i}\in R^{n}$. This is equivalent to saying that every vector in $R^{n}$ is an outer product (hence the acronym OP). The notion of OP-ring was introduced in \cite{Lissner-OPrings}. In particular, for $n=3$ the condition on $R$ of being an OP-ring is equivalent to the condition that every trace zero matrix in $\M_{2}(R)$ is a commutator (see \cite[Section~3]{Lissner}). It is known that every Dedekind domain is an OP-ring \cite[p.~534]{Lissner-OPrings} and that every polynomial ring in one variable over a Dedekind domain is an OP-ring \cite[Theorem~1.2]{Towber}. This prompts the following problem: \begin{problem*} Let $R$ be a Dedekind domain and assume that $A\in\M_{n}(R)$, $n\geq2$, has trace zero. Is it true that $A=[X,Y]$ for some $X,Y\in\M_{n}(R)$? \end{problem*} \noindent Since Dedekind domains are OP-rings the question has an affirmative answer for $n=2$, and one could ask the same question for any OP-ring. In the setting of matrices over a Dedekind domain the methods we have used to prove Theorem~\ref{thm:Main} are of little use because they rely crucially on the underlying ring being both atomic and B\'ezout, which implies that it is a PID. \end{document}

\begin{document} \title{Super-stability in the \\ Student-Project Allocation Problem with Ties\thanks{A preliminary version of a part of this paper appeared in \cite{OM18} \begin{abstract} The \emph{Student-Project Allocation problem with lecturer preferences over Students} ({\sc spa-s}) involves assigning students to projects based on student preferences over projects, lecturer preferences over students, and the maximum number of students that each project and lecturer can accommodate. This classical model assumes that each project is offered by one lecturer and that preference lists are strictly ordered. Here, we study a generalisation of {\sc spa-s} where ties are allowed in the preference lists of students and lecturers, which we refer to as the \emph{Student-Project Allocation problem with lecturer preferences over Students with Ties} ({\sc spa-st}). We investigate stable matchings under the most robust definition of stability in this context, namely \emph{super-stability}. We describe the first polynomial-time algorithm to find a super-stable matching or to report that no such matching exists, given an instance of {\sc spa-st}. Our algorithm runs in $O(L)$ time, where $L$ is the total length of all the preference lists. Finally, we present results obtained from an empirical evaluation of the linear-time algorithm based on randomly-generated {\sc spa-st} instances. Our main finding is that, whilst super-stable matchings can be elusive when ties are present in the students' and lecturers' preference lists, the probability of such a matching existing is significantly higher if ties are restricted to the lecturers' preference lists. \keywords{Student-project allocation \and Stable matching \and Super-stability \and Polynomial-time algorithm \and Empirical evaluation} \end{abstract} \thispagestyle{empty} \setcounter{page}{1} \pagestyle{headings} \section{Introduction} \label{introduction} The \emph{Student-Project Allocation problem} ({\sc spa}) \cite{AIM07,CFG19,Man13} is a many-one matching problem which involves three sets of entities: students, projects and lecturers. Each project is proposed by one lecturer and each student is required to rank a subset of these projects that she finds acceptable, in order of preference. Further, each lecturer may have preferences over the students that find her projects acceptable and/or the projects that she offers. Typically there may be capacity constraint on the number of students that each project and lecturer can accommodate. The goal is to find a \emph{matching}, i.e., an assignment of students to projects based on the stated preferences such that each student is assigned to at most one project, and the capacity constraints on projects and lecturers are not violated. Applications of {\sc spa} can be found in many university departments, for example, the School of Computing Science, University of Glasgow \cite{KIMS15}, the Faculty of Science, University of Southern Denmark \cite{CFG19}, the Department of Computing Science, University of York \cite{Kaz02}, and elsewhere \cite{AB03,RGSA17,HSVS05}. In this work, we will concern ourselves with a variant of {\sc spa} that involves lecturer preferences over students, which is known as the \emph{Student-Project Allocation problem with lecturer preferences over Students} ({\sc spa-s}) \cite{AIM07,Man13}. This variant falls under the category of bipartite matching problem with two-sided preferences.\footnote{For further reading on the classification of matching problems, we refer the interested reader to \cite{Man13}.} In this context, it has been argued that a natural property for a matching to satisfy is that of \emph{stability} \cite{Rot84,Rot90,Rot91}. Informally, a \emph{stable matching} ensures that no student and lecturer would have an incentive to deviate from the matching by forming a private arrangement involving some project. The classical {\sc spa-s} model assumes that preferences are strictly ordered. However, this might not be achievable in practice. For instance, a lecturer may be unable or unwilling to provide a strict ordering of all the students who find her projects acceptable. Such a lecturer may be happier to rank two or more students equally in a tie, which indicates that the lecturer is indifferent between the students concerned. This leads to a generalisation of {\sc spa-s} which we refer to as the \emph{Student-Project Allocation problem with lecturer preferences over Students with Ties} ({\sc spa-st}). If we allow ties in the preference lists of students and lecturers, three different stability definitions naturally arise. Suppose $M$ is a matching in an instance of {\sc spa-st}. Informally, we say that $M$ is \emph{weakly stable, strongly stable} or \emph{super-stable} if there is no student and lecturer such that if they decide to form an arrangement outside the matching, respectively, \begin{itemize} \item[(i)] both of them would be better off, \item[(ii)] one of them would be better off and the other would be no worse off, \item[(iii)] neither of them would be worse off. \end{itemize} With respect to this informal definition, a super-stable matching is also strongly stable, and a strongly stable matching is also weakly stable. These concepts were first defined and studied by Irving \cite{Irv94} in the context of the \emph{Stable Marriage problem with Ties} ({\sc smt}), and subsequently extended to the \emph{Hospitals/Residents problem with Ties} ({\sc hrt}) \cite{IMS00,IMS03} (where {\sc hrt} is the special case of {\sc spa-st} in which each lecturer offers only one project, and the capacity of each project is the same as the capacity of the lecturer offering the project; and {\sc smt} is a restriction of {\sc hrt} where the capacity of each hospital is $1$). Considering the weakest of the three stability concepts mentioned above, every instance of {\sc spa-st} admits a weakly stable matching (this follows by breaking the ties in an arbitrary fashion and applying the stable matching algorithm described in \cite{AIM07} to the resulting {\sc spa-s} instance). However, such matchings could be of different sizes \cite{MIIMM02}. Thus opting for weak stability leads to the problem of finding a weakly stable matching that matches as many students to projects as possible -- a problem that is known to be NP-hard \cite{IMMM99,MIIMM02}, even for the so-called \emph{Stable Marriage problem with Ties and Incomplete lists} ({\sc smti}), which is an extension of {\sc smt} in which the preference lists need not be complete. However, we note that a $\frac{3}{2}$-approximation algorithm was described in \cite{CM18} for the problem of finding a maximum size weakly stable matching, given an instance of {\sc spa-st}.\footnote{This approximation algorithm finds a weakly stable matching that is at least two-thirds the size of a maximum weakly stable matching.} Although a super-stable matching can be elusive, it avoids the problem of finding a maximum size weakly stable matching, because, as we will show in this paper, analogous to the {\sc hrt} case \cite{IMS00}: (i) all super-stable matchings have the same size; (ii) finding one or reporting that none exists can be accomplished in linear-time; and (iii) if a super-stable matching $M$ exists then all weakly stable matchings are of the same size (equal to the size of $M$), and match exactly the same set of students. Furthermore, Irving \emph{et al}.~\cite{IMS00} argued that super-stability is a very natural solution concept in cases where agents have incomplete information. Central to their argument is the following proposition, stated for {\sc hrt} in \cite[Proposition 2]{IMS00}, which extends naturally to {\sc spa-st} as follows (see Section \ref{subsection:spa-st} for a proof). \begin{restatable}[]{proposition}{superstability} \label{proposition1} Let $I$ be an instance of {\sc spa-st}, and let $M$ be a matching in $I$. Then $M$ is super-stable in $I$ if and only if $M$ is stable in every instance of {\sc spa-s} obtained from $I$ by breaking the ties in some way. \end{restatable} In a practical setting, suppose that a student $s_i$ has incomplete information about two or more projects and decides to rank them equally in a tie $T$, and a super-stable matching $M$ exists in the corresponding {\sc spa-st} instance $I$. Then $M$ is stable in every instance of {\sc spa-s} (obtained from $I$ by breaking the ties) that represents the true preferences of $s_i$. Consequently, we will focus on the concept of super-stability in the {\sc spa-st} context. Unfortunately not every instance of {\sc spa-st} admits a super-stable matching. This is true, for example, in the case where there are two students, two projects and one lecturer, the capacity of each project is $1$, the capacity of the lecturer is $2$, and every preference list is a single tie of length 2; any matching will be undermined by some student $s_i$ and the lecturer involving a project that $s_i$ is not assigned to. Nonetheless, it should be clear from the discussions above that a super-stable matching should be preferred in practical applications when one does exist. \paragraph{\textbf{Related work.}} Irving \emph{et al}.~\cite{IMS00} described an algorithm to find a super-stable matching given an instance of {\sc hrt}, or to report that no such matching exists. However, merely reducing an instance of {\sc spa-st} to an instance of {\sc hrt} and applying the algorithm described in \cite{IMS00} to the resulting {\sc hrt} instance does not work in general (we explain this further in Section \ref{subsect:cloning}). Other variants of {\sc spa} in the literature involve lecturer preferences over their proposed projects \cite{IMY12,MMO18,MO08}, lecturer preferences over (student, project) pairs \cite{AM09}, and no lecturer preferences at all \cite{KIMS15} (see \cite{CFG19} for a more detailed survey in this latter case). A similar model known as the \textit{Student-Project-Resource Matching-Allocation problem} ({\sc spr}) was recently considered in \cite{IYYY19}. This model is different from {\sc spa-s} in the following ways: (i) in {\sc spa-s}, the capacity of each project is fixed by the lecturer offering it, while in {\sc spr}, the capacity of each project is determined by the resources allocated to it; (ii) in {\sc spa-s}, each lecturer has a fixed capacity on the total number of students that can be assigned to her projects, while in {\sc spr}, there is no notion of lecturer capacity. \paragraph{\textbf{Our contribution.}} In this paper, we describe the first polynomial-time algorithm to find a super-stable matching or to report that no such matching exists, given an instance of {\sc spa-st} -- thus solving an open problem given in \cite{AIM07,Man13}. Our algorithm is student-oriented because it involves the students applying to projects. Moreover, the algorithm returns the student-optimal super-stable matching, in the sense that if the given instance admits a super-stable matching then our algorithm will output a solution in which each assigned student has the best project that she could obtain in any super-stable matching that the instance admits. We also present the results of an empirical evaluation based on an implementation of our algorithm that investigates how the nature of the preference lists would affect the likelihood of a super-stable matching existing, with respect to randomly-generated {\sc spa-st} instances.\footnote{From a theoretical perspective, the likelihood of a stable matching existing has been explored for the Stable Roommates problem -- a non-bipartite generalisation of the Stable Marriage problem \cite{PI94}.} Our main finding from the empirical evaluation is that super-stable matchings are very elusive with ties in the students' and lecturers' preference lists. However, if the preference lists of the students are strictly ordered and only the lecturers express ties in their preference lists, the probability of a super-stable matching existing is significantly higher. The remainder of this paper is structured as follows. We give a formal definition of the {\sc spa-s} problem, the {\sc spa-st} variant, and the super-stability concept in Section \ref{section:definitions}. We describe our algorithm for {\sc spa-st} under super-stability in Section \ref{section:algorithm}. Further, Section \ref{section:algorithm} also presents our algorithm's correctness results and some structural properties satisfied by the set of super-stable matchings in an instance of {\sc spa-st}. In Section \ref{emprical-results}, we present the experimental results obtained from our algorithm's empirical evaluation. Finally, Section \ref{section:conclusions} presents some concluding remarks and potential direction for future work. \section{Preliminary definitions and results} \label{section:definitions} \subsection{Formal definition of {\footnotesize SPA-S}} \label{subsection:spa-s} An instance $I$ of {\sc spa-s} involves a set $\mathcal{S} = \{s_1 , s_2, \ldots , s_{n_1}\}$ of \emph{students}, a set $\mathcal{P} = \{p_1 , p_2, \ldots , p_{n_2}\}$ of \emph{projects} and a set $\mathcal{L} = \{l_1 , l_2, \ldots , l_{n_3}\}$ of \emph{lecturers}. Each student $s_i$ ranks a subset of $\mathcal{P}$ in strict order, which forms $s_i$'s preference list. We say that $s_i$ finds $p_j$ \emph{acceptable} if $p_j$ is in $s_i$'s preference list, and we denote by $A_i$ the set of projects that $s_i$ finds acceptable. Each lecturer $l_k \in \mathcal{L}$ offers a non-empty set of projects $P_k$, where $P_1, P_2, \ldots,$ $P_{n_3}$ partitions $\mathcal{P}$. Also, $l_k$ ranks in strict order of preference those students who find at least one project in $P_k$ acceptable, which forms $l_k$'s preference list. We say that $l_k$ finds $s_i$ \textit{acceptable} if $s_i$ is in $l_k$'s preference list, and we denote by $\mathcal{L}_k$ the set of students that $l_k$ finds acceptable. For any pair $(s_i, p_j) \in \mathcal{S} \times \mathcal{P}$, where $p_j$ is offered by $l_k$, we refer to $(s_i, p_j)$ as an \textit{acceptable pair} if $s_i$ and $l_k$ both find each other acceptable, i.e., if $p_j \in A_i$ and $s_i \in \mathcal{L}_k$. Each project $p_j \in \mathcal{P}$ has a capacity $c_j \in \mathbb{Z}^+$ indicating the maximum number of students that can be assigned to $p_j$. Similarly, each lecturer $l_k \in \mathcal{L}$ has a capacity $d_k \in \mathbb{Z}^+$ indicating the maximum number of students that $l_k$ is willing to supervise. We assume that for any lecturer $l_k$, $$\max\{c_j: p_j \in P_k\} \leq d_k \leq \sum \{c_j: p_j \in P_k\},$$ \noindent i.e., the capacity of $l_k$ is (i) at least the highest capacity of the projects offered by $l_k$, and (ii) at most the sum of the capacities of all the projects $l_k$ is offering. We denote by $\mathcal{L}_k^j$, the \emph{projected preference list} of lecturer $l_k$ for $p_j$, which can be obtained from $\mathcal{L}_k$ by removing those students that do not find $p_j$ acceptable (thereby retaining the order of the remaining students from $\mathcal{L}_k$). An \emph{assignment} $M$ is a subset of $\mathcal{S} \times \mathcal{P}$ such that $(s_i, p_j) \in M$ implies that $s_i$ finds $p_j$ acceptable. If $(s_i, p_j) \in M$, we say that $s_i$ \emph{is assigned to} $p_j$, and $p_j$ \emph{is assigned} $s_i$. For convenience, if $s_i$ is assigned in $M$ to $p_j$, where $p_j$ is offered by $l_k$, we may also say that $s_i$ \emph{is assigned to} $l_k$, and $l_k$ \emph{is assigned} $s_i$. For any student $s_i \in \mathcal{S}$, we let $M(s_i)$ denote the set of projects that are assigned to $s_i$ in $M$. For any project $p_j \in \mathcal{P}$, we denote by $M(p_j)$ the set of students that are assigned to $p_j$ in $M$. Project $p_j$ is \emph{undersubscribed}, \emph{full} or \emph{oversubscribed} in $M$ according as $|M(p_j)|$ is less than, equal to, or greater than $c_j$, respectively. Similarly, for any lecturer $l_k \in \mathcal{L}$, we denote by $M(l_k)$ the set of students that are assigned to $l_k$ in $M$. Lecturer $l_k$ is \emph{undersubscribed}, \emph{full} or \emph{oversubscribed} in $M$ according as $|M(l_k)|$ is less than, equal to, or greater than $d_k$, respectively. A \emph{matching} $M$ is an assignment such that each student is assigned to at most one project in $M$, each project is assigned at most $c_j$ students in $M$, and each lecturer is assigned at most $d_k$ students in $M$ (i.e., $|M(s_i)| \leq 1$ for each $s_i \in \mathcal{S}$, $|M(p_j)| \leq c_j$ for each $p_j \in \mathcal{P}$, and $|M(l_k)| \leq d_k$ for each $l_k \in \mathcal{L}$). If $s_i$ is assigned to some project in $M$, for convenience we let $M(s_i)$ denote that project. In what follows, $l_k$ is the lecturer who offers project $p_j$. \begin{definition}[Stability] \label{def:stability} Let $I$ be an instance of {\sc spa-st}, and let $M$ be a matching in $I$. We say that $M$ is \emph{stable} if it admits no blocking pair, where a \emph{blocking pair} is an acceptable pair $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$ such that (a) and (b) holds as follows: \begin{enumerate}[(a)] \item either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$; \item either (i), (ii) or (iii) holds as follows: \begin{enumerate} [(i)] \item each of $p_j$ and $l_k$ is undersubscribed in $M$; \item $p_j$ is undersubscribed in $M$, $l_k$ is full in $M$ and either \begin{enumerate}[(1)] \item $s_i \in M(l_k)$, or \item $l_k$ prefers $s_i$ to the worst student in $M(l_k)$; \end{enumerate} \item $p_j$ is full in $M$ and $l_k$ prefers $s_i$ to the worst student in $M(p_j)$. \end{enumerate} \end{enumerate} \end{definition} To find a stable matching in an instance of {\sc spa-s}, two linear-time algorithms were described in \cite{AIM07}. The stable matching produced by the first algorithm is \emph{student-optimal} (i.e., each assigned student has the best-possible project that she could obtain in any stable matching) while the one produced by the second algorithm is \emph{lecturer-optimal} (i.e., each lecturer has the best set of students that she could obtain in any stable matching). The set of stable matchings in a given instance of {\sc spa-s} satisfy several interesting properties that together form what we will call the \emph{Unpopular Projects Theorem} (analogous to the Rural Hospitals Theorem for {\scriptsize HR} \cite{IMS00}), which we state as follows. \begin{theorem}[\cite{AIM07}] \label{thrm:rural-spa-s} For a given instance of {\sc spa-s}, the following holds: \begin{enumerate} \item each lecturer is assigned the same number of students in all stable matchings; \item exactly the same students are unassigned in all stable matchings; \item a project offered by an undersubscribed lecturer is assigned the same number of students in all stable matchings. \end{enumerate} \end{theorem} As we will see later in this paper, when ties are present in the preference lists of students and lecturers, the set of super-stable matchings also satisfy each of the properties in Theorem \ref{thrm:rural-spa-s}. \subsection{Ties in the preference lists} \label{subsection:spa-st} We now define formally the generalisation of {\sc spa-s} in which the preference lists can include ties. In the preference list of lecturer $l_k\in \mathcal{L}$, a set $T$ of $r$ students forms a \emph{tie of length $r$} if $l_k$ does not prefer $s_i$ to $s_{i'}$ for any $s_i, s_{i'} \in T$ (i.e., $l_k$ is \emph{indifferent} between $s_i$ and $s_{i'}$). A tie in a student's preference list is defined similarly. For convenience, henceforth, we consider a non-tied entry in a preference list as a tie of length one. We denote by {\sc spa-st} the generalisation of {\sc spa-s} in which the preference list of each student (respectively lecturer) comprises a strict ranking of ties, each comprising one or more projects (respectively students). An example {\sc spa-st} instance $I_1$ is given in Fig.~\ref{fig:spa-st-instance-1}, which involves the set of students $\mathcal{S} = \{s_1, s_2, s_3, s_4, \\ s_5\}$, the set of projects $\mathcal{P} = \{p_1, p_2, p_3\}$ and the set of lecturers $\mathcal{L} = \{l_1, l_2\}$, with $P_1 = \{p_1, p_2\}$ and $P_2 = \{p_3\}$. Ties in the preference lists are indicated by round brackets. \begin{figure} \caption{ \small An example instance $I_1$ of {\sc spa-st}.} \label{fig:spa-st-instance-1} \end{figure} In the context of {\sc spa-st}, we assume that all notation and terminology carries over from Section \ref{subsection:spa-s} as defined for {\sc spa-s} with the exception of stability, which we now define. When ties appear in the preference lists, three levels of stability arise (as in the {\sc hrt} context \cite{IMS00,IMS03}), namely \emph{weak stability, strong stability and super-stability}. The formal definition for weak stability in {\sc spa-st} follows from the definition for stability in {\sc spa-s} (see Definition \ref{def:stability}). Moreover, the existence of a weakly stable matching in an instance $I$ of {\sc spa-st} is guaranteed by breaking the ties in $I$ arbitrarily, thus giving rise to an instance $I'$ of {\sc spa-s}. Clearly, a stable matching in $I'$ is weakly stable in $I$. Indeed a converse of sorts holds, which gives rise to the following proposition. \begin{restatable}[]{proposition}{weakstability} \label{proposition2} Let $I$ be an instance of {\sc spa-st}, and let $M$ be a matching in $I$. Then $M$ is weakly stable in $I$ if and only if $M$ is stable in some instance $I'$ of {\sc spa-s} obtained from $I$ by breaking the ties in some way. \end{restatable} \begin{proof} Let $I$ be an instance of {\sc spa-st} and let $M$ be a matching in $I$. Suppose that $M$ is weakly stable in $I$. Let $I'$ be an instance of {\sc spa-s} obtained from $I$ by breaking the ties in the following way. For each student $s_i$ in $I$ such that the preference list of $s_i$ includes a tie $T$ containing two or more projects, we order the preference list of $s_i$ in $I'$ as follows: if $s_i$ is assigned in $M$ to a project $p_j$ in $T$ then $s_i$ prefers $p_j$ to every other project in $T$; otherwise, we order the projects in $T$ arbitrarily. For each lecturer $l_k$ in $I$ such that $l_k$'s preference list includes a tie $X$, if $X$ contains students that are assigned to $l_k$ in $M$ and students that are not assigned to $l_k$ in $M$ then $l_k$'s preference list in $I'$ is ordered in such a way that each $s_i\in X\cap M(l_k)$ is preferred to each $s_{i'}\in X\setminus M(l_k)$; otherwise, we order the students in $X$ arbitrarily. Now, suppose $(s_i, p_j)$ forms a blocking pair for $M$ in $I'$. Given how the ties in $I$ were removed to obtain $I'$, this implies that $(s_i, p_j)$ forms a blocking pair for $M$ in $I$, a contradiction to our assumption that $M$ is weakly stable in $I$. Thus $M$ is stable in $I'$. Conversely, suppose $M$ is stable in some instance $I'$ of {\sc spa-s} obtained from $I$ by breaking the ties in some way. Now suppose that $M$ is not weakly stable in $I$. Then some pair $(s_i, p_j)$ forms a blocking pair for $M$ in $I$. It is then clear from the definition of weak stability and from the construction of $I'$ that $(s_i, p_j)$ is a blocking pair for $M$ in $I'$, a contradiction. \qed \end{proof} \noindent As mentioned earlier, super-stability is the most robust concept to seek. Only if no super-stable matching exists in the underlying problem instance should other forms of stability be sought in a practical setting. Thus, for the remainder of this paper, we focus on super-stability in the {\sc spa-st} context. \begin{definition}[Super-stability] \label{definition:super-stability} Let $I$ be an instance of {\sc spa-st}, and let $M$ be a matching in $I$. We say that $M$ is \emph{super-stable} if it admits no blocking pair, where a \emph{blocking pair} is an acceptable pair $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$ such that (a) and (b) holds as follows: \begin{enumerate}[(a)] \item either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them; \item either (i), (ii), or (iii) holds as follows: \begin{enumerate} [(i)] \item each of $p_j$ and $l_k$ is undersubscribed in $M$; \item $p_j$ is undersubscribed in $M$, $l_k$ is full in $M$ and either \begin{enumerate}[(1)] \item $s_i \in M(l_k)$, or \item $l_k$ prefers $s_i$ to the worst student/s in $M(l_k)$ or is indifferent between them; \end{enumerate} \item $p_j$ is full in $M$ and $l_k$ prefers $s_i$ to the worst student/s in $M(p_j)$ or is indifferent between them. \end{enumerate} \end{enumerate} \end{definition} It may be verified that the matching $M = \{(s_3, p_2), (s_4, p_3), (s_5, p_1)\}$ is super-stable in Fig.~\ref{fig:spa-st-instance-1}. Clearly, a super-stable matching is also weakly stable. Moreover, the super-stability definition gives rise to Proposition \ref{proposition1}, which can be regarded as an analogue of Proposition \ref{proposition2} for super-stability, restated as follows. \superstability* \begin{proof} Let $I$ be an instance of {\sc spa-st} and let $M$ be a matching in $I$. Suppose that $M$ is super-stable in $I$. We want to show that $M$ is stable in every instance of {\sc spa-s} obtained from $I$ by breaking the ties in some way. Now, let $I'$ be an arbitrary instance of {\sc spa-s} obtained from $I$ by breaking the ties in some way, and suppose $M$ is not stable in $I'$. This implies that $M$ admits a blocking pair $(s_i, p_j)$ in $I'$. Since $I'$ is an arbitrary {\sc spa-s} instance obtained from $I$ by breaking the ties in some way, it follows that in $I$: (i) if $s_i$ is assigned in $M$ then $s_i$ either prefers $p_j$ to $M(s_i)$ or is indifferent between them, (ii) if $p_j$ is full in $M$ then $l_k$ either prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them, and (iii) if $l_k$ is full in $M$ then either $s_i \in M(l_k)$ or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or is indifferent between them. This implies that $(s_i, p_j)$ forms a blocking pair for $M$ in $I$, a contradiction to the super-stability of $M$. Conversely, suppose $M$ is stable in every instance of {\sc spa-s} obtained from $I$ by breaking the ties in some way. Now suppose $M$ is not super-stable in $I$. This implies that $M$ admits a blocking pair $(s_i, p_j)$ in $I$. We construct an instance $I'$ of {\sc spa-s} from $I$ by breaking the ties in the following way: (i) if $s_i$ is assigned in $M$ and $s_i$ is indifferent between $p_j$ and $M(s_i)$ in $I$ then $s_i$ prefers $p_j$ to $M(s_i)$ in $I'$; otherwise we break the ties in $s_i$'s preference list arbitrarily, and (ii) if some student, say $s_{i'}$, different from $s_i$ is assigned to $l_k$ in $M$ such that $l_k$ is indifferent between $s_i$ and $s_{i'}$ in $I$ then $l_k$ prefers $s_i$ to $s_{i'}$ in $I'$; otherwise we break the ties in $l_k$'s preference list arbitrarily. Thus $(s_i, p_j)$ forms a blocking pair for $M$ in $I'$, i.e., $M$ is not stable in $I'$, a contradiction to the fact that $M$ is stable in every instance of {\sc spa-s} obtained from $I$ by breaking the ties in some way. \qed\end{proof} \noindent The following proposition, which is a consequence of Propositions \ref{proposition1} and \ref{proposition2}, and Theorem \ref{thrm:rural-spa-s}, tells us that if a super-stable matching $M$ exists in $I$ then all weakly stable matchings in $I$ are of the same size (equal to the size of $M$) and match exactly the same set of students. \begin{restatable}[]{proposition}{allinone} \label{proposition3} Let $I$ be an instance of {\sc spa-st}, and suppose that $I$ admits a super-stable matching $M$. Then the Unpopular Projects Theorem holds for the set of weakly stable matchings in $I$. \end{restatable} \begin{proof} Let $I$ be an instance of {\sc spa-st}. Let $M$ be a super-stable matching in $I$ and let $M'$ be a weakly stable matching in $I$. Then by Proposition \ref{proposition2}, $M'$ is stable in some instance $I'$ of {\sc spa-s} obtained from $I$ by breaking the ties in some way. Also $M$ is stable in $I'$ by Proposition \ref{proposition1}. By Theorem \ref{thrm:rural-spa-s}, each lecturer is assigned the same number of students in $M$ and $M'$, exactly the same students are unassigned in $M$ and $M'$, and a project offered by an undersubscribed lecturer is assigned the same number of students in $M$ and $M'$. Hence, the Unpopular Projects Theorem holds for the set of weakly stable matchings in $I$. \qed \end{proof} \subsection{Cloning from {\sc spa-st} to {\sc hrt} does not work in general} \label{subsect:cloning} As mentioned earlier, Irving \textit{et al.}~\cite{IMS00} described a polynomial-time algorithm to find a super-stable matching or report that no such matching exists, given an instance of {\sc hrt}. The authors referred to their algorithm as Algorithm {\sf HRT-Super-Res}. One might assume that reducing a given instance of {\sc spa-st} to an instance of {\sc hrt} (using a ``cloning'' technique) and subsequently applying Algorithm {\sf HRT-Super-Res} to the resulting instance would solve our problem. However, this is not always true. In what follows, we describe an obvious method to clone an instance of {\sc spa-st} to an instance of {\sc hrt}, and we show that applying the super-stable matching algorithm described in \cite{IMS00} to the resulting {\sc hrt} instance does not work in general. A method to derive an instance $I'$ of {\sc hrt} from an instance $I$ of {\sc spa-st} was described by Cooper and Manlove \cite{CM18a}. We explain this method as follows. The students and projects involved in $I$ are converted into residents and hospitals respectively in $I'$, i.e., each $s_i \in \mathcal{S}$ becomes $r_i$ in the cloned instance, and each $p_j \in \mathcal{P}$ becomes $h_j$. Residents inherit their preference lists naturally from students, i.e., if $r_i$ corresponds to $s_i$ then the preference list of $r_i$ in $I'$ is $A_i$, with each project in $A_i$ being replaced by the associated hospital. Hospitals inherit their preference lists from the projected preference list of the associated project according to the lecturer offering the project, i.e., if $p_j$ corresponds to $h_j$ (where $p_j$ is offered by $l_k$) then the preference list of $h_j$ in $I'$ is $\mathcal{L}_k^j$, with each student in $\mathcal{L}_k^j$ being replaced by the associated resident. Each hospital also inherits its capacity from the project, i.e., for each $h_j$ associated with $p_j$, the capacity of $h_j$ is $c_j$. Let $l_k$ be an arbitrary lecturer in $I$. In order to translate $l_k$'s capacity into the {\sc hrt} instance, we create $n$ \emph{dummy residents}\footnote{The dummy residents created for each hospital will offset the difference between the corresponding lecturer capacity and the total capacity of her proposed projects.} for each hospital $h_j$ corresponding to a project $p_j \in P_k$, where $n$ is the difference between the sum of the capacities of all the projects in $P_k$ and the capacity of $l_k$ (recall that $\sum_{p_j \in P_k} c_j \geq d_k$). The preference list for each of these dummy residents will be a single tie consisting of all the hospitals corresponding to a project in $P_k$. Further, the preference list for each hospital corresponding to a project in $P_k$ will include a tie in its first position consisting of all the dummy residents associated with $l_k$. Next, we describe how to map between matchings in $I$ and in $I'$. Let $M$ and $M'$ be a matching in $I$ and $I'$ respectively. Let $r_i$ be the resident associated with $s_i$ and let $h_j$ be the hospital associated with $p_j$. If $s_i$ is assigned in $M$ to project $p_j$, then $r_i$ is assigned in $M'$ to hospital $h_j$. To illustrate the cloning technique described above, we give an example instance $I$ of {\sc spa-st} in Fig.~\ref{fig:super-instance-2} as well as the corresponding cloned {\sc hrt} instance $I'$ in Fig.~\ref{fig:super-instance-2-cloned}. Also, we give an intuition as to why this technique will not work in general. \begin{figure}\label{fig:super-instance-2} \end{figure} \begin{figure}\label{fig:super-instance-2-cloned} \end{figure} With respect to Figs.~\ref{fig:super-instance-2} and \ref{fig:super-instance-2-cloned}, each resident $r_1, r_2$ and $r_3$ in $I'$ corresponds to student $s_1, s_2$ and $s_3$ in $I$, respectively; and the preference list of each resident is adapted from the preference list of the associated student. Also, each hospital $h_1, h_2$ and $h_3$ in $I'$ corresponds to project $p_1, p_2$ and $p_3$ in $I$, respectively. The preference list of hospitals $h_1$ and $h_2$ is $\mathcal{L}_1^1$ and $\mathcal{L}_1^2$ respectively, since $l_1$ is the lecturer that offers both $p_1$ and $p_2$. Similarly, the preference list of hospital $h_3$ is $\mathcal{L}_2^3$, since $l_2$ is the lecturer that offers $p_3$. Further, for lecturer $l_1$ who offers both $p_1$ and $p_2$, since $c_1 + c_2 = 2 > 1 = d_1$, we add one dummy resident $r_{d_1}$ to the cloned instance. The preference list of $r_{d_1}$ is a single tie consisting of $h_1$ and $h_2$; and the preference list of both $h_1$ and $h_2$ includes $r_{d_1}$ in first position. The reader can easily verify that matching $M = \{(s_1, p_1), (s_3, p_3)\}$ is super-stable in the {\sc spa-st} instance $I$ illustrated in Fig.~\ref{fig:super-instance-2}. Now, following our description of how to map between matchings in $I$ and in $I'$, a matching in $I'$ is $M' = \{(r_{d_1}, h_2), (r_1, h_1), (r_3, h_3)\}$, with $(s_1, p_1) \in M$ corresponding to $(r_1, h_1) \in M'$ and $(s_3, p_3) \in M$ corresponding to $(r_3, h_3) \in M'$. Clearly, $M'$ is not super-stable in $I'$ as $(r_{d_1}, h_1)$ forms a blocking pair. In fact, the {\sc hrt} instance $I'$ admits no super-stable matching. The justification for this is as follows: irrespective of the hospital that the dummy resident $r_{d_1}$ is assigned to in any matching obtained from $I'$, $r_{d_1}$ will block this matching via the other hospitals tied in her preference list (since the hospital would be better off taking on $r_{d_1}$, and $r_{d_1}$ would be no worse off). One way to avoid this problem would be to strictly order the hospitals in $r_{d_1}$'s preference list; however, the order in which the hospitals appear will lead to different possibilities. For instance: if $r_{d_1}$ prefers $h_1$ to $h_2$, the reader can verify that the corresponding {\sc hrt} instance admits no super-stable matching; however, if $r_{d_1}$ prefers $h_2$ to $h_1$, again the reader can verify that the corresponding {\sc hrt} instance admits the super-stable matching $\{(r_{d_1}, h_2), (r_1, h_1), (r_3, h_3)\}$. The downside of this strategy is that there is no obvious reason as to why $r_{d_1}$ should prefer $h_2$ to $h_1$ in the cloned {\sc hrt} instance in Fig.~\ref{fig:super-instance-2-cloned} by merely looking at the original {\sc spa-st} instance in Fig.~\ref{fig:super-instance-2}. Hence, in order to make this technique work in general, we will need to generate every {\sc hrt} instance obtained by ordering the dummy residents' preference lists in some way. This is exponential in the problem instance. \section{An algorithm for {\small SPA-ST} under super-stability} \label{section:algorithm} In this section we present our algorithm for {\sc spa-st} under super-stability, which we will refer to as Algorithm {\sf SPA-ST-super}. Before we proceed, we briefly describe Algorithm {\sf HRT-Super-Res} \cite{IMS00}. The algorithm involves a sequence of proposals from the residents to the hospitals. Each resident proposes in turn to all of the hospitals tied together at the head of her preference list, and all proposals are provisionally accepted. If a hospital $h$ becomes oversubscribed then none of $h$'s worst assignees nor any resident tied with these assignees in $h$'s preference list can be assigned to $h$ in any super-stable matching -- such pairs $(r, h)$ are deleted from each other's preference lists. If a hospital $h$ is full then no resident strictly worse than $h$'s worst assignees can be assigned to $h$ in any super-stable matching -- again such $(r,h)$ pairs are deleted from each other's preference lists. The proposal sequence terminates once every resident is either assigned to a hospital or has an empty preference list. At this point, if the constructed assignment of residents to hospitals is super-stable in the original {\sc hrt} instance then the assignment is returned as a super-stable matching. Otherwise, the algorithm reports that no super-stable matching exists. We note that our algorithm is a non-trivial extension of Algorithm {\sf HRT-Super-Res} for {\sc hrt} \cite{IMS00}. Due to the more general setting of {\sc spa-st}, Algorithm {\sf SPA-ST-super} requires some new ideas (precisely lines 27-34 of the algorithm on page \pageref{algorithmSPA-STsuper}), and the proofs of the correctness results are more complex than for the aforementioned algorithm for {\sc hrt}. We give definitions relating to the algorithm in Section \ref{subsect:algorithm-definition}. We give a description of our algorithm in Section \ref{subsect:algorithm-description}, before presenting it in pseudocode form. In Section \ref{example-description}, we illustrate an execution of our algorithm with respect to an example {\sc spa-st} instance. We present the algorithm's correctness results in Section \ref{correctness-result}. Finally, in Section \ref{subsect:properties}, we show that the set of super-stable matchings in an instance of {\sc spa-st} satisfy analogous properties to those given in Theorem \ref{thrm:rural-spa-s}. \subsection{Definitions relating to the algorithm} \label{subsect:algorithm-definition} First, we present some definitions relating to the algorithm. In what follows, $I$ is an instance of {\sc spa-st}, $(s_i, p_j)$ is an acceptable pair in $I$ and $l_k$ is the lecturer who offers $p_j$. Further, if $(s_i,p_j)$ belongs to some super-stable matching in $I$, we call $(s_i, p_j)$ a \textit{super-stable pair}. During the execution of the algorithm, students become \textit{provisionally assigned} to projects. It is possible for a project to be provisionally assigned a number of students that exceed its capacity. This holds analogously for a lecturer. The algorithm proceeds by deleting from the preference lists certain $(s_i, p_j)$ pairs that cannot be super-stable. By the term \textit{delete} $(s_i, p_j)$, we mean the removal of $p_j$ from $s_i$'s preference list and the removal of $s_i$ from $\mathcal{L}_k^j$ (the projected preference list of lecturer $l_k$ for $p_j$). In addition, if $s_i$ is provisionally assigned to $p_j$ at this point, we break the assignment. If $s_i$ has been deleted from every projected preference list of $l_k$ that she originally belonged to, we will implicitly assume that $s_i$ has been deleted from $l_k$'s preference list. By the \textit{head} of a student's preference list at a given point, we mean the set of one or more projects, tied in her preference list after any deletions might have occurred, that she prefers to all other projects in her list. For project $p_j$, we define the \textit{tail} of $\mathcal{L}_k^j$ as the least-preferred tie in $\mathcal{L}_k^j$ after any deletions might have occurred (recalling that a tie can be of length one). In the same fashion, we define the \textit{tail} of $\mathcal{L}_k$ (the preference list of lecturer $l_k$) as the least-preferred tie in $\mathcal{L}_k$ after any deletions might have occurred. If $s_i$ is provisionally assigned to $p_j$, we define the \textit{successors} of $s_i$ in $\mathcal{L}_{k}^j$ as those students that are worse than $s_i$ in $\mathcal{L}_{k}^j$. An analogous definition holds for the successors of $s_i$ in $\mathcal{L}_k$. \subsection{Description of the algorithm} \label{subsect:algorithm-description} We now describe our algorithm, shown in pseudocode form in Algorithm~\ref{algorithmSPA-STsuper}. Algorithm {\sf SPA-ST-super} begins by initialising an empty set $M$ which will contain the provisional assignments of students to projects (and implicitly to lecturers). We remark that such assignments can subsequently be broken during the algorithm's execution. Also, each project is initially assigned to be empty (i.e., not assigned to any student). The \texttt{while} loop of the algorithm involves each student $s_i$ who is not provisionally assigned to any project in $M$ and who has a non-empty preference list applying in turn to each project $p_j$ at the head of her list. Immediately, $s_i$ becomes provisionally assigned to $p_j$ in $M$ (and to $l_k$). If, by gaining a new student, $p_j$ becomes oversubscribed, it turns out that none of the students $s_t$ at the tail of $\mathcal{L}_k^j$ can be assigned to $p_j$ in any super-stable matching -- such pairs $(s_t, p_j)$ are deleted. Similarly, if by gaining a new student, $l_k$ becomes oversubscribed, none of the students $s_t$ at the tail of $\mathcal{L}_k$ can be assigned to any project offered by $l_k$ in any super-stable matching -- the pairs $(s_t, p_u)$, for each project $p_u \in P_k$ that $s_t$ finds acceptable, are deleted. Regardless of whether any deletions occurred as a result of the two conditionals described in the previous paragraph, we have two further (possibly non-disjoint) cases in which deletions may occur. If $p_j$ becomes full, we let $s_r$ be any worst student provisionally assigned to $p_j$ (according to $\mathcal{L}_k^j$), and we delete $(s_t, p_j)$ for each successor $s_t$ of $s_r$ in $\mathcal{L}_k^j$. Similarly if $l_k$ becomes full, we let $s_r$ be any worst student provisionally assigned to $l_k$, and we delete $(s_t, p_u)$, for each successor $s_t$ of $s_r$ in $\mathcal{L}_k$ and for each project $p_u \in P_k$ that $s_t$ finds acceptable. As we will prove later, none of the (student, project) pairs that we delete is a super-stable pair. At the point where the \texttt{while} loop terminates (i.e., when every student is provisionally assigned to one or more projects or has an empty preference list), if some project $p_j$ that was previously full ends up undersubscribed, we let $s_r$ be any one of the most-preferred students (according to $\mathcal{L}_k^j$) who was provisionally assigned to $p_j$ during some iteration of the algorithm but is not assigned to $p_j$ at this point (for convenience, we henceforth refer to such $s_r$ as the most-preferred student rejected from $p_j$ according to $\mathcal{L}_k^j$). If the students at the tail of $\mathcal{L}_k$ (recalling that the tail of $\mathcal{L}_k$ is the least-preferred tie in $\mathcal{L}_k$ after any deletions might have occurred) are no better than $s_r$, it turns out that none of these students $s_t$ can be assigned to any project offered by $l_k$ in any super-stable matching -- the pairs $(s_t, p_u)$, for each project $p_u \in P_k$ that $s_t$ finds acceptable, are deleted. The \texttt{while} loop is then potentially reactivated, and the entire process continues until every student is provisionally assigned to a project or has an empty preference list, at which point the \texttt{repeat-until} loop terminates. Upon termination of the \texttt{repeat-until} loop, if the set $M$, containing the assignment of students to projects, is super-stable relative to the given instance $I$ then $M$ is output as a super-stable matching in $I$. Otherwise, the algorithm reports that no super-stable matching exists in $I$. \begin{algorithm}[htbp] \caption{Algorithm {\sf SPA-ST-super}} \label{algorithmSPA-STsuper} \begin{algorithmic}[1] \Require {{\sc spa-st} instance $I$} \Ensure{a super-stable matching $M$ in $I$ or ``no super-stable matching exists in $I$''} \State $M \gets \emptyset$ \ForEach {$p_j \in \mathcal{P}$} \State \texttt{full}($p_j$) = \texttt{false} \EndFor \Repeat{} \While {some student $s_i$ is unassigned and has a non-empty preference list} \ForEach {project $p_j$ at the head of $s_i$'s preference list} \State $l_k \gets $ lecturer who offers $p_j$ \State /* $s_i$ applies to $p_j$ */ \State $M \gets M \cup \{(s_i, p_j)\}$ /*provisionally assign $s_i$ to $p_j$ (and to $l_k$) */ \If {$p_j$ is oversubscribed} \ForEach{student $s_t$ at the tail of $\mathcal{L}_{k}^{j}$} \State delete $(s_t, p_j)$ \EndFor \ElsIf {$l_k$ is oversubscribed} \ForEach{student $s_t$ at the tail of $\mathcal{L}_{k}$} \ForEach {project $p_u \in P_k \cap A_t$} \State delete $(s_t, p_u)$ \EndFor \EndFor \EndIf \If {$p_j$ is full} \State \texttt{full}($p_j$) = \texttt{true} \State $s_r \gets $ worst student assigned to $p_j$ according to $\mathcal{L}_{k}^{j}$ \{any if $> 1$\} \ForEach{successor $s_t$ of $s_r$ on $\mathcal{L}_{k}^{j}$} \State delete $(s_t, p_j)$ \EndFor \EndIf \If {$l_k$ is full} \State $s_r \gets $ worst student assigned to $l_k$ according to $\mathcal{L}_{k}$ \{any if $> 1$\} \ForEach{successor $s_t$ of $s_r$ on $\mathcal{L}_{k}$} \ForEach{project $p_u \in P_k \cap A_t$ } \State delete $(s_t, p_u)$ \EndFor \EndFor \EndIf \EndFor \EndWhile \ForEach{$p_j \in \mathcal{P}$} \If {$p_j$ is undersubscribed and \texttt{full}($p_j$) is \texttt{true}} \State $l_k \gets $ lecturer who offers $p_j$ \State $s_r \gets $ most-preferred student rejected from $p_j$ according to $\mathcal{L}_{k}^{j}$ \{any if $> 1$\} \If{the students at the tail of $\mathcal{L}_k$ are no better than $s_r$} \ForEach{student $s_t$ at the tail of $\mathcal{L}_k$} \ForEach{project $p_u \in P_k \cap A_t$ } \State delete $(s_t, p_u)$ \label{alg:deletion-outside} \EndFor \EndFor \EndIf \EndIf \EndFor \Until {every unassigned student has an empty preference list} \If {$M$ is super-stable in $I$} \State \Return $M$ \Else \State \Return ``no super-stable matching exists in $I$'' \EndIf \end{algorithmic} \end{algorithm} \subsection{Example algorithm execution} \label{example-description} We illustrate an execution of Algorithm {\sf SPA-ST-super} with respect to the {\sc spa-st} instance shown in Fig.~\ref{fig:spa-st-instance-1} (page \pageref{fig:spa-st-instance-1}). We initialise $M = \{\}$, which will contain the provisional assignment of students to projects. For each project $p_j \in \mathcal{P}$, we set \texttt{full}($p_j$) = \texttt{false} (\texttt{full}($p_j$) will be set to \texttt{true} when $p_j$ becomes full, so that we can easily identify any project that was full during an iteration of the algorithm and ended up undersubscribed). We assume that the students become provisionally assigned to each project at the head of their list in subscript order. Table~\ref{example-illustration} illustrates how this execution of Algorithm {\sf SPA-ST-super} proceeds with respect to $I_1$. \begin{table}[htbp] \caption{\label{example-illustration} \small An execution of Algorithm {\sf SPA-ST-super} with respect to Fig.~\ref{fig:spa-st-instance-1}.} \centering \small \setlength{\tabcolsep}{0.8em} \renewcommand{1}{1.7} \begin{tabular}{p{1.6cm}p{2.4cm}p{10cm}} \hline\noalign{ } {\texttt while} loop iterations & Student applies to project & Consequence \\ \noalign{ }\hline\noalign{ } $1$ & $s_1$ applies to $p_1$ & $M=\{(s_1, p_1)\}$. \texttt{full}($p_1$) = \texttt{true}. \\ \hline $2$ & $s_2$ applies to $p_1$ & $M=\{(s_1, p_1), (s_2, p_1)\}$. $p_1$ becomes oversubscribed. The tail of $\mathcal{L}_1^1$ contains $s_1$ and $s_2$ -- thus we delete the pairs $(s_1, p_1)$ and $(s_2, p_1)$ (and we break the provisional assignments). \\ & $s_2$ applies to $p_3$ & $M=\{(s_2, p_3)\}$. \texttt{full}($p_3$) = \texttt{true}.\\ \hline $3$ & $s_3$ applies to $p_2$ & $M=\{(s_2, p_3), (s_3, p_2)\}$. \\ \hline $4$ & $s_4$ applies to $p_2$ & $M = \{(s_2, p_3), (s_3, p_2), (s_4, p_2)\}$. \texttt{full}($p_2$) = \texttt{true}. \\ \hline $5$ & $s_5$ applies to $p_3$ & $M = \{(s_2, p_3), (s_3, p_2), (s_4, p_2), (s_5, p_3)\}$. $p_3$ becomes oversubscribed. The tail of $\mathcal{L}_2^3$ contains only $s_2$ -- thus we delete the pair $(s_2, p_3)$ (and we break the provisional assignment).\\ \hline \multicolumn{3}{p{15.2cm}}{The first iteration of the \texttt{while} loop terminates since every unassigned student (i.e., $s_1$ and $s_2$) has an empty preference list. At this point, \texttt{full}($p_1$) is \texttt{true} and $p_1$ is undersubscribed. Moreover, the student at the tail of $\mathcal{L}_1$ (i.e., $s_4$) is no better than $s_1$, where $s_1$ was previously assigned to $p_1$ and $s_1$ is also the most-preferred student rejected from $p_1$ according to $\mathcal{L}_1^1$; thus we delete the pair $(s_4, p_2)$. The \texttt{while} loop is then reactivated.}\\ \hline $6$ & $s_4$ applies to $p_3$ & $M = \{(s_3, p_2), (s_5, p_3), (s_4, p_3)\}$. $p_3$ becomes oversubscribed. The tail of $\mathcal{L}_2^3$ contains only $s_5$ -- thus we delete the pair $(s_5, p_3)$.\\ \hline $7$ & $s_5$ applies to $p_1$ & $M = \{(s_3, p_2), (s_4, p_3), (s_5, p_1)\}$. \\ \hline \multicolumn{3}{p{15.2cm}}{Again, every unassigned students has an empty preference list. We also have that \texttt{full}($p_2$) is \texttt{true} and $p_2$ is undersubscribed; however no further deletion is carried out in line 34 of the algorithm, since the student at the tail of $\mathcal{L}_1$ (i.e., $s_3$) is better than $s_4$, where $s_4$ was previously assigned to $p_2$ and $s_4$ is also the most-preferred student rejected from $p_2$ according to $\mathcal{L}_1^2$. Hence, the \texttt{repeat-until} loop terminates and the algorithm outputs $M = \{(s_3, p_2), (s_4, p_3), (s_5, p_1)\}$ as a super-stable matching. It is clear that $M$ is super-stable in the original instance $I_2$.}\\ \noalign{ }\hline \end{tabular} \end{table} \subsection{Correctness of Algorithm {\sf SPA-ST-super}} \label{correctness-result} We now present a series of results concerning the correctness of Algorithm {\sf SPA-ST-super}. The first of these results deals with the fact that no super-stable pair is deleted during an execution of the algorithm. In what follows, $I$ is an instance of {\sc spa-st}, $(s_i, p_j)$ is an acceptable pair in $I$ and $l_k$ is the lecturer who offers $p_j$. \begin{restatable}[]{lemma}{nopairdeletion} \label{pair-deletion} If a pair $(s_i, p_j)$ is deleted during an execution of Algorithm {\sf SPA-ST-super}, then $(s_i, p_j)$ does not belong to any super-stable matching in $I$. \end{restatable} \noindent In order to prove Lemma \ref{pair-deletion}, we present Lemmas \ref{lemma:super-pair-deletion-within} and \ref{lemma:super-pair-deletion-outside}. \begin{lemma} \label{lemma:super-pair-deletion-within} If a pair $(s_i, p_j)$ is deleted within the \texttt{while} loop during an execution of Algorithm {\sf SPA-ST-super} then $(s_i, p_j)$ does not belong to any super-stable matching in $I$. \end{lemma} \begin{proof} Without loss of generality, suppose that the first super-stable pair to be deleted within the \texttt{while} loop during an arbitrary execution $E$ of the algorithm is $(s_i, p_j)$, which belongs to some super-stable matching, say $M^*$. Suppose that $M$ is the assignment immediately after the deletion. Let us denote this point in the algorithm where the deletion is made by $\ddagger$. During $E$, there are four cases that would lead to the deletion of any (student, project) pair within the \texttt{while} loop. \begin{enumerate}[(1)] \item \emph{$p_j$ is oversubscribed.} Suppose that $(s_i, p_j)$ is deleted because some student (possibly $s_i$) became provisionally assigned to $p_j$ during $E$, causing $p_j$ to become oversubscribed. If $p_j$ is full or undersubscribed at point $\ddagger$, since $s_{i} \in M^*(p_j) \setminus M(p_j)$ and no project can be oversubscribed in $M^*$, then there is some student $s_r \in M(p_j) \setminus M^*(p_j)$ such that $l_k$ prefers $s_r$ to $s_i$ or is indifferent between them. We note that $s_r$ cannot be assigned to a project that she prefers to $p_j$ in any super-stable matching. Otherwise, since $p_j$ must have been in the head of $s_r$'s preference list when she applied, this would mean that a super-stable pair was deleted before $(s_i, p_j)$. Thus either $s_r$ is unassigned in $M^*$ or $s_r$ prefers $p_j$ to $M^*(s_r)$ or $s_r$ is indifferent between them. Clearly, for any combination of $l_k$ and $p_j$ being full or undersubscribed in $M^*$, it follows that $(s_r, p_j)$ blocks $M^*$, a contradiction. \item \emph{$l_k$ is oversubscribed.} Suppose that $(s_i, p_j)$ is deleted because some student (possibly $s_i$) became provisionally assigned to a project offered by lecturer $l_k$ during $E$, causing $l_k$ to become oversubscribed. At point $\ddagger$, none of the projects offered by $l_k$ is oversubscribed in $M$, otherwise we will be in case (1). Similar to case (1), if $l_k$ is full or undersubscribed at point $\ddagger$, since $s_{i} \in M^*(p_{j}) \setminus M(p_{j})$ and no lecturer can be oversubscribed in $M^*$, it follows that there is some project $p_{j'} \in P_k$ and some student $s_{r} \in M(p_{j'}) \setminus M^*(p_{j'})$ such that $l_k$ prefers $s_{r}$ to $s_i$ or is indifferent between them. We consider two subcases. \begin{enumerate}[(i)] \item If $p_{j'} = p_j$ then $s_{r} \neq s_i$. Moreover, as in case (1), either $s_{r}$ is unassigned in $M^*$ or $s_{r}$ prefers $p_{j'}$ to $M^*(s_{r})$ or $s_r$ is indifferent between them. For any combination of $l_k$ and $p_{j'}$ being full or undersubscribed in $M^*$, we have that $(s_{r}, p_{j'})$ blocks $M^*$, a contradiction. \item If $p_{j'} \neq p_j$. Assume firstly that $s_{r} \neq s_i$. Then as $p_{j'}$ has fewer assignees in $M^*$ than it has provisional assignees in $M$, and as in (i) above, $(s_{r}, p_{j'})$ blocks $M^*$, a contradiction. Finally assume $s_{r} = s_i$. Then $s_i$ must have applied to $p_{j'}$ at some point during $E$ before $\ddagger$. Clearly, either $s_i$ prefers $p_{j'}$ to $p_j$ or $s_i$ is indifferent between them, since $p_{j'}$ must have been in the head of $s_i$'s preference list when $s_i$ applied. Since $s_i \in M^*(l_k)$ and $p_{j'}$ is undersubscribed in $M^*$, it follows that $(s_i, p_{j'})$ blocks $M^*$, a contradiction. \end{enumerate} \item \emph{$p_j$ is full.} Suppose that $(s_i, p_j)$ is deleted because $p_j$ became full during $E$. At point $\ddagger$, $p_j$ is full in $M$. Thus at least one of the students in $M(p_j)$, say $s_{r}$, will not be assigned to $p_j$ in $M^*$, for otherwise $p_j$ will be oversubscribed in $M^*$. This implies that either $s_{r}$ is unassigned in $M^*$ or $s_{r}$ prefers $p_j$ to $M^*(s_{r})$ or $s_{r}$ is indifferent between them. For otherwise, we obtain a contradiction to $(s_i, p_j)$ being the first super-stable pair to be deleted. Since $l_k$ prefers $s_{r}$ to $s_i$, it follows that $(s_{r}, p_j)$ blocks $M^*$, a contradiction. \item \emph{$l_k$ is full.} Suppose that $(s_i, p_j)$ is deleted because $l_k$ became full during $E$. We consider two subcases. \begin{enumerate}[(i)] \item All the students assigned to $p_j$ in $M$ at point $\ddagger$ (if any) are also assigned to $p_j$ in $M^*$. This implies that $p_j$ has one more assignee in $M^*$ than it has provisional assignees in $M$, namely $s_i$. Thus, some other project $p_{j'} \in P_k$ has fewer assignees in $M^*$ than it has provisional assignees in $M$, for otherwise $l_k$ would be oversubscribed in $M^*$. Hence there exists some student $s_{r} \in M(p_{j'}) \setminus M^*(p_{j'})$. It is clear that $s_{r} \neq s_i$, since $s_i$ plays the role of $s_t$ at some for loop iteration in line 24 of the algorithm. Also, $s_{r}$ cannot be assigned to a project that she prefers to $p_{j'}$ in $M^*$, as explained in case (1). Moreover, since $p_{j'}$ is undersubscribed in $M^*$ and $l_k$ prefers $s_{r}$ to $s_i$, it follows that $(s_{r}, p_{j'})$ blocks $M^*$, a contradiction. \item Some student, say $s_{r}$, who is assigned to $p_j$ in $M$ is not assigned to $p_j$ in $M^*$, i.e., $s_{r} \in M(p_j) \setminus M^*(p_j)$. Since $s_{r}$ cannot be assigned in $M^*$ to a project that she prefers to $p_j$ and since $l_k$ prefers $s_{r}$ to $s_i$, it follows that $(s_{r}, p_j)$ blocks $M^*$, a contradiction. \end{enumerate} \end{enumerate} \qed \end{proof} \begin{lemma} \label{lemma:super-pair-deletion-outside} If a pair $(s_i, p_j)$ is deleted in line 34 of Algorithm {\sf SPA-ST-super} then $(s_i, p_j)$ does not belong to any super-stable matching in $I$. \end{lemma} \begin{proof} Without loss of generality, suppose that the first super-stable pair to be deleted during an arbitrary execution $E$ of the algorithm is $(s_i, p_j)$, which belongs to some super-stable matching, say $M^*$. Then by Lemma \ref{lemma:super-pair-deletion-within}, $(s_i, p_j)$ was deleted in line 34 during $E$. Let $l_k$ be the lecturer who offers $p_j$. Suppose that $M$ is the assignment during the iteration of the \texttt{repeat-until} loop where $(s_i, p_j)$ was deleted. Let $p_{j'}$ be some other project offered by $l_k$ which was full during a previous \texttt{repeat-until} loop iteration and subsequently ends up undersubscribed in the current \texttt{repeat-until} loop iteration, i.e., $p_{j'}$ plays the role of $p_j$ in line 28. Suppose that $s_{i'}$ plays the role of $s_r$ in line 30, i.e., $s_{i'}$ is the most-preferred student rejected from $p_{j'}$ according to $\mathcal{L}_k^{j'}$ (possibly $s_{i'} = s_i$). Moreover $s_{i'}$ was provisionally assigned to $p_{j'}$ during a previous \texttt{repeat-until} loop iteration but $(s_{i'}, p_{j'}) \notin M$ in the current \texttt{repeat-until} loop iteration. Thus $(s_{i'}, p_{j'})$ has been deleted before the deletion of $(s_i, p_j)$ occurred; and thus, $(s_{i'}, p_{j'}) \notin M^*$, since $(s_i, p_j)$ is the first super-stable pair to be deleted. Further, $l_k$ either prefers $s_{i'}$ to $s_i$ or is indifferent between them, since $s_i$ plays the role of $s_t$ at some for loop iteration in line 32. We remark that no student who is provisionally assigned to some project in $M$ can be assigned to a project better than her current assignment in any super-stable matching. For otherwise, this would mean a super-stable pair must have been deleted before $(s_i, p_j)$, since each student who is assigned in $M$ applies to projects in the head of her preference list. So, either $s_{i'}$ is unassigned in $M^*$ or $s_{i'}$ prefers $p_{j'}$ to $M^*(s_{i'})$ or $s_i$ is indifferent between them. By the super-stability of $M^*$, $p_{j'}$ is full in $M^*$ and $l_k$ prefers every student in $M^*(p_{j'})$ to $s_{i'}$; for otherwise, $(s_{i'}, p_{j'})$ blocks $M^*$, a contradiction. Let $l_{z_0} = l_k$, $p_{t_0} = p_{j'}$ and $s_{q_0} = s_{i'}$. Just before the deletion of $(s_i, p_j)$ occurred, $p_{t_0}$ is undersubscribed in $M$. Since $p_{t_0}$ is full in $M^*$, there exists some student $s_{q_1} \in M^*(p_{t_0}) \setminus M(p_{t_0})$. We note that $l_{z_0}$ prefers $s_{q_1}$ to $s_{q_0}$; for otherwise, $(s_{i'}, p_{j'})$ blocks $M^*$, a contradiction. Let $p_{t_1} = p_{t_0}$. Since $(s_i, p_j)$ is the first super-stable pair to be deleted, $s_{q_1}$ is assigned in $M$ to a project $p_{t_2}$ such that $s_{q_1}$ prefers $p_{t_2}$ to $p_{t_1}$. For otherwise, as each student applies to projects at the head of her preference list, that would mean $(s_{q_1}, p_{t_1})$ must have been deleted before $(s_i, p_j)$, a contradiction. We note that $p_{t_2} \neq p_{t_1}$, since $(s_{q_1} , p_{t_2}) \in M$ and $(s_{q_1} , p_{t_1}) \notin M$. Let $l_{z_1}$ be the lecturer who offers $p_{t_2}$. By the super-stability of $M^*$, either (i) or (ii) holds as follows: \begin{enumerate}[(i)] \item $p_{t_2}$ is full in $M^*$ and $l_{z_1}$ prefers the worst student/s in $M^*(p_{t_2})$ to $s_{q_1}$; \item $p_{t_2}$ is undersubscribed in $M^*$, $l_{z_1}$ is full in $M^*$, $s_{q_1} \notin M^*(l_{z_1})$ and $l_{z_1}$ prefer the worst student/s in $M^*(l_{z_1})$ to $s_{q_1}$. \end{enumerate} Otherwise $(s_{q_1}, p_{t_2})$ blocks $M^*$. In case (i), there exists some student $s_{q_2} \in M^*(p_{t_2}) \setminus M(p_{t_2})$. Let $p_{t_3} = p_{t_2}$. In case (ii), there exists some student $s_{q_2} \in M^*(l_{z_1}) \setminus M(l_{z_1})$. We note that $l_{z_1}$ prefers $s_{q_2}$ to $s_{q_1}$. Now, suppose $M^*(s_{q_2}) = p_{t_3}$ (possibly $p_{t_3} = p_{t_2}$). It is clear that $s_{q_2} \neq s_{q_1}$. Applying similar reasoning as for $s_{q_1}$, $s_{q_2}$ is assigned in $M$ to a project $p_{t_4}$ such that $s_{q_2}$ prefers $p_{t_4}$ to $p_{t_3}$. Let $l_{z_2}$ be the lecturer who offers $p_{t_4}$. We are identifying a sequence $\langle s_{q_i}\rangle_{i \geq 1}$ of students, a sequence $\langle p_{t_i}\rangle_{i \geq 1}$ of projects, and a sequence $\langle l_{z_i}\rangle_{i \geq 1}$ of lecturers, such that, for each $i \geq 1$ \begin{enumerate} \item $s_{q_{i}}$ prefers $p_{t_{2i}}$ to $p_{t_{2i-1}}$, \item $(s_{q_i}, p_{t_{2i}}) \in M$ and $(s_{q_i}, p_{t_{2i - 1}}) \in M^*$, \item $l_{z_i}$ prefers $s_{q_{i+1}}$ to $s_{q_{i}}$; also, $l_{z_i}$ offers both $p_{t_{2i}}$ and $p_{t_{2i+1}}$ (possibly $p_{t_{2i}} = p_{t_{2i+1}}$). \end{enumerate} First we claim that for each new project that we identify, $p_{t_{2i}} \neq p_{t_{2i-1}}$ for $i \geq 1$. Suppose $p_{t_{2i}} = p_{t_{2i-1}}$ for some $i \geq 1$. From above $s_{q_{i}}$ was identified by $l_{z_{i-1}}$ such that $(s_{q_{i}}, p_{t_{2i-1}}) \in M^* \setminus M$. Moreover $(s_{q_{i}}, p_{t_{2i}}) \in M$. Hence we reach a contradiction. Clearly, for each student $s_{q_i}$ that we identify, for $i \geq 1$ , $s_{q_i}$ must be assigned to distinct projects in $M$ and in $M^*$. Next we claim that for each new student $s_{q_i}$ that we identify, $s_{q_i} \neq s_{q_t}$ for $1 \leq t < i$. We prove this by induction on $i$. For the base case, clearly $s_{q_2} \neq s_{q_1}$. We assume that the claim holds for some $i \geq 1$, i.e., the sequence $s_{q_{1}}, s_{q_2}, \ldots, s_{q_{i}}$ consists of distinct students. We show that the claim holds for $i+1$, i.e., the sequence $s_{q_{1}}, s_{q_2}, \ldots, s_{q_{i}}, s_{q_{i+1}}$ also consists of distinct students. Clearly $s_{q_{i+1}} \neq s_{q_{i}}$ since $l_{z_{i}}$ prefers $s_{q_{i+1}}$ to $s_{q_{i}}$. Thus, it suffices to show that $s_{q_{i+1}} \neq s_{q_{j}}$ for $1 \leq j \leq i-1$. Now, suppose $s_{q_{i+1}} = s_{q_{j}}$ for $1 \leq j \leq i-1$. This implies that $s_{q_{j}}$ was identified by $l_{z_{i}}$ and clearly $l_{z_{i}}$ prefers $s_{q_{j}}$ to $s_{q_{j-1}}$. Now since $s_{q_{i+1}}$ was also identified by $l_{z_{i}}$ to avoid the blocking pair $(s_{q_i}, p_{t_{2_i}})$ in $M^*$, it follows that either (i) $p_{t_{2i}}$ is full in $M^*$, or (ii) $p_{t_{2i}}$ is undersubscribed in $M^*$ and $l_{z_{i}}$ is full in $M^*$. We consider each cases further as follows. \begin{enumerate}[(i)] \item If $p_{t_{2i}}$ is full in $M^*$, we know that $(s_{q_{i}}, p_{t_{2i}}) \in M \setminus M^*$. Moreover $s_{q_j}$ was identified by $l_{z_{i+1}}$ because of case (i). Furthermore $(s_{q_{j-1}}, p_{t_{2i}}) \in M \setminus M^*$. In this case, $p_{t_{2i+1}} = p_{t_{2i}}$ and we have that $$(s_{q_{i}}, p_{t_{2i+1}})\in M \setminus M^* \mbox{ and } (s_{q_{i+1}}, p_{t_{2i+1}}) \in M^* \setminus M,$$ $$(s_{q_{j-1}}, p_{t_{2i+1}}) \in M \setminus M^* \mbox{ and } (s_{q_{j}}, p_{t_{2i+1}}) \in M^* \setminus M.$$ By the inductive hypothesis, the sequence $s_{q_{1}}, s_{q_2}, \ldots, s_{q_{j-1}}, $ $s_{q_j}, \ldots, s_{q_{i}}$ consists of distinct students. This implies that $s_{q_{i}} \neq s_{q_{j-1}}$. Thus since $p_{t_{2i+1}}$ is full in $M^*$, $l_{z_{i}}$ should have been able to identify distinct students $s_{q_j}$ and $s_{q_{i+1}}$ to avoid the blocking pairs $(s_{q_{j-1}}, p_{t_{2i+1}})$ and $(s_{q_{i}}, p_{t_{2i+1}})$ respectively in $M^*$, a contradiction. \item $p_{t_{2i}}$ is undersubscribed in $M^*$ and $l_{z_{i}}$ is full in $M^*$. Similarly as in case (i) above, we have that $$s_{q_{i}} \in M(l_{z_i}) \setminus M^*(l_{z_i}) \mbox{ and } s_{q_{i+1}} \in M^*(l_{z_i}) \setminus M(l_{z_i}),$$ $$s_{q_{j-1}} \in M(l_{z_i}) \setminus M^*(l_{z_i}) \mbox{ and } s_{q_{j}} \in M^*(l_{z_i}) \setminus M(l_{z_i}).$$ Since $s_{q_{i}} \neq s_{q_{j-1}}$ and $l_{z_{i}}$ is full in $M^*$, $l_{z_{i}}$ should have been able to identify distinct students $s_{q_j}$ and $s_{q_{i+1}}$ corresponding to students $s_{q_{j-1}}$ and $s_{q_{i}}$ respectively, a contradiction. \end{enumerate} This completes the induction step. As the sequence of distinct students and projects we are identifying is infinite, we reach an immediate contradiction. \qed \end{proof} Lemmas \ref{lemma:super-pair-deletion-within} and \ref{lemma:super-pair-deletion-outside} immediately give rise to Lemma \ref{pair-deletion}. The next lemma will be used as a tool in the proof of the remaining lemmas. \begin{restatable}[]{lemma}{lecturerundersubscribedtool} Let $M$ be the assignment at the termination of Algorithm {\sf SPA-ST-super} and let $M^*$ be any super-stable matching in $I$. Let $l_k$ be an arbitrary lecturer: (i) if $l_k$ is undersubscribed in $M^*$ then every student who is assigned to $l_k$ in $M$ is also assigned to $l_k$ in $M^*$; and (ii) if $l_k$ is undersubscribed in $M$ then $l_k$ has the same number of assignees in $M^*$ as in $M$. \label{lemma:super-lecturer-undersubscribed-tool} \end{restatable} \begin{proof} Let $l_k$ be an arbitrary lecturer. First, we show that (i) holds. Suppose otherwise, then there exists a student, say $s_i$, such that $s_i \in M(l_k) \setminus M^*(l_k)$. Moreover, there exists some project $p_j \in P_k$ such that $s_i \in M(p_j) \setminus M^*(p_j)$. By Lemma \ref{pair-deletion}, $s_i$ cannot be assigned to a project that she prefers to $p_j$ in $M^*$. Also, by the super-stability of $M^*$, $p_j$ is full in $M^*$ and $l_k$ prefers the worst student/s in $M^*(p_j)$ to $s_i$. Let $l_{z_0} = l_k$, $p_{t_0} = p_{j}$, and $s_{q_0} = s_{i}$. As $p_{t_0}$ is full in $M^*$ and no project is oversubscribed in $M$, there exists some student $s_{q_1} \in M^*(p_{t_0}) \setminus M(p_{t_0})$ such that $l_{z_0}$ prefers $s_{q_1}$ to $s_{q_0}$. Let $p_{t_1} = p_{t_0}$. By Lemma \ref{pair-deletion}, $s_{q_1}$ is assigned in $M$ to a project $p_{t_2}$ such that $s_{q_1}$ prefers $p_{t_2}$ to $p_{t_1}$. We note that $s_{q_1}$ cannot be indifferent between $p_{t_2}$ and $p_{t_1}$; for otherwise, as each student applies to projects at the head of her preference list, since $(s_{q_1}, p_{t_1}) \notin M$, that would mean $(s_{q_1}, p_{t_1})$ must have been deleted during the algorithm's execution, contradicting Lemma \ref{pair-deletion}. It follows that $s_{q_1} \in M(p_{t_2}) \setminus M^*(p_{t_2})$. Let $l_{z_1}$ be the lecturer who offers $p_{t_2}$. By the super-stability of $M^*$, either (i) or (ii) holds as follows: \begin{enumerate}[(i)] \item $p_{t_2}$ is full in $M^*$ and $l_{z_1}$ prefers the worst student/s in $M^*(p_{t_2})$ to $s_{q_1}$; \item $p_{t_2}$ is undersubscribed in $M^*$, $l_{z_1}$ is full in $M^*$, $s_{q_1} \notin M^*(l_{z_1})$ and $l_{z_1}$ prefers the worst student/s in $M^*(l_{z_1})$ to $s_{q_1}$. \end{enumerate} Otherwise $(s_{q_1}, p_{t_2})$ blocks $M^*$. In case (i), there exists some student $s_{q_2} \in M^*(p_{t_2}) \setminus M(p_{t_2})$. Let $p_{t_3} = p_{t_2}$. In case (ii), there exists some student $s_{q_2} \in M^*(l_{z_1}) \setminus M(l_{z_1})$. We note that $l_{z_1}$ prefers $s_{q_2}$ to $s_{q_1}$. Now, suppose $M^*(s_{q_2}) = p_{t_3}$ (possibly $p_{t_3} = p_{t_2}$). It is clear that $s_{q_2} \neq s_{q_1}$. Applying similar reasoning as for $s_{q_1}$, student $s_{q_2}$ is assigned in $M$ to a project $p_{t_4}$ such that $s_{q_2}$ prefers $p_{t_4}$ to $p_{t_3}$. Let $l_{z_2}$ be the lecturer who offers $p_{t_4}$. We are identifying a sequence $\langle s_{q_i}\rangle_{i \geq 1}$ of students, a sequence $\langle p_{t_i}\rangle_{i \geq 1}$ of projects, and a sequence $\langle l_{z_i}\rangle_{i \geq 1}$ of lecturers, such that, for each $i \geq 1$ \begin{enumerate} \item $s_{q_{i}}$ prefers $p_{t_{2i}}$ to $p_{t_{2i-1}}$, \item $(s_{q_i}, p_{t_{2i}}) \in M$ and $(s_{q_i}, p_{t_{2i - 1}}) \in M^*$, \item $l_{z_i}$ prefers $s_{q_{i+1}}$ to $s_{q_{i}}$; also, $l_{z_i}$ offers both $p_{t_{2i}}$ and $p_{t_{2i+1}}$ (possibly $p_{t_{2i}} = p_{t_{2i+1}}$). \end{enumerate} Following a similar argument as in the proof of Lemma~\ref{lemma:super-pair-deletion-outside}, we can identify an infinite sequence of distinct students and projects, a contradiction. Hence, if $l_k$ is undersubscribed in $M^*$ then every student who is assigned to $l_k$ in $M$ is also assigned to $l_k$ in $M^*$. Next, we show that (ii) holds. By the first claim, any lecturer who is full in $M$ is also full in $M^*$, and any lecturer who is undersubscribed in $M$ has as many assignees in $M^*$ as she has in $M$. Hence \begin{eqnarray} \label{ineq:undersubscribed-lecturer-1} \sum_{l_k \in \mathcal{L}}{|M(l_k)|} \leq \sum_{l_k \in \mathcal{L}}{|M^*(l_k)|} \enspace. \end{eqnarray} We note that if a student $s_{i}$ is unassigned in $M$, by Lemma \ref{pair-deletion}, $s_{i}$ is unassigned in $M^*$. Equivalently, if $s_{i}$ is assigned in $M^*$ then $s_{i}$ is assigned in $M$. Let $S_1$ denote the set of students who are assigned to at least one project in $M$, and let $S_2$ denote the set of students who are assigned to a project in $M^*$; it follows that $|S_2| \leq |S_1|$. Further, we have that \begin{eqnarray} \label{ineq:undersubscribed-lecturer-2} \sum_{l_k \in \mathcal{L}}{|M^*(l_k)|} = |S_2| \leq |S_1| \leq \sum_{l_k \in \mathcal{L}}{|M(l_k)|}, \end{eqnarray} From Inequalities \eqref{ineq:undersubscribed-lecturer-1} and \eqref{ineq:undersubscribed-lecturer-2}, it follows that $|M(l_k)| = |M^*(l_k)|$ for each $l_k \in \mathcal{L}$. \qed \end{proof} The next three lemmas deal with the case that Algorithm {\sf SPA-ST-super} reports the non-existence of a super-stable matching in $I$. \begin{restatable}[]{lemma}{studentlemma} \label{lemma-super-multi-assignment} If a student is assigned to two or more projects at the termination of Algorithm {\sf SPA-ST-super} then $I$ admits no super-stable matching. \end{restatable} \begin{proof} Let $M$ be the assignment at the termination of the algorithm. Suppose for a contradiction that there exists a super-stable matching $M^*$ in $I$. Suppose that a student is assigned to two or more projects in $M$. Then either (a) any two of these projects are offered by different lecturers or (b) all of these projects are offered by the same lecturer. Firstly, suppose (a) holds. Then some lecturer has fewer assignees in $M^*$ than in $M$. Suppose not, then \begin{eqnarray} \label{eqn-multiple-assignment-1} \sum_{l_k \in \mathcal{L}}{|M^*(l_k)|} \geq \sum_{l_k \in \mathcal{L}}{|M(l_k)|}\enspace. \end{eqnarray} Let $S_1$ and $S_2$ be as defined in the proof of Lemma \ref{lemma:super-lecturer-undersubscribed-tool}, it follows that $|S_2| \leq |S_1|$. Hence, \begin{eqnarray} \label{eqn-multiple-assignment-2} \sum_{l_k \in \mathcal{L}}{|M^*(l_k)|} = |S_2| \leq |S_1| < \sum_{l_k \in \mathcal{L}}{|M(l_k)|}, \end{eqnarray} since some student in $S_1$ is assigned in $M$ to two or more projects offered by different lecturers. Inequality \eqref{eqn-multiple-assignment-2} contradicts Inequality \eqref{eqn-multiple-assignment-1}. Hence, our claim is established. As some lecturer $l_k$ has fewer assignees in $M^*$ than in $M$, it follows that $l_k$ is undersubscribed in $M^*$, since no lecturer is oversubscribed in $M$. In particular, there exists some project $p_j \in P_k$ and some student, say $s_i$, such that $p_j$ is undersubscribed in $M^*$ and $(s_i, p_j) \in M \setminus M^*$. Since $(s_i, p_j) \in M$, then $p_j$ must have been in the head of $s_i$'s preference list when $s_i$ applied to $p_j$ during the algorithm's execution. By Lemma \ref{pair-deletion}, either $s_i$ is unassigned in $M^*$ or $s_i$ prefers $p_j$ to $M^*(s_i)$ or $s_i$ is indifferent between them. Hence $(s_i, p_j)$ blocks $M^*$, a contradiction. Next, suppose (b) holds. Then $|S_1| \leq \sum_{l_k \in \mathcal{L}} |M(l_k)|$. As in case (a), since $|S_2|\leq |S_1|$, it follows that $$\sum_{l_k \in \mathcal{L}} |M^*(l_k)| \leq \sum_{l_k \in \mathcal{L}} |M(l_k)|\enspace.$$ Suppose first that $|M^*(l_k)| < |M(l_k)|$ for some $l_k \in \mathcal{L}$. Then $l_k$ has fewer assignees in $M^*$ than in $M$, and following a similar argument as in case (a) above, we reach an immediate contradiction. Hence, $|M^*(l_k)| = |M(l_k)|$ for all $l_k \in \mathcal{L}$. For each $l_k \in \mathcal{L}$, we claim that every student who is assigned to $l_k$ in $M$ is also assigned to $l_k$ in $M^*$. Suppose otherwise. Let $l_{z_1}$ be an arbitrary lecturer in $\mathcal{L}$. Then there exists some student $s_{q_1} \in M(l_{z_1}) \setminus M^*(l_{z_1})$. Let $M(s_{q_1}) = p_{t_2}$. By Lemma \ref{pair-deletion}, $s_{q_1}$ is assigned in $M^*$ to a project $p_{t_1}$ such that $s_{q_1}$ prefers $p_{t_2}$ to $p_{t_1}$. Clearly, $p_{t_1}$ is not offered by $l_{z_1}$, since $s_{q_1} \in M(l_{z_1}) \setminus M^*(l_{z_1})$. We also note that $s_{q_1}$ cannot be indifferent between $p_{t_2}$ and $p_{t_1}$. Otherwise, the argument follows from (a), since $s_{q_1}$ is assigned in $M$ to two projects offered by different lecturers, and we reach an immediate contradiction. By the super-stability of $M^*$, either (i) or (ii) holds as follows: \begin{enumerate}[(a)] \item $p_{t_2}$ is full in $M^*$ and $l_{z_1}$ prefers every student in $M^*(p_{t_2})$ to $s_{q_1}$; \item $p_{t_2}$ is undersubscribed in $M^*$, $l_{z_1}$ is full in $M^*$ and $l_{z_1}$ prefers every student in $M^*(l_{z_1})$ to $s_{q_1}$. \end{enumerate} Otherwise, $(s_{q_1}, p_{t_2})$ blocks $M^*$. In case (i), there exists some student $s_{q_2} \in M^*(p_{t_2}) \setminus M(p_{t_2})$. Let $p_{t_3} = p_{t_2}$. In case (ii), there exists some student $s_{q_2} \in M^*(l_{z_1}) \setminus M(l_{z_1})$. We note that $l_{z_1}$ prefers $s_{q_2}$ to $s_{q_1}$, and clearly $s_{q_2} \neq s_{q_1}$. Let $M^*(s_{q_2}) = p_{t_3}$ (possibly $p_{t_3} = p_{t_2}$). Applying similar reasoning as for $s_{q_1}$, student $s_{q_2}$ is assigned in $M$ to a project $p_{t_4}$ such that $s_{q_2}$ prefers $p_{t_4}$ to $p_{t_3}$. We are identifying a sequence $\langle s_{q_i}\rangle_{i \geq 1}$ of students, a sequence $\langle p_{t_i}\rangle_{i \geq 1}$ of projects, and a sequence $\langle l_{z_i}\rangle_{i \geq 1}$ of lecturers, such that, for each $i \geq 1$ \begin{enumerate} \item $s_{q_{i}}$ prefers $p_{t_{2i}}$ to $p_{t_{2i-1}}$, \item $(s_{q_i}, p_{t_{2i}}) \in M$ and $(s_{q_i}, p_{t_{2i - 1}}) \in M^*$, \item $l_{z_i}$ prefers $s_{q_{i+1}}$ to $s_{q_{i}}$; also, $l_{z_i}$ offers both $p_{t_{2i}}$ and $p_{t_{2i+1}}$ (possibly $p_{t_{2i}} = p_{t_{2i+1}}$). \end{enumerate} Following a similar argument as in the proof of Lemma~\ref{lemma:super-pair-deletion-outside}, we can identify an infinite sequence of distinct students and projects, a contradiction. Now, let $s_i$ be an arbitrary student such that $s_i$ is assigned in $M$ to two or more projects offered by a lecturer, say $l_k$. Then $s_i \in M^*(l_k)$. Moreover, there exists some project $p_j \in P_k$ such that $(s_i, p_j) \in M \setminus M^*$. We claim that $p_j$ is undersubscribed in $M^*$. Suppose otherwise. Let $l_{z_0} = l_k$, $p_{t_0} = p_j$ and $s_{q_0} = s_i$. Then there exists some student $s_{q_1} \in M^*(p_{t_0}) \setminus M(p_{t_0})$, since $p_{t_0}$ is not oversubscribed in $M$ and $s_{q_0} \in M(p_{t_0}) \setminus M^*(p_{t_0})$. Again, by Lemma \ref{pair-deletion}, $s_{q_1}$ is assigned in $M$ to a project $p_{t_1}$ such that $s_{q_1}$ prefers $p_{t_1}$ to $p_{t_0}$. Let $l_{z_1}$ be the lecturer who offers $p_{t_1}$. Following a similar argument as in the proof of Lemma~\ref{lemma:super-pair-deletion-outside}, we can identify a sequence of distinct students and projects, and as this sequence is infinite, we reach a contradiction. Hence our claim holds, i.e., $p_j$ is undersubscribed in $M^*$. Finally, since $s_i$ cannot be assigned to any project that she prefers to $p_j$ in $M^*$ and since $(s_i, p_j) \in M^*(l_k)$, we have that $(s_i, p_j)$ blocks $M^*$, a contradiction. \qed \end{proof} \begin{restatable}[]{lemma}{lecturerlemma} \label{lemma:super-lec-full-under} If some lecturer $l_k$ becomes full during some execution of Algorithm {\sf SPA-ST-super} and $l_k$ subsequently ends up undersubscribed at the termination of the algorithm, then $I$ admits no super-stable matching. \end{restatable} \begin{proof} Let $M$ be the assignment at the termination of the algorithm. Suppose for a contradiction that there exists a super-stable matching $M^*$ in $I$. Let $l_k$ be the lecturer who became full during some execution of the algorithm and subsequently ends up undersubscribed in $M$. By Lemma \ref{lemma:super-lecturer-undersubscribed-tool}, $|M(l_k)| = |M^*(l_k)|$ and thus $l_k$ is undersubscribed in $M^*$. At the point in the algorithm where $l_k$ became full (line 22), we note that none of the projects offered by $l_k$ is oversubscribed. Since $l_k$ ended up undersubscribed in $M$, it follows that there is some project $p_j \in P_k$ that has fewer assignees in $M$ at the termination of the algorithm than it had at some point during the algorithm's execution, thus $p_j$ is undersubscribed in $M$. We claim that each project offered by $l_k$ has the same number of assignees in $M^*$ as in $M$. Suppose otherwise, then there is some project $p_t \in P_k$ such that $|M^*(p_t)| < |M(p_t)|$; thus $p_t$ is undersubscribed in $M^*$, since no project is oversubscribed in $M$. It follows that there exists some student $s_r \in M(p_t) \setminus M^*(p_t)$. By Lemma \ref{pair-deletion}, $s_r$ is either unassigned in $M^*$ or prefers $p_t$ to $M^*(s_r)$. Since $l_k$ is undersubscribed in $M^*$, $(s_r, p_t)$ blocks $M^*$, a contradiction. Hence $|M^*(p_t)| \geq |M(p_t)|$. Moreover, since $|M(l_k)| = |M^*(l_k)|$, we have that $|M(p_t)| = |M^*(p_t)|$ for all $p_t \in P_k$. Hence $p_j$ undersubscribed in $M$ implies that $p_j$ is undersubscribed in $M^*$. Moreover, there is some student $s_i$ who was provisionally assigned to $p_j$ at some point during the execution of the algorithm but $s_i$ is not assigned to $p_j$ in $M$. Thus, the pair $(s_i, p_j)$ was deleted during the algorithm's execution, so that $(s_i, p_j) \notin M^*$ by Lemma \ref{pair-deletion}. It follows that either $s_i$ is unassigned in $M^*$ or $s_i$ prefers $p_j$ to $M^*(s_i)$ or $s_i$ is indifferent between them. Hence, $(s_i, p_j)$ blocks $M^*$, a contradiction. \qed \end{proof} \begin{restatable}[]{lemma}{projectlemma} \label{lemma-super-proj-full-under} If the pair $(s_i, p_j)$ was deleted during some execution of Algorithm {\sf SPA-ST-super}, and at the termination of the algorithm $s_i$ is not assigned to a project better than $p_j$, and each of $p_j$ and $l_k$ is undersubscribed, then $I$ admits no super-stable matching. \end{restatable} \begin{proof} Suppose for a contradiction that there exists a super-stable matching $M^*$ in $I$. Let $(s_i, p_j)$ be a pair that was deleted during an arbitrary execution $E$ of the algorithm. This implies that $(s_i, p_j) \notin M^*$ by Lemma \ref{pair-deletion}. Let $M$ be the assignment at the termination of $E$. By the hypothesis of the lemma, $l_k$ is undersubscribed in $M$. This implies that $l_k$ is undersubscribed in $M^*$, by Lemma~\ref{lemma:super-lecturer-undersubscribed-tool}. Since $p_j$ is offered by $l_k$, and $p_j$ is undersubscribed in $M$, it follows from the proof of Lemma \ref{lemma:super-lec-full-under} that $p_j$ is undersubscribed in $M^*$. Further, by the hypothesis of the lemma, either $s_i$ is unassigned in $M$, or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them. By Lemma \ref{pair-deletion}, this is true for $s_i$ in $M^*$. Hence $(s_i, p_j)$ blocks $M^*$, a contradiction. \qed \end{proof} The next lemma shows that the final assignment may be used to determine the existence, or otherwise, of a super-stable matching in $I$. \begin{restatable}[]{lemma}{nosuperstablematchinglemma} \label{lemma-super-correctness} If at the termination of Algorithm {\sf SPA-ST-super}, the assignment $M$ is not super-stable in $I$ then no super-stable matching exists in $I$. \end{restatable} \begin{proof} Suppose $M$ is not super-stable in $I$. If some student $s_i$ is assigned to two or more projects in $M$ then $I$ admits no super-stable matching, by Lemma \ref{lemma-super-multi-assignment}. Hence every student is assigned to at most one project in $M$. Moreover, since no project or lecturer is oversubscribed in $M$, it follows that $M$ is a matching. Let $(s_i, p_j)$ be a blocking pair for $M$, then $s_i$ is either unassigned in $M$ or prefers $p_j$ to $M(s_i)$ or is indifferent between them. Whichever is the case, $(s_i, p_j)$ has been deleted. Let $l_k$ be the lecturer who offers $p_j$. In what follows, we will identify the point in the algorithm at which $(s_i, p_j)$ was deleted, and consequently, we will arrive at a conclusion that no super-stable matching exists. Firstly, suppose $(s_i, p_j)$ was deleted as a result of $p_j$ being full or oversubscribed (on lines 12 or 21). Suppose $p_j$ is full in $M$. Then $(s_i, p_j)$ cannot block $M$ irrespective of whether $l_k$ is undersubscribed or full in $M$, since $l_k$ prefers the worst assigned student/s in $M(p_j)$ to $s_i$. Hence $p_j$ is undersubscribed in $M$. As $p_j$ was previously full, each pair $(s_t, p_u)$, for each $s_t$ that is no better than $s_i$ at the tail of $\mathcal{L}_k$ and each $p_u \in P_k \cap A_t$, would have been deleted on line 34 of the algorithm. Thus, if $l_k$ is full in $M$ then $(s_i, p_j)$ does not block $M$. Suppose $l_k$ is undersubscribed in $M$. If $l_k$ was full at some point during the execution of the algorithm then $I$ admits no super-stable matching, by Lemma \ref{lemma:super-lec-full-under}. Hence $l_k$ was never full during the algorithm's execution. Recall that each of $p_j$ and $l_k$ is undersubscribed in $M$. As $(s_i, p_j)$ is a blocking pair of $M$, $s_i$ cannot be assigned in $M$ to a project that she prefers to $p_j$. Hence $I$ admits no super-stable matching, by Lemma \ref{lemma-super-proj-full-under}. Next, suppose $(s_i, p_j)$ was deleted as a result of $l_k$ being full or oversubscribed (on lines 16 or 26), $(s_i, p_j)$ could only block $M$ if $l_k$ is undersubscribed in $M$. If this is the case then $I$ admits no super-stable matching, by Lemma \ref{lemma:super-lec-full-under}. Finally, suppose $(s_i, p_j)$ was deleted (on line 34) because some other project $p_{j'}$ offered by $l_k$ was previously full and ended up undersubscribed on line 28. Then $l_k$ must have identified the most-preferred student, say $s_r$, who was previously assigned to $p_{j'}$ but subsequently got rejected from $p_{j'}$. At this point, $s_i$ is at the tail of $\mathcal{L}_k$ and $s_i$ is no better than $s_r$ in $\mathcal{L}_k$. Moreover, every project offered by $l_k$ that $s_i$ finds acceptable would have been deleted from $s_i$'s preference list at the for loop iteration in line 34. If $p_j$ is full in $M$ then $(s_i,p_j)$ does not block $M$. Hence $p_j$ is undersubscribed in $M$. If $l_k$ is full in $M$ then $(s_i, p_j)$ does not block $M$, since $s_i \notin M(l_k)$ and $l_k$ prefers the worst student/s in $M(l_k)$ to $s_i$. Hence $l_k$ is undersubscribed in $M$. Again by Lemma \ref{lemma-super-proj-full-under}, $I$ admits no super-stable matching. Since $(s_i, p_j)$ is an arbitrary pair, this implies that $I$ admits no super-stable matching. \qed \end{proof} The next lemma shows that Algorithm {\sf SPA-ST-super} may be implemented to run in linear time. \begin{restatable}[]{lemma}{lineartimelemma} \label{lemma-super-complexity} Algorithm {\sf SPA-ST-super} may be implemented to run in $O(L)$ time and $O(n_1n_2)$ space, where $n_1$, $n_2$, and $L$ are the number of students, number of projects, and the total length of the preference lists, respectively, in $I$. \end{restatable} \begin{proof} The algorithm's time complexity depends on how efficiently we can execute the operation of a student applying to a project and the operation of deleting a (student, project) pair, each of which occur once for any (student, project) pair. It turns out that both operations can be implemented to run in constant time, giving Algorithm {\sf SPA-ST-super} an overall complexity of $\Theta(L)$, where $L$ is the total length of all the preference lists. In what follows, we describe the non-trivial aspects of such an implementation. We remark that the data structures discussed here are inspired by, and extend, those detailed in \cite[Section 3.3]{AIM07} for Algorithm {\sf SPA}-student. For each student $s_i$, build an array $\mathit{position}_{s_i}$, where $\mathit{position}_{s_i}(p_j)$ is the position of project $p_j$ in $s_i$'s preference list. For example, if $s_i$'s preference list is $(p_2 \; p_5 \; p_3) \; p_7 \; (p_6 \; p_1)$ then $\mathit{position}_{s_i}(p_5) = 2$ and $\mathit{position}_{s_i}(p_1) = 6$. In general, position captures the order in which the projects appear in the preference list when read from left to right, ignoring any ties. Represent $s_i$'s preference list by embedding doubly linked lists in an array $\mathit{preference}_{s_i}$. For each project $p_j \in A_i$, $\mathit{preference}_{s_i}(\mathit{position}_{s_i}(p_j))$ stores the list node containing $p_j$. This node contains two next pointers (and two previous pointers) -- one to the next project in $s_i$'s preference list (after deletions, this project may not be located at the next array position), and another pointer to the next project $p_{j'}$ in $s_i$'s preference list, where $p_{j'}$ and $p_j$ are both offered by the same lecturer. Construct the latter list by traversing through $s_i$'s preference list, using a temporary array to record the last project in the list offered by each lecturer. Use virtual initialisation (described in \cite[p.~149]{BB96}) for these arrays, since the overall $O(n_1 n_3)$ initialisation may be too expensive. To represent the ties in $s_i$'s preference list, build an array $\mathit{successor}_{s_i}$. For each project $p_j$ in $s_i$'s preference list, $\mathit{successor}_{s_i}(\mathit{position}_{s_i}(p_j))$ stores the \texttt{true} boolean if $p_j$ is tied with its successor in $A_i$ and \texttt{false} otherwise. After the deletion of any (student, project) pair, update the successor booleans. As an illustration, with respect to $s_i$'s preference list given in the previous paragraph, $\mathit{successor}_{s_i}$ is the array [\texttt{true, true, false, false, true, false}]. Now, suppose $p_3$ was deleted from $s_i$'s preference list, since $\mathit{successor}_{s_i}(\mathit{position}_{s_i}(p_3))$ is \texttt{false} and $\mathit{successor}_{s_i}(\mathit{position}_{s_i}(p_5))$ is \texttt{true}, set $\mathit{successor}_{s_i}$ $(\mathit{position}_{s_i}(p_5))$ to \texttt{false} (since $p_5$ is the predecessor of $p_3$). Clearly using these data structures, we can find the next project at the head of each student's preference list, find the next project offered by a given lecturer on each student's preference list, as well as delete a project from a given student's preference list in constant time. For each lecturer $l_k$, build two arrays $\mathit{preference}_{l_k}$ and $\mathit{successor}_{l_k}$, where $\mathit{preference}_{l_k}(s_i)$ is the position of student $s_i$ in $l_k$'s preference list, and $\mathit{successor}_{l_k}$ $(\mathit{preference}_{l_k}(s_i))$ stores the position of the first strict successor (with respect to position) of $s_i$ in $\mathcal{L}_k$ or a null value if $s_i$ has no strict successor\footnote{For example, if $l_k$'s preference list is $s_5 \; (s_3 \; s_1 \; s_6) \; s_7 \; (s_2 \; s_8)$ then $\mathit{successor}_{l_k}$ is the array $[2 \; 5 \; 5 \; 5 \; 6 \; 0 \; 0]$.}. Represent $l_k$'s preference list (i.e., $\mathcal{L}_k$) by the array $\mathit{preference}_{l_k}$, with an additional pointer, $\mathit{last}_{l_k}$. Initially, $\mathit{last}_{l_k}$ stores the index of the last position in $\mathit{preference}_{l_k}$. To represent the ties in $l_k$'s preference list, build an array $\mathit{predecessor}_{l_k}$. For each $s_i \in \mathcal{L}_k$, $\mathit{predecessor}_{l_k}(\mathit{preference}_{l_k}(s_i))$ stores the \texttt{true} boolean if $s_i$ is tied with its predecessor in $\mathcal{L}_k$ and \texttt{false} otherwise. When $l_k$ becomes full, make $\mathit{last}_{l_k}$ equivalent to $l_k$'s worst assigned student through the following method. Perform a backward traversal through the array $\mathit{preference}_{l_k}$, starting at $\mathit{last}_{l_k}$, and continuing until $l_k$'s worst assigned student, say $s_{i'}$, is encountered (each student stores a pointer to their assigned project, or a special null value if unassigned). Deletions must be carried out in the preference list of each student who is worse than $s_{i'}$ on $l_k$'s preference list (precisely those students whose position in $\mathit{preference}_{l_k}$ is greater than or equal to that stored in $\mathit{successor}_{l_k}(\mathit{preference}_{l_k}(s_{i'}))$)\footnote{For efficiency, we remark that it is not necessary to make deletions from the preference lists of lecturers or projected preference lists of lecturers for each project the lecturer offers, since the while loop of Algorithm {\sf SPA-ST-super} involves students applying to projects in the head of their preference list.}. When $l_k$ becomes oversubscribed, we can find and delete the students at the tail of $l_k$ by performing a backward traversal through the array $\mathit{preference}_{l_k}$, starting at $\mathit{last}_{l_k}$, and continuing until we encounter a student, say $s_{i'}$, such that $\mathit{predecessor}_{l_k}(\mathit{preference}_{l_k}(s_{i'}))$ stores the \texttt{false} boolean. If $l_k$ becomes undersubscribed after we break the assignment of students encountered on this traversal (including $s_{i'}$) to $l_k$, rather than update $\it{last}_{l_k}$ immediately, which could be expensive, we wait until $l_k$ becomes full again. The cost of these traversals taken over the algorithm's execution is thus linear in the length of $l_k$'s preference list. For each project $p_j$ offered by $l_k$, build the arrays $\mathit{preference}_{p_j}$, $\mathit{successor}_{p_j}$ and $\mathit{predecessor}_{p_j}$ corresponding to $\mathcal{L}_k^j$, as described in the previous paragraph for $\mathcal{L}_k$. Represent the projected preference list of $l_k$ for $p_j$ (i.e., $\mathcal{L}_k^j$) by the array $\mathit{preference}_{p_j}$, with an additional pointer, $\mathit{last}_{p_j}$. These project preference arrays are used in much the same way as the lecturer preference arrays Since we only visit a student at most twice during these backward traversals, once for the lecturer and once for the project, the asymptotic running time remains linear. \qed \end{proof} \noindent Lemma \ref{pair-deletion} shows that there is an optimality property for each assigned student in any super-stable matching found by the algorithm, whilst Lemma \ref{lemma-super-correctness} establishes the correctness of Algorithm {\sf SPA-ST-super}. The following theorem collects together Lemmas \ref{pair-deletion}, \ref{lemma-super-correctness} and \ref{lemma-super-complexity}. \begin{theorem} \label{thrm:super-optimality} For a given instance $I$ of {\sc spa-st}, Algorithm {\sf SPA-ST-super} determines, in $O(L)$ time and $O(n_1n_2)$ space, whether or not a super-stable matching exists in $I$. If such a matching does exist, all possible executions of the algorithm find one in which each assigned student is assigned to the best project that she could obtain in any super-stable matching, and each unassigned student is unassigned in all super-stable matchings. \end{theorem} Given the optimality property established by Theorem \ref{thrm:super-optimality}, we define the super-stable matching found by Algorithm {\sf SPA-ST-super} to be \textit{student-optimal}. \subsection{Properties of super-stable matchings in {\sc spa-st}} \label{subsect:properties} In this section, we consider properties of the set of super-stable matchings in an instance of {\sc spa-st}. We show that the Unpopular Projects Theorem for {\sc spa-s} (see Theorem \ref{thrm:rural-spa-s}) holds for {\sc spa-st} under super-stability. \begin{restatable}[]{theorem}{upt} \label{thrm:upt} For a given instance $I$ of {\sc spa-st}, the following holds: \begin{enumerate} \item each lecturer is assigned the same number of students in all super-stable matchings; \item exactly the same students are unassigned in all super-stable matchings; \item a project offered by an undersubscribed lecturer has the same number of students in all super-stable matchings. \end{enumerate} \end{restatable} \begin{proof} Let $M$ and $M^*$ be two arbitrary super-stable matchings in $I$. Let $I'$ be an instance of {\sc spa-s} obtained from $I$ by breaking the ties in $I$ in some way. Then by Proposition \ref{proposition1}, each of $M$ and $M^*$ is stable in $I'$. Thus by Theorem \ref{thrm:rural-spa-s}, each lecturer is assigned the same number of students in $M$ and $M^*$, exactly the same students are unassigned in $M$ and $M^*$, and a project offered by an undersubscribed lecturer has the same number of students in $M$ and $M^*$. \qed\end{proof} \begin{figure} \caption{ \small Instance $I_2$ of {\sc spa-st}.} \label{fig:spa-st-instance-2} \end{figure} To illustrate this, consider the {\sc spa-st} instance $I_2$ given in Fig.~\ref{fig:spa-st-instance-2}, which admits the super-stable matchings $M_1 = \{(s_3, p_3), (s_4, p_2), (s_5, p_3),$ $(s_6, p_2)\}$ and $M_2 = \{(s_3, p_3), (s_4, p_3), $ $(s_5, p_2), (s_6, p_2)\}$. Each of $l_1$ and $l_2$ is assigned the same number of students in both $M_1$ and $M_2$, illustrating part (1) of Theorem \ref{thrm:upt}. Also, each of $s_1$ and $s_2$ is unassigned in both $M_1$ and $M_2$, illustrating part (2) of Theorem \ref{thrm:upt}. Finally, $l_2$ is undersubscribed in both $M_1$ and $M_2$, and each of $p_3$ and $p_4$ has the same number of students in both $M_1$ and $M_2$, illustrating part (3) of Theorem \ref{thrm:upt}. \section{Empirical Evaluation} \label{emprical-results} In this section, we evaluate an implementation of Algorithm {\sf SPA-ST-super}. We implemented our algorithm in Python\footnote{https://github.com/sofiatolaosebikan/spa-st-super}, and performed our experiments on a system with dual Intel Xeon CPU E5-2640 processors with 64GB of RAM, running Ubuntu 17.10. For our experiment, we were primarily concerned with the following question: how does the nature of the preference lists in a given {\sc spa-st} instance affect the existence of a super-stable matching? \subsection{Datasets} When generating random datasets, there are clearly several parameters that can be varied, such as the number of students, projects and lecturers; the lengths of the students' preference lists as well as a measure of the density of ties present in the preference lists. We denote by $t_d$, the measure of the density of ties present in the preference lists. In each student's preference list, the tie density $t_{d_s} \; (0 \leq t_{d_s} \leq 1)$ is the probability that some project is tied to its successor. The tie density $t_{d_l}$ in each lecturer's preference list is defined similarly. At $t_{d_s} = t_{d_l} = 1$, each preference list comprises a single tie while at $t_{d_s} = t_{d_l} = 0$, no tie would exist in the preference lists, thus reducing the problem to an instance of {\sc spa-s}. \subsection{Experimental Setup} For each range of values for the aforementioned parameters, we randomly generated a set of {\sc spa-st} instances, involving $n_1$ students (which we will henceforth refer to as the size of the instance), $0.5n_1$ projects, $0.2n_1$ lecturers and $1.5n_1$ total project capacity which was randomly distributed amongst the projects. The capacity for each lecturer $l_k$ was chosen uniformly at random to lie between the highest capacity of the projects offered by $l_k$ and the sum of the capacities of the projects that $l_k$ offers.\footnote{We remark that the parameter space was chosen to ensure that projects could typically accommodate more than one student, that the total capacity of the projects exceeded the number of students, and that each lecturer typically offered multiple projects, without reflecting any specific real-world application.} In each set, we measured the proportion of instances that admit a super-stable matching. It is worth mentioning that when we varied the tie density on both the students' and lecturers' preference lists between $0.1$ and $0.5$, super-stable matchings were very elusive, even with an instance size of $100$ students. Thus, for the purpose of our experiment, we decided to choose a low tie density. \subsubsection{Correctness testing} To test the correctness of our algorithm's implementation, we implemented an Integer Programming (IP) model for super-stability in {\sc spa-st} (see Appendix \ref{appendixA}) using the Gurobi optimisation solver in Python. We randomly generated $10,000$ {\sc spa-st} instances, each consisting of $100$ students and a constant ratio of projects, lecturers, project capacities and lecturer capacities as described above. Also, each student's preference list was fixed at $10$, with a tie density of $0.1$. With this setup, we verified consistency between the outcomes of our implementation of Algorithm {\sf SPA-ST-super} and our implementation of the IP-based algorithm in terms of the existence or otherwise of a super-stable matching. \subsubsection{Experiment 1} In our first experiment, we examined how the length of the students' preference lists affects the existence of a super-stable matching. We increased the number of students $n_1$ while maintaining a constant ratio of projects, lecturers, project capacities and lecturer capacities as described above. For various values of $n_1 \; (100 \leq n_1 \leq 1000)$ in increments of $100$, we varied the length of each student's preference list for various values of $x$ ($5 \leq x \leq 50$) in increments of $5$; and with each of these parameters, we randomly generated $1000$ instances. For all the preference lists, we set $t_{d_s} = t_{d_l} = 0.005$ (on average, $1$ out of $5$ students has a single tie of length $2$ in their preference list, and this holds similarly for the lecturers). The result, which is displayed in Fig.~\ref{super-experiment1}, shows that as we varied the length of the preference list, there was no significant uplift in the number of instances that admitted a super-stable matching. In most cases, we observed that the proportion of instances that admit a super-stable matching is slightly higher when the preference list length is $50$ compared to when the preference list length is $5$. The result also shows that the proportion of instances that admit a super-stable matching decreases as the number of students increases. Further, we recorded the time taken for our algorithm's implementation to terminate, and as can be seen in Table \ref{fig:super-time-table}, for an instance size of $1000$ and preference list length $50$, the algorithm terminates in approximately $0.4$ second. \begin{figure} \caption{ Proportion of instances that admit a super-stable matching as the size of the instance increases while varying the length of the preference lists with tie density fixed at $0.005$ in both the students' and lecturers' preference lists.} \label{super-experiment1} \end{figure} \begin{table}[H] \setlength{\tabcolsep}{0.4em} \renewcommand*{1}{1.2} \centering \caption{Time (in seconds) for our algorithm's implementation to terminate averaged over $1000$ for each instance size, with the length of each student's preference list fixed at $50$.} \label{fig:super-time-table} \begin{tabular}{c|cccccccccc} \hline\noalign{ } $n_1$ & $100$ & $200$ & $300$ & $400$ & $500$ & $600$ & $700$ & $800$ & $900$ & $1000$ \\ \noalign{ }\hline\noalign{ } Time & $0.017$ & $0.046$ & $0.082$ & $0.120$ & $0.160$ & $0.203$ & $0.248$ & $0.298$ & $0.349$ & $0.399$ \\ \noalign{ }\hline \end{tabular} \end{table} \subsubsection{Experiment 2} In our second experiment, we investigated how the variation in tie density in both the students' and lecturers' preference lists affects the existence of a super-stable matching. To achieve this, we varied the tie density in the students' preference lists $t_{d_s} \; (0 \leq t_{d_s} \leq 0.05)$ and the tie density in the lecturers' preference lists $t_{d_l} \; (0 \leq t_{d_l} \leq 0.05)$, both in increments of $0.005$. For each pair of tie densities in $t_{d_s} \times t_{d_l}$, we randomly-generated $1000$ {\sc spa-st} instances for various values of $n_1 \; (100 \leq n_1 \leq 1000)$ in increments of $100$. For each of these instances, we maintained the same ratio of projects, lecturers, project capacities and lecturer capacities as in Experiment 1. Considering our discussion from Experiment 1, we fixed the length of each student's preference list at $50$. The result displayed in Fig.~\ref{super-experiment2} shows that increasing the tie density in both the students' and lecturers' preference lists reduces the proportion of instances that admit a super-stable matching. In fact, this proportion reduces further as the size of the instance increases. When ties occur only in the lecturers' preference lists, we found that a significantly higher proportion of instances admit a super-stable matching -- about $74\%$ of the randomly-generated {\sc spa-st} instances involving $1000$ students admitted a super-stable matching. The confidence interval for this value is $(0.71, 0.77)$. However, the reverse is the case when ties occur only in the students' preference lists. We have no explanation for this outcome. \begin{figure} \caption{ Result for Experiment 2. Each of the coloured square boxes represents the proportion of the $1000$ randomly-generated {\sc spa-st} instances that admit a super-stable matching, with respect to the tie density in the students' and lecturers' preference lists. See the colour bar transition, as this proportion ranges from dark ($100\%$) to light ($0\%$).} \label{super-experiment2} \end{figure} \section{Discussions and Concluding Remarks} \label{section:conclusions} In this paper, we have described a linear-time algorithm to find a super-stable matching or report that no such matching exists, given an instance of {\sc spa-st}. We established that for instances that do admit a super-stable matching, our algorithm produces the student-optimal super-stable matching, in the sense that each assigned student has the best project that she could obtain in any super-stable matching. We leave open the formulation of a lecturer-oriented counterpart to our algorithm. Further, we carried out an empirical evaluation of our algorithm's implementation. The purpose of our experiments was to investigate how the nature of the preference lists affects the existence (or otherwise) of super-stable matchings in an arbitrary instance of {\sc spa-st}. Based on the instances we generated randomly, the experimental results suggest that as we increase the size of the instance and the density of ties in the preference lists, the likelihood of a super-stable matching existing decreases. There was no significant uplift in this likelihood even as we increased the length of the students' preference lists. When the ties occur only in the lecturers' preference lists, we found that a significantly higher proportion of instances admit a super-stable matching. However, the reverse is the case when the ties occur only in the students' preference lists. Given that there are typically more students than lecturers in practical applications, it could be that only lecturers are permitted to have some form of indifference over the students that they find acceptable, whilst each student might be able to provide a strict ordering over what may be a small number of projects that she finds acceptable. Further evaluation of our algorithm could investigate how other parameters (e.g., the popularity of some projects, or the position of the ties in the preference lists) affect the existence of a super-stable matching. It would also be interesting to examine the existence of super-stable matchings in real {\sc spa-st} datasets. From a theoretical perspective, the following are other directions for future work. Let $I$ be an arbitrary instance of {\sc spa-st}. \begin{enumerate} \item Can we formalise the results on the probability of a super-stable matching existing in $I$? As mentioned in Section \ref{introduction}, this question has been partially explored for the Stable Roommates problem \cite{PI94}. \item Is there a characterisation of the set of super-stable matchings in $I$ in terms of a lattice structure? It is known that the set of super-stable matchings in an instance of {\sc smt} forms a distributive lattice under the dominance relation \cite{Man02,Spi95}. To generalise this structural result for {\sc spa-st}, ideas from \cite{Man02,Spi95} would certainly be useful. \end{enumerate} \begin{subappendices} \renewcommand{\Alph{section}}{\Alph{section}} \section{An IP model for super-stability in {\small SPA-ST}} \label{appendixA} \subsection{Introduction} In this section, we describe an IP model for super-stability in {\sc spa-st}. Although a super-stable matching in an instance of {\sc spa-st} can be found in polynomial-time (as illustrated by Theorem \ref{thrm:super-optimality}), our reason for this is purely experimental. Let $I$ be an instance of {\sc spa-st} involving a set $\mathcal{S} = \{s_1, s_2, \ldots, s_{n_1}\}$ of students, a set $\mathcal{P} = \{p_1, p_2, \ldots, p_{n_2}\}$ of projects and a set $\mathcal{L} = \{l_1, l_2, \ldots, l_{n_3}\}$ of lecturers. We construct an IP model $J$ of $I$ as follows. Firstly, we create binary variables $x_{i, j} \in \{0, 1\}$ $(1 \leq i \leq n_1, 1 \leq j \leq n_2)$ for each acceptable pair $(s_i, p_j) \in \mathcal{S} \times \mathcal{P}$ such that $x_{i, j}$ indicates whether $s_i$ is assigned to $p_j$ in a solution or not. Henceforth, we denote by $S$ a solution in the IP model $J$, and we denote by $M$ the matching derived from $S$ in the following natural way: if $x_{i,j} = 1$ under $S$ then $s_i$ is assigned to $p_j$ in $M$, otherwise $s_i$ is not assigned to $p_j$ in $M$. \subsection{Constraints} In this section, we give the set of constraints to ensure that the assignment obtained from a feasible solution in $J$ is a matching, and that the matching admits no blocking pair. \paragraph{\textbf{Matching constraints.}} The feasibility of a matching can be ensured with the following three set of constraints. \begin{align} \label{ineq:spa-st-ip-studentassignment} \sum\limits_{p_{j} \in A_{i}} x_{i,j} \leq 1 &\qquad (1 \leq i \leq n_1), \\ \label{ineq:spa-st-ip-projectcapacity} \sum\limits_{i = 1}^{n_1} x_{i,j} \leq c_j & \qquad (1 \leq j \leq n_2), \\ \label{ineq:spa-st-ip-lecturercapacity} \sum\limits_{i = 1}^{n_1} \; \sum\limits_{p_{j} \in P_k} x_{i,j} \leq d_k & \qquad (1 \leq k \leq n_3)\enspace. \end{align} Note that Inequality \eqref{ineq:spa-st-ip-studentassignment} ensures that each student $s_i \in \mathcal{S}$ is not assigned to more than one project, while Inequalities \eqref{ineq:spa-st-ip-projectcapacity} and \eqref{ineq:spa-st-ip-lecturercapacity} ensure that the capacity of each project $p_j \in \mathcal{P}$ and each lecturer $l_k \in \mathcal{L}$ is not exceeded. Given an acceptable pair $(s_i, p_j)$, we define $\rank(s_i, p_j)$, the \textit{rank} of $p_j$ on $s_i$'s preference list, to be $r+1$, where $r$ is the number of projects that $s_i$ prefers to $p_j$. \label{rank} Clearly, projects that are tied together on $s_i$'s preference list have the same rank. Given a lecturer $l_k \in \mathcal{L}$ and a student $s_i \in \mathcal{L}_k$, we define $\rank(l_k, s_i)$, the \textit{rank} of $s_i$ on $l_k$'s preference list, to be $r+1$, where $r$ is the number of students that $l_k$ prefers to $s_i$. Similarly, students that are tied together on $l_k$'s preference list have the same rank. With respect to an acceptable pair $(s_i, p_j)$, we define $S_{i,j} = \{p_{j'} \in A_i: \rank(s_i, p_{j'}) < \rank(s_i, p_j)\}$, the set of projects that $s_i$ prefers to $p_j$. Let $l_k$ be the lecturer who offers $p_j$. We also define $T_{i,j,k} = \{s_{i'} \in \mathcal{L}_{k}^{j}: \rank(l_k, s_{i'}) < \rank(l_k, s_{i})\}$, the set of students that are better than $s_i$ on the projected preference list of $l_k$ for $p_j$. Finally, we define $D_{i,k} = \{s_{i'} \in \mathcal{L}_{k}: \rank(l_k, s_{i'}) < \rank(l_k, s_{i})\}$, the set of students that are better than $s_i$ on $l_k$'s preference list. In what follows, we fix an arbitrary acceptable pair $(s_i, p_j)$ and we enforce constraints to ensure that $(s_i, p_j)$ does not form a blocking pair for the matching $M$. Henceforth, $l_k$ is the lecturer who offers $p_j$. \paragraph{\textbf{Blocking pair constraints.}} First, we define $\theta_{i,j} = 1 - x_{i,j} - \sum\limits_{p_{j'} \in S_{i, j}}x_{i,j'}$. Intuitively, $\theta_{i,j} = 1$ if and only if $s_i$ is unassigned in $M$, or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them. Henceforth, if $(s_i, p_j)$ forms a blocking pair for $M$ then we refer to $(s_i, p_j)$ as a blocking pair of type (i), type (ii) or type (iii), according as $(s_i, p_j)$ satisfies condition (i), (ii), or (iii) of Definition \ref{definition:super-stability}, respectively. We describe the constraints to avoid these types of blocking pair as follows. \paragraph{\textbf{Type (i)}. \label{type-i}} First, we create a binary variable $\alpha_j$ in $J$ such that if $p_j$ is undersubscribed in $M$ then $\alpha_j = 1$. We enforce this condition by imposing the following constraint. \begin{eqnarray} \label{ineq:spa-st-ip-project-under} c_j \alpha_j \geq c_j - \sum\limits_{i' = 1}^{n_1} x_{i',j}, \end{eqnarray} where $\sum_{i' = 1}^{n_1} x_{i',j} = |M(p_j)|$. If $p_j$ is undersubscribed in $M$ then the RHS of Inequality \eqref{ineq:spa-st-ip-project-under} is at least $1$ and this implies that $\alpha_j = 1$, otherwise $\alpha_j$ is not constrained. Next, we create a binary variable $\beta_k$ in $J$ such that if $l_k$ is undersubscribed in $M$ then $\beta_k = 1$. We enforce this condition by imposing the following constraint: \begin{eqnarray} \label{ineq:spa-st-ip-lecturerunder} d_k\beta_k \geq d_k - \sum\limits_{i' = 1}^{n_1} \; \sum\limits_{p_{j'} \in P_k} x_{i',j'}, \end{eqnarray} where $\sum\limits_{i' = 1}^{n_1} \; \sum\limits_{p_{j'} \in P_k} x_{i',j'} = |M(l_k)|$. If $l_k$ is undersubscribed in $M$ then the RHS of Inequality \eqref{ineq:spa-st-ip-lecturerunder} is at least $1$ and this implies that $\beta_k = 1$, otherwise $\beta_k$ is not constrained. The following constraint ensures that $(s_i, p_j)$ does not form a type (i) blocking pair for $M$. \begin{align} \label{ineq:super-bp-type-i} \Aboxed{ \theta_{i,j} + \alpha_{j} + \beta_k \leq 2\enspace.} \end{align} \paragraph{\textbf{Type (ii)}. \label{type-ii}} We create a binary variable $\eta_{k}$ in $J$ such that if $l_k$ is full in $M$ then $\eta_{k} = 1$. We enforce this condition by imposing the following constraint. \begin{eqnarray} \label{ineq:spa-st-ip-lecturerfull} d_k\eta_{k} \geq \left(1 + \sum\limits_{i' = 1}^{n_1} \; \sum\limits_{p_{j'} \in P_k} x_{i',j'}\right) - d_k\enspace. \end{eqnarray} If $l_k$ is full in $M$ then the RHS of Constraint \eqref{ineq:spa-st-ip-lecturerfull} is at least $1$ and this implies that $\eta_k = 1$, otherwise $\eta_k$ is not constrained. Next, we create a binary variable $\delta_{i,k}$ in $J$ such that if $s_i \in M(l_k)$, or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or is indifferent between them, then $\delta_{i,k} = 1$. We enforce this condition by imposing the following constraint. \begin{eqnarray} \label{ineq:spa-st-ip-lecturerfull-student} d_k\delta_{i,k} \geq \sum\limits_{i' = 1}^{n_1} \; \sum\limits_{p_{j'} \in P_k} x_{i',j'} - \sum\limits_{s_{i'} \in D_{i,k}} \; \sum\limits_{p_{j'} \in P_k}x_{i',j'}\enspace. \end{eqnarray} Note that if $s_i \in M(l_k)$ or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or $l_k$ is indifferent between them, then the RHS of Constraint \eqref{ineq:spa-st-ip-lecturerfull-student} is at least 1 and this implies that $\delta_{i,k} = 1$, otherwise $\delta_{i,k}$ is not constrained. The following constraint ensures that $(s_i, p_j)$ does not form a type (ii) blocking pair for $M$. \begin{align} \label{ineq:super-bp-type-ii} \Aboxed{ \theta_{i,j} + \alpha_{j} + \eta_{k} + \delta_{i,k} \leq 3\enspace.} \end{align} \paragraph{\textbf{Type (iii)}. \label{type-iii}} Next we create a binary variable $\gamma_{j}$ in $J$ such that if $p_j$ is full in $M$ then $\gamma_{j} = 1$. We enforce this condition by imposing the following constraint. \begin{eqnarray} \label{ineq:spa-st-ip-projectfull} c_j\gamma_{j} \geq \left( 1 + \sum\limits_{i' = 1}^{n_1} \; x_{i',j} \right) - c_j\enspace. \end{eqnarray} where $\sum_{i' = 1}^{n_1} x_{i',j} = |M(p_j)|$. If $p_j$ is full in $M$ then the RHS of Inequality \eqref{ineq:spa-st-ip-projectfull} is at least $1$ and this implies that $\gamma_j = 1$, otherwise $\gamma_j$ is not constrained. Next, we create a binary variable $\lambda_{i,j,k}$ in $J$ such that if $l_k$ prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them, then $\lambda_{i,j,k}=1$. We enforce this condition by imposing the following constraint. \begin{eqnarray} \label{ineq:spa-st-ip-projectfull-student} c_j\lambda_{i,j,k} \geq \sum\limits_{i' = 1}^{n_1} x_{i',j} - \sum\limits_{s_{i'} \in T_{i,j,k}} x_{i',j}\enspace. \end{eqnarray} Note that if $l_k$ prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them, then the RHS of Inequality \eqref{ineq:spa-st-ip-projectfull-student} is at least 1 and this implies that $\lambda_{i,j,k} = 1$, otherwise $\lambda_{i,j,k}$ is not constrained. The following constraint ensures that $(s_i, p_j)$ does not form a type (iii) blocking pair for $M$. \begin{align} \label{ineq:super-bp-type-iii} \Aboxed{ \theta_{i,j} + \gamma_j + \lambda_{i,j,k} \leq 2\enspace.} \end{align} \subsection{Variables} \label{sect:spa-st-ip-variables} We define a collective notation for each set of variables involved in $J$ as follows: \begin{center} \begin{tabular}{p{5cm}p{0.2cm}p{6cm}} $A = \{ \alpha_{j}: 1 \leq j \leq n_2\}$, & & $\Gamma = \{ \gamma_{j}: 1 \leq j \leq n_2\}$, \\ $B = \{\beta_{k}: 1 \leq k \leq n_3\}$, & & $\Delta = \{ \delta_{i,k}: 1 \leq i \leq n_1, 1 \leq k \leq n_3\}$, \\ $N = \{\eta_{k}: 1 \leq k \leq n_3\}$, & & $X = \{ x_{i,j}: 1 \leq i \leq n_1, 1 \leq j \leq n_2\}$, \\ \multicolumn{3}{p{12cm}}{$\Lambda = \{\lambda_{i,j,k}: 1 \leq i \leq n_1, 1 \leq j \leq n_2, 1 \leq k \leq n_3 \}$\enspace.} \\ \end{tabular} \end{center} \subsection{Objective function} On one hand, all super-stable matchings are of the same size, and thus nullifies the need for an objective function. On the other hand, optimization solvers require an objective function in addition to the variables and constraints in order to produce a solution. The objective function given below involves maximising the summation of all the $x_{i,j}$ binary variables. \begin{align} \label{ineq:super-objectivefunction} \Aboxed{\max \sum\limits_{i = 1}^{n_1} \; \sum\limits_{p_j \in A_i}x_{i,j}\enspace.} \end{align} Finally, we have constructed an IP model $J$ of $I$ comprising the set of integer-valued variables $A, B, N, X, \Gamma, \Delta, \mbox{ and } \Lambda$, the set of Inequalities \eqref{ineq:spa-st-ip-studentassignment} - \eqref{ineq:super-bp-type-iii} and an objective function \eqref{ineq:super-objectivefunction}. Note that $J$ can then be used to construct a super-stable matching in $I$, should one exist. \subsection{Correctness of the IP model} Given an instance $I$ of {\sc spa-st} formulated as an IP model $J$ using the above transformation, we present the following lemmas regarding the correctness of $J$. \begin{lemma} \label{lemma:super-solution-stability} A feasible solution $S$ to $J$ corresponds to a super-stable matching $M$ in $I$. \end{lemma} \begin{proof} Assume firstly that $J$ has a feasible solution $S$. Let $M = \{(s_i, p_j) \in \mathcal{S} \times \mathcal{P}: x_{i,j} = 1\}$ be the assignment in $I$ generated from $S$. We note that Inequality \eqref{ineq:spa-st-ip-studentassignment} ensures that each student is assigned in $M$ to at most one project. Moreover, Inequalities \eqref{ineq:spa-st-ip-projectcapacity} and \eqref{ineq:spa-st-ip-lecturercapacity} ensures that the capacity of each project and lecturer is not exceeded in $M$. Thus $M$ is a matching. We will prove that Inequalities \eqref{ineq:spa-st-ip-project-under} - \eqref{ineq:super-bp-type-iii} ensures that $M$ admits no blocking pair. Suppose for a contradiction that there exists some acceptable pair $(s_i, p_j)$ that forms a blocking pair for $M$, where $l_k$ is the lecturer who offers $p_j$. This implies that either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them. Thus $\sum_{p_{j'} \in S_{i,j}} x_{{i},{j'}} = 0$. Moreover, since $s_i$ is not assigned to $p_j$ in $M$, we have that $x_{i,j} = 0$. Thus $\theta_{i,j} = 1$. Now suppose $(s_i, p_j)$ forms a type (i) blocking pair for $M$. Then each of $p_j$ and $l_k$ is undersubscribed in $M$. Thus $\sum_{i' = 1}^{n_1} x_{i',j} < c_j$ and $\sum_{i' = 1}^{n_1} \; \sum_{p_{j'} \in P_k} x_{i',j'}$ $< d_k$. This implies that the RHS of Inequality \eqref{ineq:spa-st-ip-project-under} and the RHS of Inequality \eqref{ineq:spa-st-ip-lecturerunder} is strictly greater than $0$. Moreover, since $S$ is a feasible solution to $J$, $\alpha_j = \beta_k = 1$. Hence, the LHS of Inequality \eqref{ineq:super-bp-type-i} is strictly greater than $2$, a contradiction to the feasibility of $S$. Now suppose $(s_i, p_j)$ forms a type (ii) blocking pair for $M$. Then $p_j$ is undersubscribed in $M$ and as explained above, $\alpha_j = 1$. Also, $l_k$ is full in $M$ and this implies that the RHS of Inequality \eqref{ineq:spa-st-ip-lecturerfull} is strictly greater than $0$. Since $S$ is a feasible solution, we have that $\eta_k = 1$. Furthermore, either $s_i \in M(l_k)$ or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or $l_k$ is indifferent between them. In any of these cases, the RHS of Inequality \eqref{ineq:spa-st-ip-lecturerfull-student} is strictly greater than $0$. Thus $\delta_{i,k} = 1$, since $S$ is a feasible solution. Hence the LHS of Inequality \eqref{ineq:super-bp-type-ii} is strictly greater than 3, a contradiction to the feasibility of $S$. Finally, suppose $(s_i, p_j)$ forms a type (iii) blocking pair for $M$. Then $p_j$ is full in $M$ and thus the RHS of Inequality \eqref{ineq:spa-st-ip-projectfull} is strictly greater than $0$. Since $S$ is a feasible solution, we have that $\gamma_j = 1$. In addition, $l_k$ prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them. This implies that the RHS of Inequality \eqref{ineq:spa-st-ip-projectfull-student} is strictly greater than $0$. Thus $\lambda_{i,j,k} = 1$, since $S$ is a feasible solution. Hence the LHS of Inequality \eqref{ineq:super-bp-type-iii} is strictly greater than 2, a contradiction to the feasibility of $S$. Hence $M$ admits no blocking pair; and hence, $M$ is a super-stable matching in $I$. \qed \end{proof} \begin{lemma} \label{lemma:super-stability-solution} A super-stable matching $M$ in $I$ corresponds to a feasible solution $S$ to $J$. \end{lemma} \begin{proof} Let $M$ be a super-stable matching in $I$. First we set all the binary variables involved in $J$ to $0$. For each $(s_i, p_j) \in M$, we set $x_{i,j} = 1$. Since $M$ is a matching, it is clear that Inequalities \eqref{ineq:spa-st-ip-studentassignment} - \eqref{ineq:spa-st-ip-lecturercapacity} is satisfied. For any acceptable pair $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$ such that $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them, we set $\theta_{i,j} = 1$. For any project $p_j \in \mathcal{P}$ such that $p_j$ is undersubscibed in $M$, we set $\alpha_j = 1$ and thus Inequality \eqref{ineq:spa-st-ip-project-under} is satisfied. For any lecturer $l_k \in \mathcal{L}$ such that $l_k$ is undersubscribed in $M$, we set $\beta_k = 1$ and thus Inequality \eqref{ineq:spa-st-ip-lecturerunder} is satisfied. Now, for Inequality \eqref{ineq:super-bp-type-i} not to be satisfied, its LHS must be strictly greater than 2. This would only happen if there exists some $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$, where $l_k$ is the lecturer who offers $p_j$, such that $\theta_{i,j} = 1$, $\alpha_j = 1$ and $\beta_k = 1$. This implies that either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them, and each of $p_j$ and $l_k$ is undersubscribed in $M$. Thus $(s_i, p_j)$ forms a type (i) blocking pair for $M$, a contradiction to the super-stability of $M$. Hence, Inequality \eqref{ineq:super-bp-type-i} is satisfied. For any lecturer $l_k \in \mathcal{L}$ such that $l_k$ is full in $M$, we set $\eta_k = 1$. Thus Inequality \eqref{ineq:spa-st-ip-lecturerfull} is satisfied. Let $(s_i, p_j)$ be an acceptable pair such that $p_j \in P_k$ and $(s_i, p_j) \notin M$. If $s_i \in M(l_k)$ or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or is indifferent between them, we set $\delta_{i,k} = 1$. Thus Inequality \eqref{ineq:spa-st-ip-lecturerfull-student} is satisfied. Suppose Inequality \eqref{ineq:super-bp-type-ii} is not satisfied. Then there exists $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$, where $l_k$ is the lecturer who offers $p_j$, such that $\theta_{i,j} = 1$, $\alpha_j = 1$, $\eta_k = 1$ and $\delta_{i,k} = 1$. This implies that either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them. In addition, $p_j$ is undersubscribed in $M$, $l_k$ is full in $M$ and either $s_i \in M(l_k)$ or $l_k$ prefers $s_i$ to a worst student in $M(l_k)$ or is indifferent between them. Thus $(s_i, p_j)$ forms a type (ii) blocking pair for $M$, a contradiction to the super-stability of $M$. Hence Inequality \eqref{ineq:super-bp-type-ii} is satisfied. Finally, for any project $p_j \in \mathcal{P}$ such that $p_j$ is full in $M$, we set $\gamma_j = 1$. Thus Inequality \eqref{ineq:spa-st-ip-projectfull} is satisfied. Let $l_k$ be the lecturer who offers $p_j$ and let $(s_i, p_j)$ be an acceptable pair. If $l_k$ prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them, we set $\lambda_{i,j,k} = 1$. Thus Inequality \eqref{ineq:spa-st-ip-projectfull-student} is satisfied. Suppose Inequality \eqref{ineq:super-bp-type-iii} is not satisfied. Then there exists some $(s_i, p_j) \in (\mathcal{S} \times \mathcal{P}) \setminus M$ such that $\theta_{i,j} = 1$, $\gamma_j = 1$ and $\lambda_{i,j,k} = 1$. This implies that either $s_i$ is unassigned in $M$ or $s_i$ prefers $p_j$ to $M(s_i)$ or is indifferent between them. In addition, $p_j$ is full in $M$ and $l_k$ prefers $s_i$ to a worst student in $M(p_j)$ or is indifferent between them. Thus $(s_i, p_j)$ forms a type (iii) blocking pair for $M$, a contradiction to the super-stability of $M$. Hence, Inequality \eqref{ineq:super-bp-type-iii} is satisfied. Hence $S$, comprising the above assignments of values to the variables in $A \cup B \cup N \cup X \cup \Gamma \cup \Delta \cup \Lambda$, is a feasible solution to $J$. \qed \end{proof} The following theorem is a consequence of Lemmas \ref{lemma:super-solution-stability} and \ref{lemma:super-stability-solution}. \begin{theorem} \label{theorem:super-stable-solution} Let $I$ be an instance of {\sc spa-st} and let $J$ be the IP model for $I$ as described above. A feasible solution to $J$ corresponds to a super-stable matching in $I$. Conversely, a super-stable matching in $I$ corresponds to a feasible solution to $J$. \end{theorem} \end{subappendices} \end{document}

\begin{document} \title{On semi-vector spaces and semi-algebras} \author{Giuliano G. La Guardia, Jocemar de Q. Chagas, Ervin K. Lenzi, Leonardo Pires \thanks{Giuliano G. La Guardia ({\tt \small [email protected]}), Jocemar de Q. Chagas ({\tt \small [email protected]}) and Leonardo Pires ({\tt \small [email protected]}) are with Department of Mathematics and Statistics, State University of Ponta Grossa (UEPG), 84030-900, Ponta Grossa - PR, Brazil. Ervin K. Lenzi ({\tt \small [email protected]}) is with Department of Physics, State University of Ponta Grossa (UEPG), 84030-900, Ponta Grossa - PR, Brazil. Corresponding author: Giuliano G. La Guardia ({\tt \small [email protected]}). }} \maketitle \begin{abstract} It is well-known that the theories of semi-vector spaces and semi-algebras -- which were not much studied over time -- are utilized/applied in Fuzzy Set Theory in order to obtain extensions of the concept of fuzzy numbers as well as to provide new mathematical tools to investigate properties and new results on fuzzy systems. In this paper we investigate the theory of semi-vector spaces over the semi-field of nonnegative real numbers ${\mathbb R}_{0}^{+}$. We prove several results concerning semi-vector spaces and semi-linear transformations. Moreover, we introduce in the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing in some cases how to compute them. Topological properties of semi-vector spaces such as completeness and separability are also investigated. New families of semi-vector spaces derived from semi-metric, semi-norm, semi-inner product, among others are exhibited. Additionally, some results on semi-algebras are presented. \end{abstract} \emph{keywords}: semi-vector space; semi-algebras; semi-linear operators \section{Introduction} The concept of semi-vector space was introduced by Prakash and Sertel in \cite{Prakash:1974}. Roughly speaking, semi-vector spaces are ``vector spaces" where the scalars are in a semi-field. Although the concept of semi-vector space was investigated over time, there exist few works available in the literature dealing with such spaces \cite{Radstrom:1952,Prakash:1974,Prakash:1976,Pap:1980,Gahler:1999,Janyska:2007,Milfont:2021}. This fact occurs maybe due to the limitations that such concept brings, i.e., the non-existence of (additive) symmetric for some (for all) semi-vector. A textbook in such a topic of research is the book by Kandasamy~\cite{Kandasamy:2002}. Although the seminal paper on semi-vector spaces is \cite{Prakash:1974}, the idea of such a concept was implicit in \cite{Radstrom:1952}, where Radstrom shown that a semi-vector space over the semi-field of nonnegative real numbers can be extended to a real vector space (see \cite[Theorem 1-B.]{Radstrom:1952}). In \cite{Prakash:1974}, Prakash and Sertel investigated the structure of topological semi-vector spaces. The authors were concerned with the study of the existence of fixed points in compact convex sets and also to generate min-max theorems in topological semi-vector spaces. In \cite{Prakash:1976}, Prakash and Sertel investigated properties of the topological semi-vector space consisting of nonempty compact subsets of a real Hausdorff topological vector space. In \cite{Pap:1980}, Pap investigated and formulated the concept of integrals of functions having, as counter-domain, complete semi-vector spaces. W. Gahler and S. Gahler \cite{Gahler:1999} showed that a (ordered) semi-vector space can be extended to a (ordered) vector space and a (ordered) semi-algebra can be extended to a (ordered) algebra. Moreover, they provided an extension of fuzzy numbers. Janyska et al.~\cite{Janyska:2007} developed such theory (of semi-vector space) by proving useful results and defining the semi-tensor product of (semi-free) semi-vector spaces. They were also interested to propose an algebraic model of physical scales. Canarutto~\cite{Canarutto:2012} explored the concept of semi-vector spaces to express aspects and to exploit nonstandard mathematical notions of basics of quantum particle physics on a curved Lorentzian background. Moreover, he dealt with the case of electroweak interactions. Additionally, in \cite{Canarutto:2016}, Canarutto provided a suitable formulation of the fundamental mathematical concepts with respect to quantum field theory. Such a paper presents a natural application of the concept of semi-vector spaces and semi-algebras. Recently, Bedregal et al. \cite{Milfont:2021} investigated (ordered) semi-vector spaces over a weak semi-field $K$ (i.e., both $(K, +)$ and $(K, \bullet)$ are monoids) in the context of fuzzy sets and applying the results in multi-criteria group decision-making. In this paper we extend the theory of semi-vector spaces. The semi-field of scalars considered here is the semi-field of nonnegative real numbers. We prove several results in the context of semi-vector spaces and semi-linear transformations. We introduce the concept of semi-eigenvalues and semi-eigenvectors of an operator and of a matrix, showing how to compute it in specific cases. We investigate topological properties such as completeness, compactness and separability of semi-vector spaces. Additionally, we present interesting new families of semi-vector spaces derived from semi-metric, semi-norm, semi-inner product, metric-preserving functions among others. Furthermore, we show some results concerning semi-algebras. Summarizing, we provide new results on semi-vector spaces and semi-algebras, although such theories are very difficult to be investigated due to the fact that vectors do not even have (additive) symmetrical. These new results can be possibly utilized in the theory of fuzzy sets in order to extend it or in the generation of new results concerning such a theory. The paper is organized as follows. In Section~\ref{sec2} we recall some concepts on semi-vector spaces which will be utilized in this work. In Section~\ref{sec3} we present and prove several results concerning semi-vector spaces and semi-linear transformations. We introduce naturally the concepts of eigenvalue and eigenvector of a semi-linear operator and of matrices. Additionally, we exhibit and show interesting examples of semi-vector spaces derived from semi-metric, semi-norms, metric-preserving functions among others. Results concerning semi-algebras are also presented. In Section~\ref{sec3a} we show relationships between Fuzzy Set Theory and the theory of semi-vector spaces and semi-algebras. Finally, a summary of this paper is presented in Section~\ref{sec4}. \section{Preliminaries}\label{sec2} In this section we recall important facts on semi-vector spaces necessary for the development of this work. In order to define formally such concept, it is necessary to define the concepts of semi-ring and semi-field. \begin{definition}\label{defSR} A semi-ring $(S, + , \bullet )$ is a set $S$ endowed with two binary operations, $+: S\times S\longrightarrow S$ (addition), $\bullet: S\times S\longrightarrow S$ (multiplication) such that: $\operatorname{(1)}$ $(S, +)$ is a commutative monoid; $\operatorname{(2)}$ $(S, \bullet)$ is a semigroup; $\operatorname{(3)}$ the multiplication $\bullet$ is distributive with respect to $+$: $\forall \ x, y, z \in S$, $(x + y)\bullet z = x\bullet z + y\bullet z$ and $x\bullet(y + z ) = x\bullet y + x\bullet z$. \end{definition} We write $S$ instead of writing $(S, + , \bullet )$ if there is not possibility of confusion. If the multiplication $\bullet$ is commutative then $S$ is a commutative semi-ring. If there exists $1 \in S$ such that, $ \forall \ x \in S$ one has $1\bullet x = x\bullet 1 = x$, then $S$ is a semi-ring with identity. \begin{definition}\cite[Definition 3.1.1]{Kandasamy:2002}\label{defSF} A semi-field is an ordered triple $(K, +, \bullet )$ which is a commutative semi-ring with unit satisfying the following conditions: $\operatorname{(1)}$ $\forall \ x, y \in K$, if $x+y=0$ then $x=y=0$; $\operatorname{(2)}$ if $x, y \in K$ and $x\bullet y = 0$ then $x=0$ or $y=0$. \end{definition} Before proceeding further, it is interesting to observe that in \cite{Gahler:1999} the authors considered the additive cancellation law in the definition of semi-vector space. In \cite{Janyska:2007}, the authors did not assume the existence of the zero (null) vector. In this paper we consider the definition of a semi-vector space in the context of that shown in \cite{Gahler:1999}, Sect.3.1. \begin{definition}\label{defSVS} A semi-vector space over a semi-field $K$ is a ordered triple $(V,$ $+, \cdot)$, where $V$ is a set endowed with the operations $+: V\times V\longrightarrow V$ (vector addition) and $\cdot: K\times V\longrightarrow V$ (scalar multiplication) such that: \begin{itemize} \item [ $\operatorname{(1)}$] $(V, +)$ is an abelian monoid equipped with the additive cancellation law: $\forall \ u, v, w \in V$, if $u + v = u + w$ then $v = w$; \item [ $\operatorname{(2)}$] $\forall$ $\alpha\in K$ and $\forall$ $u, v \in V$, $\alpha (u+v)=\alpha u + \beta v$; \item [ $\operatorname{(3)}$] $\forall$ $\alpha, \beta \in K$ and $\forall$ $v\in V$, $(\alpha + \beta)v= \alpha v + \beta v$; \item [ $\operatorname{(4)}$] $\forall$ $\alpha, \beta \in K$ and $\forall$ $v\in V$, $(\alpha\beta)v=\alpha (\beta v)$; \item [ $\operatorname{(5)}$] $\forall$ $v \in V$ and $1 \in K$, $1v=v$. \end{itemize} \end{definition} Note that from Item~$\operatorname{(1)}$ of Definition~\ref{defSVS}, all semi-vector spaces considered in this paper are \emph{regular}, that it, the additive cancellation law is satisfied. The zero (or null) vector of $V$, which is unique, will be denoted by $0_{V}$. Let $v \in V$, $v\neq 0 $. If there exists $u \in V$ such that $v + u =0$ then $v$ is said to be \emph{symmetrizable}. A semi-vector space $V$ is said to be \emph{simple} if the unique symmetrizable element is the zero vector $0_{V}$. In other words, $V$ is simple if it has none nonzero symmetrizable elements. \begin{definition}\cite[Definition 1.4]{Janyska:2007}\label{defSBasis} Let $V$ be a simple semi-vector space over ${\mathbb R}_{0}^{+}$. A subset $B \subset V$ is called a semi-basis of $V$ if every $v \in V$, $v\neq 0$, can be written in a unique way as $v = \displaystyle\sum_{i \in I_v}^{} v^{(i)} b_i$, where $v^{(i)} \in {\mathbb R}^{+}$, $b_i \in B$ and $I_v$ is a finite family of indices uniquely determined by $v$. The finite subset $B_v \subset B$ defined by $B_v := \{b_i \}_{i \in I_v }$ is uniquely determined by $v$. If a semi-vector space $V$ admits a semi-basis then it is said to be semi-free. \end{definition} The concept of semi-dimension can be defined in analogous way to semi-free semi-vector spaces due to the next result. \begin{corollary}\cite[Corollary 1.7]{Janyska:2007} Let $V$ be a semi-free semi-vector space. Then all semi-bases of $V$ have the same cardinality. \end{corollary} Therefore, the semi-dimension of a semi-free semi-vector space is the cardinality of a semi-basis (consequently, of all semi-bases) of $V$. We next present some examples of semi-vector spaces. \begin{example}\label{ex1} All real vector spaces are semi-vector spaces, but they are not simple. \end{example} \begin{example}\label{ex2} The set ${[{\mathbb R}_{0}^{+}]}^{n}=\underbrace{{\mathbb R}_{0}^{+} \times \ldots \times {\mathbb R}_{0}^{+}}_{n \operatorname{times}}$ endowed with the usual sum of coordinates and scalar multiplication is a semi-vector space over ${\mathbb R}_{0}^{+}$. \end{example} \begin{example}\label{ex3} The set ${\mathcal M}_{n\times m}({\mathbb R}_{0}^{+})$ of matrices $n \times m$ whose entries are nonnegative real numbers equipped with the sum of matrices and multiplication of a matrix by a scalar (in ${\mathbb R}_{0}^{+}$, of course) is a semi-vector space over ${\mathbb R}_{0}^{+}$. \end{example} \begin{example}\label{ex4} The set ${\mathcal P}_{n}[x]$ of polynomials with coefficients from ${\mathbb R}_{0}^{+}$ and degree less than or equal to $n$, equipped with the usual of polynomial sum and scalar multiplication, is a semi-vector space. \end{example} \begin{definition}\label{semi-subspace} Let $(V, +, \cdot )$ be a semi-vector space over ${\mathbb R}_{0}^{+}$. We say that a non-empty subset $W$ of $V$ is a semi-subspace of $V$ if $W$ is closed under both addition and scalar multiplication of $V$, that is, \begin{itemize} \item [ $\operatorname{(1)}$] $\forall \ w_1 , w_2 \in W \Longrightarrow w_1 + w_2 \in W$; \item [ $\operatorname{(2)}$] $\forall \ \lambda \in {\mathbb R}_{0}^{+}$ and $\forall \ w \in W \Longrightarrow \lambda w \in W$. \end{itemize} \end{definition} The uniqueness of the zero vector implies that for each $\lambda \in {\mathbb R}_{0}^{+}$ on has $\lambda 0_{V} = 0_{V}$. Moreover, if $ v \in V$, it follows that $0 v = 0 v + 0 v$; applying the regularity one obtains $0 v =0_{V}$. Therefore, from Item~$\operatorname{(2)}$, every semi-subspace contains the zero vector. \begin{example}\label{ex4a} Let ${\mathbb Q}_{0}^{+}$ denote the set of nonnegative rational numbers. The semi-vector space ${\mathbb Q}_{0}^{+}$ considered as an ${\mathbb Q}_{0}^{+}$ space is a semi-subspace of ${\mathbb R}_{0}^{+}$ considered as an ${\mathbb Q}_{0}^{+}$ space. \end{example} \begin{example}\label{ex4b} For each positive integer $ i \leq n$, the subset ${\mathcal P}_{(i)}[x]\cup \{0_{p}\}$, where ${\mathcal P}_{(i)}[x]=\{p(x); \partial (p(x))=i \} $ and $0_{p}$ is the null polynomial, is a semi-subspace of ${\mathcal P}_{n}[x]$, shown in Example~\ref{ex4}. \end{example} \begin{example}\label{ex4c} The set of diagonal matrices of order $n$ with entries in ${\mathbb R}_{0}^{+}$ is a semi-subspace of ${\mathcal M}_{n}({\mathbb R}_{0}^{+})$, where the latter is the semi-vector space of square matrices with entries in ${\mathbb R}_{0}^{+}$ (according to Example~\ref{ex3}). \end{example} \begin{definition}\cite[Definition 1.22]{Janyska:2007}\label{semilineartrans} Let $V$ and $W$ be two semi-vector spaces and $T: V\longrightarrow W$ be a map. We say that $T$ is a semi-linear transformation if: $\operatorname{(1)}$ $\forall \ v_1, v_2 \in V$, $T(v_1 + v_2) = T(v_1) + T(v_2)$; $\operatorname{(2)}$ $\forall \lambda \in {\mathbb R}_{0}^{+}$ and $\forall \ v \in V$, $T(\lambda v) =\lambda T(v)$. \end{definition} If $U$ and $V$ are semi-vector spaces then the set $\operatorname{Hom}(U, V)=\{ T:U\longrightarrow V; T \operatorname{is \ semi-linear} \}$ is also a semi-vector space. \section{The New Results}\label{sec3} We start this section with important remarks. \begin{remark}\label{mainremark} \begin{itemize} \item [ $\operatorname{(1)}$] Throughout this section we always consider that the semi-field $K$ is the set of nonnegative real numbers, i.e., $K= {\mathbb R}_{0}^{+}={\mathbb R}^{+}\cup \{0\}$. \item [ $\operatorname{(2)}$] In the whole section (except Subsection~\ref{subsec2}) we assume that the semi-vector spaces $V$ are simple, i.e., the unique symmetrizable element is the zero vector $0_{V}$. \item [ $\operatorname{(3)}$] It is well-known that a semi-vector space $(V, +, \cdot)$ can be always extended to a vector space according to the equivalence relation on $V \times V$ defined by $(u_1 , v_1 ) \sim (u_2 , v_2 )$ if and only if $u_1 + v_2 = v_1 + u_2$ (see \cite{Radstrom:1952}; see also \cite[Section 3.4]{Gahler:1999}). However, our results are obtained without utilizing such a natural embedding. In other words, if one want to compute, for instance, the eigenvalues of a matrix defined over ${\mathbb R}_{0}^{+}$ we cannot solve the problem in the associated vector spaces and then discard the negative ones. Put differently, all computations performed here are restricted to nonnegative real numbers and also to the fact that none vector (with exception of $0_V$) has (additive) symmetrical. However, we will show that, even in this case, several results can be obtained. \end{itemize} \end{remark} \begin{proposition}\label{prop1} Let $V$ be a semi-vector space over ${\mathbb R}_{0}^{+}$. Then the following hold: \begin{itemize} \item [ $\operatorname{(1)}$] let $ v \in V$, $ v \neq 0_{V}$, and $\lambda \in {\mathbb R}_{0}^{+}$; if $\lambda v = 0_{V}$ then $\lambda = 0$; \item [ $\operatorname{(2)}$] if $\alpha , \beta \in {\mathbb R}_{0}^{+}$, $v \in V$ and $ v \neq 0_{V}$, then the equality $\alpha v = \beta v$ implies that $\alpha = \beta$. \end{itemize} \end{proposition} \begin{proof} $\operatorname{(1)}$ If $\lambda \neq 0$ then there exists its multiplicative inverse ${\lambda}^{-1}$, hence $ 1 v = {\lambda}^{-1} 0_{V}= 0_{V}$, i.e., $v = 0_{V}$, a contradiction.\\ $\operatorname{(2)}$ If $\alpha \neq \beta$, assume w.l.o.g. that $\alpha > \beta$, i.e., there exists a positive real number $c$ such that $\alpha = \beta + c$. Thus, $\alpha v = \beta v$ implies $\beta v + c v = \beta v$. From the cancellation law we have $c v = 0_{V}$, and from Item~$\operatorname{(1)}$ it follows that $c = 0$, a contradiction. \end{proof} We next introduce in the literature the concept of eigenvalue and eigenvector of a semi-linear operator. \begin{definition}\label{eigenvector} Let $V$ be a semi-vector space and $T:V\longrightarrow V$ be a semi-linear operator. If there exist a non-zero vector $v \in V$ and a nonnegative real number $\lambda$ such that $T(v)=\lambda v$, then $\lambda$ is an eigenvalue of $T$ and $v$ is an eigenvector of $T$ associated with $\lambda$. \end{definition} As it is natural, the zero vector joined to the set of the eigenvectors associated with a given eigenvalue has a semi-subspace structure. \begin{proposition}\label{eigenspace} Let $V$ be a semi-vector space over ${\mathbb R}_{0}^{+}$ and $T:V\longrightarrow V$ be a semi-linear operator. Then the set $V_{\lambda} = \{ v \in V ; T(v)=\lambda v \}\cup \{0_{V}\}$ is a semi-subspace of $V$. \end{proposition} \begin{proof} From hypotheses, $V_{\lambda}$ is non-empty. Let $u, v \in V_{\lambda}$, i.e., $T(u)=\lambda u $ and $T(v)=\lambda v $. Hence, $T(u + v )= T(u) + T(v)= \lambda (u + v )$, i.e., $u + v \in V_{\lambda}$. Further, if $\alpha \in {\mathbb R}_{0}^{+}$ and $u \in V$, it follows that $T(\alpha u)=\alpha T(u)= \lambda (\alpha u)$, that is, $\alpha u \in V_{\lambda}$. Therefore, $V_{\lambda}$ is a semi-subspace of $V$. \end{proof} The next natural step would be to introduce the characteristic polynomial of a matrix, according to the standard Linear Algebra. However, how to compute $\det (A -\lambda I)$ if $-\lambda$ can be a negative real number? Based on this fact we must be careful to compute the eigenvectors of a matrix. In fact, the main tools to be utilized in computing eigenvalues/eigenvectors of a square matrix whose entries are nonnegative real numbers is the additive cancellation law in ${\mathbb R}_{0}^{+}$ and also the fact that positive real numbers have multiplicative inverse. However, in much cases, such a tools are not sufficient to solve the problem. Let us see some cases when it is possible to compute eigenvalues/eigenvectors of a matrix. \begin{example}\label{examatr1} Let us see how to obtain (if there exists) an eigenvalue/eigenvector of a diagonal matrix $A \in {\mathcal M}_{2}({\mathbb R}_{0}^{+})$, \begin{eqnarray*} A= \left[\begin{array}{cc} a & 0\\ 0 & b\\ \end{array} \right], \end{eqnarray*} where $a \neq b$ not both zeros. Let us assume first that $a, b > 0$. Solving the equation $A v = \lambda v$, that is, \begin{eqnarray*} \left[\begin{array}{cc} a & 0\\ 0 & b\\ \end{array} \right] \left[\begin{array}{c} x\\ y\\ \end{array} \right]= \left[\begin{array}{c} \lambda x\\ \lambda y\\ \end{array} \right], \end{eqnarray*} we obtain $\lambda = a$ with associated eigenvector $x(1, 0)$ and $\lambda = b$ with associated eigenvector $y(0, 1)$. If $a\neq 0$ and $b = 0$, then $\lambda = a$ with eigenvectors $x(1, 0)$. If $a = 0$ and $b \neq 0$, then $\lambda = b$ with eigenvectors $y(0, 1)$. \end{example} \begin{example}\label{examatr2} Let $A \in {\mathcal M}_{2}({\mathbb R}_{0}^{+})$ be a matrix of the form \begin{eqnarray*} A= \left[\begin{array}{cc} a & b\\ 0 & a\\ \end{array} \right], \end{eqnarray*} where $a \neq b$ are positive real numbers. Let us solve the matrix equation: \begin{eqnarray*} \left[\begin{array}{cc} a & b\\ 0 & a\\ \end{array} \right] \left[\begin{array}{c} x\\ y\\ \end{array} \right]= \left[\begin{array}{c} \lambda x\\ \lambda y\\ \end{array} \right]. \end{eqnarray*} If $ y \neq 0$, $\lambda = a$; hence $b y = 0$, which implies $b=0$, a contradiction. If $ y = 0$, $x \neq 0$; hence $\lambda = a$ with eigenvectors $(x, 0)$. \end{example} If $V$ and $W$ are semi-free semi-vector spaces then it is possible to define the matrix of a semi-linear transformation $T: V \longrightarrow W$ as in the usual case (vector spaces). \begin{definition}\label{semi-free matrix} Let $T: V \longrightarrow W$ be a semi-liner transformation between semi-free semi-vector spaces with semi-basis $B_1$ and $B_2$, respectively. Then the matrix $[T]_{B_1}^{B_2}$ is the matrix of the transformation $T$. \end{definition} \begin{theorem}\label{diagonalmatrix} Let $V$ be a semi-free semi-vector space over ${\mathbb R}_{0}^{+}$ and let $T:V\longrightarrow V$ be a semi-linear operator. Then $T$ admits a semi-basis $B = \{ v_1 , v_2 , \ldots , v_n \}$ such that ${[T]}_{B}^{B}$ is diagonal if and only if $B$ consists of eigenvectors of $T$. \end{theorem} \begin{proof} The proof is analogous to the case of vector spaces. Let $B=\{ v_1 , v_2 , \ldots ,$ $v_n \}$ be a semi-basis of $V$ whose elements are eigenvectors of $T$. We then have: \begin{eqnarray*} T(v_1)= {\lambda}_1 v_1 + 0 v_2 + \ldots + 0 v_n,\\ T(v_2)= 0 v_1 + {\lambda}_{2} v_2 + \ldots + 0 v_n,\\ \vdots\\ T(v_n)= 0 v_1 + 0 v_2 + \ldots + {\lambda}_{n} v_n, \end{eqnarray*} which implies that $[T]_{B}^{B}$ is of the form \begin{eqnarray*} [T]_{B}^{B}= \left[\begin{array}{ccccc} {\lambda}_1 & 0 & 0 & \ldots & 0\\ 0 & {\lambda}_2 & 0 & \ldots & 0\\ \vdots & \vdots & \vdots & \ldots & \vdots\\ 0 & 0 & 0 & \ldots & {\lambda}_{n}\\ \end{array} \right]. \end{eqnarray*} On the other hand, let $B^{*}= \{ w_1 , w_2 , \ldots , w_n \}$ be a semi-basis of $V$ such that $[T]_{B^{*}}^{B^{*}}$ is diagonal: \begin{eqnarray*} [T]_{B^{*}}^{B^{*}}=\left[\begin{array}{ccccc} {\alpha}_1 & 0 & 0 & \ldots & 0\\ 0 & {\alpha}_2 & 0 & \ldots & 0\\ \vdots & \vdots & \vdots & \ldots & \vdots\\ 0 & 0 & 0 & \ldots & {\alpha}_{n}\\ \end{array} \right]; \end{eqnarray*} thus,\\ \begin{eqnarray*} T(w_1)= {\alpha}_1 w_1 + 0 w_2 + \ldots + 0 w_n = {\alpha}_1 w_1,\\ T(w_2)= 0 w_1 + {\alpha}_{2} w_2 + \ldots + 0 w_n = {\alpha}_{2} w_2,\\ \vdots\\ T(w_n)= 0 w_1 + 0 w_2 + \ldots + {\alpha}_{n} w_n = {\alpha}_{2} w_{n}. \end{eqnarray*} This means that $w_i$ are eigenvectors of $T$ with corresponding eigenvalues ${\alpha}_{i}$, for all $i = 1, 2, \ldots , n$. \end{proof} \begin{definition}\label{kernel} Let $T: V \longrightarrow W$ be a semi-linear transformation. The set $\operatorname{Ker}(T)=\{ v \in V ; T(v)=0\}$ is called kernel of $T$. \end{definition} \begin{proposition}\label{subkernel} Let $T: V \longrightarrow W$ be a semi-linear transformation. Then the following hold: \begin{itemize} \item [ $\operatorname{(1)}$] $\operatorname{Ker}(T)$ is a semi-subspace of $V$; \item [ $\operatorname{(2)}$] if $T$ is injective then $\operatorname{Ker}(T) = \{0_{V}\}$; \item [ $\operatorname{(3)}$] if $V$ has semi-dimension $1$ then $\operatorname{Ker}(T) = \{0_{V}\}$ implies that $T$ is injective. \end{itemize} \end{proposition} \begin{proof} $\operatorname{(1)}$ We have $T(0_{V})= T(0_{V})+T(0_{V})$. Since $W$ is regular, it follows that $T(0_{V})=0_{W}$, which implies $\operatorname{Ker}(T) \neq \emptyset$. If $u, v \in \operatorname{Ker}(T)$ and $\lambda \in {\mathbb R}_{0}^{+}$, then $u + v \in \operatorname{Ker}(T)$ and $\lambda v \in \operatorname{Ker}(T)$, which implies that $\operatorname{Ker}(T)$ is a semi-subspace of $V$.\\ $\operatorname{(2)}$ Since $T(0_{V})=0_{W}$, it follows that $\{0_{V}\}\subseteq \operatorname{Ker}(T)$. On the other hand, let $ u \in \operatorname{Ker}(T)$, that is, $T(u)=0_{W}$. Since $T$ is injective, one has $u = 0_{V}$. Hence, $\operatorname{Ker}(T) = \{0_{V}\}$.\\ $\operatorname{(3)}$ Let $B=\{ v_0 \}$ be a semi-basis of $V$. Assume that $T(u) = T(v)$, where $u, v \in V$ are such that $u = \alpha v_0$ and $v = \beta v_0 $. Hence, $\alpha T(v_0) = \beta T(v_0 )$. Since $\operatorname{Ker}(T) = \{0_{V}\}$ and $v_0 \neq 0$, it follows that $T(v_0) \neq 0$. From Item~$\operatorname{(2)}$ of Proposition~\ref{prop1}, one has $\alpha = \beta$, i.e., $u = v$. \end{proof} \begin{definition}\label{image} Let $T: V \longrightarrow W$ be a semi-linear transformation. The image of $T$ is the set of all vectors $w \in W$ such that there exists $v \in V$ with $T(v)=w$, that is, $\operatorname{Im}(T)=\{ w \in W ; \exists \ v \in V \operatorname{with} T(v)=w\}$. \end{definition} \begin{proposition}\label{subImage} Let $T: V \longrightarrow W$ be a semi-linear transformation. Then the image of $T$ is a semi-subspace of $W$. \end{proposition} \begin{proof} The set $\operatorname{Im}(T)$ is non-empty because $T(0_{V})=0_{W}$. It is easy to see that if $w_1 , w_2 \in \operatorname{Im}(T)$ and $\lambda \in {\mathbb R}_{0}^{+}$, then $ w_1 + w_2 \in \operatorname{Im}(T)$ and $\lambda w_1 \in \operatorname{Im}(T)$. \end{proof} \begin{theorem}\label{isosemi} Let $V$ be a $n$-dimensional semi-free semi-vector space over ${\mathbb R}_{0}^{+}$. Then $V$ is isomorphic to $({\mathbb R}_{0}^{+})^{n}$. \end{theorem} \begin{proof} Let $B = \{ v_1 , v_2 , \ldots , v_n \}$ be a semi-basis of $V$ and consider the canonical semi-basis $e_i = (0, 0, \ldots , $ $0, \underbrace{1}_{i}, 0, \ldots, 0)$ of $({\mathbb R}_{0}^{+})^{n}$, where $i=1, 2, \ldots , n$. Define the map $T:V \longrightarrow ({\mathbb R}_{0}^{+})^{n}$ as follows: for each $v = \displaystyle\sum_{i=1}^{n}a_i v_i \in V$, put $T(v) = \displaystyle\sum_{i=1}^{n}a_i e_i$. It is easy to see that $T$ is bijective semi-linear transformation, i.e., $V$ is isomorphic to $({\mathbb R}_{0}^{+})^{n}$, as required. \end{proof} \subsection{Complete Semi-Vector Spaces}\label{subsec1} We here define and study complete semi-vector spaces, i.e., semi-vector spaces whose norm (inner product) induces a metric under which the space is complete. \begin{definition}\label{semiBanach} Let $V$ be a semi-vector space over ${\mathbb R}_{0}^{+}$. If there exists a norm $\| \ \|:V \longrightarrow {\mathbb R}_{0}^{+}$ on $V$ we say that $V$ is a normed semi-vector space (or normed semi-space, for short). If the norm defines a metric on $V$ under which $V$ is complete then $V$ is said to be Banach semi-vector space. \end{definition} \begin{definition}\label{semiHilbert} Let $V$ be a semi-vector space over ${\mathbb R}_{0}^{+}$. If there exists an inner product $\langle \ , \ \rangle:V\times V \longrightarrow {\mathbb R}_{0}^{+}$ on $V$ then $V$ is an inner product semi-vector space (or inner product semi-space). If the inner product defines a metric on $V$ under which $V$ is complete then $V$ is said to be Hilbert semi-vector space. \end{definition} The well-known norms on ${\mathbb R}^n$ are also norms on $[{\mathbb R}_{0}^{+}]^{n}$, as we show in the next propositions. \begin{proposition}\label{R+1} Let $V = [{\mathbb R}_{0}^{+}]^{n}$ be the Euclidean semi-vector space (over ${\mathbb R}_{0}^{+}$) of semi-dimension $n$ . Define the function $\| \ \|:V \longrightarrow {\mathbb R}_{0}^{+}$ as follows: if $x = (x_1 , x_2 , \ldots ,$ $x_n ) \in V$, put $\| x \|=\sqrt{x_1^2 + x_2^2 + \ldots + x_n^2}$. Then $\| \ \|$ is a norm on $V$, called the Euclidean norm on $V$. \end{proposition} \begin{proof} It is clear that $\| x \| = 0$ if and only if $x=0$ and for all $\alpha \in {\mathbb R}_{0}^{+}$ and $x \in V$, $\| \alpha x \| = |\alpha | \| x \|$. To show the triangle inequality it is sufficient to apply the Cauchy-Schwarz inequality in ${\mathbb R}_{0}^{+}$: if $x = (x_1 , x_2 , \ldots , x_n )$ and $y = (y_1 , y_2 , \ldots , y_n )$ are semi-vectors in $V$ then $\displaystyle\sum_{i=1}^{n} x_i y_i \leq {\left(\displaystyle\sum_{i=1}^{n} x_i^2 \right)}^{1/2} \cdot {\left(\displaystyle\sum_{i=1}^{n} y_i^2 \right)}^{1/2}$. \end{proof} In the next result we show that the Euclidean norm on $[{\mathbb R}_{0}^{+}]^{n}$ generates the Euclidean metric on it. \begin{proposition}\label{R+1a} Let $x = (x_1 , x_2 , \ldots ,x_n )$, $y = (y_1 , y_2 , \ldots , y_n )$ be semi-vectors in $V = [{\mathbb R}_{0}^{+}]^{n}$. Define the function $d:V \times V \longrightarrow {\mathbb R}_{0}^{+}$ as follows: for every fixed $i$, if $x_i = y_i$ put $c_i =0$; if $x_i \neq y_i$, put ${\varphi}_i = {\psi}_i + c_i$, where ${\varphi}_i =\max \{x_i, y_i \}$ and ${\psi}_i =\min \{ x_i , y_i\}$ (in this case, $c_i > 0$); then consider $d(x, y) = \sqrt{c_1^2 + \ldots + c_n^2}$. The function $d$ is a metric on $V$. \end{proposition} \begin{remark} Note that in Proposition~\ref{R+1a} we could have defined $c_i$ simply by the nonnegative real number satisfying $\max \{x_i, y_i \}=\min \{x_i, y_i \} + c_i$. However, we prefer to separate the cases when $c_i=0$ and $c_i > 0$ in order to improve the readability of this paper. \end{remark} \begin{proof} It is easy to see that $d(x, y)=0$ if and only if $x=y$ and $d(x, y)=d(y,x)$. We will next prove the triangle inequality. To do this, let $x = (x_1 , x_2 , \ldots ,x_n )$, $y = (y_1 , y_2 , \ldots , y_n )$ and $z = (z_1 , z_2 , \ldots , z_n )$ be semi-vectors in $V = [{\mathbb R}_{0}^{+}]^{n}$. We look first at a fixed $i$. If $x_i = y_i = z_i$ or if two of them are equal then $d(x_i , z_i ) \leq d(x_i , y_i ) + d(y_i, z_i )$. Let us then assume that $x_i$, $y_i$ and $z_i$ are pairwise distinct. We have to analyze the six cases: $\operatorname{(1)}$ $x_i < y_i < z_i$; $\operatorname{(2)}$ $x_i < z_i < y_i$; $\operatorname{(3)}$ $y_i < x_i < z_i$; $\operatorname{(4)}$ $y_i < z_i < x_i$; $\operatorname{(5)}$ $z_i < x_i < y_i$; $\operatorname{(6)}$ $z_i < y_i < x_i$. In order to verify the triangle inequality we will see what occurs in the worst cases. More precisely, we assume that for all $i=1, 2, \ldots , n$ we have $x_i < y_i < z_i$ or, equivalently, $z_i < y_i < x_i$. Since both cases are analogous we only verify the (first) case $x_i < y_i < z_i$, for all $i$. In such cases there exist positive real numbers $a_i$, $b_i$, for all $i=1, 2, \ldots , n$, such that $y_i = x_i + a_i$ and $z_i = y_i + b_i$, which implies $z_i = x_i + a_i + b_i$. We need to show that $d(x, z) \leq d(x, y) + d(y, z)$, i.e., ${\left(\displaystyle\sum_{i=1}^{n}(a_i + b_i)^2\right)}^{1/2} \leq {\left(\displaystyle\sum_{i=1}^{n} a_i^2\right)}^{1/2} + {\left(\displaystyle\sum_{i=1}^{n} b_i^2\right)}^{1/2}$. The last inequality is equivalent to the inequality $\displaystyle\sum_{i=1}^{n} (a_i + b_i)^2 \leq \displaystyle\sum_{i=1}^{n} a_i^2 + \displaystyle\sum_{i=1}^{n} b_i^2 + 2{\left(\displaystyle\sum_{i=1}^{n} a_i^2 \right)}^{1/2} \cdot {\left(\displaystyle\sum_{i=1}^{n} b_i^2\right)}^{1/2}$. Again, the last inequality is equivalent to $\displaystyle\sum_{i=1}^{n} a_i b_i \leq {\left(\displaystyle\sum_{i=1}^{n} a_i^2\right)}^{1/2}\cdot {\left(\displaystyle\sum_{i=1}^{n} b_i^2\right)}^{1/2}$, which is the Cauchy-Schwarz inequality in ${\mathbb R}_{0}^{+}$. Therefore, $d$ satisfies the triangle inequality, hence it is a metric on $V$. \end{proof} \begin{remark} Note that Proposition~\ref{R+1a} means that the Euclidean norm on $[{\mathbb R}_{0}^{+}]^{n}$ (see Proposition~\ref{R+1}) generates the Euclidean metric on $[{\mathbb R}_{0}^{+}]^{n}$. This result is analogous to the fact that every norm defined on vector spaces generates a metric on it. Further, a semi-vector space $V$ is Banach (see Definition~\ref{semiBanach}) if the norm generates a metric under which every Cauchy sequence in $V$ converges to an element of $V$. \end{remark} \begin{proposition}\label{R+1b} Let $V = [{\mathbb R}_{0}^{+}]^{n}$ and define the function $\langle \ , \ \rangle:V\times V \longrightarrow {\mathbb R}_{0}^{+}$ as follows: if $u = (x_1 , x_2 , \ldots , x_n )$ and $v = (y_1 , y_2 , \ldots , y_n )$ are semi-vectors in $V$, put $\langle u , v \rangle = \displaystyle\sum_{i=1}^{n}x_i y_i$. Then $\langle \ , \ \rangle$ is an inner product on $V$, called dot product. \end{proposition} \begin{proof} The proof is immediate. \end{proof} \begin{proposition}\label{R+1c} The dot product on $V = [{\mathbb R}_{0}^{+}]^{n}$ generates the Euclidean norm on $V$. \end{proposition} \begin{proof} If $x= (x_1 , x_2 , \ldots , x_n ) \in V$, define the norm of $x$ by $\| x \|=\sqrt{\langle x, x\rangle}$. Note that the norm is exactly the Euclidean norm given in Proposition~\ref{R+1}. \end{proof} \begin{remark} We observe that if an inner product on a semi-vector space $V$ generates a norm $\| \ \|$ and such a norm generates a metric $d$ on $V$, then $V$ is a Hilbert space (according to Definition~\ref{semiHilbert}) if every Cauchy sequence in $V$ converges w.r.t. $d$ to an element of $V$. \end{remark} \begin{proposition}\label{R+2} Let $V = [{\mathbb R}_{0}^{+}]^{n}$ and define the function ${\| \ \|}_1:V \longrightarrow {\mathbb R}_{0}^{+}$ as follows: if $x = (x_1 , x_2 , \ldots ,$ $x_n ) \in V$, ${\| x \|}_1=\displaystyle\sum_{i=1}^{n} x_i$. Then ${\| x \|}_1$ is a norm on $V$. \end{proposition} \begin{proof} The proof is direct. \end{proof} \begin{proposition}\label{R+2a} Let $x = (x_1 , x_2 , \ldots ,x_n )$, $y = (y_1 , y_2 , \ldots , y_n )$ be semi-vectors in $V = [{\mathbb R}_{0}^{+}]^{n}$. Define the function $d_1:V \times V \longrightarrow {\mathbb R}_{0}^{+}$ in the following way. For every fixed $i$, if $x_i = y_i$, put $c_i =0$; if $x_i \neq y_i$, put ${\varphi}_i = {\psi}_i + c_i$, where ${\varphi}_i =\max \{x_i, y_i \}$ and ${\psi}_i =\min \{ x_i , y_i\}$. Let us consider that $d_1 (x, y) = \displaystyle\sum_{i=1}^{n} c_i $. Then the function $d_1$ is a metric on $V$ derived from the norm ${\| \ \|}_1$ shown in Proposition~\ref{R+2}. \end{proposition} \begin{proof} We only prove the triangle inequality. To avoid stress of notation, we consider the same that was considered in the proof of Proposition~\ref{R+1a}. We then fix $i$ and only investigate the worst case $x_i < y_i < z_i$. In this case, there exist positive real numbers $a_i$, $b_i$ for all $i=1, 2 , \ldots , n$, such that $y_i = x_i + a_i$ and $z_i = y_i + b_i$, which implies $z_i = x_i + a_i + b_i$. Then, for all $i$, $d_1 (x_i , z_i) \leq d_1 (x_i , y_i ) + d_1 (y_i , z_i)$; hence, $d_1 (x, z)=\displaystyle\sum_{i=1}^{n} d_1 (x_i , z_i) = \displaystyle\sum_{i=1}^{n} (a_i + b_i ) = \displaystyle\sum_{i=1}^{n} a_i + \displaystyle\sum_{i=1}^{n} b_i = \displaystyle\sum_{i=1}^{n} d_1 (x_i , y_i ) + \displaystyle\sum_{i=1}^{n} d_1 (y_i , z_i )= d_1 (x, y) + d_1 (y, z)$. Therefore, $d_1$ is a metric on $V$. \end{proof} \begin{proposition}\label{R+3} Let $V = [{\mathbb R}_{0}^{+}]^{n}$ be the Euclidean semi-vector space of semi-dimension $n$. Define the function ${\| \ \|}_2:V \longrightarrow {\mathbb R}_{0}^{+}$ as follows: if $x = (x_1 , x_2 , \ldots ,$ $x_n ) \in V$, take ${\| x \|}_2=\displaystyle\max_{i} \{ x_i \}$. Then ${\| x \|}_2$ is a norm on $V$. \end{proposition} \begin{proposition}\label{R+3a} Keeping the notation of Proposition~\ref{R+1a}, define the function $d_2:V \times V \longrightarrow {\mathbb R}_{0}^{+}$ such that $d_2 (x, y) = \max_{i} \{ c_i \}$. Then $d_2$ is a metric on $V$. Moreover, $d_2$ is obtained from the norm ${\| \ \|}_2$ exhibited in Proposition~\ref{R+3}. \end{proposition} \begin{proposition}\label{R+4} The norms $\| \ \|$, ${\| \ \|}_1$ and ${\| \ \|}_2$ shown in Propositions~\ref{R+1},~\ref{R+2} and \ref{R+3} are equivalent. \end{proposition} \begin{proof} It is immediate to see that ${\| \ \|}_2 \leq \| \ \| \leq {\| \ \|}_1 \leq n {\| \ \|}_2$. \end{proof} In a natural way we can define the norm of a bounded semi-linear transformation. \begin{definition}\label{semibounded} Let $V$ and $W$ be two normed semi-vector spaces and let $T:V \longrightarrow W$ be a semi-linear transformation. We say that $T$ is bounded if there exists a real number $c > 0$ such that $\| T(v)\|\leq c \| v \|$. \end{definition} If $T:V \longrightarrow W$ is bounded and $v \neq 0$ we can consider the quotient $\frac{\| T(v)\|}{\| v \|}$. Since such a quotient is upper bounded by $c$, the supremum $\displaystyle \sup_{v \in V, v\neq 0}\frac{\| T(v)\|}{\| v \|}$ exists and it is at most $c$. We then define $$\| T \|= \displaystyle\sup_{v \in V, v\neq 0}\frac{\| T(v)\|}{\| v \|}.$$ \begin{proposition}\label{R+5} Let $T: V \longrightarrow W$ be a bounded semi-linear transformation. Then the following hold: \begin{itemize} \item [ $\operatorname{(1)}$] $T$ sends bounded sets in bounded sets; \item [ $\operatorname{(2)}$] $\| T \|$ is a norm, called norm of $T$; \item [ $\operatorname{(3)}$] $\| T \|$ can be written in the form $\| T \|= \displaystyle\sup_{v \in V, \| v \| = 1 } \| T(v) \|$. \end{itemize} \end{proposition} \begin{proof} Items~$\operatorname{(1)}$~and~ $\operatorname{(2)}$ are immediate. The proof of Item~$\operatorname{(3)}$ is analogous to the standard proof but we present it here to guarantee that our mathematical tools are sufficient to perform it. Let $v\neq 0$ be a semi-vector with norm $\| v \|= a \neq 0$ and set $u=(1/a)v$. Thus, $\| u \| =1$ and since $T$ is semi-linear one has $$\| T \|= \displaystyle\sup_{v \in V, v\neq 0} \frac{1}{a}\| T(v)\|=\displaystyle\sup_{v \in V, v\neq 0} \| T( (1/a) v) \|= \displaystyle\sup_{u \in V, \| u \| =1} \| T(u)\|=$$ $= \displaystyle\sup_{v \in V, \| v \| =1} \| T(v)\|$. \end{proof} \subsubsection{The Semi-Spaces ${l}_{+}^{\infty}$, ${l}_{+}^{p}$ and ${\operatorname{C}}_{+}[a, b]$}\label{subsubsec1} In this subsection we investigate topological aspects of some semi-vector spaces over ${\mathbb R}_{0}^{+}$ such as completeness and separability. We investigate the sequence spaces ${l}_{+}^{\infty}$, ${l}_{+}^{p}$, ${\operatorname{C}}_{+}[a, b]$, which will be defined in the sequence. We first study the space ${l}_{+}^{\infty}$, the set of all bounded sequences of nonnegative real numbers. Before studying such a space we must define a metric on it, since the metric in $l^{\infty}$ which is defined as $ d(x, y)=\displaystyle\sup_{i \in {\mathbb N}} | {x}_i - {y}_i |$, where $x = ({x}_i )$ and $y = ({y}_i )$ are sequences in $l^{\infty}$, has no meaning to us, because there is no sense in considering $- {y}_i$ if ${y}_i > 0$. Based on this fact, we circumvent this problem by utilizing the total order of ${\mathbb R}$ according to Proposition~\ref{R+1a}. Let $x = ({\mu}_i )$ and $y = ({\nu}_i )$ be sequences in $l_{+}^{\infty}$. We then fix $i$, and define $c_i$ as was done in Proposition~\ref{R+1a}: if ${\mu}_i = {\nu}_i $ then we put $c_i = 0$; if ${\mu}_i \neq {\nu}_i $, let ${\gamma}_i=\max \{{\mu}_i , {\nu}_i \}$ and ${\psi}_i= \min \{{\mu}_i , {\nu}_i \}$; then there exists a positive real number $c_i$ such that ${\gamma}_i = {\psi}_i + c_i$ and, in place of $| {\mu}_i - {\nu}_i |$, we put $c_i$. Thus, our metric becomes \begin{eqnarray}\label{lmetric} d(x, y) = \displaystyle\sup_{i \in {\mathbb N}} \{c_i \}. \end{eqnarray} It is clear that $d(x, y)$ shown in Eq.~(\ref{lmetric}) defines a metric. However, we must show that the tools that we have are sufficient to proof this fact, once we are working on ${\mathbb R}_{0}^{+}$. \begin{proposition}\label{metricsup} The function $d$ shown in Eq.~(\ref{lmetric}) is a metric on ${l}_{+}^{\infty}$. \end{proposition} \begin{proof} It is clear that $d(x,y)\geq 0$ and $d(x,y)= 0 \Longleftrightarrow x=y$. Let $x = ({\mu}_i )$ and $y = ({\nu}_i )$ be two sequences in $l_{+}^{\infty}$. Then, for every fixed $i \in {\mathbb N}$, if $c_i= d({\mu}_i , {\nu}_i )=0$ then ${\mu}_i = {\nu}_i$, i.e., $d({\mu}_i , {\nu}_i )=d({\nu}_i , {\mu}_i )$. If $c_i > 0$ then $c_i= d({\mu}_i , {\nu}_i )$ is computed by ${\gamma}_i = {\psi}_i + c_i$, where ${\gamma}_i=\max \{{\mu}_i , {\nu}_i \}$ and ${\psi}_i= \min \{{\mu}_i , {\nu}_i \}$. Hence, $d({\nu}_i , {\mu}_i ) = c_i^{*}$ is computed by ${\gamma}_i^{*} = {\psi}_i^{*} + c_i^{*}$, where ${\gamma}_i^{*}=\max \{{\nu}_i, {\mu}_i \}$ and ${\psi}_i^{*}= \min \{{\nu}_i, {\mu}_i \}$, which implies $d({\mu}_i , {\nu}_i ) =d({\nu}_i , {\mu}_i )$. Taking the supremum over all $i$'s we have $d(x, y) = \displaystyle\sup_{i \in {\mathbb N}} \{c_i \}= \displaystyle\sup_{i \in {\mathbb N}} \{c_i^{*} \}=d(y, x)$. To show the triangle inequality, let $x = ({\mu}_i )$, $y = ({\nu}_i )$ and $z=({\eta}_i)$ be sequences in $l_{+}^{\infty}$. For every fixed $i$, we will prove that $d({\mu}_i , {\eta}_i )\leq d({\mu}_i , {\nu}_i ) + d({\nu}_i , {\eta}_i )$. If ${\nu}_i = {\mu}_i = {\eta}_i$, the result is trivial. If two of them are equal, the result is also trivial. Assume that ${\mu}_i$, ${\nu}_i$ and ${\eta}_i$ are pairwise distinct. As in the proof of Proposition~\ref{R+1a}, we must investigate the six cases:\\ $\operatorname{(1)}$ ${\mu}_i < {\nu}_i < {\eta}_i$; $\operatorname{(2)}$ ${\mu}_i < {\eta}_i < {\nu}_i$; $\operatorname{(3)}$ ${\nu}_i < {\mu}_i < {\eta}_i$; $\operatorname{(4)}$ ${\nu}_i < {\eta}_i < {\mu}_i$; $\operatorname{(5)}$ ${\eta}_i < {\mu}_i < {\nu}_i$; $\operatorname{(6)}$ ${\eta}_i < {\nu}_i < {\mu}_i$. We only show $\operatorname{(1)}$ and $\operatorname{(2)}$. To show $\operatorname{(1)}$, note that there exist positive real numbers $c_i$ and $c_i^{'}$ such that ${\nu}_i = {\mu}_i + c_i$ and ${\eta}_i = {\nu}_i + c_i^{'}$, which implies $\eta_i = \mu_i + c_i + c_i^{'}$. Hence, $d({\mu}_i , {\eta}_i )=c_i + c_i^{'}= d({\mu}_i , {\nu}_i ) + d({\nu}_i , {\eta}_i )$. Let us show $\operatorname{(2)}$. There exist positive real numbers $b_i$ and $b_i^{'}$ such that ${\eta}_i = {\mu}_i + b_i$ and ${\nu}_i={\eta}_i + b_i^{'}$, so ${\nu}_i = {\mu}_i + b_i + b_i^{'}$. Therefore, $d({\mu}_i , {\eta}_i )=b_i < d({\mu}_i , {\nu}_i ) + d({\nu}_i , {\eta}_i )=b_i + 2b_i^{'}$. Taking the supremum over all $i$'s we have $\displaystyle\sup_{i \in {\mathbb N}} \{d({\mu}_i , {\eta}_i ) \} \leq \displaystyle\sup_{i \in {\mathbb N}} \{d({\mu}_i , {\nu}_i )\} + \displaystyle\sup_{i \in {\mathbb N}} \{d({\nu}_i , {\eta}_i ) \}$, i.e., $d(x, z) \leq d(x, y) + d(y, z)$. Therefore, $d$ is a metric on ${l}_{+}^{\infty}$. \end{proof} \begin{definition}\label{defl} The metric space ${l}_{+}^{\infty}$ is the set of all bounded sequences of nonnegative real numbers equipped with the metric $d(x, y) = \displaystyle\sup_{i \in {\mathbb N}} \{c_i \}$ given previously. \end{definition} We prove that ${l}_{+}^{\infty}$ equipped with the previous metric is complete. \begin{theorem}\label{lcomplete} The space ${l}_{+}^{\infty}$ with the metric $d(x, y) = \displaystyle\sup_{i \in {\mathbb N}} \{c_i \}$ shown above is complete. \end{theorem} \begin{proof} The proof follows the same line as the standard proof of completeness of ${l}^{\infty}$; however it is necessary to adapt it to the metric (written above) in terms of nonnegative real numbers. Let $(x_n)$ be a Cauchy sequence in ${l}_{+}^{\infty}$, where $x_i = ({\eta}_{1}^{(i)}, {\eta}_{2}^{(i)}, \ldots )$. We must show that $(x_n )$ converges to an element of ${l}_{+}^{\infty}$. As $(x_n)$ is Cauchy, given $\epsilon > 0$, there exists a positive integer $K$ such that, for all $n, m > K$, $$d(x_n, x_m)=\displaystyle\sup_{j \in {\mathbb N}} \{c_j^{(n, m)} \} < \epsilon,$$ where $c_j^{(n, m)}$ is a nonnegative real number such that, if ${\eta}_{j}^{(n)}={\eta}_{j}^{(m)}$ then $c_j^{(n, m)}=0$, and if ${\eta}_{j}^{(n)} \neq {\eta}_{j}^{(m)}$ then $c_j^{(n, m)}$ is given by $\max \{{\eta}_{j}^{(n)}, {\eta}_{j}^{(m)}\} = \min \{{\eta}_{j}^{(n)}, {\eta}_{j}^{(m)}\} +c_j^{(n, m)}$. This implies that for each fixed $j$ one has \begin{eqnarray}\label{distCauchy1} c_j^{(n, m)} < \epsilon, \end{eqnarray} where $n, m > K$. Thus, for each fixed $j$, it follows that $({\eta}_{j}^{(1)}, {\eta}_{j}^{(2)}, \ldots )$ is a Cauchy sequence in ${\mathbb R}_{0}^{+}$. Since ${\mathbb R}_{0}^{+}$ is a complete metric space, the sequence $({\eta}_{j}^{(1)}, {\eta}_{j}^{(2)}, \ldots )$ converges to an element ${\eta}_{j}$ in ${\mathbb R}_{0}^{+}$. Hence, for each $j$, we form the sequence $x$ whose coordinates are the limits ${\eta}_{j}$, i.e., $x =({\eta}_{1}, {\eta}_{2}, {\eta}_{3}, \ldots )$. We must show that $x \in {l}_{+}^{\infty}$ and $x_n \longrightarrow x$. To show that $x$ is a bounded sequence, let us consider the number $c_j^{(n, \infty)}$ defined as follows: if ${\eta}_{j} = {\eta}_{j}^{(n)}$ then $c_j^{(n, \infty)}=0$, and if ${\eta}_{j} \neq {\eta}_{j}^{(n)}$, define $c_j^{(n, \infty)}$ be the positive real number satisfying $\max \{{\eta}_{j} , {\eta}_{j}^{(n)} \}= \min \{{\eta}_{j} , {\eta}_{j}^{(n)} \} + c_j^{(n, \infty)}$. From the inequality $(\ref{distCauchy1})$ one has \begin{eqnarray}\label{distCauchy2} c_j^{(n, \infty)}\leq\epsilon . \end{eqnarray} Because ${\eta}_{j} \leq {\eta}_{j}^{(n)} + c_j^{(n, \infty)}$ and since ${\eta}_{j}^{(n)} \in l_{+}^{\infty}$, it follows that ${\eta}_{j}$ is a bounded sequence for every $j$. Hence, $x = ({\eta}_{1}, {\eta}_{2}, {\eta}_{3}, \ldots ) \in {l}_{+}^{\infty}$. From $(\ref{distCauchy2})$ we have $$\displaystyle\sup_{j \in {\mathbb N}} \{c_j^{(n, \infty)} \} \leq \epsilon,$$ which implies that $x_n \longrightarrow x$. Therefore, $l_{+}^{\infty}$ is complete. \end{proof} Although $l_{+}^{\infty}$ is a complete metric space, it is not separable. \begin{theorem}\label{lnotsep} The space ${l}_{+}^{\infty}$ with the metric $d(x, y) = \displaystyle\sup_{i \in {\mathbb N}} \{c_i \}$ is not separable. \end{theorem} \begin{proof} The proof is the same as shown in \cite[1.3-9]{Kreyszig:1978}, so it is omitted. \end{proof} Let us define the space analogous to the space $l^p$. \begin{definition}\label{deflp} Let $p \geq 1$ be a fixed real number. The set ${l}_{+}^{p}$ consists of all sequences $x =({\eta}_{1}, {\eta}_{2}, {\eta}_{3}, \ldots )$ of nonnegative real numbers such that $\displaystyle\sum_{i=1}^{\infty} ({\eta}_{i})^{p} < \infty$, whose metric is defined by $ d(x, y)={\left[\displaystyle\sum_{i=1}^{\infty} {[c_{i}]}^{p}\right]}^{1/p}$, where $y =({\mu}_{1}, {\mu}_{2}, {\mu}_{3}, \ldots )$ and $c_i$ is defined as follows: $c_i = 0$ if ${\mu}_i = {\eta}_i $, and if ${\mu}_i > {\eta}_i$ (respect. ${\eta}_i > {\mu}_i$) then $c_i > 0$ is such that ${\mu}_i = {\eta}_i + c_i$. \end{definition} \begin{theorem}\label{lp+complete} The space ${l}_{+}^{p}$ with the metric $ d(x,y)= {\left[\displaystyle\sum_{i=1}^{\infty} {[c_{i}]}^{p}\right]}^{1/p}$ exhibited above is complete. \end{theorem} \begin{proof} Recall that given two sequences $({\mu}_i)$ and $({\eta}_i )$ in ${l}_{+}^{p}$ the Minkowski inequality for sums reads as \begin{eqnarray*} {\left[\displaystyle\sum_{i=1}^{\infty} {|{\mu}_i + {\eta}_i |}^{p}\right]}^{1/p} \leq {\left[\displaystyle \sum_{j=1}^{\infty} {|{\mu}_j|}^{p}\right]}^{1/p} + {\left[\displaystyle \sum_{k=1}^{\infty} {|{\eta}_k|}^{p}\right]}^{1/p}. \end{eqnarray*} Applying the Minkowski inequality as per \cite[1.5-4]{Kreyszig:1978} with some adaptations, it follows that $d(x,y)$ is, in fact, a metric. In order to prove the completeness of ${l}_{+}^{p}$, we proceed similarly as in the proof of Theorem~\ref{lcomplete} with some adaptations. The main adaptation is performed according to the proof of completeness of $l^p$ in \cite[1.5-4]{Kreyszig:1978} replacing the last equality $x=x_m +( x - x_m) \in l^p$ (after Eq.~(5)) by two equalities in order to avoid negative real numbers. \begin{enumerate} \item [ $\operatorname{(1)}$] If the $i$-th coordinate $x^{(i)}- x_{m}^{(i)}$ of the sequence $x- x_m$ is positive, then define $c_{m}^{(i)} = x^{(i)}- x_{m}^{(i)}$ and write $x^{(i)} = x_{m}^{(i)} + c_{m}^{(i)}$. From Minkowski inequality, it follows that the sequence $(x^{(i)})_i$ is in $l_{+}^{p}$. \item [ $\operatorname{(2)}$] If $x^{(j)}- x_{m}^{(j)}$ is negative, then define $c_{m}^{(j)}= x_{m}^{(j)} - x^{(j)}$ and write $x_{m}^{(j)}= x^{(j)} + c_{m}^{(j)} $. Since $x_m \in l_{+}^{p}$, from the comparison criterion for positive series it follows that the sequence $(x^{(j)})_j$ is also in $l_{+}^{p}$. \end{enumerate} \end{proof} \begin{theorem}\label{lp+separable} The space ${l}_{+}^{p}$ is separable. \end{theorem} \begin{proof} The proof follows the same line of \cite[1.3-10]{Kreyszig:1978}. \end{proof} \begin{definition}\label{continon[a,b]} Let $I=[a, b]$ be a closed interval in ${\mathbb R}_{0}^{+}$, where $a\geq 0$ and $a < b$. Then ${\operatorname{C}}_{+}[a, b]$ is the set of all continuous nonnegative real valued functions on $I=[a, b]$, whose metric is defined by $d(f(t), g(t)) = \displaystyle\max_{t \in I} \{c(t)\}$, where $c(t)$ is given by $\max \{ f(t), g(t) \} =\min \{ f(t), g(t) \} + c(t)$. \end{definition} \begin{theorem}\label{cont[a,b]complete} The metric space $({\operatorname{C}}_{+}[a, b], d)$, where $d$ is given in Definition~\ref{continon[a,b]}, is complete. \end{theorem} \begin{proof} The proof follows the same lines as the standard one with some modifications. Let $(f_{m})$ be a Cauchy sequence in ${\operatorname{C}}_{+}[a, b]$. Given $\epsilon > 0$ there exists a positive integer $N$ such that, for all $m, n > N$, it follows that \begin{eqnarray}\label{In1} d(f_{m} , f_{n}) = \displaystyle\max_{t \in I} \{c_{m, n} (t)\} < \epsilon, \end{eqnarray} where $\max \{ f_{m} (t) , f_{n} (t) \} = \min \{ f_{m} (t) , f_{n} (t) \} + c_{m, n}(t)$. Thus, for any fixed $t_0 \in I$ we have $c_{m, n} (t_0 ) < \epsilon$, for all $m, n > N$. This means that $(f_1 (t_0 ), f_2 (t_0 ), \ldots )$ is a Cauchy sequence in ${\mathbb R}_{0}^{+}$, which converges to $f(t_0 )$ when $m \longrightarrow \infty$ since ${\mathbb R}_{0}^{+}$ is complete. We then define a function $f: [a, b] \longrightarrow {\mathbb R}_{0}^{+}$ such that for each $t \in [a, b]$, we put $f(t)$. Taking $n \longrightarrow \infty$ in (\ref{In1}) we obtain $\displaystyle\max_{t \in I} \{c_{m} (t)\} \leq \epsilon$ for all $m > N$, where $\max \{ f_{m} (t) , f(t) \} = \min \{ f_{m} (t) , f(t) \} + c_{m}(t)$, which implies $c_{m}(t)\leq \epsilon$ for all $t \in I$. This fact means that $(f_{m}(t))$ converges to $f(t)$ uniformly on $I$, i.e., $f \in {\operatorname{C}}_{+}[a, b]$ because the functions $f_{m}$'s are continuous on $I$. Therefore, ${\operatorname{C}}_{+}[a, b]$ is complete, as desired. \end{proof} \subsection{Interesting Semi-Vector Spaces}\label{subsec2} In this section we exhibit semi-vector spaces over $K= {\mathbb R}_{0}^{+}$ derived from semi-metrics, semi-metric-preserving functions, semi-norms, semi-inner products and sub-linear functionals. \begin{theorem}\label{teo1} Let $X$ be a semi-metric space and ${ \mathcal M}_{X}=\{ d: X \times X\longrightarrow {\mathbb R}; d$ $\operatorname{is \ a \ semi-metric \ on} X\}$. Then $({ \mathcal M}_{X}, +, \cdot )$ is a semi-vector space over ${\mathbb R}_{0}^{+}$, where $+$ and $\cdot$ are the addition and the scalar multiplication (in ${\mathbb R}_{0}^{+}$) pointwise, respectively. \end{theorem} \begin{proof} We first show that ${ \mathcal M}_{X}$ is closed under addition. Let $d_1 , d_2 \in { \mathcal M}_{X}$ and set $d:= d_1 + d_2$. It is clear that $d$ is nonnegative real-valued function. Moreover, for all $x, y \in X$, $d(x, y) = d(y, x)$. Let $x \in X$; $d(x, x) = d_1(x, x) + d_2 (x,x) =0$. For all $x, y, z \in X$, $d(x, z)=d_1 (x, z) + d_2 (x, z)\leq [d_1 (x, y) + d_2 (x, y)]+ [d_1 (y, z) + d_2 (y, z)]= d(x, y) + d(y, z)$. Let us show that ${ \mathcal M}_{X}$ is closed under scalar multiplication. Let $d_1 \in { \mathcal M}_{X}$ and define $d = \lambda d_1$, where $\lambda \in {\mathbb R}_{0}^{+}$. It is clear that $d$ is real-valued nonnegative and for all $x, y \in X$, $d(x, y)=d(y, x)$. Moreover, if $x \in X$, $d(x, x)=0$. For all $x, y, z \in X$, $d(x, z)=\lambda d_1 (x, z)\leq \lambda [d_1 (x, y) + d_1 (y, z)]= d(x, y) + d(y, z)$. This means that ${ \mathcal M}_{X}$ is closed under scalar multiplication. It is easy to see that $({ \mathcal M}_{X}, +, \cdot )$ satisfies the other conditions of Definition~\ref{defSVS}. \end{proof} Let $(X, d)$ be a metric space. In~\cite{Corazza:1999}, Corazza investigated interesting functions $f:{\mathbb R}_{0}^{+}\longrightarrow {\mathbb R}_{0}^{+}$ such that the composite of $f$ with $d$, i.e., $X \times X \xrightarrow{d} {{\mathbb R}_{0}^{+}} \xrightarrow{f} {{\mathbb R}_{0}^{+}}$ also generates a metric on $X$. Let us put this concept formally. \begin{definition}\label{metricprese} Let $f:{\mathbb R}_{0}^{+}\longrightarrow {\mathbb R}_{0}^{+}$ be a function. We say that $f$ is metric-preserving if for all metric spaces $(X, d)$, the composite $f \circ d$ is a metric. \end{definition} To our purpose we will consider semi-metric preserving functions as follows. \begin{definition}\label{semi-metricprese} Let $f:{\mathbb R}_{0}^{+}\longrightarrow {\mathbb R}_{0}^{+}$ be a function. We say that $f$ is semi-metric-preserving if for all semi-metric spaces $(X, d)$, the composite $f \circ d$ is a semi-metric. \end{definition} We next show that the set of semi-metric preserving functions has a semi-vector space structure. \begin{theorem}\label{teo1a} Let ${ \mathcal F}_{pres}=\{ f:{\mathbb R}_{0}^{+}\longrightarrow {\mathbb R}_{0}^{+}; f \operatorname{is \ semi-metric \ preserving} \}$. Then $({ \mathcal F}_{pres}, +, \cdot )$ is a semi-vector space over ${\mathbb R}_{0}^{+}$, where $+$ and $\cdot$ are the addition and the scalar multiplication (in ${\mathbb R}_{0}^{+}$) pointwise, respectively. \end{theorem} \begin{proof} We begin by showing that ${ \mathcal F}_{pres}$ is closed under addition and scalar multiplication pointwise. Let $f, g \in { \mathcal F}_{pres}$. Given a semi-metric space $(X, d)$, we must prove that $(f + g)\circ d$ is also semi-metric preserving. We know that $[(f + g)\circ d] (x, y ) \geq 0$ for all $x, y \in X$. Let $x \in X$; then $[(f + g)\circ d ](x, x )= f(d(x, x)) + g (d(x, x)) = 0$. It is clear that $[(f + g ) \circ d](x, y)= [(f + g ) \circ d](y, x)$. Let $x, y, z \in X$. One has: $[(f + g ) \circ d](x, y)= f(d(x, y)) + g(d(x, y))\leq [f(d(x, z))+ g(d(x, z))]+ [f(d(z, y))+ g(d(z, y))]= (f + g)(d(x, z)) + (f + g)(d(z, y))= [(f + g)\circ d](x, z) + [(f + g)\circ d](z, y) $. Here, we show that for each $f \in { \mathcal F}_{pres}$ and $ \alpha \in {\mathbb R}_{0}^{+}$, it follows that $ \alpha f \in { \mathcal F}_{pres}$. We show only the triangular inequality since the other conditions are immediate. Let us calculate: $[\alpha f \circ d](x, y)= \alpha f (d(x, y))\leq \alpha f (d(x, z)) + \alpha f (d(z, y)) = [\alpha f \circ d](x, z) + [\alpha f \circ d](z, y)$. The null vector is the null function $0_{f}:{\mathbb R}_{0}^{+}\longrightarrow {\mathbb R}_{0}^{+}$. The other conditions are easy to verify. \end{proof} \begin{theorem}\label{teo2} Let $V$ be a semi-normed real vector space and ${ \mathcal N}_{V}= \{ \| \ \|: V\longrightarrow {\mathbb R}; \| \ \|$ $\operatorname{is \ a \ semi-norm \ on} V\}$. Then $({ \mathcal N}_{V}, +, \cdot )$ is a semi-vector space over ${\mathbb R}_{0}^{+}$, where $+$ and $\cdot$ are addition and scalar multiplication (in ${\mathbb R}_{0}^{+}$) pointwise, respectively. \end{theorem} \begin{proof} From hypotheses, ${ \mathcal N}_{V}$ is non-empty. Let ${\| \ \|}_{1} , {\| \ \|}_{2} \in { \mathcal N}_{V}$ and set $\| \ \|:= {\| \ \|}_{1} + {\| \ \|}_{2}$. For all $v \in V$, $\| v \|\geq 0$. If $v \in V$ and $\alpha \in {\mathbb R}$ then $\| \alpha v \|=|\alpha| \| v \|$. For every $u, v \in V$, it follows that $\| u + v \|:= {\| u + v \|}_{1} + {\| u + v \|}_{2}\leq ({ \| u \|}_{1} + {\| u \|}_{2} ) + ({\| v \|}_{1} + {\| v \|}_{2})= \| u \| + \| v \|$. Hence, ${ \mathcal N}_{V}$ is closed under addition. We next show that ${ \mathcal N}_{V}$ is closed under scalar multiplication. Let ${\| \ \|}_{1} \in { \mathcal N}_{V}$ and define $\| \ \|:= \lambda {\| \ \|}_{1}$, where $\lambda \in {\mathbb R}_{0}^{+}$. For all $v \in V$, $\| v \|\geq 0$. If $\alpha \in {\mathbb R}$ and $ v \in V$, $ \| \alpha v \|= |\alpha |( \lambda {\| v\|}_{1})= |\alpha | \| v \|$. Let $u, v \in V$. Then $\| u + v \|\leq \lambda {\| u \|}_{1}+ \lambda {\| v \|}_{1}=\|u\| + \|v\|$. Therefore, ${ \mathcal N}_{V}$ is closed under addition and scalar multiplication over ${\mathbb R}_{0}^{+}$. The zero vector is the null function $ \textbf{0}: V \longrightarrow {\mathbb R}$. The other conditions of Definition~\ref{defSVS} are straightforward. \end{proof} \begin{remark} Note that ${ \mathcal N}_{V}^{\diamond}=\{\| \ \|: V\longrightarrow {\mathbb R}; \| \ \|$ $\operatorname{is \ a \ norm \ on} V\}$ is also closed under both function addition and scalar multiplication pointwise. \end{remark} \begin{lemma}\label{prop1} Let $T:V\longrightarrow W$ be a linear transformation. \begin{itemize} \item [ $\operatorname{(1)}$] If $\| \ \|:W\longrightarrow {\mathbb R}$ is a semi-norm on $W$ then $\| \ \|\circ T: V \longrightarrow {\mathbb R}$ is a semi-norm on $V$. \item [ $\operatorname{(2)}$] If $T$ is injective linear and $\| \ \|: W\longrightarrow {\mathbb R}$ is a norm on $W$ then $\| \ \|\circ T$ is a norm on $V$. \end{itemize} \end{lemma} \begin{proof} We only show Item~$\operatorname{(1)}$. It is clear that $[\| \ \|\circ T](v) \geq 0$ for all $v \in V$. For all $\alpha \in {\mathbb R}$ and $v \in V$, $[\| \ \|\circ T](\alpha v)= | \alpha | \| T(v) \| = | \alpha | [\| \ \|\circ T](v)$. Moreover, $ \forall \ v_1 , v_2 \in V$, $[\| \ \|\circ T](v_1 + v_2)\leq [\| \ \|\circ T](v_1 )+ [\| \ \|\circ T](v_2 )$. Therefore, $\| \ \|\circ T$ is a semi-norm on $V$. \end{proof} \begin{theorem}\label{teo2a} Let $V$ and $W$ be two semi-normed vector spaces and $T:V\longrightarrow W$ be a linear transformation. Then $${ \mathcal N}_{V_{T}}=\{ \| \ \| \circ T: V\longrightarrow {\mathbb R}; \| \ \| \operatorname{is \ a \ semi-norm \ on} W\}$$ is a semi-subspace of $({ \mathcal N}_{V}, +, \cdot )$. \end{theorem} \begin{proof} From hypotheses, it follows that ${ \mathcal N}_{V_{T}}$ is non-empty. From Item~$\operatorname{(1)}$ of Lemma~\ref{prop1}, it follows that $\| \ \|\circ T$ is a semi-norm on $V$. Let $f, g \in { \mathcal N}_{V_{T}}$, i.e., $f = {\| \ \|}_1 \circ T$ and $g = {\| \ \|}_2 \circ T$, where ${\| \ \|}_1$ and ${\| \ \|}_2$ are semi-norms on $W$. Then $f + g = [ {\| \ \|}_1 + {\| \ \|}_2 ]\circ T \in { \mathcal N}_{V_{T}}$. For every nonnegative real number $\lambda$ and $f \in { \mathcal N}_{V_{T}}$, $\lambda f = \lambda [ \| \ \|\circ T] = (\lambda \| \ \| )\circ T \in { \mathcal N}_{V_{T}}$. \end{proof} \begin{theorem}\label{teo2b} Let ${\mathcal N}$ be the class whose members are $\{{ \mathcal N}_{V}\}$, where the ${ \mathcal N}_{V}$ are given in Theorem~\ref{teo2}. Let $\operatorname{Hom}({\mathcal N})$ be the class whose members are the sets $$\operatorname{hom}({ \mathcal N}_{V}, { \mathcal N}_{W})=\{ F_T:{ \mathcal N}_{V}\longrightarrow { \mathcal N}_{W}; F_T ( {\| \ \|}_{V})= {\| \ \|}_{V} \circ T\},$$ where $T: W \longrightarrow V$ is a linear transformation and ${\| \ \|}_{V}$ is a semi-norm on $V$. Then $({\mathcal N}, \operatorname{Hom}({\mathcal N}), Id, \circ )$ is a category. \end{theorem} \begin{proof} The sets $\operatorname{hom}({ \mathcal N}_{V}, { \mathcal N}_{W})$ are pairwise disjoint. For each ${ \mathcal N}_{V}$, there exists $Id_{({ \mathcal N}_{V})}$ given by $Id_{({ \mathcal N}_{V})} ({\| \ \|}_{V})={\| \ \|}_{V}={\| \ \|}_{V}\circ Id_{(V)}$. It is clear that if ${F}_{T}:{ \mathcal N}_{V}\longrightarrow { \mathcal N}_{W}$ then ${F}_{T}\circ Id_{({ \mathcal N}_{V})} = {F}_{T}$ and $Id_{({ \mathcal N}_{W})}\circ {F}_{T} = {F}_{T}$. It is easy to see that for every $T:W\longrightarrow V$ linear transformation, the map $F_{T}$ is semi-linear, i.e., $F_{T}({\| \ \|}_{V}^{(1)} + {\| \ \|}_{V}^{(2)})= F_{T}({\| \ \|}_{V}^{(1)}) + F_{T}({\| \ \|}_{V}^{(2)})$ and $F_{T}(\lambda {\| \ \|}_{V})= \lambda F_{T}({\| \ \|}_{V})$, for every ${\| \ \|}_{V}, {\| \ \|}_{V}^{(1)}, {\| \ \|}_{V}^{(2)} \in { \mathcal N}_{V}$ and $\lambda \in {\mathbb R}_{0}^{+}$. Let ${ \mathcal N}_{U}, { \mathcal N}_{V}, { \mathcal N}_{W}, { \mathcal N}_{X} \in {\mathcal N}$ and $F_{T_1} \in \operatorname{hom}({ \mathcal N}_{U}, { \mathcal N}_{V})$, $F_{T_2} \in \operatorname{hom}({ \mathcal N}_{V}, { \mathcal N}_{W})$, $F_{T_3} \in \operatorname{hom}({ \mathcal N}_{W}, { \mathcal N}_{X})$, i.e., $${ \mathcal N}_{U}\xrightarrow{F_{T_1}} { \mathcal N}_{V}\xrightarrow{F_{T_2}} { \mathcal N}_{W} \xrightarrow{F_{T_3}} { \mathcal N}_{X}.$$ The linear transformations are of the forms $$X\xrightarrow{T_3} W\xrightarrow{T_2} V \xrightarrow{T_1} U \xrightarrow{{\| \ \|}_{U}} {\mathbb R}.$$ The associativity $(F_{T_3}\circ F_{T_2})\circ F_{T_1}=F_{T_3}\circ (F_{T_2}\circ F_{T_1})$ follows from the associativity of composition of maps. Moreover, the map $F_{T_3}\circ F_{T_2}\circ F_{T_1} \in \operatorname{Hom}({\mathcal N})$ because $F_{T_3}\circ F_{T_2}\circ F_{T_1} = ({\| \ \|}_{U})\circ (T_1\circ T_2\circ T_3)$ and $T_1\circ T_2\circ T_3$ is a linear transformation. Therefore, $({\mathcal N}, \operatorname{Hom}({\mathcal N}), Id, \circ )$ is a category, as required. \end{proof} \begin{theorem}\label{teo3} Let $V$ be a real vector space endowed with a semi-inner product and let ${ \mathcal P}_{V}=\{ \langle \ , \ \rangle: V\times V\longrightarrow {\mathbb R}; \langle \ , \ \rangle$ $\operatorname{is \ a \ semi-inner \ product \ on} V\}$. Then $({ \mathcal P}_{V}, +, \cdot )$ is a semi-vector space over ${\mathbb R}_{0}^{+}$, where $+$ and $\cdot$ are addition and scalar multiplication (in ${\mathbb R}_{0}^{+}$) pointwise, respectively. \end{theorem} \begin{proof} The proof is analogous to that of Theorems~\ref{teo1}~and~\ref{teo2}. \end{proof} \begin{proposition}\label{prop2} Let $V, W$ be two vector spaces and $T_1 , T_2:V\longrightarrow W$ be two linear transformations. Let us consider the map $T_1 \times T_2 : V \times V \longrightarrow W\times W$ given by $T_1 \times T_2 (u, v) = (T_1(u), T_2 (v))$. If $\langle \ , \ \rangle$ is a semi-inner product on $W$ then $\langle \ , \ \rangle \circ T_1 \times T_2$ is a semi-inner product on $V$. \end{proposition} \begin{proof} The proof is immediate, so it is omitted. \end{proof} Let $V$ be a real vector space. Recall that a sub-linear functional on $V$ is a functional $t: V\longrightarrow {\mathbb R}$ which is sub-additive: $\forall \ u, v \in V$, $t(u + v)\leq t(u) + t(v)$; and positive-homogeneous: $\forall \ \alpha \in {\mathbb R}_{0}^{+}$ and $\forall \ v \in V$, $t(\alpha v ) =\alpha t(v)$. \begin{theorem}\label{teo4} Let $V$ be a real vector space. Let us consider ${ \mathcal S}_{V}= \{ S: V\longrightarrow {\mathbb R};$ $S \operatorname{is} \operatorname{sub-linear} \operatorname{on} V\}$. Then $({ \mathcal S}_{V}, +, \cdot )$ is a semi-vector space on ${\mathbb R}_{0}^{+}$, where $+$ and $\cdot$ are addition and scalar multiplication (in ${\mathbb R}_{0}^{+}$) pointwise, respectively. \end{theorem} \begin{proof} The proof follows the same line of that of Theorems~\ref{teo1}~and~\ref{teo2}~and~\ref{teo3}. \end{proof} \subsection{Semi-Algebras}\label{subsec4} We start this section by recalling the definition of semi-algebra and semi-sub-algebra. For more details the reader can consult \cite{Gahler:1999}. In \cite{Olivier:1995}, Olivier and Serrato investigated relation semi-algebras, i.e., a semi-algebra being both a Boolean algebra and an involutive semi-monoid, satisfying some conditions (see page 2 in Ref.~\cite{Olivier:1995} for more details). Roy \cite{Roy:1970} studied the semi-algebras of continuous and monotone functions on compact ordered spaces. \begin{definition}\label{semialgebra} A semi-algebra $A$ over a semi-field $K$ (or a $K$-semi-algebra) is a semi-vector space $A$ over $K$ endowed with a binary operation called multiplication of semi-vectors $\bullet: A \times A\longrightarrow A$ such that, $\forall \ u, v, w \in A$ and $\lambda \in K$: \begin{itemize} \item [ $\operatorname{(1a)}$] $ u \bullet (v + w)= (u \bullet v) + (u \bullet w)$ (left-distributivity); \item [ $\operatorname{(1b)}$] $ (u + v)\bullet w= (u \bullet w) + (v \bullet w)$ (right-distributivity); \item [ $\operatorname{(2)}$] $ \lambda (u \bullet v)= (\lambda u)\bullet v = u \bullet (\lambda v)$. \end{itemize} \end{definition} A semi-algebra $A$ is \emph{associative} if $(u\bullet v)\bullet w=u\bullet (v\bullet w)$ for all $u, v, w \in A$; $A$ is said to be \emph{commutative} (or abelian) is the multiplication is commutative, that is, $\forall \ u, v \in A$, $u\bullet v= v\bullet u$; $A$ is called a semi-algebra with identity if there exists an element $1_A \in A$ such that $\forall \ u \in A$, $1_A \bullet u = u \bullet 1_A =u$; the element $1_A $ is called identity of $A$. The identity element of a semi-algebra $A$ is unique (if exists). If $A$ is a semi-free semi-vector space then the dimension of $A$ is its dimension regarded as a semi-vector space. A semi-algebra is \emph{simple} if it is simple as a semi-vector space. \begin{example}\label{ex5} The set ${\mathbb R}_{0}^{+}$ is a commutative semi-algebra with identity $e=1$. \end{example} \begin{example}\label{ex6} The set of square matrices of order $n$ whose entries are in ${\mathbb R}_{0}^{+}$, equipped with the sum of matrices, multiplication of a matrix by a scalar (in ${\mathbb R}_{0}^{+}$, of course) and by multiplication of matrices is an associative and non-commutative semi-algebra with identity $e=I_{n}$ (the identity matrix of order $n$), over ${\mathbb R}_{0}^{+}$. \end{example} \begin{example}\label{ex7} The set ${\mathcal P}_{n}[x]$ of polynomials with coefficients from ${\mathbb R}_{0}^{+}$ and degree less than or equal to $n$, equipped with the usual of polynomial sum and scalar multiplication is a semi-vector space. \end{example} \begin{example}\label{ex8} Let $V$ be a semi-vector space over a semi-field $K$. Then the set ${\mathcal L}(V, V)=\{T:V\longrightarrow V; T \operatorname{is \ a \ semi-linear \ operator}\}$ is a semi-vector space. If we define a vector multiplication as the composite of semi-linear operators (which is also semi-linear) then we have a semi-algebra over $K$. \end{example} \begin{definition}\label{subsemialgebra} Let $A$ be a semi-algebra over $K$. We say that a non-empty set $S \subseteq A$ is a semi-subalgebra if $S$ is closed under the operations of $A$, that is, \begin{itemize} \item [ $\operatorname{(1)}$] $\forall \ u, v \in A$, $u + v \in A$; \item [ $\operatorname{(2)}$] $\forall \ u, v \in A$, $u \bullet v \in A$; \item [ $\operatorname{(3)}$] $\forall \ \lambda \in K$ and $\forall u \in A$, $\lambda u \in A$. \end{itemize} \end{definition} \begin{definition}\label{A-homomorphism} Let $A$ and $B$ two semi-algebras over $K$. We say that a map $T:A\longrightarrow B$ is an $K$-semi-algebra homomorphism if, $\forall \ u, v \in A$ and $\lambda \in K$, the following conditions hold: \begin{itemize} \item [ $\operatorname{(1)}$] $T(u + v) = T(u) + T(v)$; \item [ $\operatorname{(2)}$] $T(u \bullet v) = T(u) \bullet T(v)$; \item [ $\operatorname{(3)}$] $T(\lambda v ) = \lambda T(v)$. \end{itemize} \end{definition} Definition~\ref{A-homomorphism} means that $T$ is both a semi-ring homomorphism and also semi-linear (as semi-vector space). \begin{definition}\label{isomorphic} Let $A$ and $B$ be two $K$-semi-algebras. A $K$-semi-algebra isomorphism $T:A \longrightarrow B$ is a bijective $K$-semi-algebra homomorphism. If there exists such an isomorphism, we say that $A$ is isomorphic to $B$, written $A\cong B$. \end{definition} The following results seems to be new, because semi-algebras over ${\mathbb R}_{0}^{+}$ are not much investigated in the literature. \begin{proposition}\label{propalghomo} Assume that $A$ and $B$ are two $K$-semi-algebras, where $K={\mathbb R}_{0}^{+}$ and $A$ has identity $1_A$. Let $T:A \longrightarrow B$ be a $K$-semi-algebra homomorphism. Then the following properties hold: \begin{itemize} \item [ $\operatorname{(1)}$] $T(0_A)= 0_B$; \item [ $\operatorname{(2)}$] If $ u\in A$ is invertible then its inverse is unique and $(u^{-1})^{-1}= u$; \item [ $\operatorname{(3)}$] If $T$ is surjective then $T(1_A) = 1_B$, i.e., $B$ also has identity; furthermore, $T(u^{-1})= [T(u)]^{-1}$; \item [ $\operatorname{(4)}$] If $u, v \in A$ are invertible then $(u\bullet v )^{-1}= v^{-1}\bullet u^{-1}$; \item [ $\operatorname{(5)}$] the composite of $K$-semi-algebra homomorphisms is also a $K$-semi-algebra homomorphism; \item [ $\operatorname{(6)}$] if $T$ is a $K$-semi-algebra isomorphism then also is $T^{-1}:B \longrightarrow A$. \item [ $\operatorname{(7)}$] the relation $A \sim B$ if and only if $A$ is isomorphic to $B$ is an equivalence relation. \end{itemize} \end{proposition} \begin{proof} Note that Item~$\operatorname{(1)}$ holds because the additive cancelation law holds in the definition of semi-vector spaces (see Definition\ref{defSVS}). We only show Item $\operatorname{(3)}$ since the remaining items are direct. Let $v \in B$; then there exists $u \in A$ such that $T(u)=v$. It then follows that $v \bullet T(1_A )= T(u\bullet 1_A)=v$ and $T(1_A ) \bullet v = T(1_A \bullet u)=v$; which means that $T(1_A)$ is the identity of $B$, i.e., $T(1_A) = 1_B$. We have: $T(u) \bullet T(u^{-1})= T( u \bullet u^{-1})=T(1_A)=1_B$ and $T(u^{-1}) \bullet T(u)= T( u^{-1} \bullet u)=T(1_A)=1_B$, which implies $T(u^{-1})= [T(u)]^{-1}$. \end{proof} \begin{proposition}\label{associunitsemi} If $A$ is a $K$-semi-algebra with identity $1_A$ then $A$ can be embedded in ${\mathcal L}(A, A)$, the semi-algebra of semi-linear operators on $A$. \end{proposition} \begin{proof} For every fixed $v \in A$, define $v^{*}:A \longrightarrow A$ as $v^{*}(x) = v\bullet x$. It is easy to see that $v^{*}$ is a semi-linear operator on $A$. Define $h: A \longrightarrow {\mathcal L}(A, A)$ by $h(v)= v^{*}$. We must show that $h$ is a injective $K$-semi-algebra homomorphism where the product in ${\mathcal L}(A, A)$ is the composite of maps from $A$ into $A$. Fixing $u, v \in A$, we have: $[h(u + v)](x)= (u + v)^{*}(x)= (u + v)\bullet x = u\bullet x + v \bullet x = u^{*}(x) + v^{*}(x) = [h(u)](x) + [h(v)](x)$, hence $h(u + v)= h(u) + h(v)$. For $\lambda \in K$ and $v \in A$, it follows that $[h(\lambda v)](x) = (\lambda v)^{*}(x)= (\lambda v)x = \lambda (vx)= [\lambda h(v)](x)$, i.e., $h(\lambda v)= \lambda h(v)$. For fixed $u, v \in A$, $[h(u\bullet v)](x)= (u\bullet v)^{*}(x)= (u\bullet v)\bullet x = u\bullet (v\bullet x)=u\bullet v^{*}(x)=u^{*}(v^{*}(x))=[h(u)\circ h(v)](x)$, i.e., $h(u\bullet v)= h(u) \circ h(v)$. Assume that $h(u)=h(v)$, that is, $u^{*}=v^{*}$; hence, for every $x \in A$, $u^{*}(x) = v^{*}(x)$, i.e., $u\bullet x = v\bullet x$ . Taking in particular $x=1_A$, it follows that $u = v$, which implies that $h$ is injective. Therefore, $A$ is isomorphic to $h(A)$, where $h(A)\subseteq {\mathcal L}(A, A)$. \end{proof} \begin{definition}\label{semi-Liesemialgebra} Let $A$ be a semi-vector space over a semi-field $K$. Then $A$ is said to be a Lie semi-algebra if $A$ is equipped with a product $[ \ , \ ]: A \times A\longrightarrow A$ such that the following conditions hold: \begin{itemize} \item [ $\operatorname{(1)}$] $[ \ , \ ]$ is semi-bilinear, i.e., fixing the first (second) variable, $[ \ , \ ]$ is semi-linear w.r.t. the second (first) one; \item [ $\operatorname{(2)}$] $[ \ , \ ]$ is anti-symmetric, i.e., $[v , v]=0$ $\forall \ v \in A$; \item [ $\operatorname{(3)}$] $[ \ , \ ]$ satisfies the Jacobi identity: $\forall \ u, v, w \in A$, $[u, [v,w]]+ [w, [u,v]]+ [v, [w, u]]=0$ \end{itemize} \end{definition} From Definition~\ref{semi-Liesemialgebra} we can see that a Lie semi-algebra can be non-associa-tive, i.e., the product $[ \ , \ ]$ is not always associative. Let us now consider the semi-algebra ${ \mathcal M}_n ({\mathbb R}_{0}^{+})$ of matrices of order $n$ with entries in ${\mathbb R}_{0}^{+}$ (see Example~\ref{ex6}). We know that ${ \mathcal M}_n ({\mathbb R}_{0}^{+})$ is simple, i.e., with exception of the zero matrix (zero vector), no matrix has (additive) symmetric. Therefore, the product of such matrices can be nonzero. However, in the case of a Lie semi-algebra $A$, if $A$ is simple then the unique product $[ \ , \ ]$ that can be defined over $A$ is the zero product, as it is shown in the next result. \begin{proposition}\label{semi-Lieabelian} If $A$ is a simple Lie semi-algebra over a semi-field $K$ then the semi-algebra is abelian, i.e., $[u, v]=0$ for all $u, v \in A$. \end{proposition} \begin{proof} Assume that $u, v \in A$ and $[u, v ] \neq 0$. From Items~$\operatorname{(1)}$~and~$\operatorname{(2)}$ of Definition~\ref{semi-Liesemialgebra}, it follows that $[u+v , u+v ] =[u, u] + [u, v] + [v, u] + [v, v]=0$, i.e., $[u, v] + [v, u]=0$. This means that $[u, v]$ has symmetric $[v, u]\neq 0$, a contradiction. \end{proof} \begin{definition}\label{subLiesemi} Let $A$ be a Lie semi-algebra over a semi-field $K$. A Lie semi-subalgebra $B \subseteq A$ is a semi-subspace of $A$ which is closed under $[u, v ]$, i.e., for all $u, v \in B$, $[u, v] \in B$. \end{definition} \begin{corollary} All semi-subspaces of $A$ are semi-subalgebras of $A$. \end{corollary} \begin{proof} Apply Proposition~\ref{semi-Lieabelian}. \end{proof} \section{Fuzzy Set Theory and Semi-Algebras}\label{sec3a} The theory of semi-vector spaces and semi-algebras is a natural generalization of the corresponding theories of vector spaces and algebras. Since the scalars are in semi-fields (weak semi-fields), some standard properties does not hold in this new context. However, as we have shown in Section~\ref{sec3}, even in case of nonexistence of symmetrizable elements, several results are still true. An application of the theory of semi-vector spaces is in the investigation on Fuzzy Set Theory, which was introduced by Lotfali Askar-Zadeh \cite{Zadeh:1965}. In fact, such a theory fits in the investigation/extension of results concerning fuzzy sets and their corresponding theory. Let us see an example. Let $L$ be a linearly ordered complete lattice with distinct smallest and largest elements $0$ and $1$. Recall that a fuzzy number is a function $x:{\mathbb R}\longrightarrow L$ on the field of real numbers satisfying the following items (see \cite[Sect. 1.1]{Gahler:1999}): $\operatorname{(1)}$ for each $\alpha \in L_0$ the set $x_{\alpha}= \{\varphi \in {\mathbb R} | \alpha \leq x(\varphi)\} $ is a closed interval $[x_{\alpha l} , x_{\alpha r}]$, where $L_0= \{ \alpha \in L | \alpha > 0\}$; $\operatorname{(2)}$ $\{\varphi \in {\mathbb R} | 0 < x(\varphi)\}$ is bounded. We denote the set ${\mathbb R}_L$ to be the set of all fuzzy numbers; ${\mathbb R}_L$ can be equipped with a partial order in the following manner: $x \leq y $ if and only if $x_{\alpha l} \leq y_{\alpha l}$ and $x_{\alpha r} \leq y_{\alpha r}$ for all $\alpha \in L_0$. In this scenario, Gahler et al. showed that the concepts of semi-algebras can be utilized to extend the concept of fuzzy numbers, according to the following proposition: \begin{proposition}\cite[Proposition 19]{Gahler:1999} The set ${\mathbb R}_L$ is an ordered commutative semi-algebra. \end{proposition} Thus, a direct utilization of the investigation of the structures of semi-vector spaces and semi-algebras is the possibility to generate new interesting results on the Fuzzy Set Theory. Another work relating semi-vector spaces and Fuzzy Set Theory is the paper by Bedregal et al. \cite{Milfont:2021}. In order to study the aggregation functions (geometric mean, weighted average, ordered weighted averaging, among others) w.r.t. an admissible order (a total order $\preceq$ on $L_n ([0, 1])$ such that for all $x, y \in L_n ([0, 1])$, $x \ {\leq}_{n}^{p} \ y \Longrightarrow x\preceq y$), the authors worked with semi-vector spaces over a weak semi-field. Let $L_n ([0, 1]) = \{(x_1, x_2 , \ldots , x_n ) \in {[0, 1]}^{n} | x_1 \leq x_2 \leq \ldots \leq x_n \}$ and $U= ([0, 1], \oplus , \cdot)$ be a weak semi-field defined as follows: for all $x, y \in [0, 1]$, $x \oplus y = \min\{ 1, x+y\}$ and $\cdot$ is the usual multiplication. The product order proposed by Shang et al.~\cite{Shang:2010} is given as follows: for all $x= \{(x_1, x_2 , \ldots , x_n )$ and $y= \{(y_1, y_2 , \ldots , y_n )$ vectors in $L_n ([0, 1])$, define $x \ {\leq}_{n}^{p} \ y \Longleftrightarrow {\pi}_{i}(x)\leq {\pi}_{i}(x) $ for each $i \in \{1, 2, \ldots , n\}$, where ${\pi}_i : L_n ([0, 1]) \longrightarrow [0, 1] $ is the $i$-th projection ${\pi}_i (x_1 , x_2 , \ldots , x_n ) = x_i$. With these concepts in mind, the authors showed two important results: \begin{theorem}(see \cite[Theorem 1]{Milfont:2021})\label{mil21} ${\mathcal L}_{n} ([0, 1]) = (L_n ([0, 1], \dotplus, \odot)$ is a semi-vector space over $U$, where $r \odot v = (rx_1 , \ldots , rx_n )$ and $u \dotplus v = (x_1 \oplus y_1 , \ldots , x_n \oplus y_n ) $. Moreover, $({\mathcal L}_{n} ([0, 1]), {\leq}_{n}^{p})$ is an ordered semi-vector space over $U$, where ${\leq}_{n}^{p}$ is the product order. \end{theorem} \begin{proposition}(see \cite[Propostion 2]{Milfont:2021}) For any bijection $f: \{1, 2 , \ldots , n\} \longrightarrow \{1, 2 , \ldots , n\}$, the pair\\ $({\mathcal L}_{n}([0, 1]), {\preceq}_f)$ is an ordered semi-vector space over $U$, where ${\preceq}_f$, defined in \cite[Example 1]{Milfont:2021}, is an admissible order. \end{proposition} As a consequence of the investigation made, the authors propose an algorithm to perform a multi-criteria and multi-expert decision making method. Summarizing the ideas: the better the theory of semi-vector spaces is extended and developed, the more applications and more results we will have in the Fuzzy Set Theory. Therefore, it is important to understand deeply which are the algebraic and geometry structures of semi-vector spaces, providing, in this way, support for the development of the own theory as well as other interesting theories as, for example, the Fuzzy Set Theory. \section{Summary}\label{sec4} In this paper we have extended the theory of semi-vector spaces, where the semi-field of scalars considered here is the nonnegative real numbers. We have proved several results in the context of semi-vector spaces and semi-linear transformations. We introduced the concept of eigenvalues and eigenvectors of a semi-linear operator and of a matrix and shown how to compute it in specific cases. Topological properties of semi-vector spaces such as completeness and separability were also investigated. We have exhibited interesting new families of semi-vector spaces derived from semi-metric, semi-norm, semi-inner product, among others. Additionally, some results concerning semi-algebras were presented. The results presented in this paper can be possibly utilized in the development and/or investigation of new properties of fuzzy systems and also in the study of correlated areas of research. \section*{Acknowledgment} \small \end{document}

\begin{document} \title{Proposed experiment to test fundamentally binary theories} \author{Matthias~Kleinmann} \email{[email protected]} \affiliation{Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O.~Box 644, E-48080 Bilbao, Spain} \author{Tamás~Vértesi} \email{[email protected]} \affiliation{Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O.~Box 51, Hungary} \author{Adán~Cabello} \email{[email protected]} \affiliation{Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain} \begin{abstract} Fundamentally binary theories are nonsignaling theories in which measurements of many outcomes are constructed by selecting from binary measurements. They constitute a sensible alternative to quantum theory and have never been directly falsified by any experiment. Here we show that fundamentally binary theories are experimentally testable with current technology. For that, we identify a feasible Bell-type experiment on pairs of entangled qutrits. In addition, we prove that, for any $n$, quantum $n$-ary correlations are not fundamentally $(n-1)$-ary. For that, we introduce a family of inequalities that hold for fundamentally $(n-1)$-ary theories but are violated by quantum $n$-ary correlations. \end{abstract} \maketitle \section{Introduction} Quantum theory (QT) is the most successful theory physicists have ever devised. Still, there is no agreement on which physical reasons force its formalism \cite{FS16}. It is therefore important to test ``close-to-quantum'' alternatives, defined as those which are similar to QT in the sense that they have entangled states, incompatible measurements, violation of Bell inequalities, and no experiment has falsified them, and sensible in the sense that they are in some aspects simpler than QT. Examples of these alternatives are theories allowing for almost quantum correlations \cite{NGHA15}, theories in which measurements are fundamentally binary \cite{KC16}, and theories allowing for a higher degree of incompatibility between binary measurements \cite{BHSS13}. Each of these alternatives identifies a particular feature of QT that we do not fully understand and, as a matter of fact, may or may not be satisfied by nature. For example, we still do not know which principle singles out the set of correlations in QT \cite{Cabello15}. In contrast, the set of almost quantum correlations satisfies a list of reasonable principles and is simple to characterize \cite{NGHA15}. Similarly, we do not know why in QT there are measurements that cannot be constructed by selecting from binary measurements \cite{KC16}. However, constructing the set of measurements of the theory would be simpler if this would not be the case. Finally, we do not know why the degree of incompatibility of binary measurements in QT is bounded as it is, while there are theories that are not submitted to such a limitation \cite{BHSS13}. Unfortunately, we do not yet have satisfactory answers to these questions. Therefore, it is important to test whether nature behaves as predicted by QT also in these particular aspects. However, this is not an easy task. Testing almost quantum theories is difficult because we still do not have a well-defined theory; thus, there is not a clear indication on how we should aim our experiments. Another reason, shared by theories with larger binary incompatibility, is that the only way to test them is by proving that QT is wrong, which is, arguably, very unlikely. The case of fundamentally binary theories is different. We have explicit theories \cite{KC16} and we know that fundamentally binary theories predict supraquantum correlations for some experiments but subquantum correlations for others. That is, if QT is correct, there are experiments that can falsify fundamentally binary theories \cite{KC16}. The problem is that all known cases of subquantum correlations require visibilities that escape the scope of current experiments. This is particularly unfortunate now that, after years of efforts, we have loophole-free Bell inequality tests \cite{HBD15,GVW15,SMC15,HKB16,W16}, tests touching the limits of QT \cite{PJC15,CLBGK15}, and increasingly sophisticated experiments using high-dimensional two-photon entanglement \cite{VWZ02,GJVWZ06,DLBPA11}. Therefore, a fundamental challenge is to identify a feasible experiment questioning QT beyond the local realistic theories \cite{Bell64}. The main aim of this work is to present a feasible experiment capable of excluding fundamentally binary theories. In addition, the techniques employed to identify that singular experiment will allow us to answer a question raised in Ref.~\cite{KC16}, namely, whether or not, for some $n$, quantum $n$-ary correlations are fundamentally $(n-1)$-ary. \subsection{Device-independent scenario} Consider a bipartite scenario where two observers, Alice and Bob, perform independent measurements on a joint physical system. For a fixed choice of measurements $x$ for Alice and $y$ for Bob, $P(a,b|x,y)$ denotes the joint probability of Alice obtaining outcome $a$ and Bob obtaining outcome $b$. We assume that both parties act independently in the sense that the marginal probability for Alice to obtain outcome $a$ does not depend on the choice of Bob's measurement $y$, i.e., $\sum_b P(a,b|x,y)\equiv P(a,\omitted|x,\omitted)$, and analogously $\sum_a P(a,b|x,y)\equiv P(\omitted,b|\omitted,y)$. These are the nonsignaling conditions, which are obeyed by QT whenever both observers act independently, in particular, if the operations of the observers are spacelike separated. However, QT does not exhaust all possible correlations subject to these constraints \cite{PR94}. The strength of this scenario lies in the fact that the correlations can be obtained without taking into account the details of the experimental implementation and hence it is possible to make statements that are independent of the devices used. This device-independence allows us to test nature without assuming a particular theory---such as QT---for describing any of the properties of the measurement setup. This way, it is also possible to make theory-independent statements and, in particular, to analyze the structure of any probabilistic theory that obeys the nonsignaling conditions. \subsection{Fundamentally binary theories} One key element of the structure of any probabilistic theory was identified in Ref.~\cite{KC16} and concerns how the set of measurements is constructed, depending on the number of outcomes. According to Ref.~\cite{KC16}, it is plausible to assume that a theory describing nature has, on a fundamental level, only measurements with two outcomes while situations where a measurement has more outcomes are achieved by classical postprocessing of one or several two-outcome measurements. To make this a consistent construction, it is also admissible that the classical postprocessing depends on additional classical information and, in the bipartite scenario, this classical information might be correlated between both parties. The total correlation attainable in such a scenario are the binary nonsignaling correlations, which are characterized by the convex hull of all nonsignaling correlations obeying $P(a,\omitted|x,\omitted)= 0$ for all measurements $x$ and all but two outcomes $a$, and $P(\omitted,b|\omitted,y) = 0$ for all measurements $y$ and all but two outcomes $b$. The generalization to $n$-ary nonsignaling correlations is straightforward. In Ref.~\cite{KC16}, it was shown that for no $n$ the set of $n$-ary nonlocal correlations covers all the set of quantum correlations. Albeit this being a general result, the proof in Ref.~\cite{KC16} has two drawbacks: (i) It does not provide a test which is experimentally feasible. (ii) It does not allow us to answer whether or not quantum $n$-ary correlations are still fundamentally $(n-1)$-ary. For example, the proof in Ref.~\cite{KC16} requires {10}-outcome quantum measurements for excluding the binary case. In this work, we address both problems and provide (i') an inequality that holds for all binary nonsignaling correlations, but can be violated using three-level quantum systems (qutrits) with current technology, and (ii') a family of inequalities obeyed by $(n-1)$-ary nonsignaling correlations but violated by quantum measurements with $n$ outcomes. \section{Results} \subsection{Feasible experiment to test fundamentally binary theories} We first consider the case where Alice and Bob both can choose between two measurements, $x=0,1$ and $y=0,1$, and each measurement has three outcomes $a,b=0,1,2$. For a set of correlations $P(a,b|x,y)$, we define \begin{equation} I_a=\sum_{k,x,y=0,1} (-1)^{k+x+y}P(k,k|x,y), \end{equation} where the outcomes with $k=2$ do not explicitly appear. With the methods explained in Sec.~\ref{polymeth}, we find that, up to relabeling of the outcomes, \begin{equation}\label{ineqa} I_a\le 1 \end{equation} holds for nonsignaling correlations if and only if the correlations are fundamentally binary. However, according to QT, the inequality in Eq.~\eqref{ineqa} is violated, and a value of \begin{equation}\label{qvaluea} I_a= 2(2/3)^{3/2}\approx 1.0887 \end{equation} can be achieved by preparing a two-qutrit system in the pure state \begin{equation} \ket\psi=\frac{1}{2}(\sqrt{2}\ket{00}+ \ket{11}-\ket{22}) \end{equation} and choosing the measurements $x,y=0$ as $M_{k|0}= V\proj{k}V^\dag$, and the measurements $x,y=1$ as $M_{k|1}= U\proj{k}U^\dag$, where, in canonical matrix representation, \begin{equation} V=\frac1{\sqrt{12}}\begin{pmatrix} 2 & 2 & 2 \\ -\sqrt{3}-1 & \sqrt{3}-1 & 2 \\ \sqrt{3}-1 & -\sqrt{3}-1 & 2 \end{pmatrix}, \end{equation} and $U=\diag(-1,1,1)V$. Using the second level of the Navascués--Pironio--Acín (NPA) hierarchy \cite{NPA07}, we verify that the value in Eq.~\eqref{qvaluea} is optimal within our numerical precision of $10^{-6}$. The visibility required to observe a violation of the inequality in Eq.~\eqref{ineqa} is $91.7\%$, since the value for the maximally mixed state is $I_a=0$. The visibility is defined as the minimal $p$ required to obtain a violation assuming that the prepared state is a mixture of the target state and a completely mixed state, $\rho_{\rm prepared} = p \proj\psi + (1-p) \rho_{\rm mixed}$. We show in Sec.~\ref{polymeth} that the inequality in Eq.~\eqref{ineqa} holds already if only one of the measurements of either Alice or Bob is fundamentally binary. Therefore, the violation of the inequality in Eq.~\eqref{ineqa} allows us to make an even stronger statement, namely, that none of the measurements used is fundamentally binary, thus providing a device-independent certificate of the genuinely ternary character of all measurements in the experimental setup. The conclusion at this point is that the violation of the inequality in Eq.~\eqref{ineqa} predicted by QT could be experimentally observable even achieving visibilities that have been already attained in previous Bell-inequality experiments on qutrit--qutrit systems \cite{VWZ02,GJVWZ06,DLBPA11}. It is important to point out that, in addition, a compelling experiment requires that the local measurements are implemented as measurements with three outcomes rather than measurements that are effectively two-outcome measurements. That is, there should be a detector in each of the three possible outcomes of each party. The beauty of the inequality in Eq.~\eqref{ineqa} and the simplicity of the required state and measurements suggest that this experiment could be carried out in the near future. \subsection{Quantum $n$-ary correlations are not fundamentally $(n-1)$-ary} If our purpose is to test whether or not one particular measurement is fundamentally binary (rather than all of them), then it is enough to consider a simpler scenario where Alice has a two-outcome measurement $x=0$ and a three-outcome measurement $x=1$, while Bob has three two-outcome measurements $y=0,1,2$. We show in Sec.~\ref{polymeth} that for the combination of correlations \begin{equation}\label{ieb} I_b=-P(0,\omitted|0,\omitted)+\sum_{k=0,1,2}[P(0,0|0,k)-P(k,0|1,k)], \end{equation} up to relabeling of the outcomes and Bob's measurement settings, \begin{equation}\label{ineqb} I_b\le 1 \end{equation} holds for nonsignaling correlations if and only if the correlations are fundamentally binary. According to QT, this bound can be violated with a value of \begin{equation}\label{qvalueb} I_b=\sqrt{16/15}\approx 1.0328, \end{equation} by preparing the state \begin{equation} \ket\psi=\frac1{\sqrt{(3\zeta+1)^2+2}}(\ket{00}+\ket{11}+\ket{22}+ \zeta\ket\phi\!\ket\phi), \end{equation} where $\zeta= -\frac13+\frac16\sqrt{10\sqrt{15}-38}\approx -0.19095$, $\ket\phi=\ket0+\ket1+\ket2$, and choosing Alice's measurement $x=0$ as $A_{0|0}=\openone-A_{1|0}$, $A_{1|0}=\proj{\phi}/3$, and measurement $x=1$ as $A_{k|1}=\proj k$, for $k=0,1,2$, and Bob's measurements $y=0,1,2$ as $B_{0|y}=\openone-B_{1|y}$ and $B_{1|k}=\proj{\eta_k}/\braket{\eta_k|\eta_k}$, where $\ket{\eta_k}=\ket{k}+\xi\ket\phi$, for $k=0,1,2$, and $\xi = -\frac13+\frac16\sqrt{6\sqrt{15}+22}\approx 0.78765$. [Another optimal solution is obtained by flipping the sign before the $(\frac16\sqrt{\,})$-terms in $\xi$ and $\zeta$, yielding $\xi\approx -1.4543$ and $\zeta\approx -0.47572$.] We use the third level of the NPA hierarchy to confirm that, within our numerical precision of $10^{-6}$, the value in Eq.~\eqref{qvalueb} is optimal. Notice, however, that the visibility required to observe a violation of the inequality in Eq.~\eqref{ineqb} is $96.9\%$. This contrasts with the $91.7\%$ required for the inequality in Eq.~\eqref{ineqa} and shows how a larger number of outcomes allows us to certify more properties with a smaller visibility. Nevertheless, what is interesting about the inequality in Eq.~\eqref{ineqb} is that it is a member of a family of inequalities and this family allows us to prove that, for any $n$, quantum $n$-ary correlations are not fundamentally $(n-1)$-ary, a problem left open in Ref.~\cite{KC16}. For that, we modify the scenario used for the inequality in Eq.~\eqref{ineqb}, so that now Alice's measurement $x=1$ has $n$ outcomes, while Bob has $n$ measurements with two outcomes. We let $I_b^{(n)}$ be as $I_b$ defined in Eq.~\eqref{ieb}, with the only modification that in the sum, $k$ takes values from $0$ to $n-1$. Then, \begin{equation}\label{ineqc} I_b^{(n)}\le n-2 \end{equation} is satisfied for all fundamentally $(n-1)$-ary correlations. The proof is given in Sec.~\ref{proof}. Clearly, the value $I_b^{(n)}=n-2$ can already be reached by choosing the fixed local assignments where all measurements of Alice and Bob always have outcome $a,b=0$. According to QT, it is possible to reach values of $I_b^{(n)}> (n-2)+1/(4n^3)$, as can be found by generalizing the quantum construction from above to $n$-dimensional quantum systems with $\xi=\sqrt2$ and $\zeta= -1/n+1/(\sqrt2n^2)$. Thus, the $(n-1)$-ary bound is violated already by $n$-ary quantum correlations. Note, that the maximal quantum violation is already very small for $n=4$ as the bound from the third level of the NPA hierarchy is $I_b^{(4)}<2.00959$. \section{Methods} \subsection{Restricted nonsignaling polytopes}\label{polymeth} We now detail the systematic method that allows us to obtain the inequalities in Eqs.~\eqref{ineqa}, \eqref{ineqb}, and \eqref{ineqc}. We write $S=\bisc{a_1, a_2,\dotsc, a_n}{b_1, b_2,\dotsc, b_m}$ for the case where Alice has $n$ measurements and the first measurement has $a_1$ outcomes, the second $a_2$ outcomes, etc., and similarly for Bob and his $m$ measurements with $b_1$, $b_2$,\dots, outcomes. The nonsignaling correlations for such a scenario form a polytope $C(S)$. For another bipartite scenario $S'$ we consider all correlations $P'\in C(S')$ that can be obtained by local classical postprocessing from any $P\in C(S)$. The convex hull of these correlations is again a polytope and is denoted by $C(S\rightarrow S')$. The simplest nontrivial polytope of fundamentally binary correlations is then $C(\bisc{2,2}{2,2}\rightarrow \bisc{3,3}{3,3})$. We construct the vertices of this polytope and compute the {468} facet inequalities (i.e., tight inequalities for fundamentally binary correlations) with the help of the Fourier-Motzkin elimination implemented in the software \texttt{porta} \cite{porta}. We confirm the results by using the independent software \texttt{ppl} \cite{ppl}. Up to relabeling of the outcomes, only the facet $I_a\le 1$ is not a face of the set the nonsignaling correlations $C(\bisc{3,3}{3,3})$, which concludes our construction of $I_a$. In addition, we find that \begin{equation}\label{coneq} C(\bisc{2,3}{3,3})= C(\bisc{2,2}{2,2}\rightarrow \bisc{2,3}{3,3}), \end{equation} and therefore the inequality in Eq.~\eqref{ineqa} holds for all nonsignaling correlations where at least one of the measurements is fundamentally binary. As a complementary question we consider the case where only a single measurement has three outcomes. According to Eq.~\eqref{coneq}, the smallest scenarios where such a verification is possible are $\bisc{2,3}{2,2,2}$ and $\bisc{2,2}{2,2,3}$. We first find that $C(\bisc{2,2}{3,3,3})= C(\bisc{2,2}{2,2,2}\rightarrow \bisc{2,2}{3,3,3})$, i.e., even if all of Bob's measurements would be fundamentally ternary, the correlations are always within the set of fundamentally binary correlations. Hence, we investigate the polytope $C(\bisc{2,2}{2,2,2}\rightarrow \bisc{2,3}{2,2,2})$ and its {126} facets. Up to symmetries, only the facet $I_b\le 1$ is not a face of $C(\bisc{2,3}{2,2,2})$. Our method also covers other scenarios. As an example we study the polytope $C(\bisc{2,4}{2,4}\rightarrow \bisc{2,2,2}{2,2,2})$ with its {14052} facets. In this case, the four-outcome measurements have to be distributed to two-outcome measurements (or the two-outcome measurement is used twice). Hence, this scenario is equivalent to the requirement that for each party at least two of the three measurements are compatible. The polytope has, up to relabeling, {10} facets that are not a face of $C(\bisc{2,2,2}{2,2,2})$. According to the fourth level of the NPA hierarchy, two of the facets may intersect with the quantum correlations. While for one of them the required visibility (with respect to correlations where all outcomes are equally probable) is at least $99.94\%$, the other requires a visibility of at least $97.88\%$. This latter facet is $I_c\le 0$, where \begin{multline} I_c=-P(10|00)-P(00|01)-P(00|10)-P(00|11)\\ -P(10|12)-P(01|20)-P(01|21)+P(00|22). \end{multline} For arbitrary nonsignaling correlations, $I_c\le 1/2$ is tight, while within QT, $I_c< 0.0324$ must hold. We can construct a numeric solution for two qutrits which matches the bound from the third level of the NPA hierarchy up to our numerical precision of $10^{-6}$. The required quantum visibility then computes to $97.2\%$. The quantum optimum is reached for measurements $A_{0|k}=\proj{\alpha_k}$, $A_{1|k}=\openone -A_{0|k}$, and $B_{0|k}=\proj{\beta_k}$, $B_{1|k}=\openone -B_{0|k}$, where all $\ket{\alpha_k}$ and $\ket{\beta_k}$ are normalized and $\braket{\alpha_0|\alpha_1}\approx 0.098$, $\braket{\alpha_0|\alpha_2}\approx 0.630$, $\braket{\alpha_1|\alpha_2}\approx 0.572$, and $\braket{\beta_k|\beta_\ell}\approx 0.771$ for $k\ne \ell$. A state achieving the maximal quantum value is $\ket\psi\approx 0.67931\ket{00}+0.67605\ket{11}+0.28548\ket{22}$. Note, that $I_c\approx 0.0318$ can still be reached according to QT, when Alice has only two incompatible measurements by choosing $\braket{\alpha_0|\alpha_1}= 0$. Curiously, the facet $I_c\le 0$ is equal to the inequality $M_{3322}$ in Ref.~\cite{BGS05} and a violation of it has been observed recently by using photonic qubits \cite{CLBGK15}. However, while $M_{3322}$ is the only nontrivial facet of the polytope investigated in Ref.~\cite{BGS05}, it is just one of several nontrivial facets in our case. \subsection{Proof of the inequality in Eq.~\eqref{ineqc}}\label{proof} Here, we show that for $(n-1)$-ary nonsignaling correlations, the inequality in Eq.~\eqref{ineqc} holds. We start by letting for some fixed index $0\le \ell < n$, \begin{subequations} \begin{align} F&=-\sum_b R_{0,b|0,\ell} + \sum_k [ R_{0,0|0,k}-R_{k,0|1,k} ],\\ X_{1;a|x,y}&=\sum_b(R_{a,b|x,y}-R_{a,b|x,\ell}),\\ X_{2;b|x,y}&=\sum_a(R_{a,b|x,y}-R_{a,b|0,y}), \end{align} \end{subequations} where all $R_{a,b|x,y}$ are linearly independent vectors from a real vector space $V$. Clearly, for any set of correlations, we can find a linear function $\phi\colon V\rightarrow {\mathbb R}$ with $\phi(R_{a,b|x,y})= P(a,b|x,y)$. For such a function, $I_b^{(n)}= \phi(F)$ holds and $\phi(X_\tau)= 0$ are all the nonsignaling conditions. The maximal value of $I_b^{(n)}$ for $(n-1)$-ary nonsignaling correlations is therefore given by \begin{equation}\label{prim}\begin{split} \textstyle\max_{\ell'} \max\{ \phi(F) \mid\; & \phi\colon V\rightarrow {\mathbb R} \text{, linear,}\\ &\phi(X_\tau) = 0, \text{ for all } \tau, \\ & \phi(R_{\ell',b|1,y})= 0, \text{ for all } b,y,\\ & \textstyle\sum_\upsilon \phi(R_\upsilon)= 2n, \text{ and }\\ & \phi(R_\upsilon)\ge 0, \text{ for all } \upsilon\}. \end{split}\end{equation} Since the value of the inner maximization does not depend on the choice of $\ell$, we can choose $\ell=\ell'$. Equation~\eqref{prim} is a linear program, and the equivalent dual to this program can be written as \begin{equation}\label{dual} \max_\ell \min_{t,\boldsymbol\xi, \boldsymbol\eta} \set{ t | t\ge \zeta_\upsilon \text{ for all } \upsilon}, \end{equation} where $\boldsymbol\zeta$ is the solution of \begin{equation} 2 n F - \sum_\tau \xi_\tau X_\tau -\sum_{b,y}\eta_{b,y} R_{\ell,b|1,y}= \sum_\upsilon \zeta_\upsilon R_\upsilon. \end{equation} To obtain an upper bound in Eq.~\eqref{dual}, we choose $\boldsymbol\eta\equiv 2n$ and all $\xi_\tau= 0$, but $\xi_{1;a|0,k}=4$, $\xi_{1;k|1,k}=-2n$, $\xi_{2;b|1,\ell}=-3n+2$, and $\xi_{2;b|1,k}=-(-1)^bn+2$, for $k\ne \ell$. This yields $\max_\upsilon \zeta_\upsilon= n-2$ for all $\ell$ and hence the $(n-1)$-ary nonsignaling correlations obey $I_b^{(n)}\le n-2$. \section{Conclusions} There was little chance to learn new physics from the recent loophole-free experiments of the Bell inequality \cite{HBD15,GVW15,SMC15,HKB16,W16}. Years of convincing experiments \cite{FC72,ADR82,WJSWZ98} allowed us to anticipate the conclusions: nature cannot be explained by local realistic theories \cite{Bell64}, there are measurements for which there is not a joint probability distribution \cite{Fine82}, and there are states that are not a convex combination of local states \cite{Werner89}. Here we have shown how to use Bell-type experiments to gain insights into QT. In Ref.~\cite{KC16}, it was shown that QT predicts correlations that cannot be explained by nonsignaling correlations produced by fundamentally binary measurements (including Popescu--Rohrlich boxes \cite{PR94}). We proposed a feasible experiment which will allow us to either exclude all fundamentally binary probabilistic theories or to falsify QT. If the results of the experiment violate the inequality in Eq.~\eqref{ineqa}, as predicted by QT, then we would learn that no fundamentally binary theory can possibly describe nature. In addition, it would prove that all involved measurements are genuine three-outcome measurements. If the inequality in Eq.~\eqref{ineqa} is not violated despite visibilities would \emph{a priori} lead to such a violation, then we would have evidence that QT is wrong at a fundamental level (although being subtle to detect in experiments). We have also gone beyond Ref.~\cite{KC16} by showing that, for any $n$, already $n$-ary quantum correlations are not fundamentally $(n-1)$-ary. \begin{acknowledgments} This work is supported by Project No.~FIS2014-60843-P, ``Advanced Quantum Information'' (MINECO, Spain), with FEDER funds, the FQXi Large Grant ``The Observer Observed: A Bayesian Route to the Reconstruction of Quantum Theory'', the project ``Photonic Quantum Information'' (Knut and Alice Wallenberg Foundation, Sweden), the Hungarian National Research Fund OTKA (Grants No.~K111734 and No.~KH125096), the EU (ERC Starting Grant GEDENTQOPT), and the DFG (Forschungsstipendium KL~2726/2-1). \end{acknowledgments} \end{document}

\begin{document} \selectlanguage{english} \title{Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming} \author{ Mihai Bostan \thanks{Laboratoire d'Analyse, Topologie, Probabilit\'es LATP, Centre de Math\'ematiques et Informatique CMI, UMR CNRS 7353, 39 rue Fr\'ed\'eric Joliot Curie, 13453 Marseille Cedex 13 France. E-mail : {\tt [email protected]}} , J. A. Carrillo \thanks{ICREA (Instituci\'o Catalana de Recerca i Estudis Avan\c{c}ats) and Departament de Matem\`atiques, Universitat Aut\`onoma de Barcelona, 08193 Bellaterra Spain. E-mail : {\tt [email protected]}. {\it On leave from:} Department of Mathematics, Imperial College London, London SW7 2AZ, UK.}} \date{ (\today)} \maketitle \begin{abstract} We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure. \end{abstract} \paragraph{Keywords:} Vlasov-like equations, Measure solutions, Swarming, Cucker-Smale model, Vicsek model, Laplace-Beltrami operator. \paragraph{AMS classification:} 92D50, 82C40, 92C10. \section{Introduction} \label{Intro} \indent This paper is devoted to continuum models for the dynamics of systems involving living organisms such as flocks of birds, school of fish, swarms of insects, myxobacteria... The individuals of these groups are able to organize in the absence of a leader, even when starting from disordered configurations \cite{ParEde99}. Several minimal models describing such self-organizing phenomenon have been derived \cite{VicCziBenCohSho95, GreCha04, CouKraFraLev05}. Most of these models include three basic effects: short-range repulsion, long-range attraction, and reorientation or alignment, in various ways, see \cite{HW} and particular applications to birds \cite{HCH09} and fish \cite{BTTYB,BEBSVPSS}. We first focus on populations of individuals driven by self-propelling forces and pairwise attractive and repulsive interaction \cite{LevRapCoh00, DorChuBerCha06}. We consider self-propelled particles with Rayleigh friction \cite{ChuHuaDorBer07, ChuDorMarBerCha07, CarDorPan09,UAB25}, leading to the Vlasov equation in $d=2,3$ dimensions: \begin{equation} \label{Equ1} \partial _t \fe + v \cdot \nabla _x \fe + a ^\eps (t,x) \cdot \nabla _v \fe + \frac{1}{\eps} \Divv\{\fe \abv\}= 0,\;\;(t,x,v) \in \R_+ \times \R^d \times \R^d \end{equation} where $\fe = \fe (t,x,v) \geq 0$ represents the particle density in the phase space $(x,v) \in \R^d \times \R^d$ at any time $t \in \R_+$, $a ^\eps $ stands for the acceleration \[ a^\eps (t,\cdot) = - \nabla _x U \star \rho ^\eps (t, \cdot ),\;\;\rho ^\eps (t, \cdot ) = \intv{\fe (t, \cdot, v)\;\mathrm{d}v}\, , \] and $U$ is the pairwise interaction potential modelling the repelling and attractive effects. Here, the propulsion and friction forces coefficients $\alpha ^\eps = \frac{\alpha}{\eps}>0$, $\beta ^\eps = \frac{\beta}{\eps} >0$ are scaled in such a way that for $\eps\to 0$ particles will tend to move with asymptotic speed $\sqrt{\tfrac{\alpha}\beta}$. These models have been shown to produce complicated dynamics and patterns such as mills, double mills, flocks and clumps, see \cite{DorChuBerCha06}. Assuming that all individuals move with constant speed also leads to spatial aggregation, patterns, and collective motion \cite{CziStaVic97, EbeErd03}. Another source of models arises from introducing alignment at the modelling stage. A popular choice in the last years to include this effect is the Cucker-Smale reorientation procedure \cite{CS2}. Each individual in the group adjust their relative velocity by averaging with all the others. This velocity averaging is weighted in such a way that closer individuals in space have more influence than further ones. The continuum kinetic version of them leads to Vlasov-like models of the form \eqref{Equ1} in which the acceleration is of the form \[ a^\eps (t,\cdot) = - H \star f^\eps (t, \cdot )\, , \] where $\star$ stands for the $(x,v)$-convolution, abusing a bit on the notation, with the nonnegative interaction kernel $H:\R^{2d}\longrightarrow \R^d$. In the original Cucker-Smale work, the interaction is modelled by $H(x,v)=h(x)v$, with the weight function $h$ being a decreasing radial nonnegative function. We refer to the extensive literature in this model for further details \cite{HT08,HL08,CFRT10,review,MT11}. In this work, we will consider the Vlasov equation \eqref{Equ1} where the acceleration includes the three basic effects discussed above, and then takes the form: \begin{equation}\label{accel} a^\eps (t,\cdot) = - \nabla _x U \star \rho ^\eps (t, \cdot ) - H \star f^\eps (t, \cdot )\, . \end{equation} We will assume that the interaction potential $U\in C^2_b(\R^d)$, $U$ bounded continuous with bounded continuous derivatives up to second order, and $H(x,v)=h(x)v$ with $h\in C^1_b(\R^d)$ and nonnegative. Under these assumptions the model \eqref{Equ1}-\eqref{accel} can be rigorously derived as a mean-field limit \cite{Neu77, BraHep77, Dob79,CCR10,BCC11} from the particle systems introduced in \cite{DorChuBerCha06,CS2}. We will first study in detail the linear problem, assuming that the acceleration $a = a(t,x)$ is a given global-in-time bounded smooth field. We investigate the regime $\eps \searrow 0$, that is the case when the propulsion and friction forces dominate the potential interaction between particles. At least formally we have \begin{equation} \label{EquAnsatz} \fe = f + \eps \fo + \eps ^2 f ^{(2)} + ... \end{equation} where \begin{equation} \label{Equ2} \Divv\{f \abv \} = 0 \end{equation} \begin{equation} \label{Equ3} \partial _t f + \Divx (fv) + \Divv (f a(t,x)) + \Divv\{\fo \abv \} = 0\,, \end{equation} up to first order. Therefore, to characterize the zeroth order term in the expansion we need naturally to work with solutions whose support lies on the sphere of radius $r := \sqrt{\alpha/\beta}$ denoted by $r\sphere$ with $\sphere = \{v\in \R^d : |v| = 1\}$. In turn, we need to work with measure solutions to \eqref{Equ2} which makes natural to set as functional space the set of nonnegative bounded Radon measures on $\R^d\times\R^d$ denoted by ${\cal M}_b ^+ (\R^d\times\R^d)$. We will be looking at solutions to \eqref{Equ1} which are typically continuous curves in the space ${\cal M}_b ^+ (\R^d\times\R^d)$ with a suitable notion of continuity to be discussed later on. We will denote by $\fe(t,x,v)\, \mathrm{d}(x,v)$ the integration against the measure solution $\fe(t,x,v)$ of \eqref{Equ1} at time $t$. For the sake of clarity, this is done independently of being the measure $\fe(t)$ absolutely continuous with respect to Lebesgue or not, i.e., having a $L^1(\R^d\times\R^d)$ density or not. \begin{pro}\label{Kernel} Assume that $(1+|v|^2)F \in {\cal M}_b ^+ (\R^d)$. Then $F$ is a solution to \eqref{Equ2} if and only if $\supp F \subset \{0\} \cup r \sphere$. \end{pro} The condition \eqref{Equ2} appears as a constraint, satisfied at any time $t \in \R_+$. The time evolution of the dominant term $f$ in the Ansatz \eqref{EquAnsatz} will come by eliminating the multiplier $\fo$ in \eqref{Equ3}, provided that $f$ verifies the constraint \eqref{Equ2}. In other words we are allowed to use those test functions $\psi (x,v)$ which remove the contribution of the term $\Divv\{ \fo \abv \}$ {\it i.e.,} \[ \intxv{\abv \cdot \nabla _v \psi \;\fo(t,x,v)\, \mathrm{d}(x,v) } = 0. \] Therefore we need to investigate the invariants of the field $\abv \cdot \nabla _v$. The admissible test functions are mainly those depending on $x$ and $v/|v|, v \neq 0$. The characteristic flow $(s,v) \to {\cal V}(s;v)$ associated to $\tfrac1\eps \abv \cdot \nabla _v$ \[ \frac{\mathrm{d}{\cal V}}{\mathrm{d}s} = \frac1\eps \abvs,\;\;{\cal V}(0;v) = v \] will play a crucial role in our study. It will be analyzed in detail in Section \ref{LimMod}. Notice that the elements of $\A$ are the equilibria of $\abv \cdot \nabla _v $. It is easily seen that the jacobian of this field \[ \partial _v \{ \abv \} = (\alpha - \beta |v|^2 ) I - 2 \beta v \otimes v \] is negative on $r\sphere$, saying that $r\sphere$ are stable equilibria. The point $0$ is unstable, $\partial _v \{ \abv \} |_{v = 0}=\alpha I$. When $\eps \searrow 0$ the solutions $(\fe)_\eps$ concentrate on $\xA$, leading to a limit curve of measures even if $(\fe)_\eps$ were smooth solutions. We can characterize the limit curve as solution of certain PDE whenever our initial measure does not charge the unstable point $0$. \begin{thm} \label{MainResult} Assume that $a \in \litwoix{}$, $(1 + |v|^2) \fin \in \mbxv{}$, $\supp \fin \subset \{(x,v) :|v|\geq r_0>0\}$. Then $(\fe)_\eps$ converges weakly $\star$ in $\litmbxv{}$ towards the solution of the problem \begin{equation} \label{Equ22} \partial _t f + \Divx(fv) + \Divv \left \{f \imvv a \right \} = 0 \end{equation} \begin{equation} \label{Equ23} \Divv \{f \abv \} = 0 \end{equation} with initial data $f(0) = \ave{\fin}$ defined by $$ \intxv{\psi (x,v) \ave{\fin}(x,v)\, \mathrm{d}(x,v)} = \intxv{\psi \left (x, r \vsv\right ) \fin(x,v)\, \mathrm{d}(x,v)}\,, $$ for all $\psi \in \czcxv$. \end{thm} In the rest, we will refer to $\ave{\fin}$ as the projected measure on the sphere of radius $r$ corresponding to $\fin$. Let us point out that the previous result can be equivalently written in spherical coordinates by saying that $f(t,x,\omega)$ is the measure solution to the evolution equation on $(x,\omega)\in\R ^d \times r \sphere$ given by \begin{equation*} \partial _t f + \Divx(f\omega) + \Divo \left \{f \imoo a \right \} = 0 \,. \end{equation*} These results for the linear problem, when $a(t,x,v)$ is given, can be generalized to the nonlinear counterparts where $a(t,x)$ is given by \eqref{accel}. The main result of this work is (see Section \ref{MeaSol} for the definition of $\Po$): \begin{thm} \label{MainResult2} Assume that $U\in C^2_b(\R^d)$, $H(x,v)=h(x)v$ with $h\in C^1_b(\R^d)$ nonnegative, $\fin \in \poxv{}$, $\supp \fin \subset \{(x,v) :|x| \leq L_0, r_0\leq |v| \leq R_0\}$ with $0<r_0<r<R_0<\infty$. Then for all $\delta>0$, the sequence $(\fe)_\eps$ converges in $C([\delta,\infty);\poxv)$ towards the measure solution $f(t,x,\omega)$ on $(x,\omega)\in\R ^d \times r \sphere$ of the problem \begin{equation} \label{Equ22n} \partial _t f + \Divx(f\omega) - \Divo \left \{f \imoo \left(\nabla_x U\star \rho + H\star f \right) \right \} = 0 \end{equation} with initial data $f(0) = \ave{\fin}$. Moreover, if the initial data $\fin$ is already compactly supported on $B_{L_0} \times r \sphere$, then the convergence holds in $\cztpoxv$. \end{thm} Let us mention that the evolution problem \eqref{Equ22n} on $\R ^d \times r \sphere$ was also proposed in the literature as the continuum version \cite{DM08} of the Vicsek model \cite{VicCziBenCohSho95,CouKraFraLev02} without diffusion for the particular choice $U=0$ and $H(x,v)=h(x) v$ with $h(x)$ some local averaging kernel. The original model in \cite{VicCziBenCohSho95,CouKraFraLev02} also includes noise at the particle level and was derived as the mean filed limit of some stochastic particle systems in \cite{BCC12}. In fact, previous particle systems have also been studied with noise in \cite{BCC11} for the mean-field limit, in \cite{HLL09} for studying some properties of the Cucker-Smale model with noise, and in \cite{DFL10,FL11} for analyzing the phase transition in the Vicsek model. In the case of noise, getting accurate control on the particle paths of the solutions is a complicated issue and thus, we are not able to show the corresponding rigorous results to Theorems \ref{MainResult} and \ref{MainResult2}. Nevertheless, we will present a simplified formalism, which allows us to handle more complicated problems to formally get the expected limit equations. This approach was borrowed from the framework of the magnetic confinement, where leading order charged particle densities have to be computed after smoothing out the fluctuations which correspond to the fast motion of particles around the magnetic lines \cite{BosAsyAna, BosTraEquSin, BosGuiCen3D, BosNeg09}. We apply this method to the following (linear or nonlinear) problem \begin{equation} \label{Equ31} \partial _t \fe + \Divx\{\fe v\} + \Divv \{ \fe a\} + \frac{1}{\eps} \Divv \{ \fe \abv \} = \Delta _v \fe \end{equation} with initial data $\fe (0) = \fin$ where the acceleration $a \in \litwoix{}$ and $\fin \in \mbxv{}$. By applying the projection operator $\ave{\cdot}$ to \eqref{Equ31}, we will show that the limiting equation for the evolution of $f(t,x,\omega)$ on $(x,\omega)\in\R ^d \times r \sphere$ is given by \begin{equation} \label{Equ22Diff} \partial _t f + \Divx(f\omega) + \Divo \left \{f \imoo a \right \} = \Delta_\omega f \end{equation} where $\Delta_\omega$ is the Laplace-Beltrami operator on $r \sphere$. Our paper is organized as follows. In Section \ref{MeaSol} we investigate the stability of the characteristic flows associated to the perturbed fields $v \cdot \nabla _x + a \cdot \nabla _v + \frac{1}{\eps} \abv \cdot \nabla _v $. The first limit result for the linear problem (cf. Theorem \ref{MainResult}) is derived rigorously in Section \ref{LimMod}. Section \ref{NLimMod} is devoted to the proof of the main Theorem \ref{MainResult2}. The new formalism to deal with the treatment of diffusion models is presented in Section \ref{DiffMod}. The computations to show that these models correspond to the Vicsek models, written in spherical coordinates, are presented in the Appendix \ref{A}. \section{Measure solutions} \label{MeaSol} \subsection{Preliminaries on mass transportation metrics and notations} \label{prelim} We recall some notations and result about mass transportation distances that we will use in the sequel. For more details the reader can refer to \cite{Vi1,CT}. We denote by $\Po(\R^d)$ the space of probability measures on $\R^d$ with finite first moment. We introduce the so-called \emph{Monge-Kantorovich-Rubinstein distance} in $\Po(\R^d)$ defined by \begin{equation*} W_1(f,g) = \sup \left \{ \left |\int_{\R^d} \varphi(u) (f(u)-g(u))\, \mathrm{d} u \right |, \varphi \in \mathrm{Lip}(\R^d), \mathrm{Lip}(\varphi)\leq 1 \right \} \end{equation*} where $\mathrm{Lip}(\R^d)$ denotes the set of Lipschitz functions on $\R^d$ and $\mathrm{Lip}(\varphi)$ the Lipschitz constant of a function $\varphi$. Denoting by $\Lambda$ the set of transference plans between the measures $f$ and $g$, i.e., probability measures in the product space $\R^d \times \R^d$ with first and second marginals $f$ and $g$ respectively \[ f(y) = \int_{\R^d} \pi (y,z)\,\mathrm{d}z,\;\;g(z) = \int_{\R^d} \pi (y,z)\,\mathrm{d}y \] then we have \begin{equation*} W_1(f, g) = \inf_{\pi\in\Lambda} \left\{ \int_{\R^d \times \R^d} \vert y - z \vert \, \pi(y, z)\,\mathrm{d}(y,z) \right\} \end{equation*} by Kantorovich duality. $\Po(\R^d)$ endowed with this distance is a complete metric space. Its properties are summarized below, see\cite{Vi1}. \begin{pro} \label{w2properties} The following properties of the distance $W_1$ hold: \begin{enumerate} \item[1)] {\bf Optimal transference plan:} The infimum in the definition of the distance $W_1$ is achieved. Any joint probability measure $\pi_o$ satisfying: $$ W_1(f, g) = \int_{\R^d \times \R^d} \vert y - z \vert \, \mathrm{d}\pi_o(y, z) $$ is called an optimal transference plan and it is generically non unique for the $W_1$-distance. \item[2)] {\bf Convergence of measures:} Given $\{f_k\}_{k\ge 1}$ and $f$ in $\Po(\R^d)$, the following two assertions are equivalent: \begin{itemize} \item[a)] $W_1(f_k, f)$ tends to $0$ as $k$ goes to infinity. \item[b)] $f_k$ tends to $f$ weakly $\star$ as measures as $k$ goes to infinity and $$ \sup_{k\ge 1} \int_{\vert v \vert > R} \vert v \vert \, f_k(v) \, \mathrm{d}v \to 0 \, \mbox{ as } \, R \to +\infty. $$ \end{itemize} \end{enumerate} \end{pro} Let us point out that if the sequence of measures is supported on a common compact set, then the convergence in $W_1$-sense is equivalent to standard weak-$\star$ convergence for bounded Radon measures. Finally, let us remark that all the models considered in this paper preserve the total mass. After normalization we can consider only solutions with total mass $1$ and therefore use the Monge-Kantorovich-Rubinstein distance in $\Po (\R ^d \times \R ^d)$. From now on we assume that the initial conditions has total mass $1$. \subsection{Estimates on Characteristics} In this section we investigate the linear Vlasov problem \begin{equation} \label{Equ10} \partial _t \fe + \Divx\{\fe v\} + \Divv \{ \fe a\} + \frac{1}{\eps} \Divv \{ \fe \abv \} = 0,\;\;(t,x,v) \in \R_+ \times \R^d \times \R^d \end{equation} \begin{equation} \label{Equ11} \fe (0) = \fin \end{equation} where $a \in \litwoix{}$ and $\fin \in \mbxv{}$. \begin{defi}\label{DefMeaSol} Assume that $a \in \litwoix{}$ and $\fin \in \mbxv{}$. We say that $\fe \in \litmbxv{}$ is a measure solution of \eqref{Equ10}-\eqref{Equ11} if for any test function $\varphi \in \coctxv{}$ we have \begin{align*} \inttxv{\{\partial _t + v \cdot \nabla _x + a \cdot \nabla _v + \frac{1}{\eps} \abv \cdot & \nabla _v \}\varphi \fe(t,x,v)\, \mathrm{d}(x,v)} \\ &+ \intxv{\varphi (0,x,v) \fin(x,v) \, \mathrm{d}(x,v) } = 0. \end{align*} \end{defi} We introduce the characteristics of the field $v\cdot \nabla _x + a \cdot \nabla _v + \frac{1}{\eps} \abv \cdot \nabla _v $ \begin{equation*} \frac{\mathrm{d}\Xe}{\mathrm{d}s} = \Ve(s),\;\;\frac{\mathrm{d}\Ve}{\mathrm{d}s} = a(s, \Xe(s)) + \frac{1}{\eps} \abves \end{equation*} \begin{equation*} \Xe (s=0) = x,\;\;\Ve (s = 0) = v. \end{equation*} We will prove that $(\Xe, \Ve)$ are well defined for any $(s,x,v) \in \R_+ \times \R^d \times \R^d$. Indeed, on any interval $[0,T]$ on which $(\Xe, \Ve)$ is well defined we get a bound \[ \sup _{s \in [0,T]} \{|\Xe (s) | + |\Ve (s) | \} < +\infty \] implying that the characteristics are global in positive time. For that we write \begin{equation}\label{charnew} \frac12\frac{\mathrm{d}|\Ve|^2}{\mathrm{d}s} = a(s, \Xe(s))\cdot \Ve (s) + \frac{1}{\eps} ( \alpha - \beta |\Ve (s) |^2) |\Ve (s)|^2. \end{equation} and then, we get the differential inequality \[ \frac{\mathrm{d}|\Ve |^2}{\mathrm{d}s} \leq 2\|a\|_{\linf} |\Ve (s)| + \frac{2}{\eps} ( \alpha - \beta |\Ve (s) |^2) |\Ve (s)|^2 \] for all $s\in [0,T]$, so that \[ \sup _{s \in [0,T]} |\Ve (s) | < +\infty,\;\;\sup _{s\in [0,T]} |\Xe (s) | \leq |x| + T \sup _{s\in [0,T]} |\Ve (s) | < +\infty. \] Once constructed the characteristics, it is easily seen how to obtain a measure solution for the Vlasov problem \eqref{Equ10}-\eqref{Equ11}. It reduces to push forward the initial measure along the characteristics, see \cite{CCR10} for instance. \begin{pro} For any $t \in \R_+$ we denote by $\fe (t)$ the measure given by \begin{equation}\label{EquDefMea} \intxv{\psi (x,v) \fe(t,x,v)\,\dxv} = \intxv{\psi((\Xe, \Ve)(t;0,x,v))\fin(x,v)\,\dxv}\,, \end{equation} for all $\psi \in \czcxv$. Then the application $t \to \fe (t)$, denoted $\fin \#(\Xe, \Ve)(t;0,\cdot,\cdot)$ is the unique measure solution of \eqref{Equ10}, \eqref{Equ11}, belongs to $\cztmbxv$ and satisfies $$ \intxv{\fe (t,x,v)\,\dxv} = \intxv{\fin(x,v)\,\dxv}, t \in \R_+. $$ \end{pro} \begin{proof} The arguments are straightforward and are left to the reader. We only justify that $\fe \in \cztmbxv$ meaning that for any $\psi \in \czcxv{}$ the application $t \to \intxv{\,\,\psi(x,v) \fe (t,x,v)\;\dxv}$ is continuous. Choose $\psi\in \czcxv{}$. Then, for any $0 \leq t_1 < t_2$ we have \begin{align*} \intxv{\psi(x,v) \fe (t_2,x,v) &\,\dxv } - \intxv{\psi(x,v) \fe (t_1,x,v)\,\dxv } \\ &= \intxv{\left[\psi ((\Xe, \Ve )(t_2;t_1, x, v)) - \psi (x,v)\right]\fe (t_1,x,v)\,\dxv}. \end{align*} Taking into account that $(\Xe, \Ve)$ are locally bounded (in time, position, velocity) it is easily seen that for any compact set $K \subset \R ^d \times \R^d$ there is a constant $C(K)$ such that \[ |\Xe (t_2; t_1, x, v) - x| + |\Ve (t_2; t_1, x, v) - v| \leq |t_2 - t_1 | C(K),\;\;(x,v) \in K. \] Our conclusion follows easily using the uniform continuity of $\psi$ and that $\|\fe (t_1) \|_{{\cal M}_b} = \|\fin \|_{{\cal M}_b}$. Notice also that the equality \eqref{EquDefMea} holds true for any bounded continuous function $\psi$. \end{proof} We intend to study the behavior of $(\fe)_\eps$ when $\eps $ becomes small. This will require a more detailed analysis of the characteristic flows $(\Xe, \Ve)$. The behavior of these characteristics depends on the roots of functions like $A + \frac{1}{\eps} (\alpha - \beta \rho ^2 ) \rho$, with $\rho \in \R_+$, $A \in \R$. \begin{pro}\label{NegA} Assume that $A < 0$ and $ 0 < \eps < 2\alpha r /(|A| 3 \sqrt{3})$. Then the equation $\lae (\rho) := \eps A + (\alpha - \beta \rho ^2 ) \rho = 0$ has two zeros on $\R_+$, denoted $\reo (A), \ret (A)$, satisfying \[ 0 < \reo < \frac{r}{\sqrt{3}} < \ret < r \] and \[ \lime \frac{\reo}{\eps} = \frac{|A|}{\alpha},\;\;\;\;\;\;\lime \frac{r - \ret}{\eps} = \frac{|A|}{2\alpha} \] where $r = \sqrt{\alpha/\beta}$. \end{pro} \begin{proof} It is easily seen that the function $\lae$ increases on $[0,r/\sqrt{3}]$ and decreases on $[r/\sqrt{3}, +\infty[$ with change of sign on $[0,r/\sqrt{3}]$ and $[r/\sqrt{3}, r]$. We can prove that $(\reo)_\eps, (\ret)_\eps$ are monotone with respect to $\eps >0$. Take $0 < \eps < \teps < 2\alpha r /(|A| 3 \sqrt{3})$ and observe that $\lae > \lambda ^{\teps}$. In particular we have \[ \lambda ^{\teps} (\reo) < \lae (\reo) = 0 = \lambda ^{\teps} (\rho _1 ^{\teps}) \] implying $\reo < \rho _1 ^{\teps}$, since $\lambda ^{\teps}$ is strictly increasing on $[0, r/\sqrt{3}]$. Similarly we have \[ \lambda ^{\teps} (\ret) < \lae (\ret) = 0 < \lambda ^{\teps} (\rho _2 ^{\teps}) \] and thus $\ret > \rho _2 ^{\teps}$, since $\lambda ^{\teps}$ is strictly decreasing on $[r/\sqrt{3}, r]$. Passing to the limit in $\lae (\rho _k ^\eps) = 0, k \in \{1,2\}$ it follows easily that \[ \lime \reo = 0,\;\;\lime \ret = r. \] Moreover we can write \begin{equation} \alpha = \frac{\mathrm{d}}{\mathrm{d}\rho }\{(\alpha - \beta \rho ^2 ) \rho \} |_{\rho = 0} = \lime \frac{[\alpha - \beta (\reo)^2]\reo}{\reo} = - \lime \frac{\eps A}{\reo} \nonumber \end{equation} and \begin{equation} -2 \alpha = \frac{\mathrm{d}}{\mathrm{d}\rho }\{(\alpha - \beta \rho ^2 ) \rho \} |_{\rho = r} = \lime \frac{[\alpha - \beta (\ret)^2]\ret}{\ret - r} = - \lime \frac{\eps A}{\ret - r} \nonumber \end{equation} saying that \[ \lime \frac{\reo}{\eps} = \frac{|A|}{\alpha},\;\;\lime \frac{r - \ret}{\eps} = \frac{|A|}{2\alpha}. \] \end{proof} The case $A>0$ can be treated is a similar way and we obtain \begin{pro} \label{PosA} Assume that $A > 0$ and $ \eps >0$. Then the equation $\lae (\rho) := \eps A + (\alpha - \beta \rho ^2 ) \rho = 0$ has one zero on $\R_+$, denoted $\reth (A)$, satisfying \[ \reth >r,\;\;\lime \frac{\reth - r}{\eps} = \frac{|A|}{2\alpha}. \] \end{pro} Using the sign of the function $\rho \to \eps \|a\|_{\linf{}} + (\alpha - \beta \rho ^2 ) \rho$ we obtain the following bound for the kinetic energy. \begin{pro}\label{KinBou} Assume that $a \in \litwoix{}$, $(1 + |v|^2) \fin \in \mbxv{}$ and let us denote by $\fe$ the unique measure solution of \eqref{Equ10}, \eqref{Equ11}. Then we have \[ \left \|\intxv{\,|v|^2 \fe(\cdot,x,v)\,\dxv}\right \|_{\linf (\R_+)} \leq \intxv{[(\reth)^2 + |v|^2] \fin(x,v)\,\dxv}. \] \end{pro} \begin{proof} We know that \[ \frac{\mathrm{d}}{\mathrm{d}t} |\Ve|^2 \leq 2\|a\|_{\linf{}} |\Ve(t)|+ \frac{2}{\eps} (\alpha - \beta |\Ve (t) |^2 ) |\Ve (t)|^2=\frac{2}{\eps}|\Ve (t)|\lae (|\Ve (\overline{t})| ),\;\;t \in \R_+. \] By comparison with the solutions of the autonomous differential equation associated to the righthand side, we easily deduce that \[ |\Ve (t;0,x,v)| \leq \max \{ |v|, \reth(\|a\|_{\linf{}})\}\,, \] for any $T \in \R_+, (x,v) \in \R ^d \times \R ^d$. This yields the following bound for the kinetic energy \begin{align*} \intxv{|v|^2\fe (T,x,v)\,\dxv} &= \intxv{|\Ve (T;0,x,v)|^2 \fin(x,v)\,\dxv} \\ &\leq \intxv{[(\reth)^2 + |v|^2] \fin(x,v)\,\dxv}. \end{align*} \end{proof} The object of the next result is to establish the stability of $\Ve$ around $|v| = r$. We will show that the characteristics starting at points with velocities inside an annulus of length proportional to $\eps$ around the sphere $r\sphere$ get trapped there for all positive times for small $\eps$. \begin{pro} \label{RStab} Assume that $\eps \|a\|_{\linf{}} < 2\alpha r /(3\sqrt{3})$ and that $\ret (-\|a\|_{\linf{}}) \leq |v| \leq \reth (\|a\|_{\linf{}})$. Then, for any $(t,x) \in \R_+ \times \R^d$ we have \[ \ret (-\|a\|_{\linf{}}) \leq |\Ve(t;0,x,v)| \leq \reth (\|a\|_{\linf{}}). \] \end{pro} \begin{proof} As in previous proof, we know that \[ \frac{\mathrm{d}}{\mathrm{d}t} |\Ve|^2 \leq \frac{2}{\eps}|\Ve (t)|\lae (|\Ve (\overline{t})| ),\;\;t \in \R_+\,. \] By comparison with the constant solution $\reth$ to the autonomous differential equation associated to the righthand side, we get that $\sup _{t \in \R_+} |\Ve (t;0,x,v)| \leq \reth$. Assume now that there is $T>0$ such that $|\Ve (T) | < \ret$ and we are done if we find a contradiction. Since $|\Ve (0) |= |v| \geq \ret$, we can assume that $\min _{t \in [0,T]} |\Ve (t) | > \reo>0$ by time continuity. Take now $\ot \in [0,T]$ a minimum point of $t \to |\Ve (t)|$ on $[0,T]$. Obviously $\ot >0$ since \[ |\Ve (\ot) | \leq |\Ve (T)| < \ret \leq |v| = |\Ve (0)|. \] By estimating from below in \eqref{charnew} and using that $\ot$ is a minimum point of $t \to |\Ve (t)|>0$ on $[0,T]$, we obtain \[ 0 \geq \frac{\mathrm{d}}{\mathrm{d}t} |\Ve (\ot)| \geq - \|a\|_{\linf{}} + \frac{(\alpha - \beta |\Ve (\ot)|^2)|\Ve (\ot)| }{\eps} =\frac{\lae ( |\Ve (\ot)| )}{\eps}. \] But the function $\lae$ has negative sign on $[0,\reo] \cup [\ret, +\infty[$. Since we know that $\min _{t \in [0,T]} |\Ve (t)| > \reo$, it remains that \[ \min _{t \in [0,T]} |\Ve (t)| = |\Ve (\ot)| \geq \ret \] which contradicts the assumption $|\Ve (T)| < \ret$. \end{proof} Let us see now what happens when the initial velocity is outside $[\ret (-\|a\|_{\linf{}}), \reth (\|a\|_{\linf{}})]$. In particular we prove that if initially $v \neq 0$, then $\Ve (t), t \in \R_+$ remains away from $0$. We actually show that the characteristics starting away from zero speed but inside the sphere $r\sphere$ will increase their speed with respect to its initial value while those starting with a speed outside the sphere $r\sphere$ will decrease their speed with respect to its initial value, all for sufficiently small $\eps$. \begin{pro} \label{ZeroStab} Consider $\eps >0$ such that $\eps \|a\|_{\linf{}} < 2\alpha r /(3\sqrt{3})$.\\ 1. Assume that $\reo (- \|a\|_{\linf{}}) < |v| < \ret (- \|a\|_{\linf{}})$. Then for any $(t,x) \in \R_+ ^\star \times \R ^d$ we have \[ \reo (- \|a\|_{\linf{}}) < |v| < |\Ve (t;0,x,v)|\leq\reth ( \|a\|_{\linf{}}). \] 2. Assume that $\reth ( \|a\|_{\linf{}}) < |v|$. Then for any $(t,x) \in \R_+ ^\star \times \R^d$ we have \[ \ret (- \|a\|_{\linf{}}) \leq |\Ve (t;0,x,v) | < |v|. \] \end{pro} \begin{proof} 1. Notice that if $|\Ve (T;0,x,v)| = \ret$ for some $T>0$, then we deduce by Proposition \ref{RStab} that $\ret \leq |\Ve (t) | \leq \reth$ for any $t >T$ and thus $|\Ve (t;0,x,v) | \geq \ret > |v|, t \geq T$. It remains to establish our statement for intervals $[0,T]$ such that $|\Ve (t) | < \ret$ for any $t \in [0,T]$. We are done if we prove that $t \to |\Ve (t)|$ is strictly increasing on $[0,T]$. For any $\tau \in ]0,T]$ let us denote by $\ot$ a maximum point of $t \to |\Ve (t)|>0$ on $[0,\tau]$. If $\ot \in [0,\tau[$ we have $\frac{\mathrm{d}}{\mathrm{d}t} |\Ve (\ot)| \leq 0$ and thus \[ 0 \geq \frac{\mathrm{d}}{\mathrm{d}t} |\Ve (\ot)|\geq - \|a\|_{\linf{}} + \frac{(\alpha - \beta |\Ve (\ot)|^2)|\Ve (\ot)| }{\eps} =\frac{\lae ( |\Ve (\ot)| )}{\eps}. \] By construction $|\Ve (\ot)| < \ret$ and moreover, \[ |\Ve (\ot)| = \max _{[0,\tau]} |\Ve | \geq |v| > \reo\,, \] and thus, $\lae ( |\Ve (t)| )>0$ for all $t\in [0,T]$. Consequently, we infer that $t \to |\Ve (t)|$ is strictly increasing on $[0,T]$ since \[ \frac{\mathrm{d}}{\mathrm{d}t} |\Ve (t)|\geq - \|a\|_{\linf{}} + \frac{(\alpha - \beta |\Ve (t)|^2)|\Ve (t)| }{\eps} =\frac{\lae ( |\Ve (t)| )}{\eps} >0\,. \] Therefore we have $\ot = \tau$ saying that $|\Ve (\tau)| \geq |v|$ for any $\tau \in [0,T]$. 2. As before, it is sufficient to work on intervals $[0,T]$ such that $|\Ve (t) | > \reth (\|a\|_{\linf{}})$ for any $t \in [0,T]$. We are done if we prove that $t \to |\Ve (t)|$ is strictly decreasing on $[0,T]$. We have for any $t \in [0,T]$ \[ \frac{\mathrm{d}}{\mathrm{d}t} |\Ve (t)|\leq \|a\|_{\linf{}} + \frac{(\alpha - \beta |\Ve (t)|^2)|\Ve (t)| }{\eps} =\frac{\lae ( |\Ve (t)| )}{\eps} <0 \] where for the last inequality we have used $|\Ve (t) | > \reth, t \in [0,T]$. \end{proof} \section{The limit model} \label{LimMod} We investigate now the stability of the family $(\fe)_\eps$ when $\eps$ becomes small. After extraction of a sequence $(\eps_k)_k$ converging to $0$ we can assume that $(\fek)_k$ converges weakly $\star$ in $L^\infty(\R_+;{\cal M}_b (\R^d \times \R^d))$, meaning that \[ \limk \inttxv{\varphi (t,x,v) \fek (t,x,v)\,\dxv} = \inttxv{\varphi (t,x,v) f (t,x,v)\,\dxv} \] for any $\varphi \in \lotczcxv{}$. Using the weak formulation of \eqref{Equ10}-\eqref{Equ11} with test functions $\eta (t) \varphi (x,v)$, $\eta \in C^1 _c (\R_+)$, $\varphi \in C^1 _c (\R^d \times \R^d)$ one gets \begin{align*} \inttxv{\{\eta ^{\;\prime} (t) \varphi + \eta (t) v \cdot \nabla _x \varphi + \eta (t) a \cdot \nabla _v \varphi \}\fek(t,x,v)\,\dxv&}\\ + \frac{1}{\eps _k} \inttxv{\eta (t) \abv \cdot \nabla _v \varphi \fek(t,x,v)\,\dxv &} \\ = -\intxv{\eta (0) &\varphi (x,v) \fin(x,v)\,\dxv }. \end{align*} Multiplying by $\eps _k$ and passing to the limit for $k \to +\infty$ yields \[ \inttxv{\eta (t) \abv \cdot \nabla _v \varphi f (t,x,v)\,\dxv} = 0 \] and therefore one gets for any $t \in \R_+$ and $\varphi \in \cocxv{}$ \[ \intxv{\abv \cdot \nabla _v \varphi f (t,x,v)\,\dxv} = 0. \] Under the hypothesis $(1 + |v|^2) \fin \in \mbxv{}$ we deduce by Proposition \ref{KinBou} that $( 1 + |v|^2) f(t) \in \mbxv{}$ and therefore, applying the $(x,v)$ version of Proposition \ref{Kernel} (whose proof is detailed in the sequel), we obtain \[ \supp f(t) \subset \R^d \times (\A),\;\;t \in \R_+. \] The proof of Proposition \ref{Kernel} is based on the resolution of the adjoint problem \[ - \abv \cdot \nabla _v \varphi = \psi (v),\;\;v \in \R^d \] for any smooth righthand side $\psi$ with compact support in $^c(\A)$. \begin{proof} (of Proposition \ref{Kernel}) It is easily seen that for any $F \in \mbxv{}$, $\supp F \subset \A$ and any $\varphi \in \cocv{}$ we have \[ \intv{\abv \cdot \nabla _v \varphi (v) F(v)\,\dv} = 0 \] saying that $\Divv \{F \abv \} = 0$. Assume now that $\Divv \{F \abv \} = 0$ for some $F \in \mbxv{}$ and let us prove that $\supp F \subset \A$. We introduce the flow ${\cal V} = {\cal V}(s;v)$ given by \begin{equation} \label{Equ4} \frac{\mathrm{d}{\cal V}}{\mathrm{d}s} = ( \alpha - \beta |{\cal V} (s;v) |^2 ) {\cal V } (s;v),\;\;{\cal V}(0;v) = v. \end{equation} A direct computation shows that $\vsv$ are left invariant \[ \abv \cdot \nabla _v \left ( \vsv \right ) = (\alpha - \beta |v|^2 ) \imvv \vsv = 0 \] and therefore \[ {\cal V} (s;v) = |{\cal V}(s;v)| \vsv,\;\;v \neq 0. \] Multiplying \eqref{Equ4} by ${\cal V}(s;v) / |{\cal V}(s;v)|$ yields \[ \frac{\mathrm{d}}{\mathrm{d}s}|{\cal V}| = ( \alpha - \beta |{\cal V} (s;v) |^2 ) |{\cal V } (s;v)| \] whose solution is given by \[ |{\cal V}(s;v)| = |v| \frac{r e ^{\alpha s}}{\sqrt{|v|^2 ( e ^{2\alpha s} - 1) + r^2}} \] Finally one gets \begin{equation*} {\cal V}(s;v) = \frac{r e ^{\alpha s}}{\sqrt{|v|^2 ( e ^{2\alpha s} - 1) + r^2}}\;v,\;\;s \in ]S(v),+\infty[ \end{equation*} with $S(v) = - \infty$ if $0 \leq |v| \leq r$ and $S(v) = \frac{1}{2\alpha} \ln \left ( 1 - \frac{r^2}{|v|^2} \right ) < 0$ if $|v| > r$. Notice that the characteristics ${\cal V} (\cdot;v)$ are well defined on $\R_+$ for any $v \in \R^d$ and we have \[ \lim _ {s \to +\infty} {\cal V}(s;v) = r \vsv\;\mbox{ if } v \neq 0,\;\;\lim _ {s \to +\infty} {\cal V}(s;v) =0\;\mbox{ if } v = 0 \] and \[ \lim _{s \searrow S(v)} |{\cal V}(s)| = 0\mbox{ if }0 \leq |v| < r,\;\lim _{s \searrow S(v)} |{\cal V}(s)| =r\mbox{ if } |v| = r,\;\lim _{s \searrow S(v)} |{\cal V}(s)| =+\infty\;\mbox{ if } |v| >r. \] Let us consider a $C^1$ function $\psi = \psi (v)$ with compact support in $^c (\A)$. We intend to construct a bounded $C^1$ function $\varphi = \varphi (v)$ such that \begin{equation*} - \abv \cdot \nabla _v \varphi = \psi (v),\;\;v \in \R^d. \end{equation*} Obviously, if such a function exists, we may assume that $\varphi (0) = 0$. Motivated by the equality \[ - \frac{\mathrm{d}}{\mathrm{d}s} \{\varphi ({\cal V}(s;v)) \}= \psi ({\cal V}(s;v)),\;\;0 \leq |v| < r,\;\;- \infty < s \leq 0 \] and since we know that $\lim _{s \to - \infty} {\cal V} (s;v) = 0$ for any $0 \leq |v| < r$, we define \begin{equation} \label{Equ7} \varphi (v) = - \int _{-\infty} ^ 0 \psi ( {\cal V}(\tau; v))\;\mathrm{d}\tau,\;\;0 \leq |v| < r. \end{equation} Let us check that the function $\varphi$ in \eqref{Equ7} is well defined and is $C^1$ in $|v|<r$. The key point is that $\psi $ has compact support in $^c (\A)$ and therefore there are $0 < r_1 < r_2 < r < r_3 < r_4 < +\infty$ such that $ \supp \psi \subset \{ v \in \R ^d \;:\; r_1 \leq |v| \leq r_2 \} \cup \{ v \in \R^d \;:\; r_3 \leq |v| \leq r_4\}. $ It is easily seen that $\tau \to |{\cal V} (\tau; v)|$ is strictly increasing for any $0 < |v| < r$. Therefore, for any $|v| \leq r_1$ we have $ |{\cal V} (\tau; v) | \leq |{\cal V} (0; v) | = |v| \leq r_1,\;\;\tau \leq 0 $, implying that \[ \varphi (v) = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau; v))\;\mathrm{d}\tau = 0,\;\;0 \leq |v| \leq r_1. \] For any $v$ with $r_1 < |v| < r_2$ there are $\tau _1 < 0 < \tau _2$ such that $ |{\cal V}(\tau _1; v)| = r_1 < r_2 = |{\cal V}(\tau _2; v)|. $ The time interval between $\tau _1$ and $\tau _2$ comes easily by writing \[ \frac{\frac{\mathrm{d}}{\mathrm{d}\tau}|{\cal V}(\tau) |}{(\alpha - \beta |{\cal V}(\tau)|^2)|{\cal V}(\tau) |}= 1 \] implying that \[ |\tau _2 | + |\tau _1 | = \tau _2 - \tau _1 = \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }. \] From the equality \[ \varphi (v) = - \int _{-\infty} ^{\tau _1} \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau - \int _{\tau _1} ^0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau = - \int _{\tau _1} ^0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau\,, \] we deduce that \begin{equation} \label{Equ8} |\varphi (v) | \leq |\tau _1 | \; \|\psi \|_{C^0} \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0}. \end{equation} Assume now that $r_2 \leq |v| < r$. There is $\tau _2 \geq 0$ such that $v = {\cal V} ( \tau_2 ; r_2 \vsv)$ and therefore \begin{align*} \varphi (v) & = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau + \tau _2;r_2 \vsv))\;\mathrm{d}\tau \\ & = - \int _{-\infty} ^ {-\tau _2} \psi ({\cal V}(\tau + \tau _2 ;r_2 \vsv))\;\mathrm{d}\tau = - \int _{-\infty} ^ {0} \psi ({\cal V}(\tau ;r_2 \vsv))\;\mathrm{d}\tau = \varphi \left ( r_2 \vsv \right). \end{align*} In particular, the restriction of $\varphi$ on $r_2 \leq |v| < r$ satisfies the same bound as in \eqref{Equ8} \[ |\varphi (v) | \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0},\;\;r_2 \leq |v| < r. \] It is easily seen that $\varphi $ is $C^1$ on $0 \leq |v| < r$. For that it is sufficient to consider $r_1 \leq |v| \leq r_2$. Notice that \[ \frac{\partial {\cal V}}{\partial v} (\tau; v) = \frac{|{\cal V}(\tau;v)|}{|v|} \left ( I - \frac{{\cal V}(\tau;v) \otimes {\cal V}(\tau;v)}{r^2} ( 1 - e ^ {-2\alpha \tau } ) \right) \] and therefore the gradient of $\varphi$ remains bounded on $r_1 \leq |v| \leq r_2$ \[ \nabla _v \varphi (v) = - \int _{\tau _1} ^ 0 \frac{^ t \partial {\cal V}}{\partial v }(\tau; v) \nabla \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau \] since on the interval $\tau \in [\tau _1, 0]$ we have $|{\cal V}(\tau;v)| \in [r_1, |v|] \subset [r_1, r_2]$. Taking now as definition for $|v| = r$ \[ \varphi (v) = \varphi \left ( r_2 \vsv \right )\,, \] we obtain a bounded $C^1$ function on $|v| \leq r$ satisfying \[ - \abv \cdot \nabla _v \varphi = \psi (v),\;\;|\varphi (v) | \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0},\;|v|\leq r. \] We proceed similarly in order to extend the above function for $|v| > r$. We have for any $s>0$ \[ - \varphi ({\cal V}(s;v)) + \varphi (v) = \int _0 ^s \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau,\;\;|v|> r. \] As $\lim _{s \to +\infty} {\cal V}(s;v) = r \vsv$ we must take $$ \varphi (v) = \lim _{s \to +\infty}\left \{\varphi ( {\cal V}(s;v)) + \int _0 ^s \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau \right \} = \varphi \left (r\vsv \right ) + \int _0 ^{+\infty} \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau,\;\;|v| >r.\nonumber $$ Clearly, for any $|v| > r$ the function $\tau \to |{\cal V}(\tau;v)|$ is strictly decreasing. Therefore, for any $r < |v| \leq r_3$ we have \[ \varphi (v) = \varphi \left (r\vsv \right )= \varphi \left (r_2\vsv \right ) \] since $|{\cal V}(\tau;v)|\leq |v| \leq r_3$ and $\psi ({\cal V}(\tau;v)) = 0$, $\tau \geq 0$. If $r_3 < |v| < r_4$ let us consider $\tau _4 < 0 < \tau _3$ such that $ |{\cal V}(\tau _3;v)| = r_3 < r_4 = |{\cal V}(\tau _4;v)|. $ The time interval between $\tau _4$ and $\tau _3$ is given by \[ |\tau _3 | + |\tau _4 | = \tau _3 - \tau _4 = \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } < +\infty\,, \] and therefore one gets for $r_3 < |v| < r_4$ \begin{align} |\varphi (v) | &\leq \left | \varphi \left ( r \vsv \right ) \right | + \left |\int _0 ^{\tau _3} \!\!\!\!\psi ({\cal V}(\tau;v))\;\mathrm{d}\tau \right | \nonumber \\ & \leq \left [ \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } + \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } \right ] \|\psi \|_{C^0}.\label{Equ9} \end{align} Consider now $|v|\geq r_4$. There is $\tau _4 \geq 0$ such that $r_4 \vsv = {\cal V} (\tau_4; v)$ implying that \begin{align*} \varphi (v) & = \varphi \left ( r \vsv \right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; v)) \;\mathrm{d}\tau = \varphi \left ( r \vsv \right ) + \int _{\tau _4} ^{+\infty} \psi ({\cal V} (\tau; v)) \;\mathrm{d}\tau \\ & = \varphi \left ( r \vsv \right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; {\cal V}(\tau _4;v))) \;\mathrm{d}\tau = \varphi \left ( r \vsv \right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; r_4 \vsv)) \;\mathrm{d}\tau \\ & = \varphi \left ( r_4 \vsv \right ). \end{align*} We deduce that the restriction of $\varphi $ on $\{v :|v| \geq r_4\}$ satisfies the same bound as in \eqref{Equ9}. Moreover the function $\varphi $ is $C^1$ on $\{v:|v|\geq r\}$, with bounded derivatives. Indeed, it is sufficient to consider only the case $r_3 \leq |v| \leq r_4$, observing that \begin{eqnarray} \nabla _v \varphi (v) = \frac{r_2}{|v|} \imvv \nabla _v \varphi \left ( r_2 \vsv \right ) + \int _0 ^{\tau _3} \frac{^t \partial {\cal V}}{\partial v }(\tau;v)\nabla \psi ({\cal V}(\tau;v)) \;\mathrm{d}\tau \nonumber \end{eqnarray} \[ |{\cal V} (\tau; v)| \in [r_3, |v| ] \subset [r_3, r_4],\;\tau \in [0,\tau_3],\;\;|\tau _3| + |\tau _4| = \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } < +\infty. \] By construction we have $- \abv \cdot \nabla _v \varphi = \psi (v)$, $|v| >r$. Consider a $C^1$ decreasing function on $\R_+$ such that $\chi |_{[0,1]} = 1, \chi _{[2,+\infty[} = 0$. We know that \[ \intv{\abv \cdot \nabla _v \left \{ \varphi (v) \chi \left ( \frac{|v|}{R} \right ) \right \}\,F(v)\,\dv} = 0,\;\;R>0\,, \] saying that \[ \intv{\chi \left ( \frac{|v|}{R} \right )\abv \cdot \nabla _v \varphi \;F(v)\,\dv} + \intv{(\alpha - \beta |v|^2) \varphi (v) \frac{|v|}{R} \chi ^{\;\prime} \left ( \frac{|v|}{R} \right ) \;F(v)\,\dv} = 0. \] Since $\varphi$ and $\psi = - \abv \cdot \nabla _v \varphi $ are bounded and $F$ has finite mass and kinetic energy, we can pass to the limit for $R \to +\infty$, using the dominated convergence theorem. We obtain for any $C^1$ function $\psi$, with compact support in $^c(\A)$ \[ \intv{\psi (v) F(v)\,\dv} = - \intv{\abv \cdot \nabla _v \varphi\, F(v)\,\dv} = 0. \] Actually the previous equality holds true for any continuous function $\psi$ with compact support in $^c(\A)$, since $\intv{F(v)\,\dv} < +\infty$, so that $\supp F \subset \A$. \end{proof} In order to obtain stability for $(\fek)_k$ we need to avoid the unstable equilibrium $v = 0$. For that we assume that the initial support is away from zero speed: there is $r_0 >0$ (eventually small, let us say $r_0 < r$) such that \begin{equation} \label{Equ20} \supp \fin \subset \{ (x,v)\in \R ^d \times \R^d \;:\;|v| \geq r_0\}. \end{equation} \begin{pro} \label{UnifSupp} Under the hypothesis \eqref{Equ20} we have for any $\eps >0$ small enough \[ \supp \fe (t) \subset \{ (x,v)\in \R ^d \times \R^d \;:\;|v| \geq r_0\},\;\;t \in \R_+. \] \end{pro} \begin{proof} Take $\eps >0$ such that $\eps \|a\|_{\linf{}} < 2\alpha r /(3 \sqrt{3})$ and $\reo (- \|a\|_{\linf{}}) < r_0$. For any continuous function $\psi = \psi (x,v)$ with compact support in $\R ^d \times \{v\;:\; |v| < r_0\}$ we have \begin{align*} \intxv{\psi(x,v) \fe (t,x,v)\,\dxv} & = \intxv{\psi (\Xe (t;0,x,v), \Ve (t;0,x,v))\fin(x,v)\,\dxv } \\ & = \intxv{\psi (\Xe (t;0,x,v), \Ve (t;0,x,v) ){\bf 1}_{\{|v| \geq r_0 \}}\fin(x,v)\,\dxv}. \end{align*} But for any $|v| \geq r_0 > \reo$ we know by Proposition \ref{ZeroStab} that $|\Ve (t;0,x,v)| > |v| \geq r_0$, implying that $\psi (\Xe (t), \Ve (t)) = 0$. Therefore one gets $\int _{\R^d \times \R^d}{\psi(x,v) \fe (t,x,v)\,\dxv} = 0$ saying that $\supp \fe (t) \subset \{ (x,v):|v| \geq r_0\}$. \end{proof} We are ready now to establish the model satisfied by the limit measure $f$. The idea is to use the weak formulation of \eqref{Equ10}, \eqref{Equ11} with test functions which are constant along the flow of $\abv \cdot \nabla _v$, in order to get rid of the term in $\frac{1}{\eps}$. These functions are those depending on $x$ and $\vsv$. Surely, the invariants $\vsv$ have no continuous extensions in $v = 0$, but we will see that we can use it, since our measures $\fe$ vanish around $v = 0$. \begin{proof} (of Theorem \ref{MainResult}) We already know that $f$ satisfies \eqref{Equ23}. Actually, since $\supp \fe (t) \subset \{(x,v):|v|\geq r_0\}, t \in \R_+, \eps >0$, we deduce that $\supp f(t) \subset \{(x,v):|v| \geq r_0\}$ and finally $\supp f(t) \subset \R ^d \times r \sphere, t \in \R_+$. We have to establish \eqref{Equ22} and find the initial data. Consider a $C^1$ decreasing function $\chi $ on $\R_+$ such that $\chi |_{[0,1]} = 1, \chi _{[2,+\infty[} = 0$. For any $\eta = \eta (t) \in C^1_c (\R_+)$, $\varphi = \varphi (x,v) \in \cocxv{}$ we construct the test function \[ \theta (t,x,v) = \eta (t) \left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ] \varphi \left ( x, r\vsv \right ). \] Notice that $\theta $ is $C^1$ and $\theta = 0$ for $|v| \leq \frac{r_0}{2}$. When applying the weak formulation of \eqref{Equ10}-\eqref{Equ11} with $\theta$, the term in $\frac{1}{\eps}$ vanishes. Indeed, we can write \begin{align*} \frac{1}{\eps}\inttxv{\eta (t) & \abv \cdot \nabla _v \left \{\left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ]\varphi \left ( x, r\vsv \right ) \right \}\fe(t,x,v)\,\dxv } \nonumber \\ & = \frac{1}{\eps} \int _{\R_+} \eta (t) \int _{|v|\geq r_0} \abv \cdot \nabla _v \left \{ \varphi \left ( x, r\vsv \right ) \right \}\fe(t,x,v)\,\dxv \;\mathrm{d}t = 0.\nonumber \end{align*} For the term containing $\partial _t \theta$ we obtain the following limit when $k \to +\infty$ \begin{align*} T_1 ^k := \inttxv{\partial _t \theta \fek(t,x,v)\,\dxv} \to &\inttxv{\partial _t \theta f(t,x,v)\,\dxv} \\ & = \int _{\R_+} \eta ^{\;\prime} (t) \int _{|v|\geq r_0} \varphi \left ( x, r\vsv \right ) f(t,x,v)\,\dxv \;\mathrm{d}t \\ & = \int _{\R_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, r\vsv \right ) f(t,x,v)\,\dxv \;\mathrm{d}t \\ & = \int _{\R_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, v\right ) f(t,x,v)\,\dxv \;\mathrm{d}t \\ & = \inttxv{\partial _t ( \eta \varphi )f(t,x,v)\,\dxv}. \end{align*} Similarly, one gets \begin{align*} T_2 ^k := \inttxv{ v \cdot \nabla _x \theta \fek (t,x,v)\,\dxv} \to & \inttxv{v \cdot \nabla _x \theta f (t,x,v)\,\dxv} \\ & = \inttxv{v \cdot \nabla _x ( \eta \varphi ) f(t,x,v)\,\dxv}.\nonumber \end{align*} For the term containing $a \cdot \nabla _v \theta$ notice that on the set $|v| \geq r_0$ we have \[ a \cdot \nabla _v \theta = \eta (t) a \cdot \nabla _v \left \{ \varphi \left ( x, r\vsv \right )\right \} = \eta (t) \frac{r}{|v|}a \cdot \imvv (\nabla _v \varphi ) \left ( x, r\vsv \right ) \] and therefore we obtain \begin{align*} T_3 ^k := \inttxv{& a \cdot \nabla _v \theta \fek (t,x,v)\,\dxv} \to \inttxv{a \cdot \nabla _v \theta f (t,x,v)\,\dxv} \nonumber \\ & = \int _{\R_+} \eta (t) \int _{|v|\geq r_0} \frac{r}{|v|} \imvv a \cdot (\nabla _v \varphi ) \left ( x, r\vsv \right ) f(t,x,v)\,\dxv \;\mathrm{d}t \\ & = \inttxv{\;\;\;\imvv a \cdot \nabla _v (\eta \varphi ) f(t,x,v)\,\dxv}.\nonumber \end{align*} For treating the term involving the initial condition, we write \begin{align*} T_4 : = \intxv{\theta (0,x,v) \fin(x,v)\,\dxv } &= \intxv{\eta (0) \varphi \left ( x, r \vsv \right ) \fin(x,v)\,\dxv } \\ &= \intxv{\eta (0) \varphi (x,v) \ave{\fin}(x,v)\,\dxv}. \end{align*} Passing to the limit for $k \to +\infty$ in the weak formulation $T_1 ^ k + T_2 ^ k + T_3 ^ k + T_4 = 0$ yields the problem \[ \partial _t f + \Divx\{f v \} + \Divv \left \{f \imvv a \right \} = 0,\;\;f(0) = \ave{\fin} \] as desired. \end{proof} \begin{remark} \label{ConstraintPropagation} The constraint \eqref{Equ23} is propagated by the evolution equation \eqref{Equ22}. This comes by the fact that the flow $(X,V)$ associated to the field $v \cdot \nabla _x + \imvv a \cdot \nabla _v$ leaves invariant $\R^d \times r\sphere$. Indeed, if $(X,V)$ solves \[ \frac{\mathrm{d}X}{\mathrm{d}s} = V(s),\;\;\frac{\mathrm{d}V}{\mathrm{d}s} = \left (I - \frac{V(s) \otimes V(s)}{|V(s)|^2} \right ) a(s, X(s)) \] \[ X(s;0,x,v) = x,\;\;V(s;0,x,v) = v \neq 0 \] then \[ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}s}|V(s)|^2 = \left (I - \frac{V(s) \otimes V(s)}{|V(s)|^2} \right ) a(s, X(s)) \cdot V(s) = 0 \] saying that $|V(s;0,x,v)| = |v|$ for any $(s,x) \in \R_+ \times \R^d$. In particular, for any continuous function $\psi = \psi (x,v)$ with compact support in $^c (\R ^d \times r\sphere)$ we have \begin{align*} \intxv{\psi(x,v) f(s,x,v)\,\dxv} & = \intxv{\psi (X(s;0,x,v), V(s;0,x,v)) \ave{\fin}(x,v)\,\dxv} \\ & = \int _{|v| = r} \psi (X(s;0,x,v), V(s;0,x,v)) \ave{\fin}(x,v)\,\dxv = 0 \end{align*} since $\supp \ave{\fin} \subset \R ^d \times r\sphere$. Therefore for any $s \in \R_+$ we have $\supp f(s) \subset \R ^d \times r\sphere$ implying that $\Divv \{f(s)\abv \} = 0, s \in \R_+$. \end{remark} \begin{remark} \label{Uni} By the uniqueness of the solution for \eqref{Equ22} with initial data $\ave{\fin}$, we deduce that all the family $(\fe)_\eps$ converges weakly $\star$ in $\litmbxv{}$. \end{remark} \section{The non linear problem} \label{NLimMod} Up to now we considered the stability of the linear problems \eqref{Equ10}-\eqref{Equ11} for a given smooth field $a = a(t,x) \in \litwoix{}$. We concentrate now on the non linear problem \begin{equation} \label{Equ41} \partial _t \fe + \Divx\{\fe v\} + \Divv \{ \fe a^\eps\} + \frac{1}{\eps} \{ \fe \abv \}= 0,\;\;(t,x,v) \in \R_+ \times \R ^d \times \R ^d \end{equation} with $a^\eps = - \nabla _x U \star \rho ^\eps - H \star \fe.$ The well posedness of the non linear equation \eqref{Equ41} comes by fixed point arguments in suitable spaces of measures, and it has been discussed in \cite{CCR10, BCC12} in the measure solution framework. We summarize next the properties of the solutions $(\fe)_{\eps >0}$. \begin{pro} \label{ExiUniNonLin} Assume $h \in C^1_b (\R^d), U \in C^2 _b (\R ^d)$ and $( 1 + |v|^2 ) \fin \in \mbxv{}$. For all $\eps >0$, there is a unique solution $(\fe, a ^\eps) \in C(\R_+;\poxv{}) \times \litwoix{}$ to \begin{equation} \label{Equ43} \partial _t \fe + \Divx \{\fe v \} + \Divv \{\fe \aeps\} + \frac{1}{\eps}\Divv \{ \fe \abv \} = 0,\;\;(t,x,v) \in \R _+ \times \R ^d \times \R ^d \end{equation} \begin{equation} \label{Equ44} \aeps = - \nabla _x U \star \int _{\R ^d} \fe \;\md v - H \star \fe,\;\;H(x,v) = h(x)v \end{equation} with initial data $\fe (0) = \fin$, satisfying the uniform bounds \[ \sup _{\eps >0, t \in \R_+} \intxv{|v|^2 \fe (t,x,v)\;\dxv}<+\infty \] \[ \sup _{\eps >0} \|\aeps \|_{L^\infty(\R_+;L^\infty(\R^d))} = :A <+\infty,\;\;\sup _{\eps >0} \|\nabla _x \aeps \|_{L^\infty(\R_+;L^\infty(\R^d))} = :A_1 <+\infty. \] Moreover, if the initial condition satisfies \begin{equation*} \supp \fin \subset \{ (x,v) \in \R ^d \times \R ^d \;:\;|x| \leq L_0, r_0 \leq |v| \leq R_0 \} \end{equation*} for some $L_0 >0, 0 < r_0 < r < R_0 < +\infty$, then for any $\eps >0$ small enough we have \[ \supp \fe (t) \subset \{ (x,v) \in \R ^d \times \R ^d \;:\;|x|\leq L_0 + t R_0, r_0 \leq |v| \leq R_0 \},\;\;t \in \R_+. \] \end{pro} \begin{proof} Here, we only justify the uniform bounds in $\eps$, the rest is a direct application of the results in \cite{CCR10, BCC12}. The divergence form of \eqref{Equ43} guarantees the mass conservation \[ \intxv{\,\fe (t,x,v)\;\dxv } = \intxv{\,\fin (x,v)\;\dxv},\;\;t \in \R_+. \] Notice that the term $- \Divv \{ \fe H \star \fe\}$ balances the momentum $$ \int _{\R^{2d}}{\!\! v \Divv \{ \fe H \star \fe\}\;\dxv} = \int _{\R^{4d}}{\!\! h(x-x^{\prime}) (v ^{\prime}-v) \fe (t, x^\prime, v^\prime)\fe (t,x,v)\;\md (x^\prime, v^\prime)}\dxv = 0 $$ and decreases the kinetic energy \begin{align*} \int _{\R^{2d}}{\!\!\!|v|^2 \Divv \{ \fe H \star \fe\}\;\dxv} &= 2\int _{\R^{4d}}{{\!\!\!\!h(x-x^{\prime}) (v ^{\prime}-v) \cdot v \fe (t, x^\prime, v^\prime)\fe (t,x,v)\;\md (x^\prime, v^\prime)}\dxv} \\ & = - \int _{\R^{4d}}{{\!\!h(x-x^{\prime}) |v - v ^{\prime}|^2 \fe (t, x^\prime, v^\prime)\fe (t,x,v)\;\md (x^\prime, v^\prime)}\dxv}. \end{align*} In particular, as $|v|^2 \fin \in \mbxv{}$, then the kinetic energy $\int _{\R^{2d}}{|v|^2 \fe (t,x,v)\;\dxv}$ remains bounded, uniformly in time $t \in \R_+$ and $\eps >0$. Indeed, using the continuity equation one gets \[ \intxv{v \cdot (\nabla _x U \star \rho ^\eps ) \fe (t,x,v)\;\dxv} = \frac{1}{2}\frac{\md}{\md t} \int _{\R^d}{(U \star \rho ^\eps (t))(x) \rho ^\eps (t,x)\;\md x} \] and after multiplying \eqref{Equ43} by $\frac{|v|^2}{2}$ together with \eqref{Equ44}, we obtain \begin{align} \frac{\md }{\md t} \intxv{&\,\,\left (\frac{|v|^2}{2} +\frac{U \star \rho ^\eps}{2} \right ) \fe (t,x,v) \;\dxv } - \frac{1}{\eps}\intxv{(\alpha |v|^2 - \beta |v|^4) \fe (t,x,v) \;\dxv} \nonumber \\ & = - \frac{1}{2}\int_{\R^{4d }}h(x-x^{\prime}) |v - v ^{\prime}|^2 \fe (t, x^\prime, v^\prime)\fe (t,x,v)\;\md (x^\prime, v^\prime)\dxv\leq 0\,.\label{energy} \end{align} Consider now $t ^\eps $ a maximum point on $[0,T], T>0$, of the total energy \[ W^\eps (t) = \intxv{\left (\frac{|v|^2}{2} + \frac{U \star \rho ^\eps}{2} \right ) \fe (t,x,v) \;\dxv },\;\;t \in [0,T]. \] If $t^\eps = 0$ then it is easily seen that for any $t \in [0,T]$ \[ \intxv{\frac{|v|^2}{2} \fe (t,x,v) \;\dxv} \leq \intxv{\frac{|v|^2}{2} \fin (x,v) \;\dxv} + \|U \| _{\linf} \left ( \intxv{\fin \;\dxv}\right ) ^2. \] If $t ^\eps \in ]0,T]$ then $\frac{\md }{\md t} W ^\eps (t^\eps) \geq 0$ implying from \eqref{energy} by moment interpolation in $v$ that \[ \sup _{\eps >0, T>0} \intxv{(1 + |v|^4) \fe (t ^\eps ,x,v) \;\dxv } <+\infty \] and thus the inequality $W^\eps (t) \leq W ^\eps (t ^\eps), t \in [0,T]$ yields \begin{align*} \sup _{\eps >0, t \in [0,T]} \intxv{\frac{|v|^2}{2} \fe (t,x,v) \;\dxv} \leq & \sup _{\eps >0, T>0} \intxv{\frac{|v|^2}{2} \fe (t ^\eps ,x,v) \;\dxv } \\ & + \|U \|_{\linf} \left ( \intxv{\fin \;\dxv}\right ) ^2 <+\infty. \end{align*} Therefore the kinetic energy remains bounded on $[0,T]$, uniformly with respect to $\eps >0$, and the bound does not depend on $T>0$. The uniform bounds for $\aeps$ come immediately by convolution with $\nabla _x U$ and $H$, thanks to the uniform estimate \[ \sup _{\eps >0, t \in \R_+} \intxv{|v| \fe (t,x,v)} < +\infty. \] We analyze the support of $(\fe )_{\eps >0}$. Take $\eps >0$ small enough such that $\eps A < 2 \alpha r /(3\sqrt{3})$ and $\reo (-A) < r_0,\; \reth (A) < R_0$. By Proposition \ref{UnifSupp} we already know that \[ \supp \fe (t) \subset \{ (x,v) \in \R ^d \times \R ^d \;:\;|v | \geq r_0\},\;\;t \in \R_+. \] For any continuous function $\psi = \psi (x,v)$ with compact support in $\R ^d \times \{ v\in \R ^d\;:\;|v| > R_0\}$ we have \begin{align*} \intxv{\psi(x,v) \fe(t,x,v) \;\dxv} & = \intxv{\psi (\Xe (t), \Ve (t)) \fin(x,v) \;\dxv } \\ & = \intxv{\psi (\Xe (t), \Ve (t) ) \ind{\{r_0 \leq |v| \leq R_0\}}\fin(x,v) \;\dxv}. \end{align*} We distinguish several cases:\\ 1. If $r_0 \leq |v| < \ret (-A)$ we deduce by Proposition \ref{ZeroStab} that $ |v| < |\Ve (t;0,x,v)| \leq \reth (A) < R_0,\;\;t \in \R_+, \eps >0. $ 2. If $\ret (-A) \leq |v| \leq \reth (A)$ we obtain by Proposition \ref{RStab} that $ \ret (-A) \leq |\Ve (t;0,x,v)| \leq \reth (A) < R_0,\;\;t \in \R_+, \eps >0. $ 3. If $\reth (A) < |v| \leq R_0$ one gets thanks to Proposition \ref{ZeroStab} $ \ret (-A) \leq |\Ve (t;0,x,v) | < |v| \leq R_0. $ In all cases $(\Xe, \Ve )(t;0,x,v)$ remains outside the support of $\psi$, implying that \[ \intxv{\psi (x,v) \fe (t,x,v)\;\dxv} = 0. \] Thus for any $t \in \R_+$ and $\eps >0$ small enough one gets \[ \supp \fe (t) \subset \{ (x,v) \in \R ^d \times \R ^d \;:\;r_0 \leq |v | \leq R_0\}. \] Consider $\theta \in C^1 (\R)$ non decreasing, verifying $\theta (u) = 0$ if $u \leq 0$, $\theta (u) >0$ if $u>0$. Applying the weak formulation of \eqref{Equ43}-\eqref{Equ44} with the test function $\theta (|x| - L_0 - t R_0)$ yields \begin{align*} \intxv{\theta (|x| - L_0 -& t R_0) \fe (t,x,v)\;\dxv} = \intxv{\theta (|x| - L_0 )\fin(x,v)\;\dxv} \\ & + \int _0 ^t \intxv{\theta ^{\prime}(|x| - L_0 - s R_0) \left ( v \cdot \frac{x}{|x|} - R_0\right )\fe (s,x,v)\;\dxv} \md s \leq 0 \end{align*} implying that $\supp \fe (t) \subset \{ (x,v) \in \R ^d \times \R ^d :|x| \leq L_0 + t R_0\}, t \in \R_+$. \end{proof} \ The uniform bound for the total mass allows us to extract a sequence $(\eps _k)_k \subset \R_+ ^\star$ convergent to $0$ such that $(\fek)_k$ converges weakly $\star$ in $\litmbxv{}$. The treatment of the non linear term requires a little bit more, that is convergence in $C(\R_+;\poxv{})$ or at least in $C([\delta, +\infty[;\poxv{})$ for any $\delta >0$. The key argument for establishing that is emphasized by the lemma \begin{lemma} \label{TimeEstimate} Consider $\eps >0$ small enough.\\ 1. For any $(x,v)\in \R ^d \times \R ^d$ with $r_0 \leq |v| < \ret (-A) - \eps$, the first time $t^\eps _1 = t ^\eps _1 (x,v)$ such that $|\Ve (t^\eps _1;0,x,v) | = \ret (-A) - \eps$ satisfies \[ t ^\eps _1 \leq \frac{\eps}{2\beta r_0 ^2} \ln \left ( \frac{r - r_0 }{\eps} \right ). \] 2. For any $(x,v)\in \R ^d \times \R ^d$ with $\reth (A) + \eps < |v| \leq R_0$, the first time $t^\eps _2 = t ^\eps _2 (x,v)$ such that $|\Ve (t^\eps _2;0,x,v) | = \reth (A) + \eps$ satisfies \[ t ^\eps _2 \leq \frac{\eps}{2\beta r ^2} \ln \left ( \frac{R_0 - r}{\eps} \right ). \] \end{lemma} \begin{proof} 1. During the time $[0,t ^\eps _1]$ the velocity modulus $|\Ve (t)|$ remains in $[r_0, \ret (-A) - \eps] \subset [\reo (-A), \ret (-A)]$ and we can write for any $t \in [0, t ^\eps _1]$ \[ \frac{\eps \frac{\md |\Ve |}{\md t }}{- \eps A + (\alpha - \beta |\Ve (t) |^2 ) \;|\Ve (t) |}\geq \frac{\frac{\md |\Ve |}{\md t }}{\aeps (t, \Xe (t)) \cdot \frac{\Ve (t)}{|\Ve (t)|} + \frac{1}{\eps}(\alpha - \beta |\Ve (t) |^2 ) \;|\Ve (t) |} = 1 \] since $- \eps A + (\alpha - \beta u^2)u$ is positive for $u \in [\reo (-A), \ret (-A)]$. Integrating with respect to $t \in [0, t ^\eps _1]$ yields \[ t ^\eps _1 (x,v) \leq \eps \int _{|v|} ^{\ret (-A) - \eps } \frac{\md u }{- \eps A + (\alpha - \beta u ^2 ) u } \leq \eps \int _{r_0} ^{\ret (-A) - \eps } \frac{\md u }{- \eps A + (\alpha - \beta u ^2 ) u }. \] Recall that $\ret (-A)$ is one of the roots of $u \to - \eps A + (\alpha - \beta u ^2 ) u$ and therefore a direct computation lead to \[ - \eps A + (\alpha - \beta u ^2 ) u = \beta (\ret - u ) [u ^2 + u \ret + (\ret ) ^2 - r^2] \geq 2 \beta r_0 ^2 ( \ret - u), \;\;u \in [r_0, \ret],\;\eps \;\mbox{small enough } \] implying that \[ t ^\eps _1 (x,v) \leq \frac{\eps}{2\beta r_0 ^2} \int _{r_0} ^{\ret - \eps} \frac{\md u }{ \ret - u} = \frac{\eps}{2\beta r_0 ^2}\ln \left (\frac{\ret - r_0}{\eps} \right ) \leq \frac{\eps}{2\beta r_0 ^2}\ln \left (\frac{r - r_0}{\eps} \right ). \] 2. During the time $[0,t ^\eps _2]$ the velocity modulus $|\Ve (t)|$ remains in $[\reth (A) + \eps, R_0] \subset [\reth(A), +\infty[$ and we can write for any $t \in [0, t ^\eps _2]$ \[ \frac{\eps \frac{\md |\Ve |}{\md t }}{ \eps A + (\alpha - \beta |\Ve (t) |^2 ) \;|\Ve (t) |}\geq \frac{\frac{\md |\Ve |}{\md t }}{\aeps (t, \Xe (t)) \cdot \frac{\Ve (t)}{|\Ve (t)|} + \frac{1}{\eps}(\alpha - \beta |\Ve (t) |^2 ) \;|\Ve (t) |} = 1 \] since $ \eps A + (\alpha - \beta u^2)u$ is negative for $u \in [\reth (A), +\infty[$. Integrating with respect to $t \in [0, t ^\eps _2]$ yields \[ t ^\eps _2 (x,v) \leq \eps \int _{|v|} ^{\reth (A) + \eps } \frac{\md u }{ \eps A + (\alpha - \beta u ^2 ) u } \leq \eps \int _{R_0} ^{\reth (A) + \eps } \frac{\md u }{ \eps A + (\alpha - \beta u ^2 ) u }. \] By direct computation we obtain \[ \eps A + (\alpha - \beta u ^2 ) u = - \beta (u - \reth ) [u ^2 + u \reth + (\reth ) ^2 - r^2] \leq -2 \beta r ^2 ( u - \reth ), \;\;u \geq \reth,\;\eps \;\mbox{small enough } \] implying that \[ t ^\eps _2 (x,v) \leq \frac{\eps}{2\beta r ^2} \int _{\reth + \eps} ^{R_0} \frac{\md u }{ u - \reth} = \frac{\eps}{2\beta r ^2}\ln \left (\frac{R _0 - \reth }{\eps} \right ) \leq \frac{\eps}{2\beta r ^2}\ln \left (\frac{R_0 - r}{\eps} \right ). \] \end{proof} We intend to apply Arzela-Ascoli theorem in $C(\R_+;\Po(\R ^d \times \R ^d))$ in order to extract a convergent sequence $(\fek)_k$ with $\limk \eps _k = 0$. We need to establish the uniform equicontinuity of the family $(\fe)_{\eps >0}$. The argument below is essentially similar to arguments in \cite{CCR10}. \begin{pro} \label{UnifEquiCont} 1. If the initial data is well prepared {\it i.e.,} $\supp \fin \subset \{ (x,v) \in \R ^d \times \R ^d\;:\;|x|\leq L_0, |v| = r\}$ then there is a constant $C$ (not depending on $t \in \R_+, \eps >0$) such that \[ W_1 (\fe (t), \fe (s)) \leq C | t - s|,\;\;t, s \in \R_+, \eps >0. \] 2. If $\supp \fin \subset \{ (x,v) \in \R ^d \times \R ^d\;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$ then there is a constant $C$ (not depending on $t \in \R_+, \eps >0$) such that for any $\delta >0$ we can find $\eps _\delta$ satisfying \[ W_1 (\fe (t), \fe (s)) \leq C |t - s|,\;\; t,s \geq \delta,\;\;0 < \eps < \eps _\delta. \] \end{pro} \begin{proof} 1. Consider $\varphi = \varphi (x,v)$ a Lipschitz function on $\R^d \times \R ^d$ with $\mathrm{Lip} (\varphi ) \leq 1$. For any $t, s \in \R_+, \eps >0$ we have \begin{align*} \left | \intxv{\!\!\!\varphi ( \fe (t) - \fe (s))\dxv}\right | &= \left | \intxv{\!\!\!\{ \varphi (\Xe (t), \Ve (t)) - \varphi (\Xe (s), \Ve (s))\}\fin (x,v)\dxv}\right | \\ & \leq \intxv{\{ |\Xe (t) - \Xe (s)| + |\Ve (t) - \Ve (s)|\} \ind{\{|v| = r\}} \fin \dxv}. \end{align*} Thanks to Proposition \ref{RStab} we have for any $(\tau, x, v) \in \R_+ \times \R ^d \times r \sphere $ \[ \frac{\ret (-A) - r}{\eps} \leq \frac{|\Ve (\tau;0,x,v)| - r}{\eps} \leq \frac{\reth (A) - r}{\eps} \] and it is easily seen, integrating the system of characteristics between $s$ and $t$, that \[ |\Xe (t;0,x,v) - \Xe (s;0,x,v) | = \left | \int _s ^ t \Ve (\tau;0,x,v) \;\mathrm{d}\tau\right | \leq R_0 |t-s| \] and \begin{align*} \left | \Ve (t;0,x,v) - \Ve (s;0,x,v) \right | & \leq \left | \int _s ^t \left \{ |a^\eps(\tau, \Xe (\tau))| + \frac{|\alpha - \beta |\Ve (\tau) | ^2 | \;|\Ve (\tau) |}{\eps} \right \}\md \tau \right |\nonumber \\ & \leq |t -s | \left \{ A + \beta ( r + R_0) R_0 \max \left ( \frac{\reth (A) - r}{\eps}, \frac{r - \ret (-A) }{\eps} \right ) \right \}. \end{align*} Our conclusion comes immediately by Propositions \ref{NegA}, \ref{PosA}.\\ 2. Consider $\delta >0$ and $\eps _\delta $ small enough such that $\frac{\eps}{2\beta r_0 ^2} \ln \left ( \frac{r - r_0}{\eps} \right ) < \delta$, $\frac{\eps}{2\beta r ^2} \ln \left ( \frac{R_0 - r}{\eps} \right ) < \delta$ for $0 < \eps < \eps _\delta$. For any Lipschitz function $\varphi $ with $\mathrm{Lip} (\varphi ) \leq 1$ and any $t, s \geq \delta$ we have \[ \left | \intxv{\!\!\!\!\varphi ( \fe (t) - \fe (s) ) \;\dxv}\right | \leq \!\!\intxv{\{|\Xe (t) - \Xe (s) | + |\Ve (t) - \Ve (s)| \} \ind{\{r_0 \leq |v| \leq R_0\}} \fin \;\dxv}. \] For any $(\tau, x) \in \R_+ \times \R ^d$, $\ret (-A) - \eps \leq |v| \leq \reth (A) + \eps$ we have by Propositions \ref{RStab}, \ref{ZeroStab} \[ \ret (-A) - \eps \leq |\Ve (\tau;0,x,v) | \leq \reth (A) + \eps. \] The same conclusion holds true for any $\tau \geq \delta$, $x \in \R^d$ and $|v| \in [r_0, \ret (-A) - \eps[ \cup ]\reth (A) + \eps, R_0]$, thanks to Lemma \ref{TimeEstimate}, since $\delta > \max \{ t^\eps _1 (x,v), t^\eps _2 (x,v)\}$ (after a time $\delta$, the velocity modulus $|\Ve (\tau;0,x,v)|$ is already in the set $\{w\;:\;\ret (-A) - \eps < |w| < \reth (A) + \eps \}$). Our statement follows as before, integrating the system of characteristics between $s$ and $t$. \end{proof} Applying Arzela-Ascoli theorem, we deduce that there is a sequence $(\eps _k)_k \subset \R _+ ^\star$, convergent to $0$ such that \[ \limk W_1 (\fek (t), f(t)) = 0 \mbox{ uniformly for } t \in [0,T],\;\; T>0 \] for some $f \in C(\R_+;\Po (\R ^d \times \R ^d))$ if $\supp \fin \subset \{(x,v)\in \R ^d \times \R ^d\;:\;|x| \leq L_0,|v| = r\}$ and \[ \limk W_1 (\fek (t), f(t)) = 0 \mbox{ uniformly for } t \in [\delta ,T],\;\; T>\delta >0 \] for some $f \in C(\R_+ ^\star;\Po (\R ^d \times \R ^d))$ if $\supp \fin \subset \{(x,v)\in \R ^d \times \R ^d\;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$. It is easily seen that if the initial condition is well prepared then there is a constant $C$ cf. Proposition \ref{UnifEquiCont} such that $ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \in \R_+. $ The same is true for not prepared initial conditions $\fin$. Take $\delta >0$ and $\eps _\delta$ as in Proposition \ref{UnifEquiCont}. For any $0 < \eps < \eps _\delta$ we have $ W_1 (\fe(t), \fe(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ For $k$ large enough we have $\eps _k < \eps _\delta$ and therefore $ W_1 (\fek(t), \fek(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ Passing to the limit as $k$ goes to infinity yields $ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ Since the constant $C$ does not depend on $\delta$ one gets \[ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s >0. \] In particular we deduce that $f$ has a limit as $t$ goes to $0$ since $( \Po (\R ^d \times \R ^d), W_1)$ is a complete metric space and therefore we can extend $f$ by continuity at $t = 0$. The extended function, still denoted by $f$, belongs to $C(\R_+;\Po (\R ^d \times \R ^d))$ and satisfies \[ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \in \R_+. \] The above convergence allows us to handle the non linear terms. We use the following standard argument \cite{Dob79,CCR10}. \begin{lemma} \label{NonLinTerm} Consider $f,g \in \Po (\R ^d \times \R ^d)$ compactly supported $ \supp f \cup \supp g \subset \{ (x,v) \in \R ^d \times \R ^d \;:\;|x|\leq L, |v| \leq R\}$, and let us consider \[ a_f = - \nabla _x U \star \int _{\R ^d} f \;\md v - H \star f,\;\;a_g = - \nabla _x U \star \int _{\R ^d} g \;\md v - H \star g. \] Then we have \[ \|a_f - a_g \|_{L^\infty ( \R ^3 \times B_R)} \leq \left \{\|\nabla _x ^2 U \|_{\linf} + \left (\|h \|^2 _{\linf} + 4 R ^2 \|\nabla _x h \| ^2 _{\linf} \right ) ^{1/2} \right \}W_1 (f,g) \] where $B_R$ stands for the closed ball in $\R ^d$ of center $0$ and radius $R$. \end{lemma} \begin{proof} Take $\pi $ to be a optimal transportation plan between $f$ and $g$. Then for any $x \in \R^d$ we have, using the marginals of $\pi$ \begin{align*} | (\nabla _x U \star f) (x) - (\nabla _x U \star g) (x) | &= \left | \intxv{\nabla _x U ( x - x^\prime) \{ f(\xp, \vp) - g(\xp, \vp)\}\;\dxpvp} \right | \\ & = \left | \intxv{\intxv{ [\nabla _x U (x - \xp) - \nabla _x U (x - \xs)]\md \pi (\xp, \vp, \xs, \vs)}} \right | \\ & \leq \|\nabla _x ^2 U \|_{\linf{}} \intxv{\intxv{|\xp - \xs | \;\md \pi (\xp, \vp, \xs, \vs)}} \\ & \leq \|\nabla _x ^2 U \|_{\linf{}} W_1 (f,g). \end{align*} The estimate for $H \star f - H \star g$ follows similarly observing that on the support of $\pi$, which is included in $\{(\xp, \vp, \xs, \vs)\in \R ^{4d}\;:\; |\vp|\leq R, |\vs | \leq R\}$ we have \begin{align*} |h(x- \xp) (v- \vp) - h(x- \xs) & (v- \vs)| \\ & \leq |h(x- \xp)( \vs - \vp)| + |h(x- \xp)- h(x- \xs)| \;|v - \vs | \\ & \leq \left ( \|h \|^2 _{\linf} + 4 R ^2 \|\nabla _x h \| ^2 _{\linf} \right ) ^{1/2} \left ( |\xp - \xs | ^2 + |\vp - \vs |^2 \right ) ^{1/2}. \end{align*} \end{proof} We are ready now to prove Theorem \ref{MainResult2}. \begin{proof} (of Theorem \ref{MainResult2}) The arguments are the same as those in the proof of Theorem \ref{MainResult} except for the treatment of the non linear terms. We only concentrate on it. Consider $(\fek )_k$ with $\limk \eps _k = 0$ such that $\limk W_1 (\fek (t), f(t)) = 0$ uniformly for $t \in [0,T], T>0$ if $\supp \fin \subset \{ (x,v) \;:\;|x|\leq L_0, |v| = r\}$ and $\limk W_1 (\fek (t), f(t)) = 0$ uniformly for $t \in [\delta,T], T>\delta >0$ if $\supp \fin \subset \{ (x,v) \;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$ for some function $f \in C(\R_+;\Po (\R ^d \times \R ^d))$. Thanks to Proposition \ref{Kernel} we deduce (for both prepared or not initial data) that \[ \supp f(t) \subset \{ (x,v)\in \R ^d \times \R ^d \;:\;|v| = r\},\;\;t>0. \] The previous statement holds also true at $t = 0$, by the continuity of $f$. The time evolution for the limit $f$ comes by using the particular test functions \[ \theta (t,x,v) = \eta (t) \left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ] \varphi \left ( x, r\vsv \right ) \] with $\eta \in C^1_c (\R_+)$, $\varphi \in \cocxv{}$. From now on we consider only the not prepared initial data case (the other case is simpler). We recall the notation $\aeps = - \nabla _x U \star \int _{\R^d} \fe \;\md v - H \star \fe $ and we introduce $a = - \nabla _x U \star \int _{\R^d} f \;\md v - H \star f$. Since $f$ satisfies the same bounds as $(\fe)_\eps$, we deduce that $\|a\|_{\linf{}} \leq A, \|\nabla _x a\|_{\linf{}} \leq A_1$. For any $\delta >0$ we can write \begin{align} \label{EquBil} &\left |\inttxv{\left \{ \aek \cdot \nabla _v \theta \;\fek - a \cdot \nabla _v \theta \;f\right \}\dxv\!\!} \right | \leq \left |\int _0 ^\delta \intxv{\aek \cdot \nabla _v \theta \fek \;\dxv \md t } \right | \nonumber \\ &\qquad + \left |\int _0 ^\delta \intxv{a \cdot \nabla _v \theta \;f \;\dxv \md t} \right | + \left | \int _\delta ^{+\infty} \!\!\intxv{\left \{ \aek \cdot \nabla _v \theta \;\fek - a \cdot \nabla _v \theta \;f \right \}\;\dxv \md t} \right | \nonumber \\ \leq &\, 2 A \delta \|\nabla _v \theta \|_{C^0} \intxv{\fin \;\dxv} + \left | \int _\delta ^{+\infty}\!\! \intxv{ (\aek - a) \cdot \nabla _v \theta \;\ind{\{|v|\leq R_0\}}\fek \;\dxv }\md t \right | \nonumber \\ & + \left | \int _\delta ^{+\infty}\!\! \intxv{ a \cdot \nabla _v \theta \;(\fek - f)\;\dxv}\md t \right |. \end{align} We keep $\delta >0$ fixed and we pass to the limit when $k$ goes to infinity. Lemma \ref{NonLinTerm} implies that the second term in the last right hand side can be estimated as \[ \|\aek - a \|_{\linf (\R^d \times B_{R_0})} = \| a_{\fek} - a_f \|_{\linf (\R^d \times B_{R_0})} \leq C(R_0) W_1 (\fek (t), f(t)) \to 0 \;\mbox{ when } k \to +\infty \] uniformly for $t \in [\delta, T]$, implying, for $T$ large enough \[ \left | \int _\delta ^{+\infty}\!\! \int _{|v| \leq R_0}{ (\aek - a) \cdot \nabla _v \theta \fek \;\dxv }\md t \right | \leq C(R_0)\| \theta \|_{C^1} \int _\delta ^T W_1 (\fek (t), f(t))\;\md t \to 0 \] when $k$ goes to infinity. For the third term in the right hand side of \eqref{EquBil} we use the weak $\star$ convergence $\limk \fek (t) = f(t)$ in $\mbxv{}$ for any $t\geq \delta$, cf. Proposition \ref{w2properties} \[ \limk \intxv{a \cdot \nabla _v \theta (\fek (t)- f(t)) \;\dxv } = 0,\;\;t\geq \delta \] and we conclude by the Lebesgue dominated convergence theorem \[ \limk \int _\delta ^{+\infty} \intxv{ a \cdot \nabla _v \theta (\fek (t,x,v) - f(t,x,v) )\;\dxv}\md t = 0\,. \] Passing to the limit in \eqref{EquBil} when $k$ goes to infinity, we obtain \[ \limsup _{k \to +\infty} \left |\inttxv{\left \{ \aek \cdot \nabla _v \theta \fek - a \cdot \nabla _v \theta f\right \}\;\dxv\!\!} \right | \leq 2 A \delta \|\nabla _v \theta \|_{C^0} \,. \] Sending $\delta$ to $0$ we obtain that \[ \limk \inttxv{ \aek \cdot \nabla _v \theta \; \fek\;\dxv\!\!} = \inttxv{ a \cdot \nabla _v \theta \; f\;\dxv\!\!}\,. \] \end{proof} \section{Diffusion models}\label{DiffMod} We intend to introduce a formalism which will allow us to investigate in a simpler manner the asymptotic behavior of \eqref{Equ10} and \eqref{Equ31}. This method comes from gyrokinetic models in plasma physics: when studying the magnetic confinement we are looking for averaged models with respect to the fast motion of particles around the magnetic lines. The analysis relies on the notion of gyro-average operator \cite{BosTraEquSin}, which is a projection onto the space of slow time depending functions. In other words, projecting means smoothing out the fluctuations with respect to the fast time variable, corresponding to the high cyclotronic frequency. This projection appears like a gyro-average operator. Here the arguments are developed at a formal level. We first introduce rigorously the projected measure on the sphere $r\sphere$ for general measures. Let $f \in \mbxv{}$ be a non negative bounded measure on $\R^d \times \R^d$. We denote by $\ave{f}$ the measure corresponding to the linear application \[ \psi \to \intxv{\psi(x,v)\,\ind{v = 0} f(x,v)\,\dxv } + \intxv{\psi\left ( x , r \vsv \right ) \ind{v \neq 0} f(x,v)\,\dxv }\,, \] for all $\psi \in \czcxv$, {\it i.e.,} \[ \intxv{\psi(x,v) \ave{f}(x,v)\,\dxv} = \int _{v = 0} \psi(x,v) f(x,v)\,\dxv + \int _{v \neq 0} \psi \left ( x , r \vsv \right ) f(x,v)\,\dxv\,, \] for all $\psi \in \czcxv$. Observe that $\ave{f}$ is a non negative bounded measure, $$ \intxv{\;\;\ave{f}(x,v)\,\dxv} = \intxv{\;\;f(x,v)\,\dxv}, $$ with $\supp \ave{f} \subset \R^d \times (\A)$. We have the following characterization. \begin{pro} \label{VarChar} Assume that $f$ is a non negative bounded measure on $\R^d \times \R^d$. Then $\ave{f}$ is the unique measure $F$ satisfying $\supp F \subset \R^d \times (\A)$, \[ \int _{v\neq 0} \psi \left ( x , r \vsv \right )F(x,v)\,\dxv = \int _{v \neq 0}\psi \left ( x , r \vsv \right )f(x,v)\,\dxv,\;\;\psi \in \czcxv{} \] and $F = f$ on $\R^d \times \{0\}$. \end{pro} \begin{proof} The measure $\ave{f}$ defined before satisfies the above characterization. Indeed, $\supp \ave{f} \subset \R^d \times (\A)$. Taking now $\psi (x,v) = \varphi (x) \chi (|v|/\delta)$ with $\varphi \in C^0 _c (\R^d)$ and $\delta >0$ one gets \begin{align*} \intxv{\varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) \ave{f}(x,v)\,\dxv } = &\,\int _{v = 0} \varphi (x) f(x,v)\,\dxv \\ &+ \int _{v \neq 0} \varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) f(x,v)\,\dxv. \end{align*} Passing to the limit for $\delta \searrow 0$ yields \[ \int _{v = 0} \varphi (x) \ave{f}(x,v)\,\dxv = \int _{v = 0} \varphi (x) f(x,v)\,\dxv,\;\;\varphi \in \czc{} \] meaning that $\ave{f} = f$ on $\R^d \times \{0\}$. Therefore one gets for any $\psi \in \czcxv{}$ \begin{align*} \int_{v \neq 0} \psi \left ( x , r \vsv \right )\ave{f}(x,v)\,\dxv & = \int_{|v| = r} \psi ( x , v )\ave{f}(x,v)\,\dxv \\ & = \int _{v \neq 0} \psi (x,v) \ave{f}(x,v)\,\dxv \\ & = \intxv{\psi \ave{f}}(x,v)\,\dxv- \int _{v = 0}\psi \ave{f}(x,v)\,\dxv \\ & = \intxv{\psi \ave{f}}(x,v)\,\dxv- \int _{v = 0}\psi f(x,v)\,\dxv \\ & = \int _{v \neq 0} \psi \left ( x , r \vsv \right ) f(x,v)\,\dxv. \end{align*} Conversely, let us check that the above characterization exactly defines the measure $\ave{f}$. For any $\psi \in \czcxv{}$ we have \begin{align*} \intxv{\psi (x,v) F(x,v)\,\dxv} & = \int _{v = 0} \psi F(x,v)\,\dxv + \int _{v \neq 0} \psi F(x,v)\,\dxv \\ & = \int _{v = 0} \psi (x,v) f(x,v)\,\dxv + \int _{v \neq 0} \psi \left ( x, r \vsv \right ) F(x,v)\,\dxv \\ & = \int _{v = 0} \psi (x,v) f(x,v)\,\dxv + \int _{v \neq 0} \psi \left ( x, r \vsv \right )f(x,v)\,\dxv \end{align*} saying that $F = \ave{f}$. \end{proof} By Proposition \ref{VarChar} it is clear that $\ave{\cdot}$ leaves invariant the measures with support in $\R^d \cup (\A)$. Consider $f \in \mbxv{}$. We say that $\Divv \{f \abv \} \in {\cal M}_b (\R^d \times \R^d)$ if and only if there is a constant $C>0$ such that \[ \intxv{\abv \cdot \nabla _v \psi f(x,v)\,\dxv } \leq C \|\psi \|_{\linf{}},\;\;\psi \in \cocxv{}. \] In this case there is a bounded measure $\mu$ such that \[ - \intxv{\abv \cdot \nabla _v \psi f(x,v)\,\dxv } = \intxv{\psi \mu },\;\;\psi \in \cocxv{}. \] By definition we take $\Divv \{f \abv \} = \mu$. The main motivation for the construction of the projection $\ave{\cdot}$ is the following result. \begin{pro} \label{ZeroAve} For any $f \in \mbxv{}$ such that $ \Divv \{f \abv \}\in {\cal M}_b (\R^d \times \R^d)$ we have $\ave{\Divv \{f \abv \}} = 0$. \end{pro} \begin{proof} Let us take $\Divv \{f \abv \} = \mu$. We will check that the zero measure $0$ satisfies the characterization of $\ave{\mu}$ in Proposition \ref{VarChar}. Clearly $\supp 0 = \emptyset \subset \R^d \times (\A)$. For any $\varphi (x) \in \czc{}$ we have \begin{align*} \int _{v = 0} \varphi (x) \mu(x,v)\,\dxv & = \limd \intxv{\varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) \mu(x,v)\,\dxv} \\ & = - \limd \intxv{\varphi (x) \chi ^{\;\prime} \left ( \frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta |v|^2) f(x,v)\,\dxv}= 0 \end{align*} by dominated convergence, since \[ \left | \chi ^{\;\prime} \left ( \frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta |v|^2) \right |\leq \alpha \sup _{u \geq 0} |\chi ^{\;\prime} (u) u | + \beta \delta ^2 \sup _{u \geq 0} |\chi ^{\;\prime} (u) u ^3|. \] Therefore we deduce that $\Divv \{f \abv\} = 0$ on $\R ^d \times \{0\}$. Consider now $\psi \in \cocxv{}$ and lets us compute \begin{align*} \int _{v \neq 0} \psi \left (x,r \vsv \right ) & \mu(x,v)\,\dxv = \limd \intxv{\psi \left (x,r \vsv \right ) \left ( 1 - \chi \left ( \frac{|v|}{\delta} \right ) \right ) \mu(x,v)\,\dxv} \\ & = \limd \intxv{\psi \left (x,r \vsv \right ) \chi ^{\;\prime}\left ( \frac{|v|}{\delta} \right ) \frac{|v|}{\delta} (\alpha - \beta |v|^2) f(x,v)\,\dxv} = 0 \end{align*} since $v \cdot \nabla _v \{ \psi (x, r \vsv)\} = 0$. By density, the same conclusion holds true for any $\psi \in \czcxv{}$ and thus $\ave{\Divv \{f \abv \}} = 0$. \end{proof} \begin{remark} \label{SimplerAve} When $f \in \mbxv{}$ does not charge $\R^d \times \{0\}$, $\ave{f}$ is given by \[ \supp \ave{f} \subset \R ^d \times r\sphere,\;\;\int _{v \neq 0} \psi \left ( x, r \vsv \right ) \ave{f} = \int _{v \neq 0} \psi \left ( x, r \vsv \right ) f,\;\;\psi \in \czcxv{} \] or equivalently \begin{equation} \label{Equ34} \intxv{\psi \ave{f} } = \int _{v \neq 0} \psi \left ( x, r \vsv \right ) f,\;\;\psi \in \czcxv{}. \end{equation} \end{remark} \ Using Proposition \ref{ZeroAve} we can obtain, at least formally, the limit model satisfied by $f = \lime \fe$. By \eqref{Equ2} we know that $\supp f \subset \R ^d \times (\A)$. The time evolution of $f$ comes by eliminating $\fo$ in \eqref{Equ3}. For that it is sufficient to project on the subspace of the measures satisfying the constraint \eqref{Equ2}, {\it i.e.,} to apply $\ave{\cdot}$. \begin{equation} \label{Equ35} \ave{\partial _t f } + \ave{\Divx \{f v\}} + \ave{\Divv \{ f a \}} = 0. \end{equation} It is easily seen that $\ave{\partial _t f } = \partial _t \ave{f} = \partial _t f $ since $\supp f \subset \R ^d \times (\A)$ and therefore $\ave{f} = f$. We need to compute the last two terms in \eqref{Equ35}. We show that \begin{pro}\label{TransportAve} Assume that $a = a(x)$ is a bounded continuous field. Then we have the following equalities \[ \ave{\Divx \{f v \}} = \Divx \{f v \}\;\;\mbox{ if } \;\supp f \subset \R ^d \times (\A) \] \[ \ave{\Divv \{ f a \}} = \Divv \left \{ f \imvv a \right \} \;\;\mbox{ if } \;\supp f \subset \R ^d \times r\sphere. \] As a consequence, \eqref{Equ35} yields the transport equation \eqref{Equ22} obtained rigorously in Theorems {\rm\ref{MainResult}} and {\rm\ref{MainResult2}}. \end{pro} \begin{proof} For any $\psi \in \cocxv{}$ we have \begin{align*} \intxv{\psi & \ave{\Divx\{f v \}}} = \int _{v = 0} \psi \Divx\{f v \} + \int _{v \neq 0} \psixv \Divx\{f v \}\\ & = \limd \intxv{\psi \chivd \Divx\{f v \}} + \limd \intxv{\psixv \left ( 1 - \chivd \right ) \Divx\{f v \}} \\ & = - \limd \intxv{v \cdot \nabla _x \psi \chivd f} - \limd \intxv{v \cdot \nabla _x \psixv \left ( 1 - \chivd \right ) f} \\ & = - \int _{v = 0} v \cdot \nabla _x \psi f - \int _{v \neq 0} v \cdot \nabla _x \psixv f \\ & = - \intxv{v \cdot \nabla _x \psi f } = \intxv{\psi \Divx\{f v \}} \end{align*} saying that $\ave{\Divx \{f v \}} = \Divx \{f v \}$. Assume now that $\supp f \subset \R ^d \times r\sphere$. It is easily seen that $\Divv (fa)$ does not charge $\R ^d \times \{0\}$. Indeed, for any $\psi \in \czcxv{}$ we have by dominated convergence \begin{align*} \int _{v = 0} \psi \Divv (fa) & = \limd \intxv{\psi \chivd \Divv (fa)} \\ & = - \limd \intxv{a \cdot \nabla _v \psi \chivd f} - \limd \intxv{a \cdot \frac{v}{|v|} \frac{1}{\delta} \chipvd \psi f } = 0. \end{align*} Therefore we can use \eqref{Equ34} \begin{align*} \intxv{\psi \ave{\Divv (fa)}} & = \int _{v \neq 0} \psixv \Divv (fa) \\ & = \limd \intxv{\left ( 1 - \chivd \right ) \psixv \Divv (fa) }\\ & = - \limd \intxv{\left ( 1 - \chivd \right ) \frac{r}{|v|} \imvv a \cdot (\nabla _v \psi ) \left ( x, r\vsv \right ) f } \\ & \quad + \limd \intxv{\;\;\frac{1}{\delta} \chipvd \frac{v}{|v|} \cdot a \psixv f } \\ & = - \int _{v \neq 0} \imvv a \cdot \nabla _v \psi f = \intxv{\psi \;\Divv \left \{f \imvv a \right \}}. \end{align*} \end{proof} We investigate now the limit when $\eps \searrow 0$ of the diffusion model \eqref{Equ31}. We are done if we compute $\ave{\Delta _v f}$ for a non negative bounded measure with support contained in $\R^d \times r\sphere$. As before we can check that $\Delta _v f$ does not charge $\R^d \times \{0\}$ and therefore, thanks to \eqref{Equ34}, we obtain after some computations \begin{equation} \label{Equ37} \intxv{\psi \ave{\Delta _v f}} = \int _{v \neq 0} \psixv \Delta _v f = \int _{v \neq 0} \Delta _v \left \{ \psixv \right \}f,\;\;\psi \in \ctcxv{}. \end{equation} \begin{lemma} \label{ZeroHom} For any function $\varphi \in C^2 (\R^d \setminus \{0\})$ and any $r >0$ we have \[ \Delta _v \left \{ \varphi \left ( r \vsv \right ) \right \} = \left ( \frac{r}{|v|}\right ) ^2 \imvv : \partial ^2 _v \varphi \left ( r \vsv \right ) - 2 \frac{r}{|v|} \frac{v \cdot \nabla _v \varphi \left ( r \vsv \right ) }{|v|^2},\;\;v \neq 0. \] \end{lemma} \ \noindent Combining \eqref{Equ37}, Lemma \ref{ZeroHom} and the fact that $\supp f \subset \R ^d \times r \sphere$ we obtain \begin{align*} \intxv{\psi (x,v) \ave{\Delta _v f} } & = \int _{v \neq 0} \left [ \imvv : \partial _v ^2 \psi (x,v) - 2 \frac{v \cdot \nabla _v \psi (x,v)}{|v|^2} \right]f \nonumber \\ & = \intxv{\psi (x,v) \Divv \left \{ \Divv \left [ f \imvv \right ] + 2 f \frac{v}{|v|^2} \right \}}. \nonumber \end{align*} We deduce the formula \[ \ave{\Delta _v f} = \Divv \left \{ \Divv \left [ f \imvv \right ] + 2 f \frac{v}{|v|^2} \right \} \] for any $f$ satisfying $\supp f \subset \R ^d \times r \sphere$ and the limit of the Vicsek model \eqref{Equ31} when $\eps \searrow 0$ becomes \begin{equation}\label{equnew} \partial _t f + \Divx (fv) + \Divv \left \{ f \imvv a \right \} = \Divv \left \{ \Divv \left [ f \imvv \right ] + 2 f \frac{v}{|v|^2} \right \} \end{equation} with the initial condition $f(0) = \ave{\fin}$, as stated in \eqref{Equ22Diff}. \appendix \section{Spherical coordinates and the Laplace-Beltrami operator} \label{A} In this appendix, we show the computations to relate the equations written in original variables $(x,v)$ to the equations in spherical coordinates $(x,\omega)$. Our limit densities have their support contained in $\R^d \times r \sphere$ and thus reduce to measures on $\R^d \times r\sphere$. For example, let us consider the measure on $\R^d \times r\sphere$ still denoted by $f$, given by \[ \intxo{\psi (x, \omega) f(x,\omega)\,\mathrm{d}(x,\omega)} = \int _{v \neq 0} \psixv{} f(x,v)\,\dxv \] for any function $\psi \in \czcxo{}$. In particular, to any $f \in \mbxv{}$ not charging $\R^d \times \{0\}$ it corresponds $\ave{f} \in \mbxv{}$, with $\supp \ave{f} \subset \R ^d \times r \sphere$, whose characterization is \[ \intxo{\psi (x, \omega)\ave{f}(x,\omega)\,\mathrm{d}(x,\omega)} = \int _{v \neq 0}\psixv f(x,v)\,\dxv. \] We intend to write the previous limit models (in Theorems \ref{MainResult}, \ref{MainResult2}, and \eqref{equnew}) in spherical coordinates. \begin{pro} \label{SpherCoord} Assume that $f \in \mbxv{}$, $\supp f \subset \R^d \times r \sphere$ and let us denote by $F \in \mbxo $ its corresponding measure on $\R^d \times r \sphere$. Therefore we have \[ \ave{\Divx (fv)} = \Divx (F \omega),\;\;\ave{\Divv (fa)} = \Divo \left \{F \imoo a \right \},\;\;\ave{\Delta _v f } = \Delta _\omega F. \] \end{pro} \begin{proof} Thanks to Proposition \ref{TransportAve} we have for any $\psi \in \cocxo{}$ \begin{align*} \intxo{\psi (x, \omega) \ave{\Divx (fv)}} & = \int _{v \neq 0} \psixv \Divx (fv) = - \int _{v\neq 0} v \cdot \nabla _x \psixv f\\ & = - \int _{v \neq 0} r \vsv \cdot \nabla _x \psixv f = - \intxo{\omega \cdot \nabla _x \psi (x, \omega) F} \end{align*} and thus $\ave{\Divx (fv)} = \Divx (F \omega)$. Similarly we can write \begin{align*} \intxo{\psi (x, \omega) \ave{\Divv (fa)}} & = \int _{v \neq 0} \psixv \ave{\Divv (fa)}(\mathrm{d}(x,v)) \nonumber \\ & = \int _{v \neq 0} \psixv \Divv \left \{f \imvv a\right \} \nonumber \\ & = - \int _{v \neq 0} \frac{r}{|v|}\imvv a \cdot \imvv \nabla _v\psixv f \nonumber \\ & = - \int _{v \neq 0} \imvv a \cdot \imvv \nabla _v \psixv f \nonumber \\ & = - \intxo{\imoo a \cdot \imoo \nabla _v \psi (x, \omega) F}\nonumber \\ & = - \intxo{\imoo a \cdot \nabla _\omega \psi (x, \omega) F}\nonumber \\ & = \intxo{\psi (x, \omega) \Divo \left \{ F\imoo a\right \}}\nonumber \end{align*} and therefore \[ \ave{\Divv (fa)} = \Divo \left \{F \imoo a \right \}. \] Here $\Divo $ stands for the divergence along $r\sphere$ (notice that $\imoo a$ is a tangent field of $r\sphere$) and $\nabla _\omega = \imoo \nabla _v$ is the gradient along $r\sphere$. For the last assertion we appeal to the following well known result asserting that the Laplace-Beltrami operator coincides with the Laplacian of the degree zero homogeneous extension, see also \cite{BCC12}. \begin{pro} \label{LaplaceBeltrami} Consider $\varphi = \varphi (\omega)$ a $C^2$ function on $r\sphere$ and we denote by $\Phi = \Phi (v)$ its degree zero homogeneous extension on $\R^d \setminus \{0\}$ \[ \Phi (v) = \varphi \left ( r \vsv \right ),\;\;v \neq 0. \] Therefore we have for any $\omega \in r \sphere$ \[ \Delta _\omega \varphi (\omega) = \Delta _v \Phi (\omega). \] \end{pro} Let us come back to the proof of Proposition \ref{SpherCoord}. For any $\psi \in C^2_c (\R ^d \times r \sphere)$ we introduce its degree zero homogeneous extension $\Psi (x,v) = \psixv$. Thanks to Proposition \ref{LaplaceBeltrami} we can write \begin{align*} \intxo{\psi (x,\omega) \ave{\Delta _v f} }& = \int _{v \neq 0} \psixv \ave{\Delta _v f} \nonumber = \int _{v \neq 0} \Psi (x,v) \Delta _v f = \int _{v \neq 0} \Delta _v \Psi f \\ & = \int _{|v| = r} \Delta _\omega \psi (x,v) f = \intxo{\Delta _\omega \psi (x,\omega) F} = \intxo{\psi (x,\omega) \Delta _\omega F} \end{align*} meaning that $\ave{\Delta _v f } = \Delta _\omega F$. \end{proof} \vskip 6pt For the sake of completeness, we finally write the equations in spherical coordinates in $\R^3$. We introduce the spherical coordinates $\omega = r (\cos \theta \cos \varphi, \cos \theta \sin \varphi, \sin \theta)$ with the angle variables $(\theta, \varphi ) \in ]-\pi/2, \pi/2[ \times [0,2\pi [$, and the orthogonal basis of the tangent space to $r\sphere$ \[ e_\theta = (- \sin \theta \cos \varphi, - \sin \theta \sin \varphi, \cos \theta),\;\;e_\varphi = (- \cos \theta \sin \varphi, \cos \theta \cos \varphi, 0) \] with $|e_\theta| = 1,\;|e_\varphi| = \cos \theta$. For any smooth function $u$ on $r\sphere$ we have \[ \nabla _\omega u = (\nabla _\omega u \cdot e_\theta) e_\theta + (\nabla _\omega u \cdot e _\varphi ) \frac{e_\varphi}{\cos ^2 \theta} = \frac{1}{r} \partial _\theta u \;e _\theta + \frac{1}{r\cos ^2 \theta} \partial _\varphi u \;e _\varphi \] and for any smooth tangent field $\xi = \xi _\theta e _\theta + \xi _\varphi e _\varphi $ we have \[ \Divo \xi = \frac{1}{r} \left \{\frac{1}{\cos \theta} \partial _\theta (\xi _\theta \cos \theta) + \partial _\varphi \xi _\varphi \right \}. \] The coordinates of the tangent field $\xi := F \imoo a$ are $ \xi _\theta = \xi \cdot e _\theta = F a_\theta,\;\;\xi _\varphi = \frac{\xi \cdot e _\varphi }{\cos ^2 \theta } = F a_\varphi $ and we obtain \[ \ave{\Divv (fa)} = \Divo \left \{ F \imoo a \right \} = \frac{1}{r} \left \{ \frac{1}{\cos \theta} \partial _\theta (F a_\theta \cos \theta) + \partial _\varphi ( F a_\varphi ) \right \}. \] The spherical Laplacian is given by \begin{align*} \Delta _\omega F & = \Divo (\nabla _\omega F) = \frac{1}{r} \left \{\frac{1}{\cos \theta} \frac{\partial}{\partial \theta} \left ( \frac{\cos \theta}{r} \partial _\theta F\right ) + \frac{\partial}{\partial \varphi } \left ( \frac{1}{r \cos ^2 \theta} \partial _\varphi F \right ) \right \}\\ & = \frac{1}{r^2}\left \{ \frac{1}{\cos \theta} \frac{\partial}{\partial \theta} ( \cos \theta \;\partial _\theta F ) + \frac{1}{\cos ^2 \theta} \;\partial ^2 _\varphi F \right \}. \end{align*} \begin{pro} The limit transport equation obtained in \eqref{equnew} for $\R^3$ is \[ \partial _t F + \omega \cdot \nabla _x F + \frac{1}{r} \left \{\frac{\partial _\theta (F a_\theta \cos \theta )}{\cos \theta} + \partial _\varphi ( F a _\varphi ) \right \} = \frac{1}{r^2} \left \{\frac{1}{\cos \theta} \frac{\partial}{\partial \theta} ( \cos \theta \;\partial _\theta F ) + \frac{1}{\cos ^2 \theta} \;\partial ^2 _\varphi F \right \}. \] \end{pro} We recall here the proof of Proposition \ref{LaplaceBeltrami}. It is a consequence of a more general result. \begin{pro} \label{MoreGenRes} Let us consider a function $\varphi = \varphi (v) \in C^2 (\R^d)$, $d \geq 2$ which writes in polar coordinates $ \varphi (v) = \tvarphi (\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = \vsv \in \sphere. $ Therefore for any $v \neq 0$ we have \[ \Delta _v \varphi (v) = \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho \tvarphi ) + \frac{1}{\rho ^2 } \Delta _\sigma \tvarphi (\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = \vsv. \] \end{pro} \begin{proof} Consider a smooth function $\psi = \psi (v) \in C^2$ with compact support in $\R ^N \setminus \{0\}$, which writes in polar coordinates $ \psi (v) = \tpsi (\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = \vsv \in \sphere. $ We have \[ \frac{\partial \tvarphi }{\partial \rho } = \nabla _v \varphi \cdot \sigma,\;\;\nabla _v \varphi = (\nabla _v \varphi \cdot \sigma ) \sigma + (I - \sigma \otimes \sigma) \nabla _v \varphi = \frac{\partial \tvarphi }{\partial \rho }\;\sigma + \nabla _{\omega = \rho \sigma} \tvarphi \] and \[ \frac{\partial \tpsi }{\partial \rho } = \nabla _v \psi \cdot \sigma,\;\;\nabla _v \psi = (\nabla _v \psi \cdot \sigma ) \sigma + (I - \sigma \otimes \sigma) \nabla _v \psi = \frac{\partial \tpsi }{\partial \rho }\;\sigma + \nabla _{\omega = \rho \sigma} \tpsi. \] Integrating by parts yields \begin{align*} - \intvN{\Delta _v \varphi \;\psi (v)} & = \intvN{\nabla _v \varphi \cdot \nabla _v \psi } = \int_{\R_+} \int _{S^{N-1}} \left \{ \frac{\partial \tvarphi}{\partial \rho} \frac{\partial \tpsi}{\partial \rho} + \frac{1}{\rho ^2} \nabla _\sigma \tvarphi \cdot \nabla _\sigma \tpsi \right \} \;\mathrm{d}\sigma \rho ^{N-1} \;\mathrm{d}\rho \nonumber \\ & = - \int _{S^{N-1}} \int _{\R_+} \tpsi \frac{\partial}{\partial \rho } \left ( \rho ^{N-1} \frac{\partial \tvarphi}{\partial \rho} \right ) \;\mathrm{d}\rho\;\mathrm{d}\sigma - \int _{\R_+} \frac{\rho ^{N-1}}{\rho ^2} \int _{S^{N-1}} \tpsi \;\Delta _\sigma \tvarphi \;\mathrm{d}\sigma \;\mathrm{d}\rho \nonumber \\ & = - \intvN{\psi (v) \left \{ \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho \tvarphi ) + \frac{1}{\rho ^2 } \Delta _\sigma \tvarphi \right \}} \end{align*} and therefore \[ \Delta _v \varphi (v) = \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho \tvarphi ) + \frac{1}{\rho ^2 } \Delta _\sigma \tvarphi (\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = \vsv. \] \end{proof} \begin{proof} (of Proposition \ref{LaplaceBeltrami}) The degree zero homogeneous extension $\Phi (v) = \varphi \left ( r \vsv \right )$ does not depend on the polar radius $\Phi (v) = \tilde{\Phi} (\sigma) = \varphi (\omega = r \sigma),\;\;\sigma = \vsv.$ Thanks to Proposition \ref{MoreGenRes}, we deduce $ \Delta _v \Phi = \frac{1}{\rho ^2} \Delta _\sigma \tilde{\Phi} = \frac{r^2}{\rho ^2} \Delta _\omega \varphi . $ Taking $\rho = r $, which means $v = r \sigma = \omega$ we obtain $ \Delta _v \Phi (\omega) = \Delta _\omega \varphi (\omega),\;\;\omega \in r \sphere. $ \end{proof} \subsection*{Acknowledgments} JAC was supported by projects MTM2011-27739-C04-02 and 2009-SGR-345 from Ag\`encia de Gesti\'o d'Ajuts Universitaris i de Recerca-Generalitat de Catalunya. \end{document}

\begin{document} \title{Quantitative estimates for simple zeros of $L$-functions} \author{Andrew R. Booker} \address{School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK} \email{[email protected]} \author{Micah B. Milinovich} \address{Department of Mathematics, University of Mississippi, University, MS 38677 USA} \email{[email protected]} \author{Nathan Ng} \address{Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB Canada T1K 3M4} \email{[email protected]} \thanks{Research of the first author was supported by EPSRC Grant \texttt{EP/K034383/1}. Research of the second author was supported by the NSA Young Investigator Grants \texttt{H98230-15-1-0231} and \texttt{H98230-16-1-0311}. Research of the third author was supported by NSERC Discovery Grant (RGPIN- 2015-05972). No data were created in the course of this study.} \begin{abstract} We generalize a method of Conrey and Ghosh \cite{CG88} to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor. \end{abstract} \subjclass[2010]{Primary 11F66, 11F11, 11M41} \maketitle \section{Introduction} Let $f\in S_k(\Gamma_1(N))$ be a classical holomorphic modular form of weight $k$ and level $N$. Assume that $f$ is \emph{primitive}, meaning that it is a normalized Hecke eigenform in the new subspace. Then it has a Fourier expansion of the shape $$ f(z)=\sum_{n=1}^\infty\lambda_f(n)n^{\frac{k-1}2}e^{2\pi inz}, $$ where the $\lambda_f(n)$ are multiplicative and satisfy the Ramanujan bound $|\lambda_f(n)|\le d(n)$. Let $\Lambda_f(s)=\Gamma_\mathbb{C}(s+\tfrac{k-1}2)L_f(s)$ denote the complete $L$-function of $f$, with analytic normalization, where $$ \Gamma_\mathbb{C}(s)=2(2\pi)^{-s}\Gamma(s) \quad\text{and}\quad L_f(s)=\sum_{n=1}^\infty\frac{\lambda_f(n)}{n^s}, $$ and let $$ N^s_f(T)=\#\bigl\{\rho\in\mathbb{C}:\Lambda_f(\rho)=0, \Lambda_f'(\rho)\ne0, |\Im(\rho)|\le T\bigr\} $$ be the number of simple zeros of $\Lambda_f(s)$ with imaginary part in $[-T,T]$. In \cite{MN14}, the second and third authors showed that if $\Lambda_f(s)$ satisfies the Generalized Riemann Hypothesis, then $$ N^s_f(T)\ge T(\log{T})^{-\varepsilon} $$ for any fixed $\varepsilon>0$ and all sufficiently large $T>0$. Unconditionally, when $N=1$ and $k=12$, Conrey and Ghosh \cite{CG88} showed that \begin{equation}\label{eq:cgestimate} \forall\varepsilon>0, \exists T\ge\varepsilon^{-1}\text{ such that } N^s_f(T)\ge T^{\frac16-\varepsilon}. \end{equation} Moreover, their proof works more generally for $N=1$ and arbitrary $k$, provided that $N^s_f(T)$ is not identically $0$. In light of the first author's result \cite{Boo16} that $N^s_f(T)\to\infty$ as $T\to\infty$, \eqref{eq:cgestimate} holds for all primitive $f$ of conductor $1$. In this paper we aim to prove similar unconditional quantitative estimates of simple zeros for primitive forms of arbitrary conductor $N$. However, we encounter some obstacles that are reminiscent of the well-known difficulty of extending Hecke's converse theorem to arbitrary conductor, and are not present for $N=1$. Taking inspiration from Weil's generalization \cite{Wei67} of Hecke's converse theorem, we consider character twists. For a Dirichlet character $\chi\pmod*{q}$, let $f\otimes\chi$ denote the unique primitive form such that $\lambda_{f\otimes\chi}(n)=\lambda_f(n)\chi(n)$ for all $n$ coprime to $q$. \begin{theorem}\label{thm:twist} Let $f\in S_k(\Gamma_1(N))$ be a primitive form. Then there is a Dirichlet character $\chi$ such that \eqref{eq:cgestimate} holds with $f\otimes\chi$ in place of $f$. \end{theorem} Next, for odd conductors we obtain a weaker but unconditional quantitative estimate for $N^s_f(T)$, without the twist. Moreover, we show that there is a sort of ``Deuring--Heilbronn phenomenon'' at play, so that if $N^s_f(T)$ is unexpectedly small then we can substantially improve our result for $N^s_{f\otimes\chi}(T)$. \begin{theorem}\label{thm:oddN} Let $f\in S_k(\Gamma_1(N))$ be a primitive form of odd conductor. Then $$ \forall\varepsilon>0, \exists T\ge\varepsilon^{-1}\text{ such that } N^s_f(T)\ge\begin{cases} \exp((\log T)^{\frac13-\varepsilon})&\text{if $k=1$ or $f$ is a CM form},\\ \log\log\log{T}&\text{otherwise}. \end{cases} $$ Further, if $N^s_f(T)\ll1+T^\varepsilon$ for every $\varepsilon>0$, then \begin{enumerate} \item[(i)] there is a Dirichlet character $\chi$ such that, $\forall\varepsilon>0, \exists T\ge\varepsilon^{-1}$ such that $\Lambda_{f\otimes\chi}(s)$ has at least $T^{\frac12-\varepsilon}$ simple zeros with real part $\frac12$ and imaginary part in $[-T,T]$; \item[(ii)] $\Lambda_f(s)$ has simple zeros with real part arbitrarily close to $1$. \end{enumerate} \end{theorem} \begin{remarks}\ \begin{enumerate} \item The exponent $\frac16$ in \eqref{eq:cgestimate} is related to the best known subconvexity estimate for modular form $L$-functions in the $t$ aspect; it can be replaced by any $\delta>0$ such that $L_f(\frac12+it)\ll_{f,\varepsilon}(1+|t|)^{\frac12-\delta+\varepsilon}$ holds for all primitive forms $f$ and all $\varepsilon>0$. In \cite{BMN19} we showed that $\delta=\frac16$ is admissible. Very recent work of Munshi \cite{Mun18} improves this to $\delta=\frac16+\frac1{1200}$ for forms of level $1$, with a corresponding improvement to \eqref{eq:cgestimate} in that case. \item In Theorem~\ref{thm:twist}, one can take the conductor of $\chi$ to be $1$ or a prime number bounded by a polynomial function of $N$. \item The proof of Theorem~\ref{thm:oddN} makes use of the idea originating with Conrey and Ghosh \cite{CG88} of twisting the coefficients of $L_f(s)$ by $(-1)^n$ to prevent the main terms of our estimate from cancelling out. This relies implicitly on the fact that there is no primitive Dirichlet character of conductor $2$, and is the ultimate reason for our restriction to odd $N$. \item The improved estimate in Theorem~\ref{thm:oddN} in the Galois and CM cases arises from Coleman's Vinogradov-type zero-free region for Hecke $L$-functions \cite{Col90}. \end{enumerate} \end{remarks} \section{Dirichlet series} In order to establish the existence of simple zeros it is useful to study not only $L_f(s)$, but some related Dirichlet series and their additive twists. This is one of the central ideas in \cite{CG88}. A key role is played by the series $$ D_f(s)=L_f(s)\frac{d^2}{ds^2}\log L_f(s)=\sum_{n=1}^{\infty}c_f(n)n^{-s}, $$ which has a meromorphic continuation to $\mathbb{C}$ with poles precisely at the simple zeros of $L_f(s)$ (including the trivial zeros $s=\frac{1-k}2-n$ for $n=0, 1, 2, \ldots$). For $\alpha\in\mathbb{Q}^\times$ and $\chi$ a Dirichlet character, let $$ L_f(s,\alpha)=\sum_{n=1}^\infty\lambda_f(n)e(\alpha n)n^{-s} \quad\text{and}\quad L_f(s,\chi)=\sum_{n=1}^\infty\lambda_f(n)\chi(n)n^{-s}. $$ Likewise, define $$ D_f(s,\alpha)=\sum_{n=1}^\infty c_f(n)e(\alpha n)n^{-s} \quad\text{and}\quad D_f(s,\chi)=\sum_{n=1}^\infty c_f(n)\chi(n)n^{-s}. $$ Let $\xi$ denote the nebentypus character of $f$. Set $$ Q(N)=\{1\}\cup\{q\text{ prime}:q\nmid N\}, $$ and for each $q\in Q(N)$, define the rational functions $$ P_{f,q}(x)=\begin{cases} 1&\text{if }q=1,\\ 1-\lambda_f(q)x+\xi(q)x^2&\text{otherwise} \end{cases} $$ and $$ R_{f,q}(x)=\begin{cases} 0&\text{if }q=1,\\ \frac{q\log^2{q}}{q-1} \frac{x(\lambda_f(q)-4\xi(q)x+\lambda_f(q)\xi(q)x^2)} {P_{f,q}(x)} &\text{if }q\ne1. \end{cases} $$ These are such that, if $$ \chi_0(n)=\begin{cases} 1&\text{if }(n,q)=1,\\ 0&\text{otherwise} \end{cases} $$ denotes the trivial character mod $q$, then $$ L_f(s,\chi_0)=P_{f,q}(q^{-s})L_f(s) $$ and \begin{equation}\label{Dfchi0} D_f(s,\chi_0)=P_{f,q}(q^{-s})D_f(s)-\frac{q-1}{q}R_{f,q}(q^{-s})L_f(s). \end{equation} For any $a\in\mathbb{Z}$ coprime to $q$, we define \begin{align*} D_{f,a,q}(s)&=D_f(s,\tfrac{a}q)-R_{f,q}(q^{-s})L_f(s) =\sum_{n=1}^{\infty}c_{f,a,q}(n)n^{-s},\\ D_{f,a,q}^*(s)&=D_{f,a,q}(s)+\psi'(s+\tfrac{k-1}2)L_f(s,\tfrac{a}{q}), \quad\text{where }\psi(s)=\frac{\Gamma'}{\Gamma}(s) \end{align*} and $$ D_{f,a,q}(s,\alpha)=\sum_{n=1}^{\infty}c_{f,a,q}(n)e(\alpha n)n^{-s} \quad\text{for }\alpha\in\mathbb{Q}^\times. $$ To each of $L_f$, $D_f$, $D_{f,a,q}$, $D_{f,a,q}^*$ and their twists, we define completed versions $\Lambda_f$, $\Delta_f$, $\Delta_{f,a,q}$, $\Delta_{f,a,q}^*$ obtained by multiplying by $\Gamma_\mathbb{C}(s+\frac{k-1}2)$. By the Ramanujan bound $|\lambda_f(q)|\le 2$ and \cite[Proposition~3.1]{BK11}, $\Delta_f(s,a/q)-\Delta_{f,a,q}^*(s)$ is holomorphic for $\Re(s)>0$. In turn, the analytic properties of $\Delta_{f,a,q}^*(s)$ are described by the following proposition. \begin{proposition}\label{voronoi} Let $f\in S_k(\Gamma_0(N),\xi)$ be a primitive form, $q\in Q(N)$, and $a\in\mathbb{Z}$ coprime to $q$. Then $\Delta_{f,a,q}^*(s)$ is a ratio of entire functions of finite order, has at most simple poles, all of which are contained in the critical strip $\{s\in\mathbb{C}:\Re(s)\in(0,1)\}$, and satisfies the functional equation \begin{equation}\label{dstarfunceq} \Delta_{f,a,q}^*(s)=\epsilon\xi(q)(Nq^2)^{\frac12-s} \Delta_{\bar{f},-\overline{Na},q}^*(1-s), \end{equation} where $\bar{f}\in S_k(\Gamma_0(N),\overline{\xi})$ is the dual of $f$, $\epsilon\in\mathbb{C}^\times$ is the root number of $f$ and $\overline{Na}$ denotes a multiplicative inverse of $Na\pmod*{q}$. \end{proposition} \begin{proof} For $q=1$ the result follows immediately from \cite[(3.1)]{Boo16}, so we may assume that $q$ is prime. Let $\chi$ be a Dirichlet character of conductor $q$. Then the complete twisted $L$-function $\Lambda_f(s,\chi)$ satisfies the functional equation $$ \Lambda_f(s,\chi)=\epsilon\xi(q)\chi(N)\frac{\tau(\chi)^2}{q} (Nq^2)^{\frac12-s}\Lambda_{\bar{f}}(1-s,\overline{\chi}), $$ where $\epsilon\in\mathbb{C}^\times$ is the root number of $f$. Applying \cite[(3.1)]{Boo16} to $f\otimes\chi$, we thus have \begin{equation}\label{dfunceq1} \Delta_f(s,\chi) -\epsilon\xi(q)\chi(N)\frac{\tau(\chi)^2}{q} (Nq^2)^{\frac12-s}\Delta_{\bar{f}}(1-s,\overline{\chi}) =\Lambda_f(s,\chi)\bigl(\psi'(\tfrac{k+1}2-s)-\psi'(s+\tfrac{k-1}2)\bigr). \end{equation} Next, we have $$ \Delta_f\!\left(s,\frac{a}q\right)=\Delta_f(s) -\frac{q}{q-1}\Delta_f(s,\chi_0) +\frac1{q-1}\sum_{\substack{\chi\pmod*{q}\\\chi\ne\chi_0}} \tau(\overline{\chi})\chi(a)\Delta_f(s,\chi), $$ where $\chi_0$ is the trivial character mod $q$. Combining this with \eqref{Dfchi0} we get \begin{equation}\label{Deltafaq} \begin{aligned} \Delta_{f,a,q}(s) =\left(1-\frac{q}{q-1}P_{f,q}(q^{-s})\right)\Delta_f(s) +\frac1{q-1}\sum_{\substack{\chi\;(\text{mod }q)\\\chi\ne\chi_0}} \tau(\overline{\chi})\chi(a)\Delta_f(s,\chi). \end{aligned} \end{equation} Note in particular that $\Delta_{f,a,q}(s)$ is a ratio of entire functions of finite order, and all of its poles in $\{s\in\mathbb{C}:\Re(s)>0\}$ are simple and located at simple zeros of either $\Lambda_f(s)$ or $\Lambda_f(s,\chi)$ for some $\chi\ne\chi_0$. Note that $P_{f,q}$ satisfies the functional equation $$ 1-\frac{q}{q-1}P_{f,q}(q^{-s}) =\xi(q)q^{1-2s}\left(1-\frac{q}{q-1}P_{\bar{f},q}(q^{s-1})\right), $$ and thus, by \cite[(3.1)]{Boo16}, \begin{equation}\label{dfunceq2} \begin{aligned} \left(1-\frac{q}{q-1}P_{f,q}(q^{-s})\right)&\Delta_f(s) -\epsilon\xi(q)(Nq^2)^{\frac12-s} \left(1-\frac{q}{q-1}P_{\bar{f},q}(q^{s-1})\right) \Delta_{\bar{f}}(1-s)\\ &=\left(1-\frac{q}{q-1}P_{f,q}(q^{-s})\right) \Lambda_f(s)\bigl(\psi'(\tfrac{k+1}2-s)-\psi'(s+\tfrac{k-1}2)\bigr). \end{aligned} \end{equation} Thus, replacing $f$ by $\bar{f}$, $s$ by $1-s$, $a$ by $-\overline{Na}$ and $\chi$ by $\overline{\chi}$ in \eqref{Deltafaq}, we get \begin{align*} \Delta_{\bar{f},-\overline{Na},q}(1-s) =\left(1-\frac{q}{q-1}P_{\bar{f},q}(q^{s-1})\right)\Delta_{\bar{f}}(1-s) +\frac1{q-1}\sum_{\substack{\chi\pmod*{q}\\\chi\ne\chi_0}} \tau(\chi)\chi(-Na)\Delta_{\bar{f}}(1-s,\overline{\chi}). \end{align*} Applying the functional equations \eqref{dfunceq1} and \eqref{dfunceq2}, together with the relation $\tau(\chi)\tau(\overline{\chi})=\chi(-1)q$, we thus have \begin{align*} &\Delta_{f,a,q}(s) -\epsilon\xi(q)(Nq^2)^{\frac12-s} \Delta_{\bar{f},-\overline{Na},q}(1-s)\\ &=\Biggl[ \left(1-\frac{q}{q-1}P_{f,q}(q^{-s})\right)\Lambda_f(s) +\frac1{q-1}\sum_{\substack{\chi\pmod*{q}\\\chi\ne\chi_0}} \tau(\overline{\chi})\chi(a)\Lambda_f(s,\chi)\Biggr] \bigl(\psi'(\tfrac{k+1}2-s)-\psi'(s+\tfrac{k-1}2)\bigr)\\ &=\Lambda_f(s,\tfrac{a}{q}) \bigl(\psi'(\tfrac{k+1}2-s)-\psi'(s+\tfrac{k-1}2)\bigr). \end{align*} Applying the classical Voronoi formula \cite[p.~179, (A.10)]{KMV02} $$ \Lambda_f(s,\tfrac{a}{q}) =\epsilon\xi(q)(Nq^2)^{\frac12-s} \Lambda_{\bar{f}}\bigl(1-s,-\tfrac{\overline{Na}}q\bigr), $$ we arrive at \eqref{dstarfunceq}. Finally, by \eqref{Deltafaq} and the nonvanishing of automorphic $L$-functions for $\Re(s)\ge1$ \cite{JS76}, $\Delta_{f,a,q}^*(s)$ is holomorphic for $\Re(s)\ge1$. This conclusion applies to $\Delta_{\bar{f},-\overline{Na},q}^*(s)$ as well, so by \eqref{dstarfunceq}, all poles of $\Delta_{f,a,q}^*(s)$ have real part in $(0,1)$. \end{proof} Fix, for the remainder of this section, a choice of $f,a,q$ as in Proposition~\ref{voronoi}, and $\alpha\in\mathbb{Q}^\times$. We define $$ N^s_{f,a,q}(T)=\#\bigl\{\rho\in\mathbb{C}:|\Im(\rho)|\le T, \Res{s=\rho}\Delta_{f,a,q}^*(s)\ne0\bigr\} $$ and \begin{equation}\label{Sydefn} S_{f,a,q}(y,\alpha)= \sum_{\rho\in\mathbb{C}}\Res{s=\rho}\Delta_{f,a,q}^*(s)(y-i\alpha)^{-\rho-\frac{k-1}2} \quad\text{for }y\in\mathbb{R}_{>0}, \end{equation} where $(y-i\alpha)^{-\rho-\frac{k-1}2}$ is defined in terms of the principal branch of $\log(y-i\alpha)$. Our goal is to derive the following expression for the Mellin transform of $S_{f,a,q}(y,\alpha)$, up to a holomorphic function on $\{s\in\mathbb{C}:\Re(s)>0\}$: \begin{proposition}\label{prop:Mellin} Define \begin{equation}\label{eq:Hdef} H_{f,a,q,\alpha}(s)= \Delta_{f,a,q}(s,\alpha)-\epsilon\xi(q)(i\sgn\alpha)^k (Nq^2\alpha^2)^{s-\frac12}\Delta_{\bar{f},-\overline{Na},q} \!\left(s,-\frac1{Nq^2\alpha}\right) \end{equation} and $$ I_{f,a,q,\alpha}(s)=\int_0^{|\alpha|/4} S_{f,a,q}(y,\alpha)y^{s+\frac{k-1}2}\frac{dy}{y}. $$ Then $I_{f,a,q,\alpha}(s)-H_{f,a,q,\alpha}(s)$ has analytic continuation to $\Re(s)>0$. Moreover, if $$ \int_0^{|\alpha|/4}|S_{f,a,q}(y,\alpha)| y^{\sigma+\frac{k-1}2}\frac{dy}{y}<\infty $$ for some $\sigma\ge0$, then $H_{f,a,q,\alpha}(s)$ is holomorphic for $\Re(s)>\sigma$. \end{proposition} The proof will be carried out in several lemmas, and involves the following auxiliary functions defined on $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)>0\}$: $$ F(z)=2\sum_{n=1}^\infty c_{f,a,q}(n)n^{\frac{k-1}{2}}e(nz), \quad\overline{F}(z)=2\sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}{2}}e(nz), $$ $$ A(z)=\frac1{2\pi i}\int_{\Re(s)=\frac{k}2}\Lambda_f(s,\tfrac{a}q) \big(\psi'(s+\tfrac{k-1}{2})+\psi'(s-\tfrac{k-1}{2})\big)(-iz)^{-s-\frac{k-1}2}\,ds, $$ and $$ B(z)=\frac1{2\pi i}\int_{\Re(s)=\frac{k}2}\Lambda_f(s,\tfrac{a}{q}) \frac{\pi^2}{\sin^2(\pi(s+\tfrac{k-1}2))}(-iz)^{-s-\frac{k-1}2}\,ds. $$ We first derive the following expression for $S_{f,a,q}$. \begin{lemma}\label{Sfaqz} For $z=\alpha+iy\in\mathbb{H}$, we have \begin{equation}\label{eq:Sfaqz} S_{f,a,q}(y,\alpha)=F(z) -\frac{\epsilon\xi(q)}{(-i\sqrt{N}qz)^k} \overline{F}\!\left(-\frac1{Nq^2z}\right)+A(z)-B(z). \end{equation} \end{lemma} \begin{proof} Let $0<\varepsilon<\frac{1}{2}$. For $z\in\mathbb{H}$ we define \[ I_R(z)=\frac1{2\pi i}\int_{\Re(s)=1+\varepsilon}\Delta_{f,a,q}(s)(-iz)^{-s-\frac{k-1}2}\,ds, \quad I_L(z)=\frac1{2\pi i}\int_{\Re(s)=-\varepsilon}\Delta_{f,a,q}(s)(-iz)^{-s-\frac{k-1}2}\,ds. \] For the remainder of the proof we let $z =\alpha+iy$. Since $\Delta_{f,a,q}^*(s)$ is a ratio of entire functions of finite order with at most simple poles, by the calculus of residues we have \[ \Res{s=0}\Delta_{f,a,q}(s)(-iz)^{-s-\frac{k-1}2} +S_{f,a,q}(y,\alpha)=I_R(z)-I_L(z). \] Note that the residue term at $s=0$ vanishes unless $k=1$. We have \begin{align*} I_R(z)&=\frac1{2\pi i}\int_{\Re(s)=1+\varepsilon}\Gamma_\mathbb{C}(s+\tfrac{k-1}{2}) D_{f,a,q}(s)(-iz)^{-s-\frac{k-1}2}\,ds\\ &=2(-2\pi iz)^{-\frac{k-1}2}\sum_{n=1}^{\infty} c_{f,a,q}(n) \frac1{2\pi i}\int_{\Re(s)=1+\varepsilon}\Gamma(s+\tfrac{k-1}{2}) (-2\pi inz)^{-s}\,ds. \end{align*} Using the identity $$ \frac1{2\pi i}\int_{\Re(s)=1+\varepsilon}\Gamma(s+\tfrac{k-1}{2})z^{-s}\,ds= z^{\frac{k-1}{2}}e^{-z}\quad\text{for }\Re(z)>0, $$ it follows that \begin{equation}\label{IRidentity} I_R(z)=2\sum_{n=1}^\infty c_{f,a,q}(n)n^{\frac{k-1}{2}}e(nz)=F(z). \end{equation} By the functional equation, we have $$ \Delta_{f,a,q}(s)=\epsilon\xi(q)(Nq^2)^{\frac12-s} \Delta_{\bar{f},-\overline{Na},q}(1-s)+\Lambda_f(s,\tfrac{a}{q}) \big( \psi'(\tfrac{k+1}{2}-s)-\psi'(s+ \tfrac{k-1}{2}) \big), $$ so $I_L(z)=I_{L1}(z)+I_{L2}(z)$, where \begin{align}\label{IL1} I_{L1}(z)&=\frac1{2\pi i}\int_{\Re(s)=-\varepsilon} \epsilon\xi(q)(Nq^2)^{\frac{1}2-s} \Delta_{\bar{f},-\overline{Na},q}(1-s)(-iz)^{-s-\frac{k-1}2}\,ds, \\ \label{IL2} I_{L2}(z)&=\frac1{2\pi i}\int_{\Re(s)=-\varepsilon} \Lambda_f(s,\tfrac{a}{q}) \big(\psi'(\tfrac{k+1}{2}-s)-\psi'(s+\tfrac{k-1}{2})\big) (-iz)^{-s-\frac{k-1}2}\,ds. \end{align} Making the substitution $s\mapsto 1-s$ in \eqref{IL1}, we get \begin{equation}\label{IL1identity} \begin{aligned} I_{L1}(z)&=\frac{1}{2\pi i} \int_{\Re(s)=1+\varepsilon} \epsilon\xi(q)(Nq^2)^{s-\frac{1}2} \Delta_{\bar{f},-\overline{Na},q}(s)(-iz)^{s-\frac{k+1}2}\,ds\\ &=2\epsilon\xi(q)(Nq^2)^{-\frac12}(-iz)^{-\frac{k+1}2} (2\pi)^{-\frac{k-1}2} \frac1{2\pi i}\int_{\Re(s)=1+\varepsilon} \Delta_{\bar{f},-\overline{Na},q}(s)\Big(\frac{2\pi}{-iNq^2z}\Big)^{-s}\,ds\\ &=2\epsilon\xi(q)(Nq^2)^{-\frac12}(-iz)^{-\frac{k+1}2}(2\pi)^{-\frac{k-1}2} \sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n) \Big(\frac{2\pi n}{-iNq^2z}\Big)^{\frac{k-1}{2}}e\Big(-\frac{n}{Nq^2z}\Big)\\ &=\frac{2\epsilon\xi(q)}{(-i\sqrt{N}qz)^k} \sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}{2}} e\Big(-\frac{n}{Nq^2z}\Big)\\ &=\frac{\epsilon\xi(q)}{(-i\sqrt{N}qz)^k} \overline{F}\!\left(-\frac1{Nq^2z}\right). \end{aligned} \end{equation} Next, note that the integrand in \eqref{IL2} is holomorphic for $-\frac{k-1}{2}<\Re(s)<\frac{k+1}{2}$. Moving the contour to $\Re(s)=\frac{k}2$, we get a contribution from the pole at $s=0$ (present only when $k=1$) of $$ \Res{s=0}\Lambda_f(s,\tfrac{a}{q})\psi'(s+\tfrac{k-1}2)(-iz)^{-s-\frac{k-1}2} =-\Res{s=0}\Delta_{f,a,q}(s,\tfrac{a}{q})(-iz)^{-s-\frac{k-1}2}. $$ Thus \begin{align*} I_{L2}(z)+\Res{s=0}&\Delta_{f,a,q}(s,\tfrac{a}{q})(-iz)^{-s-\frac{k-1}2}\\ &=\frac1{2\pi i} \int_{\Re(s)=\frac{k}2} \Lambda_f(s,\tfrac{a}{q}) \big(\psi'(\tfrac{k+1}2-s)-\psi'(s+\tfrac{k-1}2)\big)(-iz)^{-s-\frac{k-1}2}\,ds. \end{align*} The reflection formula for $\Gamma$ implies that $\psi'(1-s)+\psi'(s)=\frac{\pi^2}{\sin^2(\pi s)}$, so \[ \psi'(\tfrac{k+1}{2}-s) -\psi'(s+\tfrac{k-1}{2}) =\frac{\pi^2}{\sin^2(\pi(s+\tfrac{k-1}{2}))} -\psi'(s+ \tfrac{k-1}{2})-\psi'(s-\tfrac{k-1}{2}). \] Therefore $I_{L2}(z)+\Res{s=0}\Delta_{f,a,q}(s,\tfrac{a}{q})(-iz)^{-s-\frac{k-1}2} =I_{L2B}(z)-I_{L2A}(z)$, where \begin{align*} I_{L2A}(z)&=\frac1{2\pi i}\int_{\Re(s)=\frac{k}2}\Lambda_f(s,\tfrac{a}{q}) \big(\psi'(s+\tfrac{k-1}{2})+\psi'(s-\tfrac{k-1}{2})\big)(-iz)^{-s-\frac{k-1}2}\,ds,\\ I_{L2B}(z)&=\frac1{2\pi i}\int_{\Re(s)=\frac{k}2} \Lambda_f(s,\tfrac{a}{q})\frac{\pi^2}{\sin^2(\pi(s+\tfrac{k-1}2))}(-iz)^{-s-\frac{k-1}2}\,ds. \end{align*} Hence $$ S_{f,a,q}(y,\alpha)=I_R(z)-I_{L1}(z)+I_{L2A}(z)-I_{L2B}(z). $$ By applying \eqref{IRidentity}, \eqref{IL1identity} and by setting $A(z)=I_{L2A}(z)$ and $B(z)= I_{L2B}(z)$, we establish Lemma \ref{Sfaqz}. \end{proof} Next we evaluate $\int_0^{|\alpha|/4}S_{f,a,q}(y,\alpha)y^{s+\frac{k-1}2}\frac{dy}{y}$, considering each term on the right-hand side of \eqref{eq:Sfaqz} in turn. \begin{lemma}\label{Flemma} $\int_0^{|\alpha|/4}F(\alpha+iy)y^{s+\frac{k-1}2}\frac{dy}{y} -\Delta_{f,a,q}(s,\alpha)$ continues to an entire function of $s$. \end{lemma} \begin{proof} From the definition of $F$ we compute that $$ \int_0^\infty F(\alpha+iy)y^{s+\frac{k-1}2}\frac{dy}{y} =\Delta_{f,a,q}(s,\alpha). $$ Moreover, $F(\alpha+iy)$ decays exponentially as $y\to\infty$, so the contribution to the integral from $y>|\alpha|/4$ is entire. \end{proof} \begin{lemma}\label{Fbarlemma} For any $M\in\mathbb{Z}_{\ge0}$, \begin{equation}\label{eq:fbarmellin} \begin{aligned} &\int_0^{|\alpha|/4}\bigl(-i\sqrt{N}q(\alpha+iy)\bigr)^{-k} \overline{F}\!\left(-\frac1{Nq^2(\alpha+iy)}\right) y^{s+\frac{k-1}2}\frac{dy}{y}\\ &-(i\sgn\alpha)^k\sum_{m=0}^{M-1}(-i\alpha)^{-m} {{s+m-\frac{k+1}2}\choose{m}}(Nq^2\alpha^2)^{s-\frac12+m} \Delta_{\bar{f},-\overline{Na},q}\!\left(s+m,-\frac1{Nq^2\alpha}\right) \end{aligned} \end{equation} continues to a holomorphic function on $\{s\in\mathbb{C}:\Re(s)>1-M\}$. \end{lemma} \begin{proof} As the proof of this lemma is very similar to that of \cite[Lemma~3.3]{Boo16}, we just provide a sketch and refer to the appropriate parts of loc.~cit.\ for the relevant details. Fix $y\in(0,|\alpha|/4]$, and set $z=\alpha+iy$, $\beta=-1/Nq^2\alpha$, and $u=y/\alpha$. It may be checked that \[ -\frac1{Nq^2z}=\beta+i|\beta u|-\frac{\beta u^2}{1+iu}. \] Therefore \begin{align*} &(-i\sqrt{N}qz)^{-k}\overline{F}\!\left(-\frac1{N q^2 z}\right)\\ &=2(-i\sqrt{N}q\alpha)^{-k}\sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}2}e(\beta n)e^{-2\pi n|\beta u|} (1+iu)^{-k}e\Big(-\frac{n\beta u^2}{1+iu}\Big). \end{align*} It was shown in \cite[p.~820]{Boo16} that \[ (1+iu)^{-k}e\Big(-\frac{n\beta u^2}{1+iu}\Big) =\sum_{m=0}^\infty(-iu)^m\sum_{j=0}^m\binom{m+k-1}{m-j} \frac{(-2\pi n|\beta u|)^j}{j!}, \] and for $M,K\in\mathbb{Z}_{\ge 0}$ and $|u|\le\frac14$, \[ \sum_{m=M}^\infty(-iu)^m\sum_{j=0}^m\binom{m+k-1}{m-j} \frac{(-2\pi n|\beta u|)^j}{j!} \ll_{\alpha,M,K}|u|^{M-K}n^{-K}e^{2\pi n|\beta u|}. \] Thus, we obtain \begin{equation}\label{Fbaridentity} \begin{aligned} &(-i\sqrt{N}qz)^{-k}\overline{F}\!\left(-\frac1{Nq^2z}\right) =O_{M,K}\Big(y^{M-K}\sum_{n=1}^\infty|c_{\bar{f},-\overline{Na},q}(n)|n^{\frac{k-1}{2}-K}\Big) +2(-i\sqrt{N}q\alpha)^{-k}\\ &\times\sum_{m=0}^{M-1}\Big(-\frac{iy}{\alpha}\Big)^m \sum_{j=0}^m\binom{m+k-1}{m-j} \sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}2}e(\beta n) \frac1{j!}\Big(-\frac{2\pi ny}{Nq^2\alpha^2}\Big)^j e^{-\frac{2\pi ny}{Nq^2\alpha^2}}. \end{aligned} \end{equation} By the choice $K=\lfloor\frac{k-1}2\rfloor+2$, the error term converges and is $O_M(y^{M-K})$. For the other term note that \begin{align*} &2y^m\sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}2}e(\beta n) \frac1{j!}\Big(-\frac{2\pi ny}{Nq^2\alpha^2}\Big)^j e^{-\frac{2\pi ny}{Nq^2\alpha^2}}\\ &=2\frac{y^{j+m}}{j!}\frac{d^j}{dy^j} \sum_{n=1}^\infty c_{\bar{f},-\overline{Na},q}(n)n^{\frac{k-1}2}e(\beta n) e^{-\frac{2\pi ny}{Nq^2\alpha^2}}\\ &=\frac{y^{j+m}}{j!}\frac{d^j}{dy^j} \frac1{2\pi i}\int_{\Re(s)=m+2} (Nq^2\alpha^2)^{s+\frac{k-1}2}\Delta_{\bar{f},-\overline{Na},q}(s,\beta)y^{-s-\frac{k-1}2}\,ds\\ &=\frac1{2\pi i}\int_{\Re(s)=2} \binom{-s-\frac{k-1}2-m}{j} (Nq^2\alpha^2)^{s+\frac{k-1}2+m} \Delta_{\bar{f},-\overline{Na},q}(s+m,\beta)y^{-s-\frac{k-1}2}\,ds. \end{align*} Inserting this in the last term of \eqref{Fbaridentity} and using the Chu--Vandermonde identity \[ \sum_{j=0}^m\binom{m+k-1}{m-j}\binom{-s-\frac{k-1}2-m}{j}= \binom{-s+\frac{k-1}2}{m}=(-1)^m\binom{s+m-\frac{k+1}2}{m}, \] we arrive at \begin{align*} &(-i\sqrt{N}qz)^{-k}\overline{F}\!\left(-\frac1{Nq^2z}\right)\\ &=O_M(y^{M-\lfloor\frac{k+3}2\rfloor}) +(i\sgn\alpha)^k\sum_{m=0}^{M-1}\frac{(-i\alpha)^{-m}}{2\pi i}\int_{\Re(s)=2} {{s+m-\frac{k+1}2}\choose{m}} (Nq^2\alpha^2)^{s-\frac12+m}\\ &\hspace{8cm}\cdot \Delta_{\bar{f},-\overline{Na},q}\!\left(s+m,-\frac1{Nq^2\alpha}\right) y^{-s-\frac{k-1}2}\,ds. \end{align*} We multiply both sides by $y^{s+\frac{k-1}2-1}$ and integrate over $y\in(0,|\alpha|/4]$. The error term yields a holomorphic function for $\Re(s)>2-M$. As for the sum over $m$, by shifting the contour to the right, we see that each term decays rapidly as $y\to\infty$, so the integral over $(0,|\alpha|/4]$ differs from the full Mellin transform by an entire function. By Mellin inversion, it follows that \eqref{eq:fbarmellin} is holomorphic for $\Re(s)>2-M$. Finally, replacing $M$ by $M+1$ and discarding the final term of the sum concludes the proof of the lemma. \end{proof} \begin{lemma}\label{Alemma} $\Gamma_\mathbb{C}(s)^{-1}\int_0^{|\alpha|/4}A(\alpha+iy)y^s\frac{dy}y$ continues to an entire function of $s$. \end{lemma} \begin{proof} Let $\Phi(s)=\psi'(s+\tfrac{k-1}2)+\psi'(s-\tfrac{k-1}2)$. By the identity $\psi'(s)=\int_1^\infty\frac{\log x}{x-1}x^{-s}\,dx$, we have $\Phi(s)=\int_1^\infty\phi(x)x^{-s-\frac{k-1}2}\,dx$ for $\Re(s)>\frac{k-1}{2}$, where $\phi(x)=\frac{x^{k-1}+1}{x-1}\log{x}$. It follows that $$ \Phi(s)\Gamma(s+\tfrac{k-1}{2})=\int_1^\infty\phi(x)x^{-s-\frac{k-1}2}\,dx \int_0^\infty e^{-y}y^{s+\frac{k-1}2}\,dy =\int_1^\infty\phi(x)\int_0^\infty e^{-y} \Big(\frac{y}{x}\Big)^{s+\frac{k-1}{2}}\frac{dy}{y}. $$ By the variable change $y\mapsto xy$ we obtain $$ \Phi(s)\Gamma(s+\tfrac{k-1}{2})= \int_1^\infty\phi(x)\int_0^\infty e^{-xy}y^{s+\frac{k-1}{2}}\frac{dy}{y} =\int_0^\infty\Big(\int_1^\infty\phi(x)e^{-xy}\,dx\Big)y^{s+\frac{k-1}{2}}\frac{dy}{y}. $$ By Mellin inversion, \begin{equation}\label{mellininversion} \int_1^\infty\phi(x)e^{-xy}\,dx =\frac1{2\pi i}\int_{\Re(s)=2}\Phi(s)\Gamma(s+\tfrac{k-1}2)y^{-s-\frac{k-1}2}\,ds. \end{equation} Observe that $L_{\bar{f}}(s,-\frac{\overline{Na}}{q}) =\sum_{n=1}^\infty b_nn^{-s}$, where $b_n=\lambda_{\bar{f}}(n)e(-\frac{\overline{Na}}{q})$. Thus for $z\in\mathbb{H}$, \begin{align*} A(z)&=2\sum_{n=1}^\infty b_n\cdot \frac1{2\pi i}\int_{\Re(s)=2}\Phi(s) \Gamma(s+\tfrac{k-1}{2})(-2\pi inz)^{-s-\frac{k-1}2}\,ds\\ &=2\sum_{n=1}^\infty b_n\int_1^\infty\phi(x)e^{2\pi inxz}\,dx, \end{align*} where the last step follows from \eqref{mellininversion}. For $z=\alpha+iy$ this simplifies to \begin{equation}\label{eq:Aalphaiy} A(\alpha+iy)=2\sum_{n=1}^\infty b_n\int_1^\infty\phi(x) e(\alpha nx)e^{-2\pi nxy}\,dx. \end{equation} Using this expression, it follows that \begin{equation}\label{Amellintransform} \begin{aligned} &\int_0^\infty A(\alpha+iy)y^s\frac{dy}{y} =2\sum_{n=1}^\infty b_n\int_1^\infty\phi(x)e(\alpha nx) \int_0^\infty e^{-2\pi nxy}y^s\frac{dy}{y}\,dx\\ &=\Gamma_\mathbb{C}(s)\sum_{n=1}^\infty b_nn^{-s} \int_1^\infty\phi(x)e(\alpha nx)x^{-s}\,dx. \end{aligned} \end{equation} For $j=0,1,2,\ldots$, define the sequence of functions $\phi_j(x,s)$ by \[ \phi_0(x,s)=\phi(x), \quad \phi_{j+1}(x,s)=x\frac{\partial\phi_j}{\partial x}(x,s)-(s+j)\phi_j(x,s). \] Integrating by parts, \[ \int_1^\infty\phi_j(x,s)e(\alpha nx)x^{-s-j}\,dx =-\frac{e(\alpha n)\phi_j(1,s)}{2\pi i\alpha n} -\frac1{2\pi i\alpha n}\int_1^\infty\phi_{j+1}(x,s)e(\alpha nx) x^{-s-j-1}\,dx. \] Repeated application of this yields \begin{equation}\label{intparts} \begin{aligned} \int_1^\infty\phi(x)e(\alpha nx)x^{-s}\,dx &=e(\alpha n)\sum_{j=0}^{m-1}\frac{\phi_j(1,s)}{(-2\pi i\alpha n)^{j+1}}\\ &+(-2\pi i\alpha n)^{-m}\int_1^\infty\phi_m(x,s) e(\alpha nx)x^{-s-m}\,dx \end{aligned} \end{equation} for $m\in\mathbb{Z}_{\ge0}$. By \eqref{Amellintransform} and \eqref{intparts} it follows that \begin{align*} \frac1{\Gamma_\mathbb{C}(s)} \int_0^\infty A(\alpha+iy)y^s\frac{dy}{y} &=\sum_{j=0}^{m-1}\frac{\phi_j(1,s)}{(-2\pi i\alpha)^{j+1}} L_{\bar{f}}(s+j+1,-\tfrac{\overline{Na}}{q}+\alpha)\\ &+(-2\pi i\alpha)^{-m}\sum_{n=1}^\infty\frac{b_n}{n^{s+m}} \int_1^\infty\phi_m(x,s)e(\alpha nx)x^{-s-m}\,dx. \end{align*} Each term in the sum extends to an entire function of $s$, by \cite[Proposition~3.1]{BK11}. Furthermore, it may be checked that $\phi_m(x,s)\ll_{m,k}(1+|s|)^mx^{k-1}$. Therefore the last integral is holomorphic for $\Re(s)>k-m$. Letting $m\to\infty$ shows that $\Gamma_\mathbb{C}(s)^{-1}\int_0^\infty A(\alpha+iy)y^s\frac{dy}y$ continues to an entire function. Finally, from \eqref{eq:Aalphaiy} we see that $A(\alpha+iy)$ decays exponentially as $y\to\infty$, and hence $\int_{|\alpha|/4}^\infty A(\alpha+iy)y^s\frac{dy}{y}$ is entire. This completes the proof. \end{proof} \begin{lemma}\label{Blemma} $\Gamma_\mathbb{C}(s)^{-1}\int_0^{|\alpha|/4}B(\alpha+iy)y^s\frac{dy}y$ continues to an entire function of $s$. \end{lemma} \begin{proof} Following the proof of \cite[Lemma~3.4]{Boo16}, we obtain $$ B(\alpha+iy)=\sum_{j=0}^{M-1}P_j(\alpha)y^j+O_M(y^M) \quad\text{for all }M\in\mathbb{Z}_{\ge0}, y\in\bigl(0,\tfrac{|\alpha|}4\bigr], $$ where $$ P_j(\alpha)=\frac{(-i\alpha)^{-j}}{2\pi i}\int_{\Re(s)=\frac{k}2} e^{i\frac{\pi}2\sgn(\alpha)(s+\frac{k-1}2)}|\alpha|^{-s-\frac{k-1}2} \binom{-s-\frac{k-1}2}{j}\Lambda_f(s,\tfrac{a}{q}) \frac{\pi^2}{\sin^2(\pi(s+\frac{k-1}2))}\,ds. $$ Hence, $$ \int_0^{|\alpha|/4}B(\alpha+iy)y^s\frac{dy}{y} -\sum_{j=0}^{M-1}P_j(\alpha)\frac{|\alpha/4|^{s+j}}{s+j} $$ is holomorphic for $\Re(s)>-M$. Note that the sum over $j$ is entire apart from at most simple poles at the poles of $\Gamma_\mathbb{C}(s)$. Dividing by $\Gamma_\mathbb{C}(s)$ and taking $M\to\infty$ concludes the proof. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:Mellin}] Combining Lemmas~\ref{Sfaqz}--\ref{Blemma} and taking $M=1$, we see that $I_{f,a,q,\alpha}(s)-H_{f,a,q,\alpha}(s)$ has analytic continuation to $\Re(s)>0$. If $\int_0^{|\alpha|/4}|S_{f,a,q}(y,\alpha)| y^{\sigma+\frac{k-1}2}\frac{dy}{y}<\infty$ for some $\sigma\ge0$, then the integral defining $I_{f,a,q,\alpha}(s)$ converges absolutely for $\Re(s)>\sigma$, and hence $I_{f,a,q,\alpha}(s)$ is holomorphic in that region. \end{proof} \section{Estimates for $N^s_{f,a,q}(T)$} Fix $f,a,q$ as in Proposition~\ref{voronoi}, and let $\alpha\in\mathbb{Q}^\times$. In this section, we derive estimates for $N^s_{f,a,q}(T)$ based on Proposition~\ref{prop:Mellin}. \begin{lemma}\label{GLprimebound} Let $f\in S_k(\Gamma_1(N))$ be a primitive form. For $\rho=\beta+i\gamma$ a zero of $\Lambda_f(s)$, we have \begin{equation}\label{compLfncboundrho} \Lambda_f'(\rho)\ll_f (2+|\gamma|)^{\frac{k}{2}+\frac{|\beta-\frac12|}{3}-\frac16}\log^2(2+|\gamma|) e^{-\frac{\pi}{2}|\gamma|}. \end{equation} \end{lemma} \begin{proof} We begin by establishing, for $s=\sigma+it$ and $\sigma\in[\frac12,1]$, \begin{equation}\label{compLfncbound} \Gamma_\mathbb{C}(s+\tfrac{k-1}{2})L_f'(s) \ll \tau^{\frac{k}{2}-\frac{1-\sigma}{3}} e^{-\frac{\pi}{2}|t|}\log^2\tau, \end{equation} where $\tau=|t|+2$. By \cite[Theorem 1.1]{BMN19}, we have $$ L_f(\tfrac12+it)\ll\tau^{\frac13}\log\tau. $$ By the Phragm\'{e}n--Lindel\"{o}f principle, using $$ L_f\!\left(-\frac1{\log\tau}+it\right)\ll\tau\log\tau \quad\text{and}\quad L_f\!\left(1+\frac1{\log \tau}+it\right)\ll\log\tau, $$ it follows that $L_f(\sigma+it)\ll\tau^{\frac13}\log\tau$ when $|\sigma-\tfrac12|\le1/\log\tau$. An application of the Cauchy integral formula then yields $L_f'(\tfrac12+it)\ll\tau^{\frac13}\log^2\tau$. By Cauchy's inequality and Rankin's estimate $\sum_{n\le x}|\lambda_f(n)|^2\ll x$, we get $$ |L_f'(1+\varepsilon+it)| \le\sum_{n=1}^\infty\frac{|\lambda_f(n)|\log n}{n^{1+\varepsilon}} \le\Big(\sum_{n=1}^\infty\frac{|\lambda_f(n)|^2}{n^{1+\varepsilon}}\Big)^{\frac12} \zeta''(1+\varepsilon)^{\frac12} \ll\varepsilon^{-\frac12}\varepsilon^{-\frac32}=\varepsilon^{-2} $$ for $\varepsilon>0$. Another application of the Phragm\'{e}n--Lindel\"{o}f principle yields $$ L_f'(\sigma+it)\ll\tau^{\frac23(1-\sigma)}\log^2\tau $$ for $\sigma\in[\frac12,1]$. This, together with the Stirling formula estimate $$ \Gamma_\mathbb{C}(s+\tfrac{k-1}{2})\ll \tau^{\sigma+\tfrac{k}{2}-1} e^{-\frac{\pi}{2}|t|}, $$ yields \eqref{compLfncbound}. Setting $s=\rho=\beta+i\gamma$ with $\beta\ge\frac12$ in \eqref{compLfncbound} gives \begin{equation}\label{lambdafprimerhochi} \Lambda_f'(\rho) =\Gamma_\mathbb{C}(\rho+\tfrac{k-1}{2})L_f'(\rho) \ll(2+|\gamma|)^{\frac{k}{2}-\frac{1-\beta}{3}} \log^2(2+|\gamma|)e^{-\frac{\pi}{2}|\gamma|}. \end{equation} Now suppose $\beta<\frac12$. Differentiating the functional equation we obtain $$ \Lambda_f'(\rho)=-\epsilon N^{\frac12-\rho}\Lambda_{\bar{f}}'(1-\rho), $$ where $|\epsilon|=1$. Applying \eqref{lambdafprimerhochi} to $\Lambda_{\bar{f}}'(1-\rho)$ it follows that \begin{equation}\label{lambdafprimerhochi2} \Lambda_f'(\rho) \ll(2+|\gamma|)^{\frac{k}{2}-\frac{\beta}{3}} \log^2(2+|\gamma|)e^{-\frac{\pi}{2}|\gamma|}. \end{equation} Combining \eqref{lambdafprimerhochi} and \eqref{lambdafprimerhochi2} we obtain \eqref{compLfncboundrho}. \end{proof} \begin{lemma}\label{lem:Nathan} For any fixed $\varepsilon>0$ and all $\sigma\in[\varepsilon,2]$, $$ \int_0^{\frac{|\alpha|}{4}}|S_{f,a,q}(y,\alpha)|y^{\sigma+\frac{k-1}{2}}\frac{dy}{y}\ll \sum_{\substack{\rho=\beta+i\gamma\\\text{a pole of }\Delta_{f,a,q}^*(s)}} (2+|\gamma|)^{\frac{1+|\beta-\frac12|}3-\sigma}\log^2(2+|\gamma|). $$ \end{lemma} \begin{proof} Throughout this proof we let $\rho=\beta+i\gamma$ denote a pole of $\Delta_{f,a,q}^*(s)$, and we set $\tau=2+|\gamma|$. Recalling \eqref{Sydefn}, observe that $(y-i\alpha)^{-\rho-\frac{k-1}2}=e^{i\frac{\pi}{2}\sgn(\alpha)(\rho+\frac{k-1}2)} |\alpha|^{-\rho-\frac{k-1}2}(1+\frac{iy}{\alpha})^{-\rho-\frac{k-1}2}$ and \begin{equation}\label{1plusyalpha} \begin{aligned} \bigl|(1+i\tfrac{y}{\alpha})^{-(\beta+i\gamma+\frac{k-1}2)}\bigr| &=\bigl|e^{-(\frac12\log(1+(y/\alpha)^2)+i\arctan(y/\alpha))(\beta+i\gamma+\frac{k-1}2)}\bigr|\\ &=\bigl(1+(\tfrac{y}{\alpha})^2\bigr)^{-\frac{\beta}{2}-\frac{k-1}4} e^{\gamma\arctan(y/\alpha)}. \end{aligned} \end{equation} Therefore \begin{equation}\label{yalphabound} (y-i\alpha)^{-\rho-\frac{k-1}2} \ll e^{\gamma\sgn(\alpha)(\arctan(y/|\alpha|)-\frac{\pi}2)}. \end{equation} Next, we treat the residue in \eqref{Sydefn}. By \eqref{Deltafaq}, the poles of $\Delta_{f,a,q}^*(s)$ arise from poles of $\Delta_f(s)$ and $\Delta_f(s,\chi)$ with $\chi\ne\chi_0$. The contributrion of an individual term of \eqref{Deltafaq} to $\Res{s=\rho}\Delta_{f,a,q}^*(s)$, if nonzero, is of the form $$ -\Big(1-\frac{q}{q-1}P_{f,q}(q^{-\rho})\Big)\Lambda_f'(\rho) \quad\text{or}\quad -\frac{\tau(\overline{\chi})\chi(a)}{q-1}\Lambda_f'(\rho,\chi). $$ Applying Lemma~\ref{GLprimebound} (possibly replacing $f$ by $f\otimes\chi$) to each of these expressions, it follows that \begin{equation}\label{residue} \Res{s=\rho}\Delta_{f,a,q}^*(s) \ll\tau^{\frac{k}2+\frac{|\beta-\frac12|}3-\frac16}(\log^2\tau) e^{-\frac{\pi}{2}|\gamma|}. \end{equation} It follows from \eqref{Sydefn}, \eqref{yalphabound}, and \eqref{residue} that $$ S_{f,a,q}(y,\alpha)\ll\sum_\rho \tau^{\frac{k}2+\frac{|\beta-\frac12|}3-\frac16}(\log^2\tau) e^{|\gamma|[\sgn(\alpha\gamma)\arctan(y/|\alpha|)-\frac{\pi}2(1+\sgn(\alpha\gamma))]}. $$ By considering cases and using the bound $\arctan{u}\ge\frac{u}{2}$ for $0\le u\le\frac14$, we have $$ S_{f,a,q}(y,\alpha)\ll\sum_\rho \tau^{\frac{k}2+\frac{|\beta-\frac12|}3-\frac16}(\log^2\tau)e^{-c|\gamma|y} \quad\text{for }y\in\bigl(0,\tfrac{|\alpha|}4\bigr], $$ where $c=\frac1{2|\alpha|}>0$. We deduce from this $$ \int_0^{\frac{|\alpha|}{4}}|S_{f,a,q}(y,\alpha)| y^{\sigma+\frac{k-1}{2}}\frac{dy}{y} \ll\sum_\rho \tau^{\frac{k}2+\frac{|\beta-\frac12|}3-\frac16}\log^2\tau \int_0^{\frac{|\alpha|}{4}}e^{-c|\gamma|y}y^{\sigma+\frac{k-1}{2}}\frac{dy}{y}. $$ Now $$ \int_0^{\frac{|\alpha|}{4}}e^{-c|\gamma|y}y^{\sigma+\frac{k-1}{2}}\frac{dy}{y} \ll\int_0^{\frac{|\alpha|}{4}}e^{-c\tau y}y^{\sigma+\frac{k-1}{2}}\frac{dy}{y} \le\int_0^\infty e^{-c\tau y}y^{\sigma+\frac{k-1}{2}}\frac{dy}{y}. $$ By the variable change $u=c\tau y$, the last integral equals $$ \frac1{(c\tau)^{\sigma+\frac{k-1}{2}}}\int_0^\infty e^{-u}u^{\sigma+\frac{k-1}{2}}\frac{du}{u} =\frac{\Gamma(\sigma+\frac{k-1}{2})}{(c\tau)^{\sigma+\frac{k-1}{2}}} \ll\tau^{-\sigma-\frac{k-1}2}, $$ and thus $$ \int_0^{\frac{|\alpha|}{4}}S_{f,a,q}(y,\alpha)y^{\sigma+\frac{k-1}{2}}\frac{dy}{y} \ll\sum_\rho \tau^{\frac{1+|\beta-\frac12|}3-\sigma}\log^2\tau. $$ \end{proof} For a meromorphic function $h$ on $\{s\in\mathbb{C}:\Re(s)>1\}$, define $$ \Theta(h)=\inf\bigl\{\theta\ge0:h\text{ continues analytically to } \{s\in\mathbb{C}:\Re(s)>\theta\}\bigr\}. $$ We also set $$ \theta_{f,a,q}(T)=\sup\bigl(\{0\}\cup\bigl\{\Re(\rho),1-\Re(\rho): \rho\in\mathbb{C}, |\Im(\rho)|\le T, \Res{s=\rho}\Delta_{f,a,q}^*(s)\ne0\bigr\}\bigr) $$ and $$ \theta_{f,a,q}=\lim_{T\to\infty}\theta_{f,a,q}(T). $$ By Proposition~\ref{voronoi}, we have \begin{equation}\label{thetafaq} \theta_{f,a,q}=\max(\Theta(\Delta_{f,a,q}),\Theta(\Delta_{\bar{f},-\overline{Na},q})). \end{equation} \begin{proposition}\label{NfaqTlowerbound} If $\Theta(H_{f,a,q,\alpha})>0$ then $\theta_{f,a,q}\ge\frac12$ and \begin{equation}\label{eq:omega1} N^s_{f,a,q}(T)= \Omega\bigl(T^{\frac13(1-\theta_{f,a,q})+\Theta(H_{f,a,q,\alpha}) -\frac12-\varepsilon}\bigr) \quad\text{for all }\varepsilon>0. \end{equation} Further, if $\Theta(H_{f,a,q,\alpha})=\frac12$ and $H_{f,a,q,\alpha}(s)$ has a pole with real part $\frac12$, then \begin{equation}\label{eq:omega2} N^s_{f,a,q}(T)=\Omega\!\left( \frac{T^{\frac13(1-\theta_{f,a,q}(T))}}{(1-\theta_{f,a,q}(T))\log^2{T}}\right), \end{equation} and there are arbitrarily large $T>0$ such that \begin{equation}\label{eq:omega3} N^s_{f,a,q}(T)\ge\log\log\log{T}. \end{equation} \end{proposition} \begin{proof} Let $\beta_n+i\gamma_n$ run through the poles of $\Delta_{f,a,q}^*(s)$, in increasing order of $|\gamma_n|$. For brevity, we write $I(s)$, $H(s)$, $\Theta$, $S(y)$, $N(t)$, $\theta(t)$ and $\theta$ for $I_{f,a,q,\alpha}(s)$, $H_{f,a,q,\alpha}(s)$, $\Theta(H_{f,a,q,\alpha})$, $S_{f,a,q}(y,\alpha)$, $N^s_{f,a,q}(t)$, $\theta_{f,a,q}(t)$ and $\theta_{f,a,q}$, respectively. By Lemma~\ref{lem:Nathan}, we have \begin{align*} \int_0^{|\alpha|/4}|S(y)|y^{\sigma+\frac{k-1}2}\frac{dy}{y} &\ll\sum_{n\ge 1}(2+|\gamma_n|)^{\frac{1+|\beta_n-\frac12|}{3}-\sigma} \log^2(2+|\gamma_n|)\\ &\le\sum_{n\ge 1}(2+|\gamma_n|)^{\frac{\theta(|\gamma_n|)}{3} +\frac16-\sigma}\log^2(2+|\gamma_n|). \end{align*} If $\Theta>0$ then by Proposition~\ref{prop:Mellin}, the integral must diverge for sufficiently small $\sigma>0$, and thus the right-hand side has infinitely many terms. Thus $\Delta_{f,a,q}^*(s)$ has poles, so $\theta\ge\frac12$. Suppose that \eqref{eq:omega1} does not hold. Then there exists $\varepsilon\in(0,\Theta)$ such that $N(t)=o(t^{\frac13(1-\theta)+\Theta-\frac12-\varepsilon})$. Choosing $\sigma=\Theta-\frac{\varepsilon}3$ and using the estimate $\log^2(2+|\gamma_n|)\ll(2+|\gamma_n|)^{\frac{\varepsilon}{3}}$, we have \begin{align*} \int_0^{|\alpha|/4}|S(y)|y^{\Theta-\frac{\varepsilon}3+\frac{k-1}2}\frac{dy}{y} &\ll\sum_{n\ge1} (2+|\gamma_n|)^{\frac{\theta}{3}+\frac16-\Theta+\frac23\varepsilon} \ll1+\int_1^\infty t^{\frac{\theta}{3}+\frac16-\Theta+\frac23\varepsilon}\,dN(t)\\ &\ll1+\int_1^\infty t^{\frac{\theta}{3}+\frac16-\Theta+\frac23\varepsilon-1}N(t)\,dt \ll1+\int_1^\infty t^{-1-\frac{\varepsilon}3}\,dt \ll1. \end{align*} By Proposition~\ref{prop:Mellin}, it follows that $H(s)$ is holomorphic for $\Re(s)>\Theta-\frac{\varepsilon}3$. This is a contradiction, so \eqref{eq:omega1} must hold. Next suppose that $\Theta=\frac12$, and let $\rho$ be a pole of $H(s)$ with $\Re(\rho)=\frac12$. Then for sufficiently small $\delta>0$, by Proposition~\ref{prop:Mellin}, we have $$ \delta^{-1}\ll|H(\rho+\delta)|\ll1+|I(\rho+\delta)| \le1+\int_0^{|\alpha|/4}|S(y)|y^{\delta+\frac{k}2}\frac{dy}{y}, $$ where we understand the right-hand side to be $\infty$ if the integral diverges. Applying Lemma~\ref{lem:Nathan}, we thus have \begin{equation}\label{eq:gammasum} \delta^{-1} \ll1+\sum_{n\ge 1}(2+|\gamma_n|)^{\frac{\theta(|\gamma_n|)-1}{3}-\delta} \log^2(2+|\gamma_n|). \end{equation} In particular, the right-hand side must have infinitely many terms. Applying integration by parts, we get \begin{align*} \delta^{-1}&\ll1+\int_1^\infty t^{\frac{\theta(t)-1}{3}-\delta}\log^2{t}\,dN(t) =1-\int_1^\infty N(t)d(t^{\frac{\theta(t)-1}{3}-\delta}\log^2{t})\\ &\le1+\int_1^\infty N(t)\bigl(\tfrac{1-\theta(t)}{3}+\delta\bigr) t^{\frac{\theta(t)-1}{3}-\delta-1}\log^2{t}\,dt, \end{align*} where for the last inequality we have used the fact that $\theta(t)$ is nondecreasing and $$ d(t^{\frac{\theta(t)-1}{3}-\delta}\log^2{t}) =t^{\frac{\theta(t)-1}{3}-\delta-1}(\log{t}) \bigl[2-\bigl(\tfrac{1-\theta(t)}{3}+\delta\bigr)\log{t}\bigr]\,dt +\tfrac13\log^3{t}\,d\theta(t). $$ Suppose that \eqref{eq:omega2} is false, so that the function $\varepsilon(t)=N(t)t^{\frac13(\theta(t)-1)}(1-\theta(t))\log^2{t}$ satisfies $\lim_{t\to\infty}\varepsilon(t)=0$. Then we have $$ \delta^{-1}\ll1+\int_1^\infty \left(\frac13+\frac{\delta}{1-\theta(t)}\right) \varepsilon(t)t^{-1-\delta}\,dt. $$ By the standard zero-free region \cite[Theorem~5.10]{IK04}, we have $$ \frac1{1-\theta(t)}\ll\log\max(t,2), $$ so that $$ \delta^{-1}\ll1+\int_1^\infty(1+\delta\log{t})\varepsilon(t)t^{-1-\delta}\,dt =1+\delta^{-1}\int_0^\infty\varepsilon(e^{u/\delta})(1+u)e^{-u}\,du =o(\delta^{-1}). $$ This is a contradiction, so \eqref{eq:omega2} holds. Finally, suppose \eqref{eq:omega3} is false, so that $N(T)<\log\log\log{T}$ for all sufficiently large $T$. Then there exists $n_0\ge\mathbb{Z}_{>0}$ such that $|\gamma_n|>\exp\exp\exp{n}$ for all $n\ge n_0$. Since the terms from $n<n_0$ contribute a bounded amount to \eqref{eq:gammasum}, we have $$ 1+\sum_{n=n_0}^\infty|\gamma_n|^{-\delta}\log^2|\gamma_n| \gg\delta^{-1} $$ for all sufficiently small $\delta>0$. Next we claim that there are infinitely many $m\ge n_0$ such that \begin{equation}\label{eq:biggap} \log\log|\gamma_{m+1}|\ge\tfrac{13}{5}\log\log|\gamma_m|. \end{equation} If not then there exists $n_1\ge n_0$ such that \eqref{eq:biggap} fails for all $m\ge n_1$, and by induction it follows that $$ \log\log|\gamma_n|\le(\tfrac{13}{5})^{n-n_1}\log\log|\gamma_{n_1}| =c(\tfrac{13}{5})^n \quad\text{for }n\ge n_1, $$ where $c=(13/5)^{-n_1}\log\log|\gamma_{n_1}|>0$. Hence, $$ n<\log\log\log|\gamma_n|\le\log{c}+n\log\tfrac{13}{5}. $$ Since $\log\frac{13}{5}<1$, this is false for sufficiently large $n$, proving the claim. Choose a large $m\ge n_0$ satisfying \eqref{eq:biggap}, and set $\delta_m=(\log|\gamma_m|)^{-\frac{12}{5}}$. Then using the trivial bound $e^{e^e}\le|\gamma_n|\le|\gamma_m|$ for $n_0\le n\le m$, we have $$ \sum_{n=n_0}^m|\gamma_n|^{-\delta_m}\log^2|\gamma_n| \le m\log^2|\gamma_m|<(\log\log\log|\gamma_m|)\log^2|\gamma_m| \le(\log|\gamma_m|)^{\frac{11}{5}} =\delta_m^{-\frac{11}{12}}, $$ since $\log\log{x}\le x^{\frac15}$ for all $x>1$. To estimate the contribution from $n>m$ we apply integration by parts. Set $g(t)=t^{-\delta_m}\log^2{t}$. Then $g'(t)<0$ for $t>e^{2/\delta_m}=\exp(2(\log|\gamma_m|)^{12/5})$; in particular, if $m$ is sufficiently large then, by \eqref{eq:biggap}, $g'(t)<0$ for $t\ge|\gamma_{m+1}|$. Hence, we have \begin{align*} \sum_{n=m+1}^\infty g(|\gamma_n|) &=\lim_{\varepsilon\to0^+}\int_{|\gamma_{m+1}|-\varepsilon}^\infty g(t)\,dN(t) =\int_{|\gamma_{m+1}|}^\infty(-g'(t))(N(t)-m)\,dt\\ &\le\int_{|\gamma_{m+1}|}^\infty(-g'(t))(\log\log\log{t})\,dt \le\delta_m\int_{|\gamma_{m+1}|}^\infty t^{-\delta_m-1}(\log{t})^{\frac{11}{5}}\,dt\\ &=\delta_m\int_{\log|\gamma_{m+1}|}^\infty e^{-\delta_mu}u^{\frac{11}5}\,du. \end{align*} Applying integration by parts three times and using that $\delta_m\log|\gamma_{m+1}|\gg1$, we get $$ \delta_m\int_{\log|\gamma_{m+1}|}^\infty e^{-\delta_mu}u^{\frac{11}5}\,du \ll|\gamma_{m+1}|^{-\delta_m}(\log|\gamma_{m+1}|)^{\frac{11}{5}}. $$ Note that $\delta_m=(\log|\gamma_m|)^{-\frac{12}{5}} \ge(\log|\gamma_{m+1}|)^{-\frac{12}{13}}$, so $|\gamma_{m+1}|^{-\delta_m}\le\exp(-(\log|\gamma_{m+1}|)^{\frac1{13}})$. Hence, we conclude that $$ \sum_{n=m+1}^\infty g(|\gamma_n|)\ll \exp(-(\log|\gamma_{m+1}|)^{\frac1{13}})(\log|\gamma_{m+1}|)^{\frac{11}{5}} \ll 1. $$ Thus, altogether we have $$ \delta_m^{-1}\ll1+\sum_{n=n_0}^\infty|\gamma_n|^{-\delta_m}\log^2|\gamma_n| \ll1+\delta_m^{-\frac{11}{12}}. $$ This is false for sufficiently large $m$, so \eqref{eq:omega3} must hold for some arbitrarily large $T$. \end{proof} \section{Proofs of Theorems~\ref{thm:twist} and \ref{thm:oddN}} We begin with an overview of the argument. By Proposition~\ref{NfaqTlowerbound}, $N^s_{f,a,q}(T)$ is sometimes large if there exists $\alpha\in\mathbb{Q}^\times$ for which $H_{f,a,q,\alpha}(s)$ has a pole with large real part. The main obstacle to showing this is that $H_{f,a,q,\alpha}(s)$ is defined as the difference of two functions (cf.~\eqref{eq:Hdef}), whose poles could in principle cancel out. However, as we show, there are some dependencies between $H_{f,a,q,\alpha}(s)$ for various choices of $(a,q,\alpha)$, from which it follows that there is a suitable pole for at least one choice of inputs. More specifically, in Lemma~\ref{lem:holo} we exhibit a relationship between $H_{f,1,1,a/p}(s)$ and $H_{f,a,q,-a/q}(s)$, where $p$ and $q$ are primes satisfying $pq\equiv-1\pmod*{Na}$. For any prime $p\nmid N$, we show that there is some choice of $a\in\mathbb{Z}$ for which this leads to poles at the simple zeros of $\Lambda_f(s)$, and thanks to \cite[Theorem~1.1]{Boo16}, those exist in abundance. Ultimately this implies that at least one of $N^s_f(T)$, $N^s_{f,a,p}(T)$, $N^s_{f,a,q}(T)$ is large, which yields Theorem~\ref{thm:twist}. Choosing $p=2$ and appealing to the second and third conclusions of Proposition~\ref{NfaqTlowerbound} yields Theorem~\ref{thm:oddN}. Proceeding, given a prime $p$ and $a\in\mathbb{Z}$ coprime to $p$, define \begin{equation}\label{eq:Cdef} C_{f,a,p}(s)=\Delta_{f,a,p}(s)-\xi(p)p^{1-2s}\Delta_f(s). \end{equation} \begin{lemma}\label{lem:holo} Let $a\in\mathbb{Z}$, and let $p$ and $q$ be prime numbers such that $pq\equiv-1\pmod*{Na}$. Then \begin{enumerate} \item[(i)] $C_{f,a,p}(s)-\bigl(H_{f,1,1,a/p}(s)-\xi(p)p^{1-2s}H_{f,a,q,-a/q}(s)\bigr)$ is holomorphic for $\Re(s)>0$; \item[(ii)] $\displaystyle{\sum_{b=1}^{p-1}C_{f,b,p}(s)}=-P_{f,p}(p^{1-s})\Delta_f(s)$. \end{enumerate} \end{lemma} \begin{proof} We first consider $H_{f,a,q,\alpha}(s)$, where $\alpha=-a/q$. We have $$ \Delta_{f,a,q}(s,\alpha)-\Delta_f(s) =-R_{f,q}(q^{-s})\Lambda_f(s), $$ which is holomorphic for $\Re(s)>0$. Set $a'=-\frac{1+pq}{Na}$, so that $\frac{a'}{q}-\frac1{Nq^2\alpha}=-\frac{p}{Na}$. Let $r_{\bar{f},q}(j)$ be the numbers such that $$ R_{\bar{f},q}(x)=\sum_{j=1}^\infty r_{\bar{f},q}(j)x^j. $$ By Fourier inversion, we have $$ \sum_{\substack{j\ge1\\j\equiv{t}\;(\text{mod }\varphi(Na))}} r_{\bar{f},q}(j)x^j =\frac1{\varphi(Na)}\sum_{\ell=1}^{\varphi(Na)} e\!\left(-\frac{\ell{t}}{\varphi(Na)}\right) R_{\bar{f},q}\!\left(e\!\left(\frac\ell{\varphi(Na)}\right)x\right). $$ Thus, \begin{align*} \Delta_{\bar{f},a',q}&\!\left(s,-\frac1{Nq^2\alpha}\right) -\Delta_{\bar{f}}\!\left(s,-\frac{p}{Na}\right) =-\sum_{j=1}^\infty r_{\bar{f},q}(j)q^{-js} \Lambda_{\bar{f}}\!\left(s,\frac{q^{j-1}}{Na}\right)\\ &=-\sum_{t=1}^{\varphi(Na)}\Lambda_{\bar{f}}\!\left(s,\frac{q^{t-1}}{Na}\right) \sum_{\substack{j\ge1\\j\equiv{t}\;(\text{mod }\varphi(Na))}} r_{\bar{f},q}(j)q^{-js}\\ &=-\frac1{\varphi(Na)} \sum_{t=1}^{\varphi(Na)}\Lambda_{\bar{f}}\!\left(s,\frac{q^{t-1}}{Na}\right) \sum_{\ell=1}^{\varphi(Na)} e\!\left(-\frac{\ell{t}}{\varphi(Na)}\right) R_{\bar{f},q}\!\left(e\!\left(\frac\ell{\varphi(Na)}\right)q^{-s}\right), \end{align*} which is again holomorphic for $\Re(s)>0$. Hence, up to a holomorphic function, $H_{f,a,q,\alpha}(s)$ is $$ \Delta_f(s)-\epsilon\xi(q)(-i\sgn{a})^k(Na^2)^{s-\frac12} \Delta_{\bar{f}}\!\left(s,-\frac{p}{Na}\right). $$ Next note that $$ C_{f,a,p}(s)-H_{f,1,1,a/p}(s)= \epsilon(i\sgn{a})^k\left(\frac{Na^2}{p^2}\right)^{s-\frac12} \Delta_{\bar{f}}\!\left(s,-\frac{p}{Na}\right) -\xi(p)p^{1-2s}\Delta_f(s)-R_{f,q}(q^{-s})\Lambda_f(s). $$ Therefore, since $\xi(p)\xi(q)=\xi(-1)=(-1)^k$, we see that $$ C_{f,a,p}(s)-H_{f,1,1,a/p}(s)+\xi(p)p^{1-2s}H_{f,a,q,\alpha}(s) $$ is holomorphic for $\Re(s)>0$. Finally, by \eqref{Deltafaq} we have \begin{align*} \sum_{b=1}^{p-1}C_{f,b,p}(s) &=(p-1)\left[1-\frac{p}{p-1}P_{f,p}(p^{-s})-\xi(p)p^{1-2s}\right]\Delta_f(s)\\ &=-P_{f,p}(p^{1-s})\Delta_f(s). \end{align*} \end{proof} In the following we shall make frequent use of the observation that for any pair $h_1,h_2$ of meromorphic functions, \begin{equation}\label{eq:thetah1h2} \Theta(h_1+h_2)\le\max(\Theta(h_1),\Theta(h_2)), \quad\text{with equality when }\Theta(h_1)\ne\Theta(h_2). \end{equation} Fix a prime $p\nmid N$. By \cite[Theorem~1.1]{Boo16} and the functional equation, $\Delta_f(s)$ has a pole with real part $\ge\frac12$, and thus \begin{equation}\label{thetaf11lb} \Theta(\Delta_f)=\theta_{f,1,1}\ge\frac12. \end{equation} Since all zeros of $P_{f,p}(p^{1-s})$ have real part $1$, this is also true of $P_{f,p}(p^{1-s})\Delta_f(s)$. Hence, by Lemma~\ref{lem:holo}(ii), there exists $a\in\{1,\ldots,p-1\}$ such that $C_{f,a,p}(s)$ has a pole with real part $\ge\frac12$ and satisfies $\Theta(C_{f,a,p})\ge\theta_{f,1,1}$. By \eqref{eq:Cdef} and \eqref{eq:thetah1h2}, it follows that \begin{equation}\label{thetaCfap} \Theta(C_{f,a,p})=\max(\Theta(\Delta_{f,a,p}),\theta_{f,1,1}). \end{equation} Let $q$ be a prime satisfying $pq\equiv-1\pmod*{Na}$, and set $a'=-(1+pq)/(Na)$. We aim to prove that \begin{equation}\label{eq:summary} \max\bigl(N^s_f(T),N^s_{f,a,p}(T),N^s_{f,a,q}(T)\bigr) =\Omega\bigl(T^{\frac16-\varepsilon}\bigr) \quad\text{for all }\varepsilon>0. \end{equation} To that end, we will show that at least one of the following inequalities holds for some $\alpha\in\mathbb{Q}^\times$: \begin{itemize} \item[(i)] $\max(\Theta(H_{f,1,1,\alpha}),\Theta(H_{\bar{f},1,1,\alpha})) \ge\theta_{f,1,1}\ge\frac12$; \item[(ii)] $\max(\Theta(H_{f,a,p,\alpha}),\Theta(H_{\bar{f},a',p,\alpha})) \ge\theta_{f,a,p}\ge\frac12$; \item[(iii)] $\max(\Theta(H_{f,a,q,\alpha}),\Theta(H_{\bar{f},a',q,\alpha})) \ge\theta_{f,a,q}\ge\frac12$. \end{itemize} To see that this suffices, suppose for instance that (iii) holds. By Proposition~\ref{voronoi}, we have $N^s_{f,a,q}(T)=N^s_{\bar{f},a',q}(T)$ and $\theta_{f,a,q}=\theta_{\bar{f},a',q}$. Thus, applying Proposition~\ref{NfaqTlowerbound} to either $(f,a,q)$ or $(\bar{f},a',q)$, we conclude that $$ N^s_{f,a,q}(T)=\Omega(T^{\beta-\varepsilon}), \quad\text{where } \beta\ge\frac13(1-\theta_{f,a,q})+\theta_{f,a,q}-\frac12 =\frac{2\theta_{f,a,q}}{3}-\frac16\ge\frac16. $$ If, instead, (i) or (ii) holds, then by a similar argument we find that $N^s_{f,1,1}(T)=\Omega(T^{\beta-\varepsilon})$ or $N^s_{f,a,p}(T)=\Omega(T^{\beta-\varepsilon})$ for some $\beta\ge\frac16$. Hence, \eqref{eq:summary} follows in any case. Let us suppose that conditions (i) and (iii) are false for all $\alpha\in\mathbb{Q}^\times$ and show that this leads to (ii). Since (i) is false, in view of \eqref{thetaf11lb} we must have $\theta_{f,1,1}>\Theta(H_{f,1,1,a/p})$. In turn, by \eqref{thetaCfap} this implies that $\Theta(C_{f,a,p})>\Theta(H_{f,1,1,a/p})$. Hence, by Lemma~\ref{lem:holo}(i) and \eqref{eq:thetah1h2}, we have $\Theta(H_{f,a,q,-a/q})=\Theta(C_{f,a,p})$. By \eqref{thetaCfap}, this implies $\Theta(H_{f,a,q,-a/q})\ge\theta_{f,1,1}>0$, and thus $\theta_{f,a,q}\ge\frac12$, by Proposition~\ref{NfaqTlowerbound}. Next, by \eqref{thetafaq} we have $\theta_{f,a,p}=\max(\Theta(\Delta_{f,a,p}),\Theta(\Delta_{\bar{f},a',p}))$. If \begin{equation}\label{cond1} \Theta(\Delta_{\bar{f},a',p}) \le\max(\Theta(\Delta_{f,a,p}),\theta_{f,1,1}) =\Theta(H_{f,a,q,-a/q}) \end{equation} then it follows that \begin{equation}\label{ThetaHf} \Theta(H_{f,a,q,-a/q})=\max(\theta_{f,a,p},\theta_{f,1,1}). \end{equation} Suppose now that \eqref{cond1} is false. Then $\Theta(\Delta_{\bar{f},a',p}) >\max(\Theta(\Delta_{f,a,p}),\theta_{f,1,1})$, so that $$ \theta_{f,a,p}=\Theta(\Delta_{\bar{f},a',p})>\theta_{f,1,1}. $$ Since (i) is false, this implies that $\Theta(\Delta_{\bar{f},a',p}) >\max(\Theta(H_{\bar{f},1,1,a'/p}),\theta_{f,1,1})$. By \eqref{eq:Cdef} and Lemma~\ref{lem:holo}(i) with $(\bar{f},a')$ in place of $(f,a)$, it follows from \eqref{eq:thetah1h2} that \begin{equation}\label{ThetaHbarf} \Theta(H_{\bar{f},a',q,-a'/q})=\Theta(\Delta_{\bar{f},a',p}) =\max(\theta_{f,a,p},\theta_{f,1,1}). \end{equation} Therefore, since at least one of \eqref{ThetaHf} and \eqref{ThetaHbarf} must hold, we have $$ \max(\Theta(H_{f,a,q,-a/q}),\Theta(H_{\bar{f},a',q,-a'/q})) \ge\max(\theta_{f,a,p},\theta_{f,1,1}). $$ Since (iii) is false, this implies that $\theta_{f,a,q}>\max(\theta_{f,a,p},\theta_{f,1,1})$. Hence, by \eqref{thetafaq}, either \begin{equation}\label{branch} \Theta(\Delta_{f,a,q}) >\max(\theta_{f,a,p},\theta_{f,1,1}) \quad\text{or}\quad \Theta(\Delta_{\bar{f},a',q}) >\max(\theta_{f,a,p},\theta_{f,1,1}). \end{equation} Suppose that the first inequality in \eqref{branch} holds. Then by \eqref{eq:Cdef} (with $q$ in place of $p$) and \eqref{eq:thetah1h2}, we have $\Theta(C_{f,a,q})=\Theta(\Delta_{f,a,q})>\theta_{f,1,1}$. Since (i) is false, this implies $\Theta(C_{f,a,q})>\Theta(H_{f,1,1,a/q})$. On the other hand, by Lemma~\ref{lem:holo}(i) (with the roles of $p$ and $q$ reversed) and \eqref{eq:thetah1h2}, we have $$\Theta(H_{f,a,p,-a/p})=\Theta(C_{f,a,q}) =\Theta(\Delta_{f,a,q})>\theta_{f,a,p}.$$ This also implies that $\Theta(H_{f,a,p,-a/p})>0$, whence $\theta_{f,a,p}\ge\frac12$, by Proposition~\ref{NfaqTlowerbound}. If, instead, the second inequality holds in \eqref{branch}, then running through the same argument with $(\bar{f},a')$ in place of $(f,a)$, we find that $$\Theta(H_{\bar{f},a',p,-a'/p})=\Theta(C_{\bar{f},a',q}) =\Theta(\Delta_{\bar{f},a',q})>\theta_{\bar{f},a',p}\ge\frac12. $$ Hence, in either case we see that (ii) holds, and this concludes the proof of \eqref{eq:summary}. Now, by \eqref{eq:summary} and \eqref{Deltafaq}, it follows that there is a character $\chi$ of conductor $1$, $p$ or $q$ such that $N^s_{f\otimes\chi}(T)=\Omega(T^{\frac16-\varepsilon})$ for all $\varepsilon>0$. This implies Theorem~\ref{thm:twist}. For the proof of Theorem~\ref{thm:oddN}, we may assume that $N^s_f(T)\ll1+T^\varepsilon$ for all $\varepsilon>0$, since the result is trivial otherwise. To avoid contradicting Proposition~\ref{NfaqTlowerbound}, it must therefore be the case that $\max(\Theta(H_{f,1,1,\alpha}),\Theta(H_{\bar{f},1,1,\alpha})) \le\frac12$ for all $\alpha\in\mathbb{Q}^\times$. Since $N$ is odd, we can take $p=2$ and $a=1$ in the above, and choose any suitable prime $q$. Then by Lemma~\ref{lem:holo}(ii), we have $$ \Delta_{f,a,p}(s)=\bigl(\xi(p)p^{1-2s}-P_{f,p}(p^{1-s})\bigr)\Delta_f(s), $$ and it follows that $N^s_{f,a,p}(T)\leN^s_f(T)$ and $\max(\Theta(H_{f,a,p,\alpha}),\Theta(H_{\bar{f},a',p,\alpha})) \le\frac12$ for all $\alpha\in\mathbb{Q}^\times$. Thus, by \eqref{eq:summary}, $N^s_{f,a,q}(T)=\Omega(T^{\frac16-\varepsilon})$ for all $\varepsilon>0$. Therefore, by Proposition~\ref{voronoi}, at least one of $\Delta_{f,a,q}(s),\Delta_{\bar{f},a',q}(s)$ has a pole in the region $\{s\in\mathbb{C}:\Re(s)\ge\frac12\}$ that is not a pole of $\Delta_f(s)$. By \eqref{eq:Cdef} and Lemma~\ref{lem:holo}(i), the same applies to one of $H_{f,1,1,a/q}(s)$, $H_{f,a,p,-a/p}(s)$, $H_{\bar{f},1,1,a'/q}(s)$, or $H_{\bar{f},a',p,-a'/p}(s)$. Since $$ N^s_{f,a,p}(T)=N^s_{\bar{f},a',p}(T)\le N^s_{f,1,1}(T)=N^s_{\bar{f},1,1}(T), $$ whichever function has the pole, we can apply Proposition~\ref{NfaqTlowerbound} to see that $N^s_f(T)$ satisfies the second and third conclusions. In particular, $N^s_f(T)\ge\log\log\log{T}$ for some arbitrarily large $T$, and if $k=1$ or $f$ is a CM form then Coleman's theorem \cite{Col90} implies that $$ 1-\theta_{f,a,p}(T)\ge 1-\theta_{f,1,1}(T)\gg (\log{T})^{-\frac23}(\log\log{T})^{-\frac13} \quad\text{for all }T\ge3, $$ whence $N^s_f(T)=\Omega(\exp((\log{T})^{\frac13-\varepsilon}))$ for all $\varepsilon>0$. Moreover, since $N^s_f(T)\ll1+T^\varepsilon$, we must have $\theta_{f,1,1}=1$, so $\Lambda_f(s)$ has simple zeros with real part arbitrarily close to $1$. Finally, by Lemma~\ref{lem:holo}(ii) we have $\Theta(C_{f,a,p})=1$. Since $\Theta(H_{f,1,1,a/p})\le\frac12$, Lemma~\ref{lem:holo}(i) and \eqref{eq:thetah1h2} imply that $\Theta(H_{f,a,q,-a/q})=1$. Applying Proposition~\ref{NfaqTlowerbound}, it follows that $N^s_{f,a,q}(T)=\Omega(T^{\frac12-\varepsilon})$ for all $\varepsilon>0$. By Lemma~\ref{lem:holo}(i), $C_{f,a,q}(s)$ and $C_{\bar{f},a',q}(s)$ are holomorphic for $\Re(s)>\frac12$. Hence, by \eqref{eq:Cdef} and Proposition~\ref{voronoi}, all poles of $\Delta_{f,a,q}^*(s)$ that are not poles of $\Delta_f^*(s)$ lie on the line $\{s\in\mathbb{C}:\Re(s)=\frac12\}$. Since $N^s_f(T)\ll1+T^\varepsilon$, $\Delta_{f,a,q}^*(s)$ must have $\Omega(T^{\frac12-\varepsilon})$ poles with real part $\frac12$ and imaginary part in $[-T,T]$. By \eqref{Deltafaq}, the same applies to $\Delta_f(s,\chi)$ for some $\chi\pmod*{q}$. \begin{comment} Consider the residue sum $$ S(y,\alpha)=\sum_{\rho}\Res{s=\rho}\bigl( \Delta_{f,a,q}^*(s)-\xi(q)q^{1-2s}\Delta_f^*(s)\bigr)(y-i\alpha)^{-\rho} =S_{f,a,q}(y,\alpha)-q\xi(q)S_{f,1,1}(q^2y,q^2\alpha). $$ By Proposition~\ref{prop:Mellin}, $$ \int_0^{|\alpha|/4}S(y,\alpha)y^{s+\frac{k-1}2}\frac{dy}{y} -\bigl(H_{f,a,q,\alpha}(s)-\xi(q)q^{2-2s-k}H_{f,1,1,q^2\alpha}(s)\bigr) $$ is holomorphic for $\Re(s)>0$. We take $\alpha=-a/q$, so that $$ \Theta\bigl(H_{f,a,q,\alpha}(s) -\xi(q)q^{2-2s-k}H_{f,1,1,q^2\alpha}(s)\bigr)=1. $$ Hence, following the proof of \eqref{eq:omega1} with the obvious modifications, we get $$ N^s_{f,a,q}(T)+N^s_{f,1,1}(T)=\Omega(T^{\frac23-\varepsilon}). $$ Since $N^s_{f,1,1}(T)\ll1+T^\varepsilon$, we see that $\Delta_{f,a,q}^*(s)$ has $\Omega(T^{\frac23-\varepsilon})$ poles with real part $\frac12$ and imaginary part in $[-T,T]$. \end{comment} \end{document}

\begin{document} \title{Existence of Solutions in Bi-level Stochastic Linear Programming with Integer Variables} \titlerunning{Existence of Solutions for Bi-level Stochastic MILPs} \author{Johanna Burtscheidt \and Matthias Claus} \institute{Johanna Burtscheidt, Corresponding author \at University of Duisburg-Essen\\ Essen, Germany\\ [email protected] \and Matthias Claus \at University of Duisburg-Essen\\ Essen, Germany\\ [email protected] } \maketitle \begin{abstract} The addition of lower level integrality constraints to a bi-level linear program is known to result in significantly weaker analytical properties. Most notably, the upper level goal function in the optimistic setting lacks lower semicontinuity and the existence of an optimal solution cannot be guaranteed under standard assumptions. In this paper, we study a setting where the right-hand side of the lower level constraint system is affected by the leader's choice as well as the realization of some random vector. Assuming that only the follower decides under complete information, we employ a convex risk measure to assess the upper level outcome. Confining the analysis to the cases where the lower level feasible set is finite, we provide sufficient conditions for Hölder continuity of the leader's risk functional and draw conclusions about the existence of optimal solutions. Finally, we examine qualitative stability with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the leader's decision and the underlying probability measure. \end{abstract} \keywords{Bi-level Stochastic Linear Programming \and Discrete Optimization \and Existence Results \and Risk Aversion \and Stability} \subclass{90C15 \and 90C26 \and 90C31 \and 91A65} \section{Introduction} The present work deals with bi-level stochastic linear problems in which the realization of a random vector whose distribution does not depend on the upper level decision enters the lower level as an additional parameter. It is assumed that the leader must make his decision without knowing the realization of the random vector, while the follower decides under complete information. However, the variables controlled by the follower are integral, which entails that techniques from the non-integer case are not or only partially transferable to the present model. In addition, the feasible set of the follower is finite. An application of this problem type is conceivable, for example, in the transport sector: The leader decides on the expansion of a road or rail network, i.e., how procured certain road sections should be built in order to protect them from flooding or other weather effects on the one hand and to minimize costs on the other hand? After the realization of the randomness (i.e. the water level or weather), the follower solves a shortest path problem: Are the extended route sections usable, or does he have to fall back on existing cab connections or bus shuttles, the use of which causes costs for the leader in the course of customer satisfaction? Various types of price setting problems are listed in \cite{LaVi16}, where two types of network price problems use continuous decisions at the upper level and binary decisions at the lower level. Applications that make use of a bi-level knapsack problem can be found in \cite[\S 5]{Fa06}, \cite{OePrSc10} and references there. In the case of \cite{OePrSc10} even bi-level knapsack problems with stochastic right hand side are considered: They are a stochastic extension of the bi-level knapsack problem, where the upper level decision has an uncertain effect on the knapsack capacity in the lower level. In the literature, bi-level problems with integer variables in the lower level are often studied in combination with a suitable solution approach. We first review some literature on bi-level (mixed-)integer linear programming with (mixed-)integer lower level: In \cite{GuFl05}, Gümü\c{s} and Floudas give a classification of mixed-integer nonlinear bi-level programming problems and consider global solution strategies for all identified classes. Dempe and Mefo Kue, Mefo Kue as well as Vicente, Savard and Judice, respectively, also give the above classification in \cite{DeMe17}, in \cite{Me17}, or in \cite{ViSaJu96}. The latter consider equivalences between different classes of discrete linear bi-level programs and particular linear multi-level problems, and analyze some properties of the discrete linear bi-level program for different discretizations of the variable set. In \cite{YuGaZeYo19}, Yue, Gao, Zeng and You propose a decomposition algorithm through column-and-constraint generation for mixed-integer linear bi-level problems based on a projection-based single-level formulation. A mixed-integer linear bi-level problem solution algorithm was introduced in \cite{MoBa90} by Moore and Bard. Caramia and Mari propose two exact algorithms for integer linear bi-level problems based on a cutting plane method and a branch-and-cut algorithm in \cite{CaMa15}. A branch-and-cut framework and an accompanying open source solver, MibS, for the pure integer bi-level linear program is provided by DeNegre in \cite{De11}. In \cite{BrHaMa13}, Brotcorne, Hanafi, and Mansi describe a solution approach to bi-level knapsack problem in terms of an integer linear bi-level problem using dynamic programming and a reformulation into a linear model with integer variables. Kara and Verter consider the problem of designing a road network for hazardous materials transportation and present a solution methodology for its underlying binary linear bi-level problem in \cite{KaVe04}. We now summarize some existing work on problems where the leader's variables are continuous and follower's are integers. This part includes bi-level stochastic problems: Bi-level problems with a continuous upper level and with a discrete or integer lower level are also considered in the aforementioned paper \cite{GuFl05} by Gümü\c{s} and Floudas. Köppe, Queyranne and Ryan show in \cite{KoQuRy10} the existence of an algorithm for mixed-integer linear bi-level problems, where the leader’s variables are continuous and the follower solves an integer linear program, and one for integer linear bi-level problems. For the same types of problems, Dempe and Mefo Kue establish algorithms in \cite{DeMe17} based on the optimal value reformulations of the bi-level problems and the use of branch-and-cut. In her dissertation \cite{Me17}, Mefo Kue analyses bi-level problems with integer lower level and bi-level problems with a semidefinite programming problem in the lower level predominantly with regard to the existence of solutions and optimality conditions. She establishes two different solution approaches for bi-level problems with continuous upper level and integer lower level as well as an algorithm for solving the discrete linear bi-level problem where both levels are binary or bounded integer based on the optimal value reformulation. Fanghänel's dissertation \cite{Fa06} deals, among other things, with (extended) solution sets, (weak) solution functions, optimality conditions, and a solution approach for weak local solutions of bi-level problems with a continuous upper level and a discrete lower level. She applies this knowledge to bi-level problems with discrete convex lower level problems, to the problem class of linear bi-level problems with a 0-1 knapsack problem at the lower level and to the assortment pricing problem. Özalt\i n, Prokopyev, and Schaefer also consider a bi-level knapsack problem in \cite{OePrSc10}, but with stochastic right-hand side in the follower constraints. For a model with continuous leader variables and binary follower variables, they give a solvability result and formulate a solvability strategy for the case of integer leader variables based on a branch-and-backtrack algorithm and a branch-and-cut algorithm. In \cite{ZhOe21}, Zhang and Özalt\i n study structural properties of a model with continuous upper level variables and integer lower level variables, where the right-hand side in the follower constraints is stochastic, too, e.g., they show that there exists a characterization of the bi-level integer problem value function only by bi-level minimal right-hand side vectors. They design a dynamic programming algorithm as well as a two-step approach to solve the value function reformulation of the original bi-level integer problem and finally apply it to a bi-level facility interdiction problem with stochastic resource constraints. Under suitable assumptions, the optimal value function of an integer linear problem is known to be lower semicontinuous in the right-hand side of its constraint system. In an optimistic bi-level framework, a linear functional is optimized over the set of minimizers of said integer problem. Unfortunately, lower semicontinuity does not carry over to the resulting upper level goal function, which complicates the analysis. Assuming that only the leader decides nonanticipatorily, the presence of stochastic right-hand side uncertainty in the lower level gives rise to a family of random variables indexed by the leaders's decision. We study these random variables based on geometric insights from the underlying parametric problem. Assessing the random upper level outcome by convex risk measueres, we establish sufficient conditions for Hölder continuity of the resulting objective function under weak assumptions. This allows us to formulate sufficient conditions for the existence of global minimizers despite the difficulties described above. Incomplete information or the need to compute efficiently can lead to optimization models that use an approximation to the true underlying distribution. This motivates the analysis of the behavior of optimal values and local optimal solution sets under perturbations of the underlying distribution. Finally, we establish a qualitative stability result for bi-level stochastic linear problems with integer variables that holds for all law-invariant convex risk measures. The remainder of the paper is organized as follows. In Section~\ref{Sec_2}, we describe a parametric bi-level integer linear problem and characterize the domains of the feasible set mapping, as well as the goal functions of follower and leader. In the next section, we extend our model by introducing stochastic uncertainty and formulate a bi-level stochastic integer linear problem. From this, we consider the feasibility set induced by implicit constraints and determine the properties of the leader's goal function under various assumptions. Properties of the final risk-neutral or risk-averse problem, including an existence result, follow in Section~\ref{Sec_4}. Section~\ref{Sec_stability} contains the analysis of the bi-level stochastic integer linear bi-level problem in terms of joint continuity of the leader variable and the probability measure. The last section of this paper presents our conclusions. \section{Structural properties of bi-level integer programs} \label{Sec_2} Consider the optimistic bi-level linear program \begin{equation}\label{BSILP} \min_x \left\{ c^\top x + \inf_y \left\{ q^\top y \; | \; y \in \Psi(Tx+z) \right\} \; | \; x \in X \right\}, \end{equation} where $X \subseteq \mathbb{R}^n$ is a nonempty polyhedron and $z \in \mathbb{R}^s$ a parameter that enters the lower level optimal solution set mapping $\Psi: \mathbb{R}^s \rightrightarrows \mathbb{Z}^m$ defined by $$ \Psi(t) := \underset{y}{\mathrm{Argmin}} \left\{ d^\top y \; | \; Wy \leq t, \, y \in Y \cap \mathbb{Z}^m \right\}. $$ Throughout the paper we assume $Y \cap \mathbb{Z}^m \neq \emptyset$. In the set $Y \subseteq \mathbb{R}^m$ we summarize all constraints of the follower that do not depend on the parameter $t \in \mathbb{R}^s$, i.e. are independent of the decision of the leader $x$ and the parameter $z$. Typically, these are sign constraints or box constraints on the follower's variables $y$. For $W = 0$ or $q = 0$ or $d = 0$ or $q = d$ problem \eqref{BSILP} collapses to different types of single-level problems. The following example shows that the reduced upper level goal function $\varphi: \mathbb{R}^s \to \overline{\mathbb{R}} := \mathbb{R} \cup \lbrace \pm \infty \rbrace$ with $\varphi(t) := \inf_y \left\{q^\top y \; | \; y \in \Psi(t) \right\}$ is neither lower nor upper semicontinuous nor convex in general: \begin{example} \label{Ex_1} Let $\varphi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be given by \begin{align*} &\varphi(t_1, t_2)\\ &= \inf_y \left\{y_1 + y_2 \; | \; (y_1, y_2) \in \underset{y'}{\mathrm{Argmin}} \left\{y_1' - y_2' \; | \; -y_1' \leq t_1, \, y_2' \leq t_2, \, y' \in \mathbb{Z}^2\right\}\right\}\\ &= \lceil -t_1\rceil + \lfloor t_2\rfloor, \end{align*} then the restriction of $\varphi$ to the linear subspace $\lbrace (-t, t) \; | \; t \in \mathbb{R} \rbrace$ is neither upper nor lower semicontinuous nor convex at any integral point, cf. Figure~\ref{Fig_varphi(t,t)}. \end{example} We shall first establish some basic properties of $\varphi$: \begin{lemma} \label{Lemma_phimeasurable2} The extended real-valued function $\varphi$ is measurable. \end{lemma} {\it Proof} In view of the representation $\varphi(t) = \inf_{y \in Y \cap \mathbb{Z}^m} q^\top y \cdot \chi_{\Psi(t)}(y)$ with $$ \chi_{\Psi(t)}(y) = \begin{cases} 1, &\text{if} \; y \in \Psi(t) \\ \infty, &\text{else} \end{cases} $$ and \cite[Th. 1.14]{Ru86}, it is sufficient to show that $\chi_{\Psi(t)}(y)$ is measurable for any fixed $y \in Y \cap \mathbb{Z}^m$. This is equivalent to measurability of \begin{align*} &\lbrace t \; | \; y \in \Psi(t) \rbrace\\ &= \lbrace t \; | \; Wy \leq t \rbrace \cap \bigcup_{y' \in Y \cap \mathbb{Z}^m} \left[ \lbrace t \; | \; d^\top y \leq d^\top y' \rbrace \; \cup \; \bigcup_{i=1, \ldots, s} \lbrace t \; | \; e_i^\top Wy' > e_i^\top t \rbrace \right], \end{align*} which follows from countability of $Y \cap \mathbb{Z}^m$.\qed Let $\Phi: \mathbb{R}^s \rightrightarrows \mathbb{R}^m$ with $\Phi(t) := \lbrace y \in Y \cap \mathbb{Z}^m \; | \; Wy \leq t \rbrace$ denote the lower level feasible set mapping and let $\psi: \mathbb{R}^s \to \overline{\mathbb{R}}$ with $\psi(t) := \inf \{d^\top y \; | \; y \in \Phi(t)\}$ be the lower level optimal value function. This function, unlike the non-integer case, cf. \cite[Th. 3.15]{De02}, is generally non-convex: \noindent \textit{Example~\ref{Ex_1} (continued) } We get $\psi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ with \begin{align*} \psi(t_1, t_2) = \inf_y \left\{y_1' - y_2' \; | \; -y_1' \leq t_1, \, y_2' \leq t_2, \, y' \in \mathbb{R}^2 \cap \mathbb{Z}^2\right\} = -\lfloor t_1\rfloor - \lfloor t_2\rfloor, \end{align*} then the restriction of $\psi$ to the linear subspace $\{(t, t) \; | \; t \in \mathbb{R}\}$ is non-convex at any integral point, cf. Figure~\ref{Fig_varphi(t,t)2}. \begin{figure} \caption{Graph of the mapping \mbox{$t \mapsto \varphi(-t, t)$} of Example~\ref{Ex_1}} \label{Fig_varphi(t,t)} \caption{Graph of the mapping \mbox{$t \mapsto \psi(t, t)$} of Example~\ref{Ex_1}~(cont.)} \label{Fig_varphi(t,t)2} \end{figure} We collect some properties of $\psi$ that can be formulated based on \cite[Prop. 1]{ZhOe21} and \cite[Th. 4.5.2.]{5M} including their proofs: \begin{proposition} Let $Y = \mathbb{R}^m$. \begin{enumerate} \item The function $\psi$ is nonincreasing on $\mathbb{R}^s$. \item The function $\psi$ is subadditive on $\mathrm{dom} \; \Phi$. \item If the matrix $W$ has only rational elements, then the function $\psi$ is lower semicontinuous on $\mathrm{dom} \; \Phi$. \end{enumerate} \end{proposition} Similar results are found in \cite[Prop. 3.1] {DeMe17} and \cite[Prop. 5.2.3.]{Me17} followed by the proofs. Due to \cite[Rem. 1]{ZhOe21}, the following example shows that the reduced upper level goal function fails to be monotone and subadditive in general: \begin{example} \label{Ex_subadd} Let $\varphi(t_1, t_2) = \inf_y \left\{-3y_1 - 2y_2 - y_3 \; | \; y \in \Psi(t_1, t_2)\right\}$ with \begin{align*} \left.\begin{aligned} \Psi(t_1, t_2) = \underset{y}{\mathrm{Argmin}} \big\{-y_1 - 2y_2 - 2y_3 \; | \; &y_1 + 2y_2 + 2y_3 \leq t_1,\\ &2y_1 + 2y_2 + y_3 \leq t_2, \, y \in \mathbb{N}_0^3 \end{aligned}\right\}. \end{align*} Then $\Phi(1, 1) = \left\{(0, 0, 0)^\top\right\}$, $\Phi(1, 2) = \Phi(1, 1) \cup \left\{(1, 0, 0)^\top\right\}$, $\Phi(2, 2) = \Phi(1, 2) \cup \left\{(0, 1, 0)^\top, (0, 0, 1)^\top\right\} = \Phi(2, 3)$, \noindent cf. Figure~\ref{Fig_subbadd}, and we have $\varphi(1, 2) = -3 < -2 = \varphi(2, 2)$ as well as $\varphi(1, 1) + \varphi(1, 2) = 0 - 3 = -3 < -2 = \varphi(2, 3)$. \begin{figure} \caption{Graph of the feasible points with a) $t = (1, 1)^\top$ and b) $t = (2, 3)^\top$ of Example~\ref{Ex_subadd}} \label{Fig_subbadd} \end{figure} \end{example} Based on \cite[Lemma 2]{ZhOe21} including the proof, here is a property of the function $\varphi$ that we will need later: \begin{proposition} \label{Prop_phi} It is $\varphi(t') \geq \varphi(t)$ for $t', t \in \mathrm{dom}\; \Phi$ such that $t' \leq t$ and $\psi(t') = \psi(t)$. \end{proposition} We shall work with the following assumption: \begin{enumerate}[label=$\mathrm{(A\arabic*)}$,leftmargin=1cm] \item The set $Y \cap \mathbb{Z}^m$ is finite. \label{A1} \end{enumerate} \begin{lemma} \label{LemmaDomPhi} We have $\mathrm{dom} \; \Phi = \bigcup_{y \in Y \cap \mathbb{Z}^m} \lbrace Wy \rbrace \oplus \mathbb{R}^s_{\geq 0}$, where $\oplus$ denotes the Minkowski sum, i.e. the domain of $\Phi$ admits a representation as a countable union of translated copies of the nonnegative orthant. In particular, $\mathrm{dom} \; \Phi$ is polyhedral, i.e. a finite union of polyhedra, cf. \cite[Def. 3.1]{De02}, whenever \ref{A1} holds true. \end{lemma} {\it Proof} This is an immediate consequence of the representation \begin{align*} \mathrm{dom} \; \Phi = \bigcup_{y \in Y \cap \mathbb{Z}^m} \lbrace t \; | \; Wy \leq t \rbrace = \bigcup_{y \in Y \cap \mathbb{Z}^m} \lbrace Wy \rbrace \oplus \mathbb{R}^s_{\geq 0}.\tag*{$\qed$} \end{align*} If $Y$ is a polyhedron and all integrality constraints are deleted from the follower's problem, it can be shown that the domain of $\varphi$ is empty or coincides with the set of parameters for which the lower level problem is feasible, i.e. $\mathrm{dom} \; \varphi = \mathrm{dom} \; \Phi$ (cf. \cite[Lemma 2.1]{BuClDe20}). The following example illustrates that this does not hold in the presence of integer variables: \begin{example} \label{ExDomPhi} Consider the case where $Y := \lbrace (0,0)^\top \rbrace \cup \lbrace (1, k)^\top \; | \; k \in \mathbb{R} \rbrace$, \noindent $m = 2$, $s = 1$, $W = [1\ 0]$ and $d = q = (0, 1)^\top$. It is easy to check that \mbox{$\mathrm{dom} \; \varphi = [0,1)$}, even though the lower level problem is feasible for any nonnegative right-hand side, cf. Figure~\ref{Fig_(1,k)}. \end{example} Example~\ref{ExDomPhi} also shows that the domain of $\varphi$ is not closed in general. \begin{example} \label{Ex_Sqrt2} Let $Y = \left\{y \in \mathbb{R}^3 \; | \; -\sqrt{2}y_1 + y_2 \leq 0, \, y_1 \geq 1, \, y_2 \geq 0, \, y_3 = 0\right\} \cup \left\{(0, 0, 1)^\top\right\}$, $W = [0\ 0\ 1]$, and $d = (\sqrt{2}, -1, -1)^\top$, then \[ \Psi(t) = \begin{cases} \emptyset, & \text{if}\ t < 1\\ \{(0, 0, 1)^\top \}, & \text{if}\ t \geq 1 \end{cases} \] and the lower level infimal value $0$ is not attained for any $t \in [0, 1)$. In addition, we have \[ \Phi(t) = \begin{cases} \emptyset, & \text{if}\ t < 0\\ \left(Y \cap \mathbb{Z}^3\right) \setminus \{(0, 0, 1)^\top\}, & \text{if}\ t \in [0, 1)\\ \{(0, 0, 1)^\top \}, & \text{if}\ t \geq 1, \end{cases} \] cf. Figure~\ref{Fig_sqrt2}, such that $\mathrm{dom} \; \Phi = [0, \infty) \supseteq [1, \infty) = \mathrm{dom} \, \Psi$. \begin{figure} \caption{Graph of $\Phi(t)$ for \mbox{$t \in [0, 1)$} with \mbox{$y_3 = 0$} of Example~\ref{Ex_Sqrt2}} \label{Fig_(1,k)} \label{Fig_sqrt2} \end{figure} If we instead set $d = (0, 0, 0)^\top$ and $q = (\sqrt{2}, -1, -1)^\top$, it is easy to see that the infimal value of the optimization problem defining $\varphi(t)$ is finite but not attained for any $t \in [0,1)$. Here we have $\mathrm{dom} \; \Phi = [0, \infty) = \mathrm{dom} \, \Psi$, but $\mathrm{dom} \; \psi = [0, \infty) \supseteq [1, \infty) = \mathrm{dom} \, \varphi$. \end{example} In view of Lemma~\ref{LemmaDomPhi} it is desirable to identify situations in which the domains of $\varphi$ and $\Phi$ coincide: \begin{lemma} \label{LemmaA1DomainsPolyhedral} Assume \ref{A1}, then the domains of $\varphi, \psi, \Psi$ and $\Phi$ coincide and are polyhedral. \end{lemma} {\it Proof} The lower level problem as well as the optimization problem defining $\varphi$ have at most $|Y \cap \mathbb{Z}^m| < \infty$ feasible points, which implies the problems are solvable whenever they are feasible. Hence \mbox{$\mathrm{dom} \; \varphi = \mathrm{dom} \; \Psi = \mathrm{dom} \; \psi = \mathrm{dom} \; \Phi$} and the statement follows directly from Lemma~\ref{LemmaDomPhi}.\qed For the subsequent analysis, let us assume that $Y \cap \mathbb{Z}^m$ is nonempty and \ref{A1} holds, i.e. $Y \cap \mathbb{Z}^m = \left\{\bar{y}^1,\ldots,\bar{y}^N\right\}$ for some $N \in \mathbb{N}$. The following example illustrates that the domain of $\varphi$ may be nonconvex if $s \geq 2$. \begin{example} \label{Ex_2} Figure~\ref{Fig_Y} depicts a bounded set containing $N = 6$ integral points. \begin{enumerate}[leftmargin=2em, label={\alph*)}] \item For $W = [1\ 1]$, we have $s = 1$ and $\mathrm{dom}\;\varphi$ is shown in Figure~\ref{Fig_domphi_a}. \begin{figure} \caption{The set of integral points in a bounded subset $Y$ of $\mathbb{R}^2$ in Example~\ref{Ex_2}} \label{Fig_Y} \caption{The domain of $\varphi$ in Example~\ref{Ex_2}~a)} \label{Fig_domphi_a} \end{figure} \item As in Example~\ref{Ex_1}, we use $W = \bigl[\begin{smallmatrix} -1 & 0\\ 0 & 1 \end{smallmatrix}\bigr]$ and have $s = 2$. Then $\mathrm{dom}\;\varphi$ is shown in Figure~\ref{Fig_domphi_b}. \begin{figure} \caption{Graph of a) the feasible points for $t = (-0.5, 1.5)$ and $t = (-3.5, 2.5)$, and b) of $\mathrm{dom}\, \varphi$ \footnotesize{(boundaries of the sets are plotted side by side or better visibility)} of Example~\ref{Ex_2}~b)} \label{Fig_domphi_b} \end{figure} \end{enumerate} \end{example} As preliminary work for the analysis in Section~\ref{Sec_Model}, let us enumerate the nonempty subsets of $Y \cap \mathbb{Z}^m$: $\{A \subseteq Y \cap \mathbb{Z}^m \; | \; A \neq \emptyset \} = \left\{Y^1, \ldots, Y^M\right\}$ with $M = 2^N-1$, which allows us to define \allowdisplaybreaks \begin{align*} V^k := &\left\{t \; | \; \left\{y \in Y \cap \mathbb{Z}^m \; | \; Wy \leq t\right\} = Y^k\right\} \\ = &\bigcap_{y \in Y^k} \{t \; | \; Wy \leq t\} \cap \bigcap_{\hat{y} \in \left(Y \cap \mathbb{Z}^m\right) \setminus Y^k} \bigcup_{j = 1}^s \left\{t \; | \; e_j^\top W\hat{y} > t_j\right\} \end{align*} for any $k \in \lbrace 1, \ldots, M \rbrace$. The first intersection above guarantees that all points from $Y^k$ are feasible, while the second one rules out all other elements in $Y \cap \mathbb{Z}^m$. As illustrated in the continuation of Example~\ref{Ex_2} below, the sets $V^k$ form a partion of the domain of $\varphi$. This is indeed always true: \begin{lemma} \label{Lemma_partition} Assume \ref{A1}, then the family $\mathcal{V} := \{V^k \; | \; V^k \neq \emptyset, \, k = 1, \ldots, M\}$ forms a partition of $\mathrm{dom}\; \varphi$. \end{lemma} {\it Proof} Lemma~\ref{LemmaA1DomainsPolyhedral} yields $\mathrm{dom} \; \varphi = \lbrace t \; | \; \exists k \in \lbrace 1, \dots, M \rbrace \; : \; t \in V^k \rbrace$ and the statement follows directly from the fact that the sets in $\mathcal{V}$ are the equivalence classes w.r.t. the equivalence relation where $t \sim t'$ iff \begin{align*} \lbrace y \in Y \cap \mathbb{Z}^m \; | \; Wy \leq t \rbrace = \lbrace y \in Y \cap \mathbb{Z}^m \; | \; Wy \leq t' \rbrace.\tag*{$\qed$} \end{align*} By definition, the family $\mathcal{V}$ may have up to $2^{|Y \cap \mathbb{Z}^m|}$ elements. However, the following observation shows that the cardinality of $\mathcal{V}$ is usually much smaller: \begin{corollary} \label{Cor_emptyset} Assume \ref{A1} and let $k = 1, \ldots, M$ as well as $i, \iota \in \{1, \ldots, N\}$ be such that $\bar{y}^i \notin Y^k$, $\bar{y}^\iota \in Y^k$ and $W\bar{y}^i \leq W\bar{y}^\iota$, then $V^k = \emptyset$. \end{corollary} {\it Proof} Suppose that there is some $t \in V^k$, then $W\bar{y}^i \leq W\bar{y}^\iota \leq t$ implies $\bar{y}^i \notin Y^k$, which contradicts the assumptions. \qed \noindent\textit{Example~\ref{Ex_2} (continued) } For $A = Y^k \subseteq Y \cap \mathbb{Z}^2$ with $A \neq \emptyset$, we get the sets $V^k$ as its graphs are shown in a) Figure~\ref{Fig_Yk_a} and b) Figure~\ref{Fig_Yk_b}. \begin{figure} \caption{Graph of the sets $V^k$ of Example~\ref{Ex_2}~a)} \label{Fig_Yk_a} \end{figure} \begin{figure} \caption{Graph of the sets $V^k$ of Example~\ref{Ex_2}~b)} \label{Fig_Yk_b} \end{figure} To conclude this chapter, let us note that based on the sets $V^k$, the functions $\psi$ and $\varphi$ are piecewise constant: \begin{lemma} \label{Lemma_piecewiseconstant} Assume \ref{A1}, then the optimal value functions $\psi$ and $\varphi$ are piecewise constant on $\mathrm{dom}\; \Phi$. \end{lemma} {\it Proof} For all $k = 1, \ldots, M$, we define \mbox{$\kappa_\psi^k := \min_y \left\{d^\top y' \; | \; y' \in Y^k\right\} \in \mathbb{R}$} as well as $\kappa_\varphi^k := \min_y \left\{q^\top y \; | \; y \in \mathrm{Argmin}_{y'} \left\{d^\top y' \; | \; y' \in Y^k\right\}\right\} \in \mathbb{R}$. By Lemma~\ref{LemmaA1DomainsPolyhedral} and Lemma~\ref{Lemma_partition}, we have \[ \psi(t) = \begin{cases} \sum_{k = 1}^M \kappa_\psi^k 1\!\!1_{V^k}(t), &\text{if}\ t \in \mathrm{dom} \; \psi\\ \infty, &\text{else}, \end{cases} \] as well as \begin{equation} \label{sumphi} \varphi(t) = \begin{cases} \sum_{k = 1}^M \kappa_\varphi^k 1\!\!1_{V^k}(t), &\text{if}\ t \in \mathrm{dom} \; \varphi\\ \infty, &\text{else}, \end{cases} \end{equation} where $$ 1\!\!1_{V^k}(t) := \begin{cases} 1, &\text{if}\ t \in V^k \\ 0, &\text{else} \end{cases} $$ denotes the indicator function of the subset $V^k$. \qed \begin{remark} Fix $k$ with $V^k \neq \emptyset$. It is $\psi(t') = \psi(t) = \kappa_\psi^k$ and $\varphi(t') = \varphi(t) = \kappa_\varphi^k$ for all $t, t' \in V^k$. Due to Proposition~\ref{Prop_phi}, we have $\varphi(t') = \kappa_\varphi^{k'} \geq \kappa_\varphi^k = \varphi(t)$ for all $t' \in V^{k'}$ and all $t \in V^k$ with $t' \leq t$ and $\psi(t') = \kappa_\psi^{k'} = \kappa_\psi^k = \psi(t)$. \end{remark} Alternatively, in the proofs of \cite[Prop. 3.1 3.]{DeMe17} and \cite[Prop. 5.2.3. 3.]{Me17}, the lemma above is proved using the so-called \textit{regions of stability of} $y \in Y \cap \mathbb{Z}^m$: $R(y) := \left\{t \in \mathrm{dom}\; \varphi \; | \; y \in \Psi(t)\right\}$. \noindent The following example shows that the sets $R(y)$ with $y \in Y \cap \mathbb{Z}^m$ are not identical to the sets $V^k$ with $k = 1, \ldots, M$: \noindent\textit{Example~\ref{Ex_2}~(continued) } Let $d = (1, -1)^\top$. The graph of the sets $R(y)$ are shown in a) Figure~\ref{Fig_Ry_a} and b) Figure~\ref{Fig_Ry_b}. \begin{figure} \caption{Graph of the sets $R(y)$ of Example~\ref{Ex_2}~a)} \label{Fig_Ry_a} \end{figure} \begin{figure} \caption{Graph of a) the minimizers for $t = (-0.5, 1.5)$ and $t = (-3.5, 2.5)$, and b) of the sets $R(y)$ of Example~\ref{Ex_2}~b)} \label{Fig_Ry_b} \end{figure} Now we consider the leader's objective value function $f: \mathbb{R}^n \times \mathbb{R}^s \to \overline{\mathbb{R}}$ based on the notation in \eqref{BSILP}: \[ f(x, z) := c^\top x + \inf_y \left\{q^\top y \; | \; y \in \Psi(Tx + z)\right\} = c^\top x + \varphi(Tx + z) \] and continue with a property we need for the proof of Theorem~\ref{Th_stability}: \begin{corollary}\label{Cor_bounded} Assume \ref{A1}, then the functions $\psi$ and $\varphi$ are bounded, and there exist $L_f, C_f > 0$ such that $|f(x, z)| \leq L_f \|x\| + C_f$ for all $Tx + z \in \mathrm{dom}\; \Phi$. \end{corollary} {\it Proof} Based on the considerations in the proof of Lemma~\ref{Lemma_piecewiseconstant} we get \noindent $u_\psi := \max_{k = 1, \ldots, M} \kappa^k_\psi \geq \psi(t) \geq \min_{k = 1, \ldots, M} \kappa^k_\psi =: l_\psi$ for all $t \in \mathrm{dom}\; \psi$ as well as $u_\varphi := \max_{k = 1, \ldots, M} \kappa^k_\varphi \geq \varphi(t) \geq \min_{k = 1, \ldots, M} \kappa^k_\varphi =: l_\varphi$ for all $t \in \mathrm{dom}\; \varphi$. Thus, $|\psi(t)| \leq \max \left\{|u_\psi|, |l_\psi|\right\}$ and $|\varphi(t)| \leq \max \left\{|u_\varphi|, |l_\varphi|\right\}$ for all $t \in \mathrm{dom}\; \Phi$ and $\left|f(x, z)\right| \leq \left|c^\top x\right| + \left|\varphi(Tx + z)\right| \leq L_f \|x\| + C_f$ for all $Tx + z \in \mathrm{dom}\; \Phi$ with $L_f := \|c\|, \, C_f := \max \left\{|u_\varphi|, |l_\varphi|\right\} \geq 0$.\qed \section{Two-stage setting} \label{Sec_Model} Let us now consider a setting in which the parameter $z = Z(\omega)$ in \eqref{BSILP} is the realization of some random vector $Z: \Omega \to \mathbb{R}^s$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, that only the follower can observe. Imposing the information constraint that the leader has to decide in a here-and-now fashion, this gives rise to the bi-level stochastic integer linear program \begin{equation}\label{BSILP2} \min_x \left\{ c^\top x + \inf_y \left\{ q^\top y \; | \; y \in \Psi(Tx + Z(\omega)) \right\} \; | \; x \in X \right\}. \end{equation} By definition, the upper level outcome equals $\infty$ whenever the lower level problem has no optimal solution. Deciding nonanticipatorily, the leader thus has to choose $x$ such that $Tx + Z(\omega) \in \mathrm{dom} \; \Psi$ holds for any realization of the randomness. To form a mathematically sound model, we will confine the analysis to situations in which the upper level objective function value is finite as well and thus restrict the leader's choices to the so called induced feasible set given by $F_Z := \{x \; | \; Tx + z \in \mathrm{dom}\;\varphi \ \forall z \in \mathrm{supp} \; \mu_Z\}$ where $\mu_Z := \mathbb{P} \circ Z^{-1} \in \mathcal{P}(\mathbb{R}^s)$ is the Borel probability measure induced by mapping~$Z$. \begin{example} In the setting of Example~\ref{ExDomPhi}, set $n = 1$, $T = 1$ and let $\mu_Z$ be given by the Dirac measure at $0$, i.e. $Z(\omega) = 0$ with probability $1$, then $F_Z = \mathrm{dom} \; \varphi = [0,1)$. Hence, $F_Z$ is not closed in general. \end{example} \begin{theorem} \label{Th_F_Z} Assume \ref{A1} and let $\mathrm{supp} \; \mu_Z$ be polyhedral, then $F_Z$ is polyhedral. \end{theorem} {\it Proof} If the support of $\mu_Z$ is given by the union of polyhedra $Q_1, \ldots, Q_k$, we have $F_Z = \bigcap_{i = 1}^k \lbrace x \; | \; Tx + z \in \mathrm{dom}\,\varphi \ \forall z \in Q_i \rbrace$. Since any finite intersection of polyhedral sets is a polyhedral set (cf. last part of this proof), we may restrict the analysis to the case $k = 1$. Let $\mathrm{supp}\; \mu_Z$ be a nonempty polyhedron $\{z \; | \; Cz \leq c\} =: Q$ and let the domain of function $\varphi$ be a finite union of polyhedra $\mathrm{dom} \; \varphi = \bigcup_{i = 1}^L P_i$ with $P_i := \{t \; | \; A_i t \leq b_i\}$ for matrices $A_i \in \mathbb{R}^{m_i \times s}$ and vectors $b_i \in \mathbb{R}^{m_i}$. For any $x \in F_Z$, we have ($e_{li}(z) := e^\top_l A_i z - e^\top_l b_i + e^\top_l A_i Tx$) \allowdisplaybreaks \begin{align} & \ \forall\ z \in Q\quad \exists\ i \in \{1, \ldots, L\} \; : \; A_i(Tx + z) \leq b_i \\ & \Leftrightarrow \forall\ z \in Q\quad \exists\ i \in \{1, \ldots, L\} \quad \forall\ l \in \{1, \ldots, m_i\} \; : \; e^\top_l A_i z \leq e^\top_l b_i - e^\top_l A_i Tx\nonumber\\ & \Leftrightarrow \sup_{z \in Q} \min_{i \in \{1, \ldots, L\}} \max_{l \in \{1, \ldots, m_i\}} e_{li}(z) \leq 0 \nonumber\\ & \Leftrightarrow \sup_{z \in Q} \min_\lambda \left\{\sum\nolimits_{i = 1}^L \lambda_i \cdot \max_{l \in \{1, \ldots, m_i\}} e_{li}(z) \; | \; \lambda \in \{0, 1\}^L, \, \sum\nolimits_{i = 1}^L \lambda_i = 1\right\} \leq 0. \nonumber \end{align} As the feasible set of the inner minimization problem above is finite, its LP relaxation is solvable and has the same optimal value. Hence, \allowdisplaybreaks \begin{align*} & \Leftrightarrow\ \sup_{z \in Q} \min_\lambda \left\{\sum\nolimits_{i = 1}^L \lambda_i \cdot \max_{l \in \{1, \ldots, m_i\}} e_{li}(z) \; \Big| \; \lambda \in [0,1]^L, \, \sum\nolimits_{i = 1}^L \lambda_i = 1\right\} \leq 0\\ & \left.\begin{aligned} \Leftrightarrow\ \sup_{z \in Q} \max_{u, w} \bigg\{1\!\!1^\top_L u + w \; \Big| \; & u_i + w \leq \max_{l \in \{1, \ldots, m_i\}} e_{li}(z)\ \forall i \in \lbrace 1, \ldots, L\},\\ & u \leq 0 \end{aligned}\right\} \leq 0\\ & \left.\begin{aligned} \Leftrightarrow\ \max_{l_1, \ldots, l_L \in \{1, \ldots, s\}} \max_{z, u, w} \Big\{1\!\!1^\top_L u + w \; \big| \; & e^\top_{l_i} u + w \leq e_{l_ii}(z)\ \forall \; i \in \{1, \ldots, L \},\\ & u \leq 0, \, Cz \leq c \end{aligned}\right\} \leq 0 \end{align*} Since setting $u = 0$, $z = z_0$ for some fixed $z_0 \in \mathrm{supp}\; \mu_Z$, and $$ w = \min \left\{1, \min_{i \in \{1, \ldots, L\}} e_{l_ii}(z_0)\right\} $$ yields a feasible point for the modified problem, we may add the restriction $1\!\!1^\top_L u + w \leq 1$ to ensure that the inner maximization problem has an optimal solution. By linear programming duality, we may continue the previous equivalences: \allowdisplaybreaks \begin{align*} & \left.\begin{aligned} \Leftrightarrow\ \max_{l_1, \ldots, l_L \in \{1, \ldots, s\}} \max_{z, u, w} \big\{ & 1\!\!1^\top_L u + w \; \big|\\ & 1\!\!1^\top_L u + w \leq 1, \, u \leq 0, \, Cz \leq c,\\ & e^\top_{l_i} u + w \leq e_{li}(z) \ \forall \; i \in \{1, \ldots, L \} \end{aligned}\right\} \leq 0\\ & \left.\begin{aligned} \Leftrightarrow\ \max_{l_1, \ldots, l_L \in \{1, \ldots, s\}} \min_{\alpha, \beta, \gamma} \bigg\{ & \sum\nolimits_{i = 1}^L \alpha_i \left(-e^\top_{l_i} b_i + e^\top_{l_i} A_i Tx\right) + \beta^\top c + \gamma \; \big|\\ & 1\!\!1_L \gamma + \sum\nolimits_{i = 1}^L e_{l_i} \alpha_i \leq 1\!\!1_L, \, \sum\nolimits_{i = 1}^L \alpha_i + \gamma = 1,\\ & \sum\nolimits_{i = 1}^L (-A_i^\top e_{l_i})\alpha_i + C^\top \beta = 0, \, \alpha, \beta, \gamma \geq 0 \end{aligned}\right\} \leq 0. \end{align*} Denoting the finite nonempty set of vertices of the feasible set of the inner minimization problem above by $\mathcal{V}(l_1, \ldots, l_L)$ and the $i$-th biggest element in $\{1, \ldots, s\}^L$ with respect to the lexicographical order by $\mathbb{L}_i$, we may reformulate the previous condition as \allowdisplaybreaks \begin{align*} & \ x \in \bigcap_{\substack{l_1, \ldots, l_L\\\in \{1, \dots, s\}}} \bigcup_{\substack{(\alpha, \beta, \gamma)\\\in \mathcal{V}(l_1, \ldots, l_L)}} \underbrace{\left\{x' \; \Big| \; \sum\nolimits_{i = 1}^L \alpha_i \left(-e^\top_{l_i} b_i + e^\top_{l_i} A_i Tx'\right) + \beta^\top c + \gamma \leq 0\right\}}_{=: P(l_1, \ldots, l_L, \alpha, \beta, \gamma)}\\ & \Leftrightarrow x \in \bigcup_{(\rho_1, \ldots, \rho_{s^L}) \in \prod_{i = 1}^{s^L} \mathcal{V}(\mathbb{L}_i)} \bigcap_{j = 1}^{s^L} P(\mathbb{L}_j, \rho_j), \end{align*} which yields the desired representation. \qed \begin{lemma} \label{Lemma_welldefined} Assume \ref{A1}, then the function $\mathbb{F}: F_Z \to \mathcal{L}^\infty(\Omega, \mathcal{F}, \mathbb{P})$ with $\mathbb{F}(x)(\cdot) := \varphi(Tx + Z(\cdot))$ is well-defined. \end{lemma} {\it Proof} As $\varphi$ is measurable (cf. Lemma~\ref{Lemma_phimeasurable2}), we conclude that $\mathbb{F}(x)(\cdot)$ is measurable, i.e. $\mathbb{F} \in \mathcal{L}^0(\Omega, \mathcal{F}, \mathbb{P})$. Moreover, \ref{A1} implies $$ \|\mathbb{F}(x) \|_{\mathcal{L}^\infty(\Omega, \mathcal{F}, \mathbb{P})} \leq \max_y \lbrace |q^\top y| \; | \; y \in Y \cap \mathbb{Z}^m\rbrace < +\infty $$ and thus $\mathbb{F} \in \mathcal{L}^\infty(\Omega, \mathcal{F}, \mathbb{P})$. \qed \begin{theorem} \label{Th_FContinuous2} Assume \ref{A1}, $\mu_Z\left[D_\varphi(x)\right] = 0$, where $D_\varphi: F_Z \rightrightarrows \mathbb{R}^s$ with $D_\varphi(x) := \left\{z \; | \; \varphi\ \text{is discontinuous at}\ (Tx + z)\right\}$, then the mapping $\mathbb{F}$ is continuous at $x \in F_Z$ with respect to any $\mathcal{L}^p$-norm with $p \in [1, \infty)$. \end{theorem} {\it Proof} Let $x \in F_Z$ and $\{x_l\}_{l \in \mathbb{N}} \subseteq F_Z$ any sequence with $x_l \to x$ for $l \to \infty$. With the reformulation of $\varphi$ at \eqref{sumphi} and \begin{align*} |\varphi(Tx_l + z)| = \left|\sum\nolimits_{k = 1}^M \kappa_\varphi^k 1\!\!1_{V^k}(Tx_l + z) \right| \leq \underbrace{\max_{k = 1, \ldots, M} |\kappa_\varphi^k|}_{=: \bar{\kappa}} < \infty \end{align*} $\mu_Z$-almost everywhere for all $l \in \mathbb{N}$ we have a measurable majorant $\bar{\kappa}$. Based on $\mu_Z\left[D_\varphi(x)\right] = 0$, we have $\lim_{l \to \infty} \varphi(Tx_l + z) = \varphi(Tx + z)$ $\mu_Z\text{-almost everywhere}$ for the functions $\mathbb{F}(x_l) \in \mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})$ with $l \in \mathbb{N}$. We obtain \mbox{$\mathbb{F}(x) \in \mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})$} as well as \begin{align*} & \lim_{l \to \infty} \left\|\mathbb{F}(x_l) - \mathbb{F}(x)\right\|_{\mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})}\\ & = \lim_{l \to \infty} \left(\int_{\mathbb{R}^s} \left|\varphi(Tx_l + z) - \varphi(Tx + z)\right|^p ~\mu_Z(\mathrm{d}z)\right)^{1/p} = 0 \end{align*} using a majorized convergence theorem for $\mathcal{L}^p$-functions, cf. \cite[\S 12 Th. 4]{Fo12}. \qed A characterisation of the sets $D_\varphi$ is given in the proof of the next corollary. \begin{corollary} \label{Cor_FContinuous} Assume \ref{A1} and that the Borel measure $\mu_Z$ is absolutely continuous with respect to the Lebesgue measure, then the mapping $\mathbb{F}$ is continuous with respect to any $\mathcal{L}^p$-norm with $p \in [1, \infty)$. \end{corollary} {\it Proof} By \cite[Prop. 1.1 (i)]{GoJoRo03}, we have \[ \bar{V}^k := \mathrm{cl} \; V^k = \bigcap_{y \in Y^k} \{t \; | \; Wy \leq t\} \cap \bigcap_{\hat{y} \in \left(Y \cap \mathbb{Z}^m\right) \setminus Y^k} \bigcup_{j = 1}^s \left\{t \; | \; e_j^\top W\hat{y} \geq t_j\right\} \] and thus $\bar{V}^k \subseteq V^k \cup \bigcup_{\hat{y} \in \left(Y \cap \mathbb{Z}^m\right) \setminus Y^k} \bigcup_{j = 1}^s \left\{t \; | \; e_j^\top W\hat{y} = t_j\right\}$. In particular, the set \mbox{$\bar{V}^k \setminus V^k$} is contained in a finite union of hyperplanes and therefore a Lebesgue null set. We also obtain the still missing characterization of the set of discontinuity points from Theorem~\ref{Th_FContinuous2}: \begin{equation} \left(\bar{V}^k \setminus V^k\right) \oplus (-Tx) \supseteq D_\varphi(x) \quad \text{for all}\ x \in F_Z.\label{nullset} \end{equation} Based on that, the image of the setvalued mapping $D_\varphi$ is a Lebesgue null set. Due to the absolute continuity of $\mu_Z$ with respect to the Lebesgue measure, we get continuity of the function $\mathbb{F}$ at all $x \in F_Z$ with respect to any $\mathcal{L}^p$-norm with $p \in [1, \infty)$ by the help of Theorem~\ref{Th_FContinuous2}. \qed The following example shows that the assumptions of Corollary~\ref{Cor_FContinuous} are too weak to guarantee the local Lipschitz continuity of $\mathbb{F}$, even if the support of $\mu_Z$ is compact: \begin{example} \label{ExLocLipschitz} Consider the case where $n = m = s = 1$, $W = T = q = 1$, $d = -1$, and $Y = [0,1]$, cf. Figure~\ref{Fig_LocLipschitz}, then \begin{figure} \caption{Graph of the feasible points and $t \in \mathrm{dom}\; \varphi$ with a) $t \in (0, 1)$ and b) $t > 1$ of Example~\ref{ExLocLipschitz}} \label{Fig_LocLipschitz} \end{figure} $$ \varphi(t) = \begin{cases} \infty, &\text{if} \; t < 0 \\ 0, &\text{if} \; t \in [0,1) \\ 1, &\text{if} \; t \geq 1. \end{cases} $$ Moreover, let $\mu_Z \in \mathcal{P}(\mathbb{R})$ be the Borel probability measure with Lebesgue density $$ \delta_Z(z) := \begin{cases} \frac{1}{2\sqrt{z}}, &\text{if} \; z \in (0,1] \\ 0, &\text{else}, \end{cases} $$ then $$ \|\mathbb{F}(x)\|_{\mathcal{L}^1(\Omega, \mathcal{F}, \mathbb{P})} = \int\limits_{0}^1 \frac{1}{2\sqrt{z}} \varphi(x+z)~\mathrm{d}z = \int\limits_{1-x}^1 \frac{1}{2\sqrt{z}}~\mathrm{d}z = 1 - \sqrt{1-x} $$ holds for any $x \in [0,1]$. Hence, $\mathbb{F}$ is not locally Lipschitz continuous at $1$ w.r.t. any $\mathcal{L}^p$-norm with $p \in [1, \infty]$. \end{example} Furthermore, Corollary~\ref{Cor_FContinuous} cannot be extended to $p = \infty$: \begin{example} Consider the setting of Example~\ref{ExLocLipschitz}, but let $Z$ be uniformly distributed on $[0,1]$, then $$ \mathbb{F}(x)(\omega) := \begin{cases} 0, &\text{if} \; x + Z(\omega) < 1 \\ 1, &\text{else} \end{cases} $$ holds for any $x \in F_Z = [0,\infty)$. Thus, $\mathbb{F}$ is not continuous at $0$ w.r.t. the $\mathcal{L}^\infty$-norm, as it holds \begin{align*} &\left\|\mathbb{F}\left(\tfrac{1}{j}\right) - \mathbb{F}(0)\right\|_{\mathcal{L}^\infty(\Omega, \mathcal{F}, \mathbb{P})} = \underset{\omega \in \Omega}{\mathrm{ess} \sup} \left|\mathbb{F}\left(\tfrac{1}{j}\right)(\omega) - \mathbb{F}(0)(\omega)\right|\\ &= \underset{\omega \in \Omega}{\mathrm{ess} \sup} \left|\varphi\left(\tfrac{1}{j} + Z(\omega)\right) - \varphi(Z(\omega))\right| = 1 \quad\text{for any}\ j \in \mathbb{N}. \end{align*} \end{example} The subsequent analysis relies on the following additional assumption: \begin{enumerate}[label=$\mathrm{(A\arabic*)}$,leftmargin=1cm,start=2] \item $\mu_Z$ has a uniformly bounded Lebesgue density.\label{A2} \end{enumerate} \begin{theorem} \label{ThFLipschitz} Assume \ref{A1}, \ref{A2} as well as $p \in [1, \infty)$ and let the support of $\mu_Z$ be compact, then the mapping $\mathbb{F}$ is Hölder continuous with exponent $\frac{1}{p}$ with respect to the $\mathcal{L}^p$-norm. \end{theorem} {\it Proof} Based on the notation in the proof of Corollary~\ref{Cor_FContinuous}, we introduce the set-valued mapping $\Delta^{k,j}: F_Z \times F_Z \rightrightarrows \mathbb{R}^s$ defined by $\Delta^{k,j}(x, x') := \big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx')\big)$, \noindent and fix some $k, j \in \{1, \ldots, M\}$ and some $x \in F_Z$. As $\bar{V}^k$ and $\bar{V}^j$ are closed, $\Delta^{k,j}(x, \cdot)$ is outer semicontinuous (cf. \cite[Lemma A.3.]{BuClDe20}), i.e., $\lim \sup_{x' \to x_0} \Delta^{k, j}(x, x') \subseteq \Delta^{k, j}(x, x_0)$ for all $x_0 \in F_Z$. For $j \neq j'$ the sets $V^j$ and $V^{j'}$ are disjoint, cf. Lemma~\ref{Lemma_partition}, and the set $\big(\bar{V}^j \oplus (- Tx')\big) \cap \big(\bar{V}^{j'} \oplus (- Tx')\big)$ is a null set. Due to Lemma~\ref{LemmaA1DomainsPolyhedral} and Lemma~\ref{Lemma_partition}, $\bigcup_{j = 1}^M \bar{V}^j = \bigcup_{j = 1}^M V^k = \mathrm{dom}\, \varphi$ and it follows $\sum_{j = 1}^M \mu_Z\left[\Delta^{k,j}(x, x')\right] = \mu_Z\left[\bar{V}^k \oplus (- Tx)\right]$. \noindent Invoking \cite[Lemma 3.4]{Cl21}, we obtain continuity of a mapping \mbox{$M^{k,j}: F_Z^2 \to [0, 1]$} on $\{x\} \times F_Z$ defined by \mbox{$M^{k,j}(x, x') := \mu_Z\left[\Delta^{k,j}(x, x')\right]$} for any $j \in \{1, \ldots, M\}$. Fix an arbitrary point $x \in F_Z$ and consider any $x' \in F_Z$. We can write \allowdisplaybreaks \begin{align} & \|\mathbb{F}(x) - \mathbb{F}(x')\|^p_{\mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})}\\ & = \int_{\mathbb{R}^s} \left|\sum\nolimits_{k = 1}^M \kappa_\varphi^k 1\!\!1_{V^k}(Tx + z) - \sum\nolimits_{j = 1}^M \kappa_\varphi^j 1\!\!1_{V^j}(Tx' + z)\right|^p ~\mu_Z(\mathrm{d}z) \nonumber\\ & = \sum\nolimits_{k = 1}^M \sum\nolimits_{j = 1}^M \int_{\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)} |\kappa_\varphi^k - \kappa_\varphi^j|^p ~\mu_Z(\mathrm{d}z) \nonumber\\ & \leq \sum\nolimits_{k = 1}^M \sum\nolimits_{\substack{j = 1\\k \neq j}}^M \underbrace{\max_{k, j = 1, \ldots, M} |\kappa_\varphi^k - \kappa_\varphi^j|^p}_{=: \alpha < +\infty} \mu_Z\left[\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)\right] \nonumber\\ & \leq \sum\nolimits_{k = 1}^M \sum\nolimits_{\substack{j = 1\\k \neq j}}^M \alpha M^{k, j}(x, x') \label{sumMkj} \end{align} based on the absolute continuity of $\mu_Z$. Now it suffices to derive Lipschitz estimates for $M^{k, j}(x, x')$ for all $k, j = 1, \ldots, M$ and for all $x \in F_Z$ to continue estimation \eqref{sumMkj}: \begin{align} & M^{k, j}(x, x') = \mu_Z\left[\big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx')\big)\right] \nonumber\\ & = \mu_Z\left[\big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx + T(x - x'))\big)\right] \label{Mijx} \end{align} For $k \neq j$, it is $V^k \cap V^j = \emptyset$ and we get \begin{itemize} \item $\big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx)\big)$ is null set and, in particular, it is contained in a finite union of hyperplanes, and \item $\big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx)\big) \subseteq \mathrm{bd}\, \big(\bar{V}^j \oplus (- Tx)\big) = (\mathrm{bd}\,\bar{V}^j) \oplus (- Tx)$ \end{itemize} such that we receive \begin{align*} & \big(\bar{V}^k \oplus (- Tx)\big) \cap \big(\bar{V}^j \oplus (- Tx + T(x - x'))\big)\\ & \subseteq \mathrm{bd}\, \bar{V}^j \oplus (- Tx) \oplus [0, 1]\, T(x - x')\\ & = \left\{v + lT(x - x') \; | \; v \in \mathrm{bd}\, \bar{V}^j \oplus (- Tx), \, l \in [0, 1]\right\}. \end{align*} Now we can proceed with the estimation of \eqref{Mijx} by \[ \leq \delta \lambda^s\left[\left\{v + l T(x - x') \; | \; v \in \mathrm{bd}\, \bar{V}^j \oplus (- Tx), \, l \in [0, 1]\right\} \cap \mathrm{supp}\, \mu_Z\right], \] where $\delta < \infty$ is a bound for the Lebesgue density of $\mu_Z$. The boundary of $\bar{V}^j \oplus (- Tx)$ is the finite union of $K$ (independent of $k, j$, $x, x'$) hyperplanes $H_1, \ldots, H_K$. By the help of Cavalieri's principle and $\mathrm{diam}\,(\mathrm{supp}\,\mu_Z)^{s - 1} = \beta$, we finish the proof \begin{align*} \leq \delta \sum\nolimits_{l = 1}^K \lambda^{s - 1}\left[H_l \cap \mathrm{supp}\, \mu_Z\right] \cdot \|T(x - x')\| \leq \delta \cdot K \beta \cdot \|T\| \cdot \|x - x'\|.\tag*{$\qed$} \end{align*} Example~\ref{ExLocLipschitz} shows that boundedness of the Lebesgue density is essential for Theorem~\ref{ThFLipschitz}. If the support of $\mu_Z$ is not bounded, we still obtain a weaker estimate: \begin{theorem} Assume \ref{A1}, \ref{A2} and $p \in [1, \infty)$, then for any $\epsilon > 0$ there exists a real number $L(\epsilon)$ such that $\|\mathbb{F}(x) - \mathbb{F}(x')\|_{\mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})} \leq L(\epsilon) \|x - x\|^{\frac{1}{p}}$ holds for any $x, x' \in F_Z$. \end{theorem} {\it Proof} Throughout the proof, we will fix some $\epsilon > 0$ and use the notation established in the proof of Theorem~\ref{ThFLipschitz}. By construction, we may assume that $K \geq 2$. As the probability measure $\mu_Z$ is tight by \cite[Th. 1.3]{Bi99}, there is some compact set $C(\epsilon) \subset \mathbb{R}^s$ such that $$ \mu_Z \left[ \mathbb{R}^s \setminus C(\epsilon) \right] \leq \frac{\epsilon^p}{K(K-1)\alpha}. $$ By the arguments given in the proof of Theorem~\ref{ThFLipschitz}, we thus have \allowdisplaybreaks \begin{align*} & M^{k, j}(x, x')\\ & \leq \mu_Z\left[\left\{v + l T(x - x') \; | \; v \in \mathrm{bd}\, \bar{V}^j \oplus (- Tx), \, l \in [0, 1]\right\} \cap C(\epsilon) \right] + \tfrac{\epsilon^p}{K(K-1)\alpha}\\ & \leq \delta K \mathrm{diam}(C(\epsilon))^{s-1} \|T\|\cdot \|x - x'\| + \tfrac{\epsilon^p}{K(K-1)\alpha} \end{align*} and \eqref{sumMkj} yields the desired estimate \allowdisplaybreaks \begin{align*} &\|\mathbb{F}(x) - \mathbb{F}(x')\|_{\mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})} \leq \left(\delta K \mathrm{diam}(C(\epsilon))^{s-1} \|T\|\cdot \|x - x'\| + \epsilon^p \right)^{\frac{1}{p}}\\ & \leq \left( \delta K \mathrm{diam}(C(\epsilon))^{s-1} \|T\| \right)^{\frac{1}{p}} \|x - x'\|^{\frac{1}{p}} + \epsilon, \end{align*} where the second inequality follows from the fact that the function $t \to t^{\frac{1}{p}}$ is subadditive on $[0,\infty)$. \qed \section{Risk-averse approach} \label{Sec_4} As a first choice, we might evaluate the random upper level objective function based on its expected value, i.e. consider the risk neutral bi-level stochastic program $\min_{x \in F_Z} \left\{\mathbb{E} \left[\mathbb{F}(x)\right]\right\}$, which is well-defined by Lemma~\ref{Lemma_welldefined}. More in general, to allow for varying degrees of risk aversion, we take into account a mapping $\mathcal{R}\colon \mathcal{X} \to \mathbb{R}$ with $\mathcal{L}^\infty(\Omega, \mathcal{F}, \mathbb{P}) \subseteq \mathcal{X} \subseteq \mathcal{L}^0(\Omega, \mathcal{F}, \mathbb{P})$ and consider the bi-level stochastic program \begin{equation} \label{StochasticBi-levelProgram} \min_{x \in F_Z} \left\{\mathcal{R}\left[\mathbb{F}(x)\right]\right\}. \end{equation} $\mathcal{R}$ will typically be a monetary risk measure in the sense of \cite[Def. 4.1]{FoSc11} meaning it satisfies the following conditions: \begin{itemize} \item Monotonicity: $\mathcal{R}[Y_1] \leq \mathcal{R}[Y_2]$ for all $Y_1, Y_2 \in \mathcal{X}$ satisfying $Y_1 \leq Y_2$ $\mathbb{P}$-almost surely. \item Translation equivariance: $\mathcal{R}[Y + m] = \mathcal{R}[Y] + m$ for all $Y \in \mathcal{X}$ and $m \in \mathbb{R}$. \end{itemize} Moreover, we will assume the following: \begin{enumerate}[label=$\mathrm{(A\arabic*)}$,leftmargin=1cm,start=3] \item $\mathcal{R}\colon \mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P}) \to \mathbb{R}$ with some $p \in [1, \infty)$ is convex and nondecreasing as defined above.\label{A3} \end{enumerate} \begin{remark} \ref{A3} holds for any convex risk measure in the sense of \cite{FrRoGi02} and \cite{FoSc02}, i.e. for any monetary risk measure that is convex. In particular, this includes the expectation, the mean-upper semideviation of any order and the Conditional Value-at-Risk. However, as we do not assume translation equivariance, the assumption is also fulfilled for the expected excess of arbitrary order (cf. \cite[\S 6]{ShDeRu09}). \end{remark} \begin{theorem} \label{Th_QContinuous} Assume \ref{A1}, \ref{A3}, then the following statements hold true: \begin{enumerate} \item The function $\mathcal{Q}_\mathcal{R} \colon F_Z \to \mathbb{R}$ defined by $\mathcal{Q}_\mathcal{R}(x) := \mathcal{R}\left[\mathbb{F}(x)\right]$ is continuous at $x \in F_Z$ if $\mu_Z\left[D_\varphi(x)\right] = 0$. \item The function $\mathcal{Q}_\mathcal{R}$ is continuous if the Borel measure $\mu_Z$ is absolutely continuous with respect to the Lebesgue measure. In particular, the bi-level stochastic problem \eqref{StochasticBi-levelProgram} has an optimal solution whenever the induced feasible set $F_Z$ is nonempty and compact. \end{enumerate} \end{theorem} {\it Proof} As $\mathcal{R}$ is continuous by \cite[Lemma 3]{BuClCoRuSaSc21}, the results follow from Lemma~\ref{Lemma_welldefined} and Theorem~\ref{Th_FContinuous2} or Corollary~\ref{Cor_FContinuous}. \qed \begin{remark} In general, assumption \ref{A3} does not hold for a so-called certainty equivalent $\mathrm{CE}_u$ (cf. \cite[Def. 2.8]{BuClDe20}) because of the lack of convexity. However, Lipschitz continuity is guaranteed under the assumption, that $u: \mathbb{R} \to \mathbb{R}$ is strictly increasing and bi-Lipschitz, cf. \cite[Lemma 3.6]{BuClDe20}. \end{remark} The following result for bi-level linear optimization under uncertainty can be found in \cite[Prop. 1]{BuCl20}: \begin{proposition} \label{PropQRLipschitz} Assume \ref{A1}, \ref{A2} and let the support of $\mu_Z$ be compact. Then the following statements hold true for any mapping $\mathcal{R}: L^p(\Omega, \mathcal{F}, \mathbb{P}) \to \mathbb{R}$: \begin{enumerate} \item $\mathcal{Q}_\mathcal{R}$ is locally Hölder continuous with exponent $\frac{1}{p}$ if $\mathcal{R}$ is convex and continuous. \item $\mathcal{Q}_\mathcal{R}$ is locally Hölder continuous with exponent $\frac{1}{p}$ if we assume \ref{A3}. \item $\mathcal{Q}_\mathcal{R}$ is locally Hölder continuous with exponent $\frac{1}{p}$ if $\mathcal{R}$ is a convex risk measure. \item $\mathcal{Q}_\mathcal{R}$ is Hölder continuous with exponent $\frac{1}{p}$ if $\mathcal{R}$ is Lipschitz continuous. \item $\mathcal{Q}_\mathcal{R}$ is Hölder continuous with exponent $\frac{1}{p}$ if $\mathcal{R}$ is a coherent risk measure. \end{enumerate} \end{proposition} {\it Proof} See the proof of \cite[Prop. 1]{BuCl20} in combination with Theorem~\ref{ThFLipschitz}. \qed \begin{remark} We obtain (local) Lipschitz continuity for all feasible risk measures with $p = 1$, which includes, for example, the expectation, the expected excess of order $1$, and the Conditional Value-at-Risk. \end{remark} Due to the lack of convexity, Theorem~\ref{Th_QContinuous} and the subsequent proposition do not apply to the excess probability and the Value-at-Risk. However, the arguments from the proof of Theorem~\ref{Th_QContinuous} can be used another time: \begin{theorem} Assume \ref{A1} and that the Borel measure $\mu_Z$ is absolutely continuous with respect to the Lebesgue measure. Fix $\eta \in \mathbb{R}$, then the function $\mathcal{Q}_{\mathrm{EP}_\eta}(x) := \mathrm{EP}_\eta\left[\mathbb{F}(x)\right] = \mu_Z\left[\left\{\omega \in \Omega \; | \; \mathbb{F}(x)(\omega) > \eta\right\}\right]$ is continuous. Furthermore, let the induced feasible set $F_Z$ be nonempty and compact. Then $\min_{x \in F_Z} \left\{\mathcal{Q}_{\mathrm{EP}_\eta}(x)\right\}$ is solvable. \end{theorem} {\it Proof} The function $\mathcal{Q}_{\mathrm{EP}_\eta}(x)$ is real-valued on $F_Z$ due to Lemma~\ref{Lemma_welldefined}. Fix an arbitrary point $x \in F_Z$ and consider any $x' \in F_Z$. By Lemma~\ref{Lemma_partition}, we have \allowdisplaybreaks \begin{align*} &\left|\mathcal{Q}_{\mathrm{EP}_\eta}(x) - \mathcal{Q}_{\mathrm{EP}_\eta}(x')\right|\\ & = \left|\mu_Z\left[\left\{\omega \in \Omega \; | \; \varphi(Tx + Z(\omega)) > \eta\right\}\right] - \mu_Z\left[\left\{\omega \in \Omega \; | \; \varphi(Tx' + Z(\omega)) > \eta\right\}\right]\right|\\ & = \left|\mu_Z\left[\left\{\omega \in \Omega \; | \; \sum\nolimits_{k = 1}^M \kappa_\varphi^k 1\!\!1_{V^k}(Tx + Z(\omega)) > \eta\right\}\right]\right.\\ &\quad \left.- \mu_Z\left[\left\{\omega \in \Omega \; | \; \sum\nolimits_{j = 1}^M \kappa_\varphi^j 1\!\!1_{V^j}(Tx' + Z(\omega)) > \eta\right\}\right]\right|\\ &= \left|\mu_Z\left[\bigcup\nolimits_{\substack{k = 1\\\kappa_\varphi^k > \eta}}^M V^k \oplus (-Tx)\right] - \mu_Z\left[\bigcup\nolimits_{\substack{j = 1\\\kappa_\varphi^j > \eta}}^M V^j \oplus (-Tx')\right]\right|\\ & = \left|\sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k > \eta}}^M \mu_Z\left[V^k \oplus (-Tx)\right] - \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j > \eta}}^M \mu_Z\left[V^j \oplus (-Tx')\right]\right|\\ & = \left|\sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k > \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j \leq \eta}}^M \mu_Z\left[\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)\right]\right.\\ &\quad \left.- \sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k \leq \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j > \eta}}^M \mu_Z\left[\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)\right]\right| \end{align*} The additional restrictions below the sums exclude the case $k \neq j$, so that we can use the following considerations based on the notation introduced in the proof of Theorem~\ref{ThFLipschitz}: \[ \lim_{x' \to x} M^{k, j}(x, x') = M^{k, j}(x, x) = \begin{cases} 0 & \text{if}\ k \neq j,\\ \mu_Z\left[\bar{V}^k \oplus (- Tx)\right] & \text{if}\ k = j. \end{cases} \] We receive \allowdisplaybreaks \begin{align*} & \lim_{x' \to x} \left|\mathcal{Q}_{\mathrm{EP}_\eta}(x) - \mathcal{Q}_{\mathrm{EP}_\eta}(x')\right|\\ & = \lim_{x' \to x} \left|\sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k > \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j \leq \eta}}^M \mu_Z\left[\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)\right]\right.\\ &\quad \left.- \sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k \leq \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j > \eta}}^M \mu_Z\left[\big(V^k \oplus (- Tx)\big) \cap \big(V^j \oplus (- Tx')\big)\right]\right|\\ & = \lim_{x' \to x} \left|\sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k > \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j \leq \eta}}^M M^{k, j}(x, x') - \sum\nolimits_{\substack{k = 1\\\kappa_\varphi^k \leq \eta}}^M \sum\nolimits_{\substack{j = 1\\\kappa_\varphi^j > \eta}}^M M^{k, j}(x, x')\right| = 0\tag*{$\qed$} \end{align*} \begin{proposition} Assume \ref{A1}, \ref{A2} and let the support of $\mu_Z$ be compact. Fix $\alpha \in (0, 1)$, then the function $\mathcal{Q}_{\mathrm{VaR}_\alpha}(x) := \mathrm{VaR}_\alpha\left[\mathbb{F}(x)\right] = \inf\left\{\eta \in \mathbb{R} \; | \; \mu_Z\left[\left\{\omega \in \Omega \; | \; \mathbb{F}(x)(\omega) \leq \eta\right\}\right] \geq \alpha\right\}$ \noindent is continuous. Furthermore, let the induced feasible set $F_Z$ be nonempty and compact, then $\min_{x \in F_Z} \left\{\mathcal{Q}_{\mathrm{VaR}_\alpha}(x)\right\}$ is solvable. \end{proposition} {\it Proof} See the proof of \cite[Th. 2]{Iv14} in combination with Theorem~\ref{ThFLipschitz}. \qed \section{Implications for qualitative stability} \label{Sec_stability} We shall now examine the behaviour of local optimal values or optimal solutions sets of \eqref{StochasticBi-levelProgram} under perturbations of the underlying probability measure. As the measure's support may vary as well, stability analysis usually relies on the assumption of complete recourse, i.e. $\mathrm{dom}\; f = \mathbb{R}^n \times \mathbb{R}^s$ (cf. \cite{BuClDe20} for the case without integrality constraints). However, in the present setting, Lemma~\ref{LemmaDomPhi} and Lemma~\ref{LemmaA1DomainsPolyhedral} yield the existence of a polyhedral set $D \subsetneq \mathbb{R}^s$ such that $(x,z) \in \mathrm{dom} \, f$ holds if and only if $Tx+z \in D$. Thus, the domain of $f$ cannot encompass the whole space. Therefore, we shall only consider measures whose support is contained in the domain of function $f(x,\cdot)$ regardless of the leader's decision $x \in X$, i.e. confine the analysis to the set $\mathcal{M}(X) := \left\{\mu \in \mathcal{P}(\mathbb{R}^s) \; | \; X \times \mathrm{supp}\; \mu \in \mathrm{dom}\; f\right\}$. Since $f$ is measurable by Lemma~\ref{Lemma_phimeasurable2}, it follows $\left(\delta_x \otimes \mu\right) \circ f^{-1} \in \mathcal{P}(\mathbb{R})$, where $\delta_x \in \mathcal{P}(\mathbb{R}^n)$ denotes the Dirac measure at $x \in \mathbb{R}^n$. For fixed $x$, the mapping $f(x, \cdot)$ is bounded by Corollary~\ref{Cor_bounded}, so $\left(\delta_x \otimes \mu\right) \circ f^{-1}$ has a bounded support. Let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ be atomless (cf. \cite[Rem. 5.2]{BuClDe20}), then there exists some random variable $Z_{(x, \mu)} \in \mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P})$ such that $\left(\delta_x \otimes \mu\right) \circ f^{-1} = \mathbb{P} \circ Z_{(x, \mu)}^{-1}$. Let $\rho: \mathcal{L}^p(\Omega, \mathcal{F}, \mathbb{P}) \to \mathbb{R}$ with $p \in [1, \infty)$ be a convex function which is nondecreasing and law-invariant. Thus, we may consider the mapping \mbox{$\mathcal{Q}_\rho: X \times \mathcal{M}(X) \to \mathbb{R}$} with $\mathcal{Q}_\rho(x, \mu) := \rho[Z_{(x, \mu)}]$, \noindent which is well defined due to the law invariance of $\rho$. In what follows, we will endow $\mathcal{P}(\mathbb{R}^s)$ with the \textit{topology of weak convergence}, i.e., the topology where a sequence $\{\mu_n\}_{n \in \mathbb{N}} \subset \mathcal{P}(\mathbb{R}^s)$ converges weakly to $\mu \in \mathcal{P}(\mathbb{R}^s)$ if and only if $ \lim_{n \to \infty}\int_{\mathbb{R}^s} h(t)~\mu_n(\mathrm{d}t) = \int_{\mathbb{R}^s} h(t)~\mu(\mathrm{d}t)$ is valid for any bounded continuous function $h: \mathbb{R}^s \to \mathbb{R}$. A subset $\mathcal{M} \subseteq \mathcal{M}^p_s := \left\{\nu \in \mathcal{P}(\mathbb{R}^s) \; | \; \int_{\mathbb{R}^s} \|t\|^p~\mu(\mathrm{d}t) < \infty\right\}$, which denotes the set of Borel probability measures on $\mathbb{R}^s$ with finite moments of order $p \in [1, \infty)$, is called \textit{locally uniformly $\|\cdot\|^p$-integrating} if $\mu_n \to \mu \in \mathcal{M}$ implies $\lim_{n \to \infty}\int_{\mathbb{R}^s} \|t\|^p~\mu_n(\mathrm{d}t) = \int_{\mathbb{R}^s} \|t\|^p~\mu(\mathrm{d}t)$ for every sequence $\{\mu_n\}_{n \in \mathbb{N}} \subseteq \mathcal{M}$. Details and examples can be found in \cite[\S 5]{BuClDe20} and in \cite[\S 2]{ClKrSc17}. The next two sets are adapted from the corresponding results in \cite{BuClDe20} and \cite{ClKrSc17}, respectively. \begin{theorem}\label{Th_stability} All assumptions as described above. Let $\mathcal{M} \subseteq \mathcal{M}(X)$ be locally uniformly $\|\cdot\|^p$-integrating and let $(x, \mu) \in X \times \mathcal{M}$ be such that $D_\varphi(x) - \{Tx\}$ is a $\mu-$null set. Then $\mathcal{Q}_\rho|_{X \times \mathcal{M}}$ is continuous at $(x, \mu)$ with respect to the product topology of the standard topology on $\mathbb{R}^n$ and the topology of weak convergence on $\mathcal{M}$. \end{theorem} {\it Proof} Based on Lemma~\ref{Lemma_phimeasurable2} and \eqref{nullset}, \cite[Cor. 2.3]{ClKrSc17} is applicable, where the required growth condition follows directly from Corollary~\ref{Cor_bounded} with $\gamma = 1$. \qed \begin{remark} If we assume alternatively all assumptions as described above, together with $\mathcal{M} \subseteq \mathcal{M}(X)$ is locally uniformly $\|\cdot\|^p$-integrating and $\mu \in \mathcal{M}$ is absolutely continuous with respect to the Lebesgue-Borel measure on $\mathbb{R}^s$ in the above theorem, then $\mathcal{Q}_\rho|_{X \times \mathcal{M}}$ is continuous at $(x, \mu)$ for all $x \in X$ with respect to the product topology of the standard topology on $\mathbb{R}^n$ and the topology of weak convergence on $\mathcal{M}$. \end{remark} Next, we consider the parametric optimization problem \begin{align}\label{Pmu} \min_x \left\{\mathcal{Q}_\rho(x, \mu) \; | \; x \in X\right\}.\tag{$P_\mu$} \end{align} As \eqref{Pmu} is non-convex, we are interested in the sets of local optimal solutions: For any open set $V \subseteq \mathbb{R}^n$, we introduce the extended real-valued \textit{localized optimal value function} $\xi_V: \mathcal{M}(X) \to \overline{\mathbb{R}}$ with $\xi_V(\mu) := \inf_x \left\{\mathcal{Q}_\rho(x, \mu) \; | \; x \in X \cap \mathrm{cl}\; V\right\}$, \noindent and the \textit{localized optimal solution set mapping} $\Xi_V: \mathcal{M}(X) \rightrightarrows \mathbb{R}^n$ with $\Xi_V(\mu) := \mathrm{Argmin}_x\left\{\mathcal{Q}_\rho(x, \mu) \; | \; x \in X \cap \mathrm{cl}\; V\right\}$. \noindent Additional assumptions are required to study the stability of local optimal solution sets: The set $\Xi_V(\mu)$ is a so-called \textit{complete local minimizing (CLM) set} of \eqref{Pmu} with respect to $V$ if $\emptyset \neq \Xi_V(\mu) \subseteq V$ for a given $\mu \in \mathcal{M}(X)$ and an open set $V \subseteq \mathcal{R}^n$. \begin{theorem}\label{Th_stability2} All assumptions as described above. Let $\mathcal{M} \subseteq \mathcal{M}(X)$ be locally uniformly $\|\cdot\|^p$-integrating and let $\mu$ has a uniformly bounded Lebesgue density. Then the following statements hold true: \begin{enumerate} \item The restriction $\xi_{\mathbb{R}^n}|_\mathcal{M}$ is upper semicontinuous at $\mu \in \mathcal{M}$ with respect to the topology of weak convergence on $\mathcal{M}$. \end{enumerate} In addition, assume that $\mu \in \mathcal{M}$ is such that $\Xi_V(\mu)$ is a CLM set of \eqref{Pmu} with respect to some open bounded set $V \subset \mathbb{R}^n$. Then the following statements hold true: \begin{enumerate}[start=2] \item The restriction $\xi_V|_\mathcal{M}$ is continuous at $\mu \in \mathcal{M}$ with respect to the topology of weak convergence on $\mathcal{M}$. \item The restriction $\Xi_V|_\mathcal{M}$ is upper semicontinuous at $\mu$ with respect to the topology of weak convergence on $\mathcal{M}$ in the sense of Berge (cf. \cite{Be59}), i.e. for any open set $O \subseteq \mathbb{R}^n$ with $\Xi_V(\mu) \subset O$ there exists a weakly open neighborhood $N$ of $\mu$ such that $\Xi_V(\mu_Z) \subseteq O$ holds for all $\mu_Z \in N \cap \mathcal{M}$. \end{enumerate} \end{theorem} {\it Proof} Since the measure $\mu$ is endowed with a density, the assertions follow together with \cite[Cor. 2.4]{ClKrSc17}. \qed \section{Conclusions} We studied the function obtained by optimizing a linear functional over the set of minimizers of an integer linear problem and characterized its set of discontinuity points. This allowed us to formulate continuity results for the risk-averse bi-level stochastic model when the underlying Borel measure is absolutely continuous with respect to the Lebesgue measure. Moreover, we quantified the continuity by providing sufficient conditions for the Hölder continuity for a comprehensive class of risk measures. Qualitative stability results with respect to perturbations of the underlying probability measure conclude the paper. Due to the boundedness of the function $\varphi$, it is easy to verify that all of our results for the optimistic model carry over to the pessimistic setting. For risk measures defined on $L^1$, we have obtained an optimization problem with Lipschitz continuous goal function. Thus, Clarke's generalized derivatives can be used to formulate optimality conditions. However, the characterization of such derivatives was beyond the scope of the present paper and will be investigated in future work. Moreover, we will investigate algorithmic approaches based on these derivatives. While the assumption that the feasible set of the lower level problem is finite arises natuarilly in a combinatorial or binary optimization setting, we plan to extend the analysis to unbounded models in future work. \end{document}

\begin{document} \begin{abstract} We are given a finite group $H$, an automorphism $\tau$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langle\sigma\rangle$ of order $r$, and an absolutely irreducible representation $\rho\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $\tau$ on the character of $\rho$ is the same as the action of $\sigma$. Then the following are equivalent. $\bullet$ $\rho$ is equivalent to a representation $\rho'\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $\sigma$ on the entries of the matrices corresponds to the action of $\tau$ on $H$, and $\bullet$ the induced representation $\operatorname{\sf ind}_{H,H\rtimes\langle\tau\rangle}(\rho)$ has Schur index one; that is, it is similar to a representation over $K$. As examples, we discuss a three dimensional irreducible representation of $A_5$ over $\mathbb{Q}[\sqrt5]$ and a four dimensional irreducible representation of the double cover of $A_7$ over $\mathbb{Q}[\sqrt{-7}]$. \end{abstract} \title{Matrices for finite group representations that respect Galois automorphisms} \section{Introduction} This paper begins with the following question, suggested to the author by Richard Parker. The alternating group $A_5$ has a three dimensional representation over the field $\mathbb{Q}[\sqrt5]$ which induces up to the symmetric group $S_5$ to give a six dimensional irreducible that can be written over $\mathbb{Q}$. Given an involution in $S_5$ that is not in $A_5$, is it possible to write down a $3\times 3$ matrix representation of $A_5$ such that the Galois automorphism of $\mathbb{Q}[\sqrt5]$ acts on matrices in the same way as the involution acts on $A_5$ by conjugation? More generally, we are given a finite group $H$, an automorphism $\tau$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\operatorname{\sf Gal}(L/K)=\langle\sigma\rangle$ of order $[L:K]=r$, and an absolutely irreducible representation $\rho\colon H\to\operatorname{\sf GL}(n,L)$. We assume that the action of $\tau$ on the character of the representation $\rho$ is the same as the action of $\sigma$. Then the question is whether it is possible to conjugate to a representation $\rho'\colon H\to\operatorname{\sf GL}(n,L)$ with the property that the Galois automorphism $\sigma$ acts on matrices in the same way as $\tau$ acts on $H$. In other words, we are asking whether the following diagram can be made to commute. \[ \xymatrix{ H\ar[r]^(0.4){\rho'}\ar[d]_\tau&\operatorname{\sf GL}(n,L)\ar[d]^\sigma\\ H\ar[r]^(0.4){\rho'}&\operatorname{\sf GL}(n,L)} \] We answer this using the invariant $\lambda(\rho)$ in the relative Brauer group \[ \operatorname{\sf Br}(L/K)=H^2(\langle\sigma\rangle,L^\times)\cong K^\times/N_{L/K}(L^\times) \] that defines the division algebra associated to the representation obtained by inducing to the semidirect product. \begin{theorem}\label{th:main} Let $\rho\colon H \to \operatorname{\sf GL}(n,L)$ be as above. Then there is an invariant $\lambda(\rho)\in K^\times/N_{L/K}(L^\times)$ such that the following are equivalent. \begin{enumerate} \item $\lambda(\rho)=1$, \item There is a conjugate $\rho'$ of $\rho$ making the diagram above commute, \item If $G$ is the semidirect product $H\rtimes \langle \tau\rangle$ then the induced representation $\operatorname{\sf ind}_{H,G}(\rho)$ has Schur index equal to one; in other words, it can be written over $K$. \end{enumerate} More generally, the order of $\lambda(\rho)$ in $K^\times/N_{L/K}(L^\times)$ is equal to the Schur index of the induced representation, and the associated division algebra is the one determined by $\lambda(\rho)$. \end{theorem} The equivalence of (1) and (2) is proved in Section~\ref{se:Y}. The equivalence of (1) and (3) is more standard, see for example Turull~\cite{Turull:2000a}, and is proved in Section~\ref{se:ind}. Combining these gives the more interesting statement of the equivalence of (2) and (3). We end with some examples. In the case of the three dimensional representations of $A_5$, we have $\lambda(\rho)=1$, and we write down explicit matrices for $\rho'$, though they're not very pleasant. In the case of the four dimensional irreducible representations of $2A_7$, we have $\lambda(\rho)=-2$, which is not a norm from $\mathbb{Q}[\sqrt{-7}]$, and the division ring associated to the induced representation is the quaternion algebra with symbol $(-2,-7)_\mathbb{Q}$. \noindent {\bf Acknowledgement.} I would like to thank Richard Parker for suggesting this problem and the examples, and Alexandre Turull for some helpful comments. \section{The matrix $X$}\label{se:X} Consider the composite $\sigma\circ\rho\circ\tau^{-1}$: \[ H \xrightarrow{\tau^{-1}} H \xrightarrow{\rho}\operatorname{\sf GL}(n,L) \xrightarrow{\sigma}\operatorname{\sf GL}(n,L). \] This representation is equivalent to $\rho$, and so there exists a matrix $X$, well defined up to scalars in $L^\times$, such that conjugation by $X$ takes $\rho$ to $\sigma\circ\rho\circ\tau^{-1}$. Write $c_X$ for conjugation by $X$, so that $c_X(A)=XAX^{-1}$. Then we have \begin{equation}\label{eq:1} \sigma\circ\rho\circ\tau^{-1} = c_X \circ\rho. \end{equation} By abuse of notation, we shall also write $\sigma$ for the automorphism of $\operatorname{\sf GL}(n,L)$ given by applying $\sigma$ to each of its entries. Then $c_{\sigma(X)}(\sigma(A))=\sigma(X)\sigma(A)\sigma(X)^{-1}=\sigma(XAX^{-1})$, so we have \[ c_{\sigma(X)}\circ\sigma = \sigma\circ c_X. \] So equation~\eqref{eq:1} gives \begin{align*} \sigma^2\circ\rho\circ\tau^{-2}&=\sigma\circ c_X\circ\rho\circ\tau^{-1}\\ &=c_{\sigma(X)}\circ\sigma\circ\rho\circ\tau^{-1}\\ &=c_{\sigma(X)}\circ c_X \circ\rho\\ &=c_{\sigma(X).X}\circ\rho. \end{align*} Continuing this way, for any $i>0$ we have \[ \sigma^i\circ\rho\circ\tau^{-i} = c_{\sigma^{i-1}(X)\cdots\sigma(X).X}\circ\rho. \] Taking $i=r$, we have $\sigma^r=1$ and $\tau^r=1$, so \begin{equation}\label{eq:NX} \rho=c_{\sigma^{r-1}(X)\cdots\sigma(X).X}\circ\rho. \end{equation} \begin{definition} If $A$ is an $n\times n$ matrix over $L$, we define the norm of $A$ to be \[ N_{L/K}(A)=\sigma^{r-1}(A)\cdots\sigma(A).A \] as an $n\times n$ matrix over $K$. \end{definition} Equation~\eqref{eq:NX} now reads \[ \rho=c_{N_{L/K}(X)}\circ\rho. \] By Schur's lemma, it follows that the matrix $N_{L/K}(X)$ is a scalar multiple of the identity, \[ N_{L/K}(X)=\lambda I. \] Applying $\sigma$ and rotating the terms on the left, we see that $\lambda=\sigma(\lambda)$, so that $\lambda\in K^\times$. If we replace $X$ by a scalar multiple $\mu X$, then the scalar $\lambda$ gets multiplied by $\sigma^{r-1}(\mu)\cdots \sigma(\mu)\mu$, which is the norm $N_{L/K}(\mu)$. Thus the scalar $\lambda$ is well defined only up to norms of elements in $L^\times$. We define it to be the $\lambda$-invariant of $\rho$: \[ \lambda(\rho)\in K^\times/N_{L/K}(L^\times). \] Thus $\lambda(\rho)=1$ if and only if $X$ can be replaced by a multiple of $X$ to make $N_{L/K}(X)=I$. \section{The matrix $Y$}\label{se:Y} The goal is to find a matrix $Y$ conjugating $\rho$ to a representation $\rho'$ such that $\sigma\circ\rho'\circ\tau^{-1}=\rho'$. Thus we wish $Y$ to satisfy \[ \sigma\circ c_Y\circ \rho\circ\tau^{-1}=c_Y\circ\rho. \] We rewrite this in stages: \begin{align*} c_{\sigma(Y)}\circ\sigma\circ\rho\circ\tau^{-1}&=c_Y\circ\rho\\ \sigma\circ\rho\circ\tau^{-1}&=c_{\sigma(Y)^{-1}}\circ c_Y\circ\rho\\ c_X\circ\rho&=c_{\sigma(Y)^{-1}Y}\circ\rho. \end{align*} Again applying Schur's lemma, $\sigma(Y)^{-1}Y$ is then forced to be a multiple of $X$. Since $N_{L/K}(\sigma(Y)^{-1}Y)=I$, it follows that if there is such a $Y$ then $\lambda(\rho)$ is the identity element of $K^\times/N_{L/K}(L^\times)$. This proves one direction of Theorem~\ref{th:main}. The other direction is now an immediate consequence of the version of Hilbert's Theorem 90 given in Chapter~X, Proposition~3 of Serre~\cite{Serre:1979a}: \begin{theorem}\label{th:90} Let $L/K$ be a finite Galois extension with Galois group $\operatorname{\sf Gal}(L/K)$. Then $H^1(\operatorname{\sf Gal}(L/K),\operatorname{\sf GL}(n,L))=0$.\qed \end{theorem} \begin{corollary} Let $L/K$ be a Galois extension with cyclic Galois group $\operatorname{\sf Gal}(L/K)=\langle\sigma\rangle$ of order $r$. If a matrix $X\in\operatorname{\sf GL}(n,L)$ satisfies $N_{L/K}(X)=I$ then there is a matrix $Y$ such that $\sigma(Y)^{-1}Y=X$. \end{corollary} \begin{proof} This is the case of a cyclic Galois group of Theorem~\ref{th:90}. \end{proof} This completes the proof of the equivalence of (1) and (2) in Theorem~\ref{th:main}. \section{The induced representation}\label{se:ind} Let $G=H\rtimes\langle\tau\rangle$, so that for $h\in H$ we have $\tau(h)=\tau h\tau^{-1}$ in $G$. Then the induced representation $\operatorname{\sf ind}_H^G(\rho)$ is an $LG$-module with character values in $K$, but cannot necessarily be written as an extension to $L$ of a $KG$-module. So we restrict the coefficients to $K$ and examine the endomorphism ring. \begin{lemma}\label{le:r^2} $\operatorname{\sf End}_{KG}(\operatorname{\sf ind}_{H,G}(\rho|_K))$ has dimension $r^2$ over $K$. \end{lemma} \begin{proof} The representation $\rho|_K$ is an irreducible $KH$-module, whose extension to $L$ decomposes as the sum of the Galois conjugates of $\rho$, so $\operatorname{\sf End}_{KH}(\rho|_K)$ is $r$ dimensional over $K$. For the induced representation $\operatorname{\sf ind}_{H,G}(\rho|_K)=\operatorname{\sf ind}_{H,G}(\rho)|_K$, as vector spaces we then have \begin{equation*} \operatorname{\sf End}_{KG}(\operatorname{\sf ind}_{H,G}(\rho|_K))\cong \operatorname{\sf Hom}_{KH}(\rho|_K,\operatorname{\sf res}_{G,H}\operatorname{\sf ind}_{H,G}(\rho|_K))\cong r.\operatorname{\sf End}_{KH}(\rho|_K). \qedhere \end{equation*} \end{proof} \begin{proposition}\label{pr:End} The algebra $\operatorname{\sf End}_{KG}(\operatorname{\sf ind}_{H,G}(\rho|_K))$ is a crossed product algebra, central simple over $K$, with generators $m_\lambda$ for $\lambda\in L$ and an element $\xi$, satisfying \[ m_\lambda + m_{\lambda'}=m_{\lambda+\lambda'},\qquad m_\lambda m_{\lambda'}=m_{\lambda\lambda'},\qquad m_\lambda\circ\xi=\xi\circ m_{\sigma(\lambda)},\qquad \xi^r=m_{\lambda(\rho)}. \] \end{proposition} \begin{proof} We can write the representation $\operatorname{\sf ind}_{H,G}(\rho|_K)$ in terms of matrices as follows. \[ g \mapsto \begin{pmatrix}\rho(g)|_K\\&\sigma\rho\tau^{-1}(g)|_K\\ &&\ddots\\ &&&\sigma^{-1}\rho\tau(g)|_K \end{pmatrix},\qquad \tau\mapsto \begin{pmatrix}&&&I\\ I\\&\ddots\\&&I \end{pmatrix}\circ\sigma \] It is easy to check that the following are endomorphisms of this representation. \[ m_\lambda=\begin{pmatrix}\lambda I\\ &\sigma(\lambda) I\\&&\ddots\\ &&&\sigma^{-1}(\lambda) I \end{pmatrix},\qquad \xi =\begin{pmatrix} &&&&\sigma^{-1}(X)\\ X\\&\sigma(X)\\ &&\ddots\\ &&&\sigma^{-2}(X) \end{pmatrix}\] with $\lambda\in L$ and $X$ as in Section~\ref{se:X}. Since these generate an algebra of dimension $r^2$ over $K$, by Lemma~\ref{le:r^2} they generate the algebra $\operatorname{\sf End}_{KG}(\operatorname{\sf ind}_{H,G}(\rho|_K))$. The given relations are easy to check, and present an algebra which is easy to see has dimension at most $r^2$, and therefore no further relations are necessary. \end{proof} \begin{corollary} The Schur index of the induced representation $\operatorname{\sf ind}_{H,G}(\rho)$ is equal to the order of $\lambda(\rho)$ as an element of $K^\times/N_{L/K}(L^\times)$. In particular, the Schur index is one if and only if $\lambda(\rho)=1$ as an element of $K^\times/N_{L/K}(L^\times)$. \end{corollary} \begin{proof} This follows from the structure of the central simple algebra $\operatorname{\sf End}_{KG}(\rho|_K)$ given in Proposition~\ref{pr:End}, using the theory of cyclic crossed product algebras, as developed for example in Section~15.1 of Pierce~\cite{Pierce:1982a}, particularly Proposition~b of that section. \end{proof} This completes the proof of the equivalence of (1) and (3) in Theorem~\ref{th:main}. In particular, it shows that $\lambda(\rho)$ can only involve primes dividing the order of $G$. \section{Examples} Our first example is a three dimensional representation of $A_5$. There are two algebraically conjugate three dimensional irreducible representations of $A_5$ over $\mathbb{Q}[\sqrt5]$ swapped by an outer automorphism of $A_5$, and giving a six dimensional representation of the symmetric group $S_5$ over $\mathbb{Q}$. Setting $\alpha=\frac{1+\sqrt5}{2}$, $\bar\alpha=\frac{1-\sqrt5}{2}$, we can write the action of the generators on one of these three dimensional representations as follows. \[ (12)(34)\mapsto \begin{pmatrix}-1\ &0&0\\0&0&1\\0&\ 1\ &0\end{pmatrix} \qquad (153)\mapsto\begin{pmatrix}-1&1&\alpha\\ \alpha&0&-\alpha\\ -\alpha&0&1\end{pmatrix}\] Taking this for $\rho$, we find a matrix $X$ conjugating this to $\sigma\circ\rho\circ\tau^{-1}$ where $\sigma$ is the field automorphism and $\tau$ is conjugation by $(12)$. Using the fact that if $a=(12)(34)$ and $b=(153)$ then $ab^2abab^2=(253)$, we find that \[ X=\begin{pmatrix}1&-\bar\alpha\ &\bar\alpha\\ -\bar\alpha\ &1&-\bar\alpha\ \\\bar\alpha&-\bar\alpha\ &1\end{pmatrix}\] We compute that $\sigma(X).X$ is minus the identity. Now $-1$ is in the image of $N_{\mathbb{Q}[\sqrt5],\mathbb{Q}}$, namely we have $(2-\sqrt5)(2+\sqrt5)=-1$. So we replace $X$ by $(2-\sqrt5)X$ to achieve $\sigma(X).X=I$. Having done this, by Hilbert 90 there exists $Y$ with $\sigma(Y)^{-1}.Y=X$. Such a $Y$ conjugates $\rho$ to the desired form. For example we can take \[ Y = \begin{pmatrix} 1-2\sqrt5&3-2\sqrt5&-3+2\sqrt5\\ 3-2\sqrt5&1-2\sqrt5&3-2\sqrt5\\ -3+2\sqrt5&3-2\sqrt5&1-2\sqrt5 \end{pmatrix}. \] Thus we end up with the representation \[ (12)(34) \mapsto\begin{pmatrix}-1\ &0&0\\0&0&1\\0&\ 1\ &0\end{pmatrix} \qquad (153)\mapsto\frac{1}{40}\begin{pmatrix} 10-4\sqrt5&-5+19\sqrt5&25-9\sqrt5\\ -10-4\sqrt5&25+9\sqrt5&-5-19\sqrt5\\ -50&35-5\sqrt5&-35-5\sqrt5 \end{pmatrix} \] \[ (253)\mapsto \frac{1}{40}\begin{pmatrix} 10+4\sqrt5&-5-19\sqrt5&25+9\sqrt5\\ -10+4\sqrt5&25-9\sqrt5&-5+19\sqrt5\\ -50&35+5\sqrt5&-35+5\sqrt5 \end{pmatrix}. \] Denoting these matrices by $a$, $b$ and $c$, it is routine to check that $a^2=b^3=(ab)^5=1$, $a^2=c^3=(ac)^5=1$, and $c=\sigma(b)=ab^2abab^2$. More generally, if $H$ is an alternating group $A_n$ and $G$ is the corresponding symmetric group $S_n$ then all irreducible representations of $G$ are rational and so the invariant $\lambda(\rho)$ is equal to one for any irreducible character of $H$ that is not rational. So an appropriate matrix $Y$ may always be found in this case. Our second example is one with $\lambda(\rho)\ne 1$. Let $H$ be the group $2A_7$, namely a non-trivial central extension of $A_7$ by a cyclic group of order two. Let $\tau$ be an automorphism of $H$ of order two, lifting the action of a transposition in $S_7$ on $H$, and let $G$ be the semidirect product $H\rtimes\langle\tau\rangle$. Then $H$ has two Galois conjugate irreducible representations of dimension four over $\mathbb{Q}[\sqrt{-7}]$. Let $\rho$ be one of them. The induced representation is eight dimensional over $\mathbb{Q}[\sqrt{-7}]$. Restricting coefficients to $\mathbb{Q}$ produces a $16$ dimensional rational representation whose endomorphism algebra $E$ is a quaternion algebra. Thus the induced representation can be written as a four dimensional representation over $E^{\mathsf{op}}\cong E$. This endomorphism algebra was computed by Turull~\cite{Turull:1992a} in general for the double covers of symmetric groups. In this case, by Corollary~5.7 of that paper, the algebra $E$ is generated over $\mathbb{Q}$ by elements $u$ and $v$ satisfying $u^2=-2$, $v^2=-7$ and $uv=-vu$. Thus the invariant $\lambda(\rho)$ is equal to $-2$ as an element of $\mathbb{Q}^\times/N_{\mathbb{Q}[\sqrt{-7}],\mathbb{Q}}(\mathbb{Q}[\sqrt{-7}]^\times)$ in this case. \end{document}

\begin{document} \title{Large Galois images for Jacobian varieties of genus $3$ curves} \begin{abstract} Given a prime number $\ell \geq 5$, we construct an infinite family of three-dimensional abelian varieties over $\mathbb{Q}$ such that, for any $A/\mathbb{Q}$ in the family, the Galois representation $\overline{\rho}_{A,\ell} \colon G_{\mathbb{Q}} \to \mathrm{GSp}_6(\mathbb{F}_{\ell})$ attached to the $\ell$-torsion of $A$ is surjective. Any such variety $A$ will be the Jacobian of a genus $3$ curve over $\mathbb{Q}$ whose respective reductions at two auxiliary primes we prescribe to provide us with generators of $\mathrm{Sp}_6(\mathbb{F}_{\ell})$. \end{abstract} \title{Large Galois images for Jacobian varieties of genus $3$ curves} \section*{Introduction} Let $\ell$ be a prime number. This paper is concerned with realisations of the general symplectic group $\mathrm{GSp}_6(\mathbb{F}_{\ell})$ as a Galois group over $\mathbb{Q}$, arising from the Galois action on the $\ell$-torsion points of three-dimensional abelian varieties defined over $\mathbb{Q}$. More precisely, let $g \geq 1$ be an integer. One can exploit the theory of abelian varieties defined over $\mathbb{Q}$ as follows. If $A$ is an abelian variety of dimension $g$ defined over $\mathbb Q$, let $A[\ell] = A(\overline{\mathbb Q})[\ell]$ denote the $\ell$-torsion subgroup of $\overline{\mathbb Q}$-points of $A$. The natural action of the absolute Galois group $G_{\mathbb Q}=\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ on $A[\ell]$ gives rise to a continuous Galois representation $\overline{\rho}_{A,\ell}$ taking values in $\text{GL}(A[\ell]) \simeq \text{GL}_{2g}(\mathbb F_{\ell})$. If the abelian variety $A$ is moreover principally polarised, the image of $\overline{\rho}_{A,\ell}$ lies inside the general symplectic group $\text{GSp}(A[\ell])$ of $A[\ell]$ with respect to the symplectic pairing induced by the Weil pairing and the polarisation of~$A$; thus, we have a representation $$\overline{\rho}_{A,\ell} \: : \: G_{\mathbb Q} \longrightarrow \text{GSp}(A[\ell]) \simeq \text{GSp}_{2g}(\mathbb F_{\ell}),$$ providing a realisation of $\text{GSp}_{2g}(\mathbb F_{\ell})$ as a Galois group over $\mathbb Q$ if $\overline{\rho}_{A,\ell}$ is surjective. The image of Galois representations attached to the $\ell$-torsion points of abelian varieties has been widely studied. For an abelian variety $A$ defined over a number field, the classical result of Serre ensures surjectivity for almost all primes $\ell$ when $\mathrm{End}_{\overline{\mathbb Q}}(A)=\mathbb{Z}$ and the dimension of $A$ is 2, 6 or odd (cf.~\cite{OeuvresSerre}). More recently, Hall \cite{Hall11} proves a result for any dimension, with the additional condition that $A$ has semistable reduction of toric dimension 1 at some prime. This result has been further generalised to the case of abelian varieties over finitely generated fields (cf.~\cite{AGP}). We can use Galois representations attached to the torsion points of abelian varieties defined over $\mathbb{Q}$ to address the Inverse Galois Problem and its variations involving ramification conditions. For example, the Tame Inverse Galois Problem, proposed by Birch, asks if, given a finite group $G$, there exists a tamely ramified Galois extension $K/\mathbb{Q}$ with Galois group isomorphic to $G$. Arias-de-Reyna and Vila solved the Tame Inverse Galois problem for $\mathrm{GSp}_{2g}(\mathbb{F}_{\ell})$ when $g=1, 2$ and $\ell \geq 5$ is any prime number, by constructing a family of genus $g$ curves $C$ such that the Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ attached to the Jacobian variety $\mathrm{Jac}(C)$ is surjective and tamely ramified for every curve in the family (cf. \cite{SaraNuria09}, \cite{SaraNuria11}). For both $g=1$ and $g=2$, the strategy entails determining a set of local conditions at auxiliary primes, (that is to say, prescribing a finite list of congruences that the defining equation of $C$ should satisfy) which ensure the surjectivity of $\overline{\rho}_{\mathrm{Jac}(C), \ell}$, and a careful study of the ramification at $\ell$ in particularly favourable situations. In fact, the strategy of ensuring surjectivity of the Galois representation attached to the $\ell$-torsion of an abelian variety by prescribing local conditions at auxiliary primes works in great generality. Given a $g$-dimensional principally polarised abelian variety $A$ over $\mathbb{Q}$, such that the Galois representation $\overline{\rho}_{A, \ell}$ is surjective, it is always possible to find some auxiliary primes $p$ and $q$ depending on $\ell$ such that any abelian variety $B$ defined over $\mathbb{Q}$ which is ``close enough'' to $A$ with respect to the primes $p$ and $q$ (in a sense that can be made precise in terms of $p$-adic, resp.~$q$-adic, neighbourhoods in moduli spaces of principally polarised $g$-dimensional abelian varieties with full level structure) also has a surjective $\ell$-torsion Galois representation $\overline{\rho}_{B,\ell}$. This is a consequence of Kisin's results on local constancy in $p$-families of Galois representations; the reader can find a detailed explanation of this aspect in \cite[Section 4.2]{AK13}. In this paper we focus on the case $g=3$. Our aim is to find auxiliary primes $p$ and $q$ (depending on $\ell$), and explicit congruence conditions on polynomials defining genus~$3$ curves, which ensure that any curve $C$, defined by an equation over $\mathbb{Z}$ satisfying these congruences, will have the property that the image of $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ coincides with $\mathrm{GSp}_{6}(\mathbb{F}_{\ell})$. In this way we obtain many distinct realisations of $\mathrm{GSp}_6(\mathbb{F}_{\ell})$ as a Galois group over $\mathbb{Q}$. To state our main result, we introduce the following notation: we will say that a polynomial $f(x, y)$ in two variables is of \emph{3-hyperelliptic type} if it is of the form $f(x, y)=y^2-g(x)$, where $g(x)$ is a polynomial of degree $7$ or $8$ and of \emph{quartic type} if the total degree of $f(x, y)$ is $4$. \begin{thm}\label{thm:main} Let $\ell\geq 13$ be a prime number. For all odd distinct prime numbers $p,q\neq \ell$, with $q>1.82\ell^2$, there exist $f_p(x, y), f_q(x, y)\in\mathbb Z[x,y]$ of the same type ($3$-hyperelliptic or quartic), such that for any $f(x, y)\in\mathbb Z[x,y]$ of the same type as $f_p(x, y)$ and $f_q(x, y)$ and satisfying \begin{equation*}f(x, y)\equiv f_q(x, y)\pmod{q} \quad \text{ and }\quad f(x, y)\equiv f_p(x, y)\pmod{p^3}, \end{equation*} the image of the Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ attached to the $\ell$-torsion points of the Jacobian of the projective genus~$3$ curve $C$ defined over $\mathbb Q$ by the equation $f(x,y)=0$ is $\mathrm{GSp}_6(\mathbb{F}_{\ell})$. Moreover, for $\ell\in\{5,7,11\}$ there exists a prime number $q\neq \ell$ for which the same statement holds for each odd prime number $p\neq q,\ell$. \end{thm} In Section \ref{sec:4} we state and prove a refinement of this Theorem (cf.~Theorem \ref{thm:refined}). In fact, we have a very explicit control of the polynomial $f_p(x, y)$. In general we can say little about $f_q(x, y)$, but for any fixed $\ell\geq 13$ and any fixed $q \geq1.82 \ell^2$ we can find suitable polynomials $f_q(x, y)$ by an exhaustive search as follows: there exist only finitely many polynomials $\bar{f}_q(x, y)\in\mathbb{F}_q[x, y]$ of $3$-hyperelliptic or quartic type with non-zero discriminant. For each of these, we can compute the characteristic polynomial of the action of the Frobenius endomorphism on the Jacobian of the curve defined by $\bar{f}_q(x, y)=0$ by counting the $\mathbb{F}_{q^r}$-points of this curve, for $r=1, 2, 3$, and check whether this polynomial is an ordinary $q$-Weil polynomial with non-zero middle coefficient, non-zero trace modulo $\ell$, and which is irreducible modulo $\ell$. Proposition \ref{irredmodell} ensures that the search will terminate. Then, any lift of $\bar{f}_q(x, y)$, of the same type, gives us a suitable polynomial $f_q(x, y)\in \mathbb{Z}[x, y]$. In Example \ref{ex:mainthm} we present some concrete examples obtained using \textsc{Sage} and \textsc{Magma}. Note that the above result constitutes an explicit version of Proposition 4.6 of \cite{AK13} in the case of principally polarised $3$-dimensional abelian varieties. We can explicitly give the size of the neighbourhoods where surjectivity of $\overline{\rho}_{A, \ell}$ is preserved; in other words, we can give the powers of the auxiliary primes $p$ and $q$ such that any other curve defined by congruence conditions modulo these powers gives rise to a Jacobian variety with surjective $\ell$-torsion representation. The proof of Theorem \ref{thm:main} is based on two main pillars: the classification of subgroups of $\mathrm{GSp}_{2g}(\mathbb{F}_{\ell})$ containing a non-trivial transvection, and the fact that one can force the image of $\overline{\rho}_{A, \ell}$ to contain a non-trivial transvection by imposing a specific type of ramification at an auxiliary prime. This strategy goes back to Le Duff \cite{LeDuff98} in the case of Jacobians of genus $2$ hyperelliptic curves, and has been extended to the general case by Hall in \cite{Hall11}, where he obtains a surjectivity result for $\overline{\rho}_{A, \ell}$ for almost all primes $\ell$. We already followed this strategy in \cite{AAKRTV14} to formulate an explicit surjectivity result for $g$-dimensional abelian varieties (see Theorem 3.10 of loc.~cit.): let $A$ be a principally polarised $g$-dimensional abelian variety defined over $\mathbb{Q}$, such that the reduction of the N\'eron model of $A$ at some prime $p$ is semistable with toric rank 1, and the Frobenius endomorphism at some prime $q$ of good reduction for $A$ acts irreducibly and with trace $a\not=0$ on the reduction of the N\'eron model of $A$ at $q$. We proved that for each prime number $\ell\nmid 6pqa$, coprime with the order of the component group of the N\'eron model of $A$ at $p$, and such that the characteristic polynomial of the Frobenius endomorphism at $q$ is irreducible mod $\ell$, then the representation $\overline{\rho}_{A, \ell}$ is surjective. Section~\ref{sectionone} collects some notations and tools that we will use in the rest of the paper. In Section~\ref{sec:2} we address the condition of semistable reduction of toric rank $1$ at a prime $p$; we obtain a congruence condition modulo $p^3$ (cf.~Proposition~\ref{jacobianthm}). In Section~\ref{sec:3} we give conditions ensuring that the reduction of the N\'eron model of a Jacobian variety $A=\mathrm{Jac}(C)$ at a prime $q$ is an absolutely simple abelian variety over $\mathbb{F}_q$ such that the characteristic polynomial of the Frobenius endomorphism at $q$ is irreducible and has non-zero trace modulo $\ell$ (cf.~Theorem~\ref{A}). We make use of Honda-Tate Theory in the ordinary case, which relates so-called ordinary Weil polynomials to isogeny classes of ordinary abelian varieties defined over finite fields of characteristic $q$. First, we need to prove the existence of a suitable prime $q$ and a suitable ordinary Weil polynomial; this is the content of Proposition~\ref{irredmodell}, whose proof is postponed to Section~\ref{sectionirred}. This polynomial provides us with an abelian variety $A_q$ defined over $\mathbb{F}_q$; any abelian variety $A$ such that the reduction of the N\'eron model of $A$ at $q$ coincides with $A_q$ will satisfy the desired condition at $q$. At this point we use the fact that each principally polarised $3$-dimensional abelian variety over $\mathbb{F}_q$ is the Jacobian of a genus $3$ curve, which can be defined over $\mathbb{F}_q$ up to a quadratic twist. Once we have established congruence conditions at auxiliary primes $p$ and $q$, we need to check that any curve $C$ over $\mathbb{Z}$ whose defining equation satisfies these conditions will provide us with a Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ whose image is $\mathrm{GSp}_{6}(\mathbb{F}_{\ell})$. This is carried out in Section \ref{sec:4}. David Zywina communicated to us that he has recently and independently developed a method for studying the image of Galois representations $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ attached to the Jacobians of genus $3$ plane quartic curves $C$, for a large class of such curves (cf.~\cite{Zywina15}). In particular, for each prime $\ell$, he obtains a realisation of $\GSp_6(\mathbb{F}_{\ell})$ as a Galois group over $\mathbb{Q}$. Samuele Anni, Pedro Lemos and Samir Siksek also worked independently on this topic. In their paper \cite{ALS15}, they study semistable abelian varieties and provide an example of a hyperelliptic genus $3$ curve $C$ such that $\mathrm{Im}\overline{\rho}_{\mathrm{Jac}(C), \ell}=\GSp_6(\mathbb{F}_{\ell})$ for all $\ell\geq 3$. Both Zywina and Anni et al.~propose a method which, given a fixed genus $3$ curve $C$ satisfying suitable conditions, returns a finite list of primes such that the corresponding representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ is surjective for any $\ell$ outside the list, generalising the approach of \cite{Dieulefait2002} for the case of genus $2$ to genus $3$. Both methods rely on Hall's surjectivity result \cite{Hall11} for the image of Galois representations attached to the torsion points of abelian varieties as the main technical tool. In our paper, however, we fix a prime $\ell\geq 5$ and give congruence conditions such that, for any genus $3$ curve $C$ satisfying them, we can ensure surjectivity of the attached Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$. We also borrow some ideas from Hall's paper \cite{Hall11}, although formally we do not make use of his results. \section{Geometric preliminaries}\label{sectionone} In this section we recall some background from algebraic geometry and fix some notations. \subsection{Hyperelliptic curves and curves of genus~3}\label{subsec:notation} A smooth geometrically connected projective curve\footnote{In this article, we will say that a \emph{curve over a field $K$} is an algebraic variety over $K$ whose irreducible components are of dimension~$1$. (In particular, a curve can be singular.)} $C$ of genus $g\geq 1$ over a field $K$ is \emph{hyperelliptic} if there exists a degree $2$ finite separable morphism from $C_{\overline K} = C\times_{K} \overline K$ to $\mathbb P^1_{\overline K}$. If~$K$ is algebraically closed or a finite field, then such a curve $C$ has a \emph{hyperelliptic equation} defined over $K$\footnote{When $K$ is not algebraically closed nor a finite field, the situation can be more complicated (cf.~\cite[Section~4.1]{lercier_ritzenthaler}).}. That is to say, the function field of $C$ is $K(x)[y]$ under the relation $y^2+h(x)y=g(x)$ with $g(x), h(x)\in K[x]$, $\deg(g(x))\in \{2g+1, 2g+2\},$ and $\deg(h(x))\leq g$. Moreover, if $\mathrm{char}(K)\neq 2$, we can take $h(x)=0$. Indeed, in that case, the conic defined as the quotient of $C$ by the group generated by the hyperelliptic involution has a $K$-rational point, hence is isomorphic to $\mathbb P^1_{K}$ (see e.g.~\cite[Section~1.3]{lercier_ritzenthaler} for more details). The curve $C$ is the union of the two affine open schemes \begin{equation*} \begin{aligned} U &=\mathrm{Spec} \left(K[x,y]/(y^2+h(x)y-g(x))\right)\quad \text{and}\\ V&=\mathrm{Spec} \left(K[t,w]/(w^2+t^{g+1}h(1/t)y-t^{2g+2}g(1/t))\right)\\\end{aligned}\end{equation*} glued along $\mathrm{Spec}(K[x,y,1/x]/(y^2+h(x)y-g(x)))$ via the identifications $x=1/t, y=t^{-g-1}w$. If $\mathrm{char}(K)\neq 2$, then any separable polynomial $g(x)\in K[x]$ of degree $2g+1$ or $2g+2$ gives rise to a hyperelliptic curve $C$ of genus $g$ defined over $K$ by glueing the open affine schemes $U$ and $V$ (with $h(x)=0$) as above. We will say that $C$ is \emph{given by the hyperelliptic equation $y^2=g(x)$.} We will also say, as in the introduction, that a polynomial in two variables is of \emph{$g$-hyperelliptic type} if it is of the form $y^2-g(x)$ with $g(x)$ a polynomial of degree $2g+1$ or $2g+2$. In this article, we are especially interested in curves of genus~$3$. If $C$ is a smooth geometrically connected projective non-hyperelliptic curve of genus $3$ defined over a field $K$, then its canonical embedding $C\hookrightarrow \mathbb P_{K}^2$ identifies $C$ with a smooth plane quartic curve defined over $K$. This means that the curve $C$ has a model over $K$ given by $\mathrm{Proj}(K[X,Y,Z]/F(X,Y,Z))$ where $F(X, Y, Z)$ is a degree~$4$ homogeneous polynomial with coefficients in $K$. Conversely, any smooth plane quartic curve is the image by a canonical embedding of a non-hyperelliptic curve of genus $3$. If this curve is $\mathrm{Proj}(K[X,Y,Z]/F(X,Y,Z))$ where $F(X, Y, Z)$ is the homogenisation of a degree~$4$ polynomial $f(x, y)\in K[x,y]$, we will say that $C$ is the \emph{quartic plane curve defined by the affine equation $f(x,y)=0$}. We will say, as in the introduction, that a polynomial in two variables is of \emph{quartic type} if its total degree is $4$. \subsection{Semistable curves and their generalised Jacobians}\label{subsec:jacobians} We briefly recall the basic notions we need about semistable and stable curves, give the definition of the intersection graph of a curve and explain the link between this graph and the structure of their generalised Jacobian. The classical references we use are essentially \cite{Liu06} and \cite{BLR90}. For a nice overview which contains other references, the reader could also consult~\cite{romagny}. A curve $C$ over a field $k$ is said to be \emph{semistable} if the curve $C_{\overline k}=C\times_{k} \overline k$ is reduced and has at most ordinary double points as singularities. It is said to be \emph{stable} if moreover $C_{\overline k}$ is connected, projective of arithmetic genus $\geq 2$, and if any irreducible component of $C_{\overline k}$ isomorphic to $\mathbb{P}^1_{\overline k}$ intersects the other irreducible components in at least three points. A proper flat morphism of schemes $\mathcal C\to S$ is said to be \emph{semistable} (resp. \emph{stable}) if it has semistable (resp. stable) geometric fibres. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $C$ be a smooth projective geometrically connected curve over $K$. A \emph{model} of $C$ over $R$ is a normal scheme $\mathcal C/R$ such that $\mathcal C\times_{R} K\cong C$. We say that $C$ has \emph{semistable reduction} (resp. \emph{stable reduction}) if $C$ has a model $\mathcal C$ over $R$ which is a semistable (resp. stable) scheme over $R$. If such a stable model exists, it is unique up to isomorphism and we call it \emph{the stable model of $C$ over $R$} (cf.~\cite[Chap.10, Definition 3.27 and Theorem 3.34]{Liu06}). If the curve $C$ has genus $g\geq 1$, then it admits a minimal regular model $\mathcal C_{min} $ over $R$, unique up to unique isomorphism. Moreover, $\mathcal C_{min}$ is semistable if and only if $C$ has semistable reduction, and if $g\geq 2$, this is equivalent to $C$ having stable reduction (cf.~\cite[Chap. 10, Theorem 3.34]{Liu06}, or \cite[Theorem 3.1.1]{romagny} when $R$ is strictly henselian). Assume that $C$ is a smooth projective geometrically connected curve of genus $g\geq 2$ over $K$ with semistable reduction. Denote by $\mathcal C$ its stable model over $R$ and by $\mathcal C_{min}$ its minimal regular model over $R$. We know that the Jacobian variety $J=\mathrm{Jac}(C)$ of $C$ admits a N\'{e}ron model $\mathcal J$ over $R$ and the canonical morphism $\mathrm{Pic}^0_{\mathcal C/R}\to \mathcal J^0$ is an isomorphism (cf.~\cite[$\S 9.7$, Corollary 2]{BLR90}). Note that since $\mathcal C_{min}$ is also semistable, we have $\mathrm{Pic}^0_{\mathcal C_{min}/R}\cong \mathcal J^0$. Moreover, the abelian variety $J$ has semistable reduction, that is to say $\mathcal J^0_{k}\cong \mathrm{Pic}^0_{\mathcal C_{ k}/k}$ is canonically an extension of an abelian variety by a torus $T$. As we will see, the structure of the algebraic group $\mathcal J^0_{ k}$ (by which we mean the toric rank and the order of the component group of its geometric special fibre) is related to the intersection graphs of $\mathcal C_{\overline k}$ and $\mathcal C_{min,\overline k}$. Let $X$ be a curve over $\overline{k}$. Consider the \emph{intersection graph} (or \emph{dual graph}) $\Gamma(X)$, defined as the graph whose vertices are the irreducible components of $X$, where two irreducible components $X_i$ and $X_j$ are connected by as many edges as there are irreducible components in the intersection $X_i\cap X_j$. In particular, if the curve $X$ is semistable, two components $X_i$ and $X_j$ are connected by one edge if there is a singular point lying on both $X_i$ and $X_j$. Here $X_i=X_j$ is allowed. The \emph{(intersection) graph without loops}, denoted by $\Gamma'(X)$, is the graph obtained by removing from $\Gamma(X)$ the edges corresponding to~$X_i=X_j$. Next, we paraphrase \cite[$\S 9.2$, Example 8]{BLR90}, which gives the toric rank in terms of the cohomology of the graph $\Gamma(\mathcal C_{\overline k})$. \begin{pr}[\cite{BLR90}, $\S 9.2$, Ex.~8]\label{BLRExactSeq} The N\'{e}ron model $\mathcal J$ of the Jacobian of the curve $\mathcal{C}_k$ has semistable reduction. More precisely, let $X_1,\ldots, X_r$ be the irreducible components of $\mathcal C_{k}$, and let $\widetilde X_1,\dots, \widetilde X_r$ be their respective normalisations. Then the canonical extension associated to $\mathrm{Pic}^0_{\mathcal C_{k}/ k}$ is given by the exact sequence \[ 1\longrightarrow T\hookrightarrow \mathrm{Pic}^0_{\mathcal C_{k}/k}\xrightarrow{\pi^*}\prod_{i=1}^r \mathrm{Pic}^0_{\widetilde X_i/k} \longrightarrow 1 \] where the morphism $\pi^*$ is induced by the morphisms $\pi_i:\widetilde X_i\longrightarrow X_i$. The rank of the torus~$T$ is equal to the rank of the cohomology group $H^1(\Gamma(\mathcal C_{\overline{k}}),\mathbb Z). $ \end{pr} We will use the preceding result in Sections~\ref{sec:2} and \ref{sec:3}. Note that the toric rank does not change if we replace $\mathcal C$ by $\mathcal C_{min}$. The intersection graph of $\mathcal C_{min,\overline k}$ also determines the order of the component group of the geometric special fibre $\mathcal J_{\overline k}$. Indeed, the scheme $\mathcal C_{min}\times R^{sh}$, where $R^{sh} $ is the strict henselisation of $R$, fits the hypotheses of \cite[$\S 9.6$, Proposition 10]{BLR90} which gives the order of the component group in terms of the graph of $\mathcal C_{min,\overline k}$; we reproduce it here for the reader's convenience. \begin{pr}[\cite{BLR90}, $\S 9.6$, Prop.~10]\label{prop:Phi} Let $X$ be a proper and flat curve over a strictly henselian discrete valuation ring $R$ with algebraically closed residue field $\overline k$. Suppose that $X$ is regular and has geometrically irreducible generic fibre as well as a geometrically reduced special fibre $X_{\overline k}$. Assume that $X_{\overline k}$ consists of the irreducible components $X_1,\dots, X_r$ and that the local intersection numbers of the $X_i$ are $0$ or $1$ (the latter is the case if different components intersect at ordinary double points). Furthermore, assume that the intersection graph without loops $\Gamma'(X_{\overline k})$ consists of $l$ arcs of edges $\lambda_1,\dots,\lambda_l$, starting at $X_1$ and ending at $X_r$, each arc $\lambda_i$ consisting of $m_{i}$ edges. Then the component group $\mathcal J(R^{sh})/\mathcal J^0(R^{sh})$ has order $\sum_{i=1}^l \prod_{j\neq i}m_{j}$. \end{pr} We will use this result in the proof of Proposition~\ref{jacobianthm}. \section{Local conditions at $p$}\label{sec:2} Let $p>2$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and by $\mathbb{Q}_p$ the field of $p$-adic numbers. \begin{defn}\label{polyn} Let $f(x,y)\in\mathbb Z_p[x,y]$ be a polynomial with $f(0,0)=0$ or $v_p(f(0,0))> 2$. We say that $f(x, y)$ is of type: \begin{enumerate} \item[(H)] if $f(x,y)=y^2-g(x)$, where $g(x)\in\mathbb Z_p[x]$ is of degree $7$ or $8$ and such that $$g(x)\equiv x(x-p)m(x)\bmod{p^2\mathbb Z_p[x]},$$ with $m(x)\in \mathbb{Z}_p[x]$ such that all the roots of its mod $p$ reduction are simple and non-zero; \item[(Q)] if $f(x,y)$ is of total degree $4$ and such that $$f(x,y)\equiv px+x^2-y^2+x^4+y^4 \bmod{p^2\mathbb Z_p[x,y]}.$$ \end{enumerate} \end{defn} For $f(x,y) \in \mathbb Z_p[x,y]$ a polynomial of type (H) or (Q), we will consider the projective curve $C$ defined by $f(x, y)=0$ as explained in Subsection~\ref{subsec:notation} and the scheme $\mathcal C$ over $\mathbb Z_p$ defined, for each case of Definition~\ref{polyn} respectively, as follows: \begin{enumerate} \item[(H)] the union of the two affine subschemes $$U=\mathrm{Spec} (\mathbb Z_p[x,y]/(y^2- g(x))) \textrm{ and } V=\mathrm{Spec}(\mathbb Z_p[t,w]/(w^2-g(1/t)t^8))$$ glued along $\mathrm{Spec}(\mathbb Z_p[x,y,1/x]/(y^2-g(x))$ via $x=1/t, y=t^{-4}w$; \item[(Q)] the scheme $\mathrm{Proj}(\mathbb Z_p[X,Y,Z]/(F(X, Y, Z)))$, where $F(X, Y, Z)$ is the homogenisation of $f(x, y)$. \end{enumerate} This scheme has generic fibre $C$. \begin{pr}\label{prop:curve} Let $f(x,y) \in \mathbb Z_p[x,y]$ be a polynomial of type (H) or (Q) and $C$ be the projective curve defined by $f(x, y)=0$. The curve $C$ is a smooth projective and geometrically connected curve of genus $3$ over $\mathbb Q_p$ with stable reduction. Moreover, the scheme $\mathcal C$ is the stable model of $C$ over $\mathbb Z_p$ and the stable reduction is geometrically integral with exactly one singularity, which is an ordinary double point. \end{pr} \begin{proof} With the description we gave in Subsection~\ref{subsec:notation} of what we called the \emph{projective curve defined by $f$}, smoothness over $\mathbb{Q}_p$ follows from the Jacobian criterion. This implies that $C$ is a projective curve of genus $3$. The polynomials defining the affine schemes $U$ and $V$ and the quartic polynomial $F(X, Y, Z)$ are all irreducible over $\overline{\mathbb Q}_p$, hence over $\mathbb Z_p$. So the curve $C$ is geometrically integral (hence geometrically irreducible and geometrically connected) and $\mathcal C$ is integral as a scheme over $\mathbb Z_p$. It follows in particular that $\mathcal C$ is flat over $\mathbb Z_p$ (cf.~\cite[Chap.~4, Corollary 3.10]{Liu06}). Hence, $\mathcal C$ is a model of $C $ over~$\mathbb Z_p$. We will show that $\mathcal{C}_{\mathbb F_p}$ is semistable (i.e.~reduced with only ordinary double points as singularities) with exactly one singularity. Combined with flatness, semistability will imply that the scheme $\mathcal C$ is semistable over $\mathbb Z_p$. Since $C$ has genus greater than $2$, and $C=\mathcal C_{\mathbb Q_p}$ is smooth and geometrically connected, this is then equivalent to saying that $C$ has stable reduction at $p$ with stable model $\mathcal C$, as required (cf.~\cite[Theorem~3.1.1]{romagny}). In what follows, we denote by $\bar{f}$ the reduction modulo $p$ of any polynomial $f$ with coefficient in~$\mathbb Z_p$. In Case (H), $\mathcal{C}_{\overline{\mathbb F}_p}$ is the union of the two affine subschemes $U'=\mathrm{Spec}({\overline{\mathbb F}_p}[x,y]/(y^2-x^2\bar m(x)))$ and $V'=\mathrm{Spec}({\overline{\mathbb F}_p}[t,w]/(w^2-\bar m(1/t)t^6))$, glued along $\mathrm{Spec}(\overline{\mathbb F}_p[x,y,1/x]/(y^2-\bar g(x))$ via $x=1/t $ and $y=t^{-4}w$ (cf.~\cite[Chap.~10, Example 3.5]{Liu06}). In Case (Q), the geometric special fibre is $\mathrm{Proj}({\overline{\mathbb F}_p}[X,Y,Z]/(\bar{F}(X, Y, Z)))$. In both cases, the defining polynomials are irreducible over ${\overline{\mathbb F}_p}$. Hence, $\mathcal{C}_{\overline{\mathbb F}_p}$ is integral, i.e.~reduced and irreducible. Next, we prove that $\mathcal C_{\overline{\mathbb F}_p}$ has only one ordinary double point as singularity. For Case (H), see e.g.~\cite[Chap.~10, Examples 3.4, 3.5 and 3.29]{Liu06}. For Case (Q), we proceed analogously: first consider the open affine subscheme of $\mathcal{C}_{\overline{\mathbb F}_p}$ defined by $U=\mathrm{Spec}(\overline{\mathbb F}_p[x,y]/\bar{f}(x,y))$, where $\bar{f}(x,y)=x^2-y^2+x^4+y^4\in \mathbb{F}_p[x, y]$. Since $\mathcal C_{\overline{\mathbb F}_p}\backslash U$ is smooth, it suffices to prove that $U$ has only ordinary double singularities. Let $u\in U$. The Jacobian criterion shows that $U$ is smooth at $u\neq (0,0)$. So suppose that $u=(0,0)$, and note that $\bar f(x,y)=x^2(1+x^2)-y^2(1-y^2)$. Since $2\in\overline\mathbb F_p^\times$, there exist $a(x)=1+xc(x)\in {\overline{\mathbb F}_p}[[x]]$ and $b(y)=1+yd(y)\in {\overline{\mathbb F}_p}[[y]]$ such that $1+x^2=a(x)^2 $ and $1-y^2=b(y)^2$, by (\cite[Chap.~1, Exercise 3.9]{Liu06}). Then we have $$\widehat{\mathcal O}_{U,u}\cong {\overline{\mathbb F}_p}[[x,y]]/(xa(x)+yb(y))(xa(x)-yb(y))\cong {\overline{\mathbb F}_p}[[t,w]]/(tw) .$$ It follows that $\mathcal C_{\overline{\mathbb F}_p}$ has only one singularity (at $[0:0:1] $) which is an ordinary double singularity. We have thus showed that $\mathcal C$ is the stable model of $C$ over $\mathbb Z_p$ and that its special fibre is geometrically integral and has only one ordinary double singularity. \end{proof} \begin{pr}\label{jacobianthm} Let $f(x,y) \in \mathbb Z_p[x,y]$ be a polynomial of type (H) or (Q) and $C$ be the projective curve defined by $f(x, y)=0$. The Jacobian variety $\mathrm{Jac}(C)$ of the curve $C$ has a N\'eron model $\mathcal J$ over $\mathbb Z_p$ which has semi-abelian reduction of toric rank $1$. The component group of the geometric special fibre of $\mathcal J$ over $\overline \mathbb F_p$ has order $2$. \end{pr} \begin{proof} By Proposition~\ref{prop:curve}, the curve $C$ is a smooth projective geometrically connected curve of genus $3$ over $\mathbb Q_p$ with stable reduction and stable model $\mathcal C$ over $\mathbb Z_p$. Let $\mathcal{C}_{min}$ be the minimal regular model of $C$. As recalled in Subsection~\ref{subsec:jacobians}, $\mathrm{Jac}(C)$ admits a N\'eron model $\mathcal J$ over $\mathbb Z_p$ and the canonical morphism $\mathrm{Pic}^0_{\mathcal C/\mathbb Z_p}\to \mathcal J^0$ is an isomorphism. In particular, $\mathcal J$ has semi-abelian reduction and $\mathcal J^0_{\mathbb F_p}\cong\mathrm{Pic}^0_{\mathcal C_{\mathbb F_p}/{\mathbb F_p}}$. Since $\mathcal C_{min}$ is also semistable, we have $\mathrm{Pic}^0_{\mathcal C_{min}/S}\cong \mathcal J^0$. By Proposition \ref{BLRExactSeq}, the toric rank of $\mathcal J^0_{\overline\mathbb F_p}$ is equal to the rank of the cohomology group of the dual graph of $\mathcal C_{\overline \mathbb F_p}$. Since $\mathcal C_{\overline \mathbb F_p}$ is irreducible and has only one ordinary double point, the dual graph consists of one vertex and one loop, so the rank of $\mathcal J^0_{\overline \mathbb F_p} $ is $1$. To determine the order of the component group of the geometric special fibre $\mathcal J_{\overline \mathbb F_p}$, we apply Proposition \ref{prop:Phi} to the minimal regular model $\mathcal C_{min}\times \mathbb Z_p^{sh}$, where $\mathbb Z_p^{sh} $ is the strict henselisation of $\mathbb Z_p$. This is still regular and semistable over $\mathbb Z_p^{sh}$ (cf.~\cite[Chap.~10, Proposition~3.15-(a)]{Liu06}). Let $e$ denote the thickness of the ordinary double point of $\mathcal C_{\overline\mathbb F_p}$ (as defined in \cite[Chap.~10, Definition3.23]{Liu06}). Then by \cite[Chap.~10, Corollary 3.25]{Liu06}, the geometric special fibre $\mathcal C_{min,\overline \mathbb F_p} $ of $\mathcal C_{min}\times \mathbb Z_p^{sh}$ consists of a chain of $e-1$ projective lines over $\mathbb F_p$ and one component of genus $2$ (where the latter corresponds to the irreducible component $\mathcal C_{\overline \mathbb F_p}$), which meet transversally at rational points. It follows from Proposition \ref{BLRExactSeq} that the order of the component group $\mathcal J(\mathbb Z_p^{sh})/\mathcal J^0(\mathbb Z_p^{sh})$ of the geometric special fibre is equal to the thickness~$e$. We will now show that in both cases (H) and (Q), the thickness $e$ is equal to $2$, which will conclude the proof of Proposition~\ref{jacobianthm}. For this, in several places, we will use the well-known fact that every formal power series in $\mathbb Z_p[[x]]$ (resp. $\mathbb Z_p[[y]]$, $\mathbb Z_p[[x,y]]$) with constant term $1$ (or more generally a unit square in $\mathbb Z_p$) is a square in $\mathbb Z_p[[x]]$ (resp. $\mathbb Z_p[[y]]$, $\mathbb Z_p[[x,y]]$) of some invertible formal power series. Let $U$ denote the affine subscheme $\mathrm{Spec}(\mathbb Z_p[x,y]/(f(x,y)))$ which contains the ordinary double point $P=[0:0:1]$. Firstly, we claim that, possibly after a finite extension of scalars $R/\mathbb Z_p$ which splits the singularity, in both cases we may write in $R[[x,y]]$: \begin{equation}\label{formf}\pm f(x,y)=x^2a(x)^2-y^2b(y)^2+p\alpha x+p^2yg(x,y)+p^{r}\beta \end{equation} where $ a(x) \in R[[x]]^\times,b(y) \in R[[y]]^\times, g(x,y)\in\mathbb Z_p[x,y], \alpha\in\mathbb Z_p^\times$, $\beta\in\mathbb Z_p$. Moreover, from the assumptions on $f$, it follows that either $\beta=0$, or $\beta \in \mathbb Z_p^\times $ and $r=v_p(f(0,0))> 2$. We prove the claim case by case: \begin{enumerate} \item[(H)] We have $f(x,y)=y^2-g(x)=y^2-x(x-p)m(x)+p^2h(x)$ for some $h(x)\in\mathbb Z_p[x]$. Since $h(x)=h(0)+xs(x)$ for some $s(x)\in\mathbb Z_p[x]$ and $m(x)+ps(x)=m(0)+p s(0)+xt(x)$ for some $t(x)\in\mathbb Z_p[x]$, we obtain \begin{eqnarray*} f(x,y) &=& y^2-x^2m(x)+px(m(x)+ps(x))+p^2h(0)\\ &=& y^2-x^2(m(x)-pt(x))+px(m(0)+ps(0))+p^2h(0). \end{eqnarray*} Since $m(0)\neq 0\pmod p,$ we have $m(0)-pt(0)\in\mathbb Z_p^\times, $ hence if we extend the scalars to some finite extension $R$ over $\mathbb Z_p$, in which $m(0)-pt(0)$ is a square, we get that $(m(x)-pt(x))$ is a square of some $a(x)$ in $R[[x]]^\times$. Then $-f(x,y)$ has the expected form. Note that $R/\mathbb Z_p$ is unramified because $p\neq 2$ and $m(0)\neq 0\pmod p$, so we still denote the ideal of $R$ above $p\in\mathbb Z_p$ by $p$. \item[(Q)] We have $f(x,y)=x^4+y^4+x^2-y^2+px+p^2h(x,y)$ for some $h(x,y)\in\mathbb Z_p[x,y]$. We may write $h(x,y)=\delta+x\gamma+x^2s(x)+yt(x,y)$ for some $\gamma,\delta\in\mathbb Z_p$, $s(x) \in\mathbb Z_p[x]$ and $t(x,y)\in\mathbb Z_p[x,y]$. We obtain \begin{equation*}\begin{aligned} f(x,y)&=x^2(1+x^2)-y^2(1-y^2)+px+p^2(\delta+x\gamma+x^2s(x)+yt(x,y))\\ & = x^2(1+x^2+p^2 s(x))-y^2(1-y^2)+px(1+p\gamma)+p^2yt(x,y)+p^2\delta. \end{aligned}\end{equation*} Since $1+x^2+p^2 s(x)$ and $1-y^2$ have constant terms which are squares in $\mathbb Z_p^\times$, the formal power series are squares in $\mathbb Z_p[[x]]$, resp. $\mathbb Z_p[[y]]$. So $f(x,y)$ again has the desired form. \end{enumerate} Next, we show that $e=2$ for $\pm f(x,y)$ of the form \eqref{formf}. In $R[[x,y]]$, we have $$\pm f(x,y)=\left(xa(x)+p\frac{\alpha}{2a(x)}\right)^2-\left(yb(y)-p^2\frac{g(x,y)}{2b(y)}\right)^2+p^2c(x,y),$$ where $c(x,y)=p^{r-2}\beta-\frac{\alpha^2}{4a(x)^2}+p^2\frac{g(x,y)^2}{4b(y)^2}$. Since either $\beta=0$ or $r>2$ and $\frac{\alpha^2}{4a(0)^2}\not \equiv 0\pmod p$, the constant term $\gamma$ of the formal power series $c(x, y)$ belongs to $R^\times$. It follows that $\gamma^{-1}c(x,y)$ is the square of some formal power series $d(x,y)\in R[[x,y]]^\times$. Defining the variables $$u=\frac{xa(x)}{d(x,y)}+p\frac{\alpha}{2a(x)d(x,y)}-\frac{yb(y)}{d(x,y)}+p^2\frac{g(x,y)}{2b(y)d(x,y)}$$ and $$v= \frac{xa(x)}{d(x,y)}+p\frac{\alpha}{2a(x)d(x,y)}+\frac{yb(y)}{d(x,y)}-p^2\frac{g(x,y)}{2b(y)d(x,y)},$$ we get $ \widehat O_{U\times R,P}\cong R[[u,v]]/(uv\pm p^2\gamma)$. Since $ \gamma\in R^\times$, it follows that $e=2$. \end{proof} \section{Local conditions at $q$}\label{sec:3} This section is devoted to the proof of the following key result. In the statement, the two conditions on the characteristic polynomial, namely non-zero trace and irreducibility modulo $\ell$, are the ones appearing in Theorem~2.10 of~\cite{AAKRTV14} which is used to prove the main Theorem~\ref{thm:main}. \begin{thm}\label{A} Let $\ell \geq 13$ be a prime number. For every prime number $q>1.82\ell^2$, there exists a smooth geometrically connected curve $C_q$ of genus $3$ over $\mathbb F_q$ whose Jacobian variety $\mathrm{Jac}(C_q)$ is a $3$-dimensional ordinary absolutely simple abelian variety such that the characteristic polynomial of its Frobenius endomorphism is irreducible modulo $\ell$ and has non-zero trace modulo $\ell$. Moreover, for $\ell\in\{3,5,7,11\}$, there exists a prime number $q>1.82\ell^2$ such that the same statement holds. \end{thm} For any integer $g \geq 1$, a $g$-dimensional abelian variety over a finite field $k$ with $q$ elements is said to be \emph{ordinary} if its group of $\mathrm{char}(k)$-torsion points has rank $g$. The proof of Theorem~\ref{A} relies on Honda-Tate theory, which relates abelian varieties to Weil polynomials: \begin{defn}\label{weilpol} A \emph{Weil $q$-polynomial}, or simply a \emph{Weil polynomial}, is a monic polynomial $P_q(X) \in \mathbb Z[X]$ of even degree $2g$ whose complex roots are all \emph{Weil $q$-numbers}, i.e., algebraic integers with absolute value $\sqrt{q}$ under all of their complex embeddings. Moreover, a Weil $q$-polynomial is said to be \emph{ordinary} if its middle coefficient is coprime to $q$. \end{defn} In particular, for $g=3$, every Weil $q$-polynomial of degree~6 is of the form \begin{equation*} P_q(X)=X^6+aX^5+bX^4+cX^3+qbX^2+q^2aX+q^3 \end{equation*} for some integers $a$, $b$ and $c$ (cf.~\cite[Proposition 3.4]{Howe95}). Such a Weil polynomial is ordinary if, moreover, $c$ is coprime to $q$. Conversely, not every polynomial of this form is a Weil polynomial. However, we will prove in Proposition~\ref{boundqweil} that for $q > 1.82 \ell^2$, every polynomial as above with $|a|,|b|,|c| <\ell$ is a Weil $q$-polynomial. As an important example, the characteristic polynomial of the Frobenius endomorphism of an abelian variety over $\mathbb{F}_q$ is a Weil $q$-polynomial, by the Riemann hypothesis as proven by Deligne.\\ A variant of the Honda-Tate Theorem (cf.~\cite[Theorem~3.3]{Howe95}) states that the map which sends an ordinary abelian variety over $\mathbb F_q$ to the characteristic polynomial of its Frobenius endomorphism induces a bijection between the set of isogeny classes of ordinary abelian varieties of dimension $g \geq 1$ over $\mathbb F_q$ and the set of ordinary Weil $q$-polynomials of degree $2g$. Moreover, under this bijection, isogeny classes of simple ordinary abelian varieties correspond to irreducible ordinary Weil $q$-polynomials.\\ Hence, the proof of Theorem~\ref{A} consists in proving the existence of an irreducible ordinary Weil $q$-polynomial of degree 6 which gives rise to an isogeny class of simple ordinary abelian varieties of dimension 3. By Howe (cf.~\cite[Theorem~1.2]{Howe95}), such an isogeny class contains a principally polarised abelian variety $A$ over $\mathbb F_q$, which is the Jacobian variety of some curve $C_q$ defined over $\overline{\mathbb F}_q$ by results due to Oort and Ueno. If this abelian variety $A$ is moreover absolutely simple, the curve is geometrically irreducible and we can conclude by a Galois descent argument. Thus, it is a natural question whether the Weil $q$-polynomial determines if the abelian varieties in the isogeny class are absolutely simple. In \cite{HoweZhu02}, Howe and Zhu give a sufficient condition for an abelian variety over a finite field to be absolutely simple; for ordinary varieties, this condition is also necessary. Let $A$ be a simple abelian variety over a finite field, $\pi$ its Frobenius endomorphism and $m_A(X)\in\mathbb{Z}[X]$ the minimal polynomial of $\pi$. Since $A$ is simple, the subalgebra $\mathbb Q(\pi)$ of $\mathrm{End}(A)\otimes \mathbb Q$ is a field; it contains a filtration of subfields $\mathbb Q(\pi^d)$ for $d>1$. If moreover $A$ is ordinary, then the fields $\mathrm{End}(A)\otimes \mathbb Q=\mathbb Q(\pi) $ and $\mathbb Q(\pi^d)$ $(d>1)$ are all CM-fields, i.e., totally imaginary quadratic extensions of a totally real field. A slight reformulation of Howe and Zhu's criterion is the following (see Proposition~3 and Lemma~5 of \cite{HoweZhu02}): \begin{pr}[Howe-Zhu criterion for absolute simplicity]\label{howe-zhu} Let $A$ be a simple abelian variety over a finite field $k$. If $\mathbb Q(\pi^d)=\mathbb Q(\pi)$ for all integers $d>0$, then $A$ is absolutely simple. If $A$ is ordinary, then the converse is also true, and if $\mathbb Q(\pi^d)\neq\mathbb Q(\pi)$ for some $d>0$, then $A$ splits over the degree $d$ extension of $k$. Moreover, if $\mathbb Q(\pi^d)$ is a proper subfield of $\mathbb Q(\pi)$ such that $\mathbb Q(\pi^r)=\mathbb Q(\pi)$ for all $r<d$, then either $m_A(X)\in\mathbb Z[X^d] $, or $\mathbb Q(\pi)=\mathbb Q(\pi^d,\zeta_d)$ for a primitive $d$-th root of unity $\zeta_d$. \end{pr} From this criterion, Howe and Zhu give elementary conditions for a simple $2$-dimensional abelian variety to be absolutely simple, see~\cite[Theorem~6]{HoweZhu02}. Elaborating on their criterion and inspired by \cite[Theorem~6]{HoweZhu02}, we prove the following for dimension 3: \begin{pr}\label{abssimple} Let $A$ be an ordinary simple abelian variety of dimension $3$ over a finite field $k$ of odd cardinality $q$. Then either $A$ is absolutely simple or the characteristic polynomial of the Frobenius endomorphism of $A$ is of the form $X^6+cX^3+q^3$ with $c$ coprime to $q$ and $A$ splits over the degree~$3$ extension of $k$. \end{pr} \begin{proof} Let $A$ be an ordinary simple but not absolutely simple abelian variety of dimension $3$ over~$k$. Since $A$ is simple, the characteristic polynomial of $\pi$ is $m_A(X)$. We apply Proposition~\ref{howe-zhu} to $A$: Let $d$ be the smallest integer such that $\mathbb Q(\pi^d)\neq\mathbb Q(\pi)$. Either $m_A(X)\in\mathbb Z[X^d]$ or there exists a $d$-th root of unity $\zeta_d$ such that $\mathbb Q(\pi)=\mathbb Q(\pi^d,\zeta_d)$. We will prove by contradiction that $m_A(X)\in\mathbb Z[X^d]$. Since $m_A(X)$ is ordinary, the coefficient of degree $3$ is non-zero, and it will follow that $d=3$ and that $m_A(X)$ has the form $X^6+cX^3+q^3$, proving the proposition. So, suppose that $m_A(X)\not\in\mathbb Z[X^d]$. The field $K=\mathbb Q(\pi)=\mathbb Q(\pi^d,\zeta_d)$ is a CM-field of degree $6$ over $\mathbb Q$, hence its proper CM-subfield $L=\mathbb Q(\pi^d)$ has to be a quadratic imaginary field. It follows that $\phi(d)=3$ or $6$, where $\phi$ denotes the Euler totient function. However, $\phi(d)=3$ has no solution, so we must have $\phi(d)=6$, i.e. $d\in\{7,9,14,18\}$, and $K=\mathbb Q(\zeta_d)$. Note that $\mathbb Q(\zeta_7)=\mathbb Q(\zeta_{14})$ and $\mathbb Q(\zeta_9)=\mathbb Q(\zeta_{18})$, and they contain only one quadratic imaginary field; namely, $\mathbb Q(\sqrt{-7})$ for $d=7 $ (resp. $14$), and $\mathbb Q(\sqrt{-3})$ for $d=9$ (resp. $d=18$) (cf.~\cite{washington}). Let $\sigma$ be a generator of the (cyclic) group $\mathrm{Gal}(K/L)$ of order~$3$. In their proof of \cite[Lemma 5]{HoweZhu02}, Howe and Zhu show that we can choose $\zeta_d$ such that $\pi^\sigma=\zeta_d \pi$. Moreover, $\zeta_d^\sigma=\zeta_d^k$ for some integer $k$ (which can be chosen to lie in $[0,d-1]$). Since $\sigma$ is of order $3$, we have $\pi=\pi^{\sigma^3}=\zeta_d^{(k^2+k+1)}\pi$, which gives $k^2+k+1 \equiv 0\pmod d$. This rules out the case $d=9$ and $18$, because $-3$ is neither a square modulo $9$ nor a square modulo $18$. So $d=7$ or~$14$, $K=\mathbb Q(\zeta_7)$ and $\mathbb Q(\pi^d)=\mathbb Q(\sqrt{-7})$. It follows that the characteristic polynomial of $\pi^d$, which is of the form \[ X^6+\alpha X^5+\beta X^4+\gamma X^3+\beta q^d X^2+\alpha q^{2d} X+q^{3d}\in\mathbb Z[X], \] is the cube of a quadratic polynomial of discriminant $-7$. This is true if and only if \begin{equation*}\label{quadequ} \alpha^2-36q^d+63=0,\quad\alpha^2-3\beta+9q^d=0\quad\mbox{and}\quad \alpha^3-27\gamma+54\alpha q^d=0, \end{equation*} that is, \begin{equation*}\label{simplequadequ} \alpha^2=9(4q^d-7),\quad \beta=3(5q^d-7)\quad\mbox{and}\quad 3\gamma=\alpha(10q^d-7). \end{equation*} However, the first equation has no solution in $q$. Indeed, suppose that $4q^d-7$ is a square, say $u^2$ for some integer $u$. Then $u$ is odd, say $u=1+2t$ for some integer $t$, hence $4q^d=8+4t(t+1)$, so $2$ divides $q$, which contradicts the hypothesis. Hence, we obtain that $m_A(X)\in\mathbb Z[X^d]$ and Proposition~\ref{abssimple} follows. \end{proof} Finally, the proof of Theorem~\ref{A} relies on Proposition~\ref{abssimple} and the following proposition, whose proof consists on counting arguments and is postponed to Section~\ref{sectionirred}: \begin{pr}\label{irredmodell} For any prime number $\ell\geq 13$ and any prime number $q>1.82\ell^2$, there exists an ordinary Weil $q$-polynomial $P_q(X)=X^6+aX^5+bX^4+cX^3+qbX^2+q^2aX+q^3$, with $a\not\equiv 0\pmod \ell$, which is irreducible modulo $\ell$. For $\ell\in\{3,5,7,11\}$, there exists some prime number $q>1.82\ell^2$ and an ordinary Weil $q$-polynomial as above. Moreover, for all $\ell \geq 3$, the coefficients $a,b,c$ can be chosen to lie in $\mathbb Z\cap [-(\ell-1)/2,(\ell-1)/2]$. \end{pr} \begin{rmk}\label{remirred} Computations suggest that for $\ell\in\{5,7,11\}$ and \emph{any} prime number $q > 1.82\ell^2$, there still exist integers $a,b,c$ such that Proposition~\ref{irredmodell} holds. For $\ell=3$, this is no longer true: our computations indicate that if $q$ is such that $\legendre q\ell=-1$, then there are no suitable $a,b,c$, while if $q$ is such that $\legendre q\ell=1$, they indicate that there are $4$ suitable triples $(a,b,c)$. \end{rmk} We now have all the ingredients to prove Theorem~\ref{A}. \begin{proof}[Proof of Theorem~\ref{A}.] Let $\ell$ and $q$ be two distinct prime numbers as in Proposition~\ref{irredmodell} and let $P_q(X)$ be an ordinary Weil $q$-polynomial provided by this proposition. Since the polynomial $P_q(X)$ is irreducible modulo $\ell$, it is a fortiori irreducible over $\mathbb Z$. It is also ordinary and of degree $6$. Hence, by Honda-Tate theory, it defines an isogeny class $\mathcal A$ of ordinary simple abelian varieties of dimension $3$ over $\mathbb F_q$. By Proposition~\ref{abssimple}, since $a\neq 0$, the abelian varieties in $\mathcal A$ are actually absolutely simple. Moreover, according to Howe (cf.~\cite[Theorem~1.2]{Howe95}), $\mathcal A$ contains a principally polarised abelian variety $(A,\lambda)$. Now, by the results of Oort-Ueno (cf.~\cite[Theorem~4]{OortUeno73}), there exists a so-called good curve $C$ defined over $\overline{\mathbb F}_q$ such that $(A,\lambda)$ is $\overline{\mathbb F}_q$-isomorphic to $(\mathrm{Jac}(C),\mu_0)$, where $\mu_0$ denotes the canonical polarisation on $\mathrm{Jac}(C)$. A curve over $\overline{\mathbb F}_q$ is a \emph{good curve} if it is either irreducible and non-singular or a non-irreducible stable curve whose generalised Jacobian variety is an abelian variety (cf.~\cite[Definition (13.1)]{Howe95}). In particular, the curve $C$ is stable, and so semi-stable. Since the generalised Jacobian variety $\mathrm{Jac}(C)\cong\mathrm{Pic}^0_{C}$ is an abelian variety, the torus appearing in the short exact sequence of Proposition~\ref{BLRExactSeq} is trivial. Hence, there is an isomorphism $\mathrm{Jac}(C) \cong \prod_{i=1}^r \mathrm{Pic}^0_{\widetilde{X_i}}$, where $\widetilde{X_1},\ldots,\widetilde{X_r}$ denote the normalisations of the irreducible component of $C$ over $\overline{\mathbb F}_q$. Since $\mathrm{Jac}(C)$ is absolutely simple, we conclude that $r=1$, i.e., the curve $C$ is irreducible, hence smooth. We can therefore apply Theorem~9 of the appendix by Serre in~\cite{Lauter01} (see also the reformulation in \cite[Theorem~1.1]{Ritzenthaler10}) and conclude that the curve $C$ descends to $\mathbb F_q$. Indeed, there exists a smooth and geometrically irreducible curve $C_q$ defined over $\mathbb F_q$ which is isomorphic to $C$ over $\overline{\mathbb F}_q$. Moreover, either $(A,\lambda)$ or a quadratic twist of $(A,\lambda)$ is isomorphic to $(\mathrm{Jac}(C_q), \mu)$ over $\mathbb F_q$, where $\mu$ denotes the canonical polarisation of $\mathrm{Jac}(C_q)$. The characteristic polynomial of $\mathrm{Jac}(C_q)$ is $P_q(X)$ or $P_q(-X)$, since the twist may replace the Frobenius endomorphism with its negative. Note that the polynomial $P_q(-X)$ is still an ordinary Weil polynomial which is irreducible modulo $\ell$ with non-zero trace, and $\mathrm{Jac}(C_q)$ is still ordinary and absolutely simple. This proves Theorem~\ref{A}. \end{proof} \begin{rmk}\label{sek} In the descent argument above, the existence of a non-trivial quadratic twist may occur in the non-hyperelliptic case only. This obstruction for an abelian variety over $\overline{\mathbb{F}}_q$ to be a Jacobian over $\mathbb F_q$ was first stated by Serre in a Harvard course~\cite{Serre85}; it was derived from a precise reformulation of Torelli's theorem that Serre attributes to Weil~\cite{Weil57}. Note that Sekiguchi investigated the descent of the curve in~\cite{Sekiguchi81} and~\cite{Sekiguchi86}, but, as Serre pointed out to us, the non-hyperelliptic case was incorrect. According to MathSciNet review MR1002618 (90d:14032), together with Sekino, Sekiguchi corrected this error in~\cite{SekiSeki88}. \end{rmk} \section{Proof of the main theorem}\label{sec:4} The goal of this section is to prove Theorem \ref{thm:main}, by collecting together the results from Sections \ref{sec:2} and \ref{sec:3}. We keep the notation introduced in Subsection \ref{subsec:notation}; in particular, we will consider genus $3$ curves defined by polynomials which are of $3$-hyperelliptic or quartic type. We will prove the following refinement of Theorem \ref{thm:main}: \begin{thm}\label{thm:refined} Let $\ell\geq 13$ be a prime number. For each prime number $q>1.82 \ell^2$, there exists $\bar{f}_q(x, y)\in \mathbb{F}_q[x, y]$ of $3$-hyperelliptic or quartic type, such that if $f(x, y)\in \mathbb{Z}[x, y]$ is a lift of $\bar{f}_q(x, y)$, of the same type, satisfying the following two conditions for some prime number $p\not\in\{2, q, \ell\}$: \begin{enumerate} \item $f(0, 0)=0$ or $v_p(f(0, 0))>2$; \item $f(x,y)$ is congruent modulo $p^2$ to: $$\begin{cases} y^2 - x(x - p)m(x) &\text{ if } \bar{f}_q(x, y) \text{ is of hyperelliptic type}\\ x^4 + y^4 + x^2 -y^2 + px &\text{ if } \bar{f}_q(x, y) \text{ is of quartic type}\\ \end{cases}$$ for some $m(x) \in \mathbb{Z}_p[x]$ of degree 5 or 6 with simple non-zero roots modulo $p$; \end{enumerate} \noindent then the projective curve $C$ defined over $\mathbb{Q}$ by the equation $f(x, y)=0$ is a smooth projective geometrically irreducible genus $3$ curve, such that the image of the Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ attached to the $\ell$-torsion of $\mathrm{Jac}(C)$ coincides with $\mathrm{GSp}_6(\mathbb{F}_{\ell})$. Moreover, if $\ell\in \{5, 7, 11\}$, the statement is true, replacing ``For each prime number $q$'' by ``There exists an odd prime number $q$''. \end{thm} \begin {rmk}\label{rem:CRT} Let $\ell\geq 5$ be a prime number. Note that it is easy to construct infinitely many polynomials $f(x,y)$ satisfying the conclusion of Theorem~\ref{thm:refined}: choose a polynomial $f_p(x, y)$ satisfying the conditions in Definition~\ref{polyn}. Choose a prime $q>1.82 \ell^2$, and find a polynomial $\bar{f}_q(x,y)$ that satisfies the conditions in Proposition \ref{irredmodell} (e.g.~by a computer search based on the method suggested after Theorem 0.1). Then it suffices to choose each coefficient of $f(x, y)$ as a lift of the corresponding coefficient of $\bar{f}_q(x,y)$ to an element of $\mathbb{Z}$, which is congruent mod $p^3$ to the corresponding coefficient of $f_p(x, y)$. This also proves that Theorem~\ref{thm:main} follows from Theorem~\ref{thm:refined}. \end {rmk} \begin{exmp}\label{ex:mainthm} \begin{enumerate} \item For $\ell=13$, we choose $p=7$, $q=313$. A computer search produces the polynomial $\bar{f}_q(x,y)=y^2-(x^7+x-1)$, which defines a hyperelliptic genus $3$ curve over $\mathbb{F}_q$. Let $f_p(x,y)=y^2-x(x-7)(x-1)(x-2)(x-3)(x-4)(x-5)$. Using the Chinese Remainder Theorem we construct the hyperelliptic curve over $\mathbb{Q}$ with equation $f(x,y)=0$, where \begin{multline*} f(x,y)=y^2 -( x^7-14085 x^6 + 33804x^5 -27231 x^4 \\ + 27231x^3 -35995 x^2 -33803x + 25039). \end{multline*} \item For $\ell=5$, we choose $p=3$, $q=97$. Through a computer search we find the quartic polynomial $\bar{f}_q(x,y)=x^4 + y^3+ x^3 y + x y^2 + 1\in\mathbb{F}_q[x, y]$. Take $f_p(x,y)=x^4+y^4+x^2-y^2+3x$. Then we obtain the plane quartic curve over $\mathbb{Q}$ with equation $f(x,y)=0$, where \begin{equation*} f(x,y)=x^4 + 486 x^3 y + y^4 + 486 x y^2 - 485 x^2 + 485 y^2 - 1455 x + 486. \end{equation*} \end{enumerate} \end{exmp} The rest of the section is devoted to the proof of Theorem \ref{thm:refined}. For the convenience of the reader, we recall the contents of Theorem 3.10 from \cite{AAKRTV14}: Let $A$ be a principally polarised $n$-dimensional abelian variety defined over $\mathbb{Q}$. Assume that $A$ has semistable reduction of toric rank $1$ at some prime number $p$. Denote by $\Phi_p$ the group of connected components of the N\'eron model of $A$ at $p$. Let $q$ be a prime of good reduction of $A$ and $P_q(X)=X^{2n} + aX^{2n-1} + \cdots + q^n\in \mathbb{Z}[X]$ the characteristic polynomial of the Frobenius endomorphism acting on the reduction of $A$ at $q$. Then for all primes $\ell$ which do not divide $6pqa\vert \Phi_p\vert $ and such that the reduction of $P_q(X)$ mod $\ell$ is irreducible in $\mathbb{F}_{\ell}$, the image of $\overline{\rho}_{A, \ell}$ coincides with $\GSp_{2n}(\mathbb{F}_{\ell})$. \begin{proof}[Proof of Theorem \ref{thm:refined}] Fix a prime $\ell\geq 5$. Let $q$ and $C_{q}$ be a prime, respectively a genus $3$ curve over $\mathbb{F}_{q}$, provided by Theorem~\ref{A}. The curve $C_q$ is either a plane quartic or a hyperelliptic curve. More precisely, it is defined by an equation $\bar{f}_q(x, y)=0$, where $\bar{f}_q(x, y)\in \mathbb{F}_q[x, y]$ is a quartic type polynomial in the first case and a $3$-hyperelliptic type polynomial otherwise (cf.~Subsection \ref{subsec:notation}). Note that if $f(x, y)\in\mathbb{Z}[x, y]$ is a quartic (resp.~$3$-hyperelliptic type) polynomial which reduces to $\bar{f}_q(x, y)$ modulo $q$, then it defines a smooth projective genus $3$ curve over $\mathbb{Q}$ which is geometrically irreducible. Let now $p\not\in \{2, q, \ell\}$ be a prime. Assume that $f(x, y)\in \mathbb{Z}[x, y]$ is a polynomial of the same type as $\bar{f}_q(x,y)$ which is congruent to $\bar{f}_q(x, y)$ modulo $q$ and also satisfies the two conditions of the statement of Theorem \ref{thm:refined} for this $p$. We claim that the curve $C$ defined over $\mathbb{Q}$ by the equation $f(x, y)=0$ satisfies all the conditions of the explicit surjectivity result of (\cite[Theorem 3.10]{AAKRTV14}). Namely, Proposition \ref{prop:curve} implies that $C$ is a smooth projective geometrically connected curve of genus $3$ with stable reduction. Moreover, according to Proposition~\ref{jacobianthm}, the Jacobian $\mathrm{Jac}(C)$ is a principally polarised $3$-dimensional abelian variety over $\mathbb{Q}$, and its N\'eron model has semistable reduction at $p$ with toric rank equal to $1$. Furthermore, the component group $\Phi_p$ of the N\'eron model of $\mathrm{Jac}(C)$ at $p$ has order $2$. Finally, by the choice of $q$ and $C_q$ provided by Theorem~\ref{A}, $q$ is a prime of good reduction of $\mathrm{Jac}(C)$ such that the Frobenius endomorphism of the special fibre at $q$ has Weil polynomial $P_q(X)= X^6 + a X^5 + b X^4 + c X^3 + qbX^2 + q^2aX + q^3$, which is irreducible modulo $\ell$. Since the prime $\ell$ does not divide $6pqa\vert \Phi_p\vert $, we conclude that the image of the Galois representation $\overline{\rho}_{\mathrm{Jac}(C), \ell}$ attached to the $\ell$-torsion of $\mathrm{Jac}(C)$ coincides with $\mathrm{GSp}_6(\mathbb{F}_{\ell})$ by Theorem~3.10 from \cite{AAKRTV14}. \end{proof} \section{Counting irreducible Weil polynomials of degree~$6$}\label{sectionirred} In this section, we will prove Proposition~\ref{irredmodell} stated in Section~\ref{sec:3}. At the end of the section we present some examples. This proof is based on Proposition~\ref{boundqweil} as well as Lemmas \ref{nnsquaresix} and \ref{redsix} below. Let $\ell$ and $q$ be distinct prime numbers. Consider a polynomial of the form \begin{equation}\label{weilpolyn} P_q(X)=X^6+a X^5+bX^4+cX^3+qbX^2+q^2aX+q^3 \in\mathbb Z[X]. \tag{$\ast$} \end{equation} Proposition~\ref{boundqweil} ensures that for $q\gg\ell^2$, every polynomial \eqref{weilpolyn} with coefficients in $]-\ell,\ell[$ is a Weil polynomial. Then Lemmas~\ref{nnsquaresix} and \ref{redsix} allow us to show that the number of such polynomials which are irreducible modulo $\ell$ is strictly positive. \begin{pr}\label{boundqweil}Let $\ell$ and $q$ be two prime numbers. \begin{enumerate} \item Suppose that $q>1.67\ell^2$. Then every polynomial $$X^4+uX^3+vX^2+uqX+q^2 \in \mathbb Z[X]$$ with integers $u,v$ of absolute value $<\ell$ is a Weil $q$-polynomial. \item Suppose that $q>1.82\ell^2$. Then every polynomial \begin{equation*} P_q(X)=X^6+a X^5+bX^4+cX^3+qbX^2+q^2aX+q^3 \in\mathbb Z[X], \end{equation*} with integers $a,b,c$ of absolute value $<\ell$, is a Weil $q$-polynomial. \end{enumerate} \end{pr} \begin{rmk} The power in $\ell$ is optimal, but the constants $1.67$ and $1.82$ are not. \end{rmk} Let $D_6^{*-}$ be the number of polynomials of the form $P_q(X)=X^6+a X^5+bX^4+cX^3+qbX^2+q^2aX+q^3 \in\mathbb Z[X]$ with $a,b,c$ in $[-(\ell-1)/2,(\ell-1)/2]$, $a,c\neq 0$ and whose discriminant $\Delta_{P_q}$ is not a square modulo $\ell$, and $R_6$ the number of such polynomials which are Weil polynomials and are reducible modulo $\ell$. Denoting by $\legendre{.}{\ell}$ the Legendre symbol, we have: \begin{lm}\label{nnsquaresix} Let $\ell>3,$ then $D_6^{*-}\geq \frac 12 (\ell-1)^2\left(\ell-1-\legendre q\ell\right) +\frac 12 (\ell-1)\legendre q\ell\left(1-\legendre{-1}\ell\right)-\ell(\ell-1).$ \end{lm} \begin{lm}\label{redsix} Let $\ell>3,$ then $R_6\leq \frac 38\ell^3 - \frac 5 8\ell^2\legendre q\ell - \ell^2 + \frac 32\ell\legendre q\ell + \frac 58\ell - \frac 38\legendre q\ell - \frac 12$. \end{lm} We postpone the proofs of Proposition~\ref{boundqweil} as well as Lemmas \ref{nnsquaresix} and \ref{redsix} to the following subsections but now use those statements to prove Proposition~\ref{irredmodell}. Before that, let us recall a result of Stickelberger, as proven by Carlitz in \cite{Carlitz}, which will also be useful for proving Lemmas~\ref{nnsquaresix} and~\ref{redsix}: For any monic polynomial $P(X)$ of degree $n$ with coefficients in $\mathbb Z$, and any odd prime number $\ell$ not dividing its discriminant $\Delta_P$, the number $s$ of irreducible factors of $P(X)$ modulo $\ell$ satisfies \begin{equation}\label{eq:stickelberger} \left( \frac{\Delta_P}{\ell} \right) = (-1)^{n-s}. \end{equation} \begin{proof}[Proof of Proposition~\ref{irredmodell}] Let $\ell>3$ be a prime number. It follows from Stickelberger's result that if $P_q(X)$ as in \eqref{weilpolyn} is irreducible modulo $\ell$, then $\left( \frac{\Delta_{P_q}}{\ell} \right) = -1.$ Hence by Proposition~\ref{boundqweil}, when $q>1.82\ell^2$, we find that $(D^{*-}_6-R_6)$ is exactly the number of degree $6$ ordinary Weil polynomials which have non-zero trace modulo $\ell$ and are irreducible modulo $\ell$. By Lemmas~\ref{nnsquaresix} and \ref{redsix}, we have $$ D_6^{*-}-R_6\geq \frac 18 \ell^3 + \frac 18 \ell^2\legendre{q}\ell - \frac 12\ell\legendre{-q}\ell - \frac 32\ell^2 + \frac 12\legendre{-q}\ell+ \frac{15}8\ell - \frac 58\legendre q\ell,$$ which is strictly positive for all $q$, provided that $\ell\geq 13.$ For $\ell=3,5,7$ or $11$, direct computations of $(D_6^{*-}-R_6)$ using \textsc{Sage} show that $q=19$ for $\ell=3$, $q= 47$ for $\ell = 5,\; q=97 $ for $\ell=7, \; q=223$ for $\ell=11$ will answer to the conditions of Proposition~\ref{irredmodell}. Actually, computations indicate that for $\ell=5,7,11,$ $(D_6^{*-}-R_6)$ should be strictly positive for any prime number $q$ and for $\ell=3$, it should be strictly positive for all prime numbers $ q$ which are not squares modulo $\ell$ (see Remark~\ref{remirred}). \end{proof} \subsection{Proof of Proposition~\ref{boundqweil}} Recall that $\ell$ and $q$ are two prime numbers. We first consider degree $4$ polynomials. One can prove that a polynomial $X^4+uX^3+vX^2+uqX+q^2 \in \mathbb Z[X]$ is a $q$-Weil polynomial if and only if the integers $u,v$ satisfy the following inequalities: \begin{enumerate}[(1)] \item\label{ineq1} $|u|\leq 4\sqrt{q}$, \item\label{ineq2} $2|u|\sqrt q-2q\leq v\leq \frac{u^2}4+2q$. \end{enumerate} Let $q>1.67\ell^2 $ and $Q(X)=X^4+uX^3+vX^2+uqX+q^2 \in \mathbb Z[X]$ with $|u|<\ell,|v|<\ell$. Then $q \geq \frac 1{16}\ell^2$ and, since $\ell\geq 2$, we have $q\geq \frac 14\ell^2\geq \frac12\ell$ so \eqref{ineq1} and the right hand side inequality in \eqref{ineq2} are satisfied. Finally, $q\geq \left(1+\frac{1}{2\sqrt 3}\right)^2\ell^2$ so $\sqrt q\geq \left(1+\frac 1{2\sqrt q}\right)\ell$ and the left hand side inequality in \eqref{ineq2} is satisfied. This proves that $Q(X)$ is a Weil polynomial and the first part of the proposition. Now we turn to degree $6$ polynomials. The proof is similar to the degree $4$ case. According to Haloui \cite[Theorem~1.1]{Haloui10}, a degree $6$ polynomial of the form \eqref{weilpolyn} is a Weil polynomial if its coefficients satisfy the following inequalities: \begin{enumerate}[(1)] \item\label{condhaloui1} $|a|<6\sqrt q$, \item\label{condhaloui2} $4\sqrt q |a|-9q<b\leq \frac{a^2}3+3q$, \item\label{condhaloui3} $-\frac{2a^3}{27}+\frac{ab}3+qa-\frac{2}{27}(a^2-3b^2+9q)^{\frac 32} \leq c \leq -\frac{2a^3}{27}+\frac{ab}3+qa + \frac{2}{27}(a^2-3b^2+9q)^{\frac 32} $, \item\label{condhaloui4} $-2qa-2\sqrt q b-2q\sqrt q <c<-2qa+2\sqrt qb+2q\sqrt q$. \end{enumerate} Let $q>1.82\ell^2$ and $P_q(X)$ a polynomial of the form \eqref{weilpolyn} with $|a|,|b|,|c|<\ell$. Then we note: \begin{itemize} \item We have $q>\frac{1}{36}\ell^2$, so $\ell < 6\sqrt q$ and \eqref{condhaloui1} is satisfied. \item The right hand side inequality of \eqref{condhaloui2} is satisfied since $\ell\leq 3q$. Moreover we have $q>(1+\sqrt{17/8})\ell^2 \geq 4\ell^2(1+\sqrt{1+9/4\ell})^2/81$. Hence $9q-4\ell\sqrt q-\ell>0$ and the left hand inequality of \eqref{condhaloui2} is satisfied. \item A sufficient condition to have both inequalities in \eqref{condhaloui3} is $$ 2\ell^3+9\ell^2+27q\ell-2(-3\ell^2+9q)^{3/2}+27\ell\leq 0. $$ A computation shows that this inequality is equivalent to $A\leq B$, with \begin{align*} A=\ell^6\left(\frac{28}{729}+\frac{1}{81\ell}+\frac {7}{108\ell^2}+\frac 1{6\ell^3}+\frac1{4\ell^4}\right) \mbox{ and } B= q^3\left(1-\frac54\frac{\ell^2}{q}+\frac{\ell^4}{q^2}\left(\frac 8{27}-\frac 1{6\ell}-\frac{1}{2\ell^2}\right)\right). \end{align*} Since $\ell\geq 2$, we have $A\leq \frac{4537}{46656} \ell^6$ and $B\geq q^3\left(1-\frac 54\frac{\ell^2}{q}+\frac{19}{216}\frac{\ell^4}{q^2}\right)$. Furthermore, since the polynomial $$\frac{4537}{46656}X^3-\frac{19}{216}X^2+\frac54 X-1$$ has only one real root with approximate value $0.805$, we find that $A\leq B$, because $q\geq 1.243 \ell^2.$ \item Since $q>1.82\ell^2$ and $\ell\geq 2$, we have $ \ell \left(\frac{1}{2q}+\frac{1}{\sqrt q}+1\right)\leq \ell\left(\frac 1{22}+\frac 1{\sqrt{11}}+1\right) <\sqrt q $. Hence, $-2q\ell - 2\sqrt q \ell + 2q\sqrt q-\ell >0 $ and \eqref{condhaloui4} is satisfied. \end{itemize} This proves that $P_q(X)$ is a Weil polynomial and the second part of the proposition. $\qed$ \subsection{Proofs of Lemmas~\ref{nnsquaresix} and \ref{redsix}} In this section, $\ell>2$, $q\neq \ell$ are prime numbers and we, somewhat abusively, denote with the same letter an integer in $[-(\ell-1)/2,(\ell-1)/2]$ and its image in $\mathbb F_\ell$. We will repeatedly use the following elementary lemma. \begin{lm}\label{prelimun} Let $D\in\mathbb F_\ell^*$ and $\varepsilon\in\{-1,1\}.$ We have $$ \sharp \left\{x\in\mathbb F_\ell ;\legendre{x^2-D}{\ell} =\varepsilon\right\}=\frac 12\left(\ell-1-\varepsilon -\legendre D\ell\right); $$ and $$\sharp\left\{(x,y)\in\mathbb F_\ell^2; \legendre{x^2-Dy^2}\ell=\varepsilon\right\}=\frac{1}{2}(\ell-1)\left(\ell-\legendre D\ell\right).$$ \end{lm} \subsubsection{Estimates on the number of degree 4 Weil polynomials modulo $\ell$} \begin{pr}\label{weilfour} \begin{enumerate} \item For $\varepsilon\in\{-1,1\}$, we denote by $D_4^\varepsilon$ the number of degree $4$ polynomials of the form $X^4+uX^3+vX^2+uqX+q^2\in\mathbb F_\ell[X]$ with discriminant $\Delta$ such that $\legendre \Delta \ell=\varepsilon$. Then $$ D_4^-=\frac 12(\ell-1)\left(\ell-\legendre q\ell\right)\quad \mbox { and } \quad D_4^+=\frac 12(\ell-3)\left(\ell-\legendre q\ell\right)+1.$$ \item The number $N_4$ of degree $4$ Weil polynomials with coefficients in $[-(\ell-1)/2,(\ell-1)/2]$ which are irreducible modulo $\ell$ satisfies \begin{equation}\label{nfour} N_4\leq\frac 14 (\ell+1)(\ell-1). \end{equation} \item The number $T_4$ of degree $4$ Weil polynomials with coefficients in $[-(\ell-1)/2,(\ell-1)/2]$ with exactly two irreducible factors modulo $\ell$ satisfies \begin{equation}\label{tfour} T_4\leq\frac 14(\ell-3)\left(\ell-\legendre q\ell\right)+\frac 18(\ell-1)(\ell+1). \end{equation} \end{enumerate} \noindent Moreover, if $q>1.67\ell^2$, Inequalities (\ref{nfour}) and (\ref{tfour}) are equalities. \end{pr} \begin{proof} \begin{enumerate} \item First, we compute $D_4^{\varepsilon}$. The polynomial $Q(X)=X^4+uX^3+vX^2+uqX+q^2$ has discriminant $$\Delta = q^2 \kappa ^2 \delta \quad \mbox{ where}\quad \kappa= -u^2 +4(v-2q) \quad\mbox{ and }\quad\delta =(v+2q)^2-4qu^2.$$ Since $q\in \mathbb F_\ell^*$, we have $\legendre{\Delta}{\ell}=\legendre\kappa\ell^2\legendre\delta\ell$. Moreover, notice that if $\kappa=0$ then $\delta=(v-6q)^2$. It follows that $$D_4^- =\sharp\left\{(u,v)\in\mathbb F_\ell^2; \legendre{\delta}{\ell}=-1\right\}$$ and $$D_4^+ =\sharp\left\{(u,v)\in\mathbb F_\ell^2; \legendre{\delta}{\ell}=1\right\}-\sharp\left\{(u,v)\in\mathbb F_\ell^2; v\neq 6q \mbox{ and } u^2=4(v-2q)\right\}.$$ Since the map $(u,v)\mapsto (v+2q,2u)$ is a bijection on $\mathbb F_\ell^2$ (because $\ell\neq 2$), by Lemma~\ref{prelimun} we have $$\sharp\left\{(u,v)\in\mathbb F_\ell^2; \legendre{\delta}\ell =\varepsilon\right\}= \sharp\left\{(x,y)\in\mathbb F_\ell^2; \legendre{x^2-qy^2}\ell=\varepsilon\right\}=\frac{(\ell-1)}2\left(\ell-\legendre q\ell\right)$$ for any $\varepsilon\in\{\pm 1\}.$ This gives the result for $D_4^-$. The result for $D_4^+$ follows from: \begin{eqnarray*} \sharp\left\{(u,v); \; v\neq 6q \mbox{ and } u^2=4(v-2q)\right\}&=& \sharp\left\{(u,v);\; u^2=4(v-2q)\right\}-\sharp\{u\in\mathbb F_\ell; u^2=16q\}\\ &=&\ell-1-\legendre q\ell. \end{eqnarray*} \item Next, we bound the quantity $N_4$. By Stickelberger's result (see \eqref{eq:stickelberger}), a monic degree $4$ polynomial in $\mathbb Z[X]$ has non-square discriminant modulo $\ell$ if and only if it has one or three distinct irreducible factors in $\mathbb F_\ell[X]$. In the latter case, the polynomial has the form $$(X-\alpha')(X-q/\alpha')(X^2-B'X+q)$$ with $X^2-B'X+q$ irreducible in $\mathbb F_\ell[X]$ and $\alpha'\neq q/\alpha'$ in $\mathbb F_\ell^*$. By Lemma~\ref{prelimun}, there are $$\frac 14\left(\ell-2-\legendre{q}{\ell}\right) \left(\ell-\legendre q\ell\right)$$ such polynomials with three irreducible factors. It follows that \[ N_4\leq D_4^- - \frac 14\left(\ell-2-\legendre{q}{\ell}\right) \left(\ell-\legendre q\ell\right)\leq\frac 14(\ell-1)(\ell+1). \] \item Finally, we bound the quantity $T_4$. As above, Stickelberger's result implies that a degree $4$ Weil polynomial $Q(X)$ in $\mathbb Z[X]$ has exactly two distinct irreducible factors modulo $\ell$ if and only if $\legendre {\Delta_Q}{\ell}=1$ and $Q(X) \pmod \ell$ does not have four distinct roots in $\mathbb F_\ell$. By Lemma~\ref{prelimun}, there are $$\frac 18\left(\ell-\legendre q\ell-2\right)\left(\ell-\legendre q\ell-4\right)$$ Weil polynomials with coefficients in $[-(\ell-1)/2,(\ell-1)/2]$ whose reduction modulo $\ell$ has four distinct roots in $\mathbb F_\ell$. It follows that \begin{eqnarray*} T_4&\leq&D_4^+ - \frac 18 \left(\ell-\legendre q\ell-2\right)\left(\ell-\legendre q\ell-4\right)\\ &\leq& \frac 14(\ell-3)\left(\ell-\legendre q\ell\right)+\frac 18(\ell-1)(\ell+1). \end{eqnarray*} \end{enumerate} When $q>1.67\ell^2$, these upper bounds for $N_4$ and $T_4$ are equalities, since in this case, by Proposition~\ref{boundqweil}, every polynomial of the form $X^4+uX^3+vX^2+uqX+q^2$ with $|u|,|v|<\ell$ is a Weil polynomial. \end{proof} \subsubsection{Proof of Lemma~\ref{redsix}} Recall that $R_6$ denotes the number of Weil polynomials $P_q(X)=X^6+aX^5+bX^4+cX^3+qbX^2+q^2aX+q^3$ with coefficients in $[-(\ell-1)/2,(\ell-1)/2]$, $a,c\neq 0$, non-square discriminant modulo $\ell$ and which are reducible modulo $\ell$. We may drop the conditions $a\neq 0, c\neq 0$ to bound $R_6$. By Stickelberger's result (see \eqref{eq:stickelberger}), a monic degree $6$ polynomial in $\mathbb Z[X]$ with non-square discriminant modulo $\ell$ has $1,\ 3$ or $5$ distinct irreducible factors in $\mathbb F_\ell[X]$. Hence, the factorisation in $\mathbb F_{\ell}[X]$ of a polynomial $P_q(X)$ as above is of one of the following types (note that a root $\alpha$ of $P_q(X)$ in $\overline{\mathbb F}_\ell$ is in $\mathbb F_\ell$ if and only $q/\alpha$ is also in $\mathbb F_\ell$): \begin{enumerate} \item $P_q (X)\equiv (X-\alpha)(X-\frac q\alpha)(X-\beta)(X-\frac q\beta)(X^2-CX+q)$, with $C^2-4q$ non-square modulo $\ell$ and $\alpha\neq q/\alpha$, $\beta\neq q/\beta$ and $\{\alpha,q/\alpha\}\neq\{\beta,q/\beta\}$; equivalently $P_q(X)\equiv (X^2-AX+q)(X^2-BX+q)(X^2-CX+q)$ where the first two quadratic polynomials are distinct and both reducible and the third one is irreducible;\item $P_q (X)\equiv (X-\alpha)(X-\frac q\alpha) Q(X)$, where $\alpha\neq q/\alpha$ and the irreducible factor $Q(X)$ is the reduction of a degree $4$ Weil polynomial; \item $P_q(X)$ is the product of three distinct irreducible quadratic polynomials, i.e., $P_q(X) \equiv (X^2-CX+q)Q(X)$ where $X^2-CX+q$ is irreducible and $Q(X)$ is the reduction of a degree $4$ Weil polynomial which has two distinct irreducible factors, both of which are distinct from $X^2-CX+q$. \end{enumerate} We will count the number of polynomials of each type. \noindent\textbf{Type 1.} By Lemma~\ref{prelimun}, there are $\frac 12\left(\ell-\legendre q\ell\right)$ irreducible quadratic polynomials $X^2-CX+q$. Also by Lemma~\ref{prelimun}, there are $\frac 12\left(\ell-2-\legendre q\ell\right)$ choices for reducible $X^2-AX+q$ without a double root and then there are $\frac 12\left(\ell-2-\legendre q\ell\right)-1$ choices for reducible $X^2-BX+q$ without a double root and distinct from $X^2-AX+q$. It follows that there are $ \frac 1{16}\left(\ell-\legendre q\ell\right)\left(\ell-\legendre q\ell-2\right)\left(\ell-\legendre q\ell -4\right) $ such polynomials. \noindent\textbf{Type 2.} By Proposition~\ref{weilfour} and Lemma~\ref{prelimun}, the number of polynomials with decomposition of this type is $$\frac 12\left(\ell-\legendre q\ell-2\right)N_4 \leq \frac 18 (\ell+1)(\ell-1)\left(\ell-\legendre q\ell-2\right).$$ \noindent\textbf{Type 3.} Proposition~\ref{weilfour} and Lemma~\ref{prelimun} imply that there are $$ \leq \frac 12\left(\ell-\legendre q\ell\right) T_4 \leq \frac 18\left(\ell-\legendre q\ell\right)^2(\ell-3)+\frac 1{16}(\ell-1)(\ell+1)\left(\ell-\legendre q\ell\right) $$ polynomials of this type. \footnote{The first inequality is due to the fact that we do not take into account that $X^2-CX+q$ has to be distinct from the factors of $Q(X)$.} Summing these three upper bounds yields the lemma. $\qed$ \subsubsection{Proof of Lemma~\ref{nnsquaresix}} The discriminant of $P_q(X)$ is $\Delta_{P_q}= q^6\Gamma^2\delta $, where $$\Gamma= 8q a^4 + 9q^2a^2 - 42qa^2 b + a^2b^2 - 4a^3c + 108q^3 - 108q^2b + 36qb^2 - 4b^3 + 54qac + 18abc - 27c^2$$ and $\delta=(c+2aq)^2-4q(b+q)^2.$ Hence, we have \begin{eqnarray*} D_6^{*-}&=&\sharp\left\{(a,b,c); a,c\neq 0, \Gamma\not\equiv 0\bmod \ell \mbox{ and } \legendre \delta\ell=-1\right\}\\ &=&\sharp\left\{(a,b,c); a,c\neq 0, \legendre \delta\ell=-1\right\} -\sharp\left\{(a,b,c); a,c\neq0, \Gamma\equiv 0\bmod \ell \mbox{ and } \legendre \delta\ell=-1\right\}\\ &\geq&M-W, \end{eqnarray*} where $ M = \sharp\left\{(a,b,c); a,c\neq 0, \legendre \delta\ell=-1\right\}$ and $W = \sharp\left\{(a,b,c); a\neq0, \Gamma\equiv 0\bmod \ell \right\}$. \paragraph{Computation of $M$.} Since $\ell >2$ and $q\in\mathbb F_\ell^*$, for any fixed $c\in\mathbb F_\ell^\times$, the map $(a,b)\mapsto (c+2aq,b+q)$ is a bijection from $\mathbb F_\ell^*\times\mathbb F_\ell$ to $\mathbb F_\ell\backslash\{c\}\times\mathbb F_\ell$. From this and Lemma~\ref{prelimun} we deduce that \begin{eqnarray*} M&=& \sum_{c\in\mathbb F_\ell^*} \sharp\left\{(x,y)\in\mathbb F_\ell^2; x\neq c,\; \legendre{x^2-4qy^2}{\ell}=-1\right\}\\ &=& \sum_{c\in\mathbb F_\ell^*} \sharp\left\{(x,y)\in\mathbb F_\ell^2; \legendre{x^2-4qy^2}{\ell}=-1\right\} - \sum_{c\in\mathbb F_\ell^*} \sharp\left\{y\in\mathbb F_\ell ; \legendre{c^2-4qy^2}{\ell}=-1\right\}\\ &=& \frac 12(\ell-1)^2\left(\ell-\legendre q\ell\right) - \sum_{c\in\mathbb F_\ell^*} M'_c, \end{eqnarray*} where \begin{eqnarray*} M'_c &=& \sharp\left\{y\in\mathbb F_\ell ; \legendre{c^2-4qy^2}{\ell}=-1\right\}\\ &=&\sharp\left\{y\in\mathbb F_\ell ; \legendre{y^2-(c^2/4q)}{\ell}=-\legendre{-q}\ell\right\} \\ &=& \frac 12\left(\ell-1-\legendre q\ell+\legendre{-q}\ell\right), \end{eqnarray*} the last equality following from Lemma~\ref{prelimun}. This gives $$ M = \frac 12 (\ell-1)^2\left(\ell-1-\legendre q\ell\right) +\frac 12 (\ell-1)\legendre q\ell\left(1-\legendre{-1}\ell\right). $$ \paragraph{Computation of $W=\sharp\left\{(a,b,c)\in\mathbb F_\ell^3; a\neq0, \Gamma=0 \right\}.$} The discriminant of $\Gamma$ viewed as a\hfil \newline quadratic polynomial\footnote{More precisely, we have $ \Gamma=-27c^2+G_1 c+G_0,\ (G_0,G_1\in \mathbb F_\ell[a,b])$ with $G_1(a,b)=-2a (2a^2 - 27q -9b) $ and $G_0(a,b)=8qa^4 + 9q^2a^2 - 42qa^2b + a^2b^2 + 108q^3 - 108q^2b + 36qb^2 - 4b^3.$} in $c$ is $\gamma=16(a^2-3(b-3q))^3.$ It follows that \begin{eqnarray*} W&=& 2\cdot\sharp\left\{(a,b)\in\mathbb F_\ell^2; a\neq 0, \legendre \gamma\ell=1\right\}+ \sharp\left\{(a,b)\in\mathbb F_\ell^2; a\neq 0, \gamma=0 \right\}\\ &=& 2\cdot\sharp\left\{(a,b)\in\mathbb F_\ell^2; a\neq 0, \legendre {a^2-3(b-3q)}\ell=1\right\}+ \sharp\left\{(a,b)\in\mathbb F_\ell^2; a\neq 0, a^2=3(b-3q) \right\}. \end{eqnarray*} Moreover, since $\ell>3$, the map $b\mapsto 3(b-3q)$ is a bijection on $\mathbb F_\ell$. So we have \begin{eqnarray*} W &=& 2\cdot\sharp\left\{(x,y)\in\mathbb F_\ell^2; x\neq 0, \legendre {x^2-y}\ell=1\right\}+ \sharp\left\{(x,y)\in\mathbb F_\ell^2; x\neq 0, x^2=y \right\}\\ &=& 2\cdot\sum_{y\in\mathbb F_\ell}\sharp\left\{x\in\mathbb F_\ell; \legendre{x^2-y}\ell=1\right\}-2\cdot\sharp\left\{y\in\mathbb F_\ell; \legendre{-y}{\ell}=1\right\}+\sum_{y\in\mathbb F_\ell^*}\sharp\{x\in\mathbb F_\ell^*; x^2=y\} \\ &=& \sum_{y\in\mathbb F_\ell^*}\left(\ell-2-\legendre y\ell\right)+2(\ell-1) - (\ell-1) + (\ell-1), \end{eqnarray*} using Lemma~\ref{prelimun} (the second term is the contribution of $y=0$). This yields $W=\ell(\ell-1)$ and computing $M-W$ concludes the proof. \qed \subsection{Examples}\label{ex:section5} This section contains examples of Weil polynomials satisfying the conditions in Proposition \ref{irredmodell}. They were obtained using \textsc{Sage}. \begin{itemize} \item $\ell=3$, $q=19$: $P_q(X)=X^6 + X^5 + X^3 + 361X + 6859$; \item $\ell=5$, $q=47$: $P_q(X)=X^6 + X^5 + X^4 + X^3 + 47X^2 + 2209X + 103823$; \item $\ell=7$, $q=97$: $P_q(X)=X^6 + X^5 + 3X^3 + 9409X + 912673$; \item $\ell=11$, $q=223$: $P_q(X)=X^6 + X^5 + 5X^3 + 49729X + 11089567$; \item $\ell=13$: \begin{itemize} \item[] $q=311$: $P_q(X)=X^6 + X^5 + 3X^3 + 96721X + 30080231$; \item[] $q=313$: $P_q(X)=X^6 + X^5 + 4X^3 + 97969X + 30664297$; \item[] $q=317$: $P_q(X)=X^6 + X^5 + X^3 + 100489X + 31855013$; \item[] $q=331$: $P_q(X)=X^6 + X^5 + 3X^3 + 109561X + 36264691$. \end{itemize} \end{itemize} \end{document}

\begin{document} \title{From Unsupervised to Few-shot \\Graph Anomaly Detection: A Multi-scale Contrastive Learning Approach} \raggedbottom \author{Yu Zheng, Ming Jin, Yixin Liu, Lianhua Chi*, Khoa T. Phan, Shirui Pan, Yi-Ping Phoebe Chen \thanks{Y. Zheng, L. Chi, K. T. Phan, and Y-P. P. Chen are with Department of Computer Science and Information Technology, La Trobe University, Melbourne Australia. E-mail: \{yu.zheng, l.chi, k.phan, phoebe.chen\}@latrobe.edu.au. } \thanks{M. Jin, Y. Liu, S. Pan are with the Department of Data Science and AI, Faculty of IT, Monash University, Clayton, VIC 3800, Australia. E-mail: \{ming.jin, yixin.liu, shirui.pan\}@monash.edu. } \thanks {Y. Zheng and M. Jin contributed equally to this work.} \thanks {* Corresponding Author} \thanks{Manuscript received Jan 3, 2022; revised xx xx, 202x.} } \markboth{Journal of \LaTeX\ Class Files,~Vol.~14, No.~8, Jan~2022} {Zheng \MakeLowercase{\textit{et al.}}: Multi-scale contrastive learning} \maketitle \begin{abstract} Anomaly detection from graph data is an important data mining task in many applications such as social networks, finance, and e-commerce. Existing efforts in graph anomaly detection typically only consider the information in a single scale (view), thus inevitably limiting their capability in capturing anomalous patterns in complex graph data. To address this limitation, we propose a novel framework, graph \underline{\textbf{AN}}omaly d\underline{\textbf{E}}tection framework with \underline{\textbf{M}}ulti-scale c\underline{\textbf{ON}}trastive l\underline{\textbf{E}}arning (\texttt{ANEMONE}\xspace in short). By using a graph neural network as a backbone to encode the information from multiple graph scales (views), we learn better representation for nodes in a graph. In maximizing the agreements between instances at both the patch and context levels concurrently, we estimate the anomaly score of each node with a statistical anomaly estimator according to the degree of agreement from multiple perspectives. To further exploit a handful of ground-truth anomalies (few-shot anomalies) that may be collected in real-life applications, we further propose an extended algorithm, \texttt{ANEMONE-FS}\xspace, to integrate valuable information in our method. We conduct extensive experiments under purely unsupervised settings and few-shot anomaly detection settings, and we demonstrate that the proposed method \texttt{ANEMONE}\xspace and its variant \texttt{ANEMONE-FS}\xspace consistently outperforms state-of-the-art algorithms on six benchmark datasets. \end{abstract} \begin{IEEEkeywords} Anomaly detection, self-supervised learning, graph neural networks (GNNs), unsupervised learning, few-shot learning \end{IEEEkeywords} \IEEEpeerreviewmaketitle \section{Introduction} \label{sec:introduction} \IEEEPARstart{A}{S} a general data structure to represent inter-dependency between objects, graphs have been widely used in many domains including social networks, biology, physics, and traffic, etc. Analyzing graph data for various tasks --- detecting anomalies from graph data in particular --- has drawn increasing attention in the research community due to its wide and critical applications in e-commence, cyber-security, and finance. For instance, by using anomaly detection algorithms in e-commerce, we can detect fraudulent sellers by jointly considering their properties and behaviors \cite{pourhabibi2020fraud}. Similarly, we can detect abnormal accounts (social bots) which spread rumors in social networks with graph anomaly detection systems \cite{latah2020detection}. Different from conventional anomaly detection approaches for tabular/vector data where the attribute information is the only factor to be considered, graph anomaly detection requires collectively exploiting both graph structure as well as attribute information associated with each node. This complexity has imposed significant challenges to this task. Existing research to address the challenges can be roughly divided into two categories: (1) shallow methods, and (2) deep methods. The early shallow methods typically exploit mechanisms such as ego-network analysis \cite{amen_perozzi2016scalable}, residual analysis \cite{radar_li2017radar} or CUR decomposition \cite{anomalous_peng2018anomalous}. These methods are conceptually simple but they may be not able to learn nonlinear representation from complex graph data, leading to a sub-optimal anomaly detection performance. The deep model approaches, such as graph autoencoder (GAE) \cite{dominant_ding2019deep, li2019specae}, learn the non-linear hidden representation, and estimate the anomaly score for each node based on the reconstruction error. These methods have considerable improvements over shallow methods. However, they do not well capture the contextual information (e.g., subgraph around a node) for anomaly detection and still suffer from unsatisfactory performance. Very recently, a contrastive learning mechanism has been used for graph anomaly detection \cite{cola_liu2021anomaly}, which shows promising performance in graph anomaly detection. The key idea used in the proposed algorithm, namely CoLA \cite{cola_liu2021anomaly}, is to construct pairs of instances (e.g., a subgraph and a target node) and to employ a contrastive learning method to learn the representation. Based on contrastive learning, anomaly scores can be further calculated according to predictions of pairs of instances. Despite its success, only a single scale of information is considered in CoLA. In practice, due to the complexity of graph data, anomalies are often hidden in different scales (e.g., node and subgraph levels). For example, in the e-commerce application, fraudulent sellers may interact with only a small number of other users or items (i.e., local anomalies); in contrast, other cheaters may hide in larger communities (i.e., global anomalies). Such complex scales require more fine-grained and intelligent anomaly detection systems. Another limitation of existing graph anomaly detection methods is that these methods are designed in a purely unsupervised manner. In scenarios where the ground-truth anomalies are unknown, these methods can make an important role. However, in practice, we may collect a handful of samples (i.e., a few shots) of anomalies. As anomalies are typically rare in many applications, these few-shot samples provide valuable information and should be incorporated into anomaly detection systems \cite{pang2019deep}. Unfortunately, most of existing graph anomaly detection methods \cite{dominant_ding2019deep, li2019specae,cola_liu2021anomaly} failed to exploit these few-shot anomalies in their design, leading to a significant information loss. To overcome these limitations, in this paper, we propose a graph \underline{\textbf{AN}}omaly d\underline{\textbf{E}}tection framework with \underline{\textbf{M}}ulti-scale c\underline{\textbf{ON}}trastive l\underline{\textbf{E}}arning (\texttt{ANEMONE}\xspace in short) to detect anomalous nodes in graphs. Our theme is to construct multi-scales (views) from the original graphs and employ contrastive learning at both patch and context levels simultaneously to capture anomalous patterns hidden in complex graphs. Specifically, we first employ two graph neural networks (GNNs) as encoders to learn the representation for each node and a subgraph around the target node. Then we construct pairs of positive and negative instances at both patch levels and context levels, based on which the contrastive learning maximizes the similarity between positive pairs and minimize the similarity between negative pairs. The anomaly score of each node is estimated via a novel anomaly estimator by leveraging the statistics of multi-round contrastive scores. Our framework is a general and flexible framework in the sense that it can easily incorporate ground-truth anomalies (few shot anomalies). Concretely, the labeled anomalies are seamlessly integrated into the contrastive learning framework as additional negative pairs for both patch level and context level contrastive learning, leading to a new algorithm, \texttt{ANEMONE-FS}\xspace, for few-shot graph anomaly detection. Extensive experiments on six benchmark datasets validate the effectiveness of our algorithm \texttt{ANEMONE}\xspace for unsupervised graph anomaly detection and the effectiveness of \texttt{ANEMONE-FS}\xspace for few-shot settings. The main contributions of this work are summarized as follows: \begin{itemize} \item We propose a general framework, \texttt{ANEMONE}\xspace, based on contrastive learning for graph anomaly detection. Our method exploits multi-scale information at both patch level and context level to capture anomalous patterns hidden in complex graphs. \item We present a simple approach based on the multi-scale framework, \texttt{ANEMONE-FS}\xspace, to exploit the valuable few-shot anomalies at hand. Our method essentially enhances the flexibility of contrastive learning for anomaly detection and facilitate broader applications. \item We conduct extensive experiments on six benchmark datasets to demonstrate the superiority of our \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace for both unsupervised and few-shot graph anomaly detection. \end{itemize} The reminder of the paper is structured as follows. We review the related works in Section \ref{sec:rw} and give the problem definition in Section \ref{sec:PD}. The proposed method \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace are described in Section \ref{sec:methodology}. The experimental results are shown in Section \ref{sec:experiments}. We conclude this paper in Section \ref{sec:conclusion}. \section{Related Works} \label{sec:rw} In this section, we survey the representative works in three related topics, including graph neural networks, graph anomaly detection, and contrastive learning. \subsection{Graph Neural Networks} In recent years, graph neural networks (GNNs) have achieved significant success in dealing with graph-related machine learning problems and applications \cite{gcn_kipf2017semi,gat_velivckovic2018graph,gnn_survey_wu2021comprehensive,wu2021beyond}. Considering both attributive and structural information, GNNs can learn low-dimensional representation for each node in a graph. Current GNNs can be categorized into two types: spectral and spatial methods. The former type of methods was initially developed on the basis of spectral theory \cite{bruna2013spectral,defferrard2016convolutional,gcn_kipf2017semi}. Bruna et al. \cite{bruna2013spectral} first extend convolution operation to graph domain using spectral graph filters. Afterward, ChebNet \cite{defferrard2016convolutional} simplifies spectral GNNs by introducing Chebyshev polynomials as the convolution filter. GCN \cite{gcn_kipf2017semi} further utilizes the first-order approximation of Chebyshev filter to learn node representations more efficiently. The second type of methods adopt the message-passing mechanism in the spatial domain, which propagates and aggregates local information along edges, to perform convolution operation \cite{hamilton2017inductive,gat_velivckovic2018graph,xu2019how}. GraphSAGE \cite{hamilton2017inductive} learns node representations by sampling and aggregating neighborhoods. GAT \cite{gat_velivckovic2018graph} leverages the self-attention mechanism to assign a weight for each edge when performing aggregation. GIN \cite{xu2019how} introduces a summation-based aggregation function to ensure that GNN is as powerful as the Weisfeiler-Lehman graph isomorphism test. For a thorough review, we please refer the readers to the recent survey \cite{gnn_survey_wu2021comprehensive}. \subsection{Graph Anomaly Detection} Anomaly detection is a conventional data mining problem aiming to identify anomalous data samples that deviate significantly from others \cite{gad_survey_ma2021comprehensive}. Compared to detecting anomalies from text/image data \cite{deepsad_ruff2019deep}, anomaly detection on graphs is often more challenging since the correlations among nodes should be also considered when measuring the abnormality of samples. To tackle the challenge, some traditional solutions use shallow mechanisms like ego-network analysis (e.g., AMEN\cite{amen_perozzi2016scalable}), residual analysis (e.g., Radar\cite{radar_li2017radar}), and CUR decomposition (e.g., ANOMALOUS \cite{anomalous_peng2018anomalous}) to model the anomalous patterns in graph data. Recently, deep learning becomes increasingly popular for graph anomaly detection \cite{gad_survey_ma2021comprehensive}. As an example of unsupervised methods, DOMINANT \cite{dominant_ding2019deep} employs a graph autoencoder model to reconstruct attribute and structural information of graphs, and the reconstruction errors are leveraged to measure node-level abnormality. CoLA \cite{cola_liu2021anomaly} considers a contrastive learning model that models abnormal patterns via learning node-subgraph agreements. Among semi-supervised methods, SemiGNN \cite{semignn_wang2019semi} is a representative method that leverages hierarchical attention to learn from multi-view graphs for fraud detection. GDN \cite{gdn_ding2021few} adopts a deviation loss to train GNN for few-shot node anomaly detection. Apart from the aforementioned methods for attributed graphs, some recent works also target to identity anomalies from dynamic graphs \cite{dyn_gad_wang2019detecting,taddy_liu2021anomaly}. \subsection{Graph Contrastive Learning} Originating from visual representation learning \cite{he2020momentum,chen2020simple,grill2020bootstrap}, contrastive learning has become increasingly popular in addressing self-supervised representation learning problems in various areas. In graph deep learning, recent works based on contrastive learning show competitive performance on graph representation learning scenario \cite{gssl_survey_liu2021graph,zheng2021towards}. DGI \cite{dgi_velickovic2019deep} learns by maximizing the mutual information between node representations and a graph-level global representation, which makes the first attempt to adapt contrastive learning in GNNs. GMI \cite{peng2020graph} jointly utilizes edge-level contrast and node-level contrast to discover high-quality node representations. GCC \cite{qiu2020gcc} constructs a subgraph-level contrastive learning model to learn structural representations. GCA \cite{zhu2021graph} introduces an adaptive augmentation strategy to generate different views for graph contrastive learning. MERIT \cite{Jin2021MultiScaleCS} leverages bootstrapping mechanism and multi-scale contrastiveness to learn informative node embeddings for network data. Apart from learning effective representations, graph contrastive learning is also applied various applications, such as drug-drug interaction prediction \cite{wang2021multi} and social recommendation \cite{yu2021self}. \section{Problem Definition} \label{sec:PD} In this section, we introduce and define the problem of unsupervised and few-shot graph anomaly detection. Throughout the paper, we use bold uppercase (e.g., $\mathbf{X}$), calligraphic (e.g., $\mathbfcal{V}$), and lowercase letters (e.g., $\mathbf{x}^{(i)}$) to denote matrices, sets, and vectors, respectively. We also summarize all important notations in Table \ref{table:notation}. In this work, we mainly focus on the anomaly detection tasks on attributed graphs that are widely existed in real world. Formally speaking, we define attributed graphs and graph neural networks (GNNs) as follows: \begin{definition}[Attributed Graphs] Given an attributed graph $\mathcal{G}=(\mathbf{X},\mathbf{A})$, we denote its node attribute (i.e., feature) and adjacency matrices as $\mathbf{X} \in \mathbb{R}^{N \times D}$ and $\mathbf{A} \in \mathbb{R}^{N \times N}$, where $N$ and $D$ are the number of nodes and feature dimensions. An attribute graph can also be defined as $\mathcal{G}=(\mathbfcal{V}, \mathbfcal{E}, \mathbf{X})$, where $\mathbfcal{V}=\{v_1, v_2, \cdots, v_N\}$ and $\mathbfcal{E}=\{e_1, e_2, \cdots, e_M\}$ are node and edge sets. Thus, we have $N=|\mathbfcal{V}|$, the number of edges $M=|\mathbfcal{E}|$, and define $\mathbf{x}_i \in \mathbb{R}^{D}$ as the attributes of node $v_i$. To represent the underlying node connectivity, we let $\mathbf{A}_{ij}=1$ if there exists an edge between $v_i$ and $v_j$ in $\mathbfcal{E}$, otherwise $\mathbf{A}_{ij}=0$. In particular, given a node $v_i$, we define its neighborhood set as $\mathbfcal{N}(v_i)=\{v_j \in \mathbfcal{V} | \mathbf{A}_{ij} \neq 0 \}$. \end{definition} \begin{definition}[Graph Neural Networks] Given an attributed graph $\mathcal{G}=(\mathbf{X},\mathbf{A})$, a parameterized graph neural network $GNN(\cdot)$ aims to learn the low-dimensional embeddings of $\mathbf{X}$ by considering the topological information $\mathbf{A}$, denoted as $\mathbf{H} \in \mathbb{R}^{N \times D'}$, where $D'$ is the embedding dimensions and we have $D' \ll D$. For a specific node $v_i \in \mathbfcal{V}$, we denote its embedding as $\mathbf{h}^{(i)} \in \mathbb{R}^{D'}$ where $\mathbf{h}^{(i)} \in \mathbf{H}$. \end{definition} In this paper, we focus on two different anomaly detection tasks on attributed graphs, namely unsupervised and few-shot graph anomaly detection. Firstly, we define the problem of unsupervised graph anomaly detection as follows: \begin{definition}[Unsupervised Graph Anomaly Detection] Given an unlabeled attribute graph $\mathcal{G}=(\mathbf{X},\mathbf{A})$, we intend to train and evaluate a graph anomaly detection model $\mathcal{F}(\cdot): \mathbb{R}^{N \times D} \to \mathbb{R}^{N \times 1}$ across all nodes in $\mathbfcal{V}$, where we use $\mathbf{y}$ to denote the output node anomaly scores, and $y^{(i)}$ is the anomaly score of node $v_i$. \end{definition} For graphs with limited prior knowledge (i.e., labeling information) on the underlying anomalies, we define the problem of few-shot graph anomaly detection as below: \begin{definition}[Few-shot Graph Anomaly Detection] For an attribute graph $\mathcal{G}=(\mathbf{X},\mathbf{A})$, we have a small set of labeled anomalies $\mathbfcal{V}^L$ and the rest set of unlabeled nodes $\mathbfcal{V}^U$, where $|\mathbfcal{V}^L| \ll |\mathbfcal{V}^U|$ since it is relatively expensive to label anomalies in the real world so that only a very few labeled anomalies are typically available. Thus, our goal is to learn a model $\mathcal{F}(\cdot): \mathbb{R}^{N \times D} \to \mathbb{R}^{N \times 1}$ on $\mathbfcal{V}^L \cup \mathbfcal{V}^U$, which measures node abnormalities by calculating their anomaly scores $\mathbf{y}$. It is worth noting that during the evaluation, the well-trained model $\mathcal{F}^*(\cdot)$ is only tested on $\mathbfcal{V}^U$ to prevent the potential information leakage. \end{definition} \begin{table}[t] \small \centering \caption{Summary of important notations.} \begin{tabular}{ p{75 pt}<{\centering} | p{155 pt}} \toprule[1.0pt] Symbols & Description \\ \cmidrule{1-2} $\mathcal{G}=(\mathbf{X}, \mathbf{A})$ & An attributed graph \\ $\mathbfcal{V}, \mathbfcal{E}$ & The node and edge set of $\mathcal{G}$ \\ $\mathbfcal{V^L}$ & The labeled node set where $\mathbfcal{V^L} \in \mathbfcal{V}$ \\ $\mathbfcal{V^U}$ & The unlabeled node set where $\mathbfcal{V^U} \in \mathbfcal{V}$ \\ $\mathbf{A} \in \mathbb{R}^{N \times N}$ & The adjacency matrix of $\mathcal{G}$ \\ $\mathbf{X} \in \mathbb{R}^{N \times D}$ & The node feature matrix of $\mathcal{G}$ \\ $\mathbf{x}^{(i)} \in \mathbb{R}^{D}$ & The feature vector of $v_i$ where $ \mathbf{x}^{(i)} \in \mathbf{X}$ \\ $\mathbfcal{N}(v_i)$ & The neighborhood set of node $v_i \in \mathcal{V}$ \\ \cmidrule{1-2} $\mathcal{G}^{(i)}_p$, $\mathcal{G}^{(i)}_c$ & Two generated subgraphs of $v_i$ \\ $\mathbf{A}_{view}^{(i)} \in \mathbb{R}^{K \times K}$ & The adjacency matrix of $\mathcal{G}^{(i)}_{view}$ where $view \in \{p, c\}$ \\ $\mathbf{X}_{view}^{(i)} \in \mathbb{R}^{K \times D}$ & The node feature matrix of $\mathcal{G}^{(i)}_{view}$ where $view \in \{p, c\}$ \\ $y^{(i)}$ & The anomaly score of $v_i$ \\ \cmidrule{1-2} $\mathbf{H}^{(i)}_p \in \mathbb{R}^{K \times D'}$ & The node embedding matrix of $\mathcal{G}^{(i)}_p$ \\ $\mathbf{H}^{(i)}_c \in \mathbb{R}^{K \times D'}$ & The node embedding matrix of $\mathcal{G}^{(i)}_c$ \\ $\mathbf{h}^{(i)}_p \in \mathbb{R}^{1 \times D'}$ & The node embeddings of masked node $v_i$ in $\mathbf{H}^{(i)}_p$ \\ $\mathbf{h}^{(i)}_c \in \mathbb{R}^{1 \times D'}$ & The contextual embeddings of $\mathcal{G}^{(i)}_c$ \\ $\mathbf{z}^{(i)}_p, \mathbf{z}^{(i)}_c \in \mathbb{R}^{1 \times D'}$ & The node embeddings of $v_i$ in patch-level and context-level networks \\ $s^{(i)}_p, \tilde{s}^{(i)}_p \in \mathbb{R}$ & The positive and negative patch-level contrastive scores of $v_i$ \\ $s^{(i)}_c, \tilde{s}^{(i)}_c \in \mathbb{R}$ & The positive and negative context-level contrastive scores of $v_i$ \\ $\mathbf{\Theta}, \mathbf{\Phi} \in \mathbb{R}^{D \times D'}$ & The trainable parameter matrices of two graph encoders \\ $\mathbf{W}_{p},\mathbf{W}_{c} \in \mathbb{R}^{D' \times D'}$ & The trainable parameter matrices of two bilinear mappings \\ \cmidrule{1-2} $N^L, N^U, N$ & The number of labeled, unlabeled, and all nodes in $\mathcal{G}$ \\ $K$ & The number of nodes in subgraphs \\ $D$ & The dimension of node attributes in $\mathcal{G}$ \\ $D'$ & The dimension of node embeddings \\ $R$ & The number of evaluation rounds in anomaly scoring \\ \bottomrule[1.0pt] \end{tabular} \label{table:notation} \end{table} \section{Methodology} \label{sec:methodology} In this section, we introduce the proposed \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace algorithms in detecting node-level graph anomalies in an unsupervised and few-shot supervised manner. The overall frameworks of our methods are shown in Figure \ref{fig:framework1} and \ref{fig:framework2}, which consist of four main components, namely the \textit{augmented subgraphs generation}, \textit{patch-level contrastive network}, \textit{context-level contrastive network}, and \textit{statistical graph anomaly scorer}. Firstly, given a target node from the input graph, we exploit its contextual information by generating two subgraphs associated with it. Then, we propose two general yet powerful contrastive mechanisms for graph anomaly detection tasks. Specifically, for an attributed graph without any prior knowledge on the underlying anomalies, the proposed \texttt{ANEMONE}\xspace method learns the patch-level and context-level agreements by maximizing (1) the mutual information between node embeddings in the patch-level contrastive network and (2) the mutual information between node embeddings and their contextual embeddings in the context-level contrastive network. The underlying intuition is that there are only a few anomalies, and thus our well-trained model can identify the salient attributive and structural mismatch between an abnormal node and its surrounding contexts by throwing a significantly higher contrastive score. On the other hand, if few labeled anomalies are available in an attributed graph, the proposed \texttt{ANEMONE-FS}\xspace variant can effectively utilize the limited labeling information to further enrich the supervision signals extracted by \texttt{ANEMONE}\xspace. This intriguing capability is achieved by plugging in a different contrastive route, where the aforementioned patch-level and context-level agreements are minimized for labeled anomalies while still maximized for unlabeled nodes as same as in \texttt{ANEMONE}\xspace. Finally, we design a universal graph anomaly scorer to measure node abnormalities by statistically annealing the patch-level and context-level contrastive scores at the inference stage, which shares and works on both unsupervised and few-shot supervised scenarios. In the rest of this section, we introduce the four primary components of \texttt{ANEMONE}\xspace in Subsection \ref{subsec: augmented subg generation}, \ref{subsec: patch-level}, \ref{subsec: context-level}, and \ref{subsec: scorer}. Particularly, in Subsection \ref{subsec: few-shot}, we discuss how \texttt{ANEMONE}\xspace can be extended to more competitive \texttt{ANEMONE-FS}\xspace in detail to incorporate the available supervision signals provided by a few labeled anomalies. In Subsection \ref{subsec: optimization}, we present and discuss the training objective of \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace, as well as their algorithms and time complexity. \begin{figure*} \caption{ The conceptual framework of \texttt{ANEMONE}\xspace. Given an attributed graph $\mathcal{G}$, we first sample a batch of target nodes, where their associated anonymized subgraphs are generated and fed into two contrastive networks. Then, we design a multi-scale (i.e., patch-level and context-level) contrastive network to learn agreements between node and contextual embeddings from different perspectives. During the model inference, two contrastive scores are statistically annealed to obtain the final anomaly score of each node in $\mathcal{G}$. } \label{fig:framework1} \end{figure*} \subsection{Augmented Subgraphs Generation} \label{subsec: augmented subg generation} Graph contrastive learning relies on effective discrimination pairs to extract supervision signals from the rich graph attributive and topological information \cite{gssl_survey_liu2021graph}. Recently, graph augmentations, such as attribute masking, edge modification, subgraph sampling, and graph diffusion, are widely applied to assist contrastive models in learning expressive graph representations \cite{Jin2021MultiScaleCS, zheng2021towards}. However, not all of them are directly applicable to anomaly detection tasks. For example, edge modification and graph diffusion can distort the original topological information, thus hindering the model to distinguish an abnormal node from its surrounding contexts effectively. To avoid falling into this trap, we adopt an \textit{anonymized subgraph sampling} mechanism to generate graph views for our contrastive networks, which based on two motivations: (1) The agreement between a node and its surrounding contexts (i.e., subgraphs) is typically sufficient to reflect the abnormality of this node \cite{jin2021anemone, zheng2021generative}; (2) Subgraph sampling provides adequate diversity of node surrounding contexts for robust model training and statistical anomaly scoring. Specifically, we explain the details of the proposed augmentation strategy for graph anomaly detection as follows: \begin{enumerate} \item \textbf{Target node sampling.} As this work mainly focuses on node-level anomaly detection, we first sample a batch of target nodes from a given attributed graph. It is worth noting that for a specific target node, it may associate with the label or not, which results in different contrastive routes as shown in the middle red and green dashed boxes in Figures \ref{fig:framework1} and \ref{fig:framework2}. We discuss this in detail in the following subsections. \item \textbf{Surrounding context sampling.} Although several widely adopted graph augmentations are available \cite{gssl_survey_liu2021graph}, most of them are designed to slightly attack the original graph attributive or topological information to learn robust and expressive node-level or graph-level representations, which violate our motivations mentioned above and introduce extra anomalies. Thus, in this work, we employ the subgraph sampling as the primary augmentation strategy to generate augmented graph views for each target nodes based on the random walk with restart (RWR) algorithm \cite{tong2006fast}. Taking a target node $v_i$ for example, we generate its surrounding contexts by sampling subgraphs centred at it with a fixed size $K$, denoted as $\mathcal{G}^{(i)}_p = (\mathbf{A}^{(i)}_p, \mathbf{X}^{(i)}_p)$ and $\mathcal{G}^{(i)}_c = (\mathbf{A}^{(i)}_c, \mathbf{X}^{(i)}_c)$ for patch-level and context-level contrastive networks. In particular, we let the first node in $\mathcal{G}^{(i)}_p$ and $\mathcal{G}^{(i)}_c$ as the starting (i.e., target) node. \item \textbf{Target node anonymization.} Although the above-generated graph views can be directly fed into the contrastive networks, there is a critical limitation: The attributive information of target nodes involves calculating their patch-level and context-level embeddings, which results in information leakage during the multi-level contrastive learning. To prevent this issue and construct harder pretext tasks to boost model training \cite{gssl_survey_liu2021graph}, we anonymize target nodes in their graph views by completely masking their attributes, i.e., $\mathbf{X}^{(i)}_p[1,:] \rightarrow \overrightarrow{0}$ and $\mathbf{X}^{(i)}_c[1,:] \rightarrow \overrightarrow{0}$. \end{enumerate} \subsection{Patch-level Contrastive Network} \label{subsec: patch-level} The objective of patch-level contrastiveness is to learn the local agreement between the embedding of masked target node $v_i$ in its surrounding contexts $\mathcal{G}^{(i)}_p$ and the embedding of $v_i$ itself. The underlying intuition of the proposed patch-level contrastive learning is that the mismatch between a node and its directly connected neighbors is an effective measurement to detect \textit{local anomalies}, which indicates the anomalies that are distinguishable from their neighbors. For example, some e-commerce fraudsters are likely to transact with unrelated users directly, where patch-level contrastiveness is proposed to detect such anomalies in an attributed graph. As shown in Figure \ref{fig:framework1}, our patch-level contrastive network consists of two main components: Graph encoder and contrastive module. \noindent \textbf{Graph encoder.} The patch-level graph encoder takes a target node and one of its subgraph (i.e., surrounding context $\mathcal{G}^{(i)}_p$) as the input. Specifically, the node embeddings of $\mathcal{G}^{(i)}_p$ are calculated in below: \begin{equation} \begin{aligned} \mathbf{H}^{(i)}_{p} &= GNN_{\theta}\left(\mathcal{G}^{(i)}_p\right) = GCN\left(\mathbf{A}^{(i)}_p, \mathbf{X}^{(i)}_p ; \mathbf{\Theta} \right)\\ &= \sigma\left(\widetilde{{\mathbf{D}}^{(i)}_p}^{-\frac{1}{2}} \widetilde{{\mathbf{A}}^{(i)}_p} \widetilde{{\mathbf{D}}^{(i)}_p}^{-\frac{1}{2}} \mathbf{X}^{(i)}_p \mathbf{\Theta} \right), \end{aligned} \label{eq:gnn} \end{equation} where $\mathbf{\Theta} \in \mathbb{R}^{D \times D'}$ denotes the set of trainable parameters of patch-level graph neural network $GNN_{\theta}(\cdot)$. For simplicity and follow \cite{jin2021anemone}, we adopt a single layer graph convolution network (GCN) \cite{gcn_kipf2017semi} as the backbone encoder, where $\sigma(\cdot)$ denotes the ReLU activation in a typical GCN layer, $\widetilde{{\mathbf{A}}^{(i)}_p} = \mathbf{A}^{(i)}_p + \mathbf{I}$, and $\widetilde{{\mathbf{D}}^{(i)}_p}$ is the calculated degree matrix of $\mathcal{G}^{(i)}_p$ by row-wise summing $\widetilde{{\mathbf{A}}^{(i)}_p}$. Alternatively, one may also replace GCN with other off-the-shelf graph neural networks to aggregate messages from nodes' neighbors to calculate $\mathbf{H}^{(i)}_{p}$. In patch-level contrastiveness, our discrimination pairs are the masked and original target node embeddings (e.g., $\mathbf{h}^{(i)}_{p}$ and $\mathbf{z}^{(i)}_{p}$ for a target node $v_i$), where the former one can be easily obtained via $\mathbf{h}^{(i)}_{p} = \mathbf{H}^{(i)}_{p}[1,:]$. To calculate the embeddings of original target nodes, e.g., $\mathbf{z}^{(i)}_{p}$, we only have to fed $\mathbf{x}^{(i)} = \mathbf{X}[i,:]$ into $GNN_{\theta}(\cdot)$ without the underlying graph structure since there is only a single node $v_i$. In such a way, $GNN_{\theta}(\cdot)$ degrades to a MLP that is parameterized with $\mathbf{\Theta}$. We illustrate the calculation of $\mathbf{z}^{(i)}_{p}$ as follows: \begin{equation} \mathbf{z}^{(i)}_{p} = GNN_{\theta}\left(\mathbf{x}^{(i)}\right) = \sigma\left(\mathbf{x}^{(i)} \mathbf{\Theta} \right), \label{eq:mlp} \end{equation} where adopting $\mathbf{\Theta}$ ensures $\mathbf{z}^{(i)}_{p}$ and $\mathbf{h}^{(i)}_{p}$ are mapped into the same latent space to assist the following contrasting. \noindent \textbf{Patch-level Contrasting.} To measure the agreement between $\mathbf{h}^{(i)}_{p}$ and $\mathbf{z}^{(i)}_{p}$, we adopt a bilinear mapping to compute the similarity between them (i.e., the positive score in \texttt{ANEMONE}\xspace), denoted as $\mathbf{s}^{(i)}_{p}$: \begin{equation} \mathbf{s}^{(i)}_{p} = Bilinear\left( \mathbf{h}^{(i)}_{p}, \mathbf{z}^{(i)}_{p} \right) = \sigma\left(\mathbf{h}^{(i)}_{p} \mathbf{W}_p {\mathbf{z}^{(i)}_{p}}^\top \right), \label{eq:patch-level anemone positive score} \end{equation} where $W_p \in \mathbb{R}^{D' \times D'}$ is a set of trainable weighting parameters, and $\sigma(\cdot)$ denotes the Sigmoid activation in this equation. Also, there is a \textit{patch-level negative sampling} mechanism to assist model training and avoid it being biased by merely optimizing on positive pairs. Specifically, we first calculate $\mathbf{h}^{(j)}_{p}$ based on the subgraph centred at an irrelevant node $v_j$, then we calculate the similarity between $\mathbf{h}^{(j)}_{p}$ and $\mathbf{z}^{(i)}_{p}$ (i.e., negative score) with the identical bilinear mapping: \begin{equation} \tilde{\mathbf{s}}^{(i)}_{p} = Bilinear\left( \mathbf{h}^{(j)}_{p}, \mathbf{z}^{(i)}_{p} \right) = \sigma\left(\mathbf{h}^{(j)}_{p} \mathbf{W}_p {\mathbf{z}^{(i)}_{p}}^\top \right). \label{eq:patch-level anemone negative score} \end{equation} In practice, we train the model in a mini-batch manner as mentioned in Subsection \ref{subsec: augmented subg generation}. Thus, $\mathbf{h}^{(j)}_{p}$ can be easily acquired by using other masked target node embeddings in the same mini-batch with size $B$. Finally, the patch-level contrastive objective of \texttt{ANEMONE}\xspace (under the context of unsupervised graph anomaly detection) can be formalized with the Jensen-Shannon divergence \cite{dgi_velickovic2019deep}: \begin{equation} \mathcal{L}_{p}=-\frac{1}{2n}\sum_{i=1}^{B}\left(log\left(\mathbf{s}^{(i)}_{p}\right)+log\left(1-\tilde{\mathbf{s}}^{(i)}_{p}\right)\right). \label{eq:patch-level loss} \end{equation} \begin{figure*} \caption{ The conceptual framework of \texttt{ANEMONE-FS}\xspace, which shares the similar pipeline of \texttt{ANEMONE}\xspace. Given an attributed graph $\mathcal{G}$, we first sample a batch of target nodes. After this, we design a different multi-scale contrastive network equipped with two contrastive routes, where the agreements between node and contextual embeddings are maximized for unlabeled node while minimized for labeled anomalies in a mini-batch. At the inference stage, it has the same graph anomaly detector to estimate the anomlay socre of each node in $\mathcal{G}$.} \label{fig:framework2} \end{figure*} \subsection{Context-level Contrastive Network} \label{subsec: context-level} Different from the patch-level contrastiveness, the objective of context-level contrasting is to learn the global agreement between the contextual embedding of a masked target node $v_i$ in $\mathcal{G}^{(i)}_c$ and the embedding of itself by mapping its attributes to the latent space. The intuition behind this is to capture the \textit{global anomalies} that are difficult to be distinguished by directly comparing with the closest neighbors. For instance, the fraudsters are also likely to camouflage themselves in large communities, resulting in a more challenging anomaly detection task. To enable the model to detect these anomalies, we propose the context-level contrastiveness with a multi-scale (i.e., node versus graph) contrastive learning schema. In the middle part of Figure \ref{fig:framework1}, we illustrate the conceptual design of our context-level contrastive network, which has three main components: graph encoder, readout module, and contrastive module. \noindent \textbf{Graph encoder and readout module.} The context-level graph encoder shares the identical neural architecture of the patch-level graph encoder, but it has a different set of trainable parameters $\mathbf{\Phi}$. Specifically, given a subgraph $\mathcal{G}^{(i)}_c$ centred at the target node $v_i$, we calculate the node embeddings of $\mathcal{G}^{(i)}_c$ in a similar way: \begin{equation} \mathbf{H}^{(i)}_{c} = GNN_{\phi}\left(\mathcal{G}^{(i)}_c\right) = \sigma\left(\widetilde{{\mathbf{D}}^{(i)}_c}^{-\frac{1}{2}} \widetilde{{\mathbf{A}}^{(i)}_c} \widetilde{{\mathbf{D}}^{(i)}_c}^{-\frac{1}{2}} \mathbf{X}^{(i)}_c \mathbf{\Phi} \right). \label{eq:gnn2} \end{equation} The main difference between the context-level and patch-level contrastiveness is that the former aims to contrast target node embeddings with subgraph embeddings (i.e., node versus subgraph), while the aforementioned patch-level contrasting learns the agreements between the masked and original target node embeddings (i.e., node versus node). To obtain the contextual embedding of target node $v_i$ (i.e., $\mathbf{h}^{(i)}_{c}$), we aggregate all node embeddings in $\mathbf{H}^{(i)}_{c}$ with an average readout function: \begin{equation} \mathbf{h}^{(i)}_{c} = readout \left( \mathbf{H}^{(i)}_{c} \right) = \frac{1}{K}\sum_{j=1}^{K}\mathbf{H}^{(i)}_{c}[j,:], \label{eq:readout} \end{equation} where $K$ denotes the number of nodes in a contextual subgraph. Similarly, we can also obtain the embedding of $v_i$ via a non-linear mapping: \begin{equation} \mathbf{z}^{(i)}_{c} = GNN_{\phi}\left(\mathbf{x}^{(i)}\right) = \sigma\left(\mathbf{x}^{(i)} \mathbf{\Phi} \right), \label{eq:mlp2} \end{equation} where $\mathbf{z}^{(i)}_{c}$ and $\mathbf{h}^{(i)}_{c}$ are projected to the same latent space with a shared set of parameters $\mathbf{\Phi}$. \noindent \textbf{Context-level contrasting.} We measure the similarity between $\mathbf{h}^{(i)}_{c}$ and $\mathbf{z}^{(i)}_{c}$ (i.e., the positive score in \texttt{ANEMONE}\xspace) with a different parameterized bilinear function: \begin{equation} \mathbf{s}^{(i)}_{c} = Bilinear\left( \mathbf{h}^{(i)}_{c}, \mathbf{z}^{(i)}_{c} \right) = \sigma\left(\mathbf{h}^{(i)}_{c} \mathbf{W}_c {\mathbf{z}^{(i)}_{c}}^\top \right), \label{eq:context-level anemone positive score} \end{equation} where $W_c \in \mathbb{R}^{D' \times D'}$ and $\sigma(\cdot)$ is the Sigmoid activation. Similarly, there is a \textit{context-level negative sampling} mechanism to avoid model collapse, where negatives $\mathbf{h}^{(j)}_{c}$ are obtained from other irrelevant surrounding contexts $\mathcal{G}^{(j)}_c$ where $j \neq i$. Thus, the negative score can be obtained via: \begin{equation} \tilde{\mathbf{s}}^{(i)}_{c} = Bilinear\left( \mathbf{h}^{(j)}_{c}, \mathbf{z}^{(i)}_{c} \right) = \sigma\left(\mathbf{h}^{(j)}_{c} \mathbf{W}_c {\mathbf{z}^{(i)}_{c}}^\top \right). \label{eq:context-level anemone negative score} \end{equation} Finally, the context-level contrastiveness is ensured by optimizing the following objective: \begin{equation} \mathcal{L}_{c}=-\frac{1}{2n}\sum_{i=1}^{B}\left(log\left(\mathbf{s}^{(i)}_{c}\right)+log\left(1-\tilde{\mathbf{s}}^{(i)}_{c}\right)\right). \label{eq:context-level loss} \end{equation} \subsection{Few-shot Multi-scale Contrastive Network} \label{subsec: few-shot} The above patch-level and context-level contrastiveness are conceptually designed to detect attributive and structural graph anomalies in an unsupervised manner (i.e., the proposed \texttt{ANEMONE}\xspace method in Algorithm \ref{algo: anemone}). However, how to incorporate limited supervision signals remains unknown if there are a few available labeled anomalies. To answer this question, we proposed an extension of \texttt{ANEMONE}\xspace named \texttt{ANEMONE-FS}\xspace to perform graph anomaly detection in a few-shot supervised manner without drastically changing the overall framework (i.e., Figure \ref{fig:framework2}) and training objective (i.e., Equation \ref{eq:patch-level loss}, \ref{eq:context-level loss}, and \ref{eq:loss}). This design further boosts the performance of \texttt{ANEMONE}\xspace significantly with only a few labeled anomalies, which can be easily acquired in many real-world applications. Specifically, given a mini-batch of target nodes $\mathbfcal{V}_B=\{\mathbfcal{V}^L_B, \mathbfcal{V}^U_B\}$ with labeled anomalies and unlabeled nodes, we design and insert a different contrastive route in the above patch-level and context-level contrastiveness, as the bottom dashed arrows, i.e., the so-called negative pairs, in the red and green dashed boxes shown in the middle part of Figure \ref{fig:framework2}. \noindent \textbf{Few-shot patch-level contrasting.} For nodes in $\mathbfcal{V}^U_B$, we follow Equation \ref{eq:patch-level anemone positive score} and \ref{eq:patch-level anemone negative score} to compute positive and negative scores as in \texttt{ANEMONE}\xspace. However, for a target node $v_k$ in $\mathbfcal{V}^L_B$, we minimize the mutual information between its masked node embedding $\mathbf{h}^{(k)}_{p}$ and original node embedding $\mathbf{z}^{(k)}_{p}$, which equivalents to enrich the patch-level negative set with an additional negative pair: \begin{equation} \tilde{\mathbf{s}}^{(k)}_{p} = Bilinear\left( \mathbf{h}^{(k)}_{p}, \mathbf{z}^{(k)}_{p} \right) = \sigma\left(\mathbf{h}^{(k)}_{p} \mathbf{W}_p {\mathbf{z}^{(k)}_{p}}^\top \right). \label{eq:patch-level anemone-fs negative score} \end{equation} The behind intuitions are in two-folds. Firstly, for most unlabeled nodes, we assume that most of them are not anomalies so that the mutual information between masked and original target node embeddings should be maximized for the model to distinguish normal nodes from a few anomalies in an attributed graph. Secondly, for a labeled target node (i.e., an anomaly), this mutual information should be minimized for the model to learn how anomalies should be different from their surrounding contexts. As a result, there are two types of patch-level negative pairs in \texttt{ANEMONE-FS}\xspace: (1) $\mathbf{h}^{(j)}_{p}$ and $\mathbf{z}^{(i)}_{p}$ where $v_i \in \mathbfcal{V}^U_B$ and $i \neq j$; (2) $\mathbf{h}^{(k)}_{p}$ and $\mathbf{z}^{(k)}_{p}$ where $v_k \in \mathbfcal{V}^L_B$. \noindent \textbf{Few-shot context-level contrasting.} Similarly, for a node $v_k$ in $\mathbfcal{V}^L_B$, we minimize the mutual information between its contextual embedding $\mathbf{h}^{(k)}_{c}$ and the embedding of itself $\mathbf{h}^{(k)}_{c}$ by treating them as a negative pair: \begin{equation} \tilde{\mathbf{s}}^{(k)}_{c} = Bilinear\left( \mathbf{h}^{(k)}_{c}, \mathbf{z}^{(k)}_{c} \right) = \sigma\left(\mathbf{h}^{(k)}_{c} \mathbf{W}_c {\mathbf{z}^{(k)}_{c}}^\top \right). \label{eq:context-level anemone-fs negative score} \end{equation} The behind intuition is the same as in few-shot patch-level contrasting, and we also have two types of negatives in this module to further assist the model in obtaining richer supervision signals from various perspectives. \\ In general, this proposed plug-and-play extension enhances the original self-supervise contrastive anomaly detection mechanism in \texttt{ANEMONE}\xspace by constructing extra negative pairs with the available limited labeled anomalies, enabling the model to achieve better performance in more realistic application scenarios. \subsection{Statistical Graph Anomaly Scorer} \label{subsec: scorer} So far, we have introduced two contrastive mechanisms in \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace for different graph anomaly detection tasks. After the model is well-trained, we propose an universal statistical graph anomaly scorer for both \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace to calculate the anomaly score of each node in $\mathcal{G}$ during the model inference. Specifically, we first generate $R$ subgraphs centred at a target node $v_i$ for both contrastive networks. Then, we calculate patch-level and context-level contrastive scores accordingly, i.e., $[\mathbf{s}^{(i)}_{p,1}, \cdots, \mathbf{s}^{(i)}_{p,R}, \mathbf{s}^{(i)}_{c,1}, \cdots, \mathbf{s}^{(i)}_{c,R},\tilde{\mathbf{s}}^{(i)}_{p,1}, \cdots, \tilde{\mathbf{s}}^{(i)}_{p,R}, \tilde{\mathbf{s}}^{(i)}_{c,1}, \cdots, \tilde{\mathbf{s}}^{(i)}_{c,R}]$. After this, we define the base patch-level and context-level anomaly scores of $v_i$ as follows: \begin{equation} b^{(i)}_{view,j} = \tilde{\mathbf{s}}^{(i)}_{view,j} - \mathbf{s}^{(i)}_{view,j}, \label{eq:base score} \end{equation} where $j \in \{1, \cdots, R\}$, and ``view" corresponds to $p$ or $c$ to denote the patch-level or context-level base score, respectively. If $v_i$ is a normal node, then $\mathbf{s}^{(i)}_{view,j}$ and $\tilde{\mathbf{s}}^{(i)}_{view,j}$ are expected to be close to 1 and 0, leading $b^{(i)}_{view,j}$ close to -1. Otherwise, if $v_i$ is an anomaly, then $\mathbf{s}^{(i)}_{view,j}$ and $\tilde{\mathbf{s}}^{(i)}_{view,j}$ are close to 0.5 due to the mismatch between $v_i$ and its surrounding contexts, resulting in $b^{(i)}_{view,j} \rightarrow 0$. Therefore, we have $b^{(i)}_{view,j}$ in the range of $[-1, 0]$. Although we can directly use the base anomaly score $b^{(i)}_{view,j}$ to indicate whether $v_i$ is an anomaly, we design a more sophisticated statistical abnormality scorer to calculate the final patch-level and context-level anomaly scores $y^{(i)}_{view}$ of $v_i$ based on $b^{(i)}_{view,j}$: \begin{equation} \begin{aligned} \bar{b}^{(i)}_{view} &={\sum_{j=1}^{R} b^{(i)}_{view,j} }/{R}, \\ y^{(i)}_{view} &= \bar{b}^{(i)}_{view} + \sqrt{{\sum_{j=1}^{R}\left(b^{(i)}_{view,j} - \bar{b}^{(i)}_{view}\right)^{2}}/{R}}. \end{aligned} \label{eq:final score v1} \end{equation} The underlying intuitions behind the above equation are: (1) An abnormal node usually has a larger base anomaly score; (2) The base scores of an abnormal node are typically unstable (i.e., with a larger standard deviation) under $R$ evaluation rounds. Finally, we anneal $y^{(i)}_{p}$ and $y^{(i)}_{c}$ to obtain the final anomaly score of $v_i$ with a tunable hyper-parameter $\alpha \in [0, 1]$ to balance the importance of two anomaly scores at different scales: \begin{equation} y^{(i)} = \alpha y^{(i)}_{c} + (1 - \alpha) y^{(i)}_{p}. \label{eq:final score v2} \end{equation} \begin{algorithm}[t] \caption{The Proposed \texttt{ANEMONE}\xspace Algorithm} \label{algo: anemone} \textbf{Input}: Attributed graph $\mathcal{G}$ with a set of unlabeled nodes $\mathbfcal{V}$; Maximum training epochs $E$; Batch size $B$; Number of evaluation rounds $R$. \\ \textbf{Output}: Well-trained graph anomaly detection model $\mathcal{F}^{*}(\cdot)$. \\ \begin{algorithmic}[1] \STATE Randomly initialize the trainable parameters $\mathbf{\Theta}$, $\mathbf{\Phi}$, $\mathbf{W}_p$, and $\mathbf{W}_c$; \STATE $/*$ {\it Model training} $*/$ \FOR{$e \in 1,2,\cdots,E$} \STATE $\mathbfcal{B} \leftarrow$ Randomly split $\mathbfcal{V}$ into batches with size $B$; \FOR{batch $\widetilde{\mathbfcal{B}}=(v_{1},\cdots,v_{B}) \in \mathbfcal{B}$} \STATE Sample two anonymized subgraphs for each node in $\widetilde{\mathbfcal{B}}$, i.e., $\{\mathcal{G}_p^{(1)},\cdots,\mathcal{G}_p^{(B)}\}$ and $\{\mathcal{G}_c^{(1)},\cdots,\mathcal{G}_c^{(B)}\}$; \STATE Calculate the masked and original node embeddings via Eq. \eqref{eq:gnn}, \eqref{eq:mlp}; \STATE Calculate the masked node and its contextual embeddings via Eq. \eqref{eq:gnn2}, \eqref{eq:readout}, and \eqref{eq:mlp2}; \STATE Calculate the patch-level positive and negative scores for for each node in $\widetilde{\mathbfcal{B}}$ via Eq. \eqref{eq:patch-level anemone positive score} and \eqref{eq:patch-level anemone negative score}; \STATE Calculate the context-level positive and negative scores for for each node in $\widetilde{\mathbfcal{B}}$ via Eq. \eqref{eq:context-level anemone positive score} and \eqref{eq:context-level anemone negative score}; \STATE Calculate the loss $\mathcal{L}$ via Eq. \eqref{eq:patch-level loss}, \eqref{eq:context-level loss}, and \eqref{eq:loss}; \STATE Back propagate to update trainable parameters $\mathbf{\Theta}$, $\mathbf{\Phi}$, $\mathbf{W}_p$, and $\mathbf{W}_c$; \ENDFOR \ENDFOR \STATE $/*$ {\it Model inference} $*/$ \FOR{$v_i \in \mathcal{V}$} \FOR{evaluation round $r \in 1,2,\cdots,R$} \STATE Calculate $b^{(i)}_p$ and $b^{(i)}_c$ via Eq. \eqref{eq:base score}; \ENDFOR \STATE Calculate the final patch-level and context-level anomaly scores $y^{(i)}_p$ and $y^{(i)}_c$ over $R$ evaluation rounds via Eq. \eqref{eq:final score v1}; \STATE Calculate the final anomaly score $y^{(i)}$ of $v_i$ via Eq. \eqref{eq:final score v2}; \ENDFOR \end{algorithmic} \end{algorithm} \begin{algorithm}[t] \caption{The Proposed \texttt{ANEMONE-FS}\xspace Algorithm} \label{algo: anemone-fs} \textbf{Input}: Attributed graph $\mathcal{G}$ with a set of labeled and unlabeled nodes $\mathbfcal{V}=\{\mathbfcal{V}^L, \mathbfcal{V}^U\}$ where $|\mathbfcal{V}^L| \ll |\mathbfcal{V}^U|$; Maximum training epochs $E$; Batch size $B$; Number of evaluation rounds $R$. \\ \textbf{Output}: Well-trained graph anomaly detection model $\mathcal{F}^{*}(\cdot)$. \\ \begin{algorithmic}[1] \STATE Randomly initialize the trainable parameters $\mathbf{\Theta}$, $\mathbf{\Phi}$, $\mathbf{W}_p$, and $\mathbf{W}_c$; \STATE $/*$ {\it Model training} $*/$ \FOR{$e \in 1,2,\cdots,E$} \STATE $\mathbfcal{B} \leftarrow$ Randomly split $\mathbfcal{V}$ into batches with size $B$; \FOR{batch $\widetilde{\mathbfcal{B}}=\{\mathbfcal{V}^L_B, \mathbfcal{V}^U_B\}=(v_{1},\cdots,v_{B}) \in \mathbfcal{B}$} \STATE Sample two anonymized subgraphs for each node in $\widetilde{\mathbfcal{B}}$, i.e., $\{\mathcal{G}_p^{(1)},\cdots,\mathcal{G}_p^{(B)}\}$ and $\{\mathcal{G}_c^{(1)},\cdots,\mathcal{G}_c^{(B)}\}$; \STATE Calculate the masked and original node embeddings via Eq. \eqref{eq:gnn} and \eqref{eq:mlp}; \STATE Calculate the masked node and its contextual embeddings via Eq. \eqref{eq:gnn2}, \eqref{eq:readout}, and \eqref{eq:mlp2}; \STATE Calculate the patch-level positive and negative scores for each node in $\mathbfcal{V}^U_B$ via Eq. \eqref{eq:patch-level anemone positive score} and \eqref{eq:patch-level anemone negative score}; \STATE Calculate the patch-level extra negative scores for each node in $\mathbfcal{V}^L_B$ via Eq. \eqref{eq:patch-level anemone-fs negative score}; \STATE Calculate the context-level positive and negative scores for each node in $\mathbfcal{V}^U_B$ via Eq. \eqref{eq:context-level anemone positive score} and \eqref{eq:context-level anemone negative score}; \STATE Calculate the context-level extra negative scores for each node in $\mathbfcal{V}^L_B$ via Eq. \eqref{eq:context-level anemone-fs negative score}; \STATE Calculate the loss $\mathcal{L}$ via Eq. \eqref{eq:patch-level loss}, \eqref{eq:context-level loss}, and \eqref{eq:loss}; \STATE Back propagate to update trainable parameters $\mathbf{\Theta}$, $\mathbf{\Phi}$, $\mathbf{W}_p$, and $\mathbf{W}_c$; \ENDFOR \ENDFOR \STATE $/*$ {\it Model inference} $*/$ \FOR{$v_i \in \mathbfcal{V}^U$} \FOR{evaluation round $r \in 1,2,\cdots,R$} \STATE Calculate $b^{(i)}_p$ and $b^{(i)}_c$ via Eq. \eqref{eq:base score}; \ENDFOR \STATE Calculate the final patch-level and context-level anomaly scores $y^{(i)}_p$ and $y^{(i)}_c$ over $R$ evaluation rounds via Eq. \eqref{eq:final score v1}; \STATE Calculate the final anomaly score $y^{(i)}$ of $v_i$ via Eq. \eqref{eq:final score v2}; \ENDFOR \end{algorithmic} \end{algorithm} \subsection{Model Training and Algorithms} \label{subsec: optimization} \noindent \textbf{Model training.} By combining the patch-level and context-level contrastive losses defined in Equation \ref{eq:patch-level loss} and \ref{eq:context-level loss}, we have the overall training objective by minimizing the following loss: \begin{equation} \mathcal{L}= \alpha \mathcal{L}_{c} + (1 - \alpha) \mathcal{L}_{p}, \label{eq:loss} \end{equation} where $\alpha$ is same as in Equation \ref{eq:final score v2} to balance the importance of two contrastive modules. The overall procedures of \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace are in Algorithms \ref{algo: anemone} and \ref{algo: anemone-fs}. Specifically, in \texttt{ANEMONE}\xspace, we first sample a batch of nodes from the input attributed graph (line 5). Then, we calculate the positive and negative contrastive scores for each node (lines 6-10) to obtain the multi-scale contrastive losses, which is adopted to calculate the overall training loss (line 11) to update all trainable parameters (line 12). For \texttt{ANEMONE-FS}\xspace, the differences are in two-folds. Firstly, it takes an attributed graph with a few available labeled anomalies as the input. Secondly, for labeled anomalies and unlabeled nodes in a batch, it has different contrastive routes in patch-level and context-level contrastive networks (lines 9-12). During the model inference, \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace shares the same anomaly scoring mechanism, where the statistical anomaly score for each node in $\mathbfcal{V}$ or$\mathbfcal{V}^U$ is calculated (lines 16-22 in Algorithm \ref{algo: anemone} and lines 18-24 in Algorithm \ref{algo: anemone-fs} ). \noindent \textbf{Complexity analysis.} We analyse the time complexity of \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace algorithms in this subsection. For the shared anonymized subgraph sampling module, the time complexity of using RWR algorithm to sample a subgraph centred at $v_i$ is $\mathcal{O}(Kd)$, where $K$ and $d$ are the number of nodes in a subgraph and the average node degree in $\mathcal{G}$. Regarding the two proposed contrastive modules, their time complexities are mainly contributed by the underlying graph encoders, which are $\mathcal{O}(K^2)$. Thus, given $N$ nodes in $\mathbfcal{V}$, the time complexity of model training is $\mathcal{O}\big(NK(d+K)\big)$ in both \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace. During the model inference, the time complexity of \texttt{ANEMONE}\xspace is $\mathcal{O}\big(RNK(d+K)\big)$, where $R$ denotes the total evaluation rounds. For \texttt{ANEMONE-FS}\xspace, its inference time complexity is $\mathcal{O}\big(RN^UK(d+K)\big)$, where $N^U = |\mathbfcal{V^U}|$ denotes the number of unlabeled nodes in $\mathcal{G}$. \section{Experiments} \label{sec:experiments} \begin{table}[t] \centering \caption{The statistics of the datasets. The upper two datasets are social networks, and the remainders are citation networks.} \begin{tabular}{@{}c|c|c|c|c@{}} \toprule \textbf{Dataset} & \textbf{Nodes} & \textbf{Edges} & \textbf{Features} & \textbf{Anomalies} \\ \midrule \textbf{Cora} \cite{sen2008collective} & 2,708 & 5,429 & 1,433 & 150 \\ \textbf{CiteSeer} \cite{sen2008collective} & 3,327 & 4,732 & 3,703 & 150 \\ \textbf{PubMed} \cite{sen2008collective} & 19,717 & 44,338 & 500 & 600 \\ \textbf{ACM} \cite{tang2008arnetminer} & 16,484 & 71,980 & 8,337 & 600 \\ \textbf{BlogCatalog} \cite{tang2009relational} & 5,196 & 171,743 & 8,189 & 300 \\ \textbf{Flickr} \cite{tang2009relational} & 7,575 & 239,738 & 12,407 & 450 \\ \bottomrule \end{tabular} \label{table:dataset} \end{table} In this section, we conduct a series of experiments to evaluate the anomaly detection performance of the proposed \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace on both unsupervised and few-shot learning scenarios. Specifically, we address the following research questions through experimental analysis: \begin{figure*} \caption{The comparison of ROC curves on four datasets in unsupervised learning scenario.} \label{subfig:parameter} \label{fig:roc} \end{figure*} \begin{table*}[!htbp] \small \centering \caption{The comparison of anomaly detection performance (i.e., AUC) in unsupervised learning scenario. The best performance is highlighted in \textbf{bold}.} { \begin{tabular}{p{85 pt}<{}|p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}} \toprule Method & Cora & CiteSeer & PubMed & ACM & BlogCatalog & Flickr \\ \midrule AMEN \cite{amen_perozzi2016scalable} & 0.6266 & 0.6154 & 0.7713 & 0.5626 & 0.6392 & 0.6573 \\ Radar \cite{radar_li2017radar} & 0.6587 & 0.6709 & 0.6233 & 0.7247 & 0.7401 & 0.7399 \\ ANOMALOUS \cite{anomalous_peng2018anomalous} & 0.5770 & 0.6307 & 0.7316 & 0.7038 & 0.7237 & 0.7434 \\ \midrule DGI \cite{dgi_velickovic2019deep} & 0.7511 & 0.8293 & 0.6962 & 0.6240 & 0.5827 & 0.6237 \\ DOMINANT \cite{dominant_ding2019deep} & 0.8155 & 0.8251 & 0.8081 & 0.7601 & 0.7468 & 0.7442 \\ CoLA \cite{cola_liu2021anomaly} & 0.8779 & 0.8968 & 0.9512 & 0.8237 & 0.7854 & 0.7513 \\ \midrule \texttt{ANEMONE}\xspace & \textbf{0.9057} & \textbf{0.9189} & \textbf{0.9548} & \textbf{0.8709} & \textbf{0.8067} & \textbf{0.7637} \\ \bottomrule \end{tabular} } \label{table:overall_unsup} \end{table*} \begin{table*}[!htbp] \small \centering \caption{The comparison of anomaly detection performance (i.e., AUC) in few-shot learning scenario. The best performance is highlighted in \textbf{bold}.} { \begin{tabular}{p{85 pt}<{}|p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}p{45 pt}<{\centering}} \toprule Method & Cora & CiteSeer & PubMed & ACM & BlogCatalog & Flickr \\ \midrule AMEN \cite{amen_perozzi2016scalable} & 0.6257&0.6103&0.7725&0.5632&0.6358&0.6615\\ Radar \cite{radar_li2017radar} &0.6589&0.6634&0.6226&0.7253&0.7461&0.7357\\ ANOMALOUS \cite{anomalous_peng2018anomalous} &0.5698&0.6323&0.7283&0.6923&0.7293&0.7504\\ \midrule DGI \cite{dgi_velickovic2019deep} &0.7398&0.8347&0.7041&0.6389&0.5936&0.6295\\ DOMINANT \cite{dominant_ding2019deep} &0.8202&0.8213&0.8126&0.7558&0.7391&0.7526\\ CoLA \cite{cola_liu2021anomaly} &0.8810&0.8878&0.9517&0.8272&0.7816&0.7581\\ \midrule DeepSAD \cite{deepsad_ruff2019deep} & 0.4909& 0.5269& 0.5606& 0.4545& 0.6277& 0.5799\\ SemiGNN \cite{semignn_wang2019semi} & 0.6657& 0.7297& OOM& OOM& 0.5289& 0.5426\\ GDN \cite{gdn_ding2021few} &0.7577&0.7889&0.7166&0.6915&0.5424&0.5240\\ \midrule \texttt{ANEMONE}\xspace & {0.8997} & {0.9191} & {0.9536} & {0.8742} & {0.8025} & {0.7671} \\ \texttt{ANEMONE-FS}\xspace & \textbf{0.9155} & \textbf{0.9318} & \textbf{0.9561} & \textbf{0.8955} & \textbf{0.8124} & \textbf{0.7781} \\ \bottomrule \end{tabular} } \label{table:overall_fs} \end{table*} \begin{itemize} \item \textit{RQ1:} How do the proposed \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace perform in comparison to state-of-the-art graph anomaly detection methods? \item \textit{RQ2:} How does the performance of \texttt{ANEMONE-FS}\xspace change by providing different numbers of labeled anomalies? \item \textit{RQ3:} How do the contrastiveness in patch-level and context-level influence the performance of \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace? \item \textit{RQ4:} How do the key hyper-parameters impact the performance of \texttt{ANEMONE}\xspace? \end{itemize} \subsection{Datasets} We conduct experiments on six commonly used datasets for graph anomaly detection, including four citation network datasets \cite{sen2008collective,tang2008arnetminer} (i.e., Cora, CiteSeer, PubMed, and ACM) and two social network datasets \cite{tang2009relational} (i.e., BlogCatalog and Flickr). Dataset statistics are summarized in Table \ref{table:dataset}. Since ground-truth anomalies are inaccessible for these datasets, we follow previous works \cite{dominant_ding2019deep,cola_liu2021anomaly} to inject two types of synthetic anomalies (i.e., structural anomalies and contextual anomalies) into the original graphs. For structural anomaly injection, we use the injection strategy proposed by \cite{anoinj_s_ding2019interactive}: several groups of nodes are randomly selected from the graph, and then we make the nodes within one group fully linked to each other. In this way, such nodes can be regarded as structural anomalies. To generate contextual anomalies, following \cite{anoinj_c_song2007conditional}, a target node along with $50$ auxiliary nodes are randomly sampled from the graph. Then, we replace the features of the target node with the features of the farthest auxiliary node (i.e., the auxiliary node with the largest features' Euclidean distance to the target node). By this, we denote the target node as a contextual anomaly. We inject two types of anomalies with the same quantity and the total number is provided in the last column of Table \ref{table:dataset}. \subsection{Baselines} We compare our proposed \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace with three types of baseline methods, including (1) shallow learning-based unsupervised methods (i.e., AMEN \cite{amen_perozzi2016scalable}, Radar \cite{radar_li2017radar}), (2) deep learning-based unsupervised methods (i.e., ANOMALOUS \cite{anomalous_peng2018anomalous}, DOMINANT \cite{dominant_ding2019deep}, and CoLA \cite{cola_liu2021anomaly}), and (3) semi-supervised methods (i.e., DeepSAD \cite{deepsad_ruff2019deep}, SemiGNN \cite{semignn_wang2019semi}, and GDN \cite{gdn_ding2021few}). Details of these methods are introduced as following: \begin{itemize} \item \textbf{AMEN} \cite{amen_perozzi2016scalable} is an unsupervised graph anomaly detection method which detects anomalies by analyzing the attribute correlation of ego-network of nodes. \item \textbf{Radar} \cite{radar_li2017radar} identifies anomalies in graphs by residual and attribute-structure coherence analysis. \item \textbf{ANOMALOUS} \cite{anomalous_peng2018anomalous} is an unsupervised method for attributed graphs, which performs anomaly detection via CUR decomposition and residual analysis. \item \textbf{DGI} \cite{dgi_velickovic2019deep} is an unsupervised contrastive learning method for representation learning. In DGI, we use the score computation module in \cite{cola_liu2021anomaly} to estimate nodes' abnormality. \item \textbf{DOMINANT} \cite{dominant_ding2019deep} is a deep graph autoencoder-based unsupervised method that detects anomalies by evaluating the reconstruction errors of each node. \item \textbf{CoLA} \cite{cola_liu2021anomaly} is a contrastive learning-based anomaly detection method which captures anomalies with a GNN-based contrastive framework. \item \textbf{DeepSAD} \cite{deepsad_ruff2019deep} is a deep learning-based anomaly detection method for non-structured data. We take node attributes as the input of DeepSAD. \item \textbf{SemiGNN} \cite{semignn_wang2019semi} is a semi-supervised fraud detection method that use attention mechanism to model the correlation between different neighbors/views. \item \textbf{GDN} \cite{gdn_ding2021few} is a GNN-based model that detects anomalies in few-shot learning scenarios. It leverages a deviation loss to train the detection model in an end-to-end manner. \end{itemize} \subsection{Experimental Setting} \mysubsubtitle{Evaluation Metric} We employ a widely used metric, AUC-ROC \cite{dominant_ding2019deep,cola_liu2021anomaly}, to evaluate the performance of different anomaly detection methods. The ROC curve indicates the plot of true positive rate against false positive rate, and the AUC value is the area under the ROC curve. The value of AUC is within the range $[0,1]$ and a larger value represents a stronger detection performance. To reduce the bias caused by randomness and compare fairly \cite{cola_liu2021anomaly}, for all datasets, we conduct a $5$-run experiment and report the average performance.\\ \mysubsubtitle{Dataset Partition} In an unsupervised learning scenario, we use graph data $\mathcal{G}=(\mathbf{X},\mathbf{A})$ to train the models, and evaluate the anomaly detection performance on the full node set $\mathbfcal{V}$ (including all normal and abnormal nodes). In few-shot learning scenario, we train the models with $\mathcal{G}$ and $k$ anomalous labels (where $k$ is the size of labeled node set $|\mathbfcal{V}^L|$), and use the rest set of nodes $\mathbfcal{V}^U$ to measure the models' performance. \\ \mysubsubtitle{Parameter Settings} In our implementation, the size $K$ of subgraph and the dimension of embeddings are fixed to $4$ and $64$, respectively. The trade-off parameter $\alpha$ is searched in $\{0.2, 0.4, 0.6, 0.8, 1\}$. The number of testing rounds of anomaly estimator is set to $256$. We train the model with Adam optimizer with a learning rate $0.001$. For Cora, Citeseer, and Pubmed datasets, we train the model for $100$ epochs; for ACM, BlogCatalog, and Flickr datasets, the numbers of epochs are $2000$, $1000$, and $500$, respectively. \subsection{Performance Comparison (RQ1)} We evaluate \texttt{ANEMONE}\xspace in unsupervised learning scenario where labeled anomaly is unavailable, and evaluate \texttt{ANEMONE-FS}\xspace in few-shot learning scenario ($k=10$) where $10$ annotated anomalies are known during model training. \\ \mysubsubtitle{Performance in Unsupervised Learning Scenario} In unsupervised scenario, we compared \texttt{ANEMONE}\xspace with $6$ unsupervised baselines. The ROC curves on $4$ representative datasets are illustrated in Fig. \ref{fig:roc}, and the comparison of AUC value on all $6$ datasets is provided in Table \ref{table:overall_unsup}. From these results, we have the following observations. \begin{itemize} \item \texttt{ANEMONE}\xspace consistently outperforms all baselines on six benchmark datasets. The performance gain is due to (1) the two-level contrastiveness successfully capturing anomalous patterns in different scales and (2) the well-designed anomaly estimator effectively measuring the abnormality of each node. \item The deep learning-based methods significantly outperform the shallow learning-based methods, which illustrates the capability of GNNs in modeling data with complex network structures and high-dimensional features. \item The ROC curves by \texttt{ANEMONE}\xspace are very close to the points in the upper left corner, indicating our method can precisely discriminate abnormal samples from a large number of normal samples. \end{itemize} \begin{table}[t] \small \centering \caption{Few-shot performance analysis of \texttt{ANEMONE-FS}\xspace.} { \begin{tabular}{l|cccc} \toprule Setting & Cora & CiteSeer & ACM & BlogCatalog \\ \midrule Unsup. & 0.8997&0.9191&0.8742&0.8025\\ \midrule 1-shot &0.9058&0.9184&0.8858&0.8076\\ 3-shot &0.9070&0.9199&0.8867&0.8125\\ 5-shot &0.9096&0.9252&0.8906&0.8123\\ 10-shot &0.9155&0.9318&0.8955&0.8124\\ 15-shot &0.9226&0.9363&0.8953&0.8214\\ 20-shot &0.9214&0.9256&0.8965&0.8228\\ \bottomrule \end{tabular} } \label{table:fewshot} \end{table} \mysubsubtitle{Performance in Few-shot Learning Scenario} In few-shot learning scenario, we consider both unsupervised and semi-supervised baseline methods for the comparison with our methods. The results are demonstrated in Table \ref{table:overall_fs}. As we can observe, \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace achieve consistently better performance than all baselines, which validates that our methods can handle few-shot learning setting as well. Also, we find that \texttt{ANEMONE-FS}\xspace has better performance than \texttt{ANEMONE}\xspace, meaning that our proposed solution for few-shot learning can further leverage the knowledge from a few numbers of labeled anomalies. In comparison, the semi-supervised learning methods (i.e., DeepSAD, SemiGNN, and GDN) do not show a competitive performance, indicating their limited capability in exploiting the label information. \subsection{Few-shot Performance Analysis (RQ2)} \begin{figure} \caption{Anomaly detection performance with different selection of trade-off parameter $\alpha$ in unsupervised and few-shot learning scenarios.} \label{fig:ablation} \end{figure} \begin{figure*} \caption{Parameter sensitivities of \texttt{ANEMONE}\xspace w.r.t. three hyper-parameters on six benchmark datasets.} \label{subfig:round} \label{subfig:subg} \label{subfig:dim} \label{fig:param} \end{figure*} In order to verify the effectiveness of \texttt{ANEMONE-FS}\xspace in different few-shot anomaly detection settings, we change the number $k$ of anomalous samples for model training to form $k$-shot learning settings for evaluation. We perform experiments on four datasets (i.e., Cora, CiteSeer, ACM, and BlogCatalog) and select $k$ from $\{1,3,5,10,15,20\}$. The experimental results are demonstrated in Table \ref{table:fewshot} where the performance in unsupervised setting (denoted as ``Unsup.'') is also reported as a baseline. The results show that the \texttt{ANEMONE-FS}\xspace can achieve good performance even when only one anomaly node is provided (i.e., 1-shot setting). A representative example is the results of ACM dataset where a $1.16\%$ performance gain is brought by one labeled anomaly. Such an observation indicates that \texttt{ANEMONE-FS}\xspace can effectively leverage the knowledge from scarce labeled samples to better model the anomalous patterns. Another finding is that the anomaly detection performance generally increases following the growth of $k$ especially when $k\leq15$. This finding demonstrates that \texttt{ANEMONE-FS}\xspace can further optimize the anomaly detection model when more labeled anomalies are given. \subsection{Ablation Study (RQ3)} In this experiment, we investigate the contribution of patch- and context- level contrastiveness to the anomaly detection performance of \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace. In concrete, we adjust the value of trade-off parameter $\alpha$ and the results are illustrated in Fig. \ref{fig:ablation}. Note that $\alpha = 0$ and $\alpha = 1$ mean that the model only considers patch- and context level contrastive learning, respectively. As we can observe in Fig. \ref{fig:ablation}, \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace can achieve the highest AUC values when $\alpha$ is between $0.2$ and $0.8$, and the best selections of $\alpha$ for each dataset are quite different. Accordingly, we summarize that jointly considering the contrastiveness in both levels always brings the best detection performance. We also notice that in some datasets (i.e., Cora, CiteSeer, and PubMed) context-level contrastive network performs better than the patch-level one, while in the rest datasets the patch-level contrastiveness brings better results. It suggests that two types of contrastiveness have unique contributions in identifying anomalies from network data with diverse properties. \subsection{Parameter Sensitivity (RQ4)} To study how our method is impacted by the key hyper-parameters, we conduct experiments for \texttt{ANEMONE}\xspace with different selections of evaluation rounds $R$, subgraph size $K$, and hidden dimension $D'$. \mysubsubtitle{Evaluation Rounds} To explore the sensitivity of \texttt{ANEMONE}\xspace to evaluation rounds $R$, we tune the $R$ from $1$ to $512$ on six datasets, and the results are demonstrated in Fig. \ref{subfig:round}. As we can find in the figure, the detection performance is relatively poor when $R<4$, indicating that too few evaluation rounds are insufficient to represent the abnormality of each node. When $R$ is between $4$ and $256$, we can witness a significant growing trade of AUC following the increase of $R$, which demonstrates that adding evaluation rounds within certain ranges can significantly enhance the performance of \texttt{ANEMONE}\xspace. An over-large $R$ ($R=512$) does not boost the performance but brings heavier computational cost. Hence, we fix $R=256$ in our experiments to balance performance and efficiency. \mysubsubtitle{Subgraph Size} In order to investigate the impact of subgraph size, we search the node number of contextual subgraph $K$ in the range of $\{2,3,\cdots,10\}$. We plot the results in Fig. \ref{subfig:subg}. We find that \texttt{ANEMONE}\xspace is not sensitive to the choice of $K$ on datasets except Flickr, which verifies the robustness of our method. For citation networks (i.e., Cora, CiteSeer, PubMed, and ACM), a suitable subgraph size between $3$ and $5$ results in the best performance. Differently, BlogCatalog requires a larger subgraph to consider more contextual information, while Flickr needs a smaller augmented subgraph for contrastive learning. \mysubsubtitle{Hidden Dimension} In this experiment, we study the selection of hidden dimension $D'$ in \texttt{ANEMONE}\xspace. We alter the value of $D'$ from $2$ to $256$ and the effect of $D'$ on AUC is illustrated in Fig. \ref{subfig:dim}. As shown in the figure, when $D'$ is within $[2,64]$, there is a significant boost in anomaly detection performance with the growth of $D'$. This observation indicates that node embeddings with a larger length can help \texttt{ANEMONE}\xspace capture more complex information. We also find that the performance gain becomes light when $D'$ is further enlarged. Consequently, we finally set $D'=64$ in our main experiments. \section{Conclusion} \label{sec:conclusion} In this paper, we investigate the problem of graph anomaly detection. By jointly capturing anomalous patterns from multiple scales with both patch level and context level contrastive learning, we propose a novel algorithm, \texttt{ANEMONE}\xspace, to learn the representation of nodes in a graph. With a statistical anomaly estimator to capture the agreement from multiple perspectives, we predict an anomaly score for each node so that anomaly detection can be conducted subsequently. As a handful of ground-truth anomalies may be available in real applications, we further extend our method as \texttt{ANEMONE-FS}\xspace, a powerful method to utilize labeled anomalies to handle the settings of few-shot graph anomaly detection. Experiments on six benchmark datasets validate the performance of the proposed \texttt{ANEMONE}\xspace and \texttt{ANEMONE-FS}\xspace. \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/yuzheng.jpg}}]{Yu Zheng} received the B.S. and M.S. degrees in computer science from Northwest A\&F University, China, in 2008 and 2011, respectively. She is currently pursuing her Ph.D. degree in computer science at La Trobe University, Melbourne, Australia. Her research interests include image classification, data mining, and machine learning. \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/mingjin.jpg}}]{Ming Jin} received the B.Eng. degree from the Hebei University of Technology, Tianjin, China, in 2017, and M.Inf.Tech. degree from the University of Melbourne, Melbourne, Australia, in 2019. He is currently pursuing his Ph.D. degree in computer science at Monash University, Melbourne, Australia. His research focuses on graph neural networks (GNNs), time series analyse, data mining, and machine learning. \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/yixin-bio.jpg}}]{Yixin Liu} received the B.S. degree and M.S. degree from Beihang University, Beijing, China, in 2017 and 2020, respectively. He is currently pursuing his Ph.D. degree in computer science at Monash University, Melbourne, Australia. His research concentrates on data mining, machine learning, and deep learning on graphs. \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/lianhua.jpg}}]{Lianhua Chi} received the dual Ph.D. degrees in computer science from the University of Technology Sydney, Australia, and the Huazhong University of Science and Technology, Wuhan, China, in 2015. She was a Post-Doctoral Research Scientist in IBM Research Melbourne. Dr. Chi was a recipient of the Best Paper Award in PAKDD in 2013. Currently, she is a Lecturer with the Department of Computer Science and Information Technology at La Trobe University since 2018. Her current research interests include data mining, machine learning and big data hashing. \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/phan.jpg}}]{Khoa T. Phan}received the B.Eng. degree in telecommunications (First Class Hons.) from the University of New South Wales (UNSW), Sydney, NSW, Australia, in 2006, the M.Sc. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2008, and California Institute of Technology (Caltech), Pasadena, CA, USA, in 2009, respectively, and the Ph.D. degree in electrical engineering from McGill University, Montreal, QC, Canada in 2017. He is currently a Senior Lecturer and Australia Research Council (ARC) Discovery Early Career Researcher Award (DECRA) Fellow with the Department of Computer Science and Information Technology, La Trobe University, Victoria, Australia. His current research interests are broadly design, control, optimization, and operation of 5G mobile communications networks with applications in the Internet of Things (IoT), satellite communications, machine-type communications (MTC), smart grids, and cloud computing. \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{figs/shirui-bio.jpg}}]{Shirui Pan} received a Ph.D. in computer science from the University of Technology Sydney (UTS), Ultimo, NSW, Australia. He is an ARC Future Fellow (2022-2025) and Senior Lecturer with the Faculty of Information Technology, Monash University, Australia. His research interests include data mining and machine learning. To date, Dr Pan has published over 130 research papers in top-tier journals and conferences, including TPAMI, TKDE, TNNLS, ICML, NeurIPS, KDD, AAAI, IJCAI, WWW, and ICDM. His research has attracted over 7600 citations. He is a recipient of the Best Student Paper Award of IEEE ICDM 2020. His survey paper on \textit{``A Comprehensive Survey on Graph Neural Networks"} in TNNLS-21 has been cited over 2500 times. He is recognised as one of the AI 2000 AAAI/IJCAI Most Influential Scholars in Australia (2021). \end{IEEEbiography} \begin{IEEEbiography}[{\includegraphics[width=1in,clip,keepaspectratio]{figs/phoebe.jpg}}]{Yi-Ping Phoebe Chen} received the B.Inf.Tech. (First Class Hons.) and the Ph.D. degrees in Computer Science from the University of Queensland, Brisbane, Australia. She is currently a Professor and Chair of the Department of Computer Science and Information Technology, La Trobe University, Melbourne, Australia. She is also the Chief Investigator of the ARC Center of Excellence in Bioinformatics. She is the Steering Committee Chair of the Asia Pacific Bioinformatics Conference (founder) and Multimedia Modeling. She has been involved in research on bioinformatics, health informatics, multimedia, and artificial intelligence. She has published over 250 research papers, many of them appeared in top journals and conferences. \end{IEEEbiography} \end{document}

\begin{document} \pagestyle{plain} \title{The Lie Algebra of S-unitary Matrices, Twisted Brackets and Quantum Channels} \author{Clarisson Rizzie Canlubo} \maketitle \begin{abstract} A dimension formula was given in \cite{caalim} in order to partially classify the Lie algebras of $S$-unitary type. The natural question of when $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ are isomorphic is left unanswered. In this article, we will give an answer to this question using the notion of quantum channels and their Kraus representation. In line with this, we will also discuss linearly twisted versions of the usual commutator bracket and its relation to the standard Lie algebra structure on $M_{n}(\mathbb{C})$. Finally, we will mention some problems that are still unanswered in relation to $S$-unitary type matrices and twisted brackets. \end{abstract} \section{Introduction} \label{intro} Let $S\in M_{n}(\mathbb{C})$. Then, the subspace $\mathfrak{u}_{S}=\left\{ X\in M_{n}(\mathbb{C}) | SX^{\ast}=-XS \right\}$ is a Lie algebra with respect to the usual bracket of matrices, given as $[A,B]=AB-BA$ for any $A,B\in \mathfrak{u}_{S}$. The subset $U_{S}=\left\{ X\in M_{n}(\mathbb{C}) | SX^{\ast}=X^{-1}S \right\}$ of $M_{n}(\mathbb{C})$ is a Lie subgroup of $GL_{n}(\mathbb{C})$ whose Lie algebra is $\mathfrak{u}_{S}$. In \cite{caalim}, the dimension of $\mathfrak{u}_{S}$ is given in terms of the spectral properties of $S$. Hence, if $S$ and $T$ are unitarily similar the Lie algebras $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ have the same dimension. Although this is not enough to conclude whether $\mathfrak{u}_{S}$ are isomorphic to $\mathfrak{u}_{T}$ as Lie algebras, this turns out to be the case according to the following proposition. \begin{prop}\label{P1} If $S$ and $T$ are unitarily similar then $\mathfrak{u}_{S}\cong\mathfrak{u}_{T}$ as Lie algebras. \end{prop} \begin{prf} Suppose $S=V^{\ast}TV$ for some unitary $V$. Then, for any $X\in\mathfrak{u}_{S}$ we have \[ V^{\ast}TVX^{\ast}=SX^{\ast}=-XS=-XV^{\ast}TV \] \noindent and so, we have \[ T(VXV^{\ast})^{\ast}=-(VXV^{\ast})T. \] \noindent Thus, $\mathfrak{u}_{S}\stackrel{\phi_{V}}{\longrightarrow}\mathfrak{u}_{T}, X\mapsto VXV^{-1}$ gives the desired isomorphism. $\blacksquare$ \end{prf} The isomorphism $\phi_{V}$ given in the proof of Proposition (\ref{P1}) turns out to be the most general one as indicated in the following theorem. \begin{thm}\label{T1} If $\mathfrak{u}_{S}\stackrel{\phi}{\longrightarrow}\mathfrak{u}_{T}$ is a Lie algebra isomorphism then $\phi(X)=VXV^{-1}$ for some invertible $V$. \end{thm} Also, a partial converse of Proposition (\ref{P1}) is a corollary of Theorem (\ref{P1}) as indicated in the next theorem. \begin{thm}\label{T2} If the Lie algebras $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ are isomorphic then the stabilizers of $S$ and $T$ under the conjugation action of $GL_{n}(\mathbb{C})$ on $M_{n}(\mathbb{C})$ are conjugate subgroups. \end{thm} \noindent We will prove Theorems (\ref{T1}) and (\ref{T2}) in section [\ref{proof}]. Note that although the entries of the matrices in $\mathfrak{u}_{S}$ are complex numbers, the Lie algebra $\mathfrak{u}_{S}$ is strictly a \textit{real} Lie algebra. Whether $\mathfrak{u}_{S}$ is a complex Lie algebra depends on the existence of a complex structure $J$ (an endomorphism $J$ such that $J^{2}=-I$) which bilinearly commutes with $[,]$, i.e. $[J(X),Y]=J[X,Y]=[X,J(Y)]$ for any $X,Y\in \mathfrak{u}_{S}$. \section{Twisted Lie Brackets}\label{twisted} Using a linear map $M_{n}(\mathbb{C})\stackrel{\psi}{\longrightarrow}M_{n}(\mathbb{C})$, one can define a bilinear form $[,]_{\psi}$ on $M_{n}(\mathbb{C})$ as follows. For any $X,Y\in M_{n}(\mathbb{C})$, define $[,]_{\psi}$ by $[X,Y]_{\psi}=X\psi(Y)-Y\psi(X)$. Clearly, $[,]_{\psi}$ is skew-symmetric for any linear map $\psi$. In the event that $[,]_{\psi}$ defines a Lie bracket on $M_{n}(\mathbb{C})$, we will call ${,}_{\psi}$ the $\psi$-twisted Lie bracket on $M_{n}(\mathbb{C})$. However, the Jacobi identity is satisfied only for certain linear maps $\psi$. In general, any linear map $M_{n}(\mathbb{C})\stackrel{\psi}{\longrightarrow}M_{n}(\mathbb{C})$ takes the form $\psi(X)=\sum\limits_{i=1}^{m}A_{i}XB_{i}^{\ast}$ for some matrices $A_{i}, B_{i}\in M_{n}(\mathbb{C})$. If $B_{i}=I$ for all $i=1,\dots, m$ then $\psi(X)=AX$ for all $X\in M_{n}(\mathbb{C})$, where $A=\sum\limits_{i=1}^{m}A_{i}$. In this case, we have \begin{eqnarray*} \sum\limits_{\circlearrowleft} [X,[Y,Z]_{\psi}]_{\psi} &=& X\psi(Y\psi(Z))-X\psi(Z\psi(Y))-Y\psi(Z)\psi(X)+Z\psi(Y)\psi(X)\\ &+& Y\psi(Z\psi(X))-Y\psi(X\psi(Z))-Z\psi(X)\psi(Y)+X\psi(Z)\psi(Y)\\ &+& Z\psi(X\psi(Y))-Z\psi(Y\psi(X))-X\psi(Y)\psi(Z)+Y\psi(X)\psi(Z)\\ & & \\ &=& XAYAZ-XAZAY-YAZAX+ZAYAX\\ &+& YAZAX-YAXAZ-ZAXAY+XAZAY\\ &+& ZAXAY-ZAYAX-XAYAZ+YAXAZ\\ &=& 0 \end{eqnarray*} \noindent where the leftmost sum indicates the cyclic sum over $X,Y$ and $Z$. Thus, we have proven the following proposition. \begin{prop} If $\psi(X)=AX$ for some $A\in M_{n}(\mathbb{C})$ then $[,]_{\psi}$ defines a Lie bracket on $M_{n}(\mathbb{C})$. \end{prop} The essential ideas of the Kraus representation of a linear map $M_{n}(\mathbb{C})\stackrel{\psi}{\longrightarrow}M_{n}(\mathbb{C})$ says that for any $X\in M_{n}(\mathbb{C})$, we have \[ \psi(X)=\sum\limits_{i=1}^{m}A_{i}XB_{i}, \] \noindent the case when the Choi matrix $J(\psi)$ of $\psi$ is of rank $m$. In the event that the rank of $J(\psi)$ is one, i.e. the sum above consists of only one summand, the following proposition gives a necessary condition when $[,]_{\psi}$ defines a Lie bracket on $M_{n}(\mathbb{C})$. \begin{prop} Let $\psi(X)=AXB$ for some $A,B\in M_{n}(\mathbb{C})$. Suppose the image of the map $M_{n}(\mathbb{C})\longrightarrow M_{n}(\mathbb{C}), X\mapsto AX$ is a subspace of $C(B)$, the commuting ring of $B$. Then $[,]_{\psi}$ defines a Lie bracket on $M_{n}(\mathbb{C})$. In particular, if $A$ is invertible then $\psi$ is of the form $\psi(X)=MX$ for some invertible matrix $M$. \end{prop} \begin{prf} For any $X\in M_{n}(\mathbb{C})$, the assumption that $AX$ belongs to $C(B)$ implies that $(AX)B=B(AX)$. Thus, we have \begin{eqnarray*} \sum\limits_{\circlearrowleft} [X,[Y,Z]_{\psi}]_{\psi} &=& X\psi(Y\psi(Z))-X\psi(Z\psi(Y))-Y\psi(Z)\psi(X)+Z\psi(Y)\psi(X)\\ &+& Y\psi(Z\psi(X))-Y\psi(X\psi(Z))-Z\psi(X)\psi(Y)+X\psi(Z)\psi(Y)\\ &+& Z\psi(X\psi(Y))-Z\psi(Y\psi(X))-X\psi(Y)\psi(Z)+Y\psi(X)\psi(Z)\\ & & \\ &=& XAY(AZB)B-XAZ(AYB)B-Y(AZB)(AXB)\\ &+& Z(AYB)(AXB)+YAZ(AXB)B-YAX(AZB)B\\ &-& Z(AXB)(AYB)+X(AZB)(AYB)+ZAX(AYB)B\\ &-& ZAY(AXB)B-X(AYB)(AZB)+Y(AXB)(AZB)\\ & & \\ &=& XAY(AZB-BAZ)B-XAZ(AYB-BAY)B)\\ &+& YAZ(AXB-BAX)B-YAX(AZB-BAZ)B\\ &+& ZAX(AYB-BAY)B-ZAY(AXB-BAX)B\\ &=& 0 \end{eqnarray*} \noindent for any $X,Y,Z\in M_{n}(\mathbb{C})$. This proves the first claim. Now, if $A$ is invertible then the image of $X\mapsto AX$ is $M_{n}(\mathbb{C})$. This implies that $B$ is necessarily a scalar matrix. Taking $M=BA$ proves the second claim. $\blacksquare$ \end{prf} The canonical bracket on $M_{n}(\mathbb{C})$ restricts to a Lie bracket on the subspace $\mathfrak{u}_{S}$ described in section [\ref{intro}]. The following proposition gives a necessary condition when this is true for the twisted Lie bracket $[,]_{\psi}$. \begin{prop}\label{P2} Let $\psi(X)=AX$ for some $A\in M_{n}(\mathbb{C})$. Then, $[,]_{\psi}$ restricts to a Lie bracket on $\mathfrak{u}_{S}$ if and only if $A$ is $S$-Hermitian. \end{prop} \begin{prf} Suppose $A$ is $S$-Hermitian. Then, for any $X,Y\in\mathfrak{u}_{S}$, we have \begin{eqnarray*} S[X,Y]_{\psi}^{\ast} &=& S(X\psi(Y)-Y\psi(X))^{\ast} \\ &=& SY^{\ast}A^{\ast}X^{\ast}-SX^{\ast}A^{\ast}Y^{\ast}\\ &=& -YSA^{\ast}X^{\ast}+XSA^{\ast}Y^{\ast}\\ &=& -YASX^{\ast}+XASY^{\ast}\\ &=& YAXS-XAYS\\ &=& (YAX-XAY)S\\ &=& -[X,Y]_{\psi}S \end{eqnarray*} \noindent Thus, $-[X,Y]_{\psi}\in \mathfrak{u}_{S}$, and so $[,]_{\psi}$ restricts to a Lie bracket on $\mathfrak{u}_{S}$. Conversely, suppose $[,]_{\psi}$ restricts to a Lie bracket on $\mathfrak{u}_{S}$. Then, for any $X,Y\in \mathfrak{u}_{S}$, we have \[ -YSA^{\ast}X^{\ast} + XSA^{\ast}Y^{\ast} = -YASX^{\ast} + XASY^{\ast} \] \noindent from the above computation. Thus, we have \[ Y(AS-SA^{\ast})X^{\ast} = X(AS-SA^{\ast})Y^{\ast} \] \noindent for any $X,Y\in \mathfrak{u}_{S}$. Taking $Y=iI\in\mathfrak{u}_{S}$, we see that \[ (AS-SA^{\ast})X^{\ast}=-X(AS-SA^{\ast}) \] \noindent and taking $X=I$, we get $SA^{\ast}=AS$. $\blacksquare$ \end{prf} It is a curiosity to know which linear maps $\psi$ whose brackets $[,]_{\psi}$ induce Lie algebra structures on $M_{n}(\mathbb{C})$ isomorphic to the canonical one. We partially answer this in the next proposition. \begin{prop}\label{P5} If the Lie bracket $[,]_{\psi}$, with $\psi(X)=AXB^{\ast}$ for all $X\in M_{n}(\mathbb{C})$, coincides with the canonical Lie bracket on $M_{n}(\mathbb{C})$ then $B^{\ast}=A^{-1}$. \end{prop} \begin{prf} For any $X\in M_{n}(\mathbb{C})$ we have \[ 0= [X,I] = [X,I]_{\psi} = X\psi(I)-I\psi(X). \] \noindent And so, we have $\psi(X)=X\psi(I)$ for all $X\in M_{n}(\mathbb{C})$. Since $[I,X]=0$, we also have $\psi(X)=\psi(I)X$ for all $X\in M_{n}(\mathbb{C})$. Thus, $\psi(I)=AB^{\ast}$ is central. Thus, for any $X,Y\in M_{n}(\mathbb{C})$ we have \[ [X,Y] = [X,Y]_{\psi} = X\psi(Y)-Y\psi(X) = X\psi(I)Y-Y\psi(I)X = \psi(I)[X,Y] \] \noindent and so, $\psi(I)=I$ from which the conclusion immediately follows. $\blacksquare$ \end{prf} Using the scalar matrix $iI$ in place of $I$ in the proof of Proposition (\ref{P5}) we get the following corollary. \begin{cor}\label{C6} If the Lie bracket $[,]_{\psi}$, with $\psi(X)=AX$ for all $X\in M_{n}(\mathbb{C})$, coincides with the canonical Lie bracket on $\mathfrak{u}_{S}$ then $A=I$. \end{cor} \section{Quantum Channels and Kraus Operators} \label{kraus} Quantum channels play prominent role in quantum information theory, see \cite{chuang} for more details. They are used to encode operations in the set-up of quantum theory. Quantum channels, in the finite dimensional case, are completely positive maps $M_{n}(\mathbb{C})\stackrel{\Phi}{\longrightarrow}M_{k}(\mathbb{C})$. In some literature, quantum channels are required to be trace-preserving. In this section, we will discuss aspects of quantum channels important for the purpose of this article. A linear map $A\stackrel{\Phi}{\longrightarrow}B$ between $C^{\ast}$-algebras is said to be $m$-\textit{positive} if the induced map \[ \Phi_{m}:=I_{m}\otimes\Phi:M_{m}(\mathbb{C})\otimes A\longrightarrow M_{m}(\mathbb{C})\otimes B \] \noindent sends positive elements to positive elements relative to the natural $C^{\ast}$-algebra structures on the involved tensor products. If $\Phi$ is $m$-positive for all natural numbers $m$ then $\Phi$ is said to be \textit{completely positive}. Note that this notion makes sense for a general map $A\stackrel{\Phi}{\longrightarrow}B$ between $C^{\ast}$-algebras since the minimal and maximal tensor products coincide in the case of tensoring with $M_{n}(\mathbb{C})$. In the case when $A$ and $B$ are finite-dimensional matrix algebras, complete positivity is equivalent to $m$-positivity for some natural number $m$. This is a consequence of Choi's Theorem as stated below. For the proof, see for example \cite{choi} and \cite{mosonyi}. \begin{thm}{(Choi's Theorem)}\\ \label{cho} Let $M_{n}(\mathbb{C})\stackrel{\Phi}{\longrightarrow}M_{k}(\mathbb{C})$ be linear. Then, the following are equivalent: \begin{enumerate} \item[(a)] $\Phi$ is completely positive \item[(b)] $\Phi$ is $n$-positive \item[(c)] There exists $A_{1},\dots,A_{r}\in M_{k,n}(\mathbb{C})$ such that \[ \Phi(X)=\sum\limits_{i=1}^{r}A_{i}XA_{i}^{\ast} \] \noindent for all $X\in M_{n}(\mathbb{C})$. Moreover, the matrices $A_{1},\dots,A_{r}$ satisfy $\sum\limits_{i=1}^{r}A_{i}A_{i}^{\ast}=I$. \end{enumerate} \end{thm} \noindent The representation given in part $(c)$ of the above theorem is called a \textit{Kraus representation} of $\Phi$ and the matrices $A_{1},\dots,A_{r}$ are called the associated \textit{Kraus operators}. The Kraus representation of a linear operator $\Phi$ is far from unique. However, the Kraus representations of a given quantum channel $\Phi$ satisfy certain transitivity relation as stated by the following theorem. \begin{thm}\label{unitary} Let $A_{1},\dots,A_{r}$ and $B_{1},\dots,B_{s}$ be two sets of Kraus operators associated to two Kraus representations of a quantum channel $\Phi$. Then, there is a unitary $(U_{ij})\in M_{max\left\{r,s\right\}}(\mathbb{C})$ such that $B_{j}=\sum\limits_{i=1}^{r}U_{ji}A_{i}$. \end{thm} \noindent For a proof, see pg. 95 of \cite{mosonyi}. Linear maps that are not necessarily completely positive have similar representations as that of a Kraus representation of a quantum channel. In the general case, a linear map $M_{n}(\mathbb{C})\stackrel{\Phi}{\longrightarrow}M_{k}(\mathbb{C})$ can be represented as \[ \Phi(X)=\sum\limits_{i=1}^{r} A_{i}XB_{i}^{\ast} \] \noindent for some matrices $A_{1},\dots,A_{r},B_{1},\dots,B_{r}\in M_{k,n}(\mathbb{C})$. \section{Proof of Theorems 1 and 2} \label{proof} \textsc{Proof of Thereom 1:} Let $\Phi$ be a linear automorphism extending the Lie algebra isomorphism $\phi$ on the whole $M_{n}(\mathbb{C})$. Let $\Psi$ be its inverse. Then, using the Kraus representation for linear maps $M_{n}(\mathbb{C})\stackrel{\Phi,\Psi}{\longrightarrow}M_{n}(\mathbb{C})$, there are $r$ matrices $A_{i},B_{i}$ and $s$ matrices $A^{\prime}_{j},B^{\prime}_{j}$ such that \[ \Phi(X)=\sum\limits_{i=1}^{r} A_{i}XB_{i}^{\ast} \hspace{.5in} \text{and} \hspace{.5in} \Psi(X)=\sum\limits_{j=1}^{s} A^{\prime}_{i}X(B^{\prime}_{i})^{\ast} \] \noindent for all $X\in M_{n}(\mathbb{C})$. Without loss of generality, we can assume the matrices $A_{i}$ and $B_{i}$ form linearly independent sets of matrices. We assume the same for the matrices $A^{\prime}_{i}$ and $B^{\prime}_{i}$. Then, $\Phi\circ \Psi=id$ gives \[ id(X)=\sum\limits_{i,j} A_{i}A^{\prime}_{j}X(B^{\prime}_{j})^{\ast}B^{\ast}_{i}.\] \noindent Complete positivity implies that the matrices $A_{i}A^{\prime}_{j}$ and $B_{i}B^{\prime}_{j}$ constitutes a set of Kraus operators for $id$. However, $id(X)=X$ is also a Kraus representation for $id$. Since the Kraus operators appearing in different Kraus representations of the same quantum channel are related by a unitary according to Theorem (\ref{unitary}), we must have $r=s=1$ and so, $\Phi(X)=AXB^{\ast}$. Since $\Phi$ is an isomorphism, the matrices $A$ and $B$ are necessarily invertible. The restriction $\phi$ of $\Phi$ on $\mathfrak{u}_{S}$ is given by $\phi(X)=AXB^{\ast}$. Thus, for any $X,Y\in \mathfrak{u}_{S}$ we have \[ AXYB^{\ast}-AYXB^{\ast}=\phi[X,Y]=[\phi(X),\phi(Y)]=AXB^{\ast}AYB^{\ast}-AYB^{\ast}AXB^{\ast} \] \noindent from which we immediately see that \[ [X,Y]=XY-YX=XB^{\ast}AY-YB^{\ast}AX=[X,Y]_{\psi} \] \noindent where $\psi(X)=B^{\ast}AX$ for all $X\in\mathfrak{u}_{S}$. Hence, by Corollary (\ref{C6}) we have $B^{\ast}A=I$ and so, $A^{-1}=B^{\ast}$. Taking $V=A$ proves the theorem. $\blacksquare$ \noindent \textsc{Proof of Theorem 2:} If $X\in \mathfrak{u}_{S}$ then \[ T\phi(X)^{\ast}=-\phi(X)T \Longleftrightarrow V^{-1}TV^{-\ast}X^{\ast}=-XV^{-1}TV^{-\ast}. \] \noindent Thus, $\mathfrak{u}_{S}= \mathfrak{u}_{V^{-1}TV^{-\ast}}$. This implies that the Lie groups $U_{S}$ and $U_{V^{-1}TV^{-\ast}}$ associated to $\mathfrak{u}_{S}$ and $\mathfrak{u}_{V^{-1}TV^{-\ast}}$, respectively, are the same. That is, \[ SX^{\ast}=X^{-1}S \Longleftrightarrow V^{-1}TV^{-\ast}X^{\ast}=X^{-1}V^{-1}TV^{-\ast} \] \noindent or equivalently, \[ XSX^{\ast}=S \Longleftrightarrow (VXV^{-1})T(VXV^{-1})^{\ast}=T \] \noindent for all $X\in U_{S}$. Thus, \[ stab(S)=U_{S}=V^{-1}\cdot U_{T}\cdot V=V^{-1}\cdot stab(T)\cdot V \] \noindent showing that $stab(S)$ and $stab(T)$ are conjugate subgroups of $GL_{n}(\mathbb{C})$. $\blacksquare$ \section{Unanswered Questions} \label{problems} Relative to the usual commutator bracket $[,]$, the subspaces $\mathfrak{u}_{S}$ are Lie subalgebras of $M_{n}(\mathbb{C})$ for any $S\in M_{n}(\mathbb{C})$. However, as we have seen in Proposition (\ref{P2}), not all $\mathfrak{u}_{S}$ are Lie subalgebras of $M_{n}(\mathbb{C})$ relative to the twisted bracket $[,]_{\psi}$. \begin{que} What are the Lie subalgebras of $M_{n}(\mathbb{C})$ under the Lie bracket $[,]_{\psi}$? \end{que} The bracket $[,]_{\psi}$ for a linear map $M_{n}(\mathbb{C})\stackrel{\psi}{\longrightarrow}M_{n}(\mathbb{C})$ is a special case of the class of brackets of the form $[X,Y]_{B}:=B(X,Y)-B(Y,X)$ for some bilinear form $B$ on $M_{n}(\mathbb{C})$. \begin{que} When is the bracket $[X,Y]_{B}=B(X,Y)-B(Y,X)$ a Lie bracket on $M_{n}(\mathbb{C})$? And on $\mathfrak{u}_{S}$? In particular, when does $[,]_{B}$ satisfy the Jacobi identity? \end{que} In Theorem (\ref{T2}), a necessary condition for $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ to be isomorphic is the (unitary) conjugacy of the stabilizers subgroups of the matrices $S$ and $T$. This is much weaker than $S$ and $T$ being unitarily similar. \begin{que} Using the orbit-stabilizer theorem for the $\ast$-conjugation action of $GL_{n}(\mathbb{C})$ on $M_{n}(\mathbb{C})$, what can we say about the matrices $S$ and $T$ if $stab(S)=stab(T)$? Or if they are only conjugate subgroups? \end{que} The main goal of this article is to determine when the Lie algebras $\mathfrak{u}_{S}$ and $\mathfrak{u}_{T}$ are abstractly isomorphic. Another natural inquiry is to understand the lattice structure of the Lie algebras $\mathfrak{u}_{S}$ in terms of inclusions. In line with this, we have the following question. \begin{que} If $\mathfrak{u}_{S}\leqslant\mathfrak{u}_{T}$, is it the case that $stab(T)$ is conjugate to a (possibly trivial) subgroup of $stab(S)$? \end{que} More importantly, the author is very much interested with the following question. \begin{que} Let $\psi(X)=AX$ for some $A\in M_{n}(\mathbb{C})$. What is the Lie group structure on $GL_{n}(\mathbb{C})$ so that its Lie algebra is $M_{n}(\mathbb{C})$ with bracket $[,]_{\psi}$? By Ado's Theorem, every finite dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a Lie subgroup $G$ of $GL_{R}(\mathbb{C})$ for some $R$. Since the bracket $[,]_{\psi}$ is a 'twist' of the usual commutator on the same vector space, it is reasonable to expect that its Lie group has the same underlying manifold as that of $GL_{n}(\mathbb{C})$ but with a 'twisted' multiplication. See \cite{procesi}. \end{que} \hspace{1in} \noindent\textsc{Clarisson Rizzie P. Canlubo}\\ University of the Philippines$-$Diliman\\ Quezon City, Philippines 1101\\ [email protected] \end{document}

\begin{document} \pagestyle{plain} \title{Representations of the Infinite-Dimensional Affine Group} \date{} \author{ \textbf{Yuri Kondratiev}\\ Department of Mathematics, University of Bielefeld, \\ D-33615 Bielefeld, Germany,\\ Dragomanov University, Kyiv, Ukraine\\ } \begin{abstract} We introduce an infinite-dimensional affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions. \end{abstract} \maketitle \vspace*{3cm} {\bf Key words: } affine group; configurations; Poisson measure; ergodicity {\bf MSC 2010}. Primary: 22E66. Secondary: 60B15. \section{Introduction} Given a vector space $V$ the affine group can be described concretely as the semidirect product of $V$ by $\mathrm{GL}(V)$, the general linear group of $V$: $$ \mathrm{Aff} (V)=V \rtimes \mathrm{ GL} (V). $$ The action of $\mathrm{GL}(V)$ on $V$ is the natural one (linear transformations are automorphisms), so this defines a semidirect product. Affine groups play important role in the geometry and its applications, see, e.g., \cite{Ar,Ly}. Several recent papers \cite{AJO,AK,EH,GJ,Jo,Ze} are devoted to representations of the real, complex and $p$-adic affine groups and their generalizations, as well as diverse applications, from wavelets and Toeplitz operators to non-Abelian pseudo-differential operators and $p$-adic quantum groups. In the particular case of field $V= \X$ the group $\mathrm{Aff}(\X)$ defined as following. Consider a function $b:\X \to \X$ which is a step function on $\X$. Take another matrix valued function $A:\X\to L(\X) $ s.t. $A(x)=\mathrm{Id} +A_0(x)$, $A(x)$ is invertible, $A_0$ is a matrix valued step function on $\X$. Introduce an infinite dimensional affine group $\Aff (\X)$ that is the set of all pairs $g=(A,b)$ with component satisfying assumptions above. Define the group operation $$ g_2 g_1= (A_2,b_2) (A_1, b_1) = (A_1 A_2, b_1 +A_1 b_2). $$ The unity in this group is $e=(\mathrm{Id} ,0)$. For $g\in \Aff(\X)$ holds $g^{-1}= (A^{-1}, -A^{-1}b)$. It is clear that for step mappings we use these definitions are correct. Our aim is to construct irreducible representations of $\Aff (\X)$. As a rule, only special classes of irreducible representations can be constructed for infinite-dimensional groups. For various classes of such groups, special tools were invented; see \cite{Is,Ko} and references therein. We will follow an approach by Vershik-Gefand -Graev \cite{VGG75} proposed in the case of the group of diffeomorphisms. A direct application of this approach meets certain difficulties related with the absence of the possibility to define the action of the group $\Aff (\X)$ on a phase space similar to \cite{VGG75}. A method to overcome this problem is the main technical step in the present paper. We wold like to mention that a similar approach was already used in \cite{PAFF} for the construction of the representation for p-adic infinie dimensional affine group. \maketitle \section{Infinite dimensional affine group} In our definitions and studies of vector and matrix valued functions on $\X$ we will use as basic functional spaces collections of step mappings. It means that each such mapping is a finite sum of indicator functions with measurable bounded supports with constant vector/matrix coefficients. Such spaces of functions on $\X$ are rather unusual in the framework of infinite dimensional groups but we will try to show that their use is natural for the study of affine groups. For $x\in\X$ consider the section $G_x= \{g(x)\; |\; g\in \Aff(\X)\}$. It is an affine group with constant coefficients. Note that for a ball $B_N (0) \subset \X$ with the radius $N$ centered at zero we have $g(x)= (1,0), x\in B^c_N(0)$. Define the action of $g$ on a point $x\in\X$ as $$ gx= g(x)x = A(x)^{-1} (x+b(x)). $$ Denote the orbit $O_x=\{gx| g\in G_x\}\subset \X$. Actually, as a set $O_x=\X$ but elements of this set are parametrized by $g\in G_x$. For any element $y\in O_x$ and $h\in G_x$ we can define $hy= h(gx)= (hg)x\in O_x$. It means that we have the group $G_x$ action on the orbit $O_x$. It gives $$ (g_1g_2)(x) x= g_1(x)( g_2(x)x) $$ that corresponds to the group multiplication $$ g_2 g_1= (A_2,b_2) (A_1, b_1) = (A_1 A_2, b_1 +A_1 b_2) $$ considered in the given point $x$. \begin{Remark} The situation we have is quite different w.r.t. the standard group of motions on a phase space. Namely, we have one fixed point $x\in\X$ and the section group $G_x$ associated with this point. Then we have the motion of $x$ under the action of $G_x$. It gives the group action on the orbit $O_x$. \end{Remark} We will use the configuration space $\Ga(\X)$, i.e., the set of all locally finite subsets of $\X$. Each configuration may be identified with the measure $$ \gamma(dx) = \sum_{x\in\gamma} \delta_x $$ which is a positive Radon measure on $\X$: $\gamma\in \M(\X)$. We define the vague topology on $\Ga(\X)$ as the weakest topology for which all mappings $$ \Ga(\X) \ni \ga \mapsto <f,\gamma>\in \R,\;\; f\in C_0(\X) $$ are continuous. The Borel $\sigma$-algebra for this topology denoted $\B(\Ga(\X))$. For $\ga\in \Ga(\X)$, $\ga=\{x\}\subset \X$ define $g\gamma$ as a motion of the measure $\ga$: $$ g\ga=\sum_{x\gamma} \delta_{g(x)x}\in \M(\X). $$ Here we have the group action of $\Aff(\X)$ produced by individual transformations of points from the configuration. Again, as above, we move a fixed configuration using previously defined actions of $G_x$ on $x\in\ga$. Note that $g\gamma$ is not more a configuration. More precisely, for some $B_N(0) $ the set $(g\ga)_N= g\ga\cap B_N^c(0)$ is a configuration in $B^c_N(0) $ but the finite part of $g\ga$ may include multiple points. For any $f\in \mathcal D(\X,\C)$ we have corresponding cylinder function on $\Ga(\X)$: $$ L_f(\ga)= <f,\ga > = \int_{\X} f(x)\ga(dx) = \sum_{x\in \ga} f(x). $$ Denote ${\mathcal P}_{cyl}$ the set of all cylinder polynomials generated by such functions. More generally, consider functions of the form \begin{equation} \label{cyl} F(\ga)= \psi(<f_1,\ga>,\dots, <f_n,\ga>),\; \ga\in\Ga(\X), f_j\in \mathcal D(\X), \psi\in C_b(\R^n). \end{equation} These functions form the set $\mathcal F_b(\Ga(\X))$ of all bounded cylinder functions. For any clopen set $\Lambda \in \mathcal{O}_b(\X)$ (also called a finite volume) denote $\Ga(\Lambda)$ the set of all (with necessity finite) configurations in $\La$. We have as before the vague topology on this space and the Borel $\sigma$-algebra $\B(\Ga(\La))$ is generated by functions $$ \Ga(\La)\ni\ga \mapsto <f,\ga>\in\R $$ for $f\in C_0 (\La)$. For any $\La\in \mathcal{O}_b(\X)$ and $T\in \B(\Ga(\La))$ define a cylinder set $$ C(T)=\{\ga\in\Ga(\X)\;|\; \ga_{\La}=\ga \cap \La \in T\}. $$ Such sets form a $\sigma$-algebra $\B_{\La}(\Ga(\X))$ of cylinder sets for the finite volume $\La$. The set of bounded functions on $\Ga(\X)$ measurable w.r.t. $\B_{\La}(\Ga(\X))$ we denote $B_{\La}(\Ga(\X))$. That is a set of cylinder functions on $\Ga(\X)$. As a generating family for this set we can use the functions of the form $$ F(\ga)= \psi(<f_1,\ga>,\dots, <f_n,\ga>),\; \ga\in\Ga(\X), f_j\in C_0(\La), \psi\in C_b(\R^n). $$ For so-called one-particle functions $f:\X\to\R, f\in\mathcal D(\X)$ consider $$ (gf)(x)= f(g(x) x), x\in \X. $$ Then $gf\in \mathcal D(\X)$. Thus, we have the group action $$ \mathcal D(\X)\in f \mapsto gf\in \mathcal D(\X),\;\;g\in\Aff $$ of the infinite dimensional group $\Aff$ in the space of functions $\mathcal D(\X)$. Note that due to our definition, we have $$ <f, g\ga> = <gf,\ga> $$ and it is reasonable to define for cylinder functions (\ref{cyl}) the action of the group $\Aff$ as $$ (V_g F)(\ga)= \psi(<gf_1,\ga>,\dots <gf_n,\ga>. $$ Obviously $V_g: \mathcal F_b (\Ga(\X))\to \mathcal F_b(\Ga(\X))$. Denote $m(dx)$ the Haar measure on $\X$. The dual transformation to one-particle motion is defined via the following relation $$ \int_{\X} f(g(x)x) m(dx)=\int_{\X} f(x) g^\ast m(dx) $$ if exists such measure $g^\ast m$ on $\X$. \begin{Lemma} \label{gm} For each $g\in \Aff$ $$ g^\ast m(dx)= \rho_{g}(x) m(dx) $$ where $\rho_g = 1_{B_R^c(0) } + r_g^0,\;\; r_g^0\in \mathcal D(\X,\R_+).$ Here as above $$ B_R^c(0)= \{x\in\X\;|\; |x|_p \geq R\}. $$ \end{Lemma} \begin{proof} We have following representations for coefficients of $g(x)$: $$ b(x)= \sum_{k=1}^{n} b_k 1_{B_k}(x) , $$ $$ a(x)= \sum_{k=1}^{n} a_k 1_{B_k}(x) + 1_{B^c_R(0)}(x) $$ where $B_k$ are certain balls in $\X$. Then $$ \int_{\X} f(g(x)x) m(dx)= \sum_{k=1}^n \int_{B_k} f(\frac{x+b_k}{a_k}) m(dx) + \int_{B^c_R (0)} f(x) m(dx) = $$ $$ \sum_{k=1}^{n} \int_{C_k} f(y) |a_k|_p m(dy) + \int_{B^c_R(0)} f(y) m(dy), $$ where $$ C_k= a_k^{-1}(B_k + b_k). $$ Therefore, $$g^\ast m= (\sum_{k=1}^n |a_k|_p 1_{C_k} + 1_{B^c_R(0)}) m. $$ Note that informally we can write $$ (g^\ast m)(dx) = dm(g^{-1}x). $$ \end{proof} Note that by the duality we have the group action on the Lebesgue measure. Namely, for $f\in \mathcal D(\X)$ and $g_1, g_2\in \Aff$ $$ \int_{\X} (g_2 g_1) f(x) m(dx)= \int_{\X} g_1 f (x) (g_2^\ast m) (dx) = $$ $$ \int_{\X} f(x) (g_1^\ast g_2^\ast m)(dx)= \int_{\X} f(x) ((g_2 g_1)^\ast m)(dx). $$ In particular $$ (g^{-1})^\ast (g^\ast m)= m. $$ \begin{Lemma} Let $F\in B_\La (\Ga(\X))$ and $g\in\Aff $ has the form $g(x)=(1, h1_{B}(x))$ with certain $h\in \X$ and $B\in \mathcal{O}_b(\X)$ s.t. $\La\subset B$. Then $$ V_gF\in B_{\La -h} (\Ga(\X)). $$ \end{Lemma} \begin{proof} Due to the formula for the action $V_gF$ we need to analyze the support of functions $f_j (x+h1_B(x))$ for $\supp f_\subset \La$. If $x\in B^c$ then $x\in \La^c$ and therefore $f_j (x+h1_B(x))=f_j(x)=0$. For $x\in B$ we have $f_j(x+h)$ and only for $x+h\in \La$ this value may be nonzero, i.e., $\supp g f_j \subset \La- h$. \end{proof} Denote $\pi_m$ the Poisson measure on $\Ga(\X)$ with the intensity measure $m$. \begin{Lemma} \label{V} For all $F \in {\mathcal P}_{cyl}$ or $F\in \mathcal F_b (\Ga(\X))$ and $g\in \Aff $ holds $$ \int_{\Ga(\X)} V_g F d\pi_m = \int_{\Ga(\X)} Fd\pi_{g^\ast m} . $$ \end{Lemma} \begin{proof} It is enough to show this equality for exponential functions $$ F(\ga)= e^{<f,\ga>},\;\; f\in\mathcal D(\X). $$ We have $$ \int_{\Ga(\X)} V_g F d\pi_m = \int_{\Ga(\X)} e^{<gf, \ga>} d\pi_m(\ga)= $$ $$ \exp[ \int_{\X} (e^{gf(x)} -1) dm(x)] = \exp[ \int_{\X} (e^{f(x)} -1) d(g^{\ast} m)(x)= $$ $$ \int_{\Ga(\X)} F d\pi_{g^\ast m }. $$ \end{proof} \begin{Remark} For all functions $F,G\in \mathcal F(\Ga(\X))$ a similar calculation shows $$ \int_{\Ga(\X)} V_g F \; Gd\pi_m = \int_{\Ga(\X)} F \; V_{g^{-1}} G d\pi_{g^\ast m} . $$ \end{Remark} Let $\pi_m$ be the Poisson measure on $\Ga(\X)$ with the intensity measure $m$. For any $\La\in \mathcal{O}_b(\X)$ consider the distribution $\pi_m^\La$ of $\pi_m$ in $\Ga(\La)$ corresponding the projection $\ga\to \ga_\La$. It is again a Poisson measure $\pi_{m_\La}$ in $\Ga(\La)$ with the intensity $m_\La$ which is the restriction of $m$ on $\La$. Infinite divisibility of $\pi_m$ gives for $F_j\in B_{\La_j}(\Ga(\X)), j=1,2$ with $\La_1\cap \La_2=\emptyset$ $$ \int_{\Ga(\X)} F_1(\ga) F_2(\ga) d\pi_m(\ga)= \int_{\Ga(\X)} F_1(\ga) d\pi_m(\ga) \int_{\Ga(\X)} F_2(\ga) d\pi_m(\ga)= $$ $$ \int_{\Ga(\La_1)} F_1 d\pi^{\La_1}_m \int_{\Ga(\La_2)} F_2 d\pi^{\La_2}_m. $$ \begin{Lemma} For any $F\in B_\La(\Ga(\X)$ and $g=(1, h1_B)\in \Aff $ with $\La \cap (B+h)=\emptyset$ holds $$ \int_{\Ga(\X)} (V_g F)(\ga) d\pi_m(\ga)= \int_{\Ga(\X)} F(\ga)d\pi_m(\ga). $$ \end{Lemma} \begin{proof} Due to our calculations above we have $$ \int_{\Ga(\X)} (V_gF)(\ga) d\pi_m(\ga)= \int_{\Ga(\X)} F(\ga) d\pi_{g^{\ast}m}(\ga)= $$ $$ \int_{\Ga(\La)} F(\eta) d\pi^{\La}_{g^{\ast}m} (\eta) =\int_{\Ga(\La)} F(\eta) d\pi_{ (g^{\ast}m)_\La} (\eta). $$ But we have shown $$ (g^{\ast}m)(dx)= (1+ 1_{B+h}(x)) m(dx) = m(dx) $$ for $x\in \La$, i.e., $(g^{\ast}m)_\La =m$. \end{proof} \begin{Lemma} \label{prod} For any $F_1,F_2 \in \mathcal F_b(\Ga(\X))$ there exists $g\in\Aff$ such that $$ \int_{\Ga(\X)} F_1 \; V_g F_2 d\pi_m = \int_{\Ga(\X)} F_1 d\pi_m \int_{\Ga(\X)} F_2 d\pi_m . $$ \end{Lemma} \begin{proof} By the definition, $F_j\in B_{\La_j}(\Ga(\X)), j=1,2$ for some $\La_1,\La_2 \in \mathcal{O} (\X)$. Let us take $g=(1, h1_B)$ with the following assumptions: $$ \La_2\subset B,\;\; \La_1\cap (\La_2-h) =\emptyset,\;\; \Lambda_2\cap (B+h) =\emptyset. $$ Then accordingly to previous lemmas $$ \int_{\Ga(\X)} F_1 V_g F_2 d\pi_m = \int_{\Ga(\X)} F_1 d\pi_m \int_{\Ga(\X)} F_2 d\pi_m . $$ \end{proof} \section{$\Aff$ and Poisson measures} For $F\in {\mathcal P}_{cyl} $ or $F\in \mathcal F_b (\Ga(\X))$, we consider the motion of $F$ by $g\in \Aff$ given by the operator $V_g$. Operators $V_g$ have the group property defined point-wisely: for any $\ga \in \Ga(\X) $ $$ (V_h (V_gF))(\ga)= (V_{hg} F) (\ga). $$ This equality is the consequence of our definition of the group action of $\Aff$ on cylinder functions. As above, consider $\pi_m$, the Poisson measure on $\Ga(\X)$ with the intensity measure $m$. For the transformation $V_g$ the dual object is defined as the measure $V^\ast_g \pi_m$ on $\Ga(\X)$ given by the relation $$ \int_{\Ga(\X)} (V_gF) (\ga) d\pi_m(\ga) =\int_{\Ga(\X)} F(\ga) d(V^\ast_g \pi_m)(\ga), $$ where $V^\ast_g \pi_m= \pi_{g^\ast m}$, see Lemma \ref{V}. \begin{Corollary} For any $g\in \Aff$ the Poisson measure $V_g^\ast \pi_m$ is absolutely continuous w.r.t. $\pi_m$ with the Radon-Nykodim derivative $$ R(g,\ga)= \frac{d\pi_{g^\ast m}(\ga)}{d\pi_{ m} (\ga)} \in L^1(\pi_m). $$. \end{Corollary} \begin{proof} Note that density $\rho_g = 1_{B_R^c(0) } + r_g^0,\;\; r_g^0\in \mathcal D(\X,\R_+)$ of $g^\ast m$ w.r.t. $m$ may be equal zero on some part of $\X$ and, therefore, the equivalence of of considered Poisson measures is absent. Due to \cite{LS03}, the Radon-Nykodim derivative $$ R(g,\ga)= \frac{d\pi_{g^\ast m}(\ga)}{d\pi_{ m} (\ga)} $$ exists if $$ \int_{\X} |\rho_g(x)-1| m(dx)= \int_{B_R(0)} |1-r_g^0 (x)| m(dx) <\infty. $$ \end{proof} \begin{Remark} As in the proof of Proposition 2.2 from \cite{AKR} we have an explicit formula for $R(g,\ga)$: $$ R(g,\ga)= \prod_{x\in\ga} \rho_g (x) \exp(\int_{\X} (1-\rho_g(x)) m(dx). $$ The point-wise existence of this expression is obvious. \end{Remark} This fact gives us the possibility to apply the Vershik-Gelfand-Graev approach realized by these authors for the case of diffeomorphism group. Namely, for $F\in {\mathcal P}_{cyl}$ or $F\in {\mathcal P}_{cyl}(\Ga(\X)$ and $g\in \Aff$ introduce operators $$ (U_g F)(\ga) = (R(g^{-1} ,\ga) )^{1/2} (V_gF)(\ga). $$ \begin{Theorem} Operators $U_g,\; g\in \Aff$ are unitary in $L^2 (\Ga(\X), \pi_m)$ and give an irreducible representation of $\Aff$. \end{Theorem} \begin{proof} Let us check the isometry property of these operators. We have using Lemmas \ref{V}, \ref{gm} $$ \int_{\Ga(\X)} |U_g|^2 d\pi_m = \int_{\Ga(\X)} |V_g F|^2(\ga) d\pi_{(g^{-1})^\ast m} (\ga)= $$ $$ \int_{\Ga(\X)} |F(\ga)|^2 d\pi_{(gg^{-1})\ast m}(\ga)= \int_{\Ga(\X)} |F(\ga)|^2 d\pi_{ m}(\ga). $$ From Lemma \ref{V} follows that $U_g^\ast = U_{g^{-1}}.$ We need only to check irreducibility that shall follow from the ergodicity of Poisson measures \cite{VGG75}. But to this end we need first of all to define the action of the group $\Aff$ on sets from $\B(\Ga(\X)$. As we pointed out above, we can not define this action point-wisely. But we can define the action of operators $V_g$ on the indicators $1_A(\ga)$ for $A\in \B(\Ga(Q))$. Namely, for given $A$ we take a sequence of cylinder sets $A_n, n\in \N$ such that $$ \pi_{m}(A\Delta A_n) \to 0, n\to \infty. $$ Then $$ U_g 1_{A_n} =V_g 1_{A_n} (R(g^{-1} ,\cdot) )^{1/2} \to G (R(g^{-1} ,\cdot) )^{1/2} \in L^2(\pi_m), n\to\infty $$ in $L^2(\pi_m)$. Each $V_g 1_{A_n} $ is an indicator of a cylinder set and $$ V_g 1_{A_n} \to G \;\; \pi_m - a.s., n\to \infty. $$ Therefore, $G=1$ or $G=0$ $\pi_m$-a.s. We denote this function $V_g 1_A$. For the proof of the ergodicity of the measure $\pi_m$ w.r.t. $\Aff$ we need to show the following fact: for any $A\in \B(\Ga(\X))$ such that $\forall g\in\Aff\;\; V_g 1_A = 1_A\; \pi_m- a.s.$ holds $\pi_m(A)= 0$ or $\pi_m(A)= 1$. Fist of all, we will show that for any pair of sets $A_1, A_2 \in \B(\Ga(Q))$ with $\pi_m(A_1)>0,\;\; \pi_m(A_2) >0$ there exists $g\in\Aff$ such that \begin{equation} \label{ineq} \int_{\Ga(\X)} 1_{A_1} V_g 1_{A_2} d\pi_m \geq \frac{1}{2} \pi_m(A_1) \pi_m(A_2). \end{equation} Because any Borel set may be approximated by cylinder sets, it is enough to show this fact for cylinder sets. But for such sets due to Lemma \ref{prod} we can choose $g\in \Aff$ such that $$ \int_{\Ga(\X)} 1_{A_1} V_g 1_{A_2} d\pi_m = \pi_m(A_1) \pi_m(A_2). $$ Then using an approximation we will have (\ref{ineq}). To finish the proof of the ergodicity, we consider any $A\in\B(\Ga(\X)$ such that $$ \forall g\in \Aff\; V_g1_A = 1_A \;\;\pi_m - a.s.,\;\; \pi_m(A)>0. $$ We will show that then $\pi_m(A)= 1$. Assume $\pi_m(\Ga\setminus A) >0$. Due to the statement above, there exists $g\in \Aff$ such that $$ \int_{\Ga(\X)} 1_{\Ga\setminus A} V_g 1_A >0. $$ But due to the invariance of $1_A$ it means $$ \int_{\Ga(\X)} 1_{\Ga\setminus A} 1_A d\pi_m >0 $$ that is impossible. \end{proof} \end{document}

\begin{document} \author[M. El Bachraoui and J. S\'{a}ndor]{Mohamed El Bachraoui and J\'{o}zsef S\'{a}ndor} \address{Dept. Math. Sci, United Arab Emirates University, PO Box 15551, Al-Ain, UAE} \email{[email protected]} \address{Babes-Bolyai University, Department of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania} \email{[email protected]} \keywords{$q$-trigonometric functions; $q$-digamma function; transcendence.} \subjclass{33B15, 11J81, 33E05, 11J86} \begin{abstract} We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities. \end{abstract} \date{\textit{\today}} \maketitle \section{Introduction}\label{sec-introduction} Throughout we let $\tau$ be a complex number in the upper half plane and let $q=e^{\pi i\tau}$. Note that the assumption $\mathrm{Im}(\tau)>0$ implies that $|q|<1$. The $q$-shifted factorials of a complex number $a$ are defined by \[ (a;q)_0= 1,\quad (a;q)_n = \prod_{i=0}^{n-1}(1-a q^i),\quad (a;q)_{\infty} = \lim_{n\to\infty}(a;q)_n. \] For convenience we write \[ (a_1,\ldots,a_k;q)_n = (a_1;q)_n\cdots (a_k;q)_n,\quad (a_1,\ldots,a_k;q)_{\infty} = (a_1;q)_{\infty} \cdots (a_k;q)_{\infty}. \] The $q$-gamma function is given by \[ \Gamma_q(z) = \dfrac{(q;q)_\infty}{(q^{z};q)_\infty} (1-q)^{1-z} \quad (|q|<1) \] and it is well-known that $\Gamma_q (z)$ is a $q$-analogue for the gamma function $\Gamma (z)$, see \cite{Andrews-Askey-Roy, Askey, Gasper-Rahman, Jackson-1, Jackson-2} for details on the function $\Gamma_q(z)$. The digamma function $\psi(z)$ and the $q$-digamma function $\psi_q (z)$ are given by \[ \psi(z) = \big(\log\Gamma(z)\big)' = \frac{\Gamma'(z)}{\Gamma(z)} \quad\text{and\quad} \psi_q (z) = \big(\log\Gamma_q(z)\big)' = \frac{\Gamma_q '(z)}{\Gamma_q (z)}. \] By Krattenthaler and Srivastava~\cite{Krattenthaler-Srivastava} one has $\lim_{q\to 1} \psi_q(z) = \psi(z)$, showing that the function $\psi_q(z)$ is the $q$-analogue for the function $\psi (z)$. Jacobi first theta function is defined as follows: \[ \theta_1(z \mid \tau) = 2\sum_{n=0}^{\infty}(-1)^n q^{(2n+1)^2/4}\sin(2n+1)z = i q^{\frac{1}{4}}e^{-iz} (q^2 e^{-2iz},e^{2iz},q^2; q^2)_{\infty}. \] Jacobi theta functions have been extensively studied by mathematicians during the last two centuries with hundreds of properties and formulas as a result. Standard references on theta functions include Lawden~\cite{Lawden} and Whittaker~and~Watson~\cite{Whittaker-Watson}. Among the well-known properties of the function $\theta_1(z\mid\tau)$ which we need in this paper we have \begin{equation}\label{theta-cot} \frac{\theta_1'(z|\tau)}{\theta_1(z|\tau)} = \cot z + 4\sum_{n=1}^{\infty}\frac{q^{2n}}{1-q^{2n}} \sin (2nz). \end{equation} Gosper~\cite{Gosper} introduced $q$-analogues of $\sin z$ and $\cos z$ as follows \begin{equation}\label{sine-cosine-q-gamma} \begin{split} \sin_q \pi z &= q^{\frac{1}{4}} \Gamma_{q^2}^2\left(\frac{1}{2}\right) \frac{q^{z(z-1)}}{\Gamma_{q^2}(z) \Gamma_{q^2}(1-z)} \\ \cos_q \pi z &= \Gamma_{q^2}^2\left(\frac{1}{2}\right) \frac{q^{z^2}}{\Gamma_{q^2}\left(\frac{1}{2}-z \right) \Gamma_{q^2}\left(\frac{1}{2}+z\right)}. \end{split} \end{equation} and proved that \begin{equation}\label{sine-cosine-theta} \begin{split} \sin_q (z) = \frac{\theta_1(z\mid \tau')}{\theta_1\left( \frac{\pi}{2}\bigm| \tau' \right)} \qquad \text{and \quad} \cos_q (z) = \frac{\theta_1\left( z+\frac{\pi}{2} \bigm| \tau' \right)} {\theta_1 \left( \frac{\pi}{2} \bigm| \tau' \right)} \quad \quad (\tau' = \frac{-1}{\tau}). \end{split} \end{equation} It can be shown that $\lim_{q\to 1}\sin_q z = \sin z$ and $\lim_{q\to 1}\cos_q z = \cos z$. Moreover, from (\ref{sine-cosine-q-gamma}), one can easily verify by differentiating logarithms that $\sin_q' (z)$ is the $q$-analogue of $\sin' (z) = \cos z$ and that $\cos_q' (z)$ is the $q$-analogue of $\cos' (z) = -\sin z$. We mention that there are known other examples of $q$-analogues for the functions $\sin z$ and $\cos z$, see for instance the book by Gasper~and~Rahman~\cite{Gasper-Rahman}. A function which is very important for our current purpose is \begin{equation}\label{Cotan-q} \Ct_q(z) = \frac{\sin_q' z}{\sin_q z} \end{equation} for which we clearly have $\lim_{q\to 1} \Ct_q(z) = \cot z$. In addition, by taking in (\ref{sine-cosine-q-gamma}) logarithms and differentiating with respect to $z$ we get \begin{equation}\label{reflection} \psi_{q^2}(z)-\psi_{q^2}(1-z) = (2z-1)\log q - \pi\Ct_q (\pi z), \end{equation} which is the $q$-analogue of the well-known reflection formula \[ \psi(z)-\psi(1-z) = -\pi \cot(\pi z). \] Jacobi~\cite{Jacobi} proved that \begin{equation}\label{MainProd} \frac{(q^{2n};q^{2n})_{\infty}}{(q^2;q^2)_{\infty}^n} \prod_{k=-\frac{n-1}{2}}^{\frac{n-1}{2}}\theta_1 \left(z+\frac{k\pi}{n} \bigm| \tau \right) = \theta_1(nz \mid n\tau), \end{equation} see also Enneper~\cite[p. 249]{Enneper}. This formula turns out to be equivalent to the following $q$-trigonometric identity of Gosper~\cite[p. 92]{Gosper}: \begin{equation}\label{SineProd} \prod_{k=0}^{n-1}\sin_{q^n}\pi \left(z+\frac{k}{n} \right) = q^{\frac{(n-1)(n+1)}{12}} \frac{(q;q^2)_{\infty}^2}{(q^n;q^{2n})_{\infty}^{2n}} \sin_q n\pi z \end{equation} which he apparently was not aware of as he stated the identity without proof or reference. Unlike many of Jacobi's results, the formula (\ref{MainProd}) seems not to have received much attention by mathematicians. This is probably due to the lack of applications. The authors recently in~\cite{Bachraoui-Sandor} offered a new proof for~(\ref{SineProd}) and as an application they established a $q$-analogue for the Gauss multiplication formula for the gamma function as well as for an identity of S\'{a}ndor~and~T\'{o}th~\cite{Sandor-Toth} for a short product on Euler gamma function. Our purpose in this note is to apply~(\ref{SineProd}) in order to evaluate finite and infinite sums involving the function $\Ct_q (z)$ along with the functions $h_{q,M,a}(k)$ and $f_{q,M,a}(k)$ both defined on integers $k$ as follows: \begin{equation}\label{h-f} \begin{split} h_{q,M,a}(k) &= \frac{1}{\pi}\Big( (\log q)\frac{2k+a-2M}{2M}-\psi_q \big(\frac{2k+a}{2M} \big) - \psi_q \big(1-\frac{2k+a}{2M} \big) \Big) \\ f_{q,M,a}(k) &= \sum_{n=1}^{\infty}\frac{q^{\frac{2n}{M}}}{1-q^{\frac{2n}{M}}}\sin \frac{(2k+a)n\pi}{M}. \end{split} \end{equation} More specifically, we shall prove the following main results which are new, up to the authors' best knowledge. \begin{theorem}\label{thm-main-1} Let $M>1$ be an integer and let $a$ be an odd integer. Then \noindent \emph{(a)\ } \[ \sum_{n=1}^{\infty} \frac{1}{n} \Ct_q\Big(\frac{(2n+a)\pi}{2M}\Big) = -\frac{1}{M} \sum_{k=1}^M \Ct_q\Big(\frac{(2k+a)\pi}{2M}\Big) \psi\big(\frac{k}{M}\big). \] \noindent \emph{(b)\ } The function $h_{q,M,a}(k)$ is periodic with period $M$ and we have \[ \sum_{n=1}^{\infty} \frac{h_{q,M,a}(n)}{n} = -\frac{1}{M} \sum_{k=1}^M h_{q,M,a}(k) \psi\big(\frac{k}{M}\big). \] \noindent \emph{(c)\ } The function $f_{q,M,a}(k)$ is periodic with period $M$ and we have \[ \sum_{n=1}^{\infty} \frac{f_{q,M,a}(n)}{n} = -\frac{1}{M} \sum_{k=1}^M f_{q,M,a}(k) \psi\big(\frac{k}{M}\big). \] \end{theorem} \begin{theorem}\label{thm-main-2} Let $M$ be a positive integer and let $a$ be an odd integer. Then \begin{align*} \emph{(a)\quad } & \sum_{k=1}^M \Big( \psi_q \big( \frac{2k+a}{2M} \big) - \psi_q \big( 1- \frac{2k+a}{2M} \big) \Big) = \frac{a+1}{2} \log q. \\ \emph{(b)\quad } & \sum_{k=1}^{M}\Big( \psi_{q}\big( \frac{4k+a}{4M} \big) - \psi_{q}\big( 1-\frac{4k+a}{4M} \big) \Big) \\ & \quad = \frac{(a+2)\log q}{4} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{4}\big) \\ & \quad = \begin{cases} \frac{(a+2)\log q}{4} -\frac{\log q}{4}\frac{\Pi_{q^{1/(4M)}}^2}{\Pi_{q^{1/(2M)}}} & \text{if\ } a\equiv 1,-3 \pmod{8} \\ \frac{(a+2)\log q}{4} +\frac{\log q}{4}\frac{\Pi_{q^{1/(4M)}}^2}{\Pi_{q^{1/(2M)}}} & \text{if\ } a\equiv -1,3 \pmod{8}, \end{cases} \\ \emph{(c)\quad } & \sum_{k=1}^{M}\Big( \psi_{q}\big( \frac{6k+a}{6M} \big) - \psi_{q}\big( 1-\frac{6k+a}{6M} \big) \Big) \\ & \quad = \frac{(a+3)\log q}{6} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{6}\big) \\ & \quad = \begin{cases} \frac{(a+3)\log q}{6} +\frac{\log q}{3}\frac{\Pi_{q^{1/(6M)}}^{3/2}}{\Pi_{q^{1/(2M)}}^{1/2}} & \text{if\ } a\equiv 1,-5 \pmod{12} \\ \frac{(a+3)\log q}{6} -\frac{\log q}{3}\frac{\Pi_{q^{1/(6M)}}^{3/2}}{\Pi_{q^{1/(2M)}}^{1/2}} & \text{if\ } a\equiv -1,5 \pmod{12}. \\ \end{cases} \end{align*} \end{theorem} \begin{remark}\label{rmk-main-1} By letting $q\to 1$ in Theorem~\ref{thm-main-1} and Theorem~\ref{thm-main-2} one gets related sums for the functions $\cot z$ and $\psi(z)$. For instance, from Theorem~\ref{thm-main-1}(a) we obtain for $M>1$ and odd integer $a$ \begin{equation}\label{q-to-1} \sum_{n=1}^{\infty} \frac{1}{n} \cot\Big(\frac{(2n+a)\pi}{2M}\Big) = -\frac{1}{M} \sum_{k=1}^M \cot \Big(\frac{(2k+a)\pi}{2M}\Big) \psi\big(\frac{k}{M}\big) \end{equation} and from Theorem~\ref{thm-main-2}(c) we deduce \[ \sum_{k=1}^M \Big( \psi \big( \frac{6k+a}{6M} \big) - \psi \big( 1- \frac{6k+a}{6M} \big) \Big) = - M\pi \cot\big(\frac{a\pi}{6}\big) \] \[ = \qquad \begin{cases} - \sqrt{3} M\pi & \text{if\ } a\equiv 1,-5 \pmod{12} \\ \sqrt{3} M\pi & \text{if\ } a\equiv -1,5 \pmod{12} \\ 0 & \text{if\ } a\equiv -3,3 \pmod{12}. \end{cases} \] \end{remark} \begin{remark}\label{rmk-transcendence} By the well-known fact that $\cot r\pi$ is an algebraic number for any rational number $r$ and the relation (\ref{q-to-1}) we deduce by a result of Adhikari~\emph{et al.}~\cite{Adhikari-et-al} that the sum $\sum_{n=1}^{\infty} \frac{1}{n} \cot\Big(\frac{(2n+a)\pi}{2M}\Big)$ is either zero or transcendental. A similar statement can be made about the $q$-analogue of the sum given in Theorem~\ref{thm-main-1}(b). \end{remark} \noindent Blagouchine~\cite{Blagouchine} recently evaluated a variety of finite sums involving the digamma function and the trigonometric functions. For instance, he proved that for any positive integer $M$ \[ \sum_{k=1}^{M-1} \big(\cot\frac{k\pi}{M}\big) \psi\big(\frac{k}{M}\big) = -\frac{\pi(M-1)(M-2)}{6}. \] We have the following related contribution. \begin{theorem}\label{thm-main-3} For any integer $M>1$ and any odd integer $a$ we have \[ \sum_{k=1}^{M-1} \Big( \cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big) \psi\big(\frac{k}{M}\big) = - \sum_{k=1}^{M-1}\big(\cot\frac{k\pi}{M}\big) \cot\frac{(2k+a)\pi}{2M}. \] \end{theorem} \noindent The rest of the paper is organized as follows. In Section~\ref{Sec:q-trig} we review Gosper's $q$-trigonometry and collect the facts which are needed for our discussion. In Section~\ref{sec:proof-main-1} we give the proof of Theorem~\ref{thm-main-1}, Section~\ref{sec:proof-main-2} is devoted to the proof for Theorem~\ref{thm-main-2}, and Section~\ref{sec:proof-main-3} is devoted to the proof of Theorem~\ref{thm-main-3}. \section{Facts on Gosper's $q$-trigonometry}\label{Sec:q-trig} \noindent Just as for the function $\sin z$ and $\cos z$, it is easy to verify that \begin{align}\label{sine-cos-basics} \sin_q (\frac{\pi}{2}-z)=\cos_q z,\ \sin_q \pi = 0,\ \sin_q \frac{\pi}{2} = \cos_q 0= 1, \\ \sin_q(z+\pi)= -\sin_q z=\sin_q(-z), \ \text{and\ } -\cos_q(z+\pi) = \cos_q z = \cos_q(-z), \nonumber \end{align} from which it follows that for any odd integer $a$, \begin{equation}\label{sin-cos-aux} \begin{split} \sin_{q}\frac{a\pi}{4} &= \begin{cases} \sin_q\frac{\pi}{4} & \text{if\ } a\equiv 1, 3 \pmod{8} \\ - \sin_q \frac{\pi}{4}& \text{if\ } a\equiv -1, -3 \pmod{8}, \end{cases} \\ \sin_{q}\frac{a\pi}{6} &= \begin{cases} \sin_q\frac{\pi}{6} & \text{if\ } a\equiv 1, 5 \pmod{12} \\ -\sin_q\frac{\pi}{6} & \text{if\ } a\equiv -1, -5 \pmod{12} \\ \\ 1 & \text{if\ } a\equiv 3 \pmod{12} \\ -1 & \text{if\ } a\equiv -3 \pmod{12}. \end{cases} \end{split} \end{equation} Also, by using (\ref{sine-cos-basics}) we have \begin{align}\label{special-deriv} \sin_q'\big(\frac{\pi}{2}-z \big)= -\cos_q' z,\ \cos_q'\big(z-\frac{\pi}{2}\big)= \sin_q' z,\\ - \sin_q'(\pi-z) = \sin_q' z,\ \text{and\ } -\cos_q'(\pi-z) = \cos_q' z \nonumber \end{align} where the derivatives here and in what follows are with respect to $z$. We can easily see from (\ref{special-deriv}) that for any odd integer $a$ we have \begin{equation}\label{q-sine-derive-2} \sin_q' \frac{a\pi}{2} = \cos_q' 0 = 0. \end{equation} The following $q$-constant appears frequently in Gosper's manuscript~\cite{Gosper} \[ \Pi_q = q^{\frac{1}{4}} \frac{(q^2;q^2)_{\infty}^2}{(q;q^2)_{\infty}^2}. \] Gosper stated many identities involving $\sin_q z$ and $\cos_q z$ which easily follow just from the definition and basic properties of other related functions. To mention an example, he derived that \begin{equation}\label{q-sine-derive-1} \sin_q' 0 =- \cos_q'\frac{\pi}{2} = \frac{-2 \log q}{\pi}\Pi_q. \end{equation} On the other hand, Gosper~\cite{Gosper} using the computer facility \emph{MACSYMA} stated without proof a variety of identities involving $\sin_q z$ and $\cos_q z$ and he asked the natural question whether his formulas hold true. For instance, based on his conjectures, he stated \[ \label{q-Double-2} \tag{$q$-Double$_2$} \sin_q(2z) = \frac{\Pi_q}{\Pi_{q^2}} \sin_{q^2} z \cos_{q^2} z, \] \[ \label{q-Double-3} \tag{$q$-Double$_3$} \cos_q(2z) = (\cos_{q^2} z)^2 - (\sin_{q^2} z)^2, \] \[ \label{q-Triple-2} \tag{$q$-Triple$_2$} \sin_q(3z) = \frac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3} z)^2 \sin_{q^3}z - (\sin_{q^3}z)^3, \] and \[ \label{q-Double-5} \tag{$q$-Double$_5$} \cos_q(2z) = (\cos_{q}z)^4- (\sin_{q}z)^4. \] \noindent A proof for (\ref{q-Double-2}) can be found in Mez\H{o}~\cite{Mezo-1} and proofs for (\ref{q-Double-2}) (\ref{q-Triple-2}), and (\ref{q-Double-5}) were obtained in~\cite{Bachraoui-1, Bachraoui-2, Bachraoui-3}. Proofs for other identities of Gosper can be found in~\cite{Touk-Houchan-Bachraoui, He-Zhai, He-Zhang}. Furthermore, Gosper deduced the following special values: \begin{equation}\label{sin-cos-values} \begin{split} \sin_{q^2} \frac{\pi}{4} &= \cos_{q^2} \frac{\pi}{4} = \frac{\Pi_{q^2}^{\frac{1}{2}}}{\Pi_{q}^{\frac{1}{2}}} \\ \left(\sin_{q^3}\frac{\pi}{3} \right)^3 &= \left(\cos_{q^3}\frac{\pi}{6} \right)^3 = \frac{ \left(\frac{\Pi_q}{\Pi_{q^3}} \right)^{\frac{3}{2}}}{\left(\frac{\Pi_q}{\Pi_{q^3}} \right)^2 -1} \\ \left(\sin_{q^3}\frac{\pi}{6} \right)^3 &= \left(\cos_{q^3}\frac{\pi}{3} \right)^3 = \frac{ 1}{\left(\frac{\Pi_q}{\Pi_{q^3}} \right)^2 -1}. \end{split} \end{equation} \noindent As to special values for derivatives we have the following list. \begin{lemma} \label{lem-1-special} Let $a$ be an odd integer. Then we have \[ \begin{split} \emph{(a)\ } & \sin_{q^2}'\frac{a\pi}{4} = \begin{cases} \frac{\log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^2}^{\frac{1}{2}}} & \text{if\ } a\equiv -1, 1 \pmod{8} \\ - \frac{\log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^2}^{\frac{1}{2}}} & \text{if\ } a\equiv -3, 3 \pmod{8}. \end{cases} \\ \emph{(b)\ } & \sin_{q^3}'\frac{a\pi}{3} = \begin{cases} \frac{\log q}{\pi} \frac{\Pi_{q^3}^{\frac{1}{3}} (\Pi_q^2 - \Pi_{q^3}^2)^{\frac{2}{3}} (3\Pi_{q^3}^2-\Pi_q^2)} {\Pi_{q}^2 - \Pi_{q^3}^2} & \text{if\ } a\equiv -1, 1 \pmod{6} \\ -\frac{2\log q}{\pi} \Pi_q & \text{if\ } a\equiv 3 \pmod{6} \end{cases} \\ \emph{(c)\ } & \sin_{q^3}'\frac{a\pi}{6} = \begin{cases} -\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}} \Pi_{q^3}^{\frac{1}{6}} }{ (\Pi_q^2- \Pi_{q^3}^2)^{\frac{1}{3}}} & \text{if\ } a\equiv 1, -5 \pmod{12} \\ \frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}} \Pi_{q^3}^{\frac{1}{6}} }{ (\Pi_q^2- \Pi_{q^3}^2)^{\frac{1}{3}}} & \text{if\ } a\equiv -1, 5 \pmod{12} \\ 0 & \text{if\ } a\equiv -3, 3 \pmod{12}. \end{cases} \end{split} \] \end{lemma} \begin{proof} (a)\ From (\ref{special-deriv}) we have \begin{equation}\label{q-sine-derive-quarter} \sin_{q}' \frac{\pi}{4} = - \cos_{q}' \frac{\pi}{4}. \end{equation} On the other hand, from (\ref{q-Double-3}) we have \[ 2\cos_q' 2z = 2(\cos_{q^2} z) \cos_{q^2}' z - 2(\sin_{q^2} z) \sin_{q^2}'z, \] where if we let $z=\frac{\pi}{4}$ and use (\ref{q-sine-derive-quarter}) we deduce \[ 2\cos_q'\frac{\pi}{2} = -4 \sin_{q^2}\frac{\pi}{4} \sin_{q^2}'\frac{\pi}{4}. \] Now, combine the previous identity with (\ref{q-sine-derive-1}), (\ref{sin-cos-values}), and (\ref{q-sine-derive-quarter}) to obtain the desired identity for $\sin_{q^2}'\frac{\pi}{4}$. Finally, note that by (\ref{sine-cos-basics}) we have \[ \sin_q'\frac{a\pi}{4} = \begin{cases} \sin_q'\frac{\pi}{4} & \text{if\ } a\equiv \pm 1 \pmod{8}, \\ -\sin_q'\frac{\pi}{4} & \text{if\ } a\equiv \pm 3 \pmod{8} \end{cases} \] to complete the proof of part (a). \noindent As to part (b), from (\ref{special-deriv}), we easily find \begin{equation}\label{sixth-third} \cos_q'\frac{\pi}{6} = -\sin_q'\frac{\pi}{3} \ \text{and\ } \cos_q'\frac{\pi}{3} = -\sin_q'\frac{\pi}{6}. \end{equation} Now differentiating (\ref{q-Double-5}) we have \[ 2\cos_q' 2z = 4 (\cos_q z)^3 \cos_q'z - 4 (\sin_q z)^3 \sin_q'z. \] Then taking $z=\frac{\pi}{6}$ in the previous identity, using (\ref{sixth-third}), and simplifying yield \[ \big(2(\sin_q\frac{\pi}{6})^3 - 1 \big) \sin_q'\frac{\pi}{6} = 2 (\cos_q\frac{\pi}{6})^3 \cos_q'\frac{\pi}{6}, \] in other words, \begin{equation}\label{help1-lem-2-special} \cos_q'\frac{\pi}{6} = \frac{2(\sin_q\frac{\pi}{6})^3 - 1}{2 (\cos_q\frac{\pi}{6})^3} \sin_q'\frac{\pi}{6}. \end{equation} On the other hand, differentiate (\ref{q-Triple-2}) to derive \[ 3 \sin_q 3z = \frac{\Pi_q}{\Pi_{q^3}}\big(\sin_{q^3}'z (\cos_{q^3} z)^2 + 2\sin_{q^3}z\cos_{q^3}z\cos_{q^3}'z \big) - 3 (\sin_{q^3} z)^2 \sin_{q^3}'z, \] which for $z=\frac{\pi}{3}$ and after simplification gives \[ -\sin_q'0 = \Big(\frac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3}\frac{\pi}{3})^2 - 3(\sin_{q^3}\frac{\pi}{3})^2 \Big) \sin_{q^3}'\frac{\pi}{3} + 2 \frac{\Pi_q}{\Pi_{q^3}} \sin_{q^3}\frac{\pi}{3}\cos_{q^3}\frac{\pi}{3}\cos_{q^3}'\frac{\pi}{3}. \] It follows by virtue of (\ref{help1-lem-2-special}) and with the help of (\ref{sin-cos-values}) that \[ \frac{6 \log q}{\pi}\Pi_q =\Big(\frac{\Pi_q}{\Pi_{q^3}} (\cos_{q^3}\frac{\pi}{3})^2 -3(\sin_{q^3}\frac{\pi}{3})^2 + \frac{4 \frac{\Pi_q}{\Pi_{q^3}} \big(\sin_{q^3}\frac{\pi}{3}\big)^4 \cos_{q^3}\frac{\pi}{3}} {2 \big(\sin_{q^3}\frac{\pi}{3}\big)^3 - 1} \Big) \sin_{q^3}'\frac{\pi}{3}. \] Now solving in the previous identity for $\sin_{q^3}'\frac{\pi}{3}$ and using (\ref{sin-cos-values}), after a long but straightforward calculation, we derive the desired formula for $\sin_{q^3}'\frac{\pi}{3}$. Finally, note from (\ref{sine-cos-basics}) that \[ \sin_q'\frac{a\pi}{3} = \begin{cases} \sin_q'\frac{\pi}{3} & \text{if\ } a\equiv \pm 1 \pmod{6}, \\ -\sin_q' 0 & \text{if\ } a\equiv \pm 3 \pmod{6} \end{cases} \] to complete the proof of part (b). The proof for part (c) is similar to the previous parts and it is therefore omitted. \end{proof} \noindent By a combination of Lemma~\ref{lem-1-special} with (\ref{sin-cos-aux}) and (\ref{sin-cos-values}), we arrive at the main result of this section. \begin{corollary}\label{cor-special-C} Let $a$ be an odd integer. Then we have \[ \begin{split} \emph{(a)\quad } & \Ct_{q^2} \big(\frac{a\pi}{4}\big) = \begin{cases} \frac{\log q}{\pi} \frac{\Pi_{q}^2}{\Pi_{q^2}} & \text{if\ } a\equiv 1, -3 \pmod{8} \\ -\frac{\log q}{\pi} \frac{\Pi_{q}^2}{\Pi_{q^2}} & \text{if\ } a\equiv -1, 3 \pmod{8}. \end{cases} \\ \emph{(b)\quad } & \Ct_{q^3} \big(\frac{a\pi}{6}\big) = \begin{cases} -\frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^3}^{\frac{1}{2}}} & \text{if\ } a\equiv 1, -5 \pmod{12} \\ \frac{2 \log q}{\pi} \frac{\Pi_{q}^{\frac{3}{2}}}{\Pi_{q^3}^{\frac{1}{2}}} & \text{if\ } a\equiv -1, 5 \pmod{12} \\ 0 & \text{if\ } a\equiv -3, 3 \pmod{12}. \end{cases} \end{split} \] \end{corollary} \section{Proof of Theorem~\ref{thm-main-1}}\label{sec:proof-main-1} \noindent We need the following result of Ram~Murty~and~Saradha~\cite{Murty-Saradha} which we record as a lemma. \begin{lemma}\label{lem-MurtSara} Let $f$ be any function defined on the integers and with period $M>1$. \\ \noindent The infinite series $\sum_{n=1}^{\infty} \frac{f(n)}{n}$ converges if and only if $\sum_{k=1}^M f(k) = 0$. In case of convergence, we have \[ \sum_{n=1}^{\infty} \frac{f(n)}{n} = -\frac{1}{M} \sum_{k=1}^M f(k) \psi\big(\frac{k}{M}\big). \] \end{lemma} \noindent \emph{Proof of Theorem~\ref{thm-main-1}(a)}\ Let \[ f(k) = \Ct_q \Big( \frac{(2k+a)\pi}{2M} \Big) \] which is clearly well-defined on the integers and it is periodic with period $M$. Then based on Lemma~\ref{lem-MurtSara}, all we need is prove that $\sum_{k=1}^{M} f(k) = 0$. To do so, note that from (\ref{SineProd}) and the fact that $\sin_q(z+\pi) = -\sin_q z$ we find \begin{equation}\label{SineProd-2} \prod_{k=1}^{M}\sin_{q^M}\pi \left(z+\frac{k}{M} \right) = - q^{\frac{(M-1)(M+1)}{12}} \frac{(q;q^2)_{\infty}^2}{(q^M;q^{2M})_{\infty}^{2M}} \sin_q M\pi z. \end{equation} Take logarithms and differentiate with respect to $z$ to derive \[ \pi \sum_{k=1}^M \Ct_{q^M}\Big(z+\frac{k}{M}\Big) = M\pi\Ct_q(M\pi z)= M\pi \frac{\sin_q' M\pi z}{\sin_q M\pi z}. \] Now replace $q^M$ with $q$, let $z=\frac{a}{2M}$, and use (\ref{q-sine-derive-2}) to deduce that that \begin{equation}\label{sum-Ctq-zero} \pi \sum_{k=1}^M \Ct_q \Big( \frac{(2k+a)\pi}{2M} \Big) = 0, \end{equation} or equivalently, \[ \sum_{k=1}^{M} f(k) = 0, \] as desired. \noindent \emph{Proof of Theorem~\ref{thm-main-1}(b)}\ By the relation (\ref{sine-cosine-q-gamma}) we have \[ \sin_q\pi\big(z+\frac{k}{M}\big) = q^{\frac{1}{4}}\Gamma_{q^2}^2\left(\frac{1}{2}\right) \frac{q^{(z+\frac{k}{M})(z+\frac{k}{M} -1)}}{\Gamma_{q^2}\big(z+\frac{k}{M} \big) \Gamma_{q^2}\big(1-z-\frac{k}{M}\big)} \] which after taking logarithms and differentiating with respect to $z$ gives \[ \pi \Ct_q\pi\big(z+\frac{k}{M}\big) = (\log q) \big( 2\big(z+\frac{k}{M}\big)-1 \big) - \psi_{q^2}\big(z+\frac{k}{M}\big) - \psi_{q^2}\big(1-z-\frac{k}{M}\big). \] Replacing in the previous relation $q^2$ by $q$ and letting $z=\frac{a}{2M}$, we get \[ \Ct_{q^{1/2}}\big(\frac{(2k+a)\pi}{2M}\big) = \frac{1}{\pi}\Big( (\log q)\frac{2k+a-2M}{2M} - \psi_{q}\big(\frac{2k+a}{2M}\big) - \psi_{q}\big(1-\frac{2k+a}{2M}\big) \Big). \] As the left-hand side of the previous identity is evidently periodic with period $M$, the same holds for its right-hand side which is nothing else but $h_{q,M,a}(k)$. We now claim that $\sum_{k=1}^M h_{q,M,a}(k) = 0$. Indeed, apply~(\ref{sine-cosine-q-gamma}) to the factors in identity~(\ref{SineProd-2}), then take logarithms and finally differentiate with respect to $z$ to obtain \begin{align}\label{q-psi-key} M(\log q) & \Big(\sum_{k=1}^M 2\big(z+ \frac{k}{M}\big) -1 \Big) - \sum_{k=1}^M \Big(\psi_{q^{2M}}\big(z+\frac{k}{M}\big) - \psi_{q^{2M}}\big(1-z-\frac{k}{M}\big) \Big) \nonumber \\ & = M\pi \Ct_q(M\pi z) . \end{align} Next replace $q^{2M}$ by $q$, let $z=\frac{a}{2M}$, and use (\ref{q-sine-derive-2}) to deduce that \begin{equation}\label{help1-cor-psiq-1} \sum_{k=1}^M \Big( (\log q)\frac{2k+a-M}{2M} - \psi_{q}\big(\frac{2k+a}{2M}\big) - \psi_{q}\big(1-\frac{2k+a}{2M}\big) \Big) = 0. \end{equation} That is, \[ \sum_{k=1}^M h_{q,M,a}(k) = 0, \] and the claim is confirmed. Finally, apply Lemma~\ref{lem-MurtSara} to the function $h_q(M,a,k)$ to complete the proof of part (b). \noindent \emph{Proof of Theorem~\ref{thm-main-1}(c)}\ Note that the function $f_{q,M,a}(k)$ is clearly periodic with period $M$. By virtue of~(\ref{sine-cosine-theta}) and~(\ref{SineProd-2}) and after taking logarithm and differentiating we find \[ \pi \sum_{k=1}^M \frac{\theta_1'\big(\pi z +\frac{k\pi}{M} | \frac{\tau'}{M}\big)}{\theta_1\big(\pi z +\frac{k\pi}{M} | \frac{\tau'}{M}\big)} = M\pi \Ct_q(M\pi z), \] which upon substituting $z$ by $\frac{a}{2M}$ and $\tau'$ by $\tau$ yields \[ \sum_{k=1}^M \frac{\theta_1'\big(\frac{(2k+a)\pi}{2M} | \frac{\tau}{M}\big)}{\theta_1\big(\frac{(2k+a)\pi}{2M} | \frac{\tau}{M}\big)} = 0. \] Now combine the foregoing identity with (\ref{theta-cot}) and the $q$-analogue of (\ref{sum-Ctq-zero}) to derive \[ \sum_{k=1}^M f_{q,M,a}(k) = \sum_{k=1}^M \sum_{n=1}^{\infty}\frac{q^{\frac{2n}{M}}}{1-q^{\frac{2n}{M}}}\sin \frac{(2k+a)n\pi}{M} =0. \] Finally apply Lemma~\ref{lem-MurtSara} to the function $f_{q,M,a}(k)$ to complete the proof. \section{Proof of Theorem~\ref{thm-main-2}}\label{sec:proof-main-2} \noindent (a)\ If $M=1$, then the desired formula \[ \psi_q\big(1+\frac{a}{2}\big) - \psi_q\big(-\frac{a}{2}\big) = \frac{(a+1)\log q}{2} \] follows by virtue of (\ref{reflection}). If $M>1$, then an immediate consequence of (\ref{SineProd-2}) is \[ \sum_{k=1}^M \Big(\psi_{q}\big(\frac{2k+a}{2M}\big) - \psi_{q}\big(1-\frac{2k+a}{2M}\big)\Big) = \frac{(a+1) \log q}{2}, \] which is the desired relation. \noindent (b)\ If $M=1$, the statement follows by (\ref{reflection}). Now suppose that $M>1$. Then by letting in (\ref{q-psi-key}) $z=\frac{a}{4M}$ and after simplification we find \[ \frac{(a+2)M\log q}{2} - \sum_{k=1}^M \Big( \psi_{q^{2M}}\big(\frac{4k+a}{4M}\big) - \psi_{q^{2M}}\big(\frac{4(M-k)-a}{4M}\big)\Big) = M\pi \Ct_q\big(\frac{a\pi}{4}\big). \] Then uopn replacing $q^{2M}$ by $q$ and rearranging becomes \[ \sum_{k=1}^M \Big(\psi_{q}\big(\frac{4k+a}{4M}\big) - \psi_{q}\big(\frac{4(M-k)-a}{4M}\big)\Big) = \frac{(a+2)\log q}{4} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{4}\big), \] which is the first identity of this part. As to the second formula, simply use Corollary~\ref{cor-special-C} to evaluate the right-hand-side of the previous identity and rearrange in the appropriate way. \noindent (c)\ If $M=1$, the statement follows by (\ref{reflection}). Now suppose that $M>1$. Let in (\ref{q-psi-key}) $z=\frac{a}{6M}$ and simplify to obtain \[ \frac{(a+3)M\log q}{3} - \sum_{k=1}^M \Big(\psi_{q^{2M}}\big(\frac{6k+a}{6M}\big) - \psi_{q^{2M}}\big(\frac{6(M-k)-a}{6M}\big)\Big) = M\pi \Ct_q\big(\frac{a\pi}{6}\big). \] Then replace in the foregoing formula $q^{2M}$ by $q$ and rearrange to get \[ \sum_{k=1}^M \Big(\psi_{q}\big(\frac{6k+a}{6M}\big) - \psi_{q}\big(\frac{6(M-k)-a}{6M}\big)\Big) = \frac{(a+3)\log q}{6} - M\pi \Ct_{q^{1/(2M)}}\big(\frac{a\pi}{6}\big). \] This establishes the first formula of this part. Finally apply Corollary~\ref{cor-special-C} to complete the proof. \section{Proof of Theorem~\ref{thm-main-3}}\label{sec:proof-main-3} \noindent In our proof we shall make an appeal to a result of Weatherby~in~\cite{Weatherby} for which we need the following notation. For any real number $\alpha$ and any positive integer $l$, let \[ A_{\alpha,l} := \frac{(-1)^{l-1} \big(\pi\cot(\pi \alpha)\big)^{(l-1)}}{\pi^l(l-1)!} \] and let \[ Z(l) = \begin{cases} 0 & \text{if $l$\ is odd,} \\ \frac{\zeta(l)}{\pi^l} & \text{otherwise.} \end{cases} \] Notice that $Z(l)\in\mathbb{Q}$ for all positive integer $l$. \begin{lemma}\label{lem-Weatherby} Let $f$ be an algebraic valued function defined on the integers with period $M>1$ and let $l$ be a positive integer. Then \[ \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{f(n)}{n^l} = \Big(\frac{\pi}{M}\Big)^l \Big( \sum_{k=1}^{M-1} f(k) A_{\frac{k}{M},l}+ 2f(M) Z(l) \Big). \] \end{lemma} \noindent \emph{Proof of Theorem~\ref{thm-main-3}.\ } Let \[ f(k) = \cot \frac{(2k+a)\pi}{2M}. \] Clearly $f(k)$ is well-defined on the integers since is $a$ is odd and it is periodic with period $M$. It is a well-known fact that for any rational number $r$ we have that $\cot\pi r$ is an algebraic number. Then by virtue of Lemma~\ref{lem-Weatherby} we get \begin{equation}\label{key-sum-main-3} \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot\pi\big(\frac{2n+a}{2M}\big)}{n^l} =\Big(\frac{\pi}{M}\Big)^l \Big( \sum_{k=1}^{M-1} \big(\cot\pi\frac{2k+a}{2M}\big) A_{\frac{k}{M},l}+ 2f(M) Z(l) \Big), \end{equation} which for $l=1$ reduces to \begin{equation}\label{help2-cor-cot-1} \begin{split} \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n} &= \frac{\pi}{M}\sum_{k=1}^{M-1}A_{\frac{k}{M},1}\cot\frac{(2k+a)\pi}{2M} \\ &= \frac{\pi}{M}\sum_{k=1}^{M-1} \frac{1}{\pi} \big(\cot\frac{k\pi}{M} \big) \cot\frac{(2k+a)\pi}{2M}. \end{split} \end{equation} On the other hand, with the help of the $q$-analogue of Theorem~\ref{thm-main-1}(a) we deduce \[ \sum_{n=1}^{\infty}\frac{\cot \frac{(-2n+a)\pi}{2M}}{-n} = \sum_{n=1}^{\infty}\frac{\cot \frac{(2n-a)\pi}{2M}}{n} = -\frac{1}{M} \sum_{k=1}^{M} \Big(\cot\frac{(2k-a)\pi}{2M}\Big)\psi\big(\frac{k}{M}\big), \] which implies that \[ \begin{split} \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n} &= \sum_{n=1}^{\infty}\frac{\cot \frac{(2n+a)\pi}{2M}}{n} + \sum_{n=1}^{\infty}\frac{\cot \frac{(2n-a)\pi}{2M}}{n} \\ &= -\frac{1}{M} \sum_{k=1}^{M} \psi\big(\frac{k}{M}\big)\Big(\cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big). \end{split} \] As \[\cot\frac{(2M+a)\pi}{2M} + \cot\frac{(2M-a)\pi}{2M} = \frac{\sin 2\pi} {\frac{1}{2}\big(\cos\frac{a\pi}{M} - \cos\frac{2M\pi}{M} \big)} = 0, \] the $M$-th term in the last summation vanishes and so we get \begin{equation}\label{help1-cor-cot-1} \sum_{n\in\mathbb{Z}\setminus\{0\}} \frac{\cot \frac{(2n+a)\pi}{2M}}{n} = -\frac{1}{M} \sum_{k=1}^{M-1} \psi\big(\frac{k}{M}\big)\Big(\cot\frac{(2k+a)\pi}{2M} + \cot\frac{(2k-a)\pi}{2M} \Big). \end{equation} Now combine (\ref{help1-cor-cot-1}) and (\ref{help2-cor-cot-1}) to obtain the desired formula. \end{document}

\begin{document} \title{Minimal set of local measurements and classical communication for two-mode Gaussian state entanglement quantification} \author{Luis F. Haruna, Marcos C. de Oliveira, and Gustavo Rigolin} \affiliation{Instituto de F\'\i sica ``Gleb Wataghin'', Universidade Estadual de Campinas, 13083-970, Campinas, S\~ao Paulo, Brazil.} \begin{abstract} We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting/intensity measurements define the whole set of entangled states. In addition, conditioned on some prior information, the protocol is also useful for a complete global state reconstruction. \end{abstract} \pacs{03.67.-a, 03.67.Mn} \maketitle Quantum communication protocols extend the information theoretical notion of channel \cite{Thomas} to the quantum domain by incorporating non-local entangled states. Those channels are generated by the preparation of a pair (or more) of quantum systems in an entangled state, which are then separated to establish non-local correlations \cite{EPR}, allowing several communication tasks otherwise unattainable via classical channels \cite{nielchu}. However, for most of the quantum protocols to work properly (deterministically) one has first to be able to prepare maximally pure entangled states and then to guarantee that those states stay pure or nearly pure during all the processing time. An important problem then arises in this whole process: One has to check the ``quality'' (the amount of entanglement and purity) of the quantum channel, while usually the only available tools for that are local measurements (operations) and one (or several) classical channel. The quest for an optimal and general solution for this problem has generated a vast literature on the characterization of entangled states under local operations and classical communication (LOCC), either for qubits \cite{nielsen} or for continuous variable systems of the Gaussian type \cite{eisert1,eisert2}. Gaussian states (completely described by up to second order moments) are particularly important since they can be easily generated with radiation field modes. Moreover, operations that keep the Gaussian character (so-called Gaussian operations) are given by the transformations induced by linear (active and passive) optical devices (beam-splitters, phase-shifters, and squeezers) \cite{eisert2}. A particular result for this kind of state is that it is impossible to distill entanglement out of a set of Gaussian states through Gaussian operations \cite{eisert3}. Assuming one is left with only Gaussian local operations and a classical channel (GLOCC), how is it possible to infer the quality of a quantum channel in use? For a two-mode Gaussian state one possibility is to access directly the entanglement properties of the system after a proper manipulation of the two modes \cite{marcos2,rigolin}. This procedure requires, however, that the two parties (modes) be recombined in a beam-splitter (non-local unitary operation) in which their entanglement content are transferred to local properties of one of the output modes. Another possible way is to completely reconstruct the bipartite quantum system, a resource demanding task \cite{laurat} which also requires global operations here forbidden. In this Letter we demonstrate a minimal set of GLOCC to completely quantify the entanglement of a two-mode Gaussian state. As a bonus of this procedure one can also assess the purity of the Gaussian state and, for some particular classes of states, reconstruct the bipartite covariance matrix. The protocol consists mainly in the attainment, via local measurements, of all the symplectic invariants that allows, for example, one to test the separability of the system, to know its P-representability properties, and to quantify its entanglement content. We also show that for a particular class of Gaussian states belonging to the set of symmetric Gaussian states \cite{marcos1}, the Einstein-Podolsky-Rosen (EPR) states and general mixed squeezed states , the protocol becomes straightforward due to the relative easiness one obtains the correlation matrix elements from local measurement outcomes. Moreover, since P-representability and separability for these kind of states are equivalent, we show that for two-mode thermal squeezed states with internal noise \cite{daffer} it is possible to decide whether or not they are separable via local photon number measurements. A two-mode Gaussian state $\rho_{12}$ is characterized by its Gaussian characteristic function $C({\bm\alpha})=e^{-\frac12{\bm{\alpha}^\dagger}{\bf V}{\bm{\alpha}} }$, where $\bm{\alpha}^\dagger=\left(\alpha_1^*, \alpha_1, \alpha_2^*, \alpha_2\right)$ are complex numbers and $a_1$ ($a_1^\dagger$) and $a_2$ ($a_2^\dagger$) the annihilation (creation) operators for parties 1 and 2, respectively \cite{comment1}. The covariance matrix \textbf{V} describing all the second order moments $V_{ij}=(-1)^{i+j}\langle v_i v_j^\dagger + v_j^\dagger v_i \rangle/2$, where $(v_1,v_2,v_3,v_4)=(a_1,a_1^\dagger,a_2,a_2^\dagger)$, is given by \begin{displaymath} \textbf{V}=\left( \begin{array}{cc} \textbf{V}_1 & \textbf{C} \\ \textbf{C}^\dagger & \textbf{V}_2 \end{array} \right) =\left(\begin{array}{cccc} n_1+ \frac{1}{2} & m_1 & m_s & m_c \\ m_1^* & n_1+\frac{1}{2} & m_c^* & m_s^* \\ m_s^* & m_c & n_2+\frac{1}{2} & m_2 \\ m_c^* & m_s & m_2^* & n_2+ \frac{1}{2} \\ \end{array} \right). \end{displaymath} $\textbf{V}_1$ and $\textbf{V}_2$ are local Hermitian matrices while \textbf{C} is the correlation between the two parties. Any covariance matrix must be positive semidefinite $\textbf{V}\geq\mathbf{0}$ and the generalized uncertainty principle, $\textbf{V}+(1/2)\textbf{E}\geq\mathbf{0}$, where ${\bf E}=\text{diag}(\mathbf{Z},\mathbf{Z})$ and ${\bm{Z}}=\text{diag}(1,-1)$, must hold \cite{englert}. From local measurements on both modes of $\rho_{12}$, either through homodyne detection (see \cite{grangier} and references therein) or alternatively by employing single-photon detectors \cite{fiurasek2}, the local covariance matrices $\textbf{V}_1$ and $\textbf{V}_2$ can be reconstructed. Remark that for the reconstruction of the global matrix \textbf{V}, and therefore the joint bipartite state, one has to obtain \textbf{C}. Obviously, global joint measurements achieved through recombination of the two parties in a beam-splitter followed by local homodyne detections are forbidden. Thus one has to deal only with local measurements whose results can be sent through classical communication channels to the other party. As we now show, there are minimal operations/measurements that can be performed locally on the system to attain $|\det\textbf{C}|$ and $\det$\textbf{V}. These quantities, together with $\det\textbf{V}_1$ and $\det\textbf{V}_2$, will be shown to be all that one needs to determine whether or not a two-mode Gaussian state is entangled as well as how much it is entangled. As it will become clear, the required set of operations is minimal in the sense that only two local measurement procedures are needed - one to characterize local covariance matrices and another to locally assess the parity of one of the modes. First of all let us introduce an important result \cite{haruna1}. Given a two-mode Gaussian state with density operator $\rho_{12}$ and covariance matrix $\textbf{V}$ we can define the Gaussian operator $\sigma_1 = Tr_2\left\{e^{i\pi a_2^\dagger a_2}\rho_{12}\right\},$ whose covariance matrix ${\bf\Gamma}_1$ is the Schur complement \cite{horn} of $\textbf{V}$ relative to $\textbf{V}_2$: \begin{equation} {\bf \Gamma}_1=\textbf{V}_1-\textbf{C}\textbf{V}_2^{-1}\textbf{C}^\dagger. \label{schur_rel1} \end{equation} The meaning of $\sigma_1$ is best appreciated through a partial trace in the Fock basis: $\sigma_1=\sum_{n_{even}} \!_2\langle n|\rho_{12}|n\rangle_2 - \sum_{n_{odd}}\!_2\langle n|\rho_{12}|n\rangle_2=\rho_{1_{e}}-\rho_{1_{o}}$, being equal to the difference between Alice's mode states conditioned, respectively, to even and odd parity measurement results by Bob \cite{haruna1}. While $\rho_{1_e}$ and $\rho_{1_o}$ are not generally Gaussian, $\sigma_1$ is a Gaussian operator, and ${\bf\Gamma}_1$ can be built with only second order moments of these conditioned states. Now suppose that Alice and Bob share many copies of a two-mode Gaussian state. The protocol works as follows: ({\it i}) Firstly, in a subensemble of the copies, each party performs a set of local measurements in such a manner to obtain the covariance matrices $\textbf{V}_1$ and $\textbf{V}_2$, corresponding to the reduced operators $\rho_1=Tr_2\{\rho_{12}\}$ and $\rho_2=Tr_1\{\rho_{12}\}$; ({\it ii}) Then Bob informs Alice, via a classical communication channel, the matrix elements of $\textbf{V}_2$; ({\it iii}) After that, for the remaining copies, Bob performs parity measurements on his mode, letting Alice know to which copies does that operation correspond and the respective outcomes, i.e. even parity (eigenvalue 1) or odd parity (eigenvalue -1); ({\it iv}) Alice then separates her copies in two groups, the even ($e$) and the odd ($o$) ones. The first group ($e$) contains all the copies conditioned on an even parity measurement on Bob's copies. The other one ($o$) contains all the remaining copies, namely those conditioned on an odd parity measurement at Bob's; ({\it v}) For each group, Alice measures the respective correlation matrices $\textbf{V}_{1e}$ and $\textbf{V}_{1o}$; ({\it vi}) Finally, she obtains $\bf\Gamma_1$ (Eq. (\ref{schur_rel1})) subtracting the odd correlation matrix from the even one \cite{haruna1}: ${\bf\Gamma}_1=\textbf{V}_{1e} - \textbf{V}_{1o}$. Remarkably, with $\textbf{V}_1$, $\textbf{V}_2$ and ${\bf\Gamma}_1$ in hand Alice is able to completely characterize the Gaussian state's entanglement content as well as its purity without any global or non-local measurements. Remembering that a two-mode Gaussian state's purity $\mathcal{P}$ is equal to $1/(4\sqrt{\det\mathbf{V}})$ \cite{adesso} and using the identity \cite{horn} \begin{equation} \det\textbf{V}=\det\textbf{V}_2\det{\bf\Gamma}_1, \label{detV} \end{equation} Alice readily obtains the purity of the channel: $\mathcal{P}= 1/(4\sqrt{\det\mathbf{V}_2\det{\bf\Gamma}_1}).$ Her next task is to decide whether or not she deals with an entangled two-mode Gaussian state. Using the Simon separability \cite{simon} test she knows that it is not entangled if, and only if, \begin{equation} I_1I_2 + \left( 1/4 - |I_3|\right)^2 - I_4 \geq (I_1 + I_2)/4, \label{separabilidade} \end{equation} where $I_1=\det\mathbf{V_1}$, $I_2=\det\mathbf{V_2}$, $I_3=\det\mathbf{C}$, and $I_4=\text{tr}(\mathbf{V_1}\mathbf{Z}\mathbf{C}\mathbf{Z} \mathbf{V}_2\mathbf{Z}\mathbf{C^\dagger}\mathbf{Z})$. These four quantities are the local symplectic invariants, belonging to the $Sp(2,R) \otimes Sp(2,R)$ group \cite{simon}, that characterizes all the entanglement properties of a two-mode Gaussian state. Alice already has $I_1$ and $I_2$. We must show, however, how she can obtain $|I_3|$ and $I_4$. Since one can prove that \cite{rigolin} \begin{equation} I_4 = 2|I_3|\sqrt{I_1I_2}, \label{I4} \end{equation} we just need to show how $|I_3|$ is obtained from $I_1$, $I_2$, and $I_V=\det\mathbf{V}$, the three pieces of information locally available to Alice. To achieve this goal we first note that a direct calculation gives $I_V = I_1I_2 - I_4 + I_3^2$. Using Eq.~(\ref{I4}) we see that $|I_3|$ follows from $|I_3|^2 - 2 |I_3| \sqrt{I_1I_2} + I_1I_2 - I_V = 0.$ One of its roots is not acceptable since it implies $\mathbf{V}< 0$. Therefore, we are left with \begin{equation} |I_3|=\sqrt{I_1I_2} - \sqrt{I_V}. \label{I3} \end{equation} Hence, substituting Eqs.~(\ref{I4}) and (\ref{I3}) in Eq.~(\ref{separabilidade}), Alice is able to unequivocally tell whether or not she shares an entangled two-mode Gaussian state with Bob. Finally, if her state is entangled then $I_3<0$ \cite{simon} and, for a symmetric state ($I_1=I_2$), Alice can quantify its entanglement via the entanglement of formation ($E_f$) \cite{Gie03,Rig04}: \begin{equation} E_f(\rho_{12}) = f\left(2 \sqrt{I_1+|I_3|-\sqrt{I_4 + 2 I_1 |I_3|}}\right), \label{ef} \end{equation} where $f(x)=c_+(x)\log_2(c_+(x)) - c_-(x)\log_2(c_-(x))$ and $c_{\pm}(x)=(x^{-1/2}\pm x^{1/2})^2/4$. For arbitrary two-mode Gaussian states ($I_1\neq I_2$) Alice can work with lower bounds for $E_f$ \cite{Rig04} or calculate its negativity or logarithmic negativity \cite{vidal}. This last two quantities are the best entanglement quantifiers for non-symmetric two-mode Gaussian states and are given as analytical functions \cite{adesso,adesso2} of the four invariants here obtained from local measurements: $I_1$, $I_2$, $|I_3|=\sqrt{I_1I_2} - \sqrt{I_V}$, and $I_4=2|I_3|\sqrt{I_1I_2}$, with $I_V=\det\mathbf{V}$ given by Eq.~(\ref{detV}). It is worth mentioning that $I_1$ ($I_2$) can easily be determined by the measurement of the purity (Wigner function at the origin of the phase space) of Alice's (Bob's) mode alone \cite{fiurasek2,ban}. This measurement is less demanding than the ones required to reconstruct ${\bf V}_1$ and $\mathbf{V}_2$ \cite{rigolin}. Besides furnishing all the entanglement properties of an arbitrary two-mode Gaussian state, the previous local protocol can also be employed to reconstruct the covariance matrix for some particular types of Gaussian states. To see this, let ${\bf\Gamma}_1$ be explicitly written as \begin{equation} {\bf \Gamma}_1=\left( \begin{array}{cc} \eta_1 + \frac{1}{2}& \mu_1 \\ \mu_1^* & \eta_1 + \frac{1}{2}\end{array} \right), \end{equation} where \begin{eqnarray} &\eta_1&=\langle a_1^\dagger a_1\rangle_{e}-\langle a_1^\dagger a_1\rangle_{o}, \label{eta1} \\ \mu_1=\langle a_1^2\rangle_{e}&-&\langle a_1^2\rangle_{o}, \ \mu_1^*=\langle (a_1^\dagger)^2\rangle_{e}-\langle (a_1^\dagger)^2\rangle_{o}, \label{mi1} \end{eqnarray} being $\langle \cdot \rangle_e$ and $\langle \cdot \rangle_o$ the mean values for Alice's even and odd subensembles, respectively. From this identity it is clear that ${\bf\Gamma}_1$ does not necessarily represent a physical state since $\eta_1$ can take negative values \cite{haruna1}. From Eq. (\ref{schur_rel1}) we obtain the following two relations, \begin{eqnarray} n_1-\eta_1&=&\frac{1}{\left(n_2+\frac{1}{2}\right)^2-\vert m_2\vert^2} \left\{\!\!\left(\vert m_c\vert^2\!\!+\vert m_s\vert^2\right)\!\!\left(\!n_2\!+\!\frac{1}{2}\!\right)\right.\nonumber\\ &&\left. -2\Re e(m_2m_sm_c^*)\right\}, \label{eq1}\\ m_1-\mu_1&=&\frac{1}{\left(n_2+\frac{1}{2}\right)^2-\vert m_2\vert^2} \left\{2m_sm_c\left(n_2+\frac{1}{2}\right)\right.\nonumber\\ &&\left.-m_2^*m_c^2-m_2m_s^2\right\}. \label{eq2} \end{eqnarray} Eqs.~(\ref{eq1}) and (\ref{eq2}) give the matrix elements of $\mathbf{\Gamma_{1}}$ as a function of the matrix elements of $\mathbf{V}$. If $m_c$ and $m_s$ are real (if either $m_c$ or $m_s$ is zero) Eqs.~(\ref{eq1}) and (\ref{eq2}) can be inverted to give $m_c$ and $m_s$ (either $m_s$ or $m_c$). Let us explicitly solve the previous equations for an important case, namely the ones in which $\textbf{C}\textbf{C}^\dagger=\vert m_i\vert^2\textbf{I}$, where $i=c$ or $s$ and \textbf{I} is the identity matrix. The states comprehending this class are the ones where \textbf{C} has only diagonal or non diagonal elements, i.e., $m_s=0$ and $m_c\neq0$ or $m_c=0$ and $m_s\neq0$, reducing the unknown quantities to two, namely the absolute value and the phase of $m_s$ or $m_c$. Remark that if $i=s$ the system is separable, since $\det\textbf{C}=\vert m_s\vert^2\geq0$, i.e., the correlation between the two modes is strictly classical \cite{simon}. Otherwise, if $i=c$ the state is not necessarily separable, possibly being entangled, for in this case $\det\textbf{C}=-\vert m_c\vert^2\leq0$. This last case is more interesting since it represents a class of states that might show non-local features \cite{simon}. From Eqs. (\ref{eq1}) and (\ref{eq2}) the diagonal (off-diagonal) elements of \textbf{C}, $m_i=\vert m_i\vert e^{i\phi_i}$, for $i=s$ ($i=c$), are \begin{eqnarray} \vert m_i\vert^2&=&\frac{(n_1-\eta_1)}{n_2+1/2} \left[\left(n_2+ 1/2 \right)^2 -\vert m_2\vert^2\right], \label{mc}\\ e^{2i\phi_i}&=&\left(\frac{\mu_1-m_1}{n_1-\eta_1}\right) \frac{n_2+ 1/2}{m_{2i}}, \end{eqnarray} where $m_{2c}=m_2^*$ and $m_{2s}=m_2$. Note that whenever $m_2=0$, $\phi_i$ becomes undetermined. This problem can be solved by locally (unitary) transforming the two-mode squeezed state to a matrix $V'_2$ with $m'_2\neq 0$, where $\phi'_i$ can be determined. Then, transforming back, we get $\phi_i$. Fortunately, there are various experimentally available bipartite Gaussian states in which all the parameters are real, $m_s(m_c)=m_1=m_2=0$, and $m_c(m_s)\neq 0$. For these states, Eq. (\ref{mc}) is sufficient to determine $\mathbf{C}$. A natural and important example belonging to this class is the two-mode thermal squeezed state \cite{daffer}, which is generated in a nonlinear crystal with internal noise. Its covariance matrix is \begin{equation} \textbf{V}=\left( \begin{array}{cccc} n+ \frac{1}{2} & 0 & 0 & m_c \\ 0 & n+ \frac{1}{2} & m_c & 0 \\ 0 & m_c & n+ \frac{1}{2} & 0 \\ m_c & 0 & 0 & n+ \frac{1}{2} \\ \end{array} \right), \label{cov_esp} \end{equation} where $n$ and $m_c$ are time dependent functions having as parameters the relaxation constant of the bath as well as the nonlinearity of the crystal \cite{daffer}. In this case the protocol involves only simple local measurements, i.e., those to get $n$, $\langle a_1^\dagger a_1\rangle_{e}$ and $\langle a_1^\dagger a_1\rangle_{o}$ (or equivalently $\eta_1$) by Alice, and the parity measurements by Bob. The classical communication corresponds to Bob informing Alice the instances he performs the parity measurement in his mode and the respective outcomes. Hence, Eq.~(\ref{mc}) reduces to \begin{equation} m_c^2=(n-\eta_1)\left(n+ 1/2\right). \label{mc_esp} \end{equation} Experimentally, $n$ and $\eta_1$ (Eq.~(\ref{eta1})) are readily obtained by photodetection, while the parity measurement is related to the determination of Bob's mode Wigner function at the origin of the phase-space \cite{davidovich}, or alternatively to his mode's purity, both of which can be measured by photocounting experiments \cite{fiurasek2,ban}. We can also study the P-representability \cite{footnote2} for the state (\ref{cov_esp}), which in this case is equivalent to the Simon separability test \cite{simon,marcos1}. A two-mode Gaussian state is P-representable iff $\textbf{V}-\frac{1}{2}\textbf{I}\geq0$, where \textbf{I} is the unity matrix of dimension $4$. Explicitly, this separability condition in terms of the elements of (\ref{cov_esp}) is equivalent to $n\geq |m_c|$. From this inequality and Eq.~(\ref{mc_esp}) we see that for a given $n$ there exists a bound for $\eta_1$ below which the states are entangled (upper solid curve in Fig. \ref{fig1}): \begin{equation} -\frac{n/2}{n+1/2}\le\eta_1\le\frac{n/2}{n+1/2}. \label{ineq} \end{equation} The left bound in Eq. (\ref{ineq}) (lower solid curve in Fig. 1) is a consequence of the uncertainty principle, delimiting the set of all physical symmetric Gaussian states (SGS). This bound is marked by all the pure states and the upper curve bounds (from below) the subset of all separable (P-representable) states \cite{marcos1}. Thus, for the SGS class, photon number measurements, before and after Bob's parity measurements, are all Alice needs to discover whether or not her mode is entangled with Bob's. The exquisite symmetry of those two antagonistic bounds is quite surprising, and possibly valid only for the SGS class. There is another interesting feature for the SGS set that should be emphasized. Note that $\eta_1=0$ contains all the states where Bob has equal chances of getting even or odd outcomes for his parity measurements, delimiting two subsets (even and odd). The even subset contains all the states where Bob has greater probabilities of getting even outcomes while the odd subset contains all the states where he has greater probabilities of getting odd outcomes. The entanglement for states belonging to the SGS can be quantified through $E_f$ (Eq. (\ref{ef})) as depicted by the color scale in Fig.~\ref{fig1}. It is remarkable that the most entangled states (including the pure ones) are concentrated in the $\eta_1<0$ odd subset. \begin{figure} \caption{(Color online) Above the upper solid curve lie the separable states. Below it, entanglement is quantified via $E_f$ (Eq.~(\ref{ef})) up to the lower curve, where the pure entangled states are located. Below this curve there exist no physical states.} \label{fig1} \end{figure} In conclusion, we have presented the minimal set of local operations and classical communication that allows one to quantify the entanglement of an arbitrary two-mode Gaussian state. One important step towards the derivation of this protocol was the mathematical identity relating the two-mode covariance matrix determinant to the product of two local quantities, namely the determinants of the one-mode correlation matrix and its Schur complement. In addition, we have also shown that the Schur complement of one of the modes' covariance matrix is obtained via a set of parity measurements on the other one. We have also explicitly discussed how the protocol works for a particular class of Gaussian states belonging to the SGS set. Within this class, for states written in its standard form, we have shown that only photon number measurements (made before and after a parity measurement on the other mode) are needed to completely characterize the state's entanglement. \begin{acknowledgments} This work is supported by FAPESP and CNPq. \end{acknowledgments} \appendix \section{Erratum} Eq. (4), on page 2, of [Phys. Rev. Lett. 98, 150501 (2007)] is not so general as we have previously thought. However, our scheme does not rely on it, as we show in what follows. The experimental proposal we presented in our Letter allows one to locally obtain the matrices $\mathbf{V_1}, \mathbf{V_2}$, and $\mathbf{\Gamma_1}$, without assuming any particular form for the covariance matrix $\mathbf{V}$. We now show that with these three matrices we can determine the four invariants that completely characterize the entanglement content of a two-mode Gaussian state. The first two invariants are \begin{equation} I_1=\text{det}(\mathbf{V_1}), \hspace{1cm} I_2=\text{det}(\mathbf{V_2}). \end{equation} The third one is calculated remembering that $ \mathbf{\Gamma_1} = \mathbf{V_1} - \mathbf{C}\mathbf{V_2^{-1}}\mathbf{C^{\dagger}}. $ A simple algebra on the previous expression gives, $ \mbox{det}\left( \mathbf{V_1} - \mathbf{\Gamma_1}\right)=\mbox{det}(\mathbf{C})\mbox{det}(\mathbf{V_2^{-1}}) \mbox{det}(\mathbf{C^{\dagger}}). $ But $\mbox{det}(\mathbf{C}) = \mbox{det}(\mathbf{C^{\dagger}}) = I_3$ and $\mbox{det}(\mathbf{V_2^{-1}}) = 1/\mbox{det}(\mathbf{V_2}) = 1/I_2$. Hence \begin{eqnarray} |I_3|&=&\sqrt{I_2\,\mbox{det}\left( \mathbf{V_1} - \mathbf{\Gamma_1}\right)}. \end{eqnarray} Furthermore, $\mathbf{\Gamma_1}$ satisfies another mathematical indentity, $ I_V=\text{det}\mathbf{V} = \text{det}{\mathbf{V_2}}\text{det}{\mathbf{\Gamma_1}}. $ Therefore, since we have $\mathbf{\Gamma_1}$ and $\mathbf{V_2}$, we can also obtain $I_V$. But $I_V$ is related to the other four invariants by the following expression, $ I_V = I_1I_2 - I_4 + I_3^2. $ Thus, the fourth invariant is simply \begin{equation} I_4 = I_1I_2 + I_3^2 - I_2\,\text{det}{\mathbf{\Gamma_1}}. \end{equation} Using $I_1$, $I_2$, $|I_3|$, and $I_4$, as obtained above with the knowledge of $\mathbf{V_1}$, $\mathbf{V_2}$, and $\mathbf{\Gamma_1}$, we can apply the Simon separability test (Eq. (3) of our Letter). If a two-mode Gaussian state is entangled we know for sure that $I_3<0$ and we can, therefore, fully quantify its entanglement either via the entanglement of formation or the negativity/logarithmic negativity, as discussed in our Letter. Finally, we must emphasize that the main result of our Letter remains unchanged: it is possible to completely characterize via local operations and classical communication (LOCC) the entanglement content of an arbitrary two-mode Gaussian state. Furthermore, all the results presented in the Letter remain valid. We want to thank Yang Yang, Fu-Li Li and Hong-Rong Li for also calling our attention on the problems related to Eq. (4) of our Letter while this erratum was being formulated. \end{document}

\begin{document} \begin{abstract} Let $X$ be a topological Hausdorff space together with a continuous action of a finite group $G$. Let $R$ be the ring of integers of a number field~$F$. Let $\calE$ be a $G$-sheaf of flat $R$-modules over $X$ and let $\Phi$ be a $G$-stable paracompactifying family of supports on $X$. We show that under some natural cohomological finiteness conditions the Lefschetz number of the action of $g \in G$ on the cohomology $ \com{H}_\Phi(X,\calE) \otimes_{R} F $ equals the Lefschetz number of the $g$-action on $ \com{H}_{\Phi|X^G}(X^g, \calE_{|X^g}) \otimes_{R} F $, where $X^g$ is the set of fixed points of $g$ in $X$. More generally, the class $\sum_j (-1)^j [H^j_\Phi (X,\calE) \otimes_R F]$ in the character group equals a sum $\sum_{[H]} \sum_{\lambda \in \widehat{H}_F} m_\lambda [\ind^G_H (V_\lambda)] $ of representations induced from irreducible $F$-rational representations $\: V_\lambda \:$ of $\: H \:,$ where $[H]$ runs in the set of $G$-conjugacy classes of subgroups of $G$. The integral coefficients $m_\lambda$ are explicitly determined. \end{abstract} \maketitle \section{Introduction and main results} The most elementary classical version of the Lefschetz fixed point formula says that the Lefschetz number $\calL(g)$ of a simplicial automorphism $g$ of finite order on a finite simplicial complex $X$ equals the Euler-Poincar\'e characteristic of the fixed point set $X^g \subset X $ of the $g$-action. Here $\calL(g)$ is computed on $ H^\ast (X,\bbQ)$. Brown \cite{Brown1982} (based on Zarelua \cite{Zarelua1969}) and independently Verdier \cite{Verdier1973} have extended this formula to more general spaces under the assumption of cohomological finiteness conditions. Verdier uses cohomology with compact supports. The objective of this paper is to generalize this Lefschetz fixed point formula to Hausdorff spaces with a continuous action of a finite group $G$ and to cohomology of $G$-sheaves with a paracompactifying family of supports. For applications of the Lefschetz fixed point formula to cohomology of arithmetic groups see e.g.~\cite{Rohlfs1990}. \subsection{Notation} Throughout $F$ denotes an algebraic number field and $R$ denotes its ring of integers. Let $G$ be a finite group, then $\Gzero(F[G])$ denotes the Grothendieck group of finitely generated $F[G]$-modules. For every subgroup $H$ of $G$ there is the induction homomorphism $\ind_H^G: \Gzero(F[H]) \to \Gzero(F[G])$ which maps $[M]$ to $[F[G]\otimes_{F[H]}M]$. For $ g \in G $ the trace of the $g$-action induces a morphism $\tr(g): \Gzero(F[G]) \longrightarrow R$. Let $Y$ be a Hausdorff space and let $\Phi$ be a family of supports on $Y$. Let $k$ be a ring and $\calE$ be a sheaf of left $k$-modules on $Y$. If the cohomology $\com{H}_\Phi(Y,\calE)$ is finitely generated as $k$-module, then we say that the triple $(Y,\Phi,\calE)$ is of \emph{finite type} with respect to $k$. Given a sheaf $\calE$ of $R$-modules on $Y$, we write $\chi_\Phi(Y,\calE; F)$ for the Euler-Poincar\'e characteristic of the graded $F$-vectorspace $\com{H}_\Phi(Y,\calE)\otimes_R F$ whenever it is \mbox{finite} dimensional. Similarly, if a finite group $G$ acts continuously on $Y$ and $\calE$ is $G$-equivariant, then we denote the Euler-Poincar\'e characteristic of the graded $F[G]$-module $\com{H}_\Phi(Y,\calE)\otimes_R F$ in the Grothendieck group $\Gzero(F[G])$ by $\chi_\Phi(Y,\calE; F[G])$. The image of $\chi_\Phi (Y, \calE; F[G]) $ under the morphism $\tr(g)$ is the \emph{Lefschetz number} $ \calL_\Phi (g, \calE; F) = \sum^\infty_{j = 0} (-1)^j \tr (g| H^j _\Phi (Y ,\calE) \otimes_R F) \:.$ \subsection{Statement of results} Denote by $X$ a topological Hausdorff space together with a continuous action of a finite group $G$. We fix a paracompactifying $G$-stable family of supports $\Phi$ on $X$. For a subgroup $H \leq G$ we denote the normalizer of $H$ in $G$ by $N_G(H)$ or $N(H)$. We write $X^H$ for the set of points in $X$ which are fixed by $H$ and we write $X_H$ for the set of points in $X$ whose stabilizer is exactly the group $H$. Note that $X^H$ is closed in $X$ and $X_H$ is open in $X^H$. Let $\cla(G)$ be the set of conjugacy classes of subgroups of $G$. The paracompactifying family $\Phi$ induces paracompactifying families on the locally closed subspaces $ X_H, X^H, X_C := \bigcup_{H \in C} X_H $ for $C \in \cla(G)$ and on the quotient space $ X_H/ N(H) $. For simplicity these families will be denoted also by $\Phi$. By $ \widehat{H}_F$ we denote the set of equivalence classes of irreducible representations of $H$ on finite dimensional $F$-vectorspaces. If $\lambda \in \widehat{H}_F$, we write $V_\lambda$ for a representative of $\lambda$. We define $\deg V_\lambda := \dim_F \bigl(\Hom_{F[H]} (V_\lambda, V_\lambda )\bigr)$. Let $\calE$ be a $G$-sheaf of $R$-modules on $X$. We say that $\calE$ satisfies the finiteness condition \cF if for every subgroup $H$ of $G$ the following hold: \begin{enumerate} \item The triple $(X_H,\Phi, \calE_{|X_H}) $ is of finite type w.r.t.~$R$, and \item for any $\lambda \in \widehat{N(H)}_F$ there is an $N(H)$-invariant lattice $L_\lambda$ in $V_\lambda$ such that the triple $(X_H,\Phi,\Hom_{R[H]}\bigl(L_\lambda, \calE_{|X_H} \bigr))$ is of finite type w.r.t.~$R$. \end{enumerate} We comment on this condition in section \ref{sec:Comments}. \begin{theorem} Let $X$, $G$, $\Phi$ be as above and assume that the cohomological $\Phi$-dimension of $X$ is finite. Let $ \calE$ be a $G$-sheaf of flat $R$-modules such that condition \cF holds. Then \begin{align*} \chi_\Phi\bigl(X,\calE; F[G]\bigr) &= \sum_{[H] \in \cla(G)} \frac{|H|}{|N(H)|} \ind_H^G\Bigl(\chi_{\Phi}\bigl(X_H, \calE_{|X_H}; F[H]\bigr)\Bigr)\\ & = \sum_{[H] \in \cla(G)} \sum_{\lambda \in \widehat{H}_F}\frac{|H|\cdot e(\lambda)}{|N(H)| \cdot \deg(V_\lambda)} \ind_H^G\bigl([V_\lambda\bigr]). \end{align*} where $e(\lambda)$ denotes the Euler characteristic $\chi_{\Phi}\bigl(X_H, \Hom_{R[H]}(M_\lambda, \calE_{|X_H}); F\bigr)$ for any $H$-stable $R$-lattice $M_\lambda \subset V_\lambda$. \end{theorem} A proof of the theorem will be given in the next section. \begin{corollary}\label{cor:letg} Let $G$ be the finite cyclic group generated by an element $g$. Under the assumptions of the theorem we obtain an equality of Lefschetz numbers \begin{equation*} \calL_\Phi\bigl(g,\calE; F\bigr) = \calL_{\Phi}\bigl(g, \calE_{|X^G}; F \bigr). \end{equation*} \end{corollary} \begin{proof} Use that for $ G = \langle g \rangle $ we have $ \tr (g | \ind^G_H V) = 0 \:$ for all finite dimensional $ F[H]$-modules $V$ if $ H \neq G $ and that $ X_G = X^G $. \end{proof} \section{Proof of the Theorem} This section is devoted to the proof of the theorem. We begin with the following general Lemma. \begin{lemma}\label{lem:EulerCharModp} Let $p$ be a prime number. Let $\calE$ be a sheaf of abelian groups on $X$ and assume that the stalks of $\calE$ have no $p$-torsion. If the triple $(X,\Phi,\calE)$ is of finite type (w.r.t.~$\bbZ$) then the triple $(X,\Phi,\calE\otimes_\bbZ \bbF_p)$ is of finite type w.r.t.~$\bbF_p$. Moreover we have \begin{equation*} \chi_\Phi(X,\calE; \bbQ) = \chi_\Phi(X,\calE\otimes_\bbZ \bbF_p; \bbF_p). \end{equation*} Here $\bbF_p$ denotes the finite field with $p$ elements. \end{lemma} \begin{proof} Since $\calE$ is torsion-free, there is a short exact sequence of sheaves on $X$: \begin{equation*} 0 \longrightarrow \calE \stackrel{p}{\longrightarrow} \calE \longrightarrow \calE\otimes_\bbZ \bbF_p \longrightarrow 0. \end{equation*} Consider the associated long exact sequence. We deduce that $(X,\Phi,\calE\otimes_\bbZ \bbF_p)$ is of finite type. Further, write $H^i_\Phi(X,\calE) \cong \bbZ^{b_i} \oplus P^i \oplus T^i$, where $P^i$ is the subgroup of elements whose order is a power of $p$ and $T^i$ is the subgroup of elements of finite order prime to $p$. Let $P^i_p$ denote the elements of order exactly $p$ and let $r_i = \dim_{\bbF_p} P^i_p$. From the long exact sequence we obtain short exact sequences \begin{equation*} 0 \longrightarrow H^i_\Phi(X,\calE)\otimes_\bbZ\bbF_p \longrightarrow H^i_\Phi(X,\calE\otimes\bbF_p)\longrightarrow P^{i+1}_p \longrightarrow 0 \end{equation*} for every degree $i$. Thus $\dim_{\bbF_p} H^i_\Phi(X,\calE\otimes\bbF_p) = b_i + r_i + r_{i+1}$ and the second assertion follows via alternating summation. \end{proof} Let $\pi: X \to X/G$ be the canonical projection. Note that for a sheaf of abelian groups $\calE$ on $X$ there is a canonical isomorphism \begin{equation*} \com{H}_\Phi(X,\calE) \isomorph \com{H}_\Phi(X/G,\pi_*(\calE)). \end{equation*} In general, if $\calF$ is a sheaf of $R[G]$-modules on a space $Y$, then we write $\calF^G$ for the subsheaf of $G$-stable sections, i.e.~$\calF^G(U) = \calF(U)^G$. Let $\calE$ be a $G$-sheaf of $R$-modules on $X$. We write $\pi_*^G(\calE)$ for $\pi_*(\calE)^G$. Note that the triple $(X,\Phi, \calE)$ is of finite type w.r.t.~$R$ if and only if it is of finite type w.r.t.~$\bbZ$. In this case we simply say that $(X,\Phi, \calE)$ is of finite type. \begin{lemma}\label{lem:EulerCharCovering} Suppose that $G$ is abelian and acts freely on $X$. Let $\calE$ be a flat $G$-sheaf of $R$-modules on $X$ such that $(X,\Phi, \calE)$ is of finite type. In this case $(X/G, \Phi, \pi_*^G(\calE))$ is of finite type and \begin{equation*} \chi_\Phi(X,\calE; F) = |G| \chi_\Phi(X/G, \pi_*^G(\calE); F). \end{equation*} \end{lemma} \begin{proof} First note that $X/G$ has finite $\Phi$-dimension, since cohomological dimension is a local property (cf.~ II.~4.14.1 in \cite{Godement1958}) and $\pi$ is a covering map. Further, the triple $(X/G, \Phi , \pi_*^G(\calE))$ is of finite type due to the Grothendieck spectral sequence \begin{equation*} H^p(G, H^q_\Phi(X,\calE)) \implies H^{p+q}_\Phi(X/G,\pi_*^G(\calE)) \end{equation*} which can be obtained for paracompactifying supports just as in \cite[Thm.~5.2.1]{Grothendieck1957}. Now we prove the assertion about the Euler characteristic. It is easy to check that $[F:\bbQ] \chi_\Phi(X, \calE; F) = \chi_\Phi(X, \calE; \bbQ)$, hence we can assume $R = \bbZ$. By induction on the group structure we can assume that $G$ is finite cyclic of prime order $p$. The assertion follows from Lemma \ref{lem:EulerCharModp} and a Theorem of E.~E.~Floyd (based on a result of P. A. Smith), see \cite[Thm.~19.7]{Bredon1997} or \cite[Thm.~4.2]{Floyd1952}. Here we use that $\pi_*^G(\calE)\otimes_\bbZ \bbF_p = \pi_*^G(\calE\otimes_\bbZ \bbF_p)$. Note that we assumed the $\Phi$-dimension of $X$ to be finite, which implies in particular that $\dim_{\Phi,\bbF_p} X$ is finite in the notation of \cite{Bredon1997}. Further the pull-back sheaf $\pi^* (\pi_*^G(\calE))$ is isomorphic to $\calE$ (see~\cite[p.~199]{Grothendieck1957}). \end{proof} \begin{lemma} Let $G$ be a finite group which acts freely on $X$ and let $\calE$ be flat a $G$-sheaf of $R$-modules on $X$. We assume that $(X,\Phi, \calE)$ is of finite type. For any $g \in G$ with $g \neq 1$ the Lefschetz number vanishes. \end{lemma} \begin{proof} By taking a finite extension we can assume without loss of generality that $R$ contains all $|G|$-th roots of unity. Further we can assume that $G$ is a finite cyclic group. Let $\psi: G \to R^\times$ be a character of $G$. We can twist the $G$-sheaf $\calE$ with the character $\psi^{-1}$ to obtain a new $G$-sheaf $\calE\otimes \psi^{-1}$. This sheaf is isomorphic to $\calE$ as a sheaf of $R$-modules, but not as $G$-sheaf. Further we find that $\pi_*^G(\calE\otimes\psi^{-1})$ is the $\psi$-eigensheaf $\pi_*(\calE)_\psi$ in the sheaf $\pi_*(\calE)$ of $R[G]$-modules, this means $\pi_*(\calE)_\psi$ is the subsheaf of sections of $\pi_*(\calE)$ which transform with $\psi$ under the action of~$G$. From Lemma \ref{lem:EulerCharCovering} we deduce that all the eigensheaves $\pi_*(\calE)_\psi$ have equal Euler characteristic. However, since \begin{equation*} \calL_\Phi(g,\calE; F) = \sum_{\psi \in \widehat{G}} \psi(g) \chi_\Phi(X/G, \pi_*(\calE)_\psi; F) \end{equation*} the claim follows. \end{proof} We shall frequently use the following Lemma. \begin{lemma}\label{lem:FixInside} Let $\calE$ be a sheaf of $R[G]$-modules on a space $X$ and let $\Phi$ be a system of supports on $X$. The inclusion $\calE^G \to \calE$ induces an isomorphism of vectorspaces \begin{equation*} \com{H}_\Phi(X,\calE^G) \otimes_R F \isomorph \com{H}_\Phi(X,\calE)^G \otimes_R F. \end{equation*} \end{lemma} \begin{proof} Consider the functor $B: \calE \mapsto \calE^G$ from the category $\Sh_X(R[G])$ of sheaves of $R[G]$-modules to the category $\Sh_X(R)$ of sheaves of $R$-modules. This functor is left exact and we consider its right derived functor \begin{equation*} \RR{B}: \Der^+(\Sh_X(R[G])) \to \Der^+(\Sh_X(R)). \end{equation*} Note that $B$ takes injective sheaves of $R[G]$-modules to flabby sheaves (see Corollary to Prop.~5.1.3 in \cite{Grothendieck1957}). As in Thm.~5.2.1 in \cite{Grothendieck1957} there is a convergent spectral sequence \begin{equation*} H^p_\Phi(X,\RR{B}^q(\calE))\otimes_R F \implies H^{p+q}_\Phi(X,\calE)^G\otimes_R F, \end{equation*} where we use that $F$ is a flat $R$-module. In fact, the stalk at $x \in X$ of $\RR{B}^q(\calE)$ is the group cohomology $H^q(G,\calE_x)$ which is purely $|G|$-torsion for all $q \geq 1$. Hence the spectral sequence collapses and the claim follows. \end{proof} We obtain a refined version of Verdier's Lemma (cf.~\cite{Verdier1973}). \begin{lemma}\label{lem:VerdierLemma} Let $G$ be a finite group which acts freely on $X$ and let $\calE$ be a flat $G$-sheaf of $R$-modules on $X$. We assume that $(X,\Phi, \calE)$ is of finite type. In this case we have \begin{equation*} \chi_\Phi(X, \calE; F[G]) = \chi_\Phi(X/G,\pi^G_*(\calE); F) \cdot F[G]. \end{equation*} In particular, Lemma \ref{lem:EulerCharCovering} holds without the assumption that $G$ is abelian. \end{lemma} \begin{proof} With the same argument as in Lemma \ref{lem:EulerCharCovering} we see that $(X/G, \Phi, \pi^G_*(\calE))$ is of finite type. It suffices to compute the Lefschetz numbers of all elements of $G$ and compare them with the right hand side. The vanishing of all Lefschetz numbers for $g \neq 1$ shows that $ \chi_\Phi(X, \calE; F[G])$ is a multiple of the regular representation. The coefficient is the Euler characteristic of the graded $F$-vectorspace $\com{H}_\Phi(X,\calE)^G\otimes_{R} F$. Since $\com{H}_\Phi(X,\calE) \cong \com{H}_\Phi(X/G,\pi_*(\calE))$, we can use Lemma \ref{lem:FixInside} to deduce the claim. \end{proof} \begin{remark} Verdier uses the projection formula, the finite tor-amplitude criterion and a famous theorem of Swan to obtain this lemma for cohomology with compact supports and constant coefficients. It is possible to extend his approach to the case of families of supports using a suitable replacement for the projection formula, see~\cite{Kionke2012}. \end{remark} Finally we prove the main theorem. Recall that $X$ is a Hausdorff space with an action of a finite group $G$ and $\Phi$ is a $G$-invariant paracompactifying system of supports. \begin{proof}[Proof of the Theorem] Note that for every subgroup $H$ of $G$ the space $Y = X^H$ (resp.~$Y =X_H$) has finite $\Phi$-dimension since $Y$ is (locally) closed and $\Phi$ is paracompactifying (cf.~II.~Rem.~4.14.1 in \cite{Godement1958}). By condition \cFone the triple $(X_H,\Phi, \calE_{|X_H})$ is of finite type for every $H \leq G$. For $i=1, \dots, |G|$ we define the closed set \begin{equation*} X^i = \bigcup_{\substack{H \leq G \\ |H|\geq i}} X^H. \end{equation*} Then $X^1 = X$ and $X^{|G|} = X^G$ is the set of fixed points. An element $x\in X$ is in $X^i\setminus X^{i+1}$ exactly if it has an isotropy group with $i$ elements, hence \begin{equation*} X^i\setminus X^{i+1} = \bigcup_{\substack{H \leq G \\ |H|=i}} X_H =: \bigcup_{\substack{C \in \cla(G)_i}} X_C \end{equation*} where $\cla(G)_i$ is the set of conjugacy classes of subgroups with $i$ elements. Note that these unions are topologically disjoint. Using the long exact sequences of the pairs $(X^i,X^{i+1})$ with supports in $\Phi$ (cf.~II.~4.10.1 in \cite{Godement1958}) we obtain \begin{equation*} \chi_\Phi\bigl(X,\calE; F[G]\bigr) = \sum_{C \in \cla(G)} \chi_{\Phi}\bigl(X_C, \calE_{|X_C}; F[G]\bigr). \end{equation*} Since $X_C$ is the disjoint union $\bigcup_{H \in C} X_H$ and $G$ acts transitively on the components we obtain \begin{equation*} \chi_{\Phi}\bigl(X_C, \calE_{|X_C}; F[G]\bigr) = \ind^G_{N_G(H)}\chi_{\Phi}\bigl(X_H, \calE_{|X_H}; F[N_G(H)]\bigr) \end{equation*} for any representative $H \in C$. We are now in the specific situation where $H$ acts trivially on $X_H$ and $N_G(H)/H$ acts freely on $X_H$. For simplicity we write $N$ for the normalizer $N_G(H)$. We prove the following identity \begin{equation*} \chi_{\Phi}\bigl(X_H, \calE_{|X_H}; F[N]\bigr) = \frac{|H|}{|N|} \ind_H^N\Bigl( \chi_\Phi(X_H, \calE_{|X_H}; F[H]) \Bigr) . \end{equation*} By Frobenius reciprocity a finite dimensional $F[N]$-module $V$ is induced from the $F[H]$-module $W$ if and only if \begin{equation*} \dim_F \Hom_{F[N]}(V_\lambda, V) = \dim_F \Hom_{F[H]}\bigl((V_\lambda)_{|H}, W\bigr) \end{equation*} for all $\lambda \in \widehat{N}_F$. We use this principle in the Euler characteristic. For $\lambda \in \widehat{N}_F$ we choose some lattice $L_\lambda \subset V_\lambda$ as in condition \cFtwo. We obtain the $N$-sheaf $\Hom_R(L_\lambda, \calE_{|X_H})$ and the $N/H$-sheaf $\Hom_{R[H]}(L_\lambda, \calE_{|X_H})$. For simplicity denote the canonical map $X_H \to X_H/N$ by $\pi$ as well. Now we obtain \begin{align*} \chi\Bigl( \Hom_{F[N]}\bigl(V_\lambda, \com{H}_\Phi(X_H, &\calE_{|X_H})\otimes_R F\bigr) \Bigr) \\ &= \chi\Bigl( \Hom_{F[N]}\bigl(V_\lambda, \com{H}_\Phi(X_H/N,\pi_*(\calE_{|X_H}))\otimes_R F\bigr) \Bigr) \\ &= \chi\Bigl( \com{H}_\Phi(X_H/N,\Hom_R\bigl(L_\lambda, \pi_*(\calE_{|X_H})\bigr))^N\otimes_R F \Bigr) \\ (\text{by Lemma \ref{lem:FixInside}})\quad &= \chi\Bigl( \com{H}_\Phi(X_H/N,\pi^N_*\Hom_R\bigl(L_\lambda, \calE_{|X_H} \bigr))\otimes_R F) \Bigr)\\ &= \chi\Bigl( \com{H}_\Phi(X_H/N,\pi^{N/H}_*\Hom_{R[H]}\bigl(L_\lambda, \calE_{|X_H} \bigr))\otimes_R F) \Bigr)\\ (\text{by Lemma \ref{lem:VerdierLemma} and \cFtwo}) \quad &= \frac{|H|}{|N|}\chi\Bigl( \com{H}_\Phi(X_H,\Hom_{R[H]}\bigl(L_\lambda, \calE_{|X_H} \bigr))\otimes_R F) \Bigr)\\ (\text{by Lemma \ref{lem:FixInside}}) \quad &= \frac{|H|}{|N|}\chi\Bigl( \Hom_{F[H]}\bigl((V_\lambda)_{|H},\com{H}_\Phi(X_H, \calE_{|X_H} )\otimes_R F\bigr) \Bigr). \end{align*} This proves the first equality in the theorem. Next we decompose the $F[H]$-module $\com{H}_\Phi (X_H, \calE_{|X_H} ) \otimes_R F$ into isotypical components. Let $V_\lambda$, with $\lambda \in \widehat{H}_F $, be an irreducible module and choose some $R[H]$ stable lattice $ M_\lambda \subset V_\lambda $. Put $ D_\lambda := \Hom_{F[H]} (V_\lambda, V_\lambda)$. Then $ D_\lambda$ is a division algebra and the multiplicity $m^j_\lambda$ of $V_\lambda$ in $H^j_\Phi(X_H, \calE_{|X_H}) \otimes_R F $ equals \begin{align*} m^j_\lambda & = \dim_{D_\lambda} \Hom_{F[H]} \bigl(V_\lambda, H^j_{\Phi} (X_H, \calE_{|X_H} ) \otimes_R F \bigr) \\ & = ( \deg V_\lambda)^{-1} \dim_F \Hom_{F[H]} ( V_\lambda , H^j_{\Phi} (X_H, \calE_{|X_H} ) \otimes_R F ) \\ & = (\deg V_\lambda)^{-1} H^j_\Phi (X_H, \Hom_{R[H]} (M_\lambda , \calE_{|X_H} )) \otimes_R F . \end{align*} Recall that $\deg V_\lambda = \dim_F(D_\lambda)$. \end{proof} \section{Further comments}\label{sec:Comments} We add some remarks in order to clarify some assumptions for the main result. \subsection{The finiteness condition} If $\calE$ is a constant sheaf on $X$, then condition \cFone implies \cFtwo. In general this need not be the case. This can already be seen in examples where the group acts trivially on the space. Let $X$ be the unit disc in $\bbC$ and $G = \bbZ / 2 \bbZ$. Then there exists a sheaf $\calE$ of $\bbZ[G]$-modules on $X$ such that $\com{H}(X, \calE)$ is finitely generated but $ H^2(X, \calE^G)$ is not finitely generated as $\bbZ$-module, i.e.~\cFone holds but \cFtwo fails. It follows from a \v{C}ech cohomology argument that if $X$ and all $X^H$ are compact and homology locally connected (HLC), then condition \cF holds for every locally constant $G$-sheaf $\calE$ with finitely generated stalks (for the family of all supports). \subsection{Sheaves of vectorspaces} If we replace $R$ by $F$ and start with a $G$-sheaf $\calE_F$ of $F$-vectorspaces over $X$, then both sides of the formula in the theorem make sense under suitable cohomological finiteness assumptions. However, there are examples which show that the theorem does not hold in this situation. For instance, let $X = S^1$ be the unit circle with the nontrivial action of $ G = \bbZ / 2 \bbZ $ by rotations. There is a $G$-sheaf of $F$-vectorspaces $\calE_F$ such that $\com{H}(S^1,\calE_F)$ is finite dimensional and $\chi(S^1, \calE_F )=1$. Then clearly $| G | \chi ( X/G , \pi^G_* \calE_F ) \neq \chi (X, \calE_F )$ since the left hand side is an even number. In fact, most complications in the proof arise from the fact that we have to work with sheaves over the ring $R$. \subsection{Cohomology of arithmetic groups} We indicate that the assumptions of the theorem hold for the cohomology of arithmetic groups. Let $A$ be a reductive algebraic group defined over $\bbQ$ ($A$ is not necessarily connected). By $\Gamma \subset A (\bbQ)$ we denote an arithmetic group. Assume that $G \subset \Aut_\bbQ(A)$ is a finite subgroup which acts on $\Gamma$ and on a finite dimensional rational representation $ \rho $ of $A$ on a $\bbQ$-vectorspace $ E $ such that $g ( \rho (\eta) e) = \rho ( g(\eta)) ge$ for all $e \in E$, $\eta \in A(\bbQ)$, $g \in G$. Let $ Y $ be the symmetric space attached to $ A (\bbR)$. Then $ G $ and $ \Gamma $ act on $Y$. Put $ X := \Gamma \backslash Y $ as topological quotient and denote by $ f : Y \longrightarrow X $ the natural projection. We choose a $G$ and $\Gamma$-stable lattice $ L $ in $ E $. Put $ \calE = f ^\Gamma_\ast L_Y $, where $ L_Y $ is the constant sheaf with stalks $ L$ on $Y$. Then $ G $ acts on $ \com{H} ( X, \calE) \otimes_{\bbZ} \bbQ = \com{H}(\Gamma, L) \otimes_{\bbZ} \bbQ $. To see that the assumptions of the theorem hold one uses the Borel-Serre compactification for $ X $ and $X^H$, see \cite{BorelSerre1973}. \subsection{} For a paracompactifying family of supports $\Phi$ on $X$ there is in general no equality of the form \begin{equation*} \com{H}_\Phi(X,\calE)\otimes_R F = \com{H}_\Phi(X,\calE \otimes_R F) \end{equation*} for a sheaf of $R$-modules $\calE$ on $X$. For the family of compact supports this equality holds. For cohomology of arithmetic groups, we have for all $R[\Gamma]$-modules $M$ and all $j$ that $ H^j(\Gamma, M) \otimes_R F = H^j (\Gamma, M \otimes_R F) \:.$ This follows from the existence of a resolution $P_\bullet \longrightarrow \bbZ \longrightarrow 0 $ of $\bbZ $ by finitely generated free $\bbZ [\Gamma]$-modules and since $F$ is flat as $R$-module, see Thm.~11.4.4 in \cite{BorelSerre1973} and p.~193 in \cite{BrownBook1982}. \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\href}[2]{#2} \end{document}

\begin{document} \author[M. V. de Hoop]{Maarten V. de Hoop $^{\diamond}$} \author[T. Saksala]{Teemu Saksala $^{\diamond,\: \ast}$} \let\thefootnote\relax\footnote{ $^\diamond$ Department of Computational and Applied mathematics, Rice University, USA \\ $^\ast$ \textbf{[email protected]}} \begin{abstract} We show that the travel time difference functions, measured on the boundary, determine a compact Riemannian manifold with smooth boundary up to Riemannian isometry, if boundary satisfies a certain visibility condition. This corresponds with the inverse microseismicity problem. The novelty of our paper is a new type of a proof and a weaker assumption for the boundary than it has been presented in the literature before. We also construct an explicit smooth atlas from the travel time difference functions. \end{abstract} \maketitle \section{Introduction} \label{Se:motivation} Let $(N,g)$ be a complete, connected smooth Riemannian manifold. We split the manifold into two parts that are a closed set $M$, with non-empty interior, and the closure of the exterior $F:=\overline{N \setminus M}$. We assume that the boundary ${\partial} M$ of $M$ is a smooth co-dimension one manifold. The set $F$ is the known observation domain and $M$ is the object of interest, for instance Earth. The Riemannian metric $g$ can be seen as a proxy of the material parameters of $M$. For any $p,q \in N$ we denote by $d_N(p,q)$ the length of a distance minimizing geodesic of $(N,g)$ that connects $p$ to $q$. We assume that the wave speed in $F$ is much slower than in $M$. Especially if ${\partial} M$ is strictly convex, we may assume that distance minimizing geodesics of $(N,g)$ connecting $p$ to $q$ stay inside $M$, if $p,q\in M$. This implies \begin{equation} \label{eq:exterior_dist} d_M(p,q)=d_{N}(p,q), \quad p,q \in M, \end{equation} where $d_M(p,q)$ is the distance from $p$ to $q$ in $M$, that is given as the infimum of lengths of curves from $p$ to $q$ that stay in $M$. For a while we assume that \eqref{eq:exterior_dist} holds and we denote $d_M=d_g$. Suppose that there exists a Dirac point source $(p,s)\in M \times {\mathbb R}$ of a Riemannian wave equation, with zero Cauchy data. It follows from \cite{duistermaat1972fourier} and \cite{greenleaf1993recovering} that the singularities emitted from $(p,s)$ propagate along the geodesics of $(N,g)$ (see for instance \cite{LaSa} for more details). For every $z \in {\partial} M$ we define the \textit{arrival time} $\mathcal{T}_{p,s}(z)$ to be the infimum of times when a spherical wave emitted form $(p,s)$ is observed at $z$. Hence $\mathcal T_{p,s}(z)=d_g(p,z)+s$, and the \textit{travel time difference function} satisfy an equation \begin{equation} D_p(z_1,z_2):=d_g(p,z_1)-d_g(p,z_2) =\mathcal T_{p,s}(z_1)-\mathcal T_{p,s}(z_2), \quad z_1,z_2 \in {\partial} M. \label{eq:Relation of wave data and DDD} \end{equation} The important property of this function is that it is given as the difference of the arrival times. The knowledge of the emission time $s$ or the origin remains unknown, but the function $D_p$ can be determined without knowledge on $s$. \color{black} This paper is devoted to the study of the inverse problem of travel time difference functions. This problem can be formulated as follows. Does the collection $$ \{D_p:p \in M^{int}\}, $$ determine the Riemannian manifold $(M,g)$ up to isometry? Now we give our problem setting. Let $(M,g)$ be a compact connected $n$--dimensional Riemannian manifold with smooth boundary ${\partial} M$. Since $M$ is compact for any points $p,q \in M$ there exists a distance minimizing $C^1$--smooth curve $c$ from $p$ to $q$, see \cite{alexander1981geodesics}. Moreover for any $t_0 \in [0,d_g(p,q)]$ such that point $\gamma(t_0) $ is an interior point of $M$ there exists $\epsilon>0$ such that $c:(t_0-\epsilon, t_0 +\epsilon)$ is a geodesic. We denote the collection of all interior points of $M$ by $M^{int}$. We use the notation $SM$ for the unit sphere bundle of $(M,g)$. Therefore each $(p,v) \in SM$ determines the unique maximal unit speed geodesic $\gamma_{p,v}$ of $(M,g)$. For any $p \in M$ we define the corresponding \textit{travel time difference function}. \begin{equation} \label{eq:DDF} D_p:{\partial} M \times {\partial} M \to {\mathbb R}, \quad D_p(z_1,z_2):=d_g(p,z_1)-d_g(p,z_2). \end{equation} Notice that the function $D_p$ is continuous. We assume that the following \textit{travel time difference data} \begin{equation} \label{eq:data} ({\partial} M, \: \{D_p: \: p \in M^{int}\}), \end{equation} is given. That is we assume, that the $(n-1)$--dimensional smooth manifold ${\partial} M$ without boundary and the collection of functions $\{D_p:{\partial} M \times {\partial} M \to {\mathbb R} \: |\: p \in M^{int}\}$ are given. We emphasize that a priori the points $p$ related to $D_p$ are unknown. The aim of this paper is to prove that travel time difference data determine $(M,g)$ up to isometry. Before stating our main theorem, we describe an additional geometric property for ${\partial} M$ under which we can prove the uniqueness of the inverse problem. Let $(N,G)$ be any smooth closed Riemannian manifold that extends $(M,g)$, such that $g=G|_{M}$. We use the notation \[ \ell(x,v):=\inf \{t > 0: \gamma_{x,v}(t) \in N \setminus M\}, \quad (x,v)\in SM. \] Thus the domain of definition for $\gamma_{x,v}$ is $[-\ell(x,-v),\ell(x,v)]$. Moreover by Lemma 1 of \cite{stefanov2009}, $\ell(x,v)$ is independent of the extension. We note that $\gamma_{x,v}$ may intersect the boundary tangentially in many points. \begin{definition} \label{eq:SU-cond-2} We say that $(M,g)$ satisfies \textit{the visibility} condition, if the following holds: For every $z \in {\partial} M$ there exists $(z,\eta) \in {\partial} S M, \hbox{ such that } \ell(z,\eta) < \infty.$ Geodesic $\gamma_{z,\eta}: [0,\ell(z,\eta)] \to M$ is a distance minimizer and $\gamma_{z,\eta}(\ell(z,\eta))$ is not a cut point to $z$, $\dot{\gamma}_{z,\eta}(\ell(z,\eta))$ is tranversal to ${\partial} M$ and $\gamma_{z,\eta}((0,\ell(z,\eta))) \subset M^{int}$. \end{definition} Next, we formulate our main Theorem. Let $(M_1,g_1)$ and $(M_2,g_2)$ be two smooth compact Riemannian manifolds with smooth boundaries ${\partial} M_1$ and ${\partial} M_2$. \begin{definition} \label{de:TTDD_agree} We say that the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, if there exists a diffeomorphism $\phi:{\partial} M_1 \to {\partial} M_2$ such that \begin{equation} \label{eq:equivalent_data} \{D_p(\phi^{-1}(\cdot),\phi^{-1}(\cdot)): p \in M_1^{int}\}=\{D_q: q \in M_2^{int}\}. \end{equation} \end{definition} Then. \begin{theorem} \label{th:main} Let $(M_i,g_i),\: i=1,2$ be compact, connected $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. Suppose that $(M_1,g_1)$ satisfy the visibility condition \ref{eq:SU-cond-2}. If the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, then there exists a Riemannian isometry $\Psi:(M_1,g_1) \to (M_2,g_2)$ such that the restriction of $\Psi$ on ${\partial} M_1$ coincides with $\phi$. \end{theorem} \color{black} While preparing this paper for submission, the authors became aware that S. Ivanov very recently posted a preprint \cite{ivanov2018distance} on ArXiv with a result (Proposition 7.3.) related to the result presented here. Indeed, he proved a similar result for complete manifolds with boundary under the assumption that the boundary is nowhere concave. On the other hand by the proof of Lemma \ref{Le:jet}, the claim of Proposition 7.3. in \cite{ivanov2018distance} holds if the nowhere concave boundary condition is replaced with the visibility condition. We give a different proof for Theorem \ref{th:main} (see Section \ref{Se:outline} for the outline of our proof) compared to one given in \cite{ivanov2018distance}. The proof given in \cite{ivanov2018distance} is based on distance comparison inequalities implied by Toponogov's theorem and minimizing geodesic extension property. The latter property provides a lower bound on the length of a minimizing extension of a geodesic beyond a non-cut point in terms of the length of a minimizing extension beyond the other endpoint. We end this section by comparing the visibility condition to the nowhere concave boundary condition. Recall that the boundary ${\partial} M$ of Riemannian manifold $(M,g)$ is nowhere concave, if for every $z \in {\partial} M$ the second fundamental form of ${\partial} M$ at $z$, with respect to the inward-pointing normal vector, has at least one positive eigenvalue. If ${\partial} M$ is nowhere concave then by the proof of Proposition 3.4. of \cite{zhou2012recovery} and Section 4.1. of \cite{sharafutdinov2012integral} it holds that $(M,g)$ satisfies the visibility condition. Notice that an annulus, contained in Euclidean plane, satisfies the visibility condition, but not the nowhere concave boundary condition. Therefore the visibility condition is more general of these two. Finally we will give an example of such geometry that does not satisfy either of these boundary conditions. Let $M\subset S^2$ be a spherical cap larger than the half--sphere. If $g$ is the round metric on $M$, then $(M,g)$ does not satisfy the visibility condition, since any $g$--distance minimizing curve between boundary points lies in ${\partial} M$ and therefore it is not a geodesic of $S^2$. In this case ${\partial} M$ is not either nowhere concave. \subsection*{Background} \subsubsection{Four geometric inverse problems related to the Riemannian wave equation} In this section we assume that $N,\: M, \: F$ and $g$ are as in Section \ref{Se:motivation}. There are four different data sets that are all related to Riemannian wave equation with the Dirac point source $(p,s)\in M\times {\mathbb R}$ and zero Cauchy data. The inverse problem of travel time functions have been considered in \cite{Katchalov2001,kurylev1997multidimensional}. The authors study the properties of the map $\mathcal{R}:M \to C({\partial} M)$, in which a point $p \in M$ is mapped into the corresponding travel time function $r_p:{\partial} M \to {\mathbb R}$, given by the formula $$ r_p(z)=d_g(p,z), \quad z \in {\partial} M. $$ The authors show that the data $({\partial} M, \{r_p: p \in M\})$ determine a manifold $(M,g)$ up to isometry. They use the map $\mathcal R$ to construct an isometric copy of $M$ in $C({\partial} M)$. They don't pose any restrictions to the geometry. In \cite{LaSa} the authors prove a result related to Theorem \ref{th:main}. In this paper it is assumed that the travel time difference function is given in the \textit{observation} set $F$ with non-empty interior $$ D_p:F \times F \to {\mathbb R}. $$ In addition they assume that the Riemannian structure of $(F,g)$ is known. The proof of the main theorem in \cite{LaSa} is very similar to the proof of Theorem \ref{th:main} presented in this paper and we will often refer to it for the details that are not presented in this paper. In \cite{ivanov2018distance} S. Ivanov extends the result of \cite{LaSa} in the following set up. Let $M$ be any complete, connected Riemannian manifold without boundary. Let $F,U \subset M$ be open. If the topology and differential structure of the observation domain $F$ and $D_p, \: p \in U$ are given then these data determine the geometry of the domain $(U,g_U)$ uniquely up to a Riemannian isometry. The sets $U$ and $F$ can be faraway from each other, which is not the case in \cite{LaSa} where it is assumed that $U=M$. Furthermore S. Ivanov proves that the determination of $(M,g)$ from travel time difference functions $D_p$ is stable, if the underlying manifold has a priori bounds on its diameter, curvature, and injectivity radius. In \cite{ivanov2018distance} also a similar result to our Theorem \ref{th:main} is provided for complete manifolds with nowhere concave boundary. The inverse problem related to the set of exit directions \[ \Sigma_p=\{(\gamma_{p,v}(\ell(p,v)),\dot \gamma_{p,v}(\ell(p,v)))\in {\partial} SM: v \in S_pM\} \] of geodesics emitted from $p$ has been studied in \cite{lassas2018reconstruction}. Let $$ I(g,w,z,l):= \hbox{ number of $g$--geodesics of lenght $l $ connecting $w$ to $z$}, \quad w,z \in N, \: l >0 $$ The authors show that, if $(N,g)$ is a closed manifold such that \begin{equation} \label{eq:generic} \sup_{w,z,\ell} I(g,w,z,l) <\infty, \end{equation} $M$ is non-trapping and ${\partial} M$ is strictly convex, then the collection of exiting directions $$ \{\Sigma_p \subset {\partial} TM : p \in M^{int}\} $$ determine the manifold $(M,g)$ up to isometry. Assumption \eqref{eq:generic} is needed to show that each set $\Sigma_p$ is produced by the unique $p\in M$. To our understanding, it is not known, if \eqref{eq:generic} follows from the convexity of the boundary and non-trapping properties. On the other hand in \cite{kupka2006focal} it is shown that \eqref{eq:generic} is a generic property in the space of all Riemannian metrics of $N$. The final data set is related to a \textit{generalized sphere} of radius $r>0$, that is given by formula \[ S(p,r)=:\{\exp_p(v): v\in T_pM,\; \|v\|_g=r,\; \hbox{$\exp_p$ is not singular at $v$}\}. \] In \cite{deHoop1} the authors show that the spherical surface data $$ \{S(q,r)\cap F: q \in M, \: r >0 \} $$ determine the universal cover space of $N$. If a generalized sphere $S(p,r)$ is given the authors show that there exists a specific coordinate structure in a neighborhood of any maximal normal geodesic to $S(p,r)$ such that in these coordinates metric tensor $g$ can can be determined. However this does not determine $g$ globally. The authors provide an example of two different metric tensors which produce the same spherical surface data. \subsubsection{Microseismicity} In this paper the results in \cite{LaSa} are adapted, in a fundamental way, to data available from actual seismic surveys. The point sources are microseismic events detected in dense arrays at Earth's surface. In our theorem we show that the data determine the metric up to change of coordinates. This implies that one can locate the closest surface point and to determine the corresponding travel time to each event. For the following we assume that $M \subset {\mathbb R}^m$ and $p \in M^{int}$. Recall that the arrival time function is $\mathcal{T}_{p,s}(z)=d_g(p,z)+s$, where $z\in {\partial} M$ is a receiver point and $s\in {\mathbb R}$ is the emission time. Since $\mathcal{T}_{p,s}(z)$ is a highly non-linear function of $p$ it is traditional in seismological literature to study the linearization of $ \mathcal{T}_{p,s}(z)$ \cite{waldhauser2000double}. Let $p_0 \in M^{int}$ be a master event i.e. an event for which $d_g(p_0,z)$ is known and $d_g(\cdot,z)$ is $C^1$--smooth near $p_0$. By the Taylor series of $d_g(\cdot,z)$ we have that the linearization \[ r^z_p:=\nabla d_g(\cdot,z)\bigg|_{p_0}\cdot (p-p_0) \approx d_g(p,z)-d_g(p_0,z), \] where, $\nabla$ is the Euclidean gradient and $p$ is close to $p_0$. The \textit{double difference distance function} is $r^z_p-r^z_q$. This function is the difference of differential distances between a (receiver) point $z$ at the boundary, and two source points $p,q$ in the interior -- in which the metric is unknown -- of a manifold. The goal is to use this data to determine travel time $d_g(p,z)$ of the second event and to locate the relative distance $d_g(p,p_0)$ of the second event to the master event. The event location with this method is known as the DD earthquake location algorithm presented in \cite{waldhauser2000double}. This method assumes a flat earth model and is appropriate for local scale problems. In contrast to seismological literature we measure the difference of the arrival times $\mathcal{T}_{p,s}(z),\mathcal{T}_{p,s}(w)$ of the given event $p\in M^{int}$ to two receivers $z,w \in {\partial} M$. For our theorem it is not necessary to linearize the arrival times. The travel time difference function, given in \eqref{eq:DDF}, is closer related to applications in exploration seismology with the purpose of locating microseismic events Grechka \textit{et al.} \cite{grechka2015relative}. In this paper the authors assume that the travel time to the receivers and location of the master event is known. Notice that our result do not recover the locations of the events in Cartesian coordinates. In global seismology, the idea to decouple the earthquake doublets, that is two different events that are close to each other and produce nearly indentical waveform, to locate the events was introduced by Poupinet \textit{et al.} \cite{PoupinetEF-1984}. Zhang \& Thurber \cite{ZhangT-2006, ZhangT-2003} extend the double difference location method of Waldhauser \& Ellsworth \cite{waldhauser2000double} with an attempt to simultaneously solve for both velocity structure and seismic event locations. They develop a regional DD seismic tomography methods that deal effectively with discontinuous velocity structures without knowing them a priori. Their methods also take Earths curvature into account. \color{black} \section{Proof of the Main theorem} In this section we prove Theorem \ref{th:main}. Whenever it is not necessary to distinguish manifolds $M_1$ and $M_2$ from one other we drop the subindices. In these cases we work with the data \eqref{eq:data}. \subsection{Outline of the proof of the Main theorem} \label{Se:outline} The proof consists of three steps. First we use the data \eqref{eq:data} to construct a mapping ${\mathcal D}$ from points of $M$ to continuous functions on ${\partial} M \times {\partial} M$. We show that this mapping is a topological embedding. Then we use the diffeomorphism $\phi:{\partial} M_1 \to {\partial} M_2$ and \eqref{eq:equivalent_data} to construct a homeomorphism $\Psi:M_1 \to M_2$ as in Theorem \ref{th:main} (see \eqref{eq:map_psi} for the definition). In second part we show that this mapping is a diffeomorphism. We prove the existence of such local coordinate maps that are determined by \eqref{eq:data}. In the third part we first prove that the data \eqref{eq:data} determine the images of geodesic segments that come to the boundary ${\partial} M$. Finally we use this information to prove the uniqueness of Riemannian structure. The outline of the proof of the main theorem is similar to the proof of the main theorem of \cite{LaSa}. The proof presented in this paper contains two key differences to the earlier result. The first one is the construction of the boundary coordinate system, in the beginning of Section \ref{Se:smooth}. The determination of the boundary defining function (see \eqref{eq:func_f_p} and \eqref{eq:boundary_def_func}), only from the data \eqref{eq:data}, has not been presented in the literature before. The second difference, that is considered in the beginning of Section \ref{Se:Riemannian}, is related to the construction of metric tensor from the data \eqref{eq:data}. In order to use the similar techniques as in \cite{LaSa}, to prove that the metrics $g_1$ and $\Psi^{\ast}g_2$ coincide, we need to prove that the data \eqref{eq:data} determine the full Taylor expansion of the metric tensor on ${\partial} M$ in boundary normal coordinates. This makes it possible to extend $M_1$ to a closed manifold $N$ given with two smooth metric tensors $G$ and $\widetilde G$ that coincide in $F:=\overline{N\setminus M_1}$, $G|_{M_1}=g_1$ and $\widetilde G|_{M_1}=\Psi^{\ast}g_2$. Since we don't assume ${\partial} M$ to be strictly convex, we will need also to show that the travel time difference functions $D_p:F\times F \to {\mathbb R}, \: p\in N$ of $(N,G)$ and $(N,\widetilde G)$ coincide. For this last step we use the proof of the Proposition 7.3 of \cite{ivanov2018distance} by S. Ivanov. The visibility condition of the Definition \ref{eq:SU-cond-2} is needed to tackle these problems. \color{black} \subsection{Topology} We start first extending the data to the boundary. If $p,w \in {\partial} M$ then by the triangle inequality it holds that \begin{equation} \label{eq:boundary_distance} d_g(p,w)=\sup_{q \in M^{int}}D_q(p,w). \end{equation} Thus data \eqref{eq:data} determine $d_g:{\partial} M \times {\partial} M \to {\mathbb R}$ and the extended data \begin{equation} \label{eq:data_full} ({\partial} M, \{D_p: \: p \in M\}). \end{equation} Our first Lemma is \begin{lemma} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. If the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, then \begin{equation} \{D_p(\phi^{-1}(\cdot),\phi^{-1}(\cdot)): p \in M_1\}=\{D_q: q \in M_2\}. \label{eq:equivalent_full_data} \end{equation} \end{lemma} \begin{proof} From \eqref{eq:equivalent_data} and \eqref{eq:boundary_distance} it follows that \begin{equation} \label{eq:boundary_dist_agree} d_1(\phi^{-1}(p),\phi^{-1}(q))=d_2(p,q), \quad p,q \in {\partial} M_2. \end{equation} Here, $d_i$ is the distance function of $g_i$ for $i \in \{1,2\}$. Therefore \eqref{eq:equivalent_full_data} holds. \end{proof} We study the properties of the mapping $$ {\mathcal D}:M \to C({\partial} M \times {\partial} M), \quad {\mathcal D}(p)=D_p, $$ where the target space is equipped with the $L^\infty$--norm. \begin{lemma} \label{pr:topology} The mapping ${\mathcal D}$ is a topological embedding. \end{lemma} \begin{proof} Using triangle inequality it is easy to see that ${\mathcal D}$ is $2$--Lipschitz. Next we prove that $\mathcal{D}$ is one-to-one. To show this, assume that $x,y \in M$ are such that $D_x=D_y$. We first show that this implies that the set $\{z_x\}$ of closest boundary points of $x$ coincides with the set $\{z_y\}$ of closest boundary points of $y$. Let $w \in {\partial} M$ and define \begin{equation} \label{eq:func_f_w} f_{x,w}:{\partial} M \to {\mathbb R}, \quad f_{x,w}(z):=D_x(z,w). \end{equation} Then $\{z_x\}$ is the set of minimizers of function $f_{x,w}$. Since $f_{x,w}=f_{y,w}$, we have proven that $\{z_x\}=\{z_y\}$. We also use the function $f_{x,w}$ later when we construct a boundary defining function. Let $z_0 \in \{z_p\}$ and denote $s_x=d_g(x,z_0)$ and $s_y=d_g(y,z_0)$. Without loss of generality, we can assume that $s_x \leq s_y$. Let $\nu$ be the inward pointing unit normal vector field to ${\partial} M$. Then $\gamma_{z_0,\nu}$ is the distance minimizing geodesic from ${\partial} M$ to $x$ and $y$. Moreover \begin{equation} \label{eq:dist_from_x_to_y} x=\gamma_{z_0,\nu}(s_x), \: y=\gamma_{z_0,\nu}(s_y) \hbox{ and } d(x,y)=s_y-s_x. \end{equation} If $z \in {\partial} M\setminus \{z_0\}$ is close to $z_0$, the distance minimizing geodesic $\gamma_x$ from $z$ to $x$ is not the same geodesic as $\gamma_{z_0,\nu}$, that is, the angle $\beta$ of the curves $\gamma_x$ and $\gamma_{z_0,\nu}$ at the point $x$ is strictly between $0$ and $\pi$. Let $\gamma_y$ be a distance minimizing geodesic from $y$ to $z$. We note that $D_x(z,z_0)=D_y(z,z_0)$ and \eqref{eq:dist_from_x_to_y} yields $$ \mathcal{L}(\gamma_y)=d(y,z)=d(y,x)+d(x,z)=\mathcal{L}(\gamma_{z_x,\nu}|_{[s_x,s_y]})+\mathcal{L}(\gamma_x). $$ Thus the union $\mu$ of the curves $\gamma_{z_x,\nu}([s_x,s_y])$ and $ \gamma_x$ is a distance minimising curve from $z$ to $y$, and hence it is a geodesic. However, as the angle $\beta$, defined above, is strictly between $0$ and $\pi$, the curve $\mu$ is not smooth at $x$, and hence it is not possible that $\mu$ is a geodesic unless $x=y$. Thus $x$ and $y$ have to be equal. Since $M$ is compact and we just proved that ${\mathcal D}$ is continuous and one--to--one, we have that mapping ${\mathcal D}$ is closed. Thus the claim is proven. \end{proof} Since the mapping $\phi$, given by Definition \ref{de:TTDD_agree}, is a diffeomorphism the mapping $$ \Phi:C({\partial} M_1 \times {\partial} M_1) \to C({\partial} M_2 \times {\partial} M_2), \quad \Phi(F)=F(\phi^{-1}(\cdot),\phi^{-1}(\cdot)) $$ is an isometry. Let ${\mathcal D}_i, \: i\in \{1,2\}$ be as ${\mathcal D}$ on $(M_i,g_i)$. Now we are ready to define the mapping \begin{equation} \label{eq:map_psi} \Psi:M_1 \to M_2, \quad \Psi = {\mathcal D}_2^{-1} \circ \Phi\circ {\mathcal D}_1. \end{equation} \begin{proposition} \label{th:topology} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. If the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, then the mapping $\Psi$ given by \eqref{eq:map_psi} is a homeomorphism such that the restriction of $\Psi$ on ${\partial} M_1$ coincides with $\phi$. \end{proposition} \begin{proof} By \eqref{eq:equivalent_full_data} and the Proposition \ref{pr:topology} it holds that the map $\Psi$ is a well-defined homeomorphism. If $p \in {\partial} M_1$, then by \eqref{eq:boundary_dist_agree} for any $z, w \in {\partial} M_2$ we have $$ ({\mathcal D}_2(\phi(p))(z,w)=d_2(\phi(p),z)-d_2(\phi(p),w)=d_1(p,\phi^{-1}(z))-d_1(p,\phi^{-1}(w))=((\Phi \circ \mathcal{D}_1)(p))(z,w). $$ Applying ${\mathcal D}^{-1}_2$ for both sides of the equation above we have $\Psi(p)=\phi(p)$. \end{proof} \subsection{Smooth structure} \label{Se:smooth} In this part we show that the mapping $\Psi$ given in \eqref{eq:map_psi} is \\ a diffeomorphism. We consider separately the boundary and the interior cases. We start with the boundary case. Let $\sigma_{{\partial} M}$ be the collection of all boundary cut points, $$ \sigma_{{\partial} M}:=\{\gamma_{z,\nu}(\tau_{{\partial} M}(z)) \in M: \: z \in M\}, \quad \tau_{{\partial} M}(z):=\sup\{t>0:d_g({\partial} M, \gamma_{z,\nu}(t))=t\}. $$ By Section III.4. of \cite{sakai1996riemannian} it holds that \begin{equation} \label{eq:boundary_cut_locus} \sigma_{{\partial} M}=\overline{\{p\in M: \#\{z\in{\partial} M: d_g(p,z)=d_g(p,{\partial} M)\}\geq 2\}}. \end{equation} Choose $w \in {\partial} M$. Then by \eqref{eq:boundary_cut_locus} and the Proposition \ref{pr:topology} the data \eqref{eq:data_full} determine the set \begin{equation} \label{eq:complement_of_boundary_cut_locus} M \setminus \sigma_{{\partial} M}=\{p \in M : \hbox{ The map $f_{p,w}$ has precicely one minimizer.}\}^{int}, \end{equation} where $f_{p,w}$ is as in \eqref{eq:func_f_w}. \begin{lemma} \label{Le:equivalence_of_cut_loci} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. If the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, then $$ M_2 \setminus \sigma_{{\partial} M_2}=\Psi(M_1 \setminus \sigma_{{\partial} M_1}). $$ \end{lemma} \begin{proof} By the definition of the mapping $\Psi$ we have for any $p \in M_1$ and $w \in {\partial} M_1$ that $$ f^1_{p,w}(z)=f^2_{\Psi(p),\phi(w)}(\phi(z)), \quad z \in {\partial} M_1, $$ where $f^1_{p,w}$ and $f^2_{\Psi(p),\phi(w)}$ are defined as $f_{p,w}$ in \eqref{eq:func_f_w}. Therefore the claim follows from \eqref{eq:complement_of_boundary_cut_locus}. \end{proof} Next we construct a boundary defining function on $M \setminus \sigma_{{\partial} M}$. Let $p \in M \setminus \sigma_{{\partial} M}$ and denote by $Z(p)$ the closest boundary point of $p$. The map $x \mapsto Z(x) \in {\partial} M$ is smooth on $M \setminus \sigma_{{\partial} M}$. Define a function \begin{equation} \label{eq:func_f_p} f_p(z):=d_g(z,Z(p))-D_p(z,Z(p)), \quad z \in {\partial} M. \end{equation} Notice that this function is determined by the data \eqref{eq:data_full}, and by triangular in equality the function $f_p$ is non-negative. If $p \in {\partial} M$ then $f_p$ is a zero function. If $p \in M^{int} \setminus \sigma_{{\partial} M}$ then \begin{equation} \label{eq:func_f_p_outside_closes_bp} f_p(z)>0, \quad z \in ({\partial} M \setminus Z(p)). \end{equation} If this is not true then there exists ${\partial} M \ni z \neq Z(p)$ such that $$ d_g(p,z)=d_g(Z(p),z)+d_g(p,Z(p)). $$ Which implies that there exists a distance minimizing curve from $p$ to $z$, that goes through $Z(p)$, but is not $C^1$ at $Z(p)$. By \cite{alexander1981geodesics} this is not possible. Thus \eqref{eq:func_f_p_outside_closes_bp} holds. Therefore we have proven the following \begin{equation} \label{eq:char_of_boundary} {\partial} M=\{p \in M \setminus \sigma_{{\partial} M}: f_p \equiv 0\}. \end{equation} \begin{figure} \caption{Here is the schematic picture of the function $f_p$.} \label{Fi:f_p} \end{figure} \begin{lemma} \label{Le:smoothnes_of_dist_func} Let $(M,g)$ be a smooth Riemannian manifold with smooth boundary for which the visibility condition \ref{eq:SU-cond-2} holds. Let $p \in {\partial} M$. Then there exist $q \in {\partial} M$ and neighborhoods $U,V \subset M$ of $p$ and $q$ respectively such that $d_g:U\times V$ is smooth. The distance minimizing geodesic from $p$ to $q$ is transversal to ${\partial} M$ at $p$ and $q$. Moreover any distance minimizing geodesic $\gamma$ from $U$ to $V$ is contained $M^{int}$, if the start and end points are excluded. \end{lemma} \begin{proof} We follow the proof of Theorem 1 of \cite{stefanov2009} and show that \ref{eq:SU-cond-2} implies the following claim: There exists $\eta \in S_pM, \hbox{ that is transversal to ${\partial} M$ and } 0< \ell(p,\eta) < \infty,$ $ \gamma_{p,\eta}: [0,\ell(p,\eta)] \to N \hbox{ is distance}$ minimizer and $q:=\gamma_{p,\eta}(\ell(p,\eta))$ is not a cut point to $p$ along $\gamma_{p,\eta}$. The exit direction $\dot{\gamma}_{p,\eta}(\ell(p,\eta))$ is transversal to ${\partial} M$ and $\gamma_{p,\eta}((0,\ell(p,\eta))) \subset M^{int}$. Moreover $\ell(p,\eta)=d_{g}(p,q)$. The claim of this lemma follows from implicit function theorem. \end{proof} Let $p \in {\partial} M$. By Lemma \ref{Le:smoothnes_of_dist_func} there exists $w \in {\partial} M$ and $r>0$ such that the distance function $d_g$ is smooth in $B(p,r)\times B(w,r)$ and $B(p,r)\cap B(w,r)=\emptyset$. Let $r_{{\partial} M}>0$ be the minimum of $r$ and the boundary injectivity radius. Choose $$ z_0 \in ({\partial} M \cap (B(w,r)) \hbox{ and }\delta \in (0,r_{{\partial} M}), $$ such that $z_0$ is not the closest boundary point for any $q \in B(p,\delta)$, $Z(q) \in B(p,r) $ and the distance minimizing geodesic from $z_0$ to $p$ is not normal to ${\partial} M$ at $p$. Then \begin{equation} \label{eq:boundary_def_func} E_{z_0}:B(p,\delta) \to [0,\infty), \quad E_{z_0}(q):=f_q(z_0)=d_g(z_0,Z(q))-D_q(z_0,Z(q)) \end{equation} is well-defined and smooth. Moreover, by \eqref{eq:func_f_p_outside_closes_bp} we have that $E_{z_0}(q)=0$ if and only if $q \in B(p,\delta)\cap {\partial} M$. Thus $E_{z_0}$ is a boundary defining function. Denote $(t,Z)$ for the boundary normal coordinates in $B(p,\delta)$, where $t(q)=d_g({\partial} M, q)$ and $Z(q)$ is the closest boundary point to $q \in B(p,\delta)$. Then the map \begin{equation} \label{eq:boundary_coordinates} W_{z_0}:B(p,\delta) \to [0,\infty) \times {\partial} M, \quad W_{z_0}(q):=(E_{z_0}(q),Z(q)), \end{equation} is smooth. We show that the Jacobian of this map with respect to boundary normal coordinates is invertible at $p$. By the inverse function theorem this yields the existence of a neighborhood $V\subset M$ of $p$ such that the restriction of $W_{z_0}$ to $V$ is a coordinate map. The Jacobian of $W_{z_0}$ at $p$ is \begin{equation*} \left(\begin{array}{cc} \frac{{\partial}}{{\partial} t} E_{z_0}& \frac{{\partial}}{{\partial} t} Z \\ \\ \frac{{\partial}}{{\partial} Z} E_{z_0}&\frac{{\partial}}{{\partial} Z}Z \end{array}\right) = \left(\begin{array}{cc} \frac{{\partial}}{{\partial} t} E_{z_0}&\bar 0^T \\ \\ \frac{{\partial}}{{\partial} Z} E_{z_0}& Id_{n-1}. \end{array}\right) \end{equation*} Notice $$ \frac{{\partial}}{{\partial} t} E_{z_0}(t,Z)\bigg|_{(t,Z)=(0,p)}=1-g_p(\dot{\gamma}_{z_0,p}(d_g(p,z_0)),\nu)>0. $$ The last inequlity holds since the distance minimizing geodesic $\gamma_{z_0,p}$ from $z_0$ to $p$ is not normal to the boundary at $p$. Thus Jacobian of $W_{z_0}$ at $p$ is invertible. We use coordinates similar to $W_{z_0}$ to show that $\Psi:M_1\to M_2$ is a diffeomorphism near the boundary of $M_1$. In order to do so we first prove the following lemma. \begin{lemma} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. If the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide, then \begin{equation} \label{eq:boundary_metric} g_1|_{{\partial} M_1}=\phi^{\ast}(g_2|_{{\partial} M_2}). \end{equation} \end{lemma} \begin{proof} Since \eqref{eq:equivalent_data} implies \eqref{eq:boundary_dist_agree} the proof of this Lemma follows from the proof of Proposition 3.3. of \cite{zhou2012recovery}. \end{proof} Now we are ready to prove the following lemma. \color{black} \begin{lemma} \label{Le:boundary_coord} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$, whose travel time difference data coincide. Assume that $(M_1,g_1)$ satisfy the visibility condition \ref{de:TTDD_agree}. Let $p \in {\partial} M_1$. There exists a neighborhood $U$ of $p$ in $M_1$ and $z_0 \in {\partial} M_1$ such that on $U$ and $\Psi(U)$ the mappings $W^1_{z_0}(q_1)=(E^1_{z_0}(q_1),Z^1(q_1))$ and $W^2_{\phi(z_0)}(q_2)=(E^2_{\phi(z_0)}(q_2),Z^2(q_2))$ respectively, defined as in \eqref{eq:boundary_def_func} and \eqref{eq:boundary_coordinates}, are smooth local boundary coordinate maps. Moreover, with respect to these coordinates, the local representation of $\Psi$ is \begin{equation} \label{eq:local:rep_of_Psi_boundary} W^1_{z_0}(U)\ni (s,z) \mapsto (s,\phi(z)) \in W^2_{\phi(z_0)}(\Psi(U)). \end{equation} \end{lemma} \begin{proof} By Lemma \ref{Le:equivalence_of_cut_loci} we have for any $q \in (M_1 \setminus \sigma_{{\partial} M_1})$ that the point $z \in {\partial} M_1$ is the closest boundary to $q$ if and only if $\phi(z) \in {\partial} M_2$ is the closest boundary point to $\Psi(q) \in (M_2 \setminus \sigma_{{\partial} M_2})$. Thus $$ \phi(Z^1(q))=Z^2(\Psi(q)). $$ Therefore, using \eqref{eq:boundary_dist_agree} we have that for all $q \in (M_1 \setminus \sigma_{{\partial} M_1}), \: z \in {\partial} M_1$ \begin{equation} \label{eq:f_p_agree} f^1_q(z):=d_1(z,Z^1(q))-D_q(z,Z^1(q))=d_2(\phi(z),Z^2(\Psi(q))-D_{\Psi(q)}(\phi(z),Z^2(\Psi(q))=:f^2_{\Psi(q)}(\phi(z)). \end{equation} We choose $w \in {\partial} M_1$ neighborhoods $U'$ and $V$ for $p$ and $w$ respectively as in Lemma \ref{Le:smoothnes_of_dist_func} for $(M_1,g_1)$. Then function $(x,z) \mapsto d_1(x,z)$ is smooth in $(U' \cap {\partial} M_1) \times V\cap {\partial} M_1)$. Let $\gamma$ be the unique distance minimizing geodesic from $p$ to $w$ that is transversal to ${\partial} M_1$ at $p$ and $w$. Since $\phi$ is a diffeomorphism by \eqref{eq:boundary_dist_agree} and \eqref{eq:boundary_metric} it follows that \[ D\phi\;\bigg( \hbox{grad}'_1\;d_1(\cdot,w)\bigg|_{p}\bigg)= \hbox{grad}'_2\;d_2(\cdot,\phi(w))\bigg|_{\phi(p)} . \] Here $\hbox{grad}'_i$, $i\in \{1,2\}$ stands for the boundary gradient. Therefore, a $g_2$--distance minimizing unit speed curve $c$ from $\phi(p)$ to $\phi(w)$ is transversal to ${\partial} M_2$ at $\phi(p)$. Switching the order of $p$ and $w$ we prove also that $c$ is transversal to ${\partial} M_2$ at $\phi(w)$. Since $\gamma$ is the unique distance minimizing curve from $p$ to $w$ and $\gamma((0,d_1(p,w))) \subset M_1^{int}$ it holds by \eqref{eq:boundary_dist_agree} that $c((0,d_2(\phi(p),\phi(w)))) \subset M_2^{int}$. Therefore $c$ is a geodesic of $g_2$. Since $d_2(\phi(p),\cdot)|_{{\partial} M}$ is smooth at $\phi(w)$, $c$ is the unique distance minimizing curve of $(M_2,g_2)$ connecting $\phi(p)$ to $\phi(w)$. Moreover due to transversality of $c$ there exists a neighborhood of $\phi(w)$ such that any point in this neighborhood is connected to $\phi(p)$ via the unique distance minimizing geodesic. Since conjugate points of $\phi(p)$ in $(M_2,g_2)$ are accumulation points of those points $q\in M_2$ that can be connected to $\phi(p)$ via multiple distance minimizers, it holds that $\phi(w)$ is not either a conjugate point of $\phi(p)$ along $c$. Therefore $\phi(w)$ is not a cut point of $\phi(p)$ along $c$. This proves that also $(M_2,g_2)$ satisfies the visibility condition. By Lemma \ref{Le:smoothnes_of_dist_func} we have proved that there exists $r_{\min}>0$ smaller than the minimum of the boundary cut distances of $g_1$ and $g_2$, such that functions \[ (q,z) \mapsto d_1(q,Z^1(q)), \:d_1(q,z), \:d_1(z,Z^1(q)), \quad (q,z) \in B_1(p,r_{\min}) \times (B_1(w,r_{\min})\cap {\partial} M_1) \] and \[ (q',z') \mapsto d_2(q',Z^2(q')), \:d_2(q',z'),\: d_2(z',Z^2(q)), \quad (q',z') \in B_2(\phi(p),r_{\min}) \times (B_2(\phi(w),r_{\min})\cap {\partial} M_2) \] are smooth. Since $\Psi$ is a homeomorphism the existence of set $U$ and $z_0 \in {\partial} M_1$ as in the claim of this Lemma follow. If $q \in U$ we obtain by \eqref{eq:f_p_agree} the following equation $$ E^1_{z_0}(q)=E^2_{\phi(z_0)}(\Psi(q)). $$ Therefore we have proven that the map given in \eqref{eq:local:rep_of_Psi_boundary} and the mapping $$ W^2_{\phi(z_0)}\circ \Psi\circ (W^1_{z_0})^{-1}:W^1_{z_0}(U) \to W^2_{\phi(z_0)}(\Psi(U)) $$ coincide. \end{proof} \color{black} Next we consider the coordinates away from ${\partial} M$. Let $p \in M^{int}$ and choose any closest boundary point $z_p\in {\partial} M$ to $p$. By Lemma 2.15 of \cite{Katchalov2001} there exist neighborhoods $U \subset M^{int}$ of $p$ and $W \subset {\partial} M$ of $z_p$ such that the distance function $d_g:U\times W \to {\mathbb R}$ is smooth. Moreover for every $(q,w) \in U\times W$ the distance $d_g(q,w)$ is realized by the unique distance minimizing geodesic, contained in $M^{int}$, if the end point $w$ is excluded. We use a shorthand notation $v \in S_p M$ for the velocity $\dot \gamma_{z_p,\nu}(d_g(p,z_p))$. A similar argument as in Lemma 2.6. of \cite{LaSa} yields to an existence of a neighborhood $V \subset W$ of $z_p$ such that the set $$ \mathcal{V}=\{(z_i)_{i=1}^n \in V^n: \dim \hbox{span}((F(z_i)-v)_{i=1}^n)=n\} $$ is open and dense in $V^n:= V\times V \times \ldots \times V$. Here $F(q):=-\frac{(\exp_p)^{-1}(q)}{\|(\exp_p)^{-1}(q)\|_g}, \: q \in V$. Notice that this claims follows from Lemma 2.6. of \cite{LaSa} since $F(q)=\frac{(\exp_p)^{-1}(q')}{\|(\exp_p)^{-1}(q')\|_g}$ for some $q'\in M$ if and only if there exists $ 0<t<\tau(p,-F(q))$ such that $q'=\gamma_{p,-F(p)}(t)$. Moreover for every $(z_i)_{i=1}^n \in \mathcal{V}$ there exists an open neighborhood $U' \subset U$ of $p$ such that $$ H:U'\rightarrow {\mathbb R}^n, \quad H(q)=(d_g(q,z_i)-d_g(q,z_p))_{i=1}^n $$ is a smooth coordinate mapping. This holds, since for any $(z_i)_{i=1}^n \in \mathcal{V}$ the Jacobian of $H$ at $p$ is invertible. \begin{lemma} \label{Le:interior_coord} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$. Suppose that the travel time difference data of $(M_1,g_1)$ and $(M_2,g_2)$ coincide. Let $p \in M_1^{int}$. Let $z_p$ be any closest boundary point to $p$. There exists a neighborhood $U$ of $p$ in $M_1^{int}$ and a neighborhood $W \subset {\partial} M_1$ of $z_p$ such that the distance functions $d_1:U \times W$ of $(M_1,g_1)$ and $d_2:\Psi(U) \times \phi(W)$ of $(M_2,g_2)$ are smooth. Moreover there exists points $z_1, \ldots, z_n \in W$ and a neighborhood $V \subset U$ of $p$ such that $$ H_1:V\rightarrow {\mathbb R}^n, \quad H_1(x)=(d_1(x,z_i)-d_1(x,z_p))_{i=1}^n $$ and $$ H_2:\Psi(V)\rightarrow {\mathbb R}^n, \quad H_2(q)=(d_2(q,\phi(z_i))-d_2(q,\phi(z_p)))_{i=1}^n, $$ are smooth coordinate maps. We also have \begin{equation} \label{eq:local:rep_of_Psi_interior} H_1(V)= H_2(\Psi(V)) \hbox{ and } H_2 \circ \Psi\circ H_1 = Id_{{\mathbb R}^n}. \end{equation} \end{lemma} \begin{proof} Since $\Psi$ is a homeomorphism, the first part of the claim follows from similar construction as done before this Lemma. The proof of the latter part is a modification of the proof of Theorem 2.7. of \cite{LaSa}. \end{proof} \begin{proposition} \label{th:diffeo} Let $(M_i,g_i),\: i=1,2$ be compact $n$--dimensional Riemannian manifolds with smooth boundaries ${\partial} M_i$ whose travel time difference data coincide. If $(M_1,g_1)$ satisfy the visibility condition \ref{eq:SU-cond-2}, then mapping $\Psi:M_1 \to M_2$, given in \eqref{eq:map_psi}, is a diffeomorphism. \end{proposition} \begin{proof} The claim follows from Proposition \ref{th:topology} and lemmas \ref{Le:boundary_coord}--\ref{Le:interior_coord}. \end{proof} \subsection{Riemannian structure} \label{Se:Riemannian} As we have proven that the map $\Psi$ is diffeomorphism we can define a pull back metric $\widetilde g:=\Psi^\ast g_2$ on $M_1$. From now on we only consider manifold $M:=M_1$ with smooth boundary equipped with Riemannian metrics $g:=g_1$ and $\widetilde g$. We need to show that $g=\widetilde g$. First we notice that by the definitions of the diffeomorphism $\Psi$ and metric $\widetilde g$ on $M$ we have by the data \eqref{eq:data_full} that \begin{equation} \label{eq:dist_dif_agree} D_p(z,w)=d_g(p,z)-d_g(p,w)=d_{\widetilde g}(p,z)-d_{\widetilde g}(p,w), \quad p \in M, \: z,w \in {\partial} M. \end{equation} \begin{lemma} \label{Le:jet} Let $p \in {\partial} M$ and $(x^1,\ldots, x^n)$ be a boundary normal coordinate system of $g$ near $p$ and $\alpha\in {\mathbb N}^{n}$ any multi-index. Write $g=(g_{ij})_{i,j=1}^{n}$ and $\widetilde g=(\widetilde g_{ij})_{i,j=1}^{n}$. Then for all $i,j \in \{1, \ldots,n\}$ holds \begin{equation} \label{eq:jet} \partial^\alpha g_{ij} |_{{\partial} M}= \partial^\alpha \widetilde g_{ij} |_{{\partial} M}, \quad {\partial}^\alpha:=\prod_{k=1}^n \left(\frac{{\partial}}{{\partial} x^k}\right)^{\alpha_k}. \end{equation} \end{lemma} \begin{proof} We prove that the local lens relations $(\ell_g,\sigma_g)$ and $(\ell_{\widetilde g},\sigma_{\widetilde g})$ of $g$ and $\widetilde g$ respectively coincide at some open set $\mathcal D \subset T{\partial} M$. After this the claim follows from the proof of Theorem 1 of \cite{stefanov2009}. For the definitions of local lens relations see \cite{stefanov2009}. Choose $q \in {\partial} M$ and neighborhoods $U,V \subset M$ of $p$ and $q$ be as in Lemma \ref{Le:smoothnes_of_dist_func} for metric $g$. Let $\gamma$ be the unique geodesic of $g$ connecting $p$ to $q$. Due to \eqref{eq:boundary_dist_agree} and Lemma \ref{Le:smoothnes_of_dist_func} it holds that $d_{\widetilde g}$ is smooth on $(U \cap {\partial} M) \times (V \cap {\partial} M)$. Therefore we have for every $(x,y) \in (U \cap {\partial} M) \times (V \cap {\partial} M)$ that \begin{equation} \label{eq:boundary_gradients} \hbox{grad}'_g \; d_g(\cdot,y)\bigg|_{x}=\hbox{grad}'_{\widetilde g} \; d_{\widetilde g}(\cdot,y)\bigg|_{x} \quad \hbox{ and } \quad \hbox{grad}'_g \; d_g(\cdot,x)\bigg|_{y}=\hbox{grad}'_{\widetilde g} \; d_{\widetilde g}(\cdot,x)\bigg|_{y}. \end{equation} Denote $\dot{\gamma}(0)=:\eta$ and $\dot{\gamma}(d_g(p,q))=:v$. Then \eqref{eq:boundary_metric} and \eqref{eq:boundary_gradients} imply that $\dot{\widetilde \gamma}(0)=\eta$ and $\dot{\widetilde \gamma}(d_g(p,q))=v$, where $\widetilde \gamma$ is the unique distance minimizing geodesic of $\widetilde g$ from $p$ to $q$. By Lemma \ref{Le:smoothnes_of_dist_func} it holds that $\eta$ and $v$ are transversal to ${\partial} M$. Therefore after possibly shrinking $U$ and $V$ we have by formula (10) of \cite{stefanov2009} and formulas \eqref{eq:boundary_metric} and \eqref{eq:boundary_gradients} that the local lens relations $(\ell_g,\sigma_g)$ and $(\ell_g,\sigma_{\widetilde g})$ coincide in the set \[ \mathcal{D}:=\{ \hbox{grad}'_g \;d_g(\cdot,y)\bigg|_{x}, \: \hbox{grad}'_g \; d_g(\cdot,x)\bigg|_{y} \in T{\partial} M: \: (x,y) \in (U \cap {\partial} M) \times (V \cap {\partial} M)\}. \] The set $\mathcal D$ is open since it is an image of an open map, given by the composition of the diffeomorphism \[ W_\eta\ni (x,v) \mapsto \gamma_{x,v}(\ell(x,v)),\dot \gamma_{x,v}(\ell(x,v)) \in W_v \] and the orthogonal projection from ${\partial} SM$ to $T{\partial} M$. In the above $W_\eta\subset {\partial} SM$ is some open neighborhood of $(p,\eta)$ and $ W_v\subset {\partial} SM$ is some open neighborhood of $(q,v)$. \end{proof} \color{black} Let $(N,G)$ be a smooth closed Riemannian manifold that is a smooth extension of $(M,g)$. We write $F:= N \setminus M^{int}$, as before. By Lemma \ref{Le:jet} $(N,\widetilde G)$ is a smooth extension of $(M,\widetilde g)$, if $\widetilde G$ is a Riemannian metric defined as \begin{equation} \label{eq:def_G_tilde} \widetilde G|_F=G|_F, \quad \widetilde G|_{M^{int}}=\widetilde g. \end{equation} \begin{lemma} Let $N, F, G$ and $\widetilde G$ be as above. Then \begin{equation} \label{eq:dist_dif_on_N} d_G(p,z)-d_G(p,w)=d_{\widetilde G}(p,z)-d_{\widetilde G}(p,w) \quad p \in N, \: z,w \in F. \end{equation} The functions $d_G, d_{\widetilde G}$ are the geodesic distances of $G$ and $\widetilde G$ respectively. \end{lemma} \begin{proof} If $p \in M$, we will soon give a proof for \begin{equation} \label{eq:dist_dif_on_N_2} d_G(p,z)-d_G(p,w)=d_{\widetilde G}(p,z)-d_{\widetilde G}(p,w), \quad z,w \in F. \end{equation} This proof is an adaptation of Proposition 7.3 in \cite{ivanov2018distance}. If \eqref{eq:dist_dif_on_N_2} holds for every $p\in M$ then \eqref{eq:dist_dif_on_N_2} holds also for the case $p \in F$. The latter proof is given in Proposition 1.2. of \cite{LaSa}. Therefore equation \eqref{eq:dist_dif_on_N} holds. Let $p \in M$. Consider first the function $h_p(z)=d_{g}(p,z)-d_{\widetilde g}(p,z), \: z \in {\partial} M$. Let $w \in {\partial} M$. By \eqref{eq:dist_dif_agree} it holds that \[ h_p(z)=d_{g}(p,w)-d_{\widetilde g}(p,w). \] Thus $h_p$ is a constant function. We will prove that \begin{equation} \label{eq:proof_of_(31)} d_G(p,z)=\inf\bigg\{d_g(p,y_0)+ \bigg(\sum_{j=1}^Nd_F(y_{j-1},x_j)+d_g(x_{j},y_j)\bigg)+d_F(x_{N},z)\bigg\}, \end{equation} where $d_F$ is the distance function of the Riemannian manifold $(F,G|_F)$ and \\ $\{y_0, \ldots, y_N, x_1, \ldots, x_N\} \subset {\partial} M$. Notice that similar formula holds for $d_{\widetilde G}$, when $d_g$ is replaced with $d_{\widetilde g}$. If \eqref{eq:proof_of_(31)} holds then, it follows from equation \eqref{eq:boundary_dist_agree} that \[ d_G(p,z)-d_{\widetilde G}(p,z)=h_p(z)=\hbox{constant with respect to $z$}. \] This implies \eqref{eq:dist_dif_on_N}, in the case when $p \in M$. Finally we will prove \eqref{eq:proof_of_(31)}. Let $\epsilon>0$. Since ${\partial} M$ is a smooth co-dimension 1 submanifold of $N$, it follows from the definition of the Riemannian distance function $d_G$, that there exists a piecewise smooth curve $c$ from $p$ to $q$, that crosses the boundary finitely many times, and whose length is $\epsilon$--close to $d_G(p,z)$. Then \[ d_g(p,y_0)+ \bigg(\sum_{j=1}^Nd_F(y_{j-1},x_j)+d_g(x_{j},y_j)\bigg)+d_F(x_{N},z)\leq \mathcal{L}_G(c)\leq d_G(p,z)+\epsilon, \] where $\{y_0, \ldots, y_N, x_1, \ldots, x_N\} \subset {\partial} M$ are the points where $c$ crosses the boundary. Taking $\epsilon$ to $0$ implies \eqref{eq:proof_of_(31)}. \end{proof} Due to the previous Lemma it follows from the Section 2.4 of \cite{LaSa} that metric tensors $G$ and $\widetilde G$ coincide. We will sketch here the main ideas for this proof. First we prove that the geodesics of metrics $G$ and $\widetilde G$ agree up to reparametrization. Let $\tau_G:SN \to {\mathbb R}$ be the cut distance function of metric tensor $G$. By Lemma 2.9. of \cite{LaSa} the following equality holds for any $(z,v)\in SF^{int}$ \begin{equation} \label{eq:image_of_geo_1} \gamma^G_{z,-v}((0,\tau_G(z,-v))=\{p \in N:D_p(\cdot,z) \hbox{ is smooth at $z$ and grad$_G D_p(\cdot,z)$ at $z$ is $v$} \}. \end{equation} Where $\gamma^G_{z,-v}$ is the geodesic of $G$ with initial conditions $(z,-v)$. Since $G=\widetilde G$ on $F^{int}$, the formulas \eqref{eq:dist_dif_on_N} and \eqref{eq:image_of_geo_1} imply \begin{equation} \label{eq:image_of_geo_2} \gamma^G_{z,-v}((0,\tau_G(z,-v))=\gamma^{\widetilde G}_{z,-v}((0,\tau_{\widetilde G}(z,-v)), \quad (z,v) \in SF^{int}, \end{equation} where $\tau_{\widetilde G}$ is the cut distance function of $\widetilde G$. Therefore, for any $(z,v) \in SF^{int}$ there exists a diffeomorphism $\alpha_{z,v}:(0,\tau_G(z,-v)) \to (0,\tau_{\widetilde G}(z,-v))$ such that \begin{equation} \label{eq:image_of_geo_3} \gamma^G_{z,-v}(t)=\gamma^{\widetilde G}_{z,-v}(\alpha_{z,v}(t)), \quad t \in (0,\tau_G(z,-v)), \: (z,v) \in SF^{int}. \end{equation} Let $p \in M^{int}$. We denote the exponential map of $G$ at $p$ by $\exp_p$. Then the following set is not empty, $$ \Omega_p:=\{rv\in T_pN: r>0, \: v=\exp_p^{-1}(z), \: p \in \sigma(z,v), \: (z,v) \in SF^{int}\}^{int}, $$ and, moreover, if we denote the exponential map of $\widetilde G$ at $p$ by $\widetilde \exp_p$. In view of \eqref{eq:image_of_geo_3} we have \begin{equation} \label{eq:geodesic_in_M} \Omega_p=\{rv\in T_pN: r>0, \: v=\widetilde \exp_p^{-1}(z), \: p \in \sigma(z,v), \: (z,v) \in SF^{int}\}^{int}. \end{equation} Let $(U,x)$ be a local coordindate chart of $M^{int}$. We denote the Christoffel symbols of $G$ and $\widetilde G$ as $\Gamma$ and $\widetilde \Gamma$, respectively. By \eqref{eq:image_of_geo_3}, \eqref{eq:geodesic_in_M} and Proposition 2.13 of \cite{LaSa} there exists a smooth $1$--form $\beta$ on $U$ such that $$ \Gamma^k_{ij}(x)-\widetilde \Gamma^k_{ij}(x)=\delta^k_i\beta_j(x)+\delta^k_j\beta_i(x), $$ where $\delta^k_j$ is the Kronecker delta. This and Lemma 2.14 of \cite{LaSa} imply that the geodesics of metric tensors $G$ and $\widetilde G$ agree up to reparametrization. See also \cite{matveev2012geodesically} for the similar result. We arrive at. \begin{lemma} Suppose that $N, F, G$ and $\widetilde G$ are as above. Then $G=\widetilde{G}$ in all of $N$. \label{Le:geodesic eq -> metrics are the same} \end{lemma} \begin{proof} Since geodesics of metric tensors $G$ and $\widetilde G$ agree up to reparametrization the main result of \cite{topalov2003geodesic} shows that the function \begin{equation} \label{Matveev formula} I_0((x,v))=\bigg(\frac{\det (G(x)) }{\det(\widetilde{G}(x))}\bigg)^{\frac{2}{n+1}} \widetilde{G}(x,v), \quad (x,v) \in TN, \end{equation} where $\widetilde {G}(x,v)=\widetilde{G}_{jk}(x)v^jv^k$, is constant on the geodesic flow of $G$. Note that the function $F(x):= \frac{\text{det} (G(x)) }{\text{det}(\widetilde{G}(x))}$ is coordinate invariant. Let $\varphi_t:SN \to SN$, $t\in {\mathbb R}$ be the geodesic flow of $G$ and $\pi:TN \to N$ the projection onto the base point. Since $G=\widetilde G$ on $F^{int}$, we have $$ G(\varphi_0(z,v)) =\|v\|^2_G=I_0(\varphi_0(z,v), \quad (z,v) \in TF^{int} . $$ Therefore for any $t \in {\mathbb R}$ and for any $(z, v) \in TF^{int} \setminus \{0\}$ the following holds $$ G(\varphi_t(z,v)) =\|v\|^2_G=I_0(\varphi_t(z,v)=F(\pi(\varphi_t(z,v))\widetilde G(\varphi_t(z,v)). $$ This implies the claim. For more details, see Lemma 2.15 of \cite{LaSa}. \end{proof} We conclude that the proof of Theorem \ref{th:main} follows from Propositions \ref{th:topology}, \ref{th:diffeo} and Lemma \ref{Le:geodesic eq -> metrics are the same}. \end{document}

\begin{document} \title{On the negativity of random pure states} \author{Animesh Datta} \email{[email protected]} \affiliation{Institute for Mathematical Sciences, 53 Prince's Gate, Imperial College, London, SW7 2PG, UK} \affiliation{QOLS, The Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2BW, UK} \date{\today} \begin{abstract} This paper deals with the entanglement, as quantified by the negativity, of pure quantum states chosen at random from the invariant Haar measure. We show that it is a constant ($0.72037$) multiple of the maximum possible entanglement. In line with the results based on the concentration of measure, we find evidence that the convergence to the final value is exponentially fast. We compare the analytically calculated mean and standard deviation with those calculated numerically for pure states generated via pseudorandom unitary matrices proposed by Emerson \emph{et. al.} [Science, \textbf{302}, 3098, (2003)]. Finally, we draw some novel conclusions about the geometry of quantum states based on our result. \end{abstract} \pacs{03.67.Mn, 02.30.Gp} \maketitle \section{Introduction} Entanglement has come to be believed as one of the cornerstones of quantum information science. The necessity of entanglement in quantum computation~\cite{jl03} and information~\cite{masanes06a,pw09} tasks are well acknowledged. Substantial amounts of experimental effort is expended in the generation and manipulation of quantum entanglement. Nevertheless, the role of entanglement in quantum information science in general, and quantum computation in particular, is far from clear. Meyer has presented a version of the quantum search algorithm that requires no entanglement~\cite{meyer00a}, and instances are known of mixed-state quantum computation where exponential speedup is attained in the presence of only limited amounts of entanglement~\cite{dfc05}, and other quantities have been proposed as alternate resources for the speedup~\cite{dsc08,dg09}. Recent results have further illuminated the role of entanglement in pure-state quantum computation. It was already known, due to the Gottesman-Knill theorem~\cite{nielsen00a}, that entanglement is by no means sufficient for universal quantum computation. The new results~\cite{bmw09,gfe09} show that, in fact, almost all pure states are too entangled to be a universal resource for quantum computation. Though proved in the context to measurement-based quantum computation, and based on the geometric measure of entanglement, which is the absolute square of the inner product with the closest product state, these results drive home the point that implications on the lines of ``more entanglement implies more computational power" are fallacious~\cite{gfe09}. The strategy employed for proving these results can generally be termed as ``concentration of measure"~\cite{hlw06}, by which a typical pure state, chosen at random from the left- and right-invariant Haar measure, is almost always maximally entangled across any bipartition. Arguments based on the concentration of measure have been used to obtain average value of measures of correlations and entanglement in typical quantum states. Concentration of measure is a very powerful concept from measure theory, which puts bounds on how much the values of certain smooth (Lipshitz) functions can vary from their mean value. This is a consequence of the remarkable fact that the uniform distribution of the $k$-sphere $\mathbb{S}^k$ is concentrated largely on the equator for large $k$, and any polar cap smaller than a hemisphere has a relative volume exponentially small in $k$. Examples in quantum information theory include the entropy of the reduced density matrix, entanglement of formation, distillable common randomness~\cite{hlw06}. The entropy of reduced density matrices of typical states has also been conjectured and calculated independently~\cite{page93,s96,fk94,ruiz95}, as has been their concurrence, purity and the linear entropy~\cite{scott03}. Not much is however known of one of the most common and computable measures of entanglement, the negativity~\cite{zhsl98,vw02}, in random Haar distributed pure states. In this paper, our endeavor will be to address this question. We show that the negativity of a random pure state taken from a Haar distribution, is a constant multiple of the maximum possible. This entanglement can also be generated efficiently using two qubit gates~\cite{odp07}. We will evaluate this constant using techniques similar to those in Refs.~\cite{s96,scott03}, and confirm our results numerically using efficiently generated pseudorandom unitaries~\cite{emerson03a}. For simplicity, we will only present results for equal bipartitions, but the extensions to unequal splits is straightforward. That the negativity (defined in Eq. (\ref{E:neg})) of random pure states is less than maximal might seem to contradict the statement that random pure states in large enough Hilbert spaces are close to being maximally entangled. This is, however, not true in general. As shown in~\cite{hlw06}, for a state residing in a Hilbert space of dimension $d_A \times d_B$ with a reduced state $\rho_A= {\rm{Tr }}_B(\rho),$ and $d_B$ is a large enough multiple of $d_A\log d_A/\epsilon^2,$ then \begin{equation} \label{E:cluster} (1-\epsilon)\frac{1}{d_A}\mathbb{I} \leq \rho_A \leq (1+\epsilon)\frac{1}{d_A}\mathbb{I} \end{equation} If a state satisfies Eq.~(\ref{E:cluster}), then its negativity is evidently near-maximal. But as the condition for its validity shows, this is only true when the bipartite split is quite asymmetrical. Thus, for equal bipartite splits, which is often of interest in quantum information science, there is no \emph{a priori} reason to expect the negativity of random pure states to be close to maximal. This is the case we study here. Just to highlight the degree of asymmetry needed to have the negativity close to maximal, for $\epsilon = 0.1$ and $d_A=2,$ we require $d_B \gg 200,$ and for $d_A=16,$ $d_B \gg 6400.$ The outline of the paper is as follows. In Sec.~\ref{sec:mean}, we begin by deriving the expression of average negativity. It involves performing integrations over the probability simplex which are rewritten in terms of other nonconstrained variables, finally leaving us with a combination of hypergeometric functions. Sec.~\ref{sec:Var} derives the expressions for the variance in the negativity in terms of similar hypergeometric functions. These functions are explicitly evaluated in Sec.~\ref{sec:evals} numerically. This is necessary as the series we have is provably not summable in closed form, which we discuss in brief in Appendix~\ref{app:digress}. We obtain the final expression for the average negativity of Haar-distributed random pure states. We also compare our results with a numerical simulation using pseudorandom pure states generated from efficiently generated pseudorandom unitaries~\cite{emerson03a}, finding good agreement. We finally conclude in Sec.~\ref{sec:conclude} with discussions about the ramifications of our finding on the geometry of the set of quantum states. We also discuss the prospect of extending the present analysis to random mixed quantum states. \section{Negativity of typical pure states} \label{sec:mean} The negativity is an entanglement monotone which is based on the partial transpose test of detecting entanglement~\cite{p96}. Given a bipartite quantum state residing in $\mathcal{H}_A\otimes \mathcal{H}_B$ with dimensions $\mu$ and $\nu$, called $\rho_{AB},$ the negativity is defined as \begin{equation} \label{E:neg} \mathcal{N}(\rho_{AB}) = \frac{||\rho_{AB}^{T_A}||-1}{2}, \end{equation} where $\rho_{AB}^{T_A}$ denotes the partial transpose with respect to subsystem $A$, and $||\sigma||$ denotes the trace norm, or sum of the absolute values of the eigenvalues of $\sigma,$ when $\sigma$ is Hermitian, as is the case with $\rho_{AB}^{T_A}.$ For pure states residing in the above space, it is always possible to write a Schmidt decomposition~\cite{nielsen00a}. This paper will only deal with the scenario $\mu=\nu$, the extension to the unequal case being tedious, but straightforward. The distribution of the Schmidt coefficients is given by (for $\mu = \nu$)~\cite{lp88} \begin{equation} P(\mathbf{p}) \mathrm{d}\mathbf{p} = N \delta(1-\sum_{i=1}^\mu p_i) \prod_{1\leq i < j \leq \mu} (p_i-p_j)^2 \prod_{k=1}^\mu \mathrm{d}\mathbf{p}_k, \end{equation} where $\delta(\cdot)$ is the Dirac delta function. The negativity for pure states is \begin{equation} \mathcal{N} = \frac{1}{2}\left[\left(\sum_{i=1}^\mu\sqrt{p_i}\right)^2-1\right] =\frac{1}{2}\mathop{\sum_{i,j=1}}_{i\neq j}^{\mu}\sqrt{p_i p_j} \label{E:negdef} \end{equation} and its mean is given by \begin{equation} \avg{\mathcal{N}}= \frac{1}{2} \int \mathop{\sum_{i,j=1}}_{i\neq j}^{\mu}\sqrt{p_i p_j}P(\mathbf{p}) \mathrm{d}\mathbf{p}. \end{equation} At the outset, it helps to change variables such that $q_i=r p_i$ which removes the hurdle of integrating over the probability simplex~\cite{s96,scott03}, whereby \begin{equation} Q(\mathbf{q})\mathrm{d}\mathbf{q}\equiv \prod_{1\leq i<j\leq \mu}\left(q_i-q_j\right)^2 \prod_{k=1}^\mu e^{-q_k}\,\mathrm{d}q_k \label{Q}\\ =N\,e^{-r}r^{\mu^2-1}P(\mathbf{p})\,\mathrm{d}\mathbf{p}\,\mathrm{d}r\;. \end{equation} The new variables $q_i$ take on values independently in the range $[0,\infty),$ and $r$ is a scaling factor given by $r =\sum_iq_i$. Integrating over all the values of the new variables, we find that the normalization constant is given by $N=\overline Q/\Gamma(\mu\nu)$, where $\overline{Q}\equiv\int Q(\mathbf{q})d\mathbf{q}$. Similarly, we find that \begin{equation} \int \sqrt{q_i q_j}Q(\mathbf{q})\mathrm{d}\mathbf{q} = \overline Q\, \frac{\Gamma(\mu^2+1)}{\Gamma(\mu^2)} \int \sqrt{p_i p_j}P(\mathbf{p})\,\mathrm{d}\mathbf{p}\;, \label{QtoP} \end{equation} with $\Gamma(\mu)=(\mu-1)!.$ Notice that the first product in Eq.~(\ref{Q}) is the square of the Van der Monde determinant~\cite{s96,scott03} \begin{equation} \hspace{-2.0cm} \Delta(\mathbf{q}) \,\equiv\, \prod_{1\leq i<j\leq \mu}\left(q_i-q_j\right) = \left| \begin{array}{ccc} 1 & \ldots & 1 \\ q_1 & \ldots & q_\mu \\ \vdots & \ddots & \vdots \\ q_1^{\mu-1} & \ldots & q_\mu^{\mu-1} \end{array} \right| = \left| \begin{array}{ccc} L_0(q_1) & \ldots & L_0(q_\mu) \\ L_1(q_1) & \ldots & L_1(q_\mu) \\ \vdots & \ddots & \vdots \\ \Gamma(\mu) L_{\mu-1}(q_1) & \ldots & \Gamma(\mu)L_{\mu-1}(q_\mu) \end{array} \right|\;. \label{Van2} \end{equation} The second determinant in Eq.~(\ref{Van2}), follows from the basic property of invariance after adding a multiple of one row to another, and the polynomials $L_k(q)$ judiciously chosen to be Laguerre polynomials~\cite{gradshteyn}, satisfying the orthogonality relation \begin{equation} \int_0^\infty dq\,e^{-q} L_k(q)L_l(q) = \delta_{kl}\;. \label{L1} \end{equation} These facts in hand, we can evaluate \begin{eqnarray} \overline{Q} &=& \int\Delta(\mathbf{q})^2\prod_{k=1}^\mu e^{-q_k}\,dq_k \nonumber\\ &=&\mathop{\sum_{T,R \in \mathcal{S}_{\mu}}} (-1)^{T+R}\prod_{k=1}^\mu\Gamma(T(k))\Gamma(R(k))\int dq_k\,e^{-q_k} L_{T(k)-1}(q_{k})L_{R(k)-1}(q_k) \nonumber \\ &=&\sum_{R \in \mathcal{S}_{\mu}}(1)^R\prod_{k=1}^\mu\Gamma(R(k))^2=\mu!\prod_{k=1}^\mu\Gamma(k)^2\;, \end{eqnarray} with $T,R$ being elements of the permutation group on $\mu$ elements $\mathcal{S}_{\mu}.$ We can now calculate the integral over $\{q_1,\cdots,q_{\mu}\}$ in Eq. (\ref{QtoP}) as \begin{eqnarray} &&\mathop{\sum_{i,j=1}}_{i\neq j}^\mu \int \sqrt{q_i q_j}Q(\mathbf{q})\,\mathrm{d}\mathbf{q} \nonumber\\ &&=\mathop{\sum_{i,j=1}}_{i\neq j}^\mu \int \sqrt{q_i q_j}\prod_{m=1}^{\mu} dq_m e^{-q_m}\sum_{T,R \in \mathcal{S}_{\mu}}(-1)^{T+R}\prod_{m=1}^{\mu}\Gamma(T(m))\Gamma(R(m))L_{T(m)-1}(q_m)L_{R(m)-1}(q_m) \nonumber \\ &&=\overline{Q}\mathop{\sum_{k,l=0}}^{\mu-1}\sum_{R \in \mathcal{S}_2}(-1)^{R}\int\sqrt{q_k q_l}L_{R(k)-1}(q_k)L_{k-1}(q_k)L_{R(l)-1}(q_l)L_{l-1}(q_l)e^{-q_k-q_l}dq_kdq_l \nonumber\\ &&=\overline{Q}\sum_{k,l=0}^{\mu-1}\sum_{R \in \mathcal{S}_2} (-1)^{R}I_{k,R(k)}^{(1/2)}I_{l,R(l)}^{(1/2)}\nonumber\\ &&= \overline{Q} \sum_{k,l=0}^{\mu-1}\left|\begin{array}{cc} I_{kk}^{(1/2)} & I_{kl}^{(1/2)} \\ I_{lk}^{(1/2)} & I_{ll}^{(1/2)} \\ \end{array} \right|, \end{eqnarray} where \begin{equation} I_{kl}^{(\beta)} \equiv \int_0^{\infty}e^{-q}q^{\beta}\,L_k(q)L_l(q)\;\mathrm{d}q, \label{E:Int} \end{equation} $|\cdot|$ is the determinant and we have used the orthonormality condition in Eq. (\ref{L1}) in the first step of the evaluation. We thus have \begin{equation} \label{E:avgneg} \avg{\mathcal{N}}=\frac{1}{2\mu^2}\sum_{k,l=0}^{\mu-1}\left[ I_{kk}^{(1/2)}I_{ll}^{(1/2)}-\left(I_{kl}^{(1/2)}\right)^2\right]\;, \end{equation} except that the integral needs to be evaluated. \section{Variance in the negativity} \label{sec:Var} Having calculated the mean of the negativity for random, Haar distributed pure states, we move on to calculate its variance. Based on the definition of negativity in Eq. (\ref{E:negdef}), we obtain the expression for the variance of the negativity as ($\sigma$ is the standard deviation) \begin{equation} \sigma^2 = \frac{1}{4}\left[\avg{\left(\sum_{i=1}^\mu\sqrt{p_i}\right)^4} - \avg{\left(\sum_{i=1}^\mu\sqrt{p_i}\right)^2}^2 \right]. \end{equation} The second term has already been evaluated in the previous section, so we need con concern ourselves with the first term. We begin by expanding the fourth power above as \begin{equation} \hspace{-2.5cm} \left(\sum_{i=1}^\mu\sqrt{p_i}\right)^4=1+ 2\mathop{\sum_{i,j=1}}_{i\neq j}^{\mu} \sqrt{p_ip_j} + 2\mathop{\sum_{i,j=1}}_{i\neq j}^{\mu}p_ip_j + 4\mathop{\sum_{i,j,k=1}}_{i\neq j\neq k}^{\mu} p_i\sqrt{p_jp_k} + \mathop{\sum_{i,j,k,l=1}}_{i\neq j\neq k \neq l}^{\mu} \sqrt{p_ip_jp_kp_l}. \end{equation} Each of these terms can now be individually evaluated, and omitting the details we just present the results as \begin{eqnarray} \mathop{\sum_{i,j=1}}_{i\neq j}^{\mu}p_ip_j &=& \frac{1}{\mu^2(\mu^2+1)}\sum_{k,l=0}^{\mu-1}\left|\begin{array}{cc} I_{kk}^{(1)} & I_{kl}^{(1)} \\ I_{lk}^{(1)} & I_{ll}^{(1)} \\ \end{array} \right|, \\ \mathop{\sum_{i,j,k=1}}_{i\neq j\neq k}^{\mu} p_i\sqrt{p_jp_k} &=& \frac{1}{\mu^2(\mu^2+1)}\sum_{k,l,m=0}^{\mu-1} \left| \begin{array}{ccc} I_{kk}^{(1)} & I_{kl}^{(1/2)} & I_{km}^{(1/2)} \\ I_{lk}^{(1)} & I_{ll}^{(1/2)} & I_{lm}^{(1/2)} \\ I_{mk}^{(1)} & I_{ml}^{(1/2)} & I_{mm}^{(1/2)} \\ \end{array}\right|, \\ \mathop{\sum_{i,j,k,l=1}}_{i\neq j\neq k \neq l}^{\mu}\sqrt{p_ip_jp_kp_l}&=& \frac{1}{\mu^2(\mu^2+1)}\sum_{k,l,m,n=0}^{\mu-1} \left| \begin{array}{cccc} I_{kk}^{(1/2)} & I_{kl}^{(1/2)} & I_{km}^{(1/2)} & I_{kn}^{(1/2)} \\ I_{lk}^{(1/2)} & I_{ll}^{(1/2)} & I_{lm}^{(1/2)} & I_{ln}^{(1/2)} \\ I_{mk}^{(1/2)} & I_{ml}^{(1/2)} & I_{mm}^{(1/2)} & I_{mn}^{(1/2)} \\ I_{nk}^{(1/2)} & I_{nl}^{(1/2)} & I_{nm}^{(1/2)} & I_{nn}^{(1/2)} \\ \end{array} \right|\!. \end{eqnarray} \section{Evaluating the integrals} \label{sec:evals} Having derived formal expressions for the mean and standard deviation of the negativity of a random pure state, we now need to evaluate the integral in Eq. (\ref{E:Int}). To that end, we use the generating function for Laguerre polynomials~\cite{gradshteyn} \begin{equation} (1-z)^{-1} e^{xz/z-1} = \sum_{l=0}^{\infty} L_l(x)z^l\;\;\;\;\;\;\;\; |z|\leq1, \end{equation} and \begin{equation} \hspace{-2.0cm}\int_0^{\infty}e^{-st}t^{\beta}\,L_n^{\alpha}(t)\;\mathrm{d}t= \frac{\Gamma(\beta+1)\,\Gamma(\alpha+n+1)}{n!\,\Gamma(\alpha+1)}s^{-\beta-1}F\left(-n,\beta+1;\alpha+1,\frac{1}{s}\right), \end{equation} $F$ being the hypergeometric function such that \begin{equation} F(a,b;c;z)= \sum_{n=0}^{\infty} \frac{(a)_n\,(b)_n}{(c)_n}\frac{z^n}{n!}, \end{equation} and $(a)_n = a(a+1)(a+2)...(a+n-1)$ is the Pochhammer symbol. Note that if $a$ is a negative integer, $(a)_n = 0$ for $n > |a|$ and the hypergeometric series terminates. Then, \begin{eqnarray} \sum_{l=0}^{\infty} I_{kl}^{(\beta)} z^l &=&\int_0^{\infty}e^{-x}x^{\beta}\,L_k(x)(1-z)^{-1} e^{xz/z-1}\;\mathrm{d}x \nonumber\\ &=& s\int_0^{\infty}e^{-sx}x^{\beta}\,L_k(x)\;\mathrm{d}x\;\;\;\;\;\;\;\;\;\;\;\;\;\;s=1/(1-z) \nonumber\\ &=&s^{-\beta}\Gamma(\beta+1)F\left(-k,\beta+1;1;\frac{1}{s}\right)\nonumber\\ &=&\Gamma(\beta+1)\sum_{t=0}^{k}\frac{(-k)_t\,(\beta+1)_t}{(1)_t}\frac{1}{t!}(1-z)^{t+\beta}\nonumber\\ &=&\Gamma(\beta+1)\sum_{l=0}^{\infty}\sum_{t=0}^{k}\frac{(-1)^l}{l!}\frac{(-k)_t\,(\beta+1)_t}{(t!)^2}(t+\beta)_{\underline{l}}\,z^l, \end{eqnarray} whereby \begin{equation} I_{kl}^{(\beta)}=\Gamma(\beta+1)\frac{(-1)^l}{l!}\sum_{t=0}^{k}\frac{(-k)_t\,(\beta+1)_t}{(t!)^2}(t+\beta)_{\underline{l}}\;, \end{equation} and $(a)_{\underline{n}}=a(a-1)(a-2)...(a-n+1)$ is the `falling factorial'. Using the following identities for the Pochhammer symbols \begin{eqnarray} (x)_{\underline{n}}&=&(-1)^n (-x)_n,\\ (-x)_{n}&=& (-1)^n(x-n+1)_n, \\ (x)_n &=& \Gamma(x+n)/\Gamma(x), \end{eqnarray} we have \begin{eqnarray} \label{E:Ikl} I_{kl}^{(\beta)}&=&\frac{(-1)^l}{l!}\sum_{t=0}^{k} \left(\begin{array}{c} k \\ t \\ \end{array} \right)\frac{[\Gamma(t+\beta+1)]^2}{t!\,\Gamma(t-l+\beta+1)}\nonumber\\ &=&\frac{(-1)^l}{l!}\frac{\Gamma(1+\beta)^2}{\Gamma(1+\beta-l)} \; _3F_2\left(\{\beta+1,\beta+1,-k\};\{1,\beta+1-l\};1\right). \end{eqnarray} To get the final expression for the negativity in Eq~(\ref{E:avgneg}), we substitute the expression for the integrals from Eq~(\ref{E:Ikl}). The expressions are not very illuminating, and for the lack of an asymptotic expression, we present the numerical values in Table~(\ref{T:t1}), and plot them in Fig.~(\ref{negativitylimit}). See Appendix~\ref{app:digress} for a note on the summability of the series. Anticipating a scaling in proportion to that of a maximally entangled state, we divide the mean expressed in Eq~(\ref{E:avgneg}) by the maximum possible negativity of a $\mu\times\mu$ system as $\mathcal{N}_{max}=(\mu-1)/2.$ As can be seen from Table~(\ref{T:t1}), the average value of the negativity saturates to a constant multiple of the maximum possible. This constant is found numerically, and in the asymptotic limit of large $n$, the negativity for an equal bipartition of a randomly chosen Haar-distributed pure state is \begin{equation} \label{E:result} \avg{\mathcal{N}} \sim 0.72037\left(\frac{2^{n/2}-1}{2}\right). \end{equation} Though we have not proven this analytically, it is easily seen that the convergence is exponential. This can be concluded from the last column in the table, which shows the difference in the successive values of the third column. The value of $\Delta$ is progressively halved as the number of qubits $n$ goes up, and this shows that the negativity indeed saturates monotonically, and arguably, exponentially fast, to the value presented above. This is to be expected from the concentration of measure results~\cite{hlw06}, which means that the negativity of random states in large enough Hilbert spaces is close to their expectation value. \begin{table} \begin{center} \begin{tabular}{c|c|r@{.}l|r@{.}l} \hline $n$ & $\mu$ & \multicolumn{2}{c|}{$\avg{\mathcal{N}}/\mathcal{N}_{max}$} & \multicolumn{2}{c}{$\Delta$}\\ \hline \hline 2 & 2 & 0&589049 \\ 4 & 4 & 0&65368 & 0&0646309 \\ 6 & 8 & 0&686614 & 0&0329346 \\ 8 & 16 & 0&703378 & 0&0167641 \\ 10 & 32 & 0&711878 & 0&0084994 \\ 12 & 64 & 0&716171 & 0&0042932 \\ 14 & 128 & 0&718332 & 0&0021611 \\ 16 & 256 & 0&719417 & 0&0010851 \\ 18 & 512 & 0&719961 & 0&0005439 \\ 20 & 1024 & 0&720233 & 0&0002724 \\ 22 & 2048 & 0&72037 & 0&0001366 \\ \hline \end{tabular} \end{center} \caption{Ratio of the negativity of random pure states to the maximal negativity for Haar-distributed states of $n$ qubits. For an equipartition of $n$ qubit states, $\mu=2^{n/2}.$ $\Delta$ is the difference between successive values in the third column, providing evidence for an exponential convergence of $\avg{\mathcal{N}}/\mathcal{N}_{max}$ with $n$.} \label{T:t1} \end{table} \begin{figure}\label{negativitylimit} \end{figure} \subsection{Numerical verification} \begin{figure} \caption{Distribution of the negativity of $100000$ pseudorandom pure states, with $n=4$ (Left) and $n=8$ (Right). The pseudorandom unitaries used were generated via the techniques of \cite{emerson03a}, with $j=40$ interactions applied for each unitary. Also plotted is the gaussian distribution function with just the first two moments, as given by Eq.~(\ref{E:distfn}), as well as the analytically calculated mean (solid vertical line) and the standard deviations (dashed vertical lines). Although the convergence of the pseudorandom construction of~\cite{emerson03a} to the Haar measure is not obvious, it has been shown to do so~\cite{ell05,dop07}.} \label{F:numerics} \end{figure} As a final corroboration of our results, we test our calculations against numerically generated pure states. These are pseudorandom rather than random Haar-distributed. They are generated by applying pseudorandom unitaries presented in Ref.~\cite{emerson03a} on fiducial pure states. The negativity of these pure states is calculated and plotted as a histogram in Fig.~(\ref{F:numerics}). We compare this to an approximation of the cumulant generating function, and the probability distribution function for the negativity itself $P(\mathcal{N})d\mathcal{N}$, given by \begin{eqnarray} P(\mathcal{N})d\mathcal{N}&=&\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega \exp\left(-i\mathcal{N}\omega + \frac{\avg{\mathcal{N}}}{\mathcal{N}_{max}}i\omega + \frac{\sigma}{\mathcal{N}_{max}}\frac{(i\omega)^2}{2!}\right)d\mathcal{N} \nonumber\\ &=& \frac{1}{\sqrt{2\pi\sigma'^2}}e^{-\left(\mathcal{N}-\mathcal{N'}\right)^2/2\sigma'^2}d\mathcal{N}, \label{E:distfn} \end{eqnarray} where $\mathcal{N'}=\avg{\mathcal{N}}/\mathcal{N}_{max}$ and $\sigma'=\sigma/\mathcal{N}_{max}.$ As is evident from Fig.~(\ref{F:numerics}), the distributions are very localized, and the gaussian distribution seems quite apt. \section{Concluding Discussions} \label{sec:conclude} The negativity provides upper bounds on the teleportation capacity of a state, and its distillability, the latter via the logarithmic negativity. It is in these two contexts that our results on the negativity provides new insights, not achieved through other measures. To address the teleportation capacity, the singlet distance was introduced in~\cite{vw02}. It is defined as closest distance any quantum state can get to the singlet (the ideal resource for teleportation) while undergoing only local operations. Mathematically, \begin{equation} \Delta(\ket{\Phi},\rho) = \inf_P||\proj{\Phi}-P(\rho)||_1 \end{equation} where $P$ is the set of all local protocols, and $\ket{\Phi}$ is the singlet residing in $\mathbb{C}^m\otimes \mathbb{C}^m$. Note that in our case $m=2^{n/2} = 2\mathcal{N}_{max}+1.$ The following result, also proved in~\cite{vw02} \begin{equation} \Delta(\ket{\Phi},\rho) \geq 2\left(1-\frac{2\mathcal{N}(\rho)+1}{m}\right) \end{equation} then immediately leads to the conclusion that a pure quantum state $\ket{\psi}$, chosen at random from the Haar measure, will with high probability have a singlet distance given (all $\approx$ signs here and henceforth apply to large $n$) \begin{equation} \label{E:singletbnd} \Delta(\ket{\Phi},\ket{\psi}) \geq 2\left(1-\frac{2\avg{\mathcal{N}}+1}{2^{n/2}}\right) \approx 2\left(1-\frac{\avg{\mathcal{N}}}{\mathcal{N}_{max}}\right) \approx 0.55926, \end{equation} where we have used Eq. (\ref{E:result}), which is that $\frac{\avg{\mathcal{N}}}{\mathcal{N}_{max}} = 0.72037 =c < 1.$ This gives us a nontrivial lower bound on how close a typical pure state can be taken to a singlet by purely local operations. This can be recast in terms of an upper bound on the teleportation fidelity~\cite{hhh99,vw02} of random pure states as \begin{equation} \label{E:fid} f_{opt} \equiv \max_{P}\bra{\Phi}P(\proj{\psi})\ket{\Phi} \leq \frac{2\avg{\mathcal{N}}+1}{m} \lesssim \frac{\avg{\mathcal{N}}}{\mathcal{N}_{max}} \approx 0.72037. \end{equation} Another application of our result can be found by using the logarithmic negativity~\cite{p05} as an upper bound on the entanglement of distillation $E_D(\rho)$. It was shown~\cite{vw02} that \begin{equation} E_D(\rho) \leq E_{\mathcal{N}}(\rho) \end{equation} where $E_\mathcal{N}(\rho) = \log_2||\rho^{T_A}||_1 = \log_2(2\mathcal{N}(\rho)+1).$ Using this, we get (where c = 0.72037, as after Eq. (\ref{E:singletbnd})) \begin{equation} \avg{||\rho^{T_A}||_1} = c\;2^{n/2} + 1-c, \end{equation} whereby for a pure state $\ket{\psi}$ chosen at random from the Haar measure, we can set the upper bound of distillable entanglement to be \begin{equation} \label{E:distent} E_D(\ket{\psi}) \leq \log_2\left(\avg{||\rho^{T_A}||_1}\right) \approx \frac{n}{2} +\log_2c. \end{equation} For the constant we present in this work, this provides us with a bound that is tighter by about half an ebit ($\log_2c \approx -0.47319$). Also note that we have taken a logarithm of the average, which is always greater than or equal to the average of the logarithm. In addition to the obvious conclusions that the fidelity of teleportation and distillability of random pure states have nontrivial upper bounds, the above two mathematical results tell us a few things about the structure of the set of pure quantum states in general. Firstly, although a random pure state is very likely to be highly entangled (close to maximal), it is in no way close to the singlet state, at least in trace norm. This means that a nonzero fraction of these ``close to maximally entangled" states contain inequivalent types of entanglement which are not related by SLOCC operations to the canonical maximally entangled (singlet) state. A second, and probably stronger statement is that not only do random pure states lie in different inequivalent sets of maximally entangled states, but also that some of these classes have a greater ability to retain their entanglement under distillation protocols than others, resulting thereby in an overall lower distillation rate. This paper shows that the negativity of $n$-qubit random pure states chosen from the Haar measure is a constant multiple of maximum possible negativity, which goes as $2^{n/2}$ for an equal bipartition of the state. We also provide evidence that the convergence to the asymptotic value is monotonic and exponentially fast. The value of the constant was not evaluated in closed form, and we showed why this was the case. The expression for the negativity is a sum of hypergeometric terms, and the techniques of creative telescoping show that our particular series in not summable. Finally, we show that the results of our analytic calculation are borne out by random states generated by applying pseudorandom unitaries on fiducial states. We also show that probability distribution for the negativity is well approximated by a gaussian distribution whose mean and variance we obtain analytically. One issue that we have not addressed here is the extension of the above calculation to random quantum states that are mixed. This is made somewhat challenging by the fact that there does \emph{not} exist a unique measure on the space of mixed quantum states. Since any pure state can be generated by applying a unitary matrix on a fiducial state, a unique measure on the space of pure states can be derived from that on the space of unitary matrices, which is the rotationally invariant Haar measure. Mixed quantum states cannot be generated in a likewise manner, and therefore, it is not possible to capture the distribution of mixed states via the Haar measure. However, any mixed state can be diagonalized by a unitary matrix, and this motivates a product measures on the space of mixed states $\mathcal{M},$ which can be defined as $\mathcal{M}=\mathcal{E}\times P,$ where $P$ is the usual Haar measure that captures the distribution of eigenvectors of the states. $\mathcal{E}$ is meant to capture the distribution of eigenvalues, and there is no unique way of doing that. Attempts have been made~\cite{zs01}, and the mean entanglement, as quantified, for instance, by the purity has been calculated, as has been the logarithmic negativity for tripartite states using minimal purifications~\cite{dop07}. The calculation of the negativity for states of this form will be the subject of a future publication. This will provide us with information about the typical entanglement(negativity) content of random mixed states, which are more and more likely to be encountered as we move closer to realistic implementations of quantum technology. \appendix \section{A mathematical digression} \label{app:digress} The final expression for the negativity, though seemingly compact, is, in fact a sum of exponentially many terms. This retards the evaluation of the quantities in Table~(\ref{T:t1}) drastically, unless a closed form is found for quantity in Eq. (\ref{E:Ikl}). Consequently, it would not only be interesting, but indeed essential to have a closed form of the above expression. For some special instances of $k,l$ and $\beta$, this is possible. Unfortunately, this is not possible for general values of $k$ and $l$ (this paper deals only with $\beta = 1/2, 1$). In fact, it can be shown that there exists \emph{no} closed form solution for the sum in Eq. (\ref{E:Ikl}). The arguments leading to this `tragic' conclusion are presented next. \begin{theorem}[Zeilberger's algorithm or the method of creative telescoping~\cite{pwz97}] Let $F(n,k)$ be a proper hypergeometric term. Then F satisfies a nontrivial recurrence of the form $$ \sum_{j=0}^J a_j(n)F(n+j,k)=G(n,k+1)-G(n,k), $$ in which $G(n,k)/F(n,k)$ is a rational function of n and k. \end{theorem} That this theorem applies to the sum we have at hand is evident. The application of this algorithm to the expression in Eq.~(\ref{E:Ikl}) yields third order recurrences which can be solved using the Gosper-Petkov\v{s}ek algorithm~\cite{pwz97,petkovsek}. This algorithm (also called \texttt{Hyper}~\cite{footnote}) provides a complete solution to the problem in the sense that it either provides all the solution to the recurrence problem. On the other hand, the failure of the algorithm to come up with a solutions proves that the initial series \emph{cannot} be summed into a closed form. It is the latter that happens in our case, thereby proving that the series in Eq.~(\ref{E:Ikl}) is not summable in closed form. \section*{Acknowledgments} It is a pleasure to thank Colston Chandler, Anil Shaji, Adolfo del Campo and Miguel Navascu\'{e}s for several interesting discussions during the course of this work, and Martin B. Plenio for several comments on the manuscript. AD was supported by EPSRC (Grant No. EP/C546237/1), EPSRC QIP-IRC and the EU Integrated Project (QAP). \end{document}

\begin{document} \title{\Large Je\'{s}manowicz' conjecture and Fermat numbers} \author{\large Min Tang\thanks{Corresponding author. This work was supported by the National Natural Science Foundation of China, Grant No.10901002 and Anhui Provincial Natural Science Foundation, Grant No.1208085QA02. Email: [email protected]} and Jian-Xin Weng } \date{} \maketitle \vskip -3cm \begin{center} \vskip -1cm { \small \begin{center} School of Mathematics and Computer Science, Anhui Normal University, \end{center} \begin{center} Wuhu 241003, China \end{center} } \end{center} {\bf Abstract.} Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Je\'{s}manowicz conjectured that for any positive integer $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$. {\bf Keywords:} Je\'{s}manowicz' conjecture; Diophantine equation; Fermat numbers 2010 {\it Mathematics Subject Classification}: 11D61 \section{Introduction} Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}$ with $2\mid b.$ Clearly, for any positive integer $n$, the Diophantine equation \begin{equation}\label{eqn1}(na)^{x}+(nb)^{y}=(nc)^{z}\end{equation} has the solution $(x, y, z)=(2,2,2).$ In 1956, Sierpi\'{n}ski \cite{Sierpinski} showed there is no other solution when $n=1$ and $(a,b,c)=(3,4,5)$, and Je\'{s}manowicz \cite{Jesmanowicz} proved that when $n=1$ and $(a,b,c)=(5,12,13),(7,24,25),(9,40,41),(11,60,61),$ Eq.(\ref{eqn1}) has only the solution $(x,y,z)=(2,2,2).$ Moreover, he conjectured that for any positive integer $n,$ the Eq.(\ref{eqn1}) has no positive integer solution other than $(x,y,z)=(2,2,2).$ Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be a Fermat number. Recently, the first author of this paper and Yang \cite{Tang} proved that if $1\leq k\leq 4$, then the Diophantine equation \begin{equation}\label{eqn2}((F_k-2)n)^{x}+(2^{2^{k-1}+1}n)^{y}=(F_kn)^{z}\end{equation} has no positive integer solution other than $(x,y,z)=(2,2,2)$. For related problems, see (\cite{Deng}, \cite{Miyazaki}, \cite{Miyazaki2}). In this paper, we obtain the following result. \begin{theorem}\label{thm1} For any positive integer $n$ and Fermat number $F_k$, Eq.(\ref{eqn2}) has only the solution $(x,y,z)=(2,2,2)$. \end{theorem} Throughout this paper, let $m$ be a positive integer and $a$ be any integer relatively prime to $m$. If $h$ is the least positive integer such that $a^{h}\equiv 1 \pmod m$, then $h$ is called the order of $a$ modulo $m$, denoted by $\textnormal{ord}_{m}(a)$. \section{Lemmas} \begin{lemma}\label{lem1}(\cite{Lu}) For any positive integer $m$, the Diophantine equation $(4m^{2}-1)^{x}+(4m)^{y}=(4m^{2}+1)^{z}$ has only the solution $(x,y,z)=(2,2,2).$\end{lemma} \begin{lemma}\label{lem2}(See \cite[Lemma 2]{Deng}) If $z\geq max\{x,y\},$ then the Diophantine equation $a^{x}+b^{y}=c^{z},$ where $a,b$ and $c$ are any positive integers (not necessarily relative prime) such that $a^{2}+b^{2}=c^{2}$, has no solution other than $(x,y,z)=(2,2,2).$\end{lemma} \begin{lemma}\label{lem3} (See \cite[Corollary 1]{Le}) If the Diophantine equation $(na)^{x}+(nb)^{y}=(nc)^{z}$(with $a^2+b^2=c^2$) has a solution $(x,y,z)\neq(2,2,2),$ then $x,y,z$ are distinct.\end{lemma} \begin{lemma}\label{lem4}(See \cite[Lemma 2.3]{Deng2013}) Let $a,b,c$ be any primitive Pythagorean triple such that the Diophantine equation $a^{x}+b^{y}=c^{z}$ has the only positive integer solution $(x,y,z)=(2,2,2)$. Then (\ref{eqn1}) has no positive integer solution satisfying $x>y>z$ or $y>x>z$. \end{lemma} \begin{lemma}\label{lem5}Let $k$ be a positive integer and $F_k=2^{2^k}+1$ be a Fermat number. If $(x,y,z)$ is a solution of the Eq.(\ref{eqn2}) with $(x,y,z)\neq (2,2,2)$, then $x<z<y$. \end{lemma} \begin{proof} By Lemmas \ref{lem2}-\ref{lem4}, it is sufficient to prove that the Eq.(\ref{eqn2}) has no solution $(x,y,z)$ satisfying $y<z<x$. By Lemma \ref{lem1}, we may suppose that $n\geq2$ and the Eq.(\ref{eqn2}) has a solution $(x,y,z)$ with $y<z<x$. Then we have \begin{equation}\label{eqn9}2^{(2^{k-1}+1)y}=n^{z-y}\Big(F_k^{z}-(F_k-2)^{x}n^{x-z}\Big).\end{equation} By \eqref{eqn9} we may write $n=2^{r}$ with $r\geq1$. Noting that $$\gcd\Big(F_k^{z}-(F_k-2)^{x}2^{r(x-z)},2\Big)=1,$$ we have \begin{equation}\label{eqn10}F_k^{z}-(F_k-2)^{x}2^{r(x-z)}=1.\end{equation} Since $k\geq 1$, by (\ref{eqn10}) we have $F_k^z\equiv 1\pmod 3$, $z\equiv 0\pmod 2.$ Write $z=2z_{1},$ we have \begin{equation}\label{eqn11}\Big(\prod\limits_{i=0}^{k-1}F_i\Big)^x2^{r(x-z)}=(F_k^{z_{1}}-1)(F_k^{z_{1}}+1).\end{equation} Let $F_{k-1}=\prod\limits_{i=1}^tp_i^{\alpha_i}$ be the standard prime factorization of $F_{k-1}$ with $p_1<\cdots<p_t$. By the known Fermat primes, we know that there is the possibility of $t=1$. Moreover, \begin{equation}\label{eqn12}\textnormal{ ord}_{p_i}(2)=2^{k}, \quad i=1,\cdots,t.\end{equation} Noting that $\gcd(F_k^{z_{1}}-1,F_k^{z_{1}}+1)=2,$ we know that $p_t$ divide only one of $F_k^{z_{1}}-1$ and $F_k^{z_{1}}+1$. {\bf Case 1.} $p_t\mid F_k^{z_{1}}-1$. Then $F_k^{z_{1}}-1\equiv 2^{z_1}-1\equiv 0\pmod {p_t}$. Noting that $\textnormal{ ord}_{p_t}(2)=2^{k}$, we have $z_1\equiv 0\pmod{2^{k}}$. By (\ref{eqn12}) we have $$F_k^{z_{1}}-1\equiv 2^{z_1}-1\equiv 0\pmod {p_i}, \quad i=1,\cdots, t.$$ Since $\gcd(F_k^{z_{1}}-1,F_k^{z_{1}}+1)=2,$ by (\ref{eqn11}) we have $$F_k^{z_{1}}-1\equiv 2^{z_1}-1\equiv 0\pmod {p_i^{\alpha_ix}}, \quad i=1,\cdots, t.$$ Hence $F_{k-1}^x\mid F_k^{z_{1}}-1$. {\bf Case 2.} $p_t\mid F_k^{z_{1}}+1$. Then $F_k^{z_{1}}+1\equiv 2^{z_1}+1\equiv 0\pmod {p_t}$. Noting that $\textnormal{ ord}_{p_t}(2)=2^{k}$, we have $2^{k-1}\mid z_1$, but $2^{k}\nmid z_1$. By (\ref{eqn12}) we have $$2^{2z_1}-1=(2^{z_1}+1)(2^{z_1}-1)\equiv 0\pmod {p_i}, \quad i=1,\cdots, t.$$ Thus $$F_k^{z_{1}}+1\equiv 2^{z_1}+1\equiv 0\pmod {p_i}, \quad i=1,\cdots, t.$$ Since $\gcd(F_k^{z_{1}}-1,F_k^{z_{1}}+1)=2,$ by (\ref{eqn11}) we have $$F_k^{z_{1}}+1\equiv 2^{z_1}+1\equiv 0\pmod {p_i^{\alpha_ix}}, \quad i=1,\cdots, t.$$ Hence $F_{k-1}^x\mid F_k^{z_{1}}+1$. However, $$F_{k-1}^x=\Big(2^{2^{k-1}}+1\Big)^x>\Big(2^{2^{k-1}}+1\Big)^{2z_1}>F_k^{z_1}+1,$$ which is impossible. This completes the proof of Lemma \ref{lem5}. \end{proof} \section{Proof of Theorem \ref{thm1}} By Lemma \ref{lem1} and Lemma \ref{lem5}, we may suppose that $n\geq2$ and the Eq.(\ref{eqn2}) has a solution $(x,y,z)$ with $x<z<y$. Then \begin{equation}\label{eqn13a}\Big(\prod_{i=0}^{k-1}F_i\Big)^{x}=n^{z-x}\Big(F_k^{z}-2^{(2^{k-1}+1)y}n^{y-z}\Big).\end{equation} It is clear from \eqref{eqn13a} that $$\gcd\Big(n,\prod\limits_{i=0}^{k-1}F_i\Big)>1.$$ Let $\prod\limits_{i=0}^{k-1}F_i=\prod\limits_{i=1}^{t}p_i^{\alpha_i}$ be the standard prime factorization of $\prod\limits_{i=0}^{k-1}F_i$ and write $n=\prod\limits_{\nu=1}^{s}p_{i_\nu}^{\beta_{i_\nu}},$ where $\beta_{i_\nu}\geq1$, $\{i_1,\cdots,i_s\}\subseteq \{1,\cdots,t\}$. Let $T=\{1,2,\cdots, t\}\setminus \{i_1,\cdots,i_s\}$. If $T=\emptyset$, then let $P(k,n)=1$. If $T\neq\emptyset$, then let $$P(k,n)=\prod\limits_{i\in T}p_i^{\alpha_i}.$$ By (\ref{eqn13a}), we have \begin{equation}\label{eqn14a}P(k,n)^x=F_k^{z}-2^{(2^{k-1}+1)y}\prod\limits_{\nu=1}^{s}p_{i_\nu}^{\beta_{i_\nu}(y-z)}.\end{equation} Since $y\ge 2$, it follows that \begin{equation}\label{eqn4.3}P(k,n)^x\equiv 1\pmod{2^{2^k}}.\end{equation} If $3\mid P(k,n)$, then $P(k,n)\equiv -1\pmod 4$. This implies that $x$ is even. If $3\nmid P(k,n)$, then $P(k,n)\equiv 1\pmod 4$. Let $P(k,n)=1+2^vW$, $2\nmid W$. Then $v\ge 2$. Suppose that $x$ is odd, then $$P(k,n)^x=1+2^vW', \quad 2\nmid W'.$$ Thus $v\ge 2^k$ and $P(k,n)\ge F_k$, a contradiction with $$ P(k,n)<\prod\limits_{i=0}^{k-1}F_i=F_k-2. $$ Therefore, $x$ is even. Write $x=2^uN$ with $2\nmid N$. Then $u\geq 1$. {\bf Case 1.} $P(k,n)\equiv -1\pmod 4$. Let $P(k,n)=2^dM-1$ with $2\nmid M$. Then $d\geq 2$ and $$P(k,n)^x=1+2^{u+d}V, \quad 2\nmid V.$$ By (\ref{eqn4.3}) we have $u+d\geq 2^k$. Choose a $\nu\in\{1,\cdots,s\}$, let $p_{i_\nu}=2^rt+1$ with $r\geq 1$, $2\nmid t$. Then $$2^{d+r-1}<(2^dM-1)(2^rt+1)=P(k,n)\cdot p_{i_\nu}\leq \prod\limits_{i=0}^{k-1}F_i=2^{2^k}-1.$$ Thus $d+r\leq 2^k$. Hence $u\geq r$. By (\ref{eqn14a}) we have \begin{equation}\label{eqn3.4}P(k,n)^x\equiv 2^z\pmod{p_{i_\nu}}.\end{equation} Noting that $p_{i_\nu}-1\mid 2^ut$, we have \begin{equation}\label{eqn3.5}2^{tz}\equiv P(k,n)^{2^utN}\equiv 1\pmod {p_{i_\nu}}.\end{equation} Since $\textnormal{ord}_{p_{i_\nu}}(2)$ is even and $2\nmid t$, we have $z\equiv 0\pmod 2$. {\bf Case 2.} $P(k,n)\equiv 1\pmod 4$. Let $P(k,n)=2^{d'}M'+1$ with $2\nmid M'$. Then $d'\geq 2$ and $$P(k,n)^x=1+2^{u+d'}V', \quad 2\nmid V'.$$ By (\ref{eqn4.3}) we have $u+d'\geq 2^k$. Choose a $\mu\in\{1,\cdots,s\}$, let $p_{i_\mu}=2^{r'}t'+1$ with $r'\geq 1$, $2\nmid t'$. Then $$2^{d'+r'}<(2^{d'}M'+1)(2^{r'}{t'}+1)=P(k,n)\cdot p_{i_\mu}\leq \prod\limits_{i=0}^{k-1}F_i=2^{2^k}-1.$$ Thus $d'+r'<2^k$. Hence $u>r'$. By (\ref{eqn14a}) we have \begin{equation}\label{eqn3.6}P(k,n)^x\equiv 2^z\pmod{p_{i_\mu}}.\end{equation} Noting that $p_{i_\mu}-1\mid 2^ut'$, we have \begin{equation}\label{eqn3.7}2^{t'z}\equiv P(k,n)^{2^ut'N}\equiv 1\pmod {p_{i_\mu}}.\end{equation} Since $\textnormal{ord}_{p_{i_\mu}}(2)$ is even and $2\nmid t'$, we have $z\equiv 0\pmod 2$. Write $z=2z_{1}, x=2x_{1}$. By (\ref{eqn14a}), we have \begin{equation}\label{eqn15T}2^{(2^{k-1}+1)y}\prod\limits_{\nu=1}^{s}p_{i_\nu}^{\beta_{i_\nu}(y-z)}=\Big(F_k^{z_{1}}-P(k,n)^{x_1}\Big)\Big(F_k^{z_{1}}+P(k,n)^{x_1}\Big).\end{equation} Noting that $$\gcd\Big(F_k^{z_{1}}-P(k,n)^{x_1},F_k^{z_{1}}+P(k,n)^{x_1}\Big)=2,$$ we have \begin{equation}\label{eqn16T}2^{(2^{k-1}+1)y-1}\mid F_k^{z_{1}}-P(k,n)^{x_1},\quad 2\mid F_k^{z_{1}}+P(k,n)^{x_1},\end{equation} or \begin{equation}\label{eqn17T}2\mid F_k^{z_{1}}+P(k,n)^{x_1},\quad 2^{(2^{k-1}+1)y-1}\mid F_k^{z_{1}}-P(k,n)^{x_1}.\end{equation} However, $$2^{(2^{k-1}+1)y-1}>2^{(2^{k-1}+1)2z_1}>(F_k+F_k-2)^{z_1}>F_k^{z_{1}}+P(k,n)^{x_1},$$ a contradiction. This completes the proof of Theorem \ref{thm1}. \section{Acknowledgment} We sincerely thank Professor Yong-Gao Chen for his valuable suggestions and useful discussions. We would like to thank the referee for his/her helpful comments. \end{document}

\begin{document} \title{Sphere covering by minimal number of caps and short closed sets \thanks{{\it 1991 A M S Subject Classification.} 52A45 {\it Key words and phrases.} Sphere covering by closed sets.}} \author{A. B. N\'emeth} \maketitle \begin{abstract} A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their intersection is empty; (ii) the intersection of any proper subfamily of them is non-empty. In the case of caps (i) and (ii) are also sufficient for the family to be a covering of the sphere. \end{abstract} \section{Introduction and the main result} Denote by $\mathbb R^{n+1}$ the $n+1$-dimensional Euclidean space endowed with a Cartesian reference system, with the scalar product $\langle\cdot,\cdot\rangle$ and with the topology it generates. Denote by $S^n$ the $n$-dimensional unit sphere in $\mathbb R^{n+1}.$ A subset of the sphere $S^n$ is said \emph{short} if it is contained in an open hemisphere. The subset $C\subset S^n$ is called {\it geodesically convex} if together with any two of its points it contains the arc of minimal length of the principal circle on $S^n$ through these points. $S^n$ itself is a geodesically convex set. A short closed set which is geodesically convex is called a \emph{cap}. We use the notation $\co A$ for the convex hull of $A$ and the notation $\sco A$ for the geodesical convex hull of $A\subset S^n$ (the union of the geodesical lines with endpoints in $A$). Further $\dist(\cdot,\cdot)$ will denote the geodesical distance of points. Besides the standard notion of simplex we also use the notion of the spherical simplex $\Delta$ placed in the north hemisphere $S^+$ of $S^n$ such that their vertices are on the equator of $S^n$. In this case $\|\Delta\|=S^+$. Our main result is: \begin{theo} \begin{enumerate} \item The minimal number of short closed sets covering $S^n$ is $n+2$. \item If a family $F_1,...F_{n+2}$ of short closed sets covers $S^n$, then: (i) $\cap_{i=1}^{n+2} F_i = \emptyset$; (ii) $\cap_{i\not=j}F_i\not= \emptyset,\;\forall \; j=1,...,n+2$; (iii) if $a_j\in \cap_{i\not=j}F_i$, then the vectors $a_1,...,a_{n+2}$ are the vertices of an $n+1$-simplex containing $0$ in its interior. \end{enumerate} If the sets $F_i$ are caps, then (i) and (ii) are also sufficient for the family to be a cover of $S^n$. \end{theo} Let $\Delta$ be an $n+1$-dimensional simplex with vertices in $S^n$ containing the origin in its interior. Then the radial projection from $0$ of the closed $n$-dimensional faces of $\Delta$ into $S^n$ furnishes $n+2$ caps covering $S^n$ and satisfying (i) and (ii). A first version for caps of the above theorem is the content of the unpublished note \cite{nemeth2006}. \begin{remark} We mention the formal relation in case of caps with those in the \emph{Nerve Theorem} (\cite{hatcher2002} Corollary 4G3). If we consider \emph{"open caps"} in place of caps, then the conclusion (ii) can be deduced from the mentioned theorem. Moreover, the conclusion holds for a "good" open cover of the sphere too, i. e., an open cover with contractible members and contractible finite intersections. In our theorem the covering with caps has the properties of a "good" covering in this theorem: the members of the covering together with their nonempty intersections are contractible, but their members are closed, circumstance which seems to be rather sophisticated to be surmounted. (Thanks are due to Imre B\'ar\'any, who mentioned me this possible connection.) \end{remark} We shall use in the proofs the following (spherical) variant of Sperner's lemma (considered for simplices by Ky Fan \cite{FA}): \begin{lem}\label{sperner} If a collection of closed sets $F_1,...,F_{n+1}$ in $S^n$ covers the spherical simplex engendered by the points $a_1,...a_{n+1}\in S^n$ and $\sco \{a_1,...,a_{i-1},a_{i+1},...,a_{n+1}\}\subset F_i,\; i=1,...,n+1$ then $\cap_{i=1}^{n+1} F_i\not= \emptyset.$ \end{lem} Our first goal is to present the proof for caps. (We mention that using the methods in \cite{KN} and \cite{KN1} the proof can be carried out in a purely geometric way in contrast with the proof in \cite{nemeth2006}, where we refer to the Sperner lemma.) Using the variant for caps of the theorem and the Sperner lemma we prove then the variant for short closed sets. Except the usage of Lemma \ref{sperner}, our methods are elementary: they use repeatedly the induction with respect to the dimension. \section{The proof of the theorem for caps} 1. Consider $n+1$ caps $C^1,...,C^{n+1}$ on $S^n$. $C^i$ being a cap, can be separated strictly by a hyperplane $$H_i= \{x\in \mathbb R^{n+1}:\,\langle a_i,x\rangle +\alpha_i =0\}$$ from the origin. We can suppose without loss of generality, that the normals $a_i$ are linearly independent, since by slightly moving them we can achieve this, without affecting the geometrical picture. If the normals $a_i$ are considered oriented toward $0$, this strict separation means that $\alpha_i >0,\;i=1,...,n.$ The vectors $a_i,\,i=1,...,n+1$ engender a reference system in $\mathbb R^{n+1}$. Let $x$ be a nonzero element of the positive orthant of this reference system. Then, for $t\geq 0$, one has $\langle a_i,tx \rangle \geq 0,\;\forall \,i=1,...,n+1.$ Hence, for each $t\geq 0$, $tx$ will be a solution of the system $$\langle a_i,y\rangle +\alpha_i>0,\;i=1,...,n+1.$$ and thus $$(*)\qquad tx\in \cap_{i=1}^{n+1} H_i^+,\;\forall \, t\geq 0$$ with $$H_i^+=\{y \in R^{n+1};\,\langle a_i,y\rangle +\alpha_i > 0\}.$$ Now, if $C^1,...,C^{n+1}$ covers $S^n$, then so does the union $\cup_{i=1}^{n+1} H_i^-$ of halfspaces $$H_i^-=\{y \in R^{n+1};\,\langle a_i,y\rangle +\alpha_i \leq 0\}.$$ Since $H_i^+$ is the complementary set of $H_i^-$ and $S^n\subset \cup_{i=1}^{n+1} H_i^-$, the set $\cap_{i=1}^{n+1} H_i^+$ must be inside $S^n$ and hence bounded. But (*) shows that $tx$ with $x\not= 0$ is in this set for any $t\geq 0$. The obtained contradiction shows that the family $C^1,...,C^{n+1}$ cannot cover $S^n$. \begin{remark} The proof of this item is also consequence of the Lusternik-Schnirelmann theorem \cite{LS} which asserts that if $S^n$ is covered by the closed sets $F_1,...,F_k$ with $F_i\cap (-F_i)=\emptyset,\,i=1,...,k,$ then $k\geq n+2.$ \end{remark} 2. Let $C^1,...,C^{n+2}$ be caps covering $S^n$. (i) Then they cannot have a common point $x$, since this case $-x$ cannot be covered by any $C^i$. (No cap can contain diametrically opposite points of $S^n$.) Hence, condition (i) must hold. (ii) To prove that $\cap_{j\not= i} C^j \not= \emptyset, \;\forall \,i=1,...,n+2$ we proceed by induction. For $S^1$, the circle, $C^i$ is an arc (containing its endpoints) of length $< \pi$, $i=1,2,3$. The arcs $C^1, C^2, C^3$ cover $S^1$. Hence, they cannot have common points, and the endpoint of each arc must be contained in exactly one of the other two arcs. Hence, $C^i$ meets $C^j$ for every $j\not= i.$ If $c_i\in C^j\cap C^k,\; j\not= i\not= k\not=j$, then $c_1, \,c_2,\,c_3$ are tree pairwise different points on the circle, hence they are in general position and $0$ is an interior point of the triangle they span. Suppose the assertions (ii) and (iii) hold for $n-1$ and let us prove them for $n$. Take $C^{n+2}$ and let $H$ be a hyperplane through $0$ which does not meet $C^{n+2}.$ Then, $H$ determines the closed hemispheres $S^-$ and $S^+$. Suppose that $C^{n+2}$ is placed inside $S^-$ (in the interior of $S^-$ with respect the topology of $S^n$). Hence, $C^1,...,C^{n+1}$ must cover $S^+$ and denoting by $S^{n-1}$ the $n-1$-dimensional sphere $S^n\cap H$, these sets cover $S^{n-1}.$ Now, $D^i= C^i\cap S^{n-1},\;i=1,...,n+1$ are caps in $S^{n-1}$ which cover this sphere. Thus, the induction hypothesis works for these sets. Take the points $d_i\in \cap_{j\not= i}D^j$. Then, $d_1,...,d_{n+1}$ will be in general position and $0$ is an interior point of the simplex they span. By their definition, it follows that $d_k\in D^j, \; \forall k\not= j$ and hence $d_1,...,d_{j-1},d_{j+1}, ...,d_{n+1}\in D^j,\; j=1,2,...n+1. $ Consider the closed hemisphere $S^+$ to be endowed with a spherical simplex structure $\Delta$ whose vertices are the points $d_1,...,d_{n+1}$ . Since $C^1,...,C^{n+1}$ cover $S^+$, and $d_1,...,d_{j-1},d_{j+1}, ...,d_{n+1}\in D^j\subset C^j\cap S^+,\; j=1,2,...n+1 $, Lemma \ref{sperner} can be applied to the spherical simplex $\Delta$, yielding $$\cap_{j=1}^{n+1} C^j \supset\cap_{j=1}^{n+1} (S^+\cap C^j) \not= \emptyset.$$ This shows that each collection of $n+1$ sets $C^j$ have nonempty intersection and proves (ii) for $n$. (If we prefer a purely geometric proof of this item, we can refer to the spherical analogue of the results in \cite{KN1}.) From the geometric picture is obvious that two caps meet if and only if their convex hulls meet. Hence, from the conditions (i) and (ii) for the caps $C^i$, it follows that these conditions hold also for $A^i=\co C^i,\;i=1,...,n+2.$ Take $$a_i\in \cap_{j\not= i}A^j,\;i=1,...,n+2.$$ Let us show that for an arbitrary $k\in N$, $$a_k\not\in \aff \{a_1,...,a_{k-1},a_{k+1},...,a_{n+2}\}.$$ Assume the contrary. Denote $$H=\aff \{a_1,...,a_{k-1},a_{k+1},...,a_{n+2}\}.$$ Thus, $\dim H\leq n.$ The points $a_i$ are all in the manifold $H$. Denote $$B^i=H\cap A^i.$$ Since $a_i\in \cap_{j\not= i}A^j$ and $a_i\in H$, it follows that $$a_i\in \cap_{j\not= i}A^j\cap H=\cap_{j\not= i} B^j,\;\forall\,i.$$ This means that the family of convex compact sets $\{B^j:\,j=1,...,n+2\}$ in $H$ possesses the property that every $n+1$ of its elements have nonempty intersection. Then, by Helly's theorem, they have a common point. But this would be a point of $\cap_{i=1}^{n+2} A^i $ too, which contradicts (ii) for the sets $A^i$. Hence, every $n+2$ points $c_i\in \cap_{j\not= i} C^j \subset \cap_{j\not= i} A^j,\; i=1,...,n+2$ are in general position. Since $c_1,...,c_{i-1},c_{i+1},...,c_{n+2} \in C^i$, it follows that the open halfspace determined by the hyperplane they engender containing $0$ contains also the point $c_i$. This proves (iii). Suppose that the caps $C^1,...,C^{n+2}$ posses the properties in (i) and (ii). Then, the method in the above proof yields that the points $$c_i\in \cap_{j\not= i} C^j,\;i=1,...,n+2$$ engender an $n+1$-simplex with $0$ in its interior and $$c_1,...,c_{i-1},c_{i+1},...,c_{n+2} \in C^i,\; i=1,...,n+2.$$ The radial projections of the $n$-faces of this simplex into $S^n$ obviously cover $S^n$. The union of these projections are contained in $\cup_{i=1}^{n+2} C^i$. This completes the proof for cups. \section{The proof of the theorem for short closed sets} We carry out the proof by induction. Consider $n=1$ and suppose $F_1,F_2,F_3$ are short closed sets covering $S^1$. If $a\in \cap_{i=1}^3 F_i$, then by the above hypothesis $-a\not \in \cup_{i=1}^3 F_i$, which is impossible. Hence, $$\cap_{i=1}^3 F_i= \emptyset$$ must hold. Denote $C_3= \sco F_3$, then $C=\clo S^1\setminus C_3$ is a connected arc of $S^1$ covered by $F_1,F_2$. One must have $C\cap F_i\not=\emptyset , i=1,2$, since if for instance $C\cap F_2= \emptyset,$ then it would follow that the closed sets $F_1$ and $C_3$, both of geodesical diameter $< \pi$ cover $S^1$, which is impossible. Since $C$ is connected and $C\cap F_i,\; i=1,2$ are closed sets in $C$ covering this set, $F_1\cap F_2\supset (C\cap F_1)\cap (C\cap F_2) \not= \emptyset$ must hold. The geodesically convex sets $C_i=\sco F_i,\;i=1,2,3$ cover $S^1$, hence applying the theorem for caps to $ a_j \in \cap_{i\not= j} F_i \subset \cap_{i\not= j} C_i,\; j=1,2,3$, we conclude that these points are in general position and the simplex engendered by them must contain $0$ as an interior point. Suppose that the assertions hold for $n-1$ and prove them for $n$. Suppose that $$S^n \subset \cup_{i=1}^{n+2} F_i,\; F_i\;\textrm{short, closed},\; i=1,...,n+2.$$ The assertion (i) is a consequence of the theorem for caps applied to $C_i =\sco F_i,\; i=1,...,n+2$ (or a consequence of the Lusternik Schnirelmann theorem). Suppose that $F_{n+2}$ is contained in the interior (with respect to the topology of $S^n$) of the south hemisphere $S^-$ and denote by $S^{n-1}$ the equator of $S^n$. Now $S^{n-1} \subset \cup_{i=1}^{n+1} F'_i$ with $F'_i=(S^{n-1}\cap F_i ),\; i=1,...,n+1,$ and we can apply the induction hypothesis for $S^{n-1}$ and the closed sets $F'_i,\; i=1,...,n+1.$ Since $C'_i=\sco F'_i,\; i=1,...,n+1$ cover $S^{n-1}$, and they are caps, the theorem for caps applies and hence the points \begin{equation*} a_j \in \cap_{i=1,i\not=j}^{n+1} C'_i,\;j=1,...,n+1 \end{equation*} are in general position. The closed sets \begin{equation*} A_i = C'_i\cup(F_i\cap S^+)= C'_i\cup (F_i\cap \inter S^+),\;i=1,...,n+1 \end{equation*} cover $S^+$, the north hemisphere considered as a spherical simplex $\Delta$ engendered by $a_1,...,a_{n+1}$ $(\|\Delta\|=S^+$). (Here $\inter S^+$ is the interior of $S^+$ in the space $S^n$.) Further, $$\sco \{a_1,...,a_{k-1},a_{k+1},...,a_{n+1}\} \subset A_k,\; k=1,...,n+1.$$ Hence, we can apply Lemma \ref{sperner} to conclude that there exists a point $a$ in $\cap_{i=1}^{n+1} A_i \not= \emptyset.$ Since $$C'_i\cap (F_j\cap \inter S^+)=\emptyset,\;\forall \;i,\;j,$$ it follows that $$a \in \cap_{i=1}^{n+1} A_i= \cap_{i=1}^{n+1} C'_i \cup \cap_{i+1}^{n+1} F_i\cap \inter S^+=\cap_{i+1}^{n+1} F_i\cap \inter S^+,$$ because $\cap_{i=1}^{n+1} C'_i= \emptyset$ by the induction hypothesis and the theorem for caps. Thus, $$a\in \cap_{i=1}^{n+1} F_i,$$ and we have condition (ii) fulfilled for $n$. The condition (iii) follows from the theorem for caps applied to $$C^i= \sco F^i,\;i=1,...,n+2.$$ \begin{remark} If $S^1$ is covered by the closed sets $F_1, F_2, F_3$ with the property $F_i\cap (-F_i)=\emptyset,\; i=1,2,3 $, then $F_i\cap F_j \not= \emptyset$ $\forall \;i, j.$ Indeed, assume that $F_1\cap F_2= \emptyset$. Then $\dist (F_1,F_2)=\varepsilon >0.$ If $a_i\in F_i$ are the points in $F_i, \; i=1,2$ with $\dist (a_1,a_2)= \varepsilon,$ then the closed arc $C\subset S^1$ with the endpoints $a_1,\; a_2$ must be contained in $F_3$, and hence $-C\cap F_3= \emptyset$, and then $-C$ must be covered by $F_1\cup F_2$. Since $-a_1 \in -C$ cannot be in $F_1$, it must be in $F_2$, and $-a_2\in F_1$. Thus, $F_1\cap -C\not= \emptyset$ and $F_2\cap -C \not= \emptyset,$ while the last two sets cover $-C$. Since $-C$ is connected and the respective sets are closed, they must have a common point, contradicting the hypothesis $F_1\cap F_2= \emptyset$. This way, we obtain (ii) fulfilled for $n=1$ for this more general case. We claim that the conditions also hold for $n$, that is, if the closed sets $F_1,...,F_{n+2}$ with $F_i\cap (-F_i)=\emptyset,\; i=1,...,n+2 $ cover $S^n$, then condition (ii) holds. (Condition (i) is a consequence of the definition of the sets $F_i$.) \end{remark} \end{document}

\begin{document} \begin{abstract} We show that any two disjoint crooked planes in $\mathbb{R}^3$ are leaves of a crooked foliation. This answers a question asked by Charette and Kim \cite{foliations}. \end{abstract} \title{Foliations between crooked planes in $3$-dimensional Minkowski space} \section{Introduction} In 1983, answering a question of Milnor\cite{Milnor}, Margulis constructed the first examples of nonabelian free groups which act freely and properly discontinuously on $\mathbb{R}^3$ by affine transformations~\cite{Margulis}. In order to better understand these examples, Todd Drumm defined piecewise linear surfaces called \emph{crooked planes} which can bound fundamental domains for such actions~\cite{drummthesis}. Crooked planes have proven to be very useful in the study of affine actions. Charette-Drumm-Goldman have used them in order to obtain a complete classification for free groups of rank two \cite{CDG1,CDG2,CDG3}. In particular, they show that every free and properly discontinuous affine action of a rank two free group on $\mathbb{R}^3$ admits a fundamental domain bounded by finitely many crooked planes (the \emph{crooked plane conjecture}). A consequence of this is the \emph{tameness conjecture}, that the quotient of $\mathbb{R}^3$ by one of these actions is homeomorphic to the interior of a compact manifold with boundary. Building on this work, Danciger-Gu\'eritaud-Kassel showed in ~\cite{DGKArcComplex} that crooked planes have a natural interpretation in terms of the deformation theory of hyperbolic surfaces, and used this fact in order to prove the crooked plane conjecture in arbitrary rank, assuming that the linear part is convex cocompact in $\mathsf{O}(2,1)$. One of the key aspects of the theory of crooked planes is their intersection properties. In particular, knowing when two crooked planes are disjoint is crucial. The \emph{Drumm-Goldman inequality} provides a necessary and sufficient criterion for two crooked planes to be disjoint ~\cite{DRUMM1999323}. This criterion was later expanded upon in ~\cite{halfspaces} and reinterpreted in terms of hyperbolic geometry in ~\cite{DGKArcComplex}. As an application of the disjointness criterion, the first example of a \emph{crooked foliation}, a smooth $1$-parameter family of pairwise disjoint crooked planes, was given in ~\cite{halfspaces}. Charette-Kim \cite{foliations} investigated these foliations further and gave necessary and sufficient criteria for a one-parameter family of crooked planes to foliate a subset of $\mathbb{R}^3$. They ask the following question : given a pair of disjoint crooked planes in $\mathbb{R}^3$, can the region between them be foliated by crooked planes? We answer this question in the affirmative. \begin{thm}\label{thm:mainthm} Let $C,C'$ be a pair of disjoint crooked planes in $\mathbb{R}^3$. Then, there is a \emph{crooked foliation}, that is, a smooth family of pairwise disjoint crooked planes $C_t$, $0\leq t\leq 1$ with $C_0=C$ and $C_1=C'$. \end{thm} After recalling some definitions from the theory of crooked planes in Minkowski $3$-space in Section 2, we will prove the main theorem in Section 3.\\ We are thankful to the referee for insightful comments and for suggesting an elegant way to shorten the proof of the main theorem. \section{Definitions} \begin{defn} Lorentzian $3$-space $\mathbb{R}^{2,1}$ is the real three dimensional vector space $\mathbb{R}^3$ endowed with the following symmetric bilinear form of signature $(2,1)$: \[ \cdot : \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}\] \[ (\mathbf{u},\mathbf{v}) \mapsto u_1 v_1 + u_2 v_2 - u_3 v_3.\] We fix the orientation given by the standard basis $e_1,e_2,e_3$ and we define the \emph{Lorentzian cross product} \[\mathbf{u}\times \mathbf{v} = (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_2 v_1 - u_1 v_2)\in\mathbb{R}^{2,1},\] for $\mathbf{u},\mathbf{v}\in\mathbb{R}^{2,1}$. \end{defn} A \emph{null frame} of $\mathbb{R}^{2,1}$ is a positively oriented basis $\mathbf{u},\mathbf{u}',\mathbf{u}''$ such that $\ldot{\mathbf{u}}{\mathbf{u}}=1$, $\ldot{\mathbf{u}'}{\mathbf{u}''}=-1$ and all other products between the three vectors vanish. \begin{nota} Any unit spacelike vector $\mathbf{u}$ can be extended to a null frame. This frame is unique up to scaling $\mathbf{u}'$ and $\mathbf{u}''$ by inverse scalars. As normalization we will choose $\mathbf{u}'$ and $\mathbf{u}''$ so that their third coordinates are positive and equal. Given $\mathbf{u}$, we will denote these two null vectors by $\mathbf{u}^-$ and $\mathbf{u}^+$, respectively. \end{nota} We will denote by $\mathrm{Min}$ the pseudo-Euclidean affine space which is modeled on the vector space $\mathbb{R}^{2,1}$. In other words, $\mathrm{Min}$ is a topological space on which $\mathbb{R}^{2,1}$ acts simply transitively by homeomorphisms. For $\mathbf{v}\in \mathbb{R}^{2,1}$ and $p\in\mathrm{Min}$, we denote this action by $\mathbf{v}(p) = p + \mathbf{v}$. If $q=p+\mathbf{v}$, we will also write $q-p=\mathbf{v}$. A choice of origin $o\in \mathrm{Min}$ identifies $\mathrm{Min}$ with $\mathbb{R}^{2,1}$ via the map $\mathbf{v}\mapsto o + \mathbf{v}$. We now recall the definition of a crooked plane. First, we define a \emph{stem}, which will be one of the three linear pieces of a crooked plane. \begin{defn} Let $\mathbf{u}\in \mathbb{R}^{2,1}$ be a unit spacelike vector. The \emph{stem} $S(\mathbf{u})$ is the set of causal vectors orthogonal to $\mathbf{u}$ : \[S(\mathbf{u}) = \{ \mathbf{v} \in \mathbb{R}^{2,1} ~|~ \ldot{\mathbf{u}}{\mathbf{v}}=0 \text{ and } \ldot{\mathbf{v}}{\mathbf{v}} \leq 0 \}.\] A stem is the union of two opposite closed quadrants (see Figure \ref{fig:consistorient}). \end{defn} \begin{defn} Let $\mathbf{u}\in \mathbb{R}^{2,1}$ be a unit spacelike vector. The \emph{linear crooked plane} $C(\mathbf{u})$ is the piecewise linear surface defined by: \[ C(\mathbf{u}) := \{ \mathbf{v} \in \mathbb{R}^{2,1} ~|~ \lcross{v}{w} = k \mathbf{w} \textrm{ for some } \mathbf{w}\in S(\mathbf{u}) \text{ and } k\in \mathbb{R}_{\geq 0}\}.\] \end{defn} From this definition, we see that $S(\mathbf{u})\subset C(\mathbf{u})$ since $\mathbf{v}\times\mathbf{v}=0$ for all $\mathbf{v}\in \mathbb{R}^{2,1}$. The complement of the stem $C(\mathbf{u})-S(\mathbf{u})$ has two connected components which are called the \emph{wings} of the crooked plane. Each wing is a half-plane on which the Lorentzian bilinear form is degenerate, attached to the stem along its boundary (See Fig. \ref{fig:consistorient}). Note that $C(\mathbf{u})=C(-\mathbf{u})$. \begin{defn} Let $p\in \mathrm{Min}$ and $\mathbf{u}\in \mathbb{R}^{2,1}$ unit spacelike. The \emph{crooked plane} $C(p,\mathbf{u})$ is the set $p + C(\mathbf{u}) \subset \mathrm{Min}$. The vector $\mathbf{u}$ is called a \emph{directing vector} of the crooked plane, and $p$ its \emph{vertex}. \end{defn} In order to formally state the disjointness criteria from \cite{DRUMM1999323,foliations}, we need a normalization for pairs of unit spacelike vectors. \begin{defn} Two unit spacelike vectors $\mathbf{u}_1,\mathbf{u}_2 \in \mathbb{R}^{2,1}$ are \emph{consistently oriented} if \begin{itemize} \item $\ldot{\mathbf{u}_1}{\mathbf{u}_2}\leq -1$, and \item $\ldot{\mathbf{u}_i}{\mathbf{u}_j}^\pm\leq 0$ for $1\leq i,j\leq 2$. \end{itemize} \end{defn} Two consistently oriented unit spacelike vectors $\mathbf{u},\mathbf{u}'$ are called \emph{ultraparallel} if $\ldot{\mathbf{u}}{\mathbf{u}'}<-1$. They are called \emph{asymptotic} if $\ldot{\mathbf{u}}{\mathbf{u}'}=-1$ and $\mathbf{u}'\neq -\mathbf{u}$. Intersecting $\mathbf{u}^\perp$ and $\mathbf{u'}^\perp$ with the hyperboloid model of the hyperbolic plane defines a pair of hyperbolic geodesics, and the terminology comes from the relative position of these geodesics. Choosing one of the unit vectors $\pm\mathbf{u}$ endows the geodesic in the hyperboloid model of $\mathbb{H}^2$ defined by $\mathbf{u}^\perp$ with a transverse orientation. Two unit spacelike vectors are consistently oriented when the corresponding transversely oriented geodesics are disjoint with transverse orientations pointing away from each other (see Figure \ref{fig:consistorient}). Whenever there exists a choice of directing vectors $\mathbf{u},\mathbf{u}'$ which are consistently oriented, we will also call a pair of crooked planes $C(p,\mathbf{u}),C(p',\mathbf{u}')$ ultraparallel or asymptotic accordingly. \begin{figure} \caption{Consistent orientations, stems and crooked planes.} \label{fig:consistorient} \end{figure} We will use two disjointness criteria for crooked planes, one for pairs of crooked planes and one for foliations. Both depend on the following notion: \begin{defn} The \emph{stem quadrant} associated to a unit spacelike vector $\mathbf{u}$ is the set \[\mathsf{V}(\mathbf{u}) := \{a \mathbf{u}^- - b\mathbf{u}^+ : a,b\geq 0\}\backslash\{0\}.\] Note that $\mathsf{V}(-\mathbf{u})=-\mathsf{V}(\mathbf{u})$. \end{defn} The following disjointness criterion is a restatement of the \emph{Drumm-Goldman inequality} ~\cite{DRUMM1999323}. \begin{thm}[Burelle-Charette-Goldman \cite{halfspaces}]\label{DrummgoldmanDisjoint} Let $C=C(p,\mathbf{u})$, $C'=C(p',\mathbf{u}')$ be crooked planes and assume that $\mathbf{u},\mathbf{u}'$ are consistently oriented. Then, $C$ and $C'$ are disjoint if and only if \[p'-p \in \mathsf{A(\mathbf{u},\mathbf{u}')}:=\mathrm{int}(\mathsf{V}(\mathbf{u}')-\mathsf{V}(\mathbf{u})).\] \end{thm} \begin{rmk} It is also shown in \cite{DRUMM1999323} that if there is no choice of sign for $\mathbf{u},\mathbf{u}'$ making them consistently oriented, then $C(p,\mathbf{u})$ and $C(p',\mathbf{u}')$ necessarily intersect. Therefore, the above theorem is a characterization of disjoint crooked planes. \end{rmk} We will use the following straightforward consequence of the \emph{Charette-Kim criterion} for crooked foliations (foliations of $\mathbb{R}^{2,1}$ by crooked planes) : \begin{thm}[Charette-Kim \cite{foliations}]\label{charettekimUltrap} Let $(\mathbf{u}_t)_{t\in \mathbb{R}}$ be a path of pairwise ultraparallel or asymptotic unit spacelike vectors such that $-\mathbf{u}_t,\mathbf{u}_s$ are consistently oriented for all $t<s$. Suppose $(p_t)_{t\in \mathbb{R}}$, is a regular curve such that for every $t\in\mathbb{R}$, \[\dot{p_t} \in \mathrm{int}(\mathsf{V}(\mathbf{u}_t)).\] Then, $C(p_t,\mathbf{u}_t)$ is a crooked foliation. \end{thm} \section{Foliations between crooked planes} We now prove Theorem \ref{thm:mainthm} : there exists a crooked foliation containing any pair of disjoint crooked planes. The theorem is a consequence of the following stronger result : \begin{prop} Let $(\mathbf{u}_t)_{t\in[0,1]}$ be a smooth path of unit spacelike vectors which are pairwise ultraparallel or asymptotic. Let $p_0,p_1\in \mathrm{Min}$ such that $C(p_0,\mathbf{u}_0)$ and $C(p_1,\mathbf{u}_1)$ are disjoint crooked planes. Then, there exists a path $(p_t)_{t\in [0,1]}$ starting at $p_0$ and ending at $p_1$ such that $C(p_t,\mathbf{u}_t)$ is a smooth crooked foliation. \begin{proof} Since we assume that $\mathbf{u}_s$ are pairwise ultraparallel or asymptotic, we have that $\ldot{\mathbf{u}_t}{\mathbf{u}_s}\geq 1$ for all $t\leq s$. Changing the path $\mathbf{u}_s$ to $-\mathbf{u}_s$ if needed (both paths define the same linear crooked planes) we may also assume that $-\mathbf{u}_t,\mathbf{u}_s$ are consistently oriented for all $t<s$. For any pair of smooth functions $f,g : [0,1]\rightarrow \mathbb{R}^{>0}$, define \[\mathbf{v}_{f,g}(s) := f(s)\mathbf{u}_s^- - g(s)\mathbf{u}_s^+.\] Then, the path of vertices $p_{f,g}(t) := p_0 + \int_0^t \mathbf{v}_{f,g}(s)\,\mathrm{d} s$ satisfies the hypotheses of Theorem \ref{charettekimUltrap} since its derivative \[\dot{p}_{f,g}(t) = \mathbf{v}_{f,g}(t)\] lies in the interior of $\mathsf{V}(\mathbf{u}_t)$ by definition. Let $\mathsf{D}$ denote collection of displacement vectors $p_{f,g}(1)-p_0$ : \[\mathsf{D} = \left\{\left. \int_0^1 \mathbf{v}_{f,g}(s)\,\mathrm{d} s ~\right|~ f,g :[0,1]\rightarrow \mathbb{R}^{>0}\right\}.\] Then $\mathsf{D}$ is a convex cone since $k\mathbf{v}_{f,g}=\mathbf{v}_{kf,kg}$ for $k\in \mathbb{R}^{>0}$ and $\mathbf{v}_{f_1,g_1} + \mathbf{v}_{f_2,g_2} = \mathbf{v}_{f_1+f_2,g_1+g_2}$. Moreover, since by Theorem \ref{charettekimUltrap} the crooked planes $C(p_{f,g}(t),\mathbf{u}_t)$ define crooked foliations, the initial and final crooked planes are disjoint and so $\mathsf{D}\subset\mathsf{A}(-\mathbf{u}_0,\mathbf{u}_1)$. Since the cone $\mathsf{A}(-\mathbf{u}_0,\mathbf{u}_1)$ is the interior of the convex hull of the four rays generated by $\mathbf{u}_0^-,-\mathbf{u}_0^+,\mathbf{u}_1^-,-\mathbf{u}_1^+$, to show equality of the cones it suffices to show that these rays can be approximated by vectors in $\mathsf{D}$. Consider the sequences $f_n(s) = ne^{-n s}$ and $g_n(s)=e^{-n}$. Integrating by parts we get \[\int_0^1 f_n(s)\mathbf{u}_s^-\,\mathrm{d} s = \mathbf{u}_0^- - e^{-n}\mathbf{u}_1^- + \int_0^1 e^{-n s}\dot{\mathbf{u}}_s^- \,\mathrm{d} s.\] Therefore, as $\mathbf{u}_s$ is smooth and so $\mathbf{u}^+_s$ and $ \dot{\mathbf{u}}^-_s$ are bounded on $[0,1]$, \[\lim_{n\rightarrow \infty} \int_0^1\mathbf{v}_{f_n,g_n}(s)\,\mathrm{d} s = \mathbf{u}^-_0.\] We conclude that $\mathsf{D}$ contains vectors arbitrarily close to the ray $\mathbb{R}^{>0}\mathbf{u}_0^-$. Similarly, if $f$ is concentrated near $s=1$ and $g$ is small we can approximate the ray $\mathbf{u}_1^-$, and exchanging the roles of $f$ and $g$ we approximate the other two rays on the boundary of the convex cone $\mathsf{A}(\mathbf{u}_0,\mathbf{u}_1)$. \end{proof} \end{prop} The previous proposition has the following interpretation : given any geodesic foliation $\mathcal{F}$ of the region between two geodesics $\ell_0,\ell_1$ of $\mathbb{H}^2$ and basepoints $p_0,p_1\in \mathrm{Min}$ such that the crooked planes with vertices $p_i$ and stems corresponding to $\ell_i$ are disjoint, $\mathcal{F}$ can be lifted to a foliation by crooked planes of the region between the crooked planes. {} \end{document}

\begin{document} \begin{abstract} Long-time behavior is one of the most fundamental properties in dynamical systems. The limit behaviors of flows on surfaces are captured by the Poincar\oplus'e-Bendixson theorem using the $\omega$-limit sets. This paper demonstrates that the positive and negative long-time behaviors are not independent. In fact, we show the dependence between the $\omega$-limit sets and the $\alpha$-limit sets of points of flows on surfaces, which partially generalizes the Poincar\oplus'e-Bendixson theorem. Applying the dependency result to solve what kinds of the $\omega$-limit sets appear in the area-preserving (or, more generally, non-wandering) flows on compact surfaces, we show that the $\omega$-limit set of any non-closed orbit of such a flow on a compact surface is either a subset of singular points or a locally dense Q-set. Moreover, we show the wildness of surgeries to add totally disconnected singular points and the tameness of those to add finitely many singular points for flows on surfaces. \end{abstract} \maketitle \section{Introduction} The Poincar\oplus'e-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows and was applied to various phenomena (e.g. \cite{bhatia1966application,du2021traveling,koropecki2019poincare,hajek1968dynamical,pokrovskii2009corollary,roussarie2020topological,roussarie2021some,roussarie2021some02}). In \cite{birkhoff1927dynamical}, Birkhoff introduced the concepts of $\omega$-limit set and $\alpha$-limit set of a point. Using these concepts, one can describe the limit behaviors of orbits stated in the works of Poincar\oplus'e and Bendixson in detail. Moreover, the Poincar\oplus'e-Bendixson theorem was generalized for flows on surfaces in various ways \cite{andronov1966qualitative,aranson1996introduction,buendia2017rem,buendia2019top,ciesielski1994poincare,demuner2009poincare,gardiner1985structure,gutierrez1986smoothing,Levitt1982foliation,lopez2004accumulation,lopez2007topological,markley1969poincare,marzougui1996,marzougui1998structure,nikolaev1999flows,schwartz1963generalization,vanderschoot2003local,yano1985asymptotic,yokoyama2021poincare}, and also for foliations \cite{levitt1987differentiability,plante1973generalization}, translation lines on the sphere \cite{koropecki2019poincare}, geodesics for a meromorphic connection on Riemann surfaces \cite{abate2016poincare,abate2011poincare}, group actions \cite{hounie1981minimal}, and semidynamical systems \cite{bonotto2008limit}. Applying a generalization of the Poincar\oplus'e-Bendixson theorem, symmetric properties of long-time behaviors are studied \cite{yokoyama2023flows}. Area-preserving flows on compact surfaces are one of the basic and classic examples of dynamical systems, also known as locally Hamiltonian flows or equivalently multi-valued Hamiltonian flows. The measurable properties of such flows are studied from various aspects \cite{chaika2021singularity,conze2011cocycles,forni1997solutions,frkaczek2012ergodic,forni2002deviation,kanigowski2016ratner,kulaga2012self,ravotti2017quantitative,ulcigrai2011absence}. For instance, the study of area-preserving flows for their connection with solid-state physics and pseudo-periodic topology was initiated by Novikov \cite{novikov1982hamiltonian}. The orbits of such flows also arise in pseudo-periodic topology, as hyperplane sections of periodic manifolds (cf. \cite{arnol1991topological,zorich1999leaves}). \subsection{Statements of main results} In this paper, we discuss the dependency of the $\omega$-lilmit sets and the $\alpha$-limit sets of points and consider the following classification problem and the wildness and tameness of surgeries to add singular points of flows on surfaces. \subsubsection{On the $\omega$-limit sets of points} We show the following dependency of $\omega$-lilmit sets and the $\alpha$-limit sets, which supplements a generalization of the Poincar\oplus'e-Bendixson theorem \cite[Theorem~A]{yokoyama2021poincare}. \begin{main}\label{lem:ld} The $\omega$-limit set of any locally dense orbit for a flow with arbitrarily many singular points on a compact surface is either a nowhere dense subset of singular points or a locally dense Q-set. \end{main} The previous theorem says that the $\omega$-limit set and the $\alpha$-limit set of any point are not independent in general. For instance, the $\omega$-limit set of a point whose $\alpha$-limit set is a limit circuit is not a locally dense Q-set. The details of dependency are stated in \S 6 (see Tables~\ref{table:01} and \ref{table:02}). We apply the previous dependency result to the following question. \begin{question} What kinds of the $\omega$-limit sets of points do appear in the non-wandering, divergence-free, and Hamiltonian flows with arbitrarily many singular points on compact surfaces, respectively? \end{question} We answer that only nowhere dense subsets of singular points and locally dense Q-sets appear in such cases. Similar results hold for locally dense orbits of flows and non-closed orbits of gradient flows (see \S~\ref{sec:grad} for details). To describe precise statements of the results, we recall some concepts as follows. An orbit is {\bf closed} if it is singular or periodic, and it is {\bf locally dense} if its closure has a nonempty interior. An orbit $O$ is {\bf recurrent} if $O \subseteq \omega(O) \cup \alpha(O)$. A {\bf Q-set} is the closure of a non-closed recurrent orbit. Notice that a Q-set is also called a {\bf quasiminimal set}. We have the following dichotomy for any non-closed orbit of a non-wandering flow. \begin{main}\label{lem:nw} The $\omega$-limit set of any non-closed orbit for a non-wandering flow with arbitrarily many singular points on a compact surface is either a nowhere dense subset of singular points or a locally dense Q-set. \end{main} We have the following observation for a Hamiltonian flow with arbitrarily many singular points on a (possibly non-compact) surface. \begin{main}\label{cor:ham} The $\omega$-limit set of any non-closed orbit for a Hamiltonian flow with arbitrarily many singular points on a surface consists of singular points. In particular, there are no non-closed recurrent points of a Hamiltonian flow with arbitrarily many singular points on a surface. \end{main} \subsection{Wildness and tameness of surgeries to add singular points} We consider the following question for invariance of the Hamiltonian property by surgeries to add singular points. \begin{question}\label{adding_sing} Is the Hamiltonian property for flows on surfaces invariant under multiplying a bump function to a Hamiltonian vector field? \end{question} We answer that the Hamiltonian property for flows on compact surfaces is invariant when only finitely many singular points are added, but that the property is not invariant even if totally disconnected singular points are added. More precisely, every non-zero vector field can be deformed into a vector field with wandering domains using a bump function with totally disconnected critical points as follows. \begin{main}\label{prop:45} For any non-zero vector field $X$ on a compact surface $S$, there is a smooth function $f \colon S \to [0,1]$ such that \oplus {\rm(1)} The vector field $fX$ is not non-wandering. \oplus {\rm(2)} Every orbit of $X$ is a union of orbits of $fX$. \oplus {\rm(3)} The critical point set $f^{-1}(0)$ is totally disconnected. \oplus {\rm(4)} If the singular point set $\operatorname{Sing}(X)$ is totally disconnected, then so is $\operatorname{Sing}(fX)$. \end{main} On the other hand, the Hamiltonian property of vector fields with finitely many singular points on compact surfaces is invariant under adding finitely many singular points. To state more precisely, we recall some concepts and a fact as follows. A nonempty subset $A$ of a topological space $X$ is a {\bf level set} of a function $f \colon X \to \mathbb{R} $ if there is a value $c \in \mathbb{R} $ such that $A = f^{-1}(c)$. Recall that, for any Hamiltonian vector field $X$ on a surface $S$ with the Hamiltonian $h \colon S \to \mathbb{R} $, the set of connected components of level sets of the restriction $h|_{S - \mathop{\mathrm{Sing}}(X)}$ is the set of orbits of the restriction $X|_{S - \mathop{\mathrm{Sing}}(X)}$ and is a codimension one foliation on the surface $S - \mathop{\mathrm{Sing}}(X)$. Therefore we introduce a pre-Hamiltonian flow as follows. \begin{definition} A flow $v$ on an orientable surface is {\bf pre-Hamiltonian} if there is a continuous function $H \colon S \to \mathbb{R} $ such that the set of connected components of level sets of the restriction $H|_{S - \mathop{\mathrm{Sing}}(v)}$ is the set of orbits of the restriction $v|_{S - \mathop{\mathrm{Sing}}(v)}$ and is a codimension one foliation on the surface $S - \mathop{\mathrm{Sing}}(v)$. \end{definition} Then $H$ is called the {\bf pre-Hamiltonian} of $v$. We have the following equivalence under finiteness of singular points. \begin{main}\label{cor:characterization_ham_finite} The following are equivalent for a flow $v$ with finitely many singular points on a compact surface $S$: \oplus {\rm(1)} The flow $v$ is Hamiltonian. \oplus {\rm(2)} The flow $v$ is pre-Hamiltonian. \end{main} This implies that Hamiltonian property of flows with finitely many singular points on compact surfaces is invariant under adding finitely many singular points. The present paper consists of seven sections. In the next section, as preliminaries, we introduce fundamental concepts. In \S~3, classifications of the $\omega$-limit sets of points in the non-wandering and Hamiltonian flows on surfaces are demonstrated, and a remark is stated. In \S~4, the tameness of surgeries to add finitely many singular points is described. In \S~5, we demonstrate the wildness of surgeries to add totally disconnected singular points. In \S~6, we show the existence of non-forbidden pairs of $\alpha$-limit and $\omega$-limit sets of points. The final section remarks on Question~\ref{adding_sing}, time-reversal symmetric limit sets, and a construction like the Cherry flow box. \section{Preliminaries} \subsection{Topological notion} Denote by $\overline{A}$ the closure of a subset $A$ of a topological space and by $\partial A := \overline{A} - \mathrm{int}A$ the boundary of $A$, where $B - C$ is used instead of the set difference $B \setminus C$ when $B \subseteq C$. A {\bf curve} is a continuous mapping $C: I \to X$ where $I$ is a non-degenerate connected subset of a circle $\mathbb{S}^1$. A curve is {\bf simple} if it is injective. We also denote by $C$ the image of a curve $C$. A simple curve is a {\bf simple closed curve} if its domain is $\mathbb{S}^1$ (i.e. $I = \mathbb{S}^1$). A simple closed curve is also called a {\bf loop}. An {\bf arc} is a simple curve whose domain is an interval. An {\bf orbit arc} is an arc contained in an orbit. By a {\bf surface}, we mean a paracompact two-dimensional manifold, that does not need to be orientable. A subset of a compact surface $S$ is {\bf essential} if it is not null homotopic in $S^*$, where $S^*$ is the resulting closed surface from $S$ by collapsing all boundary components into singletons. \subsection{Notion of dynamical systems} A {\bf flow} is a continuous $\mathbb{R} $-action on a manifold. From now on, we suppose that flows are on surfaces unless otherwise stated. Let $v : \mathbb{R} \times S \to S$ be a flow on a surface $S$. For $t \in \mathbb{R} $, define $v_t : S \to S$ by $v_t := v(t, \cdot )$. For a point $x$ of $S$, we denote by $O(x)$ the orbit of $x$, $O^+(x)$ the positive orbit (i.e. $O^+(x) := \oplus{ v_t(x) \mid t > 0 \oplus}$), $O^-(x)$ the negative orbit (i.e. $O^-(x) := \oplus{ v_t(x) \mid t < 0 \oplus}$). A subset is {\bf invariant} (or saturated) if it is a union of orbits. A point $x$ of $S$ is {\bf singular} if $x = v_t(x)$ for any $t \in \mathbb{R} $ and is {\bf periodic} if there is a positive number $T > 0$ such that $x = v_T(x)$ and $x \neq v_t(x)$ for any $t \in (0, T)$. A point is {\bf closed} if it is singular or periodic. An orbit is singular (resp. periodic, closed) if it contains a singular (resp. periodic, closed) point. Denote by $\mathop{\mathrm{Sing}}(v)$ the set of singular points and by $\mathop{\mathrm{Per}}(v)$ (resp. $\mathop{\mathrm{Cl}}(v)$) the union of periodic (resp. closed) orbits. A point is {\bf wandering} if there are its neighborhood $U$ and a positive number $N$ such that $v_t(U) \cap U = \emptyset$ for any $t > N$. A point is {\bf non-wandering} if it is not wandering (i.e. for any its neighborhood $U$ and for any positive number $N$, there is a number $t \in \mathbb{R}$ with $|t| > N$ such that $v_t(U) \cap U \neq \emptyset$). Denote by $\Omega (v)$ the set of non-wandering points, called the non-wandering set. A flow is {\bf non-wandering} if any points are non-wandering. The $\omega$-limit (resp. $\alpha$-limit) set of a point $x$ is $\omega(x) := \bigcap_{n\in \mathbb{R}}\overline{\oplus{v_t(x) \mid t > n\oplus}}$ (resp. $\alpha(x) := \bigcap_{n\in \mathbb{R}}\overline{\oplus{v_t(x) \mid t < n\oplus}}$). A {\bf separatrix} is a non-singular orbit whose $\alpha$-limit or $\omega$-limit set is a singular point. A point $x$ is {\bf Poisson stable} (or strongly recurrent) if $x \in \omega(x) \cap \alpha(x)$. A point $x$ is {\bf positively} (resp. {\bf negatively}) {\bf recurrent} (or positively (resp. negatively) Poisson stable) if $x \in \omega(x)$ (resp. $x \in \alpha(x)$), and a point $x$ is {\bf recurrent} if $x \in \omega(x) \cup \alpha(x)$. Denote by $\mathrm{R}(v)$ the set of non-closed recurrent points. An orbit is recurrent if it contains a recurrent point. Recall that a {\bf Q-set} is the closure of a non-closed recurrent orbit. Notice that a Q-set is also called a {\bf quasiminimal set}, and that a Q-set need not be orientable (see such examples \cite[Theorem~1]{gutierrez1978smooth}). An orbit is {\bf proper} if it is embedded, {\bf locally dense} if its closure has a nonempty interior, and {\bf exceptional} if it is neither proper nor locally dense. A point is proper (resp. locally dense) if its orbit is proper (resp. locally dense). Denote by $\mathrm{LD}(v)$ (resp. $\mathrm{E}(v)$, $\mathrm{P}(v)$) the union of locally dense orbits (resp. exceptional orbits, non-closed proper orbits). By definitions, the union $\mathrm{P}(v)$ of non-closed proper orbits is the set of non-recurrent points, and that $\mathrm{R}(v) = \mathrm{LD}(v) \sqcup \mathrm{E}(v)$, where $\sqcup$ denotes a disjoint union (cf. \cite[\S 2.2.1]{yokoyama2021density}). Moreover, we have a decomposition $S = \mathop{\mathrm{Cl}}(v) \sqcup \mathrm{P}(v) \sqcup \mathrm{R}(v) = \mathop{\mathrm{Sing}}(v) \sqcup \mathop{\mathrm{Per}}(v) \sqcup \mathrm{P}(v) \sqcup \mathrm{LD}(v) \sqcup \mathrm{E}(v)$ for a flow $v$ on a surface $S$. Every non-wandering flow on a compact surface has no exceptional orbits (i.e. $\mathrm{E}(v) = \emptyset$) because of \cite[Lemma~2.3]{yokoyama2016topological}. \subsubsection{Quasi-circuits and quasi-Q-sets} A closed connected invariant subset is a {\bf non-trivial quasi-circuit} if it is a boundary component of an open annulus, contains a non-recurrent orbit, and consists of non-recurrent orbits and singular points. A non-trivial quasi-circuit $\gamma$ is a {\bf quasi-semi-attracting quasi-circuit} if there is a point $x \in \mathbb{A}$ with $O^+(x) \subset \mathbb{A}$ such that $\omega(x) = \gamma$. Moreover, a quasi-circuit is not a circuit in general. The transversality for a continuous flow can be defined using tangential spaces of surfaces, because each flow on a compact surface is topologically equivalent to a $C^1$-flow by Gutierrez's smoothing theorem~\cite{gutierrez1978structural}. An $\omega$-limit (resp. $\alpha$-limit) set of a point is a {\bf quasi-Q-set} if it intersects an essential closed transversal infinitely many times. A quasi-Q-set is not Q-set in general, but a Q-set is a quasi-Q-set \cite[Lemma~3.8]{yokoyama2021poincare}. \subsubsection{Types of singular points} A point $x$ is a {\bf center} if, for any its neighborhood $U$, there is an invariant open neighborhood $V \subset U$ of $x$ such that $U - \oplus{ x \oplus}$ is an open annulus that consists of periodic orbits, as in the left on Figure~\ref{multi-saddles}. A {\bf $\bm{\partial}$-$\bm{k}$-saddle} (resp. {\bf $\bm{k}$-saddle}) is an isolated singular point on (resp. outside of) $\partial S$ with exactly $(2k + 2)$-separatrices, counted with multiplicity as in Figure~\ref{multi-saddles}. \begin{figure} \caption{A center and examples of multi-saddles} \label{multi-saddles} \end{figure} A {\bf multi-saddle} is a $k$-saddle or a $\partial$-$(k/2)$-saddle for some $k \in \mathbb{Z}_{\geq 0}$. A $1$-saddle is topologically an ordinary saddle, and a $\partial$-$(1/2)$-saddle is topologically a $\partial$-saddle. The union of multi-saddles and their separatrices is called the {\bf multi-saddle connection diagram}. Any connected components of the multi-saddle connection diagram are called {\bf multi-saddle connections}. \subsubsection{Hamiltonian flows} A $C^r$ vector field $Y$ for any $r \in \mathbb{Z} _{\geq0}$ on an orientable surface $\Sigma$ is {\bf Hamiltonian} if there is a $C^{r+1}$ function $H \colon \Sigma \to \mathbb{R}$, called the {\bf Hamiltonian}, such that $dH= \omega(Y, \cdot )$ as a one-form, where $\omega$ is a volume form of $\Sigma$. In other words, locally the Hamiltonian vector field $X$ is defined by $Y = (\partial H/ \partial x_2, - \partial H/ \partial x_1)$ for any local coordinate system $(x_1,x_2)$ of a point $p \in \Sigma$. A flow is {\bf Hamiltonian} if it is topologically equivalent to a flow generated by a Hamiltonian vector field. \subsubsection{Trivial flow boxes, flow boxes, periodic annuli, and transverse annuli} A {\bf trivial flow box} is homeomorphic to $[0,1]^2$ each of whose orbit arcs correspond to $[0,1] \times \oplus{t \oplus}$ for some $t \in [0,1]$ as on the left of Figure~\ref{fig:local}. \begin{figure} \caption{Left, trivial flow box; second from left, flow box; second from right, periodic annulus; right, transverse annulus.} \label{fig:local} \end{figure} A closed subset $D$ is a {\bf flow box} with respect to a flow $v$ if there is a homeomorphism $h \colon D \to [0,1]^2$ such that every connected component of the intersection $U \cap h^{-1}([0,1] \times \oplus{t \oplus})$ for any $t \in [0,1]$ is an orbit arc of $v$ as on second from left of Figure~\ref{fig:local}, where $U := h^{-1}([0,1]^2 - [1/4,3/4]^2)$ is a neighborhood of the boundary $\partial D$ in $D$. A annulus $U$ is a {\bf periodic annulus} if there is a non-degenerate interval $I$ and there is a homeomorphism $h \colon U \to I \times \mathbb{S}^1$ such that the inverse image $h^{-1}(\oplus{ t \oplus} \times \mathbb{S}^1)$ for any $t \in I$ is a periodic orbit of $v$ as on second from right of Figure~\ref{fig:local}. A subset $U$ is an {\bf open transverse annulus} if there is a homeomorphism $h \colon U \to (0,1) \times \mathbb{S}^1$ such that the intersection $U \cap h^{-1}((0,1) \times \oplus{ \theta \oplus})$ for any $\theta \in \mathbb{S}^1$ is an orbit arc as on the right of Figure~\ref{fig:local}. \section{Proofs of main results and a remark on gradient cases} We have the following observation. \begin{lemma}\label{lem:3.0} Let $v$ be a flow with arbitrarily many singular points on a surface $S$ and $x$ a point in $S$. If $\omega(x) \subseteq \mathop{\mathrm{Sing}}(v)$ {\rm(resp.} $\alpha(x) \subseteq \mathop{\mathrm{Sing}}(v)${\rm)} then $\omega(x)$ {\rm(resp.} $\alpha(x)${\rm)} is a nowhere dense subset of singular points. \end{lemma} \begin{proof} If $x$ is singular, then $\omega(x) = \alpha (x) = \oplus{ x \oplus}$. Thus we may assume that $O(x)$ is not closed. Suppose that $\omega(x) \subseteq \mathop{\mathrm{Sing}}(v)$. By $O(x) \cap \mathop{\mathrm{Sing}}(v) = \emptyset$, we have $\omega(x) \subseteq \partial \mathop{\mathrm{Sing}}(v)$. Since the boundary $\partial \mathop{\mathrm{Sing}}(v)$ is closed and contains no interior, the $\omega$-limit set $\omega(x)$ is a nowhere dense subset of singular points. By the same argument, if $\alpha(x) \subseteq \mathop{\mathrm{Sing}}(v)$, then $\alpha(x)$ is a nowhere dense subset of singular points. \end{proof} We have the following statement, whose proof is analogous to the proof of \cite[Proposition~2.6]{yokoyama2016topological}. \begin{lemma}\label{lem:3.1} Let $v$ be a flow with arbitrarily many singular points on a compact surface $S$. Then the following statements hold for a positively recurrent point $x$ in $\mathrm{LD}(v)$ and any point $y \in \overline{O(x)}$: \oplus {\rm(1)} If $y$ is not positively recurrent, then $\omega(y) \subseteq \mathop{\mathrm{Sing}}(v)$. \oplus {\rm(2)} If $y$ is not negatively recurrent, then $\alpha(y) \subseteq \mathop{\mathrm{Sing}}(v)$. \oplus {\rm(3)} If $y \in \mathrm{P}(v)$, then $\alpha(y) \cup \omega(y) \subseteq \mathop{\mathrm{Sing}}(v)$. \end{lemma} \begin{proof} We may assume that the surface $S$ is connected. By taking a double covering of $S$ and the doubling of $S$ if necessary, we may assume that $S$ is closed and orientable. From \cite[Proposotion~2.3.11]{Hector1981foliation}, the restriction $v|_{S - \mathop{\mathrm{Sing}}(v)}$ of the flow $v$ is transversally orientable as a foliation. Let $x$ be a positively recurrent point $x$ in $\mathrm{LD}(v)$ and a point $y \in \overline{O(x)}$. Then $O(y) \subset \omega(x) = \overline{O(x)}$. If $y \in \mathop{\mathrm{Sing}}(v)$, then $\oplus{ y \oplus} = \alpha(y) = \omega(y) \subseteq \mathop{\mathrm{Sing}}(v)$. Thus we also may assume that $y \notin \mathop{\mathrm{Sing}}(v)$. \cite[Proposition~2.2 and Lemma~2.3]{yokoyama2016topological} imply that $\overline{O(x)} \cap (\mathop{\mathrm{Per}}(v) \sqcup \mathrm{E}(v)) = \emptyset$ and so that $y \in \overline{O(x)} \subseteq \mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v) \sqcup \mathrm{LD}(v)$. Therefore $y \in \mathrm{P}(v) \sqcup \mathrm{LD}(v)$. Since $\omega(y) = \omega(y_-)$ and $\alpha(y) = \alpha(y_-)$ for any point $y_- \in O^-(y)$, by replacing $y$ with a point in $O^-(y)$ if necessary, we may assume that $x \notin O^-(y) \sqcup \oplus{ y \oplus}$. Then $y \in (\mathrm{P}(v) \sqcup \mathrm{LD}(v)) - \oplus{x\oplus}$. By the flow box theorem for a continuous flow on a compact surface (cf. \cite[Theorem~1.1, p.45]{aranson1996introduction}), there is a trivial flow box centered at the non-singular point $y$ as on the left in Figure~\ref{fig:flowbox_sing}. \begin{figure} \caption{Replacement of a trivial flow box on a disk into a flow fox with a singular point, which is a fake saddle.} \label{fig:flowbox_sing} \end{figure} Replacing a trivial flow box on a disk into a flow fox with a singular point which is $y$ as on the right in Figure~\ref{fig:flowbox_sing}, denote by $v_1$ the resulting flow on the surface $S$. Notice that $O_v(z) = O_{v_1}(z)$ for any $z \notin O_v(y)$. Similarly, $O^+_v(z) = O^+_{v_1}(z)$ for any $z \notin O_v^-(y)$, and $O^-_v(z) = O^-_{v_1}(z)$ for any $z \notin O_v^+(y)$. For any $y_- \in O_v^-(y)$, we have $O_v^-(y) = O_{v_1}(y_-)$. Similarly, for any $y_+ \in O_v^+(y)$, we obtain $O_v^+(y) = O_{v_1}(y_+)$. By $x \notin O^-(y)$, we have that $O_v^+(x) = O_{v_1}^+(x)$ and so that $O_v^-(y) \subset \overline{O_v(x)} = \omega_v(x) = \omega_{v_1}(x) = \overline{O_{v_1}(x)}$. Then $x \in \mathrm{LD}(v_1)$. \cite[Proposition~2.2 and Lemma~2.3]{yokoyama2016topological} imply that $\overline{O_{v_1}(x)} \cap (\mathop{\mathrm{Per}}(v_1) \sqcup \mathrm{E}(v_1)) = \emptyset$ and so that $\overline{O_{v_1}(x)} \subseteq \mathop{\mathrm{Sing}}(v_1) \sqcup \mathrm{P}(v_1) \sqcup \mathrm{LD}(v_1)$. Moreover, $\overline{O_{v_1}(x)} \cap \mathrm{LD}(v_1) = \overline{O_{v_1}(x)} \setminus (\mathop{\mathrm{Sing}}(v_1) \sqcup \mathrm{P}(v_1)) = \oplus{ z \in S \mid \overline{O_{v_1}(x)} = \overline{O_{v_1}(z)} \oplus} = \oplus{ z \in S \mid \overline{O_{v_1}(x)} = \overline{O_{v_1}^+(z)} \text{ or } \overline{O_{v_1}(x)} = \overline{O_{v_1}^-(z)} \oplus}$ and $O_v^-(y) \subset \overline{O_{v_1}(x)} \setminus \mathop{\mathrm{Sing}}(v_1) \subseteq (\mathrm{P}(v_1) \sqcup \mathrm{LD}(v_1)) \cap \overline{\mathrm{LD}(v_1)}$. Suppose that $y$ is not positive recurrent (i.e. $y \notin \omega_v(y)$) with respect to $v$. If $x \in O_v^+(y) \sqcup \oplus{ y \oplus}$, then $y \in \overline{O_v(x)} = \omega_v(x) = \omega_v(y)$, which contradicts that $y$ is not positively recurrent with respect to $v$. If $x \in \overline{O_v^+(y)} - (O_v^+(y) \sqcup \oplus{ y \oplus})$, then $x \in \omega(y)$ and so $y \in \overline{O_v(x)} = \omega_v(x) \subseteq \omega_v(y)$, which contradicts that $y$ is not positively recurrent with respect to $v$. Thus $x \notin \overline{O_v^+(y)}$. Since $x \notin O_v^-(y) \sqcup \oplus{ y \oplus}$, we obtain $x \notin O_v(y)$. Then $O_v(x) = O_{v_1}(x)$ and so $\overline{O_{v_1}(y_+)} = \overline{O_v^+(y)} \subsetneq \overline{O_{v}(x)} = \overline{O_{v_1}(x)} = \omega_{v_1}(x)$ for any $y_+ \in O_v^+(y)$. From $x \in \mathrm{LD}(v_1)$, applying \cite[Proposition~2.2]{yokoyama2016topological} to $v_1$, we have $O_v^+(y) = O_{v_1}(y_+) \subseteq \overline{O_{v_1}(x)} \setminus (\mathop{\mathrm{Sing}}(v_1) \sqcup \mathrm{LD}(v_1)) \subseteq \mathrm{P}(v_1)$ for any $y_+ \in O_v^+(y)$. Since $O_v^+(y) \subseteq \mathrm{P}(v_1)$, applying \cite[Theorem~3.1]{marzougui1996} to the restriction of $v_1$ to the complement $S - \mathop{\mathrm{Sing}}(v_1) = S - (\mathop{\mathrm{Sing}}(v) \sqcup \oplus{y \oplus})$ which is the orientable open surface equipped with the foliation $\oplus{ O_{v_1}(z) \mid z \in S - \mathop{\mathrm{Sing}}(v_1) \oplus}$, the orbit $O_{v_1}(y_+)$ of $v_1$ for any $y_+ \in O_v^+(y)$ is closed in the complement $S - \mathop{\mathrm{Sing}}(v_1)$ and so $\omega_v(y) = \omega_{v_1}(y_+) \subset \mathop{\mathrm{Sing}}(v_1) - \oplus{ y \oplus} = \mathop{\mathrm{Sing}}(v)$, because $y \notin \omega_v(y)$. This implies assertion {\rm(1)}. Suppose that $y$ is not negatively recurrent (i.e. $y \notin \alpha_v (y)$) with respect to $v$. If $x \in \overline{O_v^-(y)} - (O_v^-(y) \sqcup \oplus{y \oplus})$, then $x \in \alpha_v(y)$ and so $y \in \overline{O_v(x)} \subseteq \alpha_v(y)$, which contradicts that $y$ is not negatively recurrent with respect to $v$. Since $x \notin O_v^-(y) \sqcup \oplus{ y \oplus}$, we obtain $x \notin \overline{O_v^-(y)}$. Then $O^+_v(x) = O^+_{v_1}(x)$ and so that $O_{v_1}(y_-) = O_v^-(y) \subsetneq \overline{O_v(x)} = \omega_{v}(x) = \omega_{v_1}(x)$ for any $y_- \in O_v^-(y)$. From $x \in \mathrm{LD}(v_1)$, applying \cite[Proposition~2.2]{yokoyama2016topological} to $v_1$, we have $O_v^-(y) = O_{v_1}(y_-) \subseteq \overline{O_{v_1}(x)} \setminus (\mathop{\mathrm{Sing}}(v_1) \sqcup \mathrm{LD}(v_1)) \subseteq \mathrm{P}(v_1)$ for any $y_- \in O_v^-(y)$. Since $O_v^-(y) \subseteq \mathrm{P}(v_1)$, applying \cite[Theorem~3.1]{marzougui1996} to the restriction of $v_1$ to the complement $S - \mathop{\mathrm{Sing}}(v_1) = S - (\mathop{\mathrm{Sing}}(v) \sqcup \oplus{y \oplus})$ which is an orientable open surface equipped with an orientable and transversally orientable foliation $\oplus{ O_{v_1}(z) \mid z \in S - \mathop{\mathrm{Sing}}(v_1) \oplus}$, the orbit $O_{v_1}(y_-)$ of $v_1$ for any $y_- \in O_v^-(y)$ is closed in the complement $S - \mathop{\mathrm{Sing}}(v_1)$ and so $\alpha_v(y) = \alpha_{v_1}(y_-) \subset \mathop{\mathrm{Sing}}(v_1) - \oplus{ y \oplus} = \mathop{\mathrm{Sing}}(v)$, because $y \notin \alpha_v(y)$. This implies assertion {\rm(2)}. Suppose that $y \in \mathrm{P}(v)$. Then $y$ is not recurrent with respect $v$. Therefore assertions {\rm(1)} and {\rm(2)} imply $\alpha_v(y) \cup \omega_v(y) \subseteq \mathop{\mathrm{Sing}}(v)$. \end{proof} We demonstrate the main results for the $\omega$-limit sets as follows. \begin{proof}[Proof of Theorem~\ref{lem:ld}] Let $v$ be a flow with arbitrarily many singular points on a compact surface $S$ and $y \in \mathrm{LD}(v)$. \cite[Theorem~VI]{cherry1937topological} implies that the Q-set $\overline{O(y)}$ contains infinitely many Poisson stable orbits $O$ with $\overline{O} = \overline{O(y)}$. Fix a Poisson stable point $x \in S$ with $\overline{O(x)} = \overline{O(y)}$. Then $x \in \mathrm{LD}(v)$ and $y \in \overline{O(x)} = \omega(x)$. If $y$ is not positively recurrent, then Lemma~\ref{lem:3.1} implies that $\omega(y) \subseteq \mathop{\mathrm{Sing}}(v)$ is a nowhere dense subset. Thus we may assume that $y$ is positively recurrent. Then $\omega(y) = \overline{O(y)}$ is a locally dense Q-set. Therefore the $\omega$-limit set $\omega(x)$ is either a nowhere dense subset of singular points or a locally dense Q-set. \end{proof} \begin{proof}[Proof of Theorem~\ref{lem:nw}] Let $v$ be a non-wandering flow with arbitrarily many singular points on a compact surface $S$. By \cite[Lemma 2.4]{yokoyama2016topological}, we have that $S = \mathop{\mathrm{Cl}}(v) \sqcup \mathrm{P}(v) \sqcup \mathrm{LD}(v) = \mathop{\mathrm{Sing}}(v) \cup \overline{\mathop{\mathrm{Per}}(v) \sqcup \mathrm{LD}(v)}$. Fix a non-closed point $x \in S$. Suppose that $x \in \mathrm{LD}(v)$. If $x \in \omega(x)$, then $\omega(x) = \overline{O(x)}$ is a locally dense Q-set. Thus we may assume that $x \notin \omega(x)$. Then $x \in \alpha(x)$. By \cite[Theorem~VI]{cherry1937topological}, there is a Poisson stable point $x' \in \mathrm{LD}(v)$ with $\overline{O(x)} = \overline{O(x')}$. Since $x \in \overline{O(x')}$ is not positively recurrent, Lemma~\ref{lem:3.1} implies that the $\omega$-limit set $\omega(x) \subseteq \mathop{\mathrm{Sing}}(v)$ is nowhere dense. Suppose that $x \notin \mathrm{LD}(v)$. Then $x \in S - (\mathop{\mathrm{Cl}}(v) \sqcup \mathrm{LD}(v)) = \mathrm{P}(v)$. From \cite[Proposition~2.6]{yokoyama2016topological}, we have $\mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v) = \oplus{ y \in S \mid \omega(y) \cup \alpha(y) \subseteq \mathop{\mathrm{Sing}}(v) \oplus}$. Then the $\omega$-limit set $\omega(x)$ consists of singular points. By Lemma~\ref{lem:3.0}, the $\omega$-limit set $\omega(x) \subseteq \mathop{\mathrm{Sing}}(v)$ is nowhere dense. \end{proof} \subsection{Characterization of the $\omega$-limit set of any non-closed orbit for any non-wandering flow on a surface} \label{sec:nw} We have the following nonexistence of no non-closed recurrent orbits for Hamiltonian flows on (possibbly non-compact) surfaces. \begin{proof}[Proof of Theorem~\ref{cor:ham}] Let $v$ be a Hamiltonian flow with arbitrarily many singular points on a surface $S$. By replacing $v$ with a flow which is topologically equivalent to $v$ and is generated by a Hamiltonian vector field, we may assume that there is a Hamiltonian $H \colon S \to \mathbb{R} $ generating $v$. Fix a non-closed point $x \in S$. Then the inverse image $H^{-1}(H(x))$ is a closed subset containing $\overline{O(x)}$. By the existence of the Hamiltonian, the point $x$ is proper and so $x \notin \omega(x) \cup \alpha(x)$. The invariance of $\omega(x)$ implies $O(x) \cap \omega(x) = \emptyset$. We claim that $\omega(x)$ consists of singular points. Indeed, assume that there is a non-singular point $y \in \omega(x)$. From $y \in \omega(x) \subseteq \overline{O(x)} \subseteq H^{-1}(H(x))$, we have $H(x) = H(y)$. By the flow box theorem for a continuous flow on a compact surface (cf. Theorem 1.1, p.45\cite{aranson1996introduction}), there is a closed disk $U$ which can be identified with $[-1,1] \times [-1,1]$ such that $y$ corresponds to the origin $(0,0)$ and the arc $C_t := [-1,1] \times \oplus{ t \oplus}$ for any $t \in [-1, 1]$ is contained in some orbit. By the definition of Hamiltonian vector field, for any $t \in [-1,1]$, there is a number $r_t \in \mathbb{R} $ with $H(C_t) = \oplus{r_t\oplus}$ and the function $r_{\cdot} \colon [-1,1] \to \mathbb{R} $ defined by $r_\cdot(t) := r_t$ is strictly increasing or decreasing. Moreover, we have $H(y) = r_0 \in \mathbb{R} $. On the other hand, since $O(x) \cap \omega(x) = \emptyset$, we obtain that $y \in \omega(x) \subseteq \overline{O(x)} - O(x)$ and so that the orbit $O^+(x)$ intersects $U - C_0 = U \setminus O(y)$. Then there is a positive number $t^+ \in \mathbb{R} _{>0}$ such that $v_{t^+}(x) \in U - C_0$. This means that $H(v_{t^+}(x)) \neq r_0 = H(y) = H(x)$, which contradicts $H(x) = H(v_{t^+}(x))$. \end{proof} \subsection{A remark on an analogous statement for gradient flows}\label{sec:grad} Notice that a similar statement holds for a gradient flow with arbitrarily many singular points on a (possibly non-compact) surface. To state more precisely, recall the definition of gradient flows. A vector field $X$ on a Riemannian manifold $(M, g)$ is a {\bf smooth gradient vector} field if there is a $C^\infty$ function $f$ on $M$ with $g(X, \cdot) = d f$. A flow is {\bf gradient} if it is topologically equivalent to a flow generated by a smooth gradient vector field. Then we have the following statement. \begin{proposition}\label{lem:grad} The $\omega$-limit set of any non-closed orbit for a gradient flow with arbitrarily many singular points on a surface consists of singular points. \end{proposition} \begin{proof} Let $v$ be a gradient flow with arbitrarily many singular points on a surface $S$. By replacing $v$ with a flow which is topologically equivalent to $v$ and is generated by a gradient vector field, we may assume that there is a height function $h \colon S \to \mathbb{R} $ generating the gradient flow $v$. By the existence of the height function $h$, any orbits of $v$ are proper and non-periodic. Then $S = \mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v)$. Fix a point $x \in S - \mathop{\mathrm{Sing}}(v) = \mathrm{P}(v)$. We claim that $\omega(x) \subseteq \mathop{\mathrm{Sing}}(v)$. Indeed, assume that there is a point $y \in \omega(x) \cap \mathrm{P}(v)$. Then there are a closed transverse arc $I$ with $y \in I$ and an increasing sequence $(t_n)_{n \in \mathbb{Z} _{>0}}$ of $t_n \in \mathbb{R} _{>0}$ with $\lim_{n \to \infty} t_n = \infty$ such that $v_{t_n}(x) \in I$ and $y = \lim_{n \to \infty} v_{t_n}(x)$. By the existence of the height function $h \colon S \to \mathbb{R} $ of $v$, the value of $y$ is finite. Since $I$ is compact, the norms of the vector field $X_h := - \operatorname{grad} h$ generated by $h$ on some compact neighborhood $U$ of $I$ are separated from zero (i.e. $\min_{z \in U}|X_h(z)| > 0$). This means that $h(y) = \lim_{n \to \infty} h(v_{t_n}(x)) = - \infty$, which contradicts $h(y) \in \mathbb{R} $. Thus $\omega(x) \subseteq S - \mathrm{P}(v) = \mathop{\mathrm{Sing}}(v)$. \end{proof} \section{On pre-Hamiltonian flows} We have the following statements. \begin{lemma}\label{prop:characterization_ham_finite} Every Hamiltonian flow on an orientable compact surface is pre-Hamiltonian. \end{lemma} \begin{proof} Let $v$ be a Hamiltonian flow on an orientable compact surface $S$. By definition of Hamiltonian flow, there is a Hamiltonian $h \colon S \to \mathbb{R} $ whose Hamiltonian vector field generates a flow $w$ which is topologically equivalent to $v$ via a topological conjugacy $k \colon S \to S$. By Theorem~\ref{cor:ham} and definition of Hamiltonian vector field, any orbits of the restriction $w|_{S - \mathop{\mathrm{Sing}}(w)}$ are closed in the surface $S -\mathop{\mathrm{Sing}}(w)$ and are connected components of level sets of the restriction $h|_{S - \mathop{\mathrm{Sing}}(w)}$. From the flow box theorem for a continuous flow on a compact surface, the set of orbits of $v|_{S - \mathop{\mathrm{Sing}}(w)}$ is a codimension one foliation on the surface $S - \mathop{\mathrm{Sing}}(w)$. Using the topological conjugacy $k \colon S \to S$ from $v$ to $w$, the composition $H := h \circ k$ is a desired continuous function $H \colon S \to \mathbb{R} $. \end{proof} \begin{lemma}\label{lem:totally_ham_finite} Let $v$ be a pre-Hamiltonian flow on an orientable compact surface $S$. If the image of $\mathop{\mathrm{Sing}}(v)$ by the pre-Hamiltonian is totally disconnected, then $v$ is non-wandering and $S = \mathop{\mathrm{Sing}}(v) \cup \overline{\mathop{\mathrm{Per}}(v)}$. \end{lemma} \begin{proof} Let $v$ be a flow on an orientable compact surface $S$ and $H \colon S \to \mathbb{R} $ the pre-Hamiltonian of $v$. Since $\mathop{\mathrm{Sing}}(v)$ is closed and so compact, the image $H(\mathop{\mathrm{Sing}}(v))$ is compact and so closed. We claim that $S - H^{-1}(H(\mathop{\mathrm{Sing}}(v)))$ consists of periodic orbits. Indeed, fix any point $x \in S$ with $H(x) \notin H(\mathop{\mathrm{Sing}}(v))$. Then the orbit $O(x)$ is a connected component of $H^{-1}(H(x))$ and so is a closed subset and so compact because $S$ is compact. Since $x$ is not a singular point, the orbit $O(x)$ is a periodic orbit. Fix any point $y \in H^{-1}(H(\mathop{\mathrm{Sing}}(v))) - \mathop{\mathrm{Sing}}(v)$. Then the orbit $O(y)$ is a connected component of $H^{-1}(H(y)) \setminus \mathop{\mathrm{Sing}}(v)$. The continuity of $H$ implies that every neighborhood of $O(y)$ in $S - \mathop{\mathrm{Sing}}(v)$ is not contained in $H^{-1}(H(y))$. By the totally disconnectivity of $H(\mathop{\mathrm{Sing}}(v))$, every neighborhood of $O(y)$ in $S - \mathop{\mathrm{Sing}}(v)$ intersects $H^{-1}(\mathbb{R} - H(\mathop{\mathrm{Sing}}(v))) = S - H^{-1}(H(\mathop{\mathrm{Sing}}(v))) \subseteq \mathop{\mathrm{Per}}(v)$. This means that $H^{-1}(H(\mathop{\mathrm{Sing}}(v))) -\mathop{\mathrm{Sing}}(v) \subset \overline{\mathop{\mathrm{Per}}(v)}$ and so that $S = \mathop{\mathrm{Sing}}(v) \cup \overline{\mathop{\mathrm{Per}}(v)}$. Therefore $v$ is non-wandering. \end{proof} The total disconnectivity of the image $H(\mathop{\mathrm{Sing}}(v))$ of the singular point set by the pre-Hamiltonian is necessary and can not be replaced by one of the singular point set $\mathop{\mathrm{Sing}}(v)$ in the previous lemma (see Corollary~\ref{cor:46}). To show Theorem~\ref{cor:characterization_ham_finite}, we recall the following concept. The {\bf extended orbit space $\bm{{S}/{v_{\mathrm{ex}}}}$} is a quotient space $S/\sim$ defined by $x \sim y$ if either $x$ and $y$ are contained in a multi-saddle connection or there is an orbit that contains $x$ and $y$ but is not contained in any multi-saddle connections. We prove Theorem~\ref{cor:characterization_ham_finite} as follows. \begin{proof}[Proof of Theorem~\ref{cor:characterization_ham_finite}] By Lemma~\ref{prop:characterization_ham_finite}, the assertion {\rm(1)} implies the assertion {\rm(2)}. Let $v$ be a pre-Hamiltonian flow on an orientable compact surface $S$, $H \colon S \to \mathbb{R} $ the pre-Hamiltonian, and $\mathcal{F} $ the foliation on $S - \mathop{\mathrm{Sing}}(v)$ induced by $H|_{S - \mathop{\mathrm{Sing}}(v)}$. By the Baire category theorem, any leaves of $\mathcal{F} $ have the empty interior. From Lemma~\ref{lem:totally_ham_finite}, the flow $v$ is non-wandering and $S = \mathop{\mathrm{Sing}}(v) \cup \overline{\mathop{\mathrm{Per}}(v)}$. By \cite[Theorem 3]{cobo2010flows}, any singular points of $v$ are either centers or multi-saddles. We claim that there are no locally dense orbits. Indeed, assume that there is a locally dense orbit $O$. Then the closure $\overline{O}$ has a nonempty interior. Put $c := H(O) \in \mathbb{R} $. Since $\overline{O} \subseteq H^{-1}(H(O)) = H^{-1}(c)$, there is a connected component $L$ of the inverse image $H^{-1}(c)$ has a nonempty interior. By definition of pre-Hamiltonian, the connected component $L$ is a leaf of the codimension one foliation $\mathcal{F} $, which contradicts that every leaf has the empty interior. We claim that the extended orbit space ${S}/{v_{\mathrm{ex}}}$ is a directed graph without directed cycles. Indeed, from \cite[Lemma 3.1]{yokoyama2021relations}, the multi-saddle connection diagram is the complement of the union of centers and $\mathop{\mathrm{Per}}(v)$, and the extended orbit space ${S}/{v_{\mathrm{ex}}}$ is a finite directed topological graph. Therefore any multi-saddle connections are connected components of $S - \mathop{\mathrm{Per}}(v)$. Since any level sets are closed, each multi-saddle connection is contained in some level set of $H$. This means that the quotient map $p_{\mathrm{ex}} \colon S \to {S}/{v_{\mathrm{ex}}}$ can be obtained by collapsing connected components of level sets of $H$ into singletons. Therefore there is an order-preserving continuous mapping $h \colon {S}/{v_{\mathrm{ex}}} \to \mathbb{R} $ with $H = h \circ p_{\mathrm{ex}}$. This implies that the directed graph ${S}/{v_{\mathrm{ex}}}$ has no directed cycles. Since $v$ is a non-wandering flow without locally dense orbits whose extended orbit space ${S}/{v_{\mathrm{ex}}}$ is a directed graph without directed cycles, from \cite[Theorem~B]{yokoyama2021relations}, the flow $v$ is Hamiltonian. \end{proof} \section{Wildness of totally disconnected singular point sets} We construct a non-trivial flow box with totally disconnected singular points that interrupts a flow. \subsection{Non-trivial flow box with totally disconnected singular points}\label{ex} Recall that the {\bf Minkowski sum $\bm{A+B}$} is defined by $A+B := \oplus{a+b \mid a \in A, b \in B\oplus}$. Steinhaus shown that $\mathcal{C} + \mathcal{C} = [0,2]$, where $\mathcal{C}$ is the Cantor ternary set \cite{steinhaus1917new}. Using the Whitney theorem, we construct the following non-trivial flow box with totally disconnected singular points. \begin{lemma}\label{lem:tatally_disconnected} Consider a vector field $X = \partial /\partial x^1 = (1,0)$ on a closed square $D := [-1,2]^2$ and let $\mathbb{D}^2 := [0,1]^2 \subset D$. Then there is a smooth function $f \colon D \to [0,1]$ with $D - [-1/2, 3/2]^2 \subset f^{-1}(1)$ satisfying the following conditions: \oplus {\rm(1)} The square $\mathbb{D}^2$ contains a wandering domain with respect to $fX$. \oplus {\rm(2)} For any point $x \in \mathbb{D}^2$, we have either $O^+_{fX}(x) \subset \mathbb{D}^2$ or $O^-_{fX}(x) \subset \mathbb{D}^2$. \oplus {\rm(3)} The intersection $\operatorname{Sing}(fX)$ is totally disconnected. \end{lemma} \begin{proof} Set $\mathcal{M} := \oplus{ (x, x + y) \mid x,y \in \frac{1}{2} \mathcal{C} \oplus} = \bigcup_{x \in \frac{1}{2} \mathcal{C}} \oplus{ x\oplus} \times \left( \oplus{ x \oplus} + \frac{1}{2}\mathcal{C} \right) \subset [0,1/2] \times [0,1] \subset \mathbb{D}^2$. By construction of $\mathcal{M}$, the subset $\mathcal{M}$ is a closed subset. By Whitney theorem \cite{Whitney1934analytic} (cf. \cite[Theorem~1.1.4]{Krantz1999geometry}), there is a $C^\infty$ function $f_0 \colon D \to [0,1]$ with $f_0^{-1}(0) = \mathcal{M}$. Using a bump function $\varphi \colon D \to [0,1]$ with $[0,1]^2 \subset \varphi^{-1}(0)$ and $D - [-1/2, 3/2]^2 \subset \varphi^{-1}(1)$, a $C^\infty$ function $f := f_0 (1-\varphi) + \varphi \colon D \to [0,1]$ satisfies $f^{-1}(0) = \mathcal{M}$ and $D - [-1/2, 3/2]^2 \subset f^{-1}(1)$. Then the vector field $fX$ is desired. \end{proof} Using such a non-trivial flow box interrupting a flow, we show the deformation to create wandering domains. \begin{proof}[Proof of Theorem~\ref{prop:45}] By Lemma~\ref{lem:tatally_disconnected}, we can replace a trivial flow box $D$ with the non-trivial one as in Lemma~\ref{lem:tatally_disconnected} by multiplying a smooth function $f \colon S \to [0,1]$ such that $f = 1$ outside of the flow box $D$, and that the restriction $f|_D$ is as in Lemma~\ref{lem:tatally_disconnected}. Thus the assertion holds. \end{proof} We have the following statement, which shows the necessity of the total disconnectivity of the image of the singular point set by the pre-Hamiltonian in Lemma~\ref{lem:totally_ham_finite}. \begin{corollary}\label{cor:46} There is a pre-Hamiltonian flow with wandering domains and totally disconnected singular points. \end{corollary} \begin{proof} Consider a Hamiltonian $h \colon \mathbb{S}^2 \to \mathbb{R} $ on the unit sphere $\mathbb{S}^2 = \oplus{ (x,y,z) \in \mathbb{R} ^3 \mid x^2+y^2+z^2 = 1 \oplus}$ defined by $h(x,y,z) = z$. Replacing a trivial flow box for the Hamiltonian vector field $X$, by Theorem~\ref{prop:45}, there is a smooth function $f \colon \mathbb{S}^2 \to [0,1]$ such that the flow $v$ generated by the vector field $fX$ is not no-wandering and the singular point set $\mathop{\mathrm{Sing}}(v)$ is totally disconnected. By construction, the function $h$ is the pre-Hamiltonian of $v$. \end{proof} \section{Existence of pairs of the $\alpha$-limit and $\omega$-limit sets} \begingroup \renewcommand{1.4}{1.4} \begin{table}[htb] \begin{center} \scalebox{0.9}{ \begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline \diagbox{$\alpha$-limit set}{$\omega$-limit set} & \begin{tabular}{c} Singular \oplus point \end{tabular} & \begin{tabular}{c} Limit\oplus circuit \end{tabular} & \begin{tabular}{c} Locally\oplus dense \oplus\mathbb{Q} -set \end{tabular} & \begin{tabular}{c} Transversely\oplus Cantor \oplus Q-set\end{tabular} \oplus \hline Singular point & 1 & 2 & 4 & 5 \oplus \hline Limit circuit & 2 & 3 & NO & 6\oplus \hline Locally dense Q-set & 4 & NO & 7 & NO \oplus \hline \begin{tabular}{c} Transversely Cantor\oplus Q-set\end{tabular} & 5 & 6 & NO & 8 \oplus \hline \end{tabular} } \end{center} \caption{``NO'' represents the non-existence of pairs of the $\alpha$-limit and $\omega$-limit sets of any non-closed orbit for a flow with finitely many singular points on a compact surface.}\label{table:01} \end{table} \endgroup \begingroup \renewcommand{1.4}{1.4} \begin{table}[htb] \begin{center} \scalebox{0.67}{ \begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline \diagbox{$\alpha$-limit set}{$\omega$-limit set } & \begin{tabular}{c} Nowhere \oplus\dense subset\oplus of \oplus $\mathop{\mathrm{Sing}}(v)$ \end{tabular} & \begin{tabular}{c} Limit\oplus cycle \end{tabular} & \begin{tabular}{c} Limit\oplus quai-\oplus circuit \end{tabular} & \begin{tabular}{c} Locally\oplus dense \oplus\mathbb{Q} -set \end{tabular} & \begin{tabular}{c} $\pitchfork$-ly\oplus Cantor \oplus Q-set\end{tabular} & \begin{tabular}{c} Quasi-Q-set in \oplus $\mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v)$\end{tabular} \oplus \hline \begin{tabular}{c} Nowhere dense subset\oplus of $\mathop{\mathrm{Sing}}(v)$ \end{tabular} & $1'$ & 2 & 2 & 4 & 5 & $5'$ \oplus \hline Limit cycle &2 & $3$ & $3'$ & NO & 6 & $6''$\oplus \hline Limit quasi-circuit & 2& $3'$ & $3'$& NO & $6'$ & $6'''$ \oplus \hline Locally dense Q-set & 4 & NO & NO & 7 & NO & NO \oplus \hline \begin{tabular}{c} $\pitchfork$-ly Cantor Q-set\end{tabular} & 5 & 6 & $6'$ & NO & 8 & $8'$ \oplus \hline \begin{tabular}{c} Quasi-Q-set in \oplus $\mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v)$ \end{tabular} &$5'$ &$6''$ &$6'''$ & NO & $8'$ & 9 \oplus \hline \end{tabular} } \end{center} \caption{``NO'' represents the non-existence of pairs of the $\alpha$-limit and $\omega$-limit sets of any non-closed orbit.}\label{table:02} \end{table} \endgroup We show the existence of non-forbidden pairs of $\alpha$-limit and $\omega$-limit sets of points as follows. \begin{proposition}\label{prop:pair} The pairs of the $\alpha$-limit and $\omega$-limit sets of points from $1$ to $9$ in Tables~\ref{table:01} and \ref{table:02} exist. \end{proposition} \begin{proof} Cases $1$ and $1'$ occur in Morse flows on closed surfaces. Cases $2$ and $3$ occur in Morse-Smale flows on closed surfaces. Replacing a limit cycle with a limit quasi-circuit consisting of a singular point (or more generally, a simply connected domain of singular points) and one non-recurrent orbit, Case $3'$ occurs in the resulting flow. From an irrational rotation on a torus, replacing an orbit with the union of a singular point and two locally dense orbits, Case $4$ occurs in the resulting flow on the torus. From the suspension flow (i.e. Denjoy flow) of the Denjoy diffeomorphism, replacing a non-recurrent orbit with the union of a singular point and two non-recurrent orbits, Case $5$ occurs in the resulting flow on the torus. From a Denjoy flow and a Morse-Smale flows with limit cycles on a sphere, replacing an open flow box consisting contained in the union of non-recurrent orbits with a flow box as in Figure~\ref{fig:cherry01} for the Denjoy flow and the Morse-Smale flow respectively, by the time reversion for one of them, Case $6$ occurs in the resulting flow on the torus. \begin{figure} \caption{A flow box on an annulus obtained from a Cherry flow box by replacing a sourc with a transverse boundary component.} \label{fig:cherry01} \end{figure} Replacing a limit cycle with a limit quasi-circuit consisting of a singular point (or more generally, a simply connected domain of singular points) and one non-recurrent orbit, Case $6'$ occurs in the resulting flow. \cite[Theorem~VI]{cherry1937topological} implies that any Q-set contains infinitely many Poisson stable orbits and so Case $7$ occurs. In particular, Case $7$ occurs in any minimal flows on a torus. From a Denjoy flow, replacing an open flow box consisting contained in the union of non-recurrent orbits with a flow box as in Figure~\ref{fig:cherry01}, taking two copies of the resulting flow and by the time reversion for one of them, Case $8$ occurs in the resulting flow on the orientable closed surface $\Sigma_2$ whose genus is two. Let $\mathcal{C}$ be the Cantor minimal set of the Denjoy diffeomorphism and $\mathcal{M}$ the transversely Cantor minimal set of the suspension flow. As in \cite[Example~5.1]{yokoyama2021poincare}, replacing the Cantor set $\oplus{0 \oplus} \times \mathcal{C}$ with singular points by using a bump function, the subset $\mathcal{M}$ becomes a quasi-Q-set $\mathcal{M}'$consisting of singular points and separatrices of the resulting flow. Replacing a copy of $\mathcal{M}$ with $\mathcal{M}'$ on the torus in Case $5$ (resp. $6$, $6'$), Case $5'$ (resp. $6''$, $6'''$) occurs in the resulting flow. Replacing a copy of $\mathcal{M}$ with $\mathcal{M}'$ on $\Sigma_2$ in Case $8$, Case $8'$ occurs in the resulting flow. Similarly, replacing $\mathcal{M}$ with $\mathcal{M}'$ for the Denjoy flow on a torus, Case $9$ occurs in the resulting flow. \end{proof} \section{Final remarks} In this section, we state three remarks. \subsection{On Question~\ref{adding_sing}} Note that the Hamiltonian property for flows on compact surfaces is invariant when only finitely many singular points are added. Indeed, consider a Hamiltonian flow $v$ on a compact surface. Adding a finitely many singular points to $v$, the resulting flow $v_1$ is pre-Hamiltonian. Theorem~\ref{cor:characterization_ham_finite} implies that the resulting flow $v_1$ is Hamiltonian. For instance, adding finitely many singular points to a closed periodic annulus, the resulting flow on the closed annulus is still Hamiltonian. On the other hand, we can break the Hamiltonian property by adding totally disconnected singular points by Theorem~\ref{prop:45}. Indeed, replacing a trivial flow box to the flow box $D$ with the vector field $fX$ constructed in Lemma~\ref{lem:tatally_disconnected}, we obtain a wandering domain for the resulting flow. For instance, consider a closed annulus $A := \mathbb{R} /3\mathbb{Z} \times [-1,2]$ and immerse $D = [-1,2]^2$ into a subset of $A$ by the canonical projection $p \colon \mathbb{R} \times [-1,2] \to A$. Equip a vector field $X := (1,0)$ on $A$ and replace $X|_D$ with $fX|_D$, where $f \colon A \to [0,1]$ is the function constructed in Lemma~\ref{lem:tatally_disconnected}. The resulting flow $v$ satisfies $A = \mathop{\mathrm{Cl}}(v) \sqcup \mathrm{P}(v)$, $\mathbb{R} /3\mathbb{Z} \times [0,1] = \mathop{\mathrm{Sing}}(v) \sqcup \mathrm{P}(v)$, and $A - (\mathbb{R} /3\mathbb{Z} \times [0,1]) = \mathop{\mathrm{Per}}(v)$ as in Figure~\ref{fig:Cantor_sing}. \begin{figure} \caption{Adding totally disconnected singular points to a trivial flow box.} \label{fig:Cantor_sing} \end{figure} \subsection{Time-reversal symmetric limit sets} In Table~\ref{table:01} and Table~\ref{table:02}, every case in the diagonal can be realized as time-reversal symmetric limit sets. Here an invariant subset $\mathcal{M}$ is a {\bf time-reversal symmetric limit set} \cite{yokoyama2023flows} if there is a point $x$ with $\mathcal{M} = \alpha(x) = \omega(x)$. More precisely, we have the following statement. \begin{proposition} For any case in the diagonal in Table~\ref{table:01} and Table~\ref{table:02}, there is a flow on a compact surface with time-reversal symmetric limit sets in the case. \end{proposition} \begin{proof} Any flows with the singular points with homoclinic separatrices on compact surfaces are desired for Case~$1$ in Table~\ref{table:01}. For Case $1'$ in Table~\ref{table:02}, we can construct flows with a time-reversal symmetric limit set which is a circle and consists of singular points. In fact, considering a Reeb component on a closed annulus as in Figure~\ref{fig:reeb_comp}, replacing the boundary into singular points, and gluing two boundary components, the resulting space is a torus and the resulting flow is continuous. By Gutierrez's smoothing theorem~\cite{gutierrez1978structural}, there is a smooth flow on a torus which is topologically equivalent to the resulting flow on the torus. \begin{figure} \caption{Modification of a Reeb component.} \label{fig:reeb_comp} \end{figure} Notice that a circle consisting of singular points for a smooth flow on a closed disk is constructed in \cite[Example~3.2]{campos1997homeomorphisms}. A flow with a time-reversal symmetric limit cycle on a torus as in Figure~\ref{fig:reeb_nontwist} is desired for Case~$3$ in Table~\ref{table:01} and Table~\ref{table:02}. \begin{figure} \caption{A toral flow with one periodic orbit and non-recurrent orbits.} \label{fig:reeb_nontwist} \end{figure} For Case~$3'$ in Table~\ref{table:02}, there is a flow with a time-reversal symmetric limit set that has an invariant Wada-Lakes-like structure. In fact, there is a time-reversal symmetric limit set for a flow on a torus which is a limit quasi-circuit and is a complement of the union of an invariant open periodic center disk and an invariant open transverse annulus (see \cite[Example~4.3]{yokoyama2023flows}). The minimal flows on a torus are desired for Case~$7$ in Table~\ref{table:01} and Table~\ref{table:02}. The Denjoy flow on a torus is desired for Case~$8$ in Table~\ref{table:01} and Table~\ref{table:02}. The flow in the proof of Proposition~\ref{prop:pair} in Case~9 is desired for Case~$9$ in Table~\ref{table:01}. \end{proof} Notice that any locally dense $\omega$-limit sets in Case~7 need be time-reversal symmetric, and that there is a more complicated example for Case $1'$. In fact, there is a non-wandering flow $v$ on an orientable closed surface consisting of singular points and locally dense orbits such that $\mathop{\mathrm{Sing}}(v)$ is a ``double of lakes of Wada continuum'' and a time-reversal symmetric limit set \cite[Example~8.2]{yokoyama2023characterizations}. This construction implies that a nowhere dense subset of singular points in Theorem~\ref{lem:ld} and Theorem~\ref{lem:nw} can become a ``double of lakes of Wada continuum''. \subsection{Construction like Cherry flow box} Using a limit quasi-circuit as above, we can construct a flow box with a $\omega$-limit set which is a limit quasi-circuit and has an invariant Wada-Lakes-like structure as on the middle in Figure~\ref{cherrybox_blowup} (see \cite[Example~4.3]{yokoyama2023flows} for details of the Wada-Lakes-like construction). \begin{figure} \caption{Upper, Blowups of a Cherry flow box into flow boxes with $\omega$-limit sets each of which has an invariant Wada-Lakes-like structure; bottom middle and bottom right, the first few steps of a Wada-Lakes-like construction.} \label{cherrybox_blowup} \end{figure} Similarly, we can also construct flow box with a $\omega$-limit set which is a subset of the singular point set and is the Lakes of Wada as on the right in Figure~\ref{cherrybox_blowup}. \noindent \textbf{Acknowledgements} The author was partially supported by JSPS Grant Number 20K03583. \end{document}

\begin{document} \title{Quantum thermal machines with single nonequilibrium environments} \author{Bruno Leggio} \affiliation{Laboratoire Charles Coulomb, UMR 5221 Universit\'{e} de Montpellier and CNRS, F- 34095 Montpellier, France} \author{Bruno Bellomo} \affiliation{Laboratoire Charles Coulomb, UMR 5221 Universit\'{e} de Montpellier and CNRS, F- 34095 Montpellier, France} \author{Mauro Antezza} \affiliation{Laboratoire Charles Coulomb, UMR 5221 Universit\'{e} de Montpellier and CNRS, F- 34095 Montpellier, France} \affiliation{Institut Universitaire de France, 103 Boulevard Saint-Michel, F-75005 Paris, France} \newcommand{\ket}[1]{\displaystyle{|#1\rangle}} \newcommand{\bra}[1]{\displaystyle{\langle #1|}} \date{\today} \begin{abstract} We propose a scheme for a quantum thermal machine made by atoms interacting with a single non-equilibrium electromagnetic field. The field is produced by a simple configuration of macroscopic objects held at thermal equilibrium at different temperatures. We show that these machines can deliver all thermodynamic tasks (cooling, heating and population inversion), and this by establishing quantum coherence with the body on which they act. Remarkably, this system allows to reach efficiencies at maximum power very close to the Carnot limit, much more than in existing models. Our findings offer a new paradigm for efficient quantum energy flux management, and can be relevant for both experimental and technological purposes. \end{abstract} \pacs{} \maketitle \section{Introduction} Recent years have seen an uprising interest in thermodynamics at atomic scale \cite{GemmerBook, Blicke2012, Brunner2012, Horodecki2013} due to the latest-generation manipulation of few, if not single, elementary quantum systems \cite{Blicke2012, Haroche2013}. In particular, out-of-equilibrium thermodynamics of quantum systems represents one of the most active research areas in the field \cite{Esposito2009, Deffner2011, Leggio2013a, Leggio2013b,Abah2014}. In this context, triggered by vast technological outcomes \cite{Scully2010, Haenggi2009}, the concept of quantum absorption thermal machine \cite{Scovil1959} has been reintroduced \cite{Linden2010, Levy2012, Correa2014, Venturelli2013, Correa2013, Brunner2014}. These machines are particularly convenient since they function without external work, extracting heat from thermal reservoirs through single atomic transitions to provide thermodynamic tasks (e.g., refrigeration). Nonetheless, fundamental issues remain unsolved. A first one is the connection a single atomic transition to given thermal reservoirs, posing serious obstacles to practical realizations of such machines. A second, more theoretical issue concerns the role of quantumness. Indeed, in typical models quantum features are not required \cite{Scovil1959,Correa2013}, and only recently the advantages of quantum properties in thermal reservoirs have been pointed out \cite{Correa2014}. The role of quantum features in the machines itself is debated \cite{Correa2013, Brunner2014}, so that the advantages of quantum machines over standard ones remains partially unclear. \begin{figure} \caption{(a) A slab of thickness $\delta$ at temperature $T_S$ is placed in the blackbody radiation of some walls at temperature $T_W$. Two atoms are placed in the resulting OTE electromagnetic field, at a distance $z$ from the slab and at a distance $r$ from each other. (b) Stationary heat fluxes between the OTE environment $E$ and each atomic transition. The OTE field also mediates for an effective atomic interaction producing an energy exchange $Q_r$ between resonant atomic transitions. Each flux contribution corresponds to a term in the atomic master equation \eqref{METQ}.} \label{model} \label{scheme} \label{system} \end{figure} In this paper we address both of these open problems by introducing a new quantum thermal machine setting, based on an out-of-thermal-equilibrium (OTE) electromagnetic bath naturally (i) coupling to each single atomic transitions, and (ii) creating quantum features in the machine. The field is produced by macroscopic objects, and acts on each atomic transition as a different thermal bath at an effective temperature, hence providing all the elements needed for quantum absorption tasks. This paper is structured as follows: the physical system is introduced in Section \ref{system}, along with the master equation governing its dynamics, while Section \ref{thermodynamics} is devoted to the introduction of thermodynamic quantities characterising the heat exchanges happening between atoms and field. In Section \ref{tasks} the first part of the results is given, concerning the action of the machines, the different tasks it can produce and their intrinsic quantum origin. The second part of the results of this work about the machine efficiency and its Carnot limit is given in Section \ref{efficiency}. Finally, remarks and conclusions are drawn in Section \ref{conclusions}. \section{Physical system}\label{system} The setup of this paper is schematically depicted in Fig. \ref{system}, where a slab of thickness $\delta$ at temperature $T_S$ is placed in the blackbody radiation emitted by some walls at temperature $T_W\neq T_S$. The total electromagnetic field embedding the space between the slab and the walls is therefore given by the sum of four contribution: the direct blackbody radiation of the walls, the radiation emitted naturally by the slab and the walls' radiation after being either reflected or transmitted by the slab. Such an OTE field has been studied in the context of Casimir-Lifshitz force and heat transfer \cite{Antezza2004, Antezza2005, Antezza2006, Messina2011}, where its properties have been characterised in terms of the field correlators through a scattering matrix approach. The slab and the walls, macroscopic objects, are here the only ones directly connected to thermal baths. In addition, a three-level atom $M$ (machine) and a two-level atom $B$ (target body) are placed at the same distance $z$ from the surface of the slab and spatially separated by a distance $r$. The atomic open system involves then four transitions: the body transition labeled as $B$ and the three machine transitions labeled as $1,2,3$. Transition $1$ connects the two lowest-lying energy eigenstates (red transition in Fig. \ref{scheme}) and transition $2$ the two highest ones (green transition in Fig. \ref{scheme}). The OTE field interacts with them through the Hamiltonian $H_I=-\sum_i\mathbf{d}_i \cdot \mathbf{E}(\mathbf{R}_i)$, where $\mathbf{d}_i$ is the dipole moment of the $i$-th transition of the atomic system and $\mathbf{E}(\mathbf{R}_i)$ is the electromagnetic field at its position $\mathbf{R}_i$. The total Hamiltonian of the system is \begin{equation} H_{tot}=H_M+H_B+H_{\mathrm{field}}+H_I, \end{equation} where $H_{\mathrm{field}}$ is the Hamiltonian of the OTE field. In the following we will not need the explicit expression of $H_{\mathrm{field}}$ since only the field correlations will enter the master equation describing the dynamics of the atoms. The free atomic Hamiltonians $H_M$ and $H_B$ have expressions \begin{eqnarray} H_B&=&\big(\omega_B +\Delta S(\omega_B)\big)\sigma^{\dag}\sigma=\widetilde{\omega}_B\sigma^{\dag}\sigma,\label{HQ}\\ H_M&=&\big(\omega_1 +\Delta S(\omega_1)+S^-(\omega_2)-S^-(\omega_3)\big)\kappa_1^{\dag}\kappa_1\nonumber \\ &+&\big(\omega_3 +\Delta S(\omega_3)+S^+(\omega_2)-S^-(\omega_1)\big)\kappa_3^{\dag}\kappa_3\nonumber\\ &=&\widetilde{\omega}_1\kappa_1^{\dag}\kappa_1+\widetilde{\omega}_3\kappa_3^{\dag}\kappa_3,\label{HT} \end{eqnarray} being $\sigma$ ($\sigma^{\dag}$) the lowering (raising) operator of the body $B$ and $\kappa_n$, $\kappa_n^{\dag}$ ($n=1,2,3$) the lowering and raising operators of $M$. $S^{\pm}(\omega_i)$ here represents a shift of the energy of each level in the $i-$th transition due to the local interaction with the field and $\Delta S(\omega)=S^+(\omega)-S^-(\omega)$. In the third equality the renormalised transition frequencies $\widetilde{\omega}_i$ for $M$ and $B$ have been introduced to account for the effects of the shifts $S^{\pm}(\omega_i)$. Throughout this work we will always assume the physical consequences of such frequency renormalisation to be negligible, such that $\widetilde{\omega}_i=\omega_i\,\,\forall i$. This assumption has been fully confirmed by extended numerical simulations, having always detected the relative error introduced by neglecting these shifts to be less than $1\%$. It is worth stressing here that, differently from previous works on atomic-scale thermal machines \cite{Linden2010, Levy2012, Correa2014, Correa2013, Brunner2014}, each atomic transition interacts here with \textit{the same} electromagnetic field, which embeds all the space where the atoms are placed. As we will show in what follows, there is then no need to conceive different environments, each interacting with a single atomic transition: a single non-equilibrium electromagnetic field is here able to produce all the physics needed for quantum thermodynamic tasks. \subsection{The master equation} In \cite{Bellomo2013} the master equation (ME) for two emitters in such a field has been derived under the Markovian limit as \begin{equation}\label{METQ} \frac{d\rho}{d t}=-\frac{i}{\hbar}\big[H_T,\rho\big]+D_{B}(\rho)+\sum_{n=1}^3D_n(\rho)+D_d(\rho), \end{equation} where $H_T=H_M+H_B+H_{MB}$. $H_{MB}=\hbar\Lambda(\omega_B)(\sigma^{\dag}\kappa_r+\sigma \kappa_r^{\dag})$ is an effective \textit{field-mediated} dipole interaction coupling resonant atomic transitions. Here we assume $B$ and $M$ to be resonant through transitions at frequency $\omega_B$, and all their dipoles to have the same magnitude and to lie along the line joining the two atoms, and oriented from $B$ to $M$. $\hbar\Lambda(\omega)$ is the effective interaction strength and $\sigma$ ($\kappa_r$) is the lowering operator of the body (of the resonant transition of the machine). $H_{MB}$ originates from the correlations of the fluctuations of atomic dipoles due to the common field. The derivation of the master equation \eqref{METQ} has been performed under the Markovian and rotating wave approximations. It involves the average photon number $n(\omega, T)=1/\big(e^{\hbar \omega/k_B T}-1\big)$ at frequency $\omega$ and temperature $T$ and the two functions $\alpha_{W(S)}$ which encompass all the properties of the environment, such as the dielectric properties of the slab and the correlation functions of the field. For their explicit expressions we refer the interested reader to \cite{Bellomo2013}. The dissipative effects due to the atom-field coupling are accounted for by the dissipators $D_k$ with expressions \begin{eqnarray} D_B(\rho)&=&\Gamma_B^+(\omega_B)\Big(\sigma\rho\sigma^{\dag}-\frac{1}{2}\big\{\sigma^{\dag}\sigma,\rho\big\}\Big)\nonumber \\ &+&\Gamma_B^-(\omega_B)\Big(\sigma^{\dag}\rho\sigma-\frac{1}{2}\big\{\sigma\sigma^{\dag},\rho\big\}\Big),\label{DQ} \end{eqnarray} \begin{eqnarray} D_n(\rho)&=&\Gamma_n^+(\omega_n)\Big(\kappa_n\rho\kappa_n^{\dag}-\frac{1}{2}\big\{\kappa_n^{\dag}\kappa_n,\rho\big\}\Big)\nonumber \\ &+&\Gamma_n^-(\omega_n)\Big(\kappa_n^{\dag}\rho\kappa_n-\frac{1}{2}\big\{\kappa_n\kappa_n^{\dag},\rho\big\}\Big),\label{DT} \end{eqnarray} \begin{eqnarray}\label{Dnonloc} D_d(\rho)&=&\Gamma_d^+(\omega_B)\Big(\kappa_{r}\rho\sigma^{\dag}-\frac{1}{2}\big\{\sigma^{\dag}\kappa_{r},\rho\big\}\Big)\nonumber \\ &+&\Gamma_d^-(\omega_B)\Big(\kappa_{r}^{\dag}\rho\sigma-\frac{1}{2}\big\{\sigma\kappa_{r}^{\dag},\rho\big\}\Big)+h.c.\label{DTQ} \end{eqnarray} where $\omega_d=\omega_B$. One recognises standard local dissipation terms ($D_B$ and $D_n$), each associated to the degrees of freedom of a well-identified atom, and non-local dissipation ($D_d$) which describes energy exchanges at frequency $\omega_B$ of the atomic system \textit{as a whole} with its OTE environment, not separable in machine or body contributions, its action involving degrees of freedom of both atoms in a symmetric way. The parameters $\Gamma_i^{\pm}(\omega_i)$ (the rates of the dissipative processes of absorption and emission of photons through local or non-local interactions) depend on local or non-local correlations of the field in the atomic positions, which in turn are functions of the temperatures $T_S$ and $T_W$ and the dielectric properties of the slab $S$ as \begin{eqnarray} \frac{\Gamma^+_i(\omega)}{\Gamma^0_i(\omega)}&=&\big[1+n(\omega,T_W)\big]\alpha_W^i(\omega)\nonumber \\ &+&\big[1+n(\omega,T_S)\big]\alpha_S^i(\omega),\\ \frac{\Gamma^-_i(\omega)}{\Gamma^0_i(\omega)}&=&n(\omega,T_W)\alpha_W^i(\omega)^*+n(\omega,T_S)\alpha_S^i(\omega)^*, \end{eqnarray} where $\Gamma^0_i(\omega)=|\mathbf{d}_i|^2\omega^3/(3\hbar \pi \varepsilon_0 c^3)$, for $i=1,2,3,B$, is the vacuum spontaneous emission rate of the $i$-th atomic transition having a dipole moment $\mathbf{d}_i$, and $\Gamma^0_d(\omega)=\sqrt{\Gamma^0_B(\omega)\Gamma_{r}^0(\omega)}$. Thanks to the functional dependence of these parameters on the frequency and on the position of the atom, and to the critical behaviour shown in correspondence to the resonance frequency $\omega_S$ of the slab material, thermodynamic tasks become achievable. To simplify the notation, in the rest of this work the explicit $\omega$-dependence in all the $\Gamma$s will be omitted. \section{Thermodynamics of the system}\label{thermodynamics} After having introduced all the dynamic effects characterizing the atomic system, we want in this Section to introduce some quantities which will characterize the machine tasks and functioning. \subsection{Environmental and population temperatures} In order to describe the machine thermodynamics, it is convenient to introduce two kind of temperatures. A first one characterizes the action of the field on the atoms: it has been shown \cite{Bellomo2012} that the atom-field interaction can be effectively rewritten as if each atomic transition felt a local \textit{equilibrium} environment whose temperature depends on the transition frequency, on the properties of the slab and on the slab-atom distance $z$. These effective \emph{environmental temperatures} depend on the rates $\Gamma_n^{\pm}$ as \begin{equation}\label{Tn} T_n=T(\omega_n)=\frac{\hbar \omega_n}{k_B \ln(\Gamma_n^+/\Gamma_n^-)}, \end{equation} with $n=1,2,3,B,d$. It is important to stress here that, despite these effective environments can be characterised by a temperature, their spectra are not simply blackbody spectra as they have their own transition-dependent Purcell factor \cite{Bellomo2012}.\linebreak In this framework we study thermodynamic effects of \textit{stationary} heat fluxes between $M$ and $B$, mediated and sustained by the OTE environment. To characterize the effects of these fluxes a second kind of temperature has to be introduced. Indeed, as much as the environmental temperatures characterise the thermodynamics of the OTE field, we need a second parameter to describe the energetics of atoms. In particular, atoms exchange energy under the form of heat with their surroundings by emitting photons through one of their transitions. This means that the possibility of such heat exchanges is related to the distribution of population in each atomic level. Note that, from the very definition of $T_n$, the environmental temperature depends on how the field tends to distribute atomic population in each pair of levels, due to the presence of the ratio $\Gamma_n^+/\Gamma_n^-$. A transition is therefore in equilibrium with its effective local environment if and only if its two levels $|a\rangle$ and $|b\rangle$ are populated such that $p_a/p_b=\Gamma^+/\Gamma^-$. If not, the field and the atom will exchange heat along such a transition until such a ratio is reached. This suggests to introduce a second temperature, hereby referred to as \textit{population temperature}, which for a transition of frequency $\omega_n$ ($n=1,2,3,B$) is defined as \begin{equation}\label{thetai} \theta_n=\frac{\hbar \omega_n}{k_B \ln(p^a_n/p^b_n)}, \end{equation} $p^a_n$ ($p^b_n$) being the stationary population of the ground (excited) state of the $n$-th transition. The result of a stationary thermodynamic task on the body, be it refrigeration, heating or population inversion, is then to modify its population temperature $\theta_B$. \subsection{Heat fluxes} The condition $T_i=\theta_i$ is satisfied only if detailed balance ($p^a_i/p^b_i=\Gamma_i^+/\Gamma_i^-$) holds. It can be proven that detailed balance can be broken in a three level atom in OTE fields. As a consequence the machine $M$ produces non-zero stationary heat fluxes with $B$ and the field environment, one for each dissipative process $D_n$ in the ME \eqref{METQ}. These fluxes, following the standard approach in the framework of Markovian open quantum systems \cite{BreuerBook}, are given as $\dot{Q}_n=\mathrm{Tr}\big[H_{at} D_n \rho\big]$, where $\rho$ is here the stationary atomic state and $H_{at}$ is a suitable atomic Hamiltonian which can be $H_M$, $H_B$ or $H_M+H_B$ depending on which part of the atomic system the heat flows into. Note that this definition implies an outgoing heat flux to be negative. Following their definition, these heat fluxes depend both on the field properties (through the structure of the dissipators $D_n$) and on the properties of the atoms through their stationary state. This dependence, for the local dissipators, can be put under the very clear thermodynamic form \begin{equation}\label{fluxtemp} \dot{Q}_n=K_n\left(e^{\frac{\hbar \omega_n}{k_B \theta_n}}-e^{\frac{\hbar \omega_n}{k_B T_n}}\right)\simeq C_n(\theta_n)\big(T_n-\theta_n\big), \end{equation} where $K_n>0$, $C_n(\theta_n)$ is a positive function of $\theta_n$ (and of other parameters such as the frequency of the transition) and the second approximated equality holds in the limit $\theta_n\simeq T_n$. Equation \eqref{fluxtemp} shows that the direction of heat flowing is uniquely determined by the sign of the difference $T_n-\theta_n$, matching the thermodynamic expectation that heat flows naturally from the hotter to the colder body and strengthening the physical meaning of $\theta_n$. There being no time-dependence in the Hamiltonian of the model, the first law of thermodynamics at stationarity for the total atomic system comprises only heat terms and assumes the form \begin{equation}\label{1stlaw} \dot{Q}_B+\sum_{n=1}^3\dot{Q}_n+\dot{Q}_d=0. \end{equation} In addition, energy is exchanged between the machine and the body thanks to their field-induced interaction $H_{MB}$. In Appendix A, following the general scheme developed in \cite{Weimer2008}, we show such an exchange to be under the form of heat. Seen by $M$ such a flux is $\dot{Q}_r=-i\mathrm{Tr}\big(H_M[H_{MB},\rho_s]\big)/\hbar$ while as expected $B$ sees the flux $-\dot{Q}_r$. By introducing the explicit expressions for $H_M$ and $H_{MB}$, one can obtain a particularly simple form for $\dot{Q}_r$ as \begin{equation}\label{Qrcorr} \dot{Q}_r=2\hbar\omega_B\Lambda(\omega_B)\langle \sigma^{\dag}\kappa_r \rangle_{-}, \end{equation} where $\langle \sigma^{\dag}\kappa_r \rangle_-=\frac{i}{2}\langle\sigma^{\dag}\kappa_r-\sigma\kappa_r^{\dag}\rangle$. In an analogous way, by employing Eq. \eqref{Dnonloc}, one can evaluate the change in internal energy of $M$ due to the non-local heat flux exchanged by the atomic system with the OTE environment, given by $\dot{Q}_d=\mathrm{Tr}\big[H_M D_d \rho\big]$. It is \begin{equation}\label{Qdexpl} \dot{Q}_d=-\hbar\omega_B\mathrm{Re}\Big\{ \langle \sigma\kappa_r^{\dag} \rangle\big[\Gamma_d^+-\left(\Gamma_d^-\right)^*\big]\Big\}. \end{equation} Finally, the change in the internal energy of $B$ due to the same effect, $\mathrm{Tr}\big[H_B D_d \rho\big]$, is given by same expression \eqref{Qdexpl}. Fig. \ref{scheme} shows the full scheme of such heat fluxes for a particular configuration of the system. The two levels of $B$ will be labeled here as $|g\rangle$ and $|e\rangle$. Despite the two-level assumption might seem specific, it has been shown in various contexts \cite{Brunner2012, DeLiberato2011} that quantum thermal machines only couple to some effective two-level subspaces in the Hilbert space of the body they are working on. A two-level system is therefore the fundamental building block of the functioning of quantum thermodynamic tasks. \section{Coherence-driven machine tasks}\label{tasks} The main result of this paper is the possibility to drive the temperature $\theta_B$ of the body outside of the range defined by the external reservoirs at $T_W$ and $T_S$. The body, without the effect of the machine, would thermalise at the local environmental temperature ($\theta_B=T_B$), corresponding to $p_e/p_g=\Gamma_B^-/\Gamma_B^+$. This temperature is necessarily constrained within the range $[T_W,T_S]$ \cite{Bellomo2012}.\linebreak Due to the particular form of the master equation \eqref{METQ}, in which all collective atomic terms involve only resonant atomic transitions, in the non-resonant subspace the collective atomic state will be diagonal in the eigenbasis of $H_B+H_M$. This is due to the fact that local dissipation in Eq. \eqref{METQ} of Section \ref{system} induces a thermalisation with respect to the free atomic Hamiltonians. On the other hand, in the resonant atomic subspace of the eigenbasis of $H_B+H_M$ spanned by the states $|g\rangle, |e\rangle$ of $B$ and the two states $|0_r\rangle, |1_r\rangle$ of the transition of $M$ at frequency $\omega_B$, the most general form of the atomic stationary state is \begin{equation}\label{rhoX} \begin{split}\begin{pmatrix} p_{e1_r} & 0 & 0 & 0\\ 0 & p_{e0_r} & c_r & 0\\ 0 & c_r^* & p_{g1_r} & 0\\ 0 & 0 & 0 & p_{g0_r}\\ \end{pmatrix}. \end{split}\end{equation} A coherence $c_r$ is present in the decoupled basis between the two atomic states $|g1_r\rangle$ and $|e0_r\rangle$ having the same energy. Note that the temperature $\theta_B$ of the body increases monotonically with the ratio $p_e/p_g$. By tracing out the machine degrees of freedom from the master equation \eqref{METQ}, one obtains a diagonal state with stationary populations $p_g$ and $p_e$ of the body $B$. Be now $\dot{Q}_r^B$ the flux $\dot{Q}_r$ seen by $B$. Then the expressions for heat fluxes exchanged by $B$ with its surroundings are \begin{eqnarray} \dot{Q}_r^B&=&-\frac{i}{\hbar}\big\langle[H_B,H_{MB}]\big\rangle,\label{dotQRQ}\\ \dot{Q}_d&=&\Gamma_{d}^+\Big[\big\langle\sigma^{\dag}H_B\kappa_{r}\big\rangle-\frac{1}{2}\big\langle\{H_B,\sigma^{\dag}\kappa_{r}\}\big\rangle\Big]\nonumber\\ &+&\Gamma_{d}^-\Big[\big\langle\sigma H_B\kappa_{r}^{\dag}\big\rangle-\frac{1}{2}\big\langle\{H_B,\sigma\kappa_{r}^{\dag}\}\big\rangle\Big]+c.c.\label{QQTQ},\\ \dot{Q}_B&=&\Gamma_B^+\Big[\big\langle \sigma^{\dag} H_B \sigma \big\rangle-\frac{1}{2}\big\langle \{ H_B,\sigma^{\dag}\sigma \} \big\rangle\Big]\nonumber \\ &+&\Gamma_B^-\Big[\big\langle \sigma H_B \sigma^{\dag} \big\rangle-\frac{1}{2}\big\langle \{ H_B,\sigma\sigma^{\dag} \} \big\rangle\Big],\label{QB} \end{eqnarray} where the mean values are evaluated over the stationary state of the total system. Exploiting its general form \eqref{rhoX}, it is just a matter of straightforward calculations to evaluate all the mean values above. Imposing the sum of \eqref{dotQRQ}, \eqref{QQTQ} and \eqref{QB} to vanish (first law for $B$, analogous to Eq. \eqref{1stlaw}), one obtains \begin{equation}\label{popobj} \frac{p_e}{p_g}=\frac{\Gamma_B^--\Delta(\omega_B)}{\Gamma_B^++\Delta(\omega_B)}, \end{equation} where \begin{equation}\label{delta} \Delta(\omega_B)=2\Lambda(\omega_B)\mathrm{Im}\{c_r\}+\mathrm{Re}\Big\{c_r \big[\Gamma_d^+-\left(\Gamma_d^-\right)^*\big]\Big\}. \end{equation} Note now that, thanks to Eq. \eqref{rhoX}, $\langle \sigma\kappa_r^{\dag} \rangle=c_r$ and $\langle \sigma^{\dag}\kappa_r \rangle_{-}=\mathrm{Im}(c_r)$, such that the first term in $\Delta(\omega_B)$ stems from the resonant heat $-\dot{Q}_r=-2\hbar\omega_B\Lambda(\omega_B)\mathrm{Im}(c_r)$ exchanged with the machine, while the second is due to the non-local heat flux $\dot{Q}_d$. Eqs. \eqref{popobj} and \eqref{delta} show that the thermal machine works \textit{only if a stationary quantum coherence $c_r$ is present}. Remarkably, it can be shown \cite{Spehner2014} that quantum discord \cite{Ollivier2001} (a key measure of purely quantum correlations) is a monotonic function of the absolute value of the coherence $c_r$ in our system. Differently from previous studies \cite{Correa2013}, here discord between $M$ and $B$ is a necessary condition for any thermodynamic task, and represents a resource the machine can use through the two different processes $\dot{Q}_r$ and $\dot{Q}_d$. Eq. \eqref{popobj} means that a quantum coherence between machine and body modifies the stationary temperature of the body with respect to $T_B$. This modification is reported in Fig. \ref{temp1}, where the behaviour of $\theta_B$ as a function of the slab-atoms distance $z$ is shown for two different slab thicknesses $\delta$. Four possible regimes can be singled out: both during refrigeration ($\theta_B<T_B$) and heating ($\theta_B>T_B$), $\theta_B$ can be either driven outside of the range $[T_W,T_S]$ (strong tasks) or kept within it (light tasks). As a limiting case of strong heating, the body can be brought to infinite temperature ($p_e=p_g$) and, further on, to negative ones, producing population inversion. \begin{figure} \caption{Stationary temperature $\theta_B$ of the body (solid black line), machine resonant temperature $\theta_M$ in the absence of $B$ (dotted pink line) and local-environment temperature $T_B$ felt by the body (dot-dashed green line) versus $z$. The slab is made of sapphire and kept at $T_S=500\,\mathrm{K}$ while $T_W=300\,\mathrm{K}$. The machine transition frequencies are $\omega_1=0.9\,\omega_S$, $\omega_B=\omega_2=0.1\,\omega_S$ and $\omega_3=\omega_S$, $\omega_S=0.81\cdot10^{14}\,\mathrm{rad}s^{-1}$ being the first resonance frequency of sapphire (the optical data for the dielectric permittivity of the slab material are taken from \cite{PalikBook}). The two atoms are placed at a distance $r=1\,\mu\mathrm{m}$ from each other. Panel (a): numerical data for a semi-infinite slab. Light and strong refrigeration are achieved in this configuration. Panel (b): same quantities for a slab of finite thickness. The plotted functions are in this case $-1/\theta_B$, $-1/\theta_M$ and $-1/T_B$ (left vertical scale), with the same color code as before. The population inversion corresponds to divergent $\theta_B$ and $\theta_M$. The corresponding value of temperature can be read on the right vertical scale. All tasks are in this case obtained.} \label{temp1} \end{figure} As one can easily see from Fig. \ref{temp1}, the physics behind the absorption tasks is enclosed in the strong sensitivity of the population temperature $\theta_B$ of the body to the population temperature $\theta_M$ of the machine along the resonant transition when the body is not present. \subsection{Optimal conditions for thermodynamic tasks}\label{optimal} It is shown in Fig. \ref{temp1} that the machine has a very high thermal inertia, such that the body, when put into thermal contact with the machine having a certain temperature $\theta_M$, thermalizes with it and $\theta_B\simeq\theta_M$. Fig. \ref{temprate} shows the mechanism the machine uses to modify its population temperature $\theta_M$ in absence of the body, thanks to the different environmental temperatures each of its transition feels. This drives $M$ out of detailed balance condition and allows $M$ to keep its resonant transition temperature almost constant. We label here the three transitions of the machine as high frequency ($\omega_h$), average frequency ($\omega_a$) and low frequency ($\omega_l$), one of which (suppose here $\omega_l$, connecting states $|0_r\rangle$ and $|1_r\rangle$) is resonant with $\omega_B$. For simplicity, let us focus on refrigeration only, which we suppose to happen either through transition 2 (connecting first and the second excited states), since in this configuration the high-frequency transition $3$ is always used by an absorption refrigerator to dissipate heat into the environment \cite{Correa2014}. As shown in Fig. \ref{temp1}, to obtain a low $\theta_B$ the resonant machine transition must be made cold. This is achieved by reducing the ratio $p_{1_r}/p_{0_r}$, which in turn happens when: \\ (a) the effective environmental temperature $T_h$ felt by the high frequency transition is very cold. In this way the environment contributes in increasing the population of the ground state of $M$ at the expenses of the population of its most energetic state. The resonant transition involves necessarily one of these two levels, and in both cases the effect of the high frequency transition helps reducing $p_{1_r}/p_{0_r}$;\\ (b) the effective environmental temperature felt by the average transition is very hot. This, following the same idea, would either mean reducing the population of $p_1$ or increasing the one of $p_0$, thus reducing $p_{1_r}/p_{0_r}$.\\ \begin{figure} \caption{Conditions for refrigeration: effective rate temperatures $T_h$, $T_a$ and $T_l$ of the local environments felt by the three machine transitions, for $\omega_h=\omega_S$, $\omega_a=0.8\,\omega_S$ and $\omega_l=0.2\,\omega_S$ ($\omega_S=0.81\cdot 10^{14}$\,rad\,\,$\mathrm{s}^{-1}$) versus the machine-slab distance $z$ in the absence of the body. The slab and walls temperatures are $T_S=200\,$K and $T_W=300\,$K, and the slab thickness is $\delta=0.05\,\mu$m. As the plot shows, the transition having the same frequency as the slab resonance is much more strongly affected by the field emitted by the slab, such that its rate temperature is kept much lower than $T_a$. This produces a mechanism, shown in the inset, according to which excitations (yellow dots) are transferred to the intermediate level of the machine and removed from its upper one. This in turn drives the population temperature $\theta_l$ of the transition at $\omega_l$ (transition 2 in the example) to values lower than the one $T_l$ of its local environment, allowing the machine to refrigerate objects. In this configuration, introducing a body $B$ at $z=1\,\mu$m from the slab and $r=1\,\mu$m from $M$, one obtains $\theta_B=160\,$K $<T_S$.} \label{temprate} \end{figure} When these two conditions are met, the machine can always redistribute its populations such that the ratio $p_{1_r}/p_{0_r}$ can be kept low and almost unaffected by the presence of another atom. The advantage of the OTE field configuration is that the effective field temperatures can be manipulated through a wide set of parameters involving $z$, $\delta$, $T_W$ and $T_S$. In particular, the role of the resonance of the slab material is crucial \cite{Bellomo2012}, as explained in the caption of Fig. \ref{temprate}. In the case $T_S<T_W$, transitions strongly affected by the field emitted by the slab feel a cold local environment. Moreover, provided $\omega_a$ is far enough from $\omega_S$, one can at the same time have $T_a\simeq T_W$. By this mechanism, $M$ can change the temperature $\theta_B$, bringing it to values far outside the range $[T_S,T_W]$. We stress here that the difference between light and strong tasks is a fundamental one: better than light tasks could in principle be done by direct connection of the body to one of the two real reservoirs at $T_S$ or $T_W$, while strong tasks can not be achieved by a simple thermal contact with anything in the system.\linebreak $\Delta(\omega_B)$ strongly depends on the slab-matter system distance $z$ and on the external temperatures through $c_r$, $\Lambda$ and $\Gamma_d^{\pm}$. One can thus engineer one or many of these regimes at will as shown in the functioning-phase diagram of the machine in Fig. \ref{phases} for a fixed thickness $\delta=0.05\,\mu\mathrm{m}$. All the strong and light functioning phases of the machine are found as a function of both $T_W-T_S$ and $z$. \begin{figure} \caption{Functioning phases of the absorption machine versus the atoms-slab distance $z$ and $\Delta T=T_W-T_S$. The sapphire slab has a thickness $\delta=0.05\,\mu\mathrm{m}$ and its temperature is continuously changed in the range $[30\,\mathrm{K},570\,\mathrm{K}]$. The walls are at $T_W=300\,\mathrm{K}$. Strong refrigeration (blue areas), strong heating (red areas), light refrigeration (cyan), light heating (orange) and population inversion (green) can all be obtained.} \label{phases} \end{figure} \section{Efficiency and Carnot limit}\label{efficiency} Consider now the refrigerating regime in which the machine extracts heat from the body through the transition $2$. The scheme of heat fluxes is then exactly the one depicted in Fig. \ref{scheme}. The efficiency of this process is \begin{equation}\label{eff} \eta_{ref}=\frac{\dot{Q}_r}{\dot{Q}_1+\dot{Q}_2}, \end{equation} due to the fact that $\dot{Q}_r$ is the power produced by the machine, which absorbs energy from its surroundings through transitions $1$ and $2$ (the equivalent of a work input) while uses transition $3$ to dissipate part of the absorbed energy after use (the equivalent of the spiral in a normal fridge). The corresponding Carnot limit $\eta^C_{ref}$ can be obtained by analysing the machine functioning in its reversible limit (zero entropy production). The instantaneous entropy production rate $\tau$ for quantum systems is defined as \cite{BreuerBook} \begin{equation}\label{secondlaw} \sigma=-\frac{\mathrm{d}}{\mathrm{d}t}S(\rho(t)||\rho^{st})\geq0, \end{equation} where $S(\rho(t)||\rho^{st})$ is the so-called relative entropy \cite{Vedral2002}, never increasing in time under a Markovian dynamics. Following \cite{Correa2014}, one can apply equation \eqref{secondlaw} term by term to each dissipator in the master equation thanks to the fact that they all are under a Markovian form. One thus obtains \begin{equation}\label{secondlawlocal} \sum_i\mathrm{Tr}\Big[(D_k\rho^{st})\ln \rho_k^{st}\Big]\geq 0, \end{equation} where $\rho_k^{st}$, $k=1,2,3,B,d$ is the kernel (stationary state) of the single dissipator $D_k$. The $3$ local dissipators $D_n$ of the machine and the local dissipator $D_B$ of the body induce stationarity under the standard Gibbs form at the effective environmental temperature, diagonal in the free atomic Hamiltonian basis. The nonlocal dissipator $D_d$, in the case studied here where the dipoles of $B$ and $M$ lie along the line connecting the two atoms (and more in general when $\Gamma_d^{\pm}\in \mathbb{R}$), has the same kernel at environmental temperature $T_d$, local in the degrees of freedom of $M$ and the $B$. Introducing these single-dissipator stationary states into equation \eqref{secondlawlocal} one obtains \begin{equation}\label{2law} \frac{\dot{Q}_1}{T_1}+\frac{\dot{Q}_2}{T_2}+\frac{\dot{Q}_3}{T_3}+\frac{\dot{Q}_B}{T_B}+\frac{2\dot{Q}_d}{T_d}\leq 0, \end{equation} which is a form of the second law at stationarity for our system. With the help of the first law in Eq. \eqref{1stlaw} of Section \ref{thermodynamics}, the known property of three-level atomic heat fluxes \cite{Scovil1959} $|\dot{Q}_n/\dot{Q}_m|=\omega_n/\omega_m$ $\forall \,m,n=1,2,3$ (where $\dot{Q}_n$ is the total flux along the $n$-th transition) and the fact that in refrigeration $T_3<T_2,T_d$ and $T_2,T_d<T_1$ (as commented in Section \ref{optimal}) and under the condition $\dot{Q}_d<0$ (other cases can be treated analogously), one obtains from \eqref{2law} another first degree inequality. This has a non-trivial solution only if $T_d>T_2$, from which a bound on the efficiency in Eq. \eqref{eff} can be obtained as shown in Appendix B. Such a bound depends only on the three frequencies of the machine and the temperatures of the effective local and non-local environments. In the case of refrigeration along transition $2$ the Carnot efficiency assumes the form \begin{equation}\label{carnot} \eta^C_{ref}= \begin{cases} \frac{\omega_2}{2\omega_1}+\frac{1}{2}\frac{T_2T_d(T_1-T_3)}{T_1T_3(T_d-T_2)}+\frac{\omega_2}{2\omega_1}\frac{T_2(T_d-T_3)}{T_3(T_d-T_2)},\,\,\mathrm{if}\,\,T_d>T_2,\\ \frac{\omega_2}{\omega_1},\,\,\mathrm{if}\,\,T_d\leq T_2. \end{cases} \end{equation} \subsection{Efficiency at maximum power} An important figure of merit for the realistic functioning of any thermal machine is how close to its Carnot limit it works when delivering maximum power (i.e., when $\dot{Q}_r$ is maximised). Many bounds are known for different setups, limiting the efficiency at maximum power $\eta^m$ to some fractions of $\eta^C$ \cite{Correa2014, Curzon1975}. Remarkably our structured OTE environment allows for refrigeration tasks with $\eta^m$ much closer to $\eta^C$ than the bound known for quantum absorption machines \cite{Correa2014} based on ideal blackbody reservoirs, reading for our system $\eta^m<0.75\,\eta^C$. This is exemplified in Fig. \ref{effmax1} for a particular configuration of the model. The blue triangles (left vertical scale) represent the ratio $\eta_{ref}/\eta^C_{ref}$, plotted versus $\omega_3$ while keeping fixed $\omega_2=\omega_B=0.1\,\omega_S$. The red dots (right vertical scale) are the power $\dot{Q}_r$ plotted versus the same quantity, while the red dashed line is the machine-body discord (right vertical scale). It is clear that the power is maximised at $\omega_3=1.05\,\omega_S$, corresponding to $\eta_{ref}^m\simeq0.89\,\eta^C_{ref}$. $\dot{Q}_r$ starts decreasing, as classically expected when the efficiency approaches $\eta^C$, around $\omega_3\simeq0.9\,\omega_S$, but suddenly increases again when $\omega_3$ approaches $\omega_S$. This behaviour is due to the fact that, when one atomic transition is resonant with the characteristic frequency of the slab material, the atomic populations are strongly affected by the field emitted by the slab. Hence the not-black-body nature of the total field become crucial (e.g., the atomic decay rate is no longer proportional to $\omega^3$), allowing to overcome bounds set by the blackbody physics. The role of discord as machine resource is clearly shown here, where discord at resonance has a sharp peak leading to the high-power performance of $M$. \begin{figure} \caption{The ratio $\eta_{ref}/\eta^C_{ref}$ (blue triangles, left vertical axis), the power of the machine ($\dot{Q}_r$, red dots, right vertical axis in units of $10^{-14}\,\mu \mathrm{J}/s$) and machine-body discord (dashed red line, right vertical scale in units of $10^{-4}$) versus the scaled machine transition frequency $\omega_3/\omega_S$. The sapphire slab is semi-infinite and at $T_S=395\,\mathrm{K}$, while $T_W=125\,\mathrm{K}$ and $z=4.8\,\mu\mathrm{m}$. The transition frequency of the body is fixed as $0.1 \omega_S$ ($\omega_S=0.81\cdot10^{14}\,\mathrm{rad}s^{-1}$), resonant with transition $2$ of the machine. The maximum power is reached for $\omega_3=1.05\,\omega_S$, corresponding to $\eta_{ref}/\eta^C_{ref}\simeq 0.89$. Remarkably, discord shows a sharp resonance peak, similarly to $\dot{Q}_r$.} \label{effmax1} \end{figure} One could wonder whether such an exceptionally high efficiency at maximum power is only seldomly attained for the kind of machines described here. To answer such a question on quantitative bases, we performed a random sampling of over $2\cdot10^4$ thermal machines, all delivering thermodynamic tasks on the same fixed body. In the simulations performed and reported in Fig. \ref{effmax2}, the machines work as a quantum refrigerator delivering strong refrigeration using a semi-infinite slab. In this sampling, the machine-slab distance $z$ has been, for each machine, randomly drawn in the range $[0.9\,\mu \mathrm{m}, 100\,\mu \mathrm{m}]$, the walls temperature has been selected randomly in $T_W\in[50\,\mathrm{K}, 500\,\mathrm{K}]$ and, for each value of $T_W$, the slab temperature has been chosen at random in $T_S\in [T_W,T_W+500\,\mathrm{K}]$. The internal structure of the body is kept fixed during the simulations, with a frequency $\omega_B=0.1 \,\omega_S$ resonant with the transition 2 of $M$. For each machine thus generated, we have then maximised the delivered power by modifying the two other machine frequencies over every possible value of $\omega_1 \in (\omega_2,\omega_S)$ and $\omega_3$ compatible with the condition $\omega_1+\omega_2=\omega_3$. Finally, once obtained the configuration corresponding to the maximum power, we have computed the efficiency of the process. Fig. \ref{effmax2} shows the histogram of the distribution of the ratio $\eta^m/\eta^C$ of efficiency at maximum power to the corresponding Carnot efficiency in the interval $[0,1]$ within these $2\cdot10^4$ random refrigerators. It is remarkable that around $50\%$ of these machines work at maximum power with efficiencies higher than the bound $0.75\,\eta^C$ in \cite{Correa2014} and that none of them have been found to work at maximum power with efficiencies lower than $0.6 \eta^C$. Moreover, as can be clearly seen in Fig. \ref{effmax2}, a small but non-negligible fraction of them can reach $\eta^m\simeq0.98\eta^C$. \begin{figure} \caption{Statistical occurrence of ratios $\eta^m/\eta^C$ for a random sampling of $2\cdot10^4$ thermal machines, always in resonance with the same body $B$. For each machine, $z$ has been randomly generated in the range $[0.9\,\mu \mathrm{m}, 100\,\mu \mathrm{m}]$, $T_W\in[50\,\mathrm{K}, 500\,\mathrm{K}]$ and, for each value of $T_W$, $T_S\in [T_W,T_W+500\,\mathrm{K}]$. The internal structure of the body is kept fixed during the simulations, with a frequency $\omega_B=0.1 \,\omega_S$ resonant with the transition 2 of $M$. The maximisation is performed over every possible value of $\omega_1 \in (\omega_2,\omega_S)$ and $\omega_3$ compatible with the condition $\omega_1+\omega_2=\omega_3$. Around $50\%$ of machines thus generated have $\eta^m>0.75\,\eta^C$.} \label{effmax2} \end{figure} \section{Conclusions}\label{conclusions} This work introduces a new realization of a quantum thermal machine using atoms interacting with single non-equilibrium electromagnetic fields. By simply connecting two thermal reservoirs to \textit{macroscopic objects}, their radiated field allows the atomic machine to achieve all quantum thermodynamic effects (heating, cooling, population inversion), without any direct external manipulation of atomic interactions. This overcomes the usual difficulty of connecting single transitions to thermal reservoirs, in a realistic and simple configuration where the field-mediated atomic interaction modifies at will stationary inter-atomic energy fluxes. Despite the environmental dissipative effects, atoms share steady quantum correlations \cite{Bellomo2013, Bellomo2014} which we showed to be necessary for one atom to deliver a thermodynamic task on the other, uncovering genuinely non-classical machine functioning. These particular features affect the tasks efficiency, which can be remarkably high also at maximum power, defying the known bounds for quantum machines based on ideal and independent blackbody reservoirs thanks to the fundamental effect of the resonance with the real material of which the slab is made. Moreover, such a remarkably high efficiency at maximum power is strongly connected to the presence of a peak in quantum correlations between the machine and the body, which represent the resource the machine uses for its tasks. These results tackle major open problems on quantum thermal machines, paving the way for an efficient quantum energy management based on the potentialities of non-equilibrium and quantum features in atomic-scale thermodynamics.\\ \section*{Acknowledgments} The authors acknowledge fruitful discussions with N. Bartolo and R. Messina, and financial support from the Julian Schwinger Foundation. \begin{appendix} \section{Resonant heat flux} In this appendix we demonstrate that the resonant energy exchange between $M$ and $B$ due to the field-mediated coherent interaction $H_{MB}$ consists only of heat. Following the approach of \cite{Weimer2008}, the dynamics of the sole $M$ induced by the Hamiltonian interaction $H_{MB}$ comprises in general an Hamiltonian and a dissipative part and can be written as \begin{equation}\label{dotrhoa} \dot{\rho}_M=-\frac{i}{\hbar}\big[H_M+H_M^{\mathrm{eff}},\rho_M\big]+D_{MB}(\rho), \end{equation} where $D_{MB}$ is a non-unitary dissipative term for $M$ due to the interaction with $B$, which depends however on the total state $\rho=\rho_{MB}$ because, in general, the two subparts are correlated. $H_M^{\mathrm{eff}}$ is a renormalised free Hamiltonian of subsystem $M$ due to the interaction with $B$. Defining the two marginals $\rho_{M(B)}=\mathrm{Tr}_{B(M)}\rho$ and the correlation operator $C_{MB}=\rho-\rho_M\otimes\rho_B$, it is shown in \cite{Weimer2008} that \begin{eqnarray} H_M^{\mathrm{eff}}&=&\mathrm{Tr}_B\Big(H_{MB}(\mathbb{I}_M\otimes\rho_B)\Big),\label{HAeff}\\ D_{MB}(\rho)&=&-i \mathrm{Tr}_B\Big([H_{MB},C_{MB}]\Big).\label{DAB} \end{eqnarray} Introducing $H_{M1}^{\mathrm{eff}}$ as the part of $H_M^{\mathrm{eff}}$ which commutes with $H_M$ and $H_{M2}^{\mathrm{eff}}$ which does not, directly from equation \eqref{dotrhoa} one has, for the internal energy of $M$ $U_M=\mathrm{Tr}\big((H_M+H_M^{\mathrm{eff}}) \rho\big)$, \begin{equation}\label{udot}\begin{split} &\dot{U}_M=\mathrm{Tr}_M\Big((H_M+H_{M1}^{\mathrm{eff}})D_{MB}(\rho)\Big)+\mathrm{Tr}_M\big(\dot{H}_{M1}^{\mathrm{eff}}\rho_M\big)\\ &-i\mathrm{Tr}_M\Big(\big[H_M+H_{M1}^{\mathrm{eff}},H_{M2}^{\mathrm{eff}}\big]\rho_M\Big). \end{split}\end{equation} It is custom to identify heat terms as the ones producing a change in the entropy of a subsystem: all the rest is identified as work $W$. Eq. \eqref{udot} can then be split in \begin{eqnarray} \dot{Q}_M&=&\mathrm{Tr}_M\Big((H_M+H_{M1}^{\mathrm{eff}})D_{MB}(\rho)\Big)\label{dotQA},\\ \dot{W}_M&=&\mathrm{Tr}_M\Big(\dot{H}_{M1}^{\mathrm{eff}}\rho_M-i\big[H_M+H_{M1}^{\mathrm{eff}},H_{M2}^{\mathrm{eff}}\big]\rho_M\Big).\label{dotWA} \end{eqnarray} Introducing the symbols $c_M^{ij}=\langle i|\rho_M|j\rangle$ ($i\neq j$) for the coherences \textit{of the marginal} $\rho_M^{st}$ (different then from the coherence $c_r$ introduced in Eq. \eqref{rhoX} of Section \ref{tasks} which is a two-atom coherence), equation \eqref{HAeff} becomes \begin{equation} H_M^{\mathrm{eff}}\propto \mathrm{Re}(c_M^{10}). \end{equation} By tracing out the machine or the body degrees of freedom from equation \eqref{rhoX}, one can prove that the two stationary marginals $\rho_M^{st}$ and $\rho_B^{st}$ are always diagonal in the eigenbases of their respective free Hamiltonians, so that $c_M^{10}=0$. No renormalisation to the machine Hamiltonian comes therefore from the interaction with $B$, which means that equation \eqref{dotWA} vanishes, proving that no work is involved in machine-body energy exchanges. As for the heat, considering that $[H_{MB},\rho_M^{st}\otimes\rho_B^{st}]=0$, Eq. \eqref{dotQA} reduces to \begin{equation} \dot{Q}_M=-i\mathrm{Tr}_M\Big(H_M\mathrm{Tr}_B\big[H_{MB},\rho^{st}\big]\Big)=\dot{Q}_r, \end{equation} with the same $\dot{Q}_r$ given in Eq. \eqref{Qrcorr}. \section{Carnot limit} In this appendix we deduce Eq. \eqref{carnot} of Section \ref{efficiency} for the Carnot efficiency in refrigeration along transition 2, and under the condition $\dot{Q}_d<0$. In addition to Eqs. \eqref{1stlaw} and \eqref{2law}, the condition $|\dot{Q}_n/\dot{Q}_m|=\omega_n/\omega_m$ gives for $n=1$ and $m=2$ the following \begin{equation}\label{ratio12} \frac{\dot{Q}_1}{\dot{Q}_2+\dot{Q}_r+\dot{Q}_d}=\frac{\omega_1}{\omega_2}. \end{equation} Solving Eqs. \eqref{1stlaw} and \eqref{ratio12} for $\dot{Q}_3$ and $\dot{Q}_d$ and using these solutions into \eqref{2law} one obtains for $\dot{Q}_r$ \begin{equation}\label{disqr} \dot{Q}_r\leq\dot{Q}_1\frac{T_d T_2}{T_d-T_2}\Bigg[\frac{1}{T_3}\Big(1+\frac{\omega_2}{\omega_1}\Big)-\frac{1}{T_1}-\frac{\omega_2}{\omega_1}\frac{1}{T_d}\Bigg]-\dot{Q}_2 \end{equation} which, used in Eq. \eqref{eff} of Section \ref{efficiency}, gives a bound on $\eta_{ref}$ as a function of $\dot{Q}_1$ and $\dot{Q}_2$. Finally, using the fact that such a bound is a decreasing function of $\dot{Q}_2$, one obtains the Carnot efficiency as the limit for $\dot{Q}_2\rightarrow 0$, which turns out to be independent on $\dot{Q}_1$ and gives ultimately the first line of Eq. \eqref{carnot}. On the other hand, in the case $T_2<T_d$, one can not obtain anything like Eq. \eqref{disqr} and the only possibility for the machine to work without producing entropy is therefore to have vanishing heat flux from/to the body. This means $\dot{Q}_2=\dot{Q}_d=0$ which, inserted in the expression for the efficiency and using again $|\dot{Q}_n/\dot{Q}_m|=\omega_n/\omega_m$ leads to the second line of Eq. \eqref{carnot}. \end{appendix} \end{document}

\begin{document} \title{Laser-beam scintillations for weak and moderate turbulence} \author{R.~A.~Baskov} \email{Email address: [email protected]} \affiliation{Institute of Physics of the National Academy of Sciences of Ukraine,\\ pr. Nauky 46, Kyiv-28, MSP 03028, Ukraine} \author{O.~O.~Chumak} \affiliation{Institute of Physics of the National Academy of Sciences of Ukraine,\\ pr. Nauky 46, Kyiv-28, MSP 03028, Ukraine} \begin{abstract} The scintillation index is obtained for the practically important range of weak and moderate atmospheric turbulence. To study this challenging range, the Boltzmann-Langevin kinetic equation, describing light propagation, is derived from first principles of quantum optics based on the technique of the photon distribution function (PDF) [G. P. Berman \textit{et al.}, Phys. Rev. A \textbf{74}, 013805 (2006)]. The paraxial approximation for laser beams reduces the collision integral for the PDF to a two-dimensional operator in the momentum space. Analytical solutions for the average value of PDF as well as for its fluctuating constituent are obtained using an iterative procedure. The calculated scintillation index is considerably greater than that obtained within the Rytov approximation even at moderate turbulence strength. The relevant explanation is proposed. \end{abstract} \maketitle \section{Introduction} Physics of light beam propagation in the Earth's atmosphere is of great interest for scientists and engineers, see \cite{Tatarskii1,Bar,Andrews,coro,fei}. This interest arises from applications in quantum and classical communications and remote sensing systems. The latest achievements in this field concern problems of quantum key distribution \cite{Capraro, Usenko}, propagation of entangled \cite{Ursin,Yin, Hosseinidehaj} and squeezed \cite{Peuntinger, Vasylyev} states, quantum nonlocality \cite{Semenov, Gumberidze}, quantum teleportation \cite{Ma, Ren}, tests of fundamental physical laws \cite{Rideout, Touboul}. In all these cases, random variations of the atmospheric refraction index distort the phase front of radiation causing intensity fluctuations (scintillations), beam wandering and increasing beam spreading. Scintillations are the most severe problem which manifests itself in a significant reduction of the signal-to-noise ratio (SNR) introducing degradation of the performance of laser communication systems. A laser beam in the Earth atmosphere is affected by turbulent eddies. Randomly distributed eddies stand for sources of local index-of-refraction fluctuations. There are numerous beam-eddies ``collisions" in the course of long-distance propagation. As a result, the radiation gradually acquires the Gaussian statistics. The scintillation index, $\sigma^2$, which is defined in classical optics as the inverse SNR, asymptotically approaches the level of $\sigma^2=1$. In this case, the intensity fluctuations are referred to as saturated \cite{Kravtsov}. Scintillations are of importance for design of reliable classical and quantum optical communication systems \cite{Andrews2, Erven}, remote sensing systems \cite{Rino,Churnside}, and adaptive optics \cite{Ribak}. This field of research has also application in atmospheric physics, geophysics, ocean acoustics, planetary physics, and astronomy \cite{Tatarskii2}. The theoretical description of scintillation phenomena faces with the increasing computational complexity when one considers the parameter region of maximal optical beam intensity fluctuations. In order to overcome this problem several phenomenological and semi-phenomenological approaches were developed, which utilize the intensity distribution functions \cite{Jakeman}, phase screens \cite{Dashen-Wang}, turbulence spectrum approximation \cite{Marians}. The existing rigorous first-principles approaches, such as the method of smooth perturbations (Rytov approximation) \cite{Tatarskii1}, the Huygens-Kirchhoff method \cite{banakh79}, the path-integral method \cite{Das}, are applicable merely to the asymptotic regimes of weak and strong optical turbulences. At the same time, maximum scintillations lay in the region of moderate turbulence. The range of moderate turbulence is the most challenging for rigorous theoretical study. First, the transition from statistics of coherent laser beam to the Gaussian statistics lies just in this region. Second, strong correlations of photon trajectories, which considerably enhance scintillations \cite{enha}, should also be taken into account here. A combined effect of these important factors can lead to maximal scintillations. Such effect clearly manifests itself in various experiments where this maximum may considerably exceed the level of saturation \cite{Kravtsov,consortini,sedin}. In the present paper we introduce for a first-principle approach for the description of weak and weak-to-moderate turbulence regimes, which remain the most challenging for the analysis. The method is based on the technique of the photon distribution function (PDF) \cite{Chu}, which is derived from the first principles of quantum optics.This method is applicable for an arbitrary quantum state of the light including coherent states, which describe laser-radiation fields. The PDF is an operator-valued function, $\hat{f}(\mathbf{r},\mathbf{q})$, of the position $\mathbf{r}$ and the wave vector $\mathbf{q}$. It retains the concept of the Wigner function \cite{Wigner} such that the integration with respect to $\mathbf{q}$ or $\mathbf{r}$ results in the field intensity operator $\hat{I}(\mathbf{r})$ or the photon-number operator $\hat{n}(\mathbf{q})$, respectively. The PDF can be found as a solution of the kinetic equation that accounts for random variations of the refractive index in the atmosphere. This approach has been originally introduced in the solid state physics (see, for example, Ref. \cite{chuZ}) and has also been successfully applied for a description of quantum radiation in waveguides \cite{sto,sto14}. Application of the PDF method to the light propagation in the turbulent atmosphere has been considered in Refs. \cite{enha,Chu, Chumak wander}. It utilizes the approximation of the smoothly varying random force and is applicable only for restricted values of the turbulence parameters. In the present paper we derive a more general kinetic equation for the PDF introducing the collision integral and Langevin source of fluctuations. An approximate solution of this equation enables us to describe the beam characteristics beyond the Rytov approximation at the moderate range of turbulence, which was unreachable with the previous techniques. To stress the significance of the present paper, it is worthwhile to recall the words of Dashen \cite{Das}. He considers ``the detailed behavior of the wave field at the boundaries between the unsaturated and saturated regimes" ``the remaining problem" in the physics of scintillation phenomena. We hope that our paper as well as the previous one \cite{enha} provide a deeper insight into physics and the theoretical description of this important region. The rest of this paper is organized as follows. In Sec. \ref{sec:pdf} we give a brief review of the method of the PDF method. In Sec. \ref{sec:ble} and Appendix \ref{sec:appendix}, we explain the derivations of the collision integral and the corresponding Langevin source. In Sec. \ref{sec:scint_ind} and Appendix \ref{sec:appendix1}, we obtain an analytical formula for the scintillation index which is represented by a many-fold integral. In Sec. \ref{sec:discussion}, the results of numerical simulations are discussed. Concluding remarks are given in Sec. \ref{sec:conclusion}. \section{Photon distribution function} \label{sec:pdf} The photon distribution function is defined in analogy to the widely used solid state physics distribution functions \cite{chuZ} (the distributions for electrons, phonons, etc). This function is given by, see Ref. \cite{UJP}, \begin{equation}\label{1threee} \hat{f}({\bf r},{\bf q},t)=\frac 1V\sum_{\bf k}e^{-i{\bf kr}}b^\dag_{{\bf q}+ {\bf k}/2}b_{{\bf q}-{\bf k}/2}, \end{equation} where $b^\dag_{\bf q}$ and $b_{\bf q}$ are bosonic creation and annihilation operators of photons with the wave vector ${\bf q}$; $V\equiv L_xL_yL_z\equiv SL_z$ is the normalizing volume. We consider the laser beam propagating along the $z$ axis in the paraxial approximation. For this case the initial polarization of the beam remains almost undisturbed for a wide range of propagation distances, cf. Ref. \cite{stroh}. The operator $\hat{f}({\bf r},{\bf q},t)$ describes the photon density in the phase space (PDF in ${\bf r}{-}{\bf q}$ space). We consider the scenario with characteristic sizes of spatial inhomogeneities of the radiation field being much greater than the optical wavelength $\lambda=(2\pi/q_0)$; here $q_0$ is the wave vector corresponding to the central frequency of the radiation, $\omega _0=cq_0$. In this case, it is reasonable to restrict the sum in Eq. (\ref{1threee}) by the range of small $k$, i.e. $k< k_0$ such that the inequality $k_0\ll q_0$ is satisfied. At the same time the value of $k_0$ should be large enough to provide a desired accuracy for the description of the beam profile. Evolution of the PDF $\hat{f}({\bf r},{\bf q},t)$ is governed by the Heisenberg equation \begin{equation}\label{2five} \partial_t \hat{f}({\bf r},{\bf q},t)=\frac 1{i\hbar }[\hat{f}({\bf r},{\bf q},t),\hat{H}], \end{equation} where \begin{equation}\label{3six} \hat{H}=\sum_{\bf q}\hbar\omega_{\bf q}b^\dag_{\bf q}b_{\bf q}-\sum_{\bf q,k}\hbar\omega_{\bf q}n_{\bf k}b^\dag_{\bf q}b_{\bf q+k} \end{equation} is the Hamiltonian of photons in a medium with a fluctuating refraction index $n({\bf r})=1+\delta n ({\bf r})$, where $\delta n ({\bf r})$ stands for fluctuating part representing atmospheric inhomogeneity. The quantities $\hbar\omega_{\bf q}=\hbar cq$ and ${\bf c_q}=\frac{\partial \omega_{\bf q}}{\partial{\bf q}}$ are the photon energy and photon velocity in vacuum, $n_{\bf k}$ is the Fourier transform of the fluctuating refraction index $\delta n({\bf r})$ is defined by \begin{equation}\label{two} n_{\bf k}=\frac 1V\int dVe^{i{\bf kr}}\delta n({\bf r}). \end{equation} By substituting the Hamiltonian \eqref{3six} into Eq. \eqref{2five}, the latter is rewritten as \begin{eqnarray}\label{5seven} \partial_t \hat{f}({\bf r},{\bf q},t)+{\bf c_q}\cdot\partial_{\bf r}\hat{f}({\bf r},{\bf q},t)-i\frac{\omega _0}{V}\sum_{{\bf k},{\bf k'}}e^{-i{\bf k\cdot r}}n_{{\bf k}^\prime}\nonumber\\ \times\big[b^\dag _{\bf q+ \frac {k}2}b_{\bf q-\frac{k}{2}+k^\prime}-b^\dag _{\bf q+ \frac {k}2-k^\prime}b_{\bf q-\frac{k}{2}} \big]=0. \end{eqnarray} The first two terms in the left-hand side describe free-space propagation of a laser beam and the last term arises from atmospheric inhomogeneity. The latter can be replaced by ${\bf F}({\bf r})\cdot\partial_{\bf q}\hat{f}({\bf r},{\bf q},t)$ if three components of the turbulence wave vectors ${\bf k}^\prime$ are much smaller than the corresponding characteristic values of ${\bf q}$, i.e. we can express the difference of functions in square brackets in Eq. (\ref{5seven}) by the corresponding derivative. The quantity ${\bf F}({\bf r})=\omega _0\partial_{\bf r}n({\bf r})$ is interpreted as a random force produced by atmospheric vortices \cite{Chu}. With this force Eq. (\ref{5seven}) takes the form of the kinetic equation \begin{equation}\label{6seven1} \partial_t \hat{f}({\bf r},{\bf q},t)+{\bf c_q}\cdot\partial_{\bf r}\hat{f}({\bf r},{\bf q},t)+ {\bf F}({\bf r})\cdot\partial_{\bf q}\hat{f}({\bf r},{\bf q},t)=0. \end{equation} This equation resembles the collisionless Boltzmann equation with a smoothly varying momentum-independent force {\bf F}({\bf r}) acting on point-like particles. The technique of the PDF (see Refs. \cite{Chu, enha, ChuSingle, Chumak wander, ChuPhase, sto, sto14}) is convenient for obtaining average parameters of the beam as well as for the description of wave-field fluctuations. The distribution function describes the photon density in the configuration-momentum phase space. A solution of the kinetic equation (\ref{6seven1}) with a smoothly varying fluctuation force has been obtained in Refs. \cite{Chu, enha, ChuSingle, Chumak wander, ChuPhase}. This simplified physical picture is justified only if the photon momentum \cite{name}, $\bf{q}$, is much greater than the inverse size of eddies. All components of ${\bf q}$ should obey this requirement. In the paraxial approximation, the perpendicular components of photon wave vector, $\bf q_\bot$, increase with the propagation time $t$ as $t^{1/2}$ \cite{Chu} and the beam inevitably reaches the region of saturated scintillations if $t{\rightarrow}\infty$. This indicates that Refs. \cite{Chu, chuZ, Chumak wander, ChuSingle, ChuPhase} consider the strong turbulence regime, including the limiting case of saturation, rather than the regime of a weak turbulence. The range, where the random force can be considered as smoothly varying function, extends towards smaller distances if the phase diffuser is used. The reason for this is that the phase diffuser increases the characteristic values of ${{\bf q}_\bot}$; see Refs. \cite{Ban55,Ban54,Chu} for more details. \section{Boltzmann-Langevin equation} \label{sec:ble} The scheme for derivation of the kinetic equation (\ref{6seven1}), outlined in previous section, can be justified when all components of photon wave vector ${\bf q}$ are sufficiently large. The corresponding situation occurs for long-distance propagation or strong turbulence (see, for example, Sec. VI in \cite{Chu}). It should be emphasized that it is just the case when the direct computer simulation of beam propagation becomes problematic \cite{Gorshkov}. In what follows, we describe more general approach which is free from this undesirable restriction. In the kinetic equation (\ref{5seven}) the last left-hand-side term describes process of photon ``collisions " with atmospheric inhomogeneities. The amplitude of this process is determined by $n_{\bf k^\prime}$ which is a random quantity with $\langle n_{\bf k^\prime}\rangle =0$. Two operators in square brackets of Eq. (\ref{5seven}) also depend on $ {\bf k}^\prime$. Their explicit dependence on the random refraction index can be obtained from the Heisenberg equations. One of them is given by \begin{eqnarray}\label{7nine} \!\left\{\partial_t - i\left(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k^\prime}}\right) \right\}b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}+k^\prime}=\nonumber\\ i\omega_0\sum_{{\bf {k}^{\prime\prime}}}n_{\bf k^{\prime\prime}}\bigg[b^\dag_{\bf q+\frac{k}{2}} b_{\bf q-\frac{k}{2}+k^{\prime}+{k}^{\prime\prime}}-b^\dag_{\bf q+\frac{k}{2}-{k}^{\prime\prime}}b_{\bf q-\frac{k}{2}+k^\prime}\bigg]. \end{eqnarray} Its solution can be written as \begin{eqnarray}\label{8ten} b^\dag_{\bf q+\frac{k}{2}}&&b_{\bf q-\frac{k}{2}+k^\prime}\bigg|_t= e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k^\prime}})(t-t_0)}\left(b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}+{k^\prime}}\right)\bigg|_{t_0}\nonumber\\ &&+i\omega_0\sum_{\bf {k}^{\prime\prime}}\int\limits_{t_0}^{t}dt^\prime e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})\left(t-t'\right)}n_{\bf {k}^{\prime\prime}}\nonumber\\ &&\times\bigg(b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}+{k'}+{k}^{\prime\prime}}-b^\dag_{\bf q+\frac{k}{2}-{k}^{\prime\prime}}b_{\bf q-\frac{k}{2}+{k'}}\bigg)\bigg|_{t^\prime}, \end{eqnarray} where the subscripts $t_0$ and $t^\prime$ indicate the dependence of the corresponding operators on time. In Eq. (\ref{8ten}) the interval $t-t_0$ is chosen to be large compared with the photon-eddy interaction time $\pi/ck^\prime$ and sufficiently short compared with the relaxation time $1/\nu$ caused by these interactions: \begin{equation}\label{9tenprime} \pi/ck^\prime \ll t- t_0\ll1/\nu . \end{equation} Here $\nu$ is the collision frequency and the quantity $1/k^\prime$ describes the characteristic length of atmospheric inhomogeneities. In other words, the time hierarchy (\ref{9tenprime}) means that the duration of photon interaction with scatterers is much shorter than the time of free flight. This is a typical criterion ensuring applicability of the Boltzmann equation for the description of many-particle systems (see, for example, Ref. \cite{chuZ}) . Substituting Eq. (\ref{8ten}) and a similar solution for the operator $b^\dag _{\bf q+ \frac {k}2-k^\prime}b_{\bf q-\frac{k}{2}}\big|_t$ into Eq. (\ref{7nine}) we obtain the kinetic equation for $\hat{f}({\bf r},{\bf q},t)$ \begin{equation}\label{10eleven} \partial_t \hat{f}({\bf r},{\bf {q}},t)+{\bf c_q}\cdot\partial_{\bf r}\hat{f}({\bf r},{\bf q},t)= \hat{K}({\bf r},{\bf q},t)- \hat{\nu}_{\bf q}\big \{ \hat{f}({\bf r},{\bf q},t)\} , \end{equation} where \begin{widetext} \begin{equation}\label{11elevven} \hat{K}({\bf r},{\bf q},t){=}\frac{i\omega_0}{V}\sum_{{\bf k,k^\prime}}e^{-i{\bf k}\cdot{\bf r}}n_{\bf k^\prime}\big[e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+k^\prime})(t-t_0)}\big(b^\dag_{\bf q{+}\frac{k}{2}}b_{\bf q{-}\frac{k}{2}{+}k'}\big){}\big|_{t=t_0}{-} e^{i(\omega_{\bf q+\frac{k}{2}-k^\prime}-\omega_{\bf q-\frac{k}{2}})(t-t_0)}\big(b^\dag_{\bf q{+}\frac{k}{2}{-}k^\prime}b_{\bf q{-}\frac{k}{2}}\big)\big|_{t=t_0}\big], \end{equation} \begin{equation}\label{12twelwe} \hat{\nu}_{\bf q}\big \{ \hat{f}({\bf r},{\bf q},t)\}=\frac{2\pi\omega_{0}^{2}}{c}\int d{\bf k'_{\bot}}\psi({\bf k'_{\bot}})\big(\hat{f}({\bf r},{\bf q},t)-\hat{f}({\bf r},{\bf q+k'_{\bot}},t)\big). \end{equation} \end{widetext} The notation $(_\bot)$ indicates components of the corresponding vector perpendicular to the $z$-axis, and $\psi ({\bf k'_{\bot}})=\frac V{(2\pi)^3}\langle|n_{\bf k'_{\bot}}|^2\rangle$. The value of $\psi ({\bf k})$ is given by the von Karman formula \begin{equation}\label{13twelwwe} \psi ({\bf k})=0.033C_n^2\frac {\exp(-(kl_0/2\pi )^2)}{(k^2+L_0^{-2})^{11/6}}, \end{equation} where the structure constant $C_n^2$ describes the strength of the index-of-refraction fluctuations, whereas $L_0$ and $l_0$ are usually referred to as the outer and inner radii of the turbulent eddies, respectively. These radii restrict a range of characteristic values of $\bf k'_{\bot}$. In atmospheric turbulence, $L_0$ may range from 1 to 100 m, and $l_0$ is on the order of few millimeters. It is seen from Eqs. (\ref{10eleven})-(\ref{13twelwwe}) that the random quantity $\hat{K}({\bf r},{\bf q},t)$ linearly depends on $n_{\bf k^\prime}$, while $\hat{\nu}_{\bf q}$ depends only on a regular variable $\langle|n_{\bf k'_{\bot}}|^2\rangle$. The contribution of fluctuating part of $n_{\bf{k}'} n_{\bf{k}''}$ can be neglected (for more details see Appendix \ref{sec:appendix}). The linear inhomogeneous equation (\ref{10eleven}) governs the evolution of photon distribution in the phase space. The term $\hat{\nu}_{\bf q}\big \{ \hat{f}({\bf r},{\bf q},t)\}$ describes dissipation of the distribution function caused by randomization of the photon wave vector ${\bf q_\bot}$. The term ``dissipation" does not mean here that the total number of photons decreases. Actually, after summing up the collision term (\ref{12twelwe}) over $\bf q$ we get zero, which indicates that the photon number is conserved. The collision frequency $\nu$ can be estimated by $\frac{2\pi\omega_0^2}c\psi (k'_{\bot})k'^2_\bot$, where $ k'_\bot$ is the characteristic value of the momentum transfer. The Langevin source of fluctuations in Eq. (\ref{10eleven}) is represented by $\hat{K}({\bf r},{\bf q},t)$. Random photon-eddy ``collisions" (see Refs. \cite{chuZ} and \cite{Kogan}) generate the Langevin source. Within the time interval, restricted by the inequality (\ref{9tenprime}), the constituents in the right-hand side of Eq. (\ref{11elevven}) have a simple oscillating dependence on time. Due to this favorable circumstance, the calculation of two-time correlation function $\langle \hat{K}({\bf r},{\bf q},t)\hat{K}({\bf r}^\prime,{\bf q}^\prime,t^\prime)\rangle$ reduces to obtaining the average value of the operator products defined at the same time, $t_0$. The source vanishes after averaging of Eq. (\ref{10eleven}). Then the remaining homogeneous equation for $\langle \hat{f}({\bf r},{\bf q},t)\rangle$ can be used for obtaining parameters of the beam at any distances.[In what follows, we use $f({\bf r},{\bf q},t)$ notation for $\langle \hat{f}({\bf r},{\bf q},t)\rangle$]. For long-distance propagation, where \begin{equation}\label{14twe} q_{\bot}\gg k'_{\bot}, \end{equation} the collision integral reduces to the differential form \begin{equation}\label{15twell} \hat{\nu}_{\bf q}\big \{ \hat{f}({\bf r},{\bf q},t)\}=-\frac{\pi\omega_{0}^{2}}{c}\int d{\bf k'_{\bot}}\psi({\bf k'_{\bot}})\bigg(\frac \partial{\partial{\bf q}} {\bf k^\prime}_\bot \bigg)^2\hat{f}({\bf r},{\bf q},t), \end{equation} which describes a diffusion-like motion in the wave vector space. The kinetic equation with $\hat{K}({\bf r},{\bf q},t)=0$ and the collision term, which is similar to (\ref{15twell}), was used in Refs. \cite{Yang51} and \cite{berman95} to investigate the propagation of relativistic charged particles through an inhomogeneous medium (for example, through a foil). The similarity arises from equivalence of the small-scattering-angle approximation, used in Refs. \cite{Yang51}, \cite{berman95}, and the paraxial approximation, used in this paper. Although the linear energy-momentum relationship holds for both the photons and ultrarelativistic particles, the microscopic scattering mechanisms are different for those cases. \section{Scintillation index} \label{sec:scint_ind} Equation (\ref{10eleven}) can be used to study the effect of photon multiple scattering on their distribution in the phase space. Summation of $\hat{f}({\bf r},{\bf q},t)$ over ${\bf q}$ results in a spatio-temporal photon distribution \begin{equation} \label{16twel} \hat{I}({\bf r},t)=\sum_{\bf q}\hat{f}({\bf r},{\bf q},t), \end{equation} which includes an average value, $\langle \hat{I}({\bf r},t)\rangle\equiv I({\bf r},t) $, and fluctuations, $\delta \hat{I}({\bf r},t)$, \begin{eqnarray} \label{17svnt} \hat{I}({\bf r},t)&{=}& I({\bf r},t)+\delta \hat{I}({\bf r},t)\nonumber\\ &{=}&\sum_{\bf q} f({\bf r},{\bf q},t)+\sum_{\bf q}\delta \hat{f}({\bf r},{\bf q},t), \end{eqnarray} where $\delta \hat{f}({\bf r},{\bf q},t)=\hat{f}({\bf r},{\bf q},t)-f({\bf r},{\bf q},t)$. To obtain $I({\bf r},t)$, one needs to solve averaged Eq. (\ref{10eleven}), accounting for the boundary conditions at the aperture plain and using $\langle \hat{K}({\bf r},{\bf q},t)\rangle=0$. The scintillation index is defined by \begin{equation}\label{18fifte} \sigma^2=\frac {\langle : \delta \hat{I}^2({\bf r}):\rangle}{ I({\bf r})^2}=\frac {\langle : \hat{I}^2({\bf r}):\rangle-I({\bf r})^2}{ I({\bf r})^2}, \end{equation} where the symbol $\{:..:\}$ means the normal ordering of the creation and annihilation operators. The definition (\ref{18fifte}) does not include contribution of shot noise. This noise enters the fluctuations of the detector counts and tends to be important in problems of quantum optics. The shot-noise term is linear in the photon density. It can be easily excluded from experimental data to facilitate the comparison with the theoretical calculation. Calculation of Eq. (\ref{18fifte}) is more intricate. It follows from Eqs. (\ref{17svnt}) and (\ref{18fifte}) that $\sigma^2$ is a quadratic form of PDF fluctuations, $\langle\delta \hat{f}({\bf r},{\bf q},t)\delta \hat{f}({\bf r}^\prime,{\bf q}^\prime,t^\prime)\rangle$. Hence, the calculation of $\sigma^2$ is possible if the correlation function of photon distributions is known. To simplify the problem, we use an approximate iterative scheme. \subsection{First order approximation} The approximation is based on the assumption that close to the transmitter aperture the collision term does not perturb significantly PDF and can be omitted. In this case, the average value of PDF satisfies the equation \begin{equation}\label{19ninn} (\partial_t+{\bf c_q}\cdot\partial_{\bf r})f_0({\bf r},{\bf q},t)=0. \end{equation} The fluctuating part of $\delta \hat{f}({\bf r},{\bf q},t)$ is governed by the similar equation supplemented with the Langevin source $\hat{K}$ \begin{equation}\label{20nint} (\partial_t+{\bf c_q}\cdot\partial_{\bf r})\delta \hat{f}({\bf r},{\bf q},t)=\hat{K}({\bf r},{\bf q},t). \end{equation} Equations (\ref{19ninn}) and (\ref{20nint}) follow from Eq. (\ref{10eleven}) after replacing $\hat{f}$ by $f_0+\delta \hat{f}$. The Langevin source linearly depends on $n_{{\bf k}_\bot}$ while the neglected collision integral is quadratic in $n_{{\bf k}_\bot}$. Therefore, Eqs. (\ref{19ninn}) and (\ref{20nint}) can be interpreted as the lowest-order expansions of Eq. (\ref{10eleven}) in powers of $n_{{\bf k}_\bot} $. The general solution of Eq. (\ref{20nint}) is represented by two terms \[\delta \hat{f}({\bf r},{\bf q},t)=\delta \hat{f}_0({\bf r_q}(t'),{\bf q},t')|_{t'=0}+\delta \hat{f}_1({\bf r,q},t),\] where ${\bf r_q}(t^\prime)={\bf r}-{\bf c_q}(t-t')$ and \begin{equation}\label{21twnt} \delta \hat{f}_1({\bf r,q},t)=\int\limits_{0}^{t}dt'\hat{K}({\bf r_q}(t'),{\bf q},t'). \end{equation} We consider the aperture plane as a starting points of photon trajectories (at $t'=0$). The paraxial approximation imposes a set of restrictions on the wave-vectors: $q_z{\sim}q_0{\gg}q_\bot,k_\bot,k_\bot^\prime$. Then $z_{\bf q}(t^\prime =0)=z-ct=0$. The term, $\delta \hat{f}_0({\bf r_q}(t'),{\bf q},t')|_{t'=0}$, describes the evolution of PDF fluctuations in vacuum. In what follows, we neglect fluctuations of the incident light. In this case $\delta \hat{f}_0({\bf r_q}(t'),{\bf q},t')|_{t'=0}=0$ and only the term, $\delta \hat{f}_1({\bf r,q},t)$, is responsible for the non-zero amount of the scintillation index, $\sigma^2$, at small propagation time $t$. It is given by \begin{widetext} \begin{equation}\label{22twnt1} \sigma^2=\frac {\sum_{\bf q, q'}\langle:\delta \hat{f}({\bf r},{\bf q},t)\delta \hat{f}({\bf r},{\bf q}^\prime,t):\rangle }{ (\sum_{\bf q}f_0({\bf r},{\bf q},t))^2}=\frac {\sum_{\bf q, q'}\int\limits_{0}^{t}\int\limits_{0}^{t}dt'dt''\langle :\hat{K}({\bf r_q}(t'),{\bf q},t')\hat{K}({\bf r_{q^\prime}}(t''),{\bf q'},t''):\rangle }{ (\sum_{\bf q}f_0({\bf r},{\bf q},t))^2}, \end{equation} where \begin{equation}\label{23twntx} f_0({\bf r},{\bf q},t)= f_0({\bf r_q}(0),{\bf q},0),\quad \sum_{\bf q} f_0({\bf r},{\bf q},t) \equiv I_0({\bf r},t)={1\over V} \sum_{\bf q,k}e^{-i{\bf k}({\bf r-c_q}t)}\langle b^\dag_{{\bf q+\frac k2}} b_{{\bf q-\frac k2}}\rangle |_{t=0}. \end{equation} \end{widetext} The first equation in (\ref{23twntx}) means that the left-hand-side term satisfies both the collisionless kinetic equation (\ref{19ninn}) and the boundary conditions at the aperture. The value of $ I_0({\bf r},t)$ is equal to photon density in the absence of turbulence. The numerator in the right-hand side of Eq. (\ref{22twnt1}) can be calculated using the explicit term (\ref{11elevven}) for $\hat{K}({\bf r},{\bf q},t)$ and meeting boundary conditions (see App. \ref{sec:appendix1}). Then the scintillation index linearly depends on $\langle|n_{\bf k_\bot}|^{2}\rangle$ and reduces to \begin{equation}\label{35twnt66} \sigma ^2=\sigma _1^2L(z,\rho_0,\rho_1), \end{equation} where $\sigma _1^2=1.23C_n^2q_0^{7/6}z^{11/6}$ is the Rytov variance, $\rho _{0,1}^2={r_{0,1}^2q_0}/z $, $r_0$ is initial radius of the beam, $r^2_1=r_0^2/(1+2r_0 ^2\lambda _c^{-2})$, the quantity $\lambda _c$ describes the effect of the phase diffuser, and $L(z,\rho_0,\rho_1)$ is the double integral \begin{equation}\label{36twnt7} L(z,\rho_0,\rho_1)=4.24\int\limits _0^1d\tau \int\limits _0^\infty d\chi \chi^{-8/3}\exp\Bigg\{-\chi^2\Bigg[\frac {q_0l_0^2}{4\pi ^2z}+ \end{equation} \[ \tau ^2\frac {\rho_0^2+\rho_1^2}{4+\rho_0^2\rho_1^2}\Bigg]\Bigg\} \sin^2\Bigg(\frac {\tau \chi^2}2-\frac {2\tau ^2\chi^2}{4+\rho _0^2\rho _1^2}\Bigg).\] Equations (\ref{35twnt66}) and (\ref{36twnt7}) were derived in \cite{Chu} using a different approach. It follows from these equations that in the limit of large initial radius of beam aperture ($\rho_0,\rho_1{\rightarrow}\,\infty$) and infinitely small inner scale of turbulence ($l_0{\rightarrow}\,0$), we have the result of Rytov theory ($\sigma ^2=\sigma _1^2$) because $L{\rightarrow}\,1$. \subsection{Collision term in average intensity} The numerator as well as the denominator in Eq. ($\ref{22twnt1}$) are derived using only first non-vanishing iterative terms. Extension of the theory towards a moderate turbulence requires accounting for the collision term $-\hat{\nu}\big \{ \hat{f}({\bf r},{\bf q},t)\}$. Following the iterative procedure, we substitute the approximate value of PDF, given by Eq. (\ref{23twntx}), into the collision term of Eq. (\ref{10eleven}). Then the right-hand side of Eq. (\ref{10eleven}) is considered as a known function. After averaging the modified equation, we obtain \begin{equation}\label{24elevenx} (\partial_t +{\bf c_q}\cdot\partial_{\bf r}) f_1({\bf r},{\bf q},t)=-\hat{\nu}_{\bf q}\big \{ f_0({\bf r},{\bf q},t)\}, \end{equation} where $ f_1$ is the first non-vanishing term generated by the collision integral. Solution of Eq. (\ref{24elevenx}), obeying zero-value boundary conditions, is given by \begin{equation}\label{25therty} f_1({\bf r},{\bf q},t)=-\int\limits _0^tdt^\prime\hat{\nu}_{\bf q}\{ f_0({\bf r_q}(t^\prime),{\bf q},t ^\prime)\}. \end{equation} The contribution of $ f_1({\bf r},{\bf q},t)$ into the total photon density is given by \begin{eqnarray}\label{26twnt} I_1({\bf r},t) \equiv \sum_{\bf q} f_1&&({\bf r},{\bf q},t)=-\frac{\omega_0^2t}{cS}\sum_{\bf q,k,k_\bot^\prime}\langle|n_{\bf k_\bot^\prime}|^2\rangle e^{-i{\bf k}({\bf r-c_q}t)}\nonumber\\ &&\times\bigg[1-\frac {\sin({\bf kc_{k_\bot ^\prime} }t)}{{\bf kc_{k_\bot ^\prime} }t}\bigg]\langle b^\dag_{{\bf q+\frac k2}} b_{{\bf q-\frac k2}}\rangle|_{t=0}. \end{eqnarray} Equation (\ref{26twnt}) accounts for the beam broadening caused by atmospheric eddies. Averaging of each factor in the sum can be performed independently because of the absence of correlations between the source fluctuations and the refractive index fluctuations. Two quantities, $ I_0({\bf r},t)$ and $ I_1({\bf r},t)$, are zeroth- and first-order terms of the development of average photon density in powers of $\langle|n_{\bf k_\bot}|^2\rangle$, respectively. \subsection{Second order $\delta \hat{f}_{2}$ and combined effect of fluctuations $\delta \hat{f}_{1}{\cdot}\delta \hat{f}_{2}$} The second iterative term for fluctuations of PDF, $\delta \hat{f}_2$, obeys the equation \begin{equation}\label{27ninty} \partial_t \delta \hat{f}_2({\bf r},{\bf q},t)+{\bf c_q}\cdot\partial_{\bf r}\delta \hat{f}_2({\bf r},{\bf q},t)=-{\hat \nu}_{\bf q}\{\delta \hat{f}_1({\bf r_q},{\bf q},t)\}, \end{equation} where the function $\delta \hat{f}_1$, given by Eq. (\ref{21twnt}), enters the collision term. Solution of Eq. (\ref{27ninty}) is \begin{equation}\label{28twnt9y} \delta \hat{f}_2({\bf r},{\bf q},t)=-\int\limits_{0}^{t}dt'\hat{\nu}_{\bf q} \{\delta \hat{f}_1({\bf r_q}(t^\prime),{\bf q},t^\prime)\}, \end{equation} were the explicit form of the collision integral is given by \begin{eqnarray}\label{29twnt9yy} &{{\hat \nu}_{\bf q}\{\delta \hat{f}_1({\bf r_q}(t^\prime),{\bf q},t')\} =\frac{L_z\omega_0^2}{c} \sum\limits_{{\bf k'_\bot}} \langle |n_{\bf k'_\bot}|^2\rangle}\nonumber\\ &\times\big[\delta \hat{f}_1({\bf r_q(t^\prime)},{\bf q},t')-\delta \hat{f}_1({\bf r_q(t^\prime)},{\bf q+k'_\bot},t')\big ]. \end{eqnarray} To proceed, let us consider a combined effect of fluctuations $\delta \hat{f}_{1,2}({\bf r},{\bf q},t)$ on $\sigma^2$. Contributions of $\delta \hat{f}_{1,2}$ into the photon density are given by $\sum\limits_{\bf q}(\delta \hat{f}_1({\bf r},{\bf q},t)+\delta \hat{f}_2({\bf r},{\bf q},t))$. This sum includes linear and cubic in $n_{\bf{k_\bot}}$ terms. The average square of this sum includes the term \begin{eqnarray}\label{37p} &\sum_{{\bf q,q}_1}&\langle\delta\hat{f}_1({\bf r},{\bf q},t){\cdot}{\delta} \hat{f}_2({\bf r},{{\bf q}_1},t)+\delta \hat{f}_2({\bf r},{\bf q},t)\cdot\delta \hat{f}_1({\bf r},{{\bf q}_1},t)\rangle\nonumber\\ &=&{2\sum_{{\bf q,q}_1}\langle\delta \hat{f}_1({\bf r},{\bf q},t)\cdot\delta \hat{f}_2({\bf r},{{\bf q}_1},t)\rangle} \end{eqnarray} which is quadratic in $\langle|n_{{\bf k}_\bot}|^2 \rangle$. For obtaining $\sigma^2$, we use this term and neglect terms of order $O(\langle|n_{\bf k_\perp}|^2\rangle^3)$. Then using Eqs. (\ref{21twnt}) and (\ref{28twnt9y}) we obtain the explicit expression for Eq. (\ref{37p}). It is given by \begin{eqnarray}\label{38p} 2&\sum\limits_{{\bf q,q}_1}&\langle\delta \hat{f}_1({\bf r},{\bf q},t)\cdot\delta \hat{f}_2({\bf r},{{\bf q}_1},t)\rangle\nonumber\\ &=&\frac{2\omega_0^4}{c^2S^2}\sum_{\substack{{\bf q,k,k^\prime} \\ {\bf q_1,k_1,k^{\prime\prime}}}}\langle|n_{{\bf k}^\prime}|^2\rangle\langle| n_{{\bf k}^{\prime\prime}}|^2\rangle\int\limits _0^td\tau\nonumber\\ &\times&\int\limits _\tau^td\tau_1 e^{-i{\bf k\cdot(r-c_q\tau)}-i{\bf k}_1\cdot({\bf r-c_q}_1 \tau_1)}\nonumber\\ &\times&\big[1-e^{-i{{\bf k\cdot c_{k^{\prime}}}\tau)}}\big]\big[1-e^{i{{\bf k\cdot c_{k^{\prime\prime}}}\tau_1)}}\big]\big[1-e^{-i{{\bf k_1\cdot c_{k^{\prime\prime}}}\tau_1)}}\big]\nonumber\\ &\times&\big\langle b^\dag_{{\bf q+k}/2} b^\dag_{{\bf q_1+k_1}/2} b_{{\bf q-k}/2+{\bf k^{\prime\prime}}} b_{{\bf q_1-k_1}/2-{\bf k^{\prime\prime}}}\big\rangle\big|_{t-\tau_1}, \end{eqnarray} where the operators in the angle brackets depend on time as in the absence of turbulence. The summation in Eq. (\ref{38p}) runs over components of vectors ${\bf q},{\bf q}_1,{\bf k},{\bf k}_1,{\bf k}^\prime,{\bf k}^{\prime\prime}$ which are perpendicular to the $z$-axis (the labels ($_\bot$) are omitted for brevity). Parallel to the $z$-axis components are given by \begin{equation}\label{39p} q_z=q_{1z}=q_0,\quad k_z=k_{1z}=k^\prime_z=k^{\prime\prime}_z=0. \end{equation} The relations (\ref{39p}) can be derived from Eq. (\ref{32twnt5xx}). The conditions $k_z^\prime =k_{z}^{\prime\prime}=0$ are consistent with the Markov approximation \cite{Tatarskii1}, \cite{Fante1} (not used here!) in which the index-of-refraction fluctuations, $\delta n({\bf r})$, are assumed to be delta-function correlated in the direction of propagation: \[\langle\delta n({\bf r}_\bot,z)\delta n({\bf r}^\prime_\bot,z^\prime)\rangle\sim \delta (z-z^\prime).\] In this case, the turbulent eddies look like flat disks oriented normally to the propagation path. At first sight, this representation of the correlation function seems unrealistic because the atmosphere is assumed to be statistically homogeneous and isotropic. The paradox is explained by the effect of relativistic length contraction (Lorentz contraction) of moving objects. The relative motion of the atmosphere towards photons results in a zero value of correlation length in the direction of motion. The effect of turbulence comes only from ``diagonal" components $\langle|n_{{\bf k}^\prime_\bot}|^2\rangle$ and $\langle|n_{{\bf k}^{\prime\prime}_\bot}|^2\rangle$ of the correlation function. As before, this is the result of statistical homogeneity of the turbulent atmosphere. The final result of this Section is represented by \begin{equation}\label{40p} \sigma^2=\frac{\sum\limits_{{\bf q},{\bf q}_1}\langle\delta \hat{f}_1({\bf r},{\bf q},t)[\delta \hat{f}_1({\bf r},{\bf q}_1,t)+2\delta \hat{f}_2({\bf r},{\bf q}_1,t) ]\rangle}{\big(\sum\limits_{\bf q} f_0({\bf r},{\bf q},t){+} f_1({\bf r},{\bf q},t)\big)^2}, \end{equation} where the numerator and denominator are defined by Eqs. (\ref{22twnt1})-(\ref{36twnt7}), (\ref{26twnt}), (\ref{37p})-(\ref{39p}) and (\ref{32twnt5xx})-(\ref{34twnt6x}). Bringing together analytical and numerical calculations, we obtain $\sigma^2$ for different experimental conditions. Also, it is possible to compare the scintillation index obtained by employing different numbers of iteration steps as described in this section. \begin{figure} \caption{(Color online) Scintillations as function of Rytov parameter. Results for different theoretical approaches qualitatively compared with experimental data. There are theoretical results of current paper (solid line) [Eq. (\ref{40p}) ], Rytov approach (dotted line) [Eq. (\ref{35twnt66}) ], asymptotic formulas for Huygens-Kirchhoff method \cite{banakh79} (dashed lines) and the results of approach that the authors developed in Ref. \cite{enha} (dash-dotted line). The inset shows the typical experimental $\sigma^2$ for the considered atmospheric conditions (adopted from Ref. \cite{consortini} for $4\,\text{mm}<l_0\leq7\,\text{mm}$ ). Parameters for theories: $l_0=6.3\,$mm, $q_0=1.29\times10^{7}\,\text{m}^{-1}$, $r_0=0.01\,\text{m}$, $z=1200\,\text{m}$. The shaded area shows the parameter region considered in the current article.} \label{fig:Experiment} \end{figure} \begin{figure} \caption{(Color online) Scintillation index for coherent beams vs. propagation distance $z$. On the upper graph dash-dotted curves are obtained using the Rytov approach [Eq. (\ref{35twnt66})]; solid curves are obtained with the account for the collision term [Eq. (\ref{40p})]; dashed curves display the results obtained in Ref. \cite{enha} (see their Fig. 1 and 2), where the correlation of photon trajectories is accounted for. Shaded area at upper graph is enlarged and depicted on lower graph. Inner turbulence scale $\frac{l_0}{2\pi}=10^{-3}\,\text{m}$ and the optical wavelength $q_0=10^{7}\,\text{m}^{-1}$.\\} \label{fig:IntCollvsCrossCorell} \end{figure} \section{Results and discussion} \label{sec:discussion} A complete theory of scintillations does not exist yet. At the same time, there are well-justified solutions in the limiting cases of weak ($\sigma_1^2\ll1$) and strong ($\sigma_1^2\gg1$) turbulences. The kinetic equation, in which a beam scattering is described by the collision integral, is applicable for any Rytov variance $\sigma_1^2$ with the exception of a very short distance equal to the typical eddy size. An exact solution of this equation is problematic. Therefore, we restrict the numerical solution to a moderate values of $\sigma_1^2$ ($\sigma_1^2\leq 0.85$, see shaded area, in Fig. \ref{fig:Experiment}, and $\sigma_1^2\leq 0.75$ for the other figures) and use the iteration scheme described in Sec. \ref{sec:scint_ind}. At the same time this parameter is appreciably greater than the range of the Rytov approach validity $\sigma_1^2<0.3$ \cite{Fante1}. Figure \ref{fig:Experiment} compares the scintillation index calculated within the Boltzmann-Langevin approach with other theoretical approaches and with the typical experimental data, adopted from Consortini \textit{et al.} \cite{consortini}. Although the original data of Ref. \cite{consortini} are collected for spherical waves while theory deals with plane waves, we propose a qualitative comparison of results to illuminate peculiarities of scintillations and advantages of our method for their description. Naturally, for small values of the Rytov parameter ( $\sigma_1^2\leq 0.25$) our result coincides with the asymptotics for the Huygens-Kirchhoff method and Rytov-like method, but differs dramatically for the larger values showing the same increasing tendency as experimental data in a weak-to-moderate turbulence regime. For the sake of completeness we also provide theoretical results from the side of large values of Rytov parameter calculated within the approach of Ref. \cite{enha} and the Huygens-Kirchhoff approach. We observe that the Huygens-Kirchhoff method presents only a limited description for strong turbulence, while results of the approach from Ref. \cite{enha} shows better description of scintillation index going deeper to the range of moderate turbulences. Moreover, the results of Ref. \cite{enha} show the tendency to mesh with the results of current paper plausibly repeating the overall behavior of scintillations in the cited experiment. \begin{figure} \caption{(Color online) Scintillation index for coherent beam vs. propagation distance $z$ for different initial radii of the beam. The rest of the parameters are the same as in Fig. \ref{fig:IntCollvsCrossCorell}. The curves from the left side are obtained using the present approach [Eq. (\ref{40p})]; the curves from the right side are obtained using the approach developed in Ref. \cite{enha}.} \label{fig:DiffRadius} \end{figure} To take a closer look at our results we provide a comparison with the results of the Rytov approach under different configurations of atmospheric channel ( Fig. \ref{fig:IntCollvsCrossCorell}). Again for small values of $\sigma^2$, there is a good agreement for data obtained within the two approaches (enlarged shaded area at lower graph) and for greater values of $\sigma^2$ we can see not only numerical inconsistencies, but also different tendencies of $\sigma^2(z,C_n^2)$ to grow for considered cases. The comparison with the results of the previous paper \cite{enha} for moderate-to-strong turbulence regime displays the tendency for matching at some intermediate region. It also demonstrates that maximum of $\sigma^2$ should be situated at shorter distances $z$ if the structure constant, $C_n^2$, is larger. This result can be easily foreseen in view of the fact that strong photon-turbulence interaction approaches the crossover to the Gaussian statistics. One more aspect taken under consideration is the dependence of scintillations on the initial radius of laser beam. Figure \ref{fig:DiffRadius} illustrates the behavior of the scintillation index in the regions adjoining the extremum of $\sigma^2$. We can see that the initial growth of $\sigma^2$ is steeper in the case of smaller initial radii $r_0$. This is due to stronger correlation of photon trajectories: the correlation is more pronounced for small $r_0$ \cite{enha}. This is easily explained since if the trajectories are closer to each other, then the probability for different photons to be scattered by the same eddy is greater. This is the case when a random scattering generates photon-photon correlations. Figure \ref{fig:popravka} can be used for explanation of the physical mechanism responsible for the increase of $\sigma^2$ in the range $\sigma_1^2\leq{0.75}$. The solid lines are obtained using Eq. (\ref{40p}). The data shown by the dash-dot line are obtained from the same expression considering $\delta \hat{f}_2=0$. There is only a small difference between the corresponding pairs of curves. Therefore, the major part of the discrepancy of our results for $\sigma^2$ from the results based on the Rytov approximation is due to the decrease of the photon density caused by the turbulence. This decrease is described by the term $f_1$ in the denominator of (\ref{40p}). \begin{figure}\label{fig:popravka} \end{figure} \section{Conclusion} \label{sec:conclusion} For decades, the description of light propagation in a turbulent atmosphere has remained a challenging theoretical problem. The interconnection between the initial and the detected signals, obtained theoretically, is not sufficient for the description of atmospheric communication system efficiency. The point is that the detected signal has a memory about random scattering events occurred in the course of propagation. Therefore, even for the statistically homogeneous and stationary atmosphere, the received signal varies (fluctuates) for different paths. The size of these fluctuations is described by the scintillation index. By definition, the scintillation index is expressed via the correlation functions of the photon distribution. The kinetic equation for the distribution function and its fluctuating part is derived here from first principles. Their solutions are obtained using the iteration procedure which is applicable for short propagation distances or small turbulence structure factors. In our analysis, we use the paraxial approximation for beams. This approximation reduces the problem to the case of a two-dimensional wave vector domain and simplifies the collision integral as well as correlation functions of the Langevin sources. Concluding, we think that further progress in the problem of scintillations lies in the improvement of our ability to carry out complex multiple integrations. \section{Acknowledgments} The authors thank A. Gabovych, G. Berman, D. Vasylyev and E. Stolyarov for useful discussions and comments. \appendix \section{The collision integral} \label{sec:appendix} \numberwithin{equation}{section} \setcounter{equation}{0} The collision integral (\ref{12twelwe}) can be derived using the standard procedure. Nevertheless, some explanations are required. The derivation of Boltzmann-like kinetic equations is based on the assumption of a negligibly short interaction time of individual particles (photons) with scatterers. \begin{widetext} The corresponding criteria are given by Eq. (\ref{9tenprime}). The other point concerns the explicit form of the scattering probability. For our case, the collision process is described by the operator \begin{equation}\label{A1} \hat{J}=-i\frac{\omega _0}{V}\sum_{{\bf k},{\bf k'}}e^{-i{\bf k\cdot r}}n_{{\bf k}^\prime} \big[b^\dag _{\bf q+ \frac {k}2}b_{\bf q-\frac{k}{2}+k^\prime}-b^\dag _{\bf q+ \frac {k}2-k^\prime}b_{\bf q-\frac{k}{2}} \big], \end{equation} (see Eq. (\ref{5seven})). Using the quantity $b^\dag _{\bf q+ \frac {k}2}b_{\bf q-\frac{k}{2}+k^\prime}$, given by Eq. (\ref{8ten}), we rewrite Eq. (\ref{A1}) as \begin{equation}\label{A2} \hat{J}=-\hat{K}({\bf r},{\bf q},t)+ \frac{\omega_0^2}V\sum_{\bf k,k^\prime, {k}^{\prime\prime}}n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}\int\limits_{t_0}^{t}dt^\prime e^{-i{\bf k \cdot r}} \bigg[e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})\left(t-t'\right)}\big(b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}+{k'}+{k}^{\prime\prime}}-b^\dag_{\bf q+\frac{k}{2}-{k}^{\prime\prime}}b_{\bf q-\frac{k}{2}+{k'}}\big) \end{equation} \[ -e^{i(\omega_{\bf q+\frac{k}{2}-k^\prime}-\omega_{\bf q-\frac{k}{2}})\left(t-t'\right)}\big(b^\dag_{\bf q+\frac{k}{2}-k^\prime}b_{\bf q-\frac{k}{2}+{k}^{\prime\prime}}-b^\dag_{\bf q+\frac{k}{2}-k^\prime-{k}^{\prime\prime}}b_{\bf q-\frac{k}{2}}\big)\bigg]\bigg|_{t^\prime}=-\hat{K}({\bf r},{\bf q},t)+\hat{\tilde{J}},\] where the second term in square brackets is derived from the first one by replacing ${\bf q} \rightarrow{\bf q-k^\prime}$ in the first one, and the interval $t-t_0$ satisfies the condition (\ref{9tenprime}). Products of $n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}$ and $b^\dag b$ in Eq. (\ref{A2}) have a fluctuating nature. In what follows, we will neglect correlations between the corresponding subsystems. In this case we may consider them separately. The quantity $n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}$ contains a nonzero average constituent and a fluctuating part. Let us consider the product $n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}$ in more details. By definition \begin{equation}\label{A3} n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}=\frac 1{V^2}\int\int d{\bf r}d{\bf r}_1e^{i[{\bf k}^\prime\cdot{\bf r}+{\bf k}^{\prime\prime}\cdot{\bf r}_1]}\delta n({\bf r})\delta n({\bf r}_1)= \frac 1{V^2}\int\int d{\bf R}d{\bf s}e^{i({\bf k}^\prime+{\bf k}^{\prime\prime})\cdot{\bf R}+i({\bf k}^{\prime}- {\bf k}^{\prime\prime})\cdot{\bf s}/2} \delta n({\bf R}+\frac {\bf s}2)\delta n({\bf R}-\frac {\bf s}2 ), \end{equation} where ${\bf R}=({\bf r}+{\bf r}_1)/2,\quad {\bf s}={\bf r}-{\bf r}_1$. The range of $s\lesssim l_{corr}$, where the correlation length $l_{corr}$ is comparable with the eddies size, provides a dominant contribution into the average part of the integral (\ref{A3}). In spatially homogeneous mediums, the quantity ${\langle\delta n({\bf R}{+}\frac {\bf s}2)\delta n({\bf R}{-}\frac {\bf s}2 )}\rangle$ does not depend on $\bf R$ and the characteristic values of $|{\bf k}^{\prime}- {\bf k}^{\prime\prime}|$ are restricted by $ 1/l_{corr}$. The characteristic value of $R$ is of the order of the system size $L$. In this case $|{\bf k}^{\prime}+ {\bf k}^{\prime\prime}|{\sim}1/L$ tends to zero if $L{\rightarrow}\infty$. This means that the relation ${{\bf k}^{\prime}=-\bf k}^{\prime\prime}$ holds at any practically important values of ${\bf k}^{\prime}$ and ${\bf k}^{\prime\prime}$. Thus we have \begin{eqnarray}\label{A4} \langle n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}\rangle &=& \frac 1V\delta_{{\bf k^\prime},-{\bf k^{\prime\prime}}}\int d{\bf s}\int \frac{d{\bf R}}Ve^{i{\bf k^\prime\cdot s}}\langle\delta n({\bf R}+\frac {\bf s}2)\delta n({\bf R}-\frac {\bf s}2 )\rangle\nonumber\\ &=&\delta_{{\bf k^\prime},-{\bf k^{\prime\prime}}}\int \frac {d{\bf s}}V e^{i{\bf k^\prime \cdot s}}\langle\delta n({\bf s})\delta n(0)\rangle\nonumber\\ &=&\delta_{{\bf k^\prime},-{\bf k^{\prime\prime}}}\langle n({\bf r})n(0)\rangle_{{\bf k}^\prime}=\delta_{{\bf k^\prime},-{\bf k^{\prime\prime}}}\langle |n_{\bf k^\prime}|^2 \rangle . \end{eqnarray} The angle brackets mean averaging over the volume $V$, which is assumed to be much greater than the correlation volume $l_{corr}^3$. Such averaging is equivalent to averaging over different configurations of turbulent atmosphere. The substitution of $ \delta_{\bf {k}^\prime,-{\bf {k}}^{\prime\prime}}\langle|n_{\bf {k}^\prime}|^2\rangle$ for $n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}$ in Eq. (\ref{A2}) transforms the second term there to \begin{eqnarray}\label{A5} {\hat{\tilde{J}}=\frac{\omega_0^2}V\sum_{\bf k,k^\prime }\langle |n_{\bf k^\prime}|^2 \rangle\int_{t_0}^{t}dt^\prime e^{-i{\bf k\cdot r}} \bigg[e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})\left(t-t'\right)}\big(b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}-b^\dag_{\bf q+\frac{k}{2}+{k}^{\prime}}b_{\bf q-\frac{k}{2}+{k'}}\big)}\nonumber\\ -e^{i(\omega_{\bf q+\frac{k}{2}-k^\prime}-\omega_{\bf q-\frac{k}{2}})\left(t-t'\right)}\big(b^\dag_{\bf q+\frac{k}{2}-k^\prime}b_{\bf q-\frac{k}{2}-{k}^{\prime}}-b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}\big)\bigg]\bigg|_{t^\prime} . \end{eqnarray} The rest of the terms with $n_{\bf {k}^{\prime}}n_{\bf {k}^{\prime\prime}}$, where $\bf {k}^{\prime}\neq-\bf {k}^{\prime\prime}$, have a random nature and should be added to the Langevin source $\hat{K}({\bf r},{\bf q},t)$. These terms contribute negligibly to $\hat{K}$ and can be neglected if Eq. (\ref{9tenprime}) holds true. For the short interval $t-t_0$ [see (\ref{9tenprime})], the distribution function does not vary significantly and the evolution of operators $b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}$ resembles the evolution in vacuum: \begin{equation}\label{A6} b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}|_{t^\prime}=e^{-i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}})\left(t-t'\right)}b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}|_{t}. \end{equation} The operators in the right side of Eq. (\ref{A6}) depend only on a fixed time $t$ and the integration in Eq. (\ref{A5}) concerns only the exponential functions \begin{eqnarray}\label{A7} \int\limits_{t_0}^{t}dt^\prime e^{-i{\bf k\cdot r}} e^{i(\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})\left(t-t'\right)}b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}|_{t^\prime} =b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}|_t\int\limits_{t_0}^{t}dt^\prime e^{i(\omega_{\bf q-\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})(t-t^\prime)}. \end{eqnarray} The condition (\ref{9tenprime}) enables the interval $t-t_0$ to be replaced by infinity \begin{eqnarray}\label{A8} {\int\limits_{t_0}^{t}dt^\prime e^{i(\omega_{\bf q-\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}})(t-t^\prime)} \approx\int\limits_0^{\infty}d\tau e^{i(\omega_{\bf q-\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}}+i\eta)\tau}} =\frac i{\omega_{\bf q-\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}}+i\eta}, \end{eqnarray} \end{widetext} where $\eta\rightarrow +0$. Similar consideration is applicable to each term in Eq. (\ref{A5}). Then Eq. (\ref{A5}) reduces to \begin{eqnarray}\label{A9} \hat{\tilde{J}}&{=}&\frac{i\omega_0^2}V\sum_{\bf k,k^\prime }\langle |n_{\bf k^\prime}|^2 \rangle e^{-i{\bf k\cdot r}}\bigg[\frac {b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}}{\omega_{\bf q-\frac{k}{2}}-\omega_{\bf q-\frac{k}{2}+{k'}}+i\eta}\nonumber\\ &&-\frac{b^\dag_{\bf q+\frac{k}{2}+{k}^{\prime}}b_{\bf q-\frac{k}{2}+{k'}}}{\omega_{\bf q+\frac{k}{2}}-\omega_{\bf q+\frac{k}{2}+{k'}}+i\eta}- \frac {b^\dag_{\bf q+\frac{k}{2}+k^\prime}b_{\bf q-\frac{k}{2}+k^\prime}}{\omega_{\bf q-\frac{k}{2}+k^\prime}-\omega_{\bf q-\frac{k}{2}}+i\eta}\nonumber\\ &&+\frac{b^\dag_{\bf q+\frac{k}{2}}b_{\bf q-\frac{k}{2}}}{\omega_{\bf q+\frac{k}{2}+k^\prime}-\omega_{\bf q+\frac{k}{2}}+i\eta}\bigg]\bigg|_t. \end{eqnarray} In the last two terms, the value of ${\bf k^\prime}$ is replaced by ${-\bf k^\prime}$. For paraxial beams, considered here, we can use the approximation $\omega_{\bf q}=cq\approx cq_z$, which implies a negligible contribution of $q_{x,y}$ components. Then, using the relation \[\frac 1{ck^\prime_z-i\eta} -\frac 1{ck^\prime_z+i\eta}=\frac{2\pi i }c\delta(k^\prime_z)\] and integration over $k^\prime_z$, Eq. (\ref{A9}) simplifies to \begin{equation}\label{A10} \hat{\tilde{J}}=\frac{2\pi\omega_{0}^{2}}{c}\int d{\bf k'_{\bot}}\psi({\bf k'_{\bot}})\big(\hat{f}({\bf r},{\bf q},t)-\hat{f}({\bf r},{\bf q+k'_{\bot}},t)\big), \end{equation} where the definition (\ref{1threee}) of PDF was used. Equation (\ref{A10}) coincides with the collision integral $\hat{\nu}_{\bf q}\big \{ \hat{f}({\bf r},{\bf q},t)\}$ represented by Eq. (\ref{12twelwe}). \section{Boundary conditions for the incident light} \label{sec:appendix1} Calculation of concrete parameters of laser radiation is possible if the boundary conditions for the incident light are specified. Usually, the Gaussian distribution of the laser field in the aperture plane is assumed \begin{equation}\label{30twnt4} \Phi({\bf r}_\bot)=(2/\pi r_0^2)^{1/2}e^{-{r^2_\bot}/{r^2_0}}, \end{equation} where $r_0$ is the aperture radius. The laser and outgoing field should match in the aperture plane. This means that \begin{equation}\label{31twnt5x} \sum_{{\bf q}_{\bot},q_z}\bigg( \frac{2\pi\hbar\omega_{\bf q}}V\bigg)^{1/2}b_{\bf q}e^{-i\omega_{\bf q}t+i{\bf q}_{\bot}\cdot{\bf r}_{\bot}}=\alpha_Lb\Phi({\bf r}_{\bot})e^{-i\omega_0t}, \end{equation} where $b$ is the amplitude of the laser mode, and the coefficient $\alpha_L$ describes penetration of this field through the aperture. As before, the paraxial approximation ($\omega_{\bf q}\approx cq_z$) can be used. Also, the requirement of synchronism of both fields, restricts the left-hand side sum with terms $q_z=\omega_0/c=q_0$. Then the explicit value for $b_{\bf q}$ follows from Eq. (\ref{31twnt5x}) \begin{equation}\label{32twnt5xx} b_{\bf q}=b\alpha_L\frac {r_0}{\sqrt{\hbar \omega_0}}\sqrt{\frac{L_z}S}e^{-q^2_{\bot}r_0^2/4}\delta_{q_z,q_0}, \end{equation} which determines the boundary value of PDF: \begin{equation}\label{33twnt5} f({\bf r_\bot},z{=}0,{\bf q},t)=\delta_ {q_z,q_0}b^\dag(t) b(t)\frac {2{\alpha_L}^2}{\pi S\hbar\omega_0}e^{-q_{\bot}^2r_0^2/2-{2r_{\bot}^2}/{r_0^2}}. \end{equation} The extension of Eq. (\ref{33twnt5}) for the case of a partially coherent beam is realized by substituting $\frac{q_\bot^2r_1 ^2}2$ for $\frac{q_\bot^2r_0^2}2$ \cite{Chu}. Here $r^2_1=r_0^2/(1+2r_0 ^2\lambda _c^{-2})$, and the quantity $\lambda _c$ describes the effect of the phase diffuser which is used for suppression of scintillations. The mentioned modification of the initial distribution expands the range of $q_\bot $ variation to the values of the order $\frac{\pi}r_1 $ and does not affect the spatial distribution in the ${\bf r}_\bot$-domain. The diffuser influence vanishes in the limit of $\lambda _c\rightarrow\infty$ because in this case ${ r}_1\rightarrow r_0$. In the case of ${\bf r}_\bot=0$, the denominator in Eq. (\ref{22twnt1}) is given by \begin{equation}\label{34twnt6x} \sum_{\bf q} f_0({\bf r},{\bf q},t)=\frac {\alpha_L^2r_1^2q_0\langle b^\dag b\rangle}{\pi^2\hbar c(4+\rho_0^2\rho_1^2)}, \end{equation} where the derivation of (\ref{34twnt6x}) was somewhat simplified by inserting $L_0^{-1}=0$ in Eq. (\ref{13twelwwe}), $\rho _{0,1}^2={r_{0,1}^2q_0}/z $. \end{document}

\begin{document} \title{Graphs with bounded tree-width and large odd-girth are almost bipartite} \author{Alexandr V. Kostochka\thanks{ Department of Mathematics, University of Illinois, Urbana, IL 61801 and Institute of Mathematics, Novosibirsk 630090, Russia. E-mail: \texttt{[email protected]}. This author's work was partially supported by NSF grant DMS-0650784 and by grant 09-01-00244-a of the Russian Foundation for Basic Research.} \and Daniel Kr{\'a}l'\thanks{ Institute for Theoretical Computer Science, Faculty of Mathematics and Physics, Charles University, Malostransk{\'e} n{\'a}m{\v e}st{\'\i} 25, 118 00 Prague, Czech Republic. E-mail: \texttt{[email protected]}. The Institute for Theoretical Computer Science (ITI) is supported by Ministry of Education of the Czech Republic as project 1M0545. This research has also been supported by the grant GACR 201/09/0197.} \and Jean-S{\'e}bastien Sereni\thanks{ CNRS (LIAFA, Universit\'e Denis Diderot), Paris, France, and Department of Applied Mathematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. E-mail: \texttt{[email protected]}.} \and Michael Stiebitz\thanks{ Technische Universit\"at Ilmenau, Institute of Mathematics, P.O.B. 100 565, D-98684 Ilmenau, Germany. E-mail: \texttt{[email protected]}.}} \date{} \maketitle \begin{abstract} We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$. \end{abstract} \section{Introduction} It has been a challenging problem to prove the existence of graphs of arbitrary high girth and chromatic number~\cite{Erd59}. On the other hand, graphs with large girth that avoid a fixed minor are known to have low chromatic number (in particular, this applies to graphs embedded on a fixed surface). More precisely, as Thomassen observed~\cite{Tho88}, a graph that avoids a fixed minor and has large girth is $2$-degenerate, and hence $3$-colorable. Further, Galluccio, Goddyn and Hell~\cite{bib-galluccio} proved the following theorem, which essentially states that graphs with large girth that avoid a fixed minor are almost bipartite. \begin{theorem}[Galluccio, Goddyn and Hell, 2001] \label{thm-ggh} For every graph $H$ and every $\varepsilon>0$, there exists an integer $g$ such that the circular chromatic number of every $H$-minor free graph of girth at least $g$ is at most $2+\varepsilon$. \end{theorem} A natural way to weaken the girth-condition is to require the graphs to have high odd-girth (the \emph{odd-girth} is the length of a shortest odd cycle). However, Young~\cite{You96} constructed $4$-chromatic projective graphs with arbitrary high odd-girth. Thus, the high odd-girth requirement is not sufficient to ensure $3$-colorability, even for graphs embedded on a fixed surface. Klostermeyer and Zhang~\cite{KlZh00}, though, proved that the circular chromatic number of every planar graph of sufficiently high odd-girth is arbitrarily close to $2$. In particular, the same is true for $K_4$-minor free graphs, i.e. graphs with tree-width at most $2$. We prove that the conclusion is still true for any class of graphs of bounded tree-width, which answers a question of Pan and Zhu~\cite[Question 6.5]{bib-pan} also appearing as Question 8.12 in the survey by Zhu~\cite{bib-zhu01}. \begin{theorem}\label{thm-main} For every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$. \end{theorem} Motivated by tree-width duality, Ne{\v s}et{\v r}il and Zhu~\cite{bib-nesetril} proved the following theorem. \begin{theorem}[Ne\v set\v ril and Zhu, 1996]\label{thm-twd} For every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph $G$ with tree-width at most $k$ and homomorphic to a graph $H$ with girth at least $g$ has circular chromatic number at most $2+\varepsilon$. \end{theorem} To see that Theorem~\ref{thm-main} implies Theorem~\ref{thm-twd}, observe that if $G$ has an odd cycle of length $g$, then $H$ has an odd cycle of length at most $g$. \section{Notation} A \emph{$(p,q)$-coloring} of a graph is a coloring of the vertices with colors from the set $\{0,\ldots,p-1\}$ such that the colors of any two adjacent vertices $u$ and $v$ satisfy $q\le |c(u)-c(v)|\le p-q$. The \emph{circular chromatic number $\chi_c(G)$} of a graph $G$ is the infimum (and it can be shown to be the minimum) of the ratios $p/q$ such that $G$ has a $(p,q)$-coloring. For every finite graph $G$, it holds that $\chi(G)=\lceil\chi_c(G)\rceil$ and there is $(p,q)$-coloring of $G$ for every $p$ and $q$ with $p/q\ge\chi_c(G)$. In particular, the circular chromatic number of $G$ is at most $2+1/k$ if and only if $G$ is homomorphic to a cycle of length $2k+1$. The reader is referred to the surveys by Zhu~\cite{bib-zhu01,bib-zhu06} for more information about circular colorings. A \emph{$p$-precoloring} is a coloring $\varphi$ of a subset $A$ of vertices of a graph $G$ with colors from $\{0,\ldots,p-1\}$, and its \emph{extension} is a coloring of the whole graph $G$ that coincides with $\varphi$ on $A$. The following lemma can be seen as a corollary of a theorem of Albertson and West~\cite[Theorem 1]{AlWe06}, and it is the only tool we use from this area. \begin{lemma} \label{lm-extend} For every $p$ and $q$ with $2<p/q$, there exists $d$ such that any $p$-precoloring of vertices with mutual distances at least $d$ of a bipartite graph $H$ extends to a $(p,q)$-coloring of $H$. \end{lemma} A \emph{$k$-tree} is a graph obtained from a complete graph of order $k+1$ by adding vertices of degree $k$ whose neighborhood is a clique. The \emph{tree-width} of a graph $G$ is the smallest $k$ such that $G$ is a subgraph of a $k$-tree. Graphs with tree-width at most $k$ are also called \emph{partial $k$-trees}. A \emph{rooted partial $k$-tree} is a partial $k$-tree $G$ with $k+1$ distinguished vertices $v_1,\ldots,v_{k+1}$ such that there exists a $k$-tree $G'$ that is a supergraph of $G$ and the vertices $v_1,\ldots,v_{k+1}$ form a clique in $G'$. We also say that the partial $k$-tree is \emph{rooted} at $v_1,\ldots,v_{k+1}$. If $G$ is a partial $k$-tree rooted at $v_1,\ldots,v_{k+1}$ and $G'$ is a partial $k$-tree rooted at $v'_1,\ldots,v'_{k+1}$, then the graph $G\oplus G'$ obtained by identifying $v_i$ and $v'_i$ is again a rooted partial $k$-tree (identify the cliques in the corresponding $k$-trees). Fix $p$ and $q$. If $G$ is a rooted partial $k$-tree, then $\F(G)$ is the set of all $p$-precolorings of the $k+1$ distinguished vertices of $G$ that can be extended to a $(p,q)$-coloring of $G$. The next lemma is a standard application of results in the area of graphs of bounded tree-width~\cite{RoSe86}. \begin{lemma} \label{lm-small} Let $k$ and $N$ be positive integers such that $N\ge k+1$. If $G$ is a partial $k$-tree with at least $3N$ vertices, then there exist partial rooted $k$-trees $G_1$ and $G_2$ such that $G$ is isomorphic to $G_1\oplus G_2$ and $G_1$ has at least $N+1$ and at most $2N$ vertices. \end{lemma} If $G$ is a partial $k$-tree rooted at $v_1,\ldots,v_{k+1}$, then its \emph{type} is a $(k+1)\times (k+1)$ matrix $M$ such that $M_{ij}$ is the length of the shortest path between the vertices $v_i$ and $v_j$. If there is no such path, $M_{ij}$ is equal to $\infty$. Any matrix $M$ that is a type of a partial rooted $k$-tree satisfies the triangle inequality (setting $\infty+x=\infty$ for any $x$). A symmetric matrix $M$ whose entries are non-negative integers and $\infty$ (and zeroes only on the main diagonal) that satisfies the triangle inequality is a \emph{type}. A type is \emph{bipartite} if $M_{ij}+M_{jk}+M_{ik}\equiv0\;\mod\; 2$ for any three finite entries $M_{ij}$, $M_{jk}$ and $M_{ik}$. Two bipartite types $M$ and $M'$ are \emph{compatible} if $M_{ij}$ and $M'_{ij}$ have the same parity whenever both of them are finite. We define a binary relation on bipartite types as follows: $M\lm M'$ if and only if $M$ and $M'$ are compatible and $M_{ij}\leq M'_{ij}$ for every $i$ and $j$. Note that the relation $\lm$ is a partial order. We finish this section with the following lemma. Its straightforward proof is included to help us in familiarizing with the just introduced notation. \begin{lemma} \label{lm-type-glue} Let $G^1$ and $G^2$ be two bipartite rooted partial $k$-trees with types $M^1$ and $M^2$ such that there exists a bipartite type $M^0$ with $M^0\lm M^1$ and $M^0\lm M^2$. Then the types $M^1$ and $M^2$ are compatible, $G^1\oplus G^2$ is a bipartite rooted partial $k$-tree and its type $M$ satisfies $M^0\lm M$. \end{lemma} \begin{proof} The types $M^1$ and $M^2$ are compatible: if both $M^1_{ij}$ and $M_{ij}^{2}$ are finite, then $M_{ij}^{0}$ is finite and has the same parity as $M^1_{ij}$ and $M_{ij}^{2}$. Hence, the entries $M^1_{ij}$ and $M^2_{ij}$ have the same parity. Let $M$ be the type of $G^1\oplus G^2$. Note that it does not hold in general that $M_{ij}=\min\{M^1_{ij},M^2_{ij}\}$. We show that $M^0\lm M$ which will also imply that $G^1\oplus G^2$ is bipartite since $M^0$ is a bipartite type. Consider a shortest path $P$ between two distinguished vertices $v_i$ and $v_{i'}$ and split $P$ into paths $P_1,\ldots,P_\ell$ delimited by distinguished vertices on $P$. Note that $\ell\le k$ since $P$ is a path. Let $j_0=i$ and let $j_i$ be the index of the end-vertex of $P_i$ for $i\in\{1,\ldots,\ell\}$. In particular, $j_\ell=i'$. Each of the paths $P_1,\ldots,P_\ell$ is fully contained in $G^1$ or in $G^2$ (possibly in both if it is a single edge). Since $M^0\lm M^1$ and $M^0\lm M^2$, the length of $P_i$ is at least $M^0_{j_{i-1}j_i}$, and it has the same parity as $M^0_{j_{i-1}j_i}$. Since $M^0$ is a bipartite type (among others, it satisfies the triangle inequality), the length of $P$, which is $M_{ii'}$, has the same parity as $M^0_{j_0j_\ell}=M^0_{ii'}$ and is at least $M^0_{ii'}$. This implies that $M^0\lm M$. \end{proof} \section{The Main Lemma} In this section, we prove a lemma which forms the core of our argument. To this end, we first prove another lemma that asserts that for every $k$, $p$ and $q$, the set of types of all bipartite rooted partial $k$-trees forbidding a fixed set of $p$-precolorings from extending (and maybe some other precolorings, too) has always a maximal element. We formulate the lemma slightly differently to facilitate its application. \begin{lemma} \label{lm-mainM} For every $k$, $p$ and $q$, there exists a finite number of (bipartite) types $M^1,\ldots,M^m$ such that for any bipartite rooted partial $k$-tree $G$ with type $M$, there exists a bipartite rooted partial $k$-tree $G'$ with type $M^i$ for some $i\in\{1,\ldots,m\}$ such that $\F(G')\subseteq\F(G)$ and $M\lm M^i$. \end{lemma} \begin{proof} Let $d\ge 2$ be the constant from Lemma~\ref{lm-extend} applied for $p$ and $q$. Let $M^1,\ldots,M^m$ be all bipartite types with entries from the set $\{1,\ldots,D^{(k+1)^2}\}\cup\{\infty\}$ where $D=4d$. Thus, $m$ is finite and does not exceed $(D^{(k+1)^2}+1)^{k(k+1)/2}$. Let $G$ be a bipartite rooted partial $k$-tree with type $M$. If $M$ is one of the types $M^1,\ldots,M^m$, then there is nothing to prove (just choose $i$ such that $M=M^i$). Otherwise, one of its entries is finite and exceeds $D^{(k+1)^2}$. For $i\in\{1,\ldots,(k+1)^2\}$, let $J^i$ be the set of all positive integers between $D^{i-1}$ and $D^i-1$ (inclusively). Let $i_0$ be the smallest integer such that no entry of $M$ is contained in $J^{i_0}$. Since $M$ has at most $k(k+1)/2$ different entries, such an index $i_0$ exists. Note that if $i_0=1$, then Lemma~\ref{lm-extend} implies that $\F(G)$ contains all possible $p$-precolorings, and the sought graph $G'$ is the bipartite rooted partial $k$-tree composed of $k+1$ isolated vertices, with the all-$\infty$ type. Two vertices $v_i$ and $v_j$ at which $G$ is rooted are \emph{close} if $M_{ij}$ is at most $D^{i_0-1}$. The relation $\approx$ of being close is an equivalence relation on $v_1,\ldots,v_{k+1}$. Indeed, it is reflexive and symmetric by the definition, and we show now that it is transitive. Suppose that $M_{ij}$ and $M_{jk}$ are both at most $D^{i_0-1}$. Then, the distance between $v_i$ and $v_k$ is at most $M_{ij}+M_{jk}\le2D^{i_0-1}-2\le D^{i_0}-1$ since $D\ge2$. Consequently, by the choice of $i_0$, the distance between $v_i$ and $v_k$ is at most $D^{i_0-1}-1$ and thus $v_i\approx v_k$. Let $C_1,\ldots,C_{\ell}$ be the equivalence classes of the relation $\approx$. Note that $C_1,\ldots,C_{\ell}$ is a finer partition than that given by the equivalence relation of being connected. Since $G$ is bipartite, we can partition its vertices into two color classes, say red and blue. For every $i\in\{1,\ldots,\ell\}$, contract the closed neighborhood of a vertex $v$ if $v$ is a blue vertex and its distance from any vertex of $C_i$ is at least $D^{i_0-1}$ and keep doing so as long as such a vertex exists. Observe that the resulting graph is uniquely defined. After discarding the components that do not contain the vertices of $C_i$, we obtain a bipartite partial $k$-tree $G_i$ rooted at the vertices of $C_i$: it is bipartite as we have always contracted closed neighborhoods of vertices of the same color (blue) to a single (red) vertex, and its tree-width is at most $k$ since the tree-width is preserved by contractions. Moreover, the distance between any two vertices of $C_i$ has not decreased since any path between them through any of the newly arising vertices has length at least $2D^{i_0-1}-2\ge D^{i_0-1}$. Now, let $G'$ be the bipartite rooted partial $k$-tree obtained by taking the disjoint union of $G_1,\ldots,G_{\ell}$. The type $M'$ of $G'$ can be obtained from the type of $G$: set $M'_{ij}$ to be $M_{ij}$ if the vertices $v_i$ and $v_j$ are close, and $\infty$ otherwise. Thus, $M'$ is one of the types $M^1,\ldots,M^m$ and $M\lm M'$. It remains to show that $\F(G')\subseteq\F(G)$. Let $c\in\F(G')$ be a $p$-precoloring that extends to $G'$, and recall that $D\ge4$. For $i\in\{1,\ldots,\ell\}$, let $A_i$ be the set of all red vertices at distance at most $D^{i_0-1}$ and all blue vertices at distance at most $D^{i_0-1}-1$ from $C_i$, and let $R_i$ be the set of all red vertices at distance $D^{i_0-1}-1$ or $D^{i_0-1}$ from $C_i$. Set $B_i=A_i\setminus R_i$ ($B_i$ is the ``interior'' of $A_i$ and $R_i$ its ``boundary''). The extension of $c$ to $G_i$ naturally defines a coloring of all vertices of $A_i$: $G_i$ is the subgraph of $G$ induced by $A_i$ with some red vertices of $R_i$ identified (two vertices of $R_i$ are identified if and only if they are in the same component of the graph $G-B_i$). Let $H$ be the following auxiliary graph obtained from $G$: remove the vertices of $B=B_1\cup\cdots\cup B_{\ell}$ and, for $i\in\{1,\ldots,\ell\}$, identify every pair of vertices of $R_i$ that are in the same component of $G-B$. Let $R$ be the set of vertices of $H$ corresponding to some vertices of $R_1\cup\cdots\cup R_{\ell}$. Precolor the vertices of $R$ with the colors given by the colorings of $G_i$ (note that two vertices of $R_i$ in the same component of $G-B_i$ are also in the same component of $G-B$, so this is well-defined). The graph $H$ is bipartite as only red vertices have been identified. The distance between any two precolored vertices is at least $d$: consider two precolored vertices $r$ and $r'$ at distance at most $d-1$. Let $i$ and $i'$ be such that $r\in R_i$ and $r'\in R_{i'}$. If $i=i'$, then $r$ and $r'$ are in the same component of $G-B$ and thus $r=r'$. If $i\not=i'$ then by the definition of $R_i$ and $R_{i'}$, the vertex $r$ is in $G$ at distance at most $D^{i_0-1}$ from some vertex $v$ of $C_i$ and $r'$ is at distance at most $D^{i_0-1}$ from some vertex $v'$ of $C_{i'}$. So, the distance between $v$ and $v'$ is at most $2D^{i_0-1}+d<D^{i_0}-1$. Since $M$ has no entry from $J^{i_0}$, the vertices $v$ and $v'$ must be close and thus $i=i'$, a contradiction. Since the distance between any two precolored vertices is at least $d$, the precoloring extends to $H$ by Lemma~\ref{lm-extend} and in a natural way it defines a coloring of $G$. We conclude that every $p$-precoloring that extends to $G'$ also extends to $G$ and thus $\F(G')\subseteq\F(G)$. \end{proof} We now prove our main lemma, which basically states that there is only a finite number of bipartite rooted partial $k$-trees that can appear in a minimal non-$(p,q)$-colorable graph with tree-width $k$ and a given odd girth. \begin{lemma} \label{lm-mainG} For every $k$, $p$ and $q$, there exist a finite number $m$ and bipartite rooted partial $k$-trees $G^1,\ldots,G^m$ with types $M^1,\ldots,M^m$ such that for any bipartite rooted partial $k$-tree $G$ with type $M$ there exists $i$ such that $\F(G^i)\subseteq \F(G)$ and $M\lm M^i$. \end{lemma} \begin{proof} Let $M^1,\ldots,M^{m}$ be the types from Lemma~\ref{lm-mainM}. We define the graph $G^i$ as follows: for every $p$-precoloring $c$ that does not extend to a bipartite partial rooted $k$-tree with type $M^i$, fix any partial rooted $k$-tree $G^i_c$ with type $M^i$ such that $c$ does not extend to $G^i_c$. Set $G^i=\bigoplus_{c} G^i_c$, where $c$ runs over all such $p$-precolorings. If the above sum of partial $k$-trees is non-empty, then the type $M$ of $G^i$ is $M^i$. Indeed, $M\lm M^i$ by the definition of $G^i$, and Lemma~\ref{lm-type-glue} implies that $M^i\lm M$. If all the $p$-precolorings of the $k+1$ vertices in the root extend to each partial $k$-tree of type $M^i$, then let $G^i$ be the graph consisting of $k+1$ isolated vertices. This happens in particular for the all-$\infty$ type. Let us verify the statement of the lemma. Let $G$ be a bipartite rooted partial $k$-tree and let $M$ be the type of $G$. If $\F(G)$ is composed of all $p$-precolorings, the sought graph $G^i$ is the one composed of $k+1$ isolated vertices. Hence, we assume that $\F(G)$ does not contain all $p$-precolorings, i.e., there are $p$-precolorings that do not extend to $G$. By Lemma~\ref{lm-mainM}, there exists a bipartite rooted partial $k$-tree $G'$ with type $M'$ such that $M\lm M'=M^i$ for some $i$ and $\F(G')\subseteq\F(G)$. For every $p$-precoloring $c$ that does not extend to $G'$ (and there exists at least one such $p$-precoloring $c$), some graph $G^i_c$ has been glued into $G^i$. Hence, $\F(G^i)\subseteq\F(G')\subseteq\F(G)$. Since the type of $G^i$ is $M^i$, the conclusion of the lemma follows. \end{proof} \section{Proof of Theorem~\ref{thm-main}} We are now ready to prove Theorem~\ref{thm-main}, which is recalled below. \begin{thm2} For every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$. \end{thm2} \begin{proof} Fix $p$ and $q$ such that $2<p/q\le 2+\varepsilon$. Let $G^1,\ldots,G^m$ be the bipartite partial $k$-trees from Lemma~\ref{lm-mainG} applied for $k$, $p$ and $q$. Set $N$ to be the largest order of the graphs $G^i$ and set $g$ to be $3N$. We assert that each partial $k$-tree with odd-girth $g$ has circular chromatic number at most $p/q$. Assume that this is not the case and let $G$ be a counterexample with the fewest vertices. The graph $G$ has at least $3N$ vertices (otherwise, it has no odd cycles and thus it is bipartite). By Lemma~\ref{lm-small}, $G$ is isomorphic to $G_1\oplus G_2$, where $G_1$ and $G_2$ are rooted partial $k$-trees and the number of vertices of $G_1$ is between $N+1$ and $2N$. By the choice of $g$, the graph $G_1$ has no odd cycle and thus it is a bipartite rooted partial $k$-tree. By Lemma~\ref{lm-mainG}, there exists $i$ such that $\F(G^i)\subseteq\F(G_1)$ and $M_1\lm M^i$ where $M_1$ is the type of $G_1$ and $M^i$ is the type of $G^i$. Let $G'$ be the partial $k$-tree $G^i\oplus G_2$. First, $G'$ has fewer vertices than $G$ since the number of vertices of $G^i$ is at most $N$ and the number of vertices of $G_1$ is at least $N+1$. Second, $G'$ has no $(p,q)$-coloring: if it had a $(p,q)$-coloring, then the corresponding $p$-precoloring of the $k+1$ vertices shared by $G^i$ and $G_2$ would extend to $G_1$ since $\F(G^i)\subseteq\F(G_1)$ and thus $G$ would have a $(p,q)$-coloring, too. Finally, $G'$ has no odd cycle of length at most $g$: if it had such a cycle, replace any path between vertices $v_j$ and $v_{j'}$ of the root of $G^i$ with a path of at most the same length between them in $G_1$ (recall that $M_1\lm M^i$). If such paths for different pairs of $v_j$ and $v_{j'}$ on the considered odd cycle intersect, take their symmetric difference. In this way, we obtain an Eulerian subgraph of $G=G_1\oplus G_2$ with an odd number of edges such that the number of its edges does not exceed $g$. Consequently, this Eulerian subgraph has an odd cycle of length at most $g$, which violates the assumption on the odd-girth of $G$. We conclude that $G'$ is a counterexample with less vertices than $G$, a contradiction. \end{proof} We end by pointing out that the approach used yields an upper bound of $3(k+1)\cdot2^{2^{p^{k+1}}((4d)^{(k+1)^2}+1)^{k^2}}$ for the smallest $g$ such that all graphs with tree-width at most $k$ and odd-girth at least $g$ have circular chromatic number at most $p/q$, whenever $p/q>2$. More precisely, the value of $N$ cannot exceed $(k+1)\cdot2^{2^{p^{k+1}}((4d)^{(k+1)^2}+1)^{k^2}}$. To see this, we consider all pairs $P=(C,M)$ where $C$ is a set of $p$-precolorings of the root and $M$ is a type such that there is a bipartite rooted partial $k$-tree of type $M$ to which no coloring of $C$ extends. Let $n_P$ be the size of a smallest such partial $k$-tree. We obtain a sequence of at most $2^{p^{k+1}}\times\left((4d)^{(k+1)^{2}}+1\right)^{k^2}$ integers. The announced bound follows from the following fact: if the sequence is sorted in increasing order, then each term is at most twice the previous one. Indeed, consider the tree-decomposition of the partial $k$-tree $G_P$ chosen for the pair $P$. If the bag containing the root has a single child, then we delete a vertex of the root, and set a vertex in the single child to be part of the root. We obtain a partial $k$-tree to which some $p$-precolorings of $C$ do not extend. Thus, $n_P\le1+n_{P'}$ for some pair $P'$ and $n_{P'}<n_P$. If the bag containing the root has more than one child, then $G_P$ can be obtained by identifying the roots of two smaller partial $k$-trees $G$ and $G'$. By the minimality of $G_P$, the orders of $G$ and $G'$ are $n_{P_1}$ and $n_{P_2}$ for two pairs $P_1$ and $P_2$ such that $n_{P_i}<n_P$ for $i\in\{1,2\}$. This yields the stated fact, which in turn implies the given bound, since the smallest element of the sequence is $k+1$. \noindent \textbf{Acknowledgment.} This work was done while the first three authors were visiting the fourth at Technische Universit\"at Ilmenau. They thank their host for providing a perfect working environment. \end{document}

\begin{document} \title{Potentiality States: Quantum versus Classical Emergence} \author{\normalsize Diederik Aerts and Bart D'Hooghe \\ \small\itshape Center Leo Apostel for Interdisciplinary Studies \\ \small\itshape Departments of Mathematics and Department of Psychology \\ \small\itshape Brussels Free University, Brussels, Belgium \\ \small Emails: \url{[email protected], [email protected]} \\ } \date{} \maketitle \begin{abstract} \noindent We identify emergence with the existence of states of potentiality related to relevant physical quantities. We introduce the concept of `potentiality state' operationally and show how it reduces to `superposition state' when standard quantum mechanics can be applied. We consider several examples to illustrate our approach, and define the potentiality states giving rise to emergence in each example. We prove that Bell inequalities are violated by the potentiality states in the examples, which, taking into account Pitowsky's theorem, experimentally indicates the presence of quantum structure in emergence. In the first example emergence arises because of the many ways water can be subdivided into different vessels. In the second example, we put forward a full quantum description of the Liar paradox situation, and identify the potentiality states, which in this case turn out to be superposition states. In the example of the soccer team, we show the difference between classical emergence as stable dynamical pattern and emergence defined by a potentiality state, and show how Bell inequalities can be violated in the case of highly contextual experiments. \end{abstract} \section{Introduction} Many everyday life examples of emergence can be given. Let us consider for instance a set of soccer players. As long as it is just a set of soccer players no emergence takes place. Suppose however that the set of players starts to practice with the aim of forming a soccer team. The co-adaptation that takes place between the different soccer players during their trainings and matches results in the emergence of a soccer team. The soccer team is a new structure that has been formed out of the set of individual soccer players. In physics emergent phenomena have been studied within the complexity and chaos approach. An example is given by the B\'{e}nard convection effect. This effect occurs when a viscous fluid is heated between two planes. In the pre-boiling stage and under suitable conditions, bubbles begin to rise and vortices --- called B\'{e}nard cells --- arise like little cylinders within which the fluid continuously streams up on the outside of the cylinder and back down through the middle of the cylinder. This motion occurs as the fluid heats up and before it starts to boil. The B\'{e}nard cells fit together in a hexagonal lattice of vortices, even without partitions to keep the boundaries between the vortices stable. The microscopical movements of the individual molecules of the fluid result in a macroscopical dynamical pattern of the fluid as a whole with the emergent B\'{e}nard cell structure. The existing complexity and chaos models are classical physics models. In this paper we want to show that a classical physics approach has to be generalized in order to describe emergence in a complete way. The reason is that the emergent structure usually contains states that have a relation of `potential' with respect to relevant observable quantities. We will call such states `potentiality states'. These emergent potentiality states cannot be described in a classical physics approach but need a quantum-like formalism. We know that the appearance of potentiality states is a basic aspect of quantum mechanics. Therefore, in this paper we will demonstrate not only the importance of potentiality states for emergence, but also the advantage of a quantum-like description of emergence versus a classical one. In the examples we will indicate in which way the quantum-like formalism appears in the description. The relevance of quantum aspects in emergence confirms earlier results in which quantum mechanical aspects in the macroscopic world are identified in a more general way \cite{aerts82,AeBroGab2000}. In physics, the state $q(t)$ of a physical entity $S$ at time $t$ represents the reality of this physical entity at that time $t$.\footnote{In physics `statistical states' are sometimes also just called states. We however will always refer with the concept state to what is called a `pure state'.} In the case of classical physics such a state is represented by a point in phase space, while for quantum physics it is represented by a unit vector in Hilbert space. In classical physics the state $q(t)$ of a physical entity $S$ determines the values for all observable quantities at this time $ t$. Hence it is common in classical physics to characterize the state $q(t)$ by the set of all values of the relevant observable quantities. For example, if the physical entity is a particle $S(classical \; particle)$, then the relevant observable quantities are its position $r(t)$ and momentum $p(t)$ at time $t$. And indeed, for each state $q(t)$ of $S(classical \; particle)$, it is the case that $r(t)$ and $p(t)$ have definite values, which makes it possible to represent $q(t)$ with the couple $(r(t),p(t))$ in phase space. In quantum physics the situation is different. The state $q(t)$ of a quantum particle $S(quantum \; particle)$ is represented by a unit vector $\psi (r,t)$ (the normalized wave function) in a Hilbert space $L_2(\mathbb{R}^3)$. Again, the relevant observable quantities are the position and the momentum. However, neither has definite values for the entity $S(quantum \; particle)$ being in a state $\psi (r,t)$. To be more specific, definite values of position exist only if the wave function is a delta function, and definite values of momentum if the wave function is a plane wave.\footnote{If one chooses as the basis of the Hilbert space an orthonormal set of eigenstates corresponding with the position operator, i.e. the state is expressed as a (wave) function $\Psi(x_1,\ldots,x_n;t)$ of variables given by the coordinates $x_i$.} Apart from the fact that in both cases the wave function is not an element of the Hilbert space $L_2(\mathbb{R}^3)$, and hence should be considered as limiting cases, they never can occur together. This means that a quantum particle never has simultaneously a definite position and a definite momentum, which is referred to as `quantum indeterminism'. It can also be expressed as follows: for a quantum particle in state $\psi (r,t)$ the values of position and momentum are potential. This means that the quantum particle has the potential of realizing them, but they are not actually realized in the state $\psi (r,t)$. Previously it has been shown that whether a specific physical entity needs a quantum-like description --- and hence should be called a quantum entity --- or a classical description, depends on the nature of the entity and the relevant observable quantities, and not on the fact that it belongs to the microworld or to the macroworld \cite{aerts82,AeBroGab2000}. Because of their importance for emergence we will concentrate on the presence of potentiality states and how without difficulties many situations can be found in the macroscopic world containing this quantum aspect. However, they can not be described by classical physical theories which identify emergent properties of a specific entity with stable dynamical patterns of the entity. The concept of potentiality state makes it possible to describe another type of emergence which is present on an ontological level and not on the dynamic level as in the case of classical emergence. In standard quantum mechanics a potentiality state related to a specific observable quantity is a superposition state of the eigenstates of this observable quantity. Hence, our potentiality states are superposition states for a situation where standard quantum physics applies. To demonstrate the more general applicability of the concept of potentiality state, we will consider situations that have both quantum and classical aspects and hence are not purely quantum. However for such situations there does not necessarily exist a Hilbert space describing the set of states, as is the case in pure quantum situations, meaning that superposition is not well defined. That is the reason that we use the name potentiality states instead of superposition states for the states that we consider. In the second example of this paper (see section 3), the liar paradox, we will see that the description is fully quantum, such that in this case the potentiality states reduce to superposition states. In the last section we give the example of a team of soccer players to illustrate the difference between emergence due to potentiality states, which is on an ontological level, and classical emergence, defined by dynamical patterns. \section{Bell Inequalities and Non-classical Emergence} In this section we consider several examples and analyze in which way the concept of potentiality state appears. We also investigate in which way the presence of potentiality states is linked to the violation of Bell inequalities. \subsection{Potentiality States in Connected Vessels of Water} Suppose that we consider two vessels of which one contains 6 liters of water and the other one 14 liters of water. We have at our disposal a third vessel that is empty, but can contain more than 20 liters of water. The entity $ S(water)$ is the water, and we consider the physical quantity which is the volume of the water. Clearly this physical quantity has a definite value for the water that is contained in the two considered vessels, namely 6 liters and 14 liters. This means that the states of water that we consider in both vessels are not potentiality states related to the observable quantity which is the volume. Suppose now that we take the two vessels and empty them in the third vessel. This third vessel will then contain 20 liters of water, which means that also for this new entity of water the physical quantity volume has a definite value. But by putting the water of the two vessels together, the new entity of water has lost the old properties of 6 liters and 14 liters as actual properties. We could however divide the water again and collect 6 liters in one vessel and 14 liters in the other vessel. This means that potentially the division in 6 liters and 14 liters is still present in the entity of 20 liters of water. But this new state of 20 liters of water contains many more possible subdivisions as potentialities. Also 8 liters and 12 liters, or 11 liters and 9 liters, or 2 liters and 18 liters,$\ldots$ are all potentialities of subdivisions. In general we can say that $x$ liters and $y$ liters of water, such that $x+y=20$ form an infinite continuous set of potential subdivisions. Therefore, from the third vessel, being in the state such that it contains 20 liters, the two subentities can be derived by dividing the 20 liters in the appropriate amounts. The state of 20 liters of water is a potentiality state related to measurements that divide the amount of water in two amounts. Taking into account the measurement that subdivides an amount of water in two, the 20 liters of water, that originated by putting together the original 6 liters and 14 liters has new emergent properties. These properties are described by the potentiality state, that allows the water to be subdivided in all these ways. If we connect the two original vessels by a tube, we get such a third vessel (see Figure~\ref{mqg01}). We can show that the new properties that this entity has are emergent properties related to measurements that divide the water up again. A criterion that we can use to show the quantum nature of these emergent properties is the violation of Bell inequalities. \begin{figure} \caption{The vessels of water example violating Bell inequalities. The entity $S$ consists of two vessels containing 20 liters of water that are connected by a tube.} \label{mqg01} \end{figure} We will recall in the next section Bell inequalities and then elaborate our connected vessels of water example with the necessary detail such that we can show that Bell inequalities are violated. The vessels of water example was introduced in \cite{aerts82} and elaborated in \cite{aerts85a,aerts85b,AeBroGab2000,aerts92}. \subsection{Bell Inequalities and the Presence of Quantum Structure} Violation of Bell inequalities related to the presence of potentiality states can happen as well in the macroworld as in the microworld, depending on the type of states and observable quantities that are considered. Let us first recall some of the most relevant historical results related to Bell inequalities, and show why the violation of Bell inequalities is an experimental indication for the presence of quantum aspects. In the seventies, a sequence of experiments was carried out to test for the presence of nonlocality in the microworld described by quantum mechanics \cite{clauser76,kas70}, culminating in decisive experiments by Aspect and his team in Paris \cite{aspect81,aspect82}. They were inspired by three important theoretical results: the EPR Paradox \cite{epr35}, Bohm's thought experiment \cite{bohm51}, and Bell's theorem \cite{bell64}. Einstein, Podolsky and Rosen believed to have shown that quantum mechanics is incomplete, in that there exist elements of reality that cannot be described by it \cite{epr35,aerts84,aerts2000}. Bohm took their insight further with a simple example: the `coupled spin-${\frac 12}$ entity' consisting of two particles with spin-${\frac 12}$, of which the spins are coupled such that the quantum spin vector is a non-product vector representing a singlet spin state \cite{bohm51}. Bohm's example inspired Bell to formulate a condition that would test experimentally for incompleteness, the Bell inequalities \cite{bell64}. Bell's theorem states that statistical results of experiments performed on a certain physical entity satisfy his inequalities if and only if the reality in which this physical entity is embedded is local. Experiments performed to test for the presence of nonlocality confirmed the results as predicted by quantum mechanics, such that it is now commonly accepted that the micro-physical world is incompatible with local realism. Bell inequalities are defined with the following experimental situation in mind. We consider a physical entity $S$, and four experiments $e_1$, $e_2$, $ e_3$, and $e_4$ that can be performed on the physical entity $S$. Each of the experiments $e_i,i\in \{1,2,3,4\}$ has two possible outcomes, respectively denoted $o_i(up)$ and $o_i(down)$. Some of the experiments can be performed together, which in principle leads to `coincidence' experiments $e_{ij},i,j\in \{1,2,3,4\}$. For example $e_i$ and $e_j$ together will be denoted $e_{ij}$. Such a coincidence experiment $e_{ij}$ has four possible outcomes, namely $(o_i(up),o_j(up))$, $(o_i(up),o_j(down))$, $ (o_i(down),o_j(up))$ and $(o_i(down),o_j(down))$. Following Bell \cite{bell64}, we define the expectation values $\hbox{$\mathbb{E}$}_{ij},i,j\in \{1,2,3,4\}$ for these coincidence experiments, as \begin{equation} \begin{array}{ll} \hbox{$\mathbb{E}$}_{ij}= & \left( +1\right) P(o_i(up),o_j(up))+\left( +1\right) P(o_i(down),o_j(down))+ \\ & \left( -1\right) P(o_i(up),o_j(down))+\left( -1\right) P(o_i(down),o_j(up)) \end{array} \end{equation} From the assumption that the outcomes are either $+1$ or $-1$, and that the correlation $\hbox{$\mathbb{E}$}_{ij}$ can be written as an integral over some hidden variable of a product of the two local outcome assignments, one derives Bell inequalities: \begin{equation} |\hbox{$\mathbb{E}$}_{13}-\hbox{$\mathbb{E}$}_{14}|+|\hbox{$\mathbb{E}$}_{23}+\hbox{$\mathbb{E}$} _{24}|\leq 2 \label{bellineq} \end{equation} To come to the point where we can use the violation of Bell inequalities as an experimental indication for the presence of quantum structure, we have to mention the work of Itamar Pitowsky. Pitowsky proved that if Bell inequalities are satisfied for a set of probabilities connected to the outcomes of the considered experiments, there exists a classical Kolmogorovian probability model. In such model the probability can be explained as due to a lack of knowledge about the precise state of the system. If however Bell inequalities are violated, Pitowsky proved that no such classical Kolmogorovian probability model exists \cite{pit89}. Hence violation of Bell inequalities shows that the probabilities that are involved are nonclassical. The only type of nonclassical probabilities that are well known in nature are the quantum probabilities. The probability structure that is present in our examples\footnote{ Except for the liar paradox example, where we derive a pure quantum description and hence the probability model is quantum.} is nonclassical and nonquantum, the classical and quantum probabilities being two special cases of this more general situation. \subsection{Violation of Bell Inequalities for the Connected Vessels of Water Entity} Let us consider again the entity $S(connected \; vessels)$ of two vessels connected by a tube and containing 20 liters of transparent water (see Figure~\ref{mqg01}). The entity is in an emergent potentiality state $s$ such that the vessel is placed in the gravitational field of the earth, with its bottom horizontal. To be able to check for the violation of Bell inequalities caused by this potentiality state we have to introduce four measurements, of which some can be performed together. Let us introduce the experiment $e_1$ that consists of putting a siphon $K_1$ in the vessel of water at the left, taking out water using the siphon, and collecting this water in a reference vessel $R_1$ placed to the left of the vessel. If we collect more than 10 liters of water, we call the outcome $ o_1(up)$, and if we collect less or equal to 10 liters, we call the outcome $ o_1(down)$. We introduce another experiment $e_2$ that consists of taking with a little spoon, from the left, a bit of the water, and determining whether it is transparent. We call the outcome $o_2(up)$ when the water is transparent and the outcome $o_2(down)$ when it is not. We introduce the experiment $e_3$ that consists of putting a siphon $K_3$ in the vessel of water at the right, taking out water using the siphon, and collecting this water in a reference vessel $R_3$ to the right of the vessel. If we collect more or equal to 10 liters of water, we call the outcome $o_3(up)$, and if we collect less than 10 liters, we call the outcome $o_3(down)$. We also introduce the experiment $e_4$ which is analogous to experiment $e_2$, except that we perform it to the right of the vessel (see Figure~\ref{mqg01}). The experiment $e_1$ can be performed together with experiments $e_3$ and $ e_4$, and we denote the coincidence experiments $e_{13}$ and $e_{14}$. Also, experiment $e_2$ can be performed together with experiments $e_3$ and $e_4$, and we denote the coincidence experiments $e_{23}$ and $e_{24}$. For the vessel in state $s$, the coincidence experiment $e_{13}$ always gives one of the outcomes $(o_1(up),o_3(down))$ or $(o_1(down),o_3(up))$, since more than 10 liters of water can never come out of the vessel at both sides. This shows that $\hbox{$\mathbb{E}$}_{13}=-1$. The coincidence experiment $e_{14}$ always gives the outcome $(o_1(up),o_4(up))$ which shows that $\hbox{$\mathbb{E}$} _{14}=+1$, and the coincidence experiment $e_{23}$ always gives the outcome $ (o_2(up),o_3(up))$ which shows that $\hbox{$\mathbb{E}$}_{23}=+1$. Clearly experiment $e_{24}$ always gives the outcome $(o_2(up),o_4(up))$ which shows that $\hbox{$\mathbb{E}$}_{24}=+1$. Let us now calculate the terms of Bell inequalities, \begin{equation} \begin{array}{ll} |\hbox{$\mathbb{E}$}_{13}-\hbox{$\mathbb{E}$}_{14}|+|\hbox{$\mathbb{E}$}_{23}+\hbox{$\mathbb{E}$} _{24}| & =|-1-1|+|+1+1| \\ & =+2+2 \\ & =+4 \end{array} \end{equation} This shows that Bell inequalities are violated. The state $s$, which is a potentiality state related to the measurements that divide up the 20 liters of water in the two reference vessels, is at the origin of this violation. \section{States of Potentiality and the Liar Paradox} \subsection{The Cognitive Entity of the Liar Paradox} Another example of an entity for which a description with potentiality states is useful, is given by the double liar paradox. In fact, it will turn out that the double liar paradox entity $S(double \; liar)$ can be given a pure quantum mechanical description. As we shall show below, the potentiality state of this entity will be given by a superposition state of the states of the sub-entities which compose the double liar entity. Before discussing the double liar paradox, let us first consider some non-paradoxical situations: \begin{itemize} \item[(S 0.1)] The sum of two and two is four. \item[(S 0.2)] This page contains 1764 letters. \item[(S 0.3)] The square root of 1764 is 42. \item[(S 0.4)] 6 times 9 yields 42. \end{itemize} \noindent Obviously, the truth-value of each sentence can be easily determined. The fact that $2+2=4$ can also be expressed by the set of following two sentences: \begin{itemize} \item[(S 1.1)] Sentence (S 1.2) is true. \item[(S 1.2)] The sum of two and two is four. \end{itemize} \noindent The truth behavior of the entity, consisting of the set of sentences (S 1.1) and (S 1.2), can also be determined rather easily. The truth values of the sentences are coupled, but we do not encounter any paradoxical situations. This happens if both sentences refer to the truth value of each other. In such case, paradoxical situations arise. Let us now consider the double liar paradox which can be presented in the following way: \begin{center} \textsl{`Double Liar'} (1) \ \ \ sentence (2) is false (2) \ \ \ sentence (1) is true \end{center} \noindent Let us describe a typical cognitive interaction that one goes through with these liar paradox sentences. Suppose we hypothesize that sentence (1) is true. We go to sentence (1) and read what is written there. It is written `sentence (2) is false'. From our hypothesis we can infer that sentence (2) is false. Let us go, using this knowledge about the status of sentence (2), to sentence (2). There is written `sentence (1) is true'. From the knowledge that sentence (2) is false, we can infer that sentence (1) is false. This means that from the hypothesis that sentence (1) is true we derive that sentence (1) is false, which is a contradiction. Similarly, starting from the hypothesis that sentence (1) is false one also obtains a contradiction. In the examples presented in the previous section the presence of potentiality states gives rise to the violation of Bell inequalities, which implies that there is no classical, i.e., Kolmogorovian, representation possible \cite{pit89}. Therefore in general, potentiality states are nonclassical states. In \cite{AeBroSme99,AeBroSme2000} the entity which is the liar sentence is studied in a similar perspective. In the description of the liar entity we interpret the interaction that a person has with this entity, as described in some detail in the forgoing sections, as a measurement. It is with the introduction of the concept of entity and of measurement (or observable quantity) in this operational way, that we can find out what the nature is of this liar entity. It turns out that the liar entity can be described using the formalism of standard quantum mechanics in a complex Hilbert space. Moreover, the self-referential circularity --- more precisely, the truth-value dynamics --- of the liar paradox can be described by the Schr\"{o}dinger equation. \subsection{Quantum Representation for the Potentiality State of the Liar Paradox} A potentiality state is necessary to represent the state of the entity defined by the liar paradox since its state is not an eigenstate of `truth' (i.e., the sentence(s) are true) neither an eigenstate of `falsehood' (i.e., the sentences are not true). Instead, the state of the liar entity can be regarded as a potentiality state related to the measurements introduced by the persons that interact cognitively with the liar entity. Since we find a full quantum description in this case, it follows that the potentiality state is a superposition state of both eigenstates, just as the singlet state in the case of two correlated spin-$\frac 12$ entities. Let us discuss the Double Liar sentences in more detail by considering following three situations: \[ \mathrm{A}\ \ \left\{ \begin{array}{ll} \mathrm{(1)\ } & \mathrm{sentence\ (2)\ is\ false} \\ \mathrm{(2)\ } & \mathrm{sentence\ (1)\ is\ true} \end{array} \right. \] \[ \mathrm{B}\ \ \left\{ \begin{array}{ll} \mathrm{(1)\ } & \mathrm{sentence\ (2)\ is\ true} \\ \mathrm{(2)\ } & \mathrm{sentence\ (1)\ is\ true} \end{array} \right. \] \[ \mathrm{C}\ \ \left\{ \begin{array}{ll} \mathrm{(1)\ } & \mathrm{sentence\ (2)\ is\ false} \\ \mathrm{(2)\ } & \mathrm{sentence\ (1)\ is\ false} \end{array} \right. \] Since truth value of a sentence is binary: either it is true or it is false, we can associate the outcomes of a dichotomic observable (e.g., the outcomes for a spin-$\frac 12$ outcome are either `spin up' or `spin down') with truth and falsehood. Let us make the convention that the `spin up' state $ \left( \begin{array}{c} 1 \\ 0 \end{array} \right) $ corresponds with truth of a sentence and `spin down' state $\left( \begin{array}{c} 0 \\ 1 \end{array} \right) $ with the falsehood of the respective sentence. It turns out that then for the paradoxes of type B and C the sentences can be represented by coupled $\Bbb{C}^2$ vectors. Indeed, since for each measurement the truth values of the two sentences are coupled (B1 true implies B2 true and B1 false implies B2 false, C1 true implies C2 false and vice versa), the dimensionality of {$\Bbb{C}$}$^2\otimes $\thinspace {$\Bbb{C}$}$^2$ is sufficient to represent these entities. Formally, the corresponding quantum mechanical representation would be by a `singlet state' and `triplet state' respectively. In the singlet state the two spin-1/2 particles are anti-alined and in an anti-symmetrical state (entity C), while in a triplet state the two spin-1/2 particles are alined and in a symmetrical state (entity B). In the specific case of (C), and taking into account the anti-symmetric spin analog, the state of the entity $\Psi $ is given by: \[ \frac 1{\sqrt{2}}\left\{ \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 1 \end{array} \right) -\left( \begin{array}{c} 0 \\ 1 \end{array} \right) \otimes \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \right\} \] Equivalently, the liar entity can be represented by other linear combinations of $\left( \begin{array}{c} 1 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 1 \end{array} \right) $ and $\left( \begin{array}{c} 0 \\ 1 \end{array} \right) \otimes \left( \begin{array}{c} 1 \\ 0 \end{array} \right) ,$ provided the coefficients have equal amplitude and the squared amplitudes add to one (such that total probability of finding the entity in one of the two possible states after a measurement equals one). In a similar manner the liar paradox in the `symmetrical' case~(B) can be represented by the triplet state: \[ \frac 1{\sqrt{2}}\left\{ \left( \begin{array}{c} 1 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 1 \\ 0 \end{array} \right) +\left( \begin{array}{c} 0 \\ 1 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 1 \end{array} \right) \right\} \] \noindent The projection operators which make sentence 1, respectively sentence 2, true are: \[ P_{1,true}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \otimes \mathbb{1}_2\ \ \ \ \ \ P_{2,true}=\mathbb{1}_1\otimes \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) \] \noindent The projection operators that make the sentences false are obtained by switching the elements $1$ and $0$ on the diagonal of the matrix. These four projection operators represent all possible `logical' interactions (measurement interactions) between the cognitive observer and the liar entity. During the measurement process carried out on the entities (B) and (C), the observer attributes truth-value to the sentences in a repetitive manner: in case of entity (B) it is a repetition of true-states (resp.~false-states) depending on whether an initial true (resp.~false) state was presupposed. In the case of entity (C) it will be an alternation between true-states and false states, no matter which state was presupposed. Finally, let us indicate the main points necessary in the derivation of a full description of the original double liar paradox, i.e., case (A), and see what pattern of truth assignments follows during the measurement (we refer to \cite{AeBroSme99} for a more detailed discussion of the derivation of the results). While for cases (B) and (C) the dimensionality of the coupled Hilbert space {$\Bbb{C}$}$^2\otimes $\thinspace {$\Bbb{C}$}$^2$ is sufficient, a space of higher dimension has to be used for case (A). This is due to the fact that no initial state can be found in the restricted space {$ \Bbb{C}$}$^2\otimes $\thinspace {$\Bbb{C}$}$^2$, such that application of the four true--false projection operators results in four orthogonal states respectively representing the four truth--falsehood states. The existence of such a superposition state --- with equal amplitudes of its components --- is required to describe the state of the entity before and after the measurement process. Since the truth-values of the two sentences in the paradox are not anymore coupled like it was the case for (B) and (C), the dynamical pattern of truth assignment by the observer is not anymore a two-step process but a four-step process. Therefore, the entity should be described in a 4 dimensional Hilbert space for each sentence. The Hilbert space needed to describe the Double Liar~(A) is therefore $\Bbb{C}^4 \otimes \Bbb{C}^4$. The initial un-measured superposition state --- $\Psi _0$ --- of the Double Liar~(A) is given by following superposition of the four true--false states: {\small \[ \frac 12\left\{ \begin{array}{c} \left( \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array} \right) +\left( \begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right) +\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right) \otimes \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) +\left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right) \end{array} \right\} \] } Each term in this superposition state is the consecutive state which is reached in the course of time, when the paradox is reasoned through. The truth--falsehood values attributed to these states, refer to the chosen measurement projectors. \noindent Making a sentence true or false in the act of measurement, is described by the appropriate projection operators in $\Bbb{C}^4 \otimes \Bbb{C}^4$. In the case we make sentence 1 (resp. sentence 2) true we get: \begin{eqnarray*} P_{1,true} &=&\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \otimes \mathbb{1}_2 \\ P_{2,true} &=&\mathbb{1}_1\otimes \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) \end{eqnarray*} The projectors for the false-states are constructed by placing the $1$ on the final diagonal place: \begin{eqnarray*} P_{1,false} &=&\left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \otimes \mathbb{1}_2 \\ P_{2,false} &=&\mathbb{1}_1\otimes \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \end{eqnarray*} \noindent Starting by making one of the sentences of (A) either true or false, by logical inference, the four consecutive states are run through repeatedly. In \cite{AeBroSme99} a continuous parameter $t$ was introduced to reflect `reasoning time' of the cognitive observer. This allowed to give a time-ordered description of the cyclic change of state present in the measurement process of the liar paradox. A Hamiltonian $H$ can be constructed such that the unitary evolution operator $U(t)$ --- with $ U(t)=e^{-iHt}$ --- describes this cyclic change. \noindent The dynamical picture of the Double Liar cognitive entity (A) is therefore as follows: when submitted to measurement, the entity starts its truth--falsehood cycle; when left un-measured the entity remains in the potentiality state. This follows immediately from the fact that the initial state $\Psi _0$ is left unchanged by the dynamical evolution $U(t)$: \[ \Psi _0(t)=\Psi _0 \] because $\Psi _0$ is a time invariant, as an eigenstate of the Hamiltonian $ H $. Because of this time-independence, the state $\Psi _0$ describes the cognitive entity of the liar paradox A, regardless of the observer. The highly contextual nature of the Double Liar (A) --- its unavoidable dynamics induced by the measurement process --- implies that intrinsically it can not expose its complete nature, analogous to the quantum entities of the micro-physical world. \section{Potentiality States versus Classical Emergence} \subsection{Potentiality State of a Soccer Team versus its Stable Classical Dynamical Pattern} Classical emergent properties of a system are defined by dynamical patterns related to its subsystems. Each of these subsystems can be described in a state space, and therefore the emergent properties which are identified with the collection of the subsystems, can be represented in essentially the same space. When potentiality states are involved, the emergent entity needs to be described in a higher dimensional state space, as the example of the liar paradox shows where the potentiality state is a superposition of tensor products of vectors representing the individual sentences. Crucial for the existence of a potentiality state is the contextuality in the measurement process, which causes a `collapse' of the potentiality state into one of the mutual exclusive possible eigenstates. Let us now clarify the main differences between a classical emergent pattern due to dynamical evolution of a complex system, and the new kind of emergence related to the existence of potentiality states, by the concrete example of a soccer team. A simple example of an entity represented with a potentiality state is given by a soccer team, e.g. the team of R.F.C. Anderlecht of Belgium (abbreviated: the A-team), consisting of eleven soccer players. The classical emergent description of such soccer team would involve analyzing the team in terms of the individual motions of the players and would reflect the dynamical pattern which is present during a game: e.g., whether the team as a whole is attacking or defending at one moment in time. The properties of the soccer team are determined by the capabilities of each player (how fast each player can run, how good they can pass the ball etc.), and on the correlation which exists between the players, i.e., how the members of the team play together as a team. Hence, the whole soccer team can be identified with the eleven members of the team and the team can be represented in terms of the descriptions of the individual players: indeed, once the movement of each player is known, the movement of the team as a whole is known too. This is what we would call the team as a `classical' emergent entity. Notice, that when the eleven players have left the field after the game, the entity of the team stops to exist. Indeed, the movements of the eleven players become incoherent after the game: each player leaves to his own home, and until they have to play the next match, they have stopped to be, in classical emergent terms, part of the larger entity `a soccer team'. For instance, one of the players of the A-team could be a foreigner who has to play a game for his native country before the next match of the A-team. Obviously, while he's playing the match for his native country, he cannot be playing a match for the A-team. Therefore, while he's playing for his home country, the classical emergent entity `the A-team' defined by the eleven soccer players does not exhibit the typical dynamical patterns of a soccer team playing a match, and therefore the entity does not exist at that moment in time. If we elaborate this example in terms of the potentiality state concept, we obtain the following. The state of the entity `the A-team' is defined by the possible instances of the team as a collection of the individual players. By this we mean that when the eleven players are on the (same) soccer field playing a game for the A-team, they are in fact actualizing a potential game, simply by the act of playing the game. The emergent concept of `the A-team' is more than just the collection of the movements of the eleven players. Even when the eleven players are not actually playing a game, they are still a `potential team'. The instance of the team playing a game is created at the moment the A-team is actually playing a game. To make this more clear, let us consider the following situation. The A-team has qualified for the final of the Belgian soccer cup in which they will have to play against the winner of the other semi-final match Bruges-Ghent which is played a day later than the A-team semi-final. Then `the A-team playing the cup final' can be considered as an emergent entity, defined by the eleven players who are preparing themselves for the final. A day later the match Bruges-Ghent will be actually played and the adversary of the A-team in the final will be known. At the moment the final is played, the instance `the A-team is playing' is activated, and potentiality of `the A-team playing a game' collapses into one of two mutually exclusive possibilities, i.e., a final against Bruges or a final against Ghent. Notice that even when the A-team is not actually playing, the emergent concept defined by the potentiality of letting the eleven players play a game is still present. As such, `the A-team playing the final' can be viewed as a potentiality state of two possible instances: one concerns the final against the (let's say defending) team of Bruges, and one against the (let's say attacking) team of Ghent. Depending on which team reaches the final, the A-team will behave differently because they have to choose between different tactics. Nevertheless, before the score of the semi-final Bruges-Ghent is determined, potentially both tactics could be followed. It is not just the state of the A-team which decides how the final will look like, also the adversary will be decisive, and the result of the game Bruges-Ghent is not influenced by the state of the A-team. The differences between classical concept of emergence and the emergent behavior which is established by the presence of potentiality states, resemble the differences in dynamical evolution of a classical system versus a quantum system. The evolution of a classical system is continuous, and the act of measurement does not influence the results, the measurements are purely non-perturbative observations. As such, the dynamical pattern of the system as a whole is identified by the dynamical evolution of the sub-systems. In other words, the state of the compound system evolves continuously in time. The dynamical evolution for a quantum particle, upon which no measurements are carried out, is given by the Schr\"{o}dinger equation, which also describes a continuous evolution of the state of the entity. However, standard quantum theory predicts a non-continuous evolution during the measurement process, such that the state of the system changes instantaneously towards one of the possible eigenstates corresponding with the observable. Similarly, the potentiality of the eleven players of the A-team to play a game against Bruges or a game against Ghent defines an emergent entity. Before the result of Bruges-Ghent is known, potentially the A-team can play a game against Ghent, but also potentially against Bruges. Nevertheless, only one of these possible finals can actually be played. Which final will be played, does not depend on the A-team alone, but also on the adversary who manages to qualify: Bruges or Ghent. This dependence on the context (i.e., the semi-final between Bruges and Ghent) causes the non-continuous evolution of `the A-team playing the final' from a potentiality of both games towards one of the specific instances. As such, one can regard the concept of potentiality state as a possible way to generalize the concept of classical emergence to cases where also contextuality is important. Eventually, this should lead to a general theory capable of describing on the one hand the continuous evolution of a system in the absence of contextual interaction and on the other hand the discontinuous evolution of the system during the interaction process with the environment during which in a highly contextual way one of the possibilities present in the potentiality state of the system is effectively actualized. \subsection{Violation of Bell Inequalities for the Soccer Teams} Let us now show that also in the case of a soccer team we can define a set of experiments for which Bell inequalities are violated, indicating the presence of a potentiality state. Depending on the context only one of mutual exclusive potential outcomes will be actually realized during the experiment. The entity that we consider is the set of 22 soccer players of two teams playing a cup final. The first experiment $e_1$ consists in letting someone give money to a player of team A such that he will cause his team to lose the cup final. If his team actually loses the cup final, the experiment $e_1$ is said to yield the outcome $o_1(up)$, if team A wins the final, the experimental outcome is $o_1(down)$. Taking into account that the final is played until one of the two teams wins (e.g., in the case of a draw the final could be decided with a number of penalties), the experiment will always give one of these two possible outcomes. The second experiment $e_2$ consists in looking whether the referee gives a player of team A a yellow card or not. If he actually gives a yellow card to at least one player of team A, the outcome is $o_2(up) $, and $o_2(down)$ otherwise. The experiment $e_3$ consists in letting someone give money to a player of team B such that he will cause his team to lose the cup final. If team B actually loses the cup final, the experiment $ e_3$ is said to yield the outcome $o_3(up)$, if team B wins the final, the experimental outcome is $o_3(down)$. The fourth and last experiment $e_4$ consists in looking whether the referee gives a player of team B a yellow card. If he actually gives a yellow card to at least one player of team B, the outcome is $o_4(up)$, and $o_4(down)$ otherwise. Finally, we assume that the referee has a bad character such that during the final he will definitely give a yellow card to at least one player of team A and one player of team B. Let us now look at the coincidence experiments $e_{13}$, $ e_{14}$, $e_{23}$ and $e_{24}$. The experiment $e_{14}$ consists in giving a player of team A money with the aim of letting team A lose the final, and looking whether a player of team B has received a yellow card. Of course, even if the other players of team A are playing very good, the bribed player can make intentionally mistakes like own-goals etc.~such that even in that case team A loses the final. Therefore, the experiment $e_{14}$ gives the outcome $(o_1(up),o_4(up))$ and the expectation value $\hbox{$\mathbb{E}$} _{14}=+1.$ Similarly, we obtain that $\hbox{$\mathbb{E}$}_{23}=+1.$ Also, due to our assumption about the bad-character referee, we can deduce that the coincidence experiment $e_{24}$ yields always the outcome $(o_2(up),o_4(up))$ such that the expectation value is given by $\hbox{$\mathbb{E}$}_{24}=+1.$ Finally, let us look at the coincidence experiment $e_{13}.$ In this case, both a player of team A and a player of team B will receive money to let their respective team lose the cup final. However, only one of the two teams can actually lose the final. Therefore, the coincidence experiment $ e_{13}$ can only yield the outcome $(o_1(up),o_3(down))$ or $ (o_1(down),o_3(up)),$ resulting in an expectation value $\hbox{$\mathbb{E}$} _{13}=-1.$ Let us now calculate the terms of Bell inequalities, \[ \begin{array}{ll} |\hbox{$\mathbb{E}$}_{13}-\hbox{$\mathbb{E}$}_{14}|+|\hbox{$\mathbb{E}$}_{23}+\hbox{$\mathbb{E}$} _{24}| & =|-1-1|+|+1+1| \\ & =+2+2 \\ & =+4 \end{array} \] and it follows that Bell inequalities are violated. This violation is due to the explicit contextuality of the experimental outcomes. To make this more clear, we could also specify in the definition of the experiments $e_1$ and $ e_3$ the amount of money which is given to the players. For instance, if we specify that in experiment $e_1$ the bribed player is very poor, and the amount of money is to be one billion dollars, then he will probably do everything possible in order to let his team lose the final. If on the other hand, the player in experiment $e_3$ is already rich, and the amount of money is only 100.000 dollars, he will probably not do such extreme things like making a lot of own-goals etc., which the other, poor player will probably do to earn his billion dollars. This shows that also in the case of deterministic coincidence experiments ($e_{13}$ always yielding the outcome $(o_1(up),o_3(down))$) one can violate Bell-inequalities. The reason why Bell-inequalities are also violated in the deterministic case is situated in the fact that only one of two possible but mutually exclusive situations can occur (i.e., only one of the two teams can lose the cup final), and that before the experiments are actually performed both experimental outcomes are possible. It is the contextual nature of the experiments which defines which of the possible experimental results will actually occur. This shows that the emergent phenomenon of two teams playing the cup final should be described with a potentiality state, such that in one of the possible cases team A loses the final, and in the other team B loses the final. Depending on which experiment is actually performed (which player is given what amount of money), the actually played final will be different. Therefore, the final cannot be regarded only as the stable dynamical pattern exhibited by the two teams of soccer players playing a game. \section{Conclusions} The above examples --- the `vessels of water', `liar paradox' and `the soccer team' --- illustrate the various ways in which potentiality states can be identified in reality. The emergent properties which the potentiality states define, depend on the particular nature of the entity. For the vessels of water example, the appearance of potentiality states indicates the presence of a particular quantum aspect, which can be demonstrated by the violation of Bell inequalities. For the liar paradox example we have worked out a full and detailed quantum description which shows the pure quantum mechanical nature of the example. The potentiality states in this case are superposition states. We mention that the example of the liar paradox shows that the reality of conceptual space is quite different from great part of the macroscopic world. It seems that it contains `Hilbert space-like' features, which makes it quantum-like. We have shown on the example of a soccer team that the emergence due to potentiality states has a quite different status than the one of the emergent dynamical patterns that are identified in classical physics. The emergent potentiality states are ontological states within the formalism, and not connected to dynamical patterns alone. Also in the case of the soccer teams playing a match, one can define experiments such that Bell inequalities are violated, indicating the contextual nature of the defined experiments for these entities. \noindent {\bf Acknowledgments} \noindent The authors would like to acknowledge the support by the Fund for Scientific Research--Flanders (Belgium)(F.W.O.--Vlaanderen). \small \end{document}

\begin{document} \title{{f On a double integral of a product of Legendre polynomials} \begin{abstract} \noindent We calculate a double integral over a product of Legendre polynomials multiplied by a binomial raised to a power. \end{abstract} During the calculation of the electromagnetic self-force of a uniformly charged spherical ball, we encountered the integral \begin{align}\label{1} I=\int_0^{\pi}d \theta \sin \theta \int_0^{\pi}d \theta' \sin \theta' (\cos \theta - \cos \theta')^{2n} P_l(\cos \theta) P_l(\cos \theta'), \end{align} where $n$ and $l$ are integer positive numbers and $P_l$ is a Legendre polynomial of order $l$. As far as we know, this integral was not calculated in closed form anywhere in the literature. We calculate it here. After changing the variables $\cos \theta \rightarrow x$, $\cos \theta' \rightarrow y$ this integral becomes \begin{align}\label{2} I=\int_{-1}^1 dx \int_{-1}^1 dy \;(x-y)^{2n} P_l(x) P_l(y). \end{align} We first perform the integral $\int_{-1}^1 dx (x-y)^{2n} P_l(x).$ For this, we can use {\bf 7.228} and {\bf 8.703} from \cite{gra}. Combining these two equations that read \begin{align} \frac{1}{2}\Gamma(1+\mu) \int_{-1}^1 P_l(x) (z-x)^{-\mu-1}= (z^2-1)^{- \frac{\mu}{2}} e^{-i \pi \mu} Q_l^{\mu}(z), l=0,1,\dots, |arg(z-1)|<\pi, \end{align} \begin{align} Q_{\nu}^{\mu}(x)= \frac{e^{i\pi \mu} \Gamma(\nu+\mu+1) \Gamma\left(\frac{1}{2}\right)}{2^{\nu+1} \Gamma\left( \nu + \frac{3}{2}\right)} (x^2-1)^{\frac{\mu}{2}} x^{-\nu-\mu-1} {}_2F_{1}\left(\frac{\nu+ \mu +2}{2}, \frac{\nu+\mu+1}{2}; \nu+\frac{3}{2}; \frac{1}{x^2}\right), \end{align} after using \[ \frac{\Gamma(l+\mu+1)}{\Gamma(\mu+1)} =(\mu +1)_l\; \text{and} \; \Gamma\left( l+\frac{3}{2}\right)= \frac{\sqrt{\pi} (l+1)_{l+1}}{2^{2l+1}},\] one obtains for $\mu=-1-2n$ \begin{align} \int_{-1}^1 dx \, (x-y)^{2n}P_l(x) = \frac{(-2n)_l \,2^{l+1} y^{2n-l}}{(l+1)_{l+1}} {}_2F_1 \left( \frac{l}{2}-n+\frac{1}{2}, \frac{l}{2}-n; l+\frac{3}{2}; \frac{1}{y^2} \right). \end{align} The same result given in Eq. (5) can be obtained by putting $a=1$, $m=0$ and $p=-2n$ in Eq. {\bf 2.17.4(5)} from \cite{pru} \begin{align} \int_{-a}^a dx \,\frac{(a^2-x^2)^{\frac{m}{2}}}{(x-y)^p} P_l^m \left( \frac{x}{a} \right)= \frac{2 (-1)^{m-1}(l+m)!}{(p-1)! (l-m)!} (y^2-a^2)^{\frac{m-p+1}{2}} Q_l^{p-m-1} \left( \frac{y}{a}\right), \end{align} although this equation is given in \cite{pru} as being valid only for $p=0,1, \dots$. The same result given in Eq.(5) can be obtained by direct calculation, by using the Rodrigues formula for Legendre polynomials \cite{rai} \begin{align} P_l(x)= \sum_{k=0}^{[l/2]} \frac{(-1)^k \left( \frac{1}{2}\right)_{l-k} (2x)^{l-2k}}{k!(l-2k)!} \end{align} and the binomial expansion for $(x-y)^{2n}$ \begin{align} (x-y)^{2n}= \sum_{p=0}^{2n} \frac{(-1)^p (2n)!}{p!(2n-p)!} x^p y^{2n-p}, \end{align} and integrating the resulting double sum term by term. For $l$ odd, after noting that the term by term integration gives non-zero result only for $p$ odd and changing the summation index $p \rightarrow 2p$, one obtains \begin{align} \int_{-1}^1 dx \,(x-y)^{2n} P_l(x)= -2^{l+1} \sum_{k=0}^{[l/2]} \sum_{p=0}^{b-1} \frac{(-1)^k \left( \frac{1}{2} \right)_{l-k}(2n)! y^{2n-2p-1}}{k! (l-2k)!2^{2k}(2p+1)!(2n-2p-1)!}\nonumber \\ \cdot \frac{1}{(l-2k+2p+2)}. \end{align} Writing all the factorials in terms of Pochhammer symbols, the above summation over $k$ can be done as follows \begin{align} &\sum_{k=0}^{\frac{l-1}{2}} \frac{(-1)^k \left( \frac{1}{2}\right)_{l-k}}{k!(l-2k)!2^{2k}(l+2p+2-2k)} \nonumber \\ &= \frac{\left( \frac{1}{2}\right)_l}{\Gamma(1+l) (l+p+2)} {}_3F_2 \left( -\frac{l}{2}+\frac{1}{2}, -\frac{l}{2}, -\frac{l}{2}-p-1; \frac{1}{2}-l, -\frac{l}{2}-p,1\right)\nonumber \\ &\frac{\left(\frac{1}{2}\right)_l \left( \frac{1-l}{2} \right)_{\frac{l-1}{2}} (-p)_{\frac{l-1}{2}}}{l! \,(l+2p+2) \left(\frac{1}{2}-l\right)_{\frac{l-1}{2}} \left(-\frac{l}{2}-p\right)_{\frac{l-1}{2}}}, \end{align} where, when we passed from the second to the third line of the above equation, we used equation {\bf 7.4.4 (81)} from \cite{pru}. We note that, because of the Pochhammer symbol $(-p)_{\frac{l-1}{2}}$, the r.h.s. of Eq. (10) is different from zero only for $p \ge \frac{l-1}{2}$. Introducing Eq. (10) in Eq. (9) and changing the summation index $p\rightarrow i, \, p-\frac{l-1}{2}=i$, the resulting summation over i can be done immediately and one obtains again the result of Eq. (5). The case $l$ even can be considered similarly. Returning now to Eq. (2), after using Eq. (5) one obtains \begin{align}\label{7} I=\frac{(-2n)_l \;2^{l+1}}{(l+1)_{l+1}} \int_{-1}^1 dy\; y^{2n-l} P_l(y)\; {}_2F_1 \left(\frac{l}{2}-n, \frac{l+1}{2}-n, l+\frac{3}{2};\frac{1}{y^2} \right). \end{align} Note that, for $l \le 2n$, the hypergeometric function in Eq.(\ref{7}) is, in fact, a finite series, because $l/2-n$ and $(l+1)/2-n$ are negative integers when $l$ is even and odd respectively. Again, we consider the cases $l$ even and $l$ odd separately. For $l$ even, using the definition of the Gauss hypergeometic function, we have \begin{align}\label{8} {}_2F_1 \left(\frac{l}{2}-n, \frac{l+1}{2}-n, l+\frac{3}{2};\frac{1}{y^2} \right)= \sum_{k=0}^{n-\frac{l}{2}} \frac{ \left( \frac{l}{2}-n \right)_k \left(\frac{l+1}{2}-n \right)_k}{k! \left( l+\frac{3}{2} \right)_k} \frac{1}{y^{2k}}. \end{align} From Eqs. (\ref{7}), (\ref{8}), one obtains \begin{align}\label{9} I= \frac{2 (-2n)_l2^{l+1}}{(l+1)_{l+1}} \sum_{k=0}^{n-\frac{l}{2}} \frac{ \left( \frac{l}{2}-n \right)_k \left(\frac{l+1}{2}-n \right)_k}{k! \left( l+\frac{3}{2} \right)_k} \int_0^1 dy \; y^{2n-2k-l} P_l(y), \end{align} where we used the fact that the integrand in the r.h.s. of Eq. (\ref{9}) is an even function, because $P_l(-y)=(-1)^l P_l(y)$ \cite{gra}. The integral in Eq. (\ref{9}) can be performed using ({\bf 2.17.1(4)} from \cite{pru} or {\bf 7.126(2)} from \cite{gra} ) \begin{align}\label{p2} \int_0^1dx \; x^{\sigma} P_{\nu}(x)= \frac{\sqrt{\pi} \;2^{-\sigma-1} \Gamma(1+\sigma)}{\Gamma \left(1+\frac{\sigma-\nu}{2} \right) \Gamma \left( \frac{\sigma+\nu+3}{2} \right)}. \end{align} One obtains \begin{align}\label{11} I=\frac{\sqrt{\pi} \;(-2n)_l \;2^{2l-2n+1}}{(l+1)_{l+1}} \sum_{k=0}^{n-\frac{l}{2}} \frac{2^{2k} \left( \frac{l}{2}-n \right)_k \left( \frac{l+1}{2}-n \right)_k \Gamma(2n-2k-l+1)}{k! \left( l+\frac{3}{2} \right)_k \Gamma(1+n-k-l) \Gamma \left( n-k+\frac{3}{2} \right)}. \end{align} Using the definition of the Pochhammer symbol \cite{pru} \[ (a)_k= \frac{\Gamma(a+k)}{\Gamma(a)}= (-1)^k \frac{\Gamma(1-a)}{\Gamma(1-a-k)}, \] and \cite{pru} \[(a)_{2k}= \left( \frac{a}{2} \right)_k \left( \frac{a+1}{2} \right)_k 2^{2k}, \] we write the Gamma functions in Eq. (\ref{11}) as follows \begin{align} & \Gamma(2n-2k-l+1)= \frac{\Gamma(2n-l+1)}{\left( \frac{l}{2}-n \right)_k \left( \frac{l}{2}-n+\frac{1}{2}\right)_k2^{2k}}, \nonumber\\ &\Gamma(1+n-l-k)= (-1)^k\frac{\Gamma(1+n-l)}{(l-n)_k},\nonumber \\ &\Gamma \left(n+\frac{3}{2}-k \right)= (-1)^k \frac{\Gamma \left( n+\frac{3}{2}\right)}{\left( -n-\frac{1}{2}\right)_k}. \end{align} Introducing (16) in (15), one obtains \begin{align} I=\frac{\sqrt{\pi} \;(-2n)_l\; 2^{2l-2n+1}(2n-l)!}{ (l+1)_{l+1} (n-l)! \left( n+\frac{1}{2} \right)!} {}_2F_1 \left(-n-\frac{1}{2}, l-n; l+\frac{3}{2}; 1 \right). \end{align} But the Gauss hypergeometric function of unit argument can be written as ({\bf 7.3.5(2)} \cite{pru}) \begin{align} {}_2F_1(a,b;c;1)=\frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a) \Gamma(c-b)}, \end{align} and we obtain \begin{align}\label{17} I=\frac{ \sqrt{\pi}\; (-2n)_l \;2^{2l-2n+1} (2n-l)! \left(l+\frac{1}{2}\right)! (2n+1)!}{ (l+1)_{l+1} (n-l)! \left(n+\frac{1}{2}\right)! (l+n+1)! \left(n+ \frac{1}{2} \right)!}, \end{align} where we use the notation $\Gamma(z)=(z-1)!$ both for integer and noninteger $z$. Using the definition of the Pochhammer symbol and \cite{pru} \begin{align} \frac{\Gamma(2z)}{\Gamma(z)}=\frac{2^{2z-1}}{\sqrt{\pi}} \Gamma \left( z+\frac{1}{2} \right), \end{align} we can write \begin{align}\label{19} &(-2n)_l= (-1)^l \frac{\Gamma(1+2n)}{\Gamma(1+2n-l)},\nonumber \\ & (l+1)_{l+1}=\frac{2^{2l+1}}{\sqrt{\pi}} \Gamma \left(l+\frac{3}{2} \right),\nonumber \\ & \frac{(2n)!}{\left( n+\frac{1}{2}\right)!}=\frac{2^{2n+1}\Gamma(n+1)}{\sqrt{\pi} (2n+1)},\\ & \frac{(2n+1)!}{\left( n+\frac{1}{2}\right)!}= \frac{2^{2n+1}\Gamma(n+1)}{\sqrt{\pi} }. \nonumber \end{align} Introducing (\ref{19}) in (\ref{17}), one obtains for $l$ even \begin{align} I= \frac{(-1)^l 2^{2n+2} (n!)^2}{(2n+1) (n-l)! (n+l+1)!}. \end{align} Note that, because of $(n-l)!$ from the denominator, this result is different from zero only for $n \ge l$. A similar calculation can be done for $l$ odd, and one obtains the same result. So, our final result for the integral (\ref{1}) is \begin{equation} I= \left\lbrace\begin{array}{c} \frac{(-1)^l 2^{2n+2} (n!)^2}{(2n+1) (n-l)! (n+l+1)!}, n \ge l\\ 0, n<l \end{array} \right.. \end{equation} \end{document}

\begin{document} \title [Chemical distance exponent] {\large On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field} \author{Yifan Gao and Fuxi Zhang} \address {Yifan Gao\\ School of Mathematical Sciences\\ Peking University\\ Beijing, China, 100871} \email{[email protected]} \address {Fuxi Zhang\\ School of Mathematical Sciences\\ Peking University\\ Beijing, China, 100871} \email{[email protected]} \subjclass[2010]{Primary 60K35, 60G60.} \keywords{Gaussian free field, percolation, chemical distance.} \begin{abstract} In this paper we introduce the two-sided level-set for the two-dimensional discrete Gaussian free field. Then we investigate the chemical distance for the two-sided level-set percolation. Our result shows that the chemical distance should have dimension strictly larger than $1$, which in turn stimulates some tempting questions about the two-sided level-set. \end{abstract} \maketitle \section{Introduction} The discrete Gaussian free field (DGFF) in $\mathbb Z^d, d\ge 3$ is a Gaussian random field with mean zero and covariance given by the Green's function. As a ``strongly'' correlated random field, the level-set percolation for the DGFF in three dimensions or higher has been extensively studied and shown to exhibit a non-trivial phase transition by a series of work \cite{MR914444,MR3053773,MR3339867,MR3843421,duminil2020equality}. More precisely, there exists a critical level $0<h_*(d)<\infty$ such that if $h<h_*(d)$, the level-set (a.k.a. excursion set, the random set of points whose value is greater than or equal to $h$) has a unique infinite cluster; if $h>h_*(d)$, the level-set has only finite clusters. In this paper, we focus on the two-dimensional DGFF (also called harmonic crystal). However, in two dimensions, we can not define the DGFF in the whole discrete plane since the two-dimensional Green's function blows up, while one can take the scaling limit (the lattice spacing is sent to $0$ while the domain is fixed) to get the continuum Gaussian free field. It is then not possible to investigate the level-set percolation in $\mathbb Z^2$ directly as that in $\mathbb Z^d, d\ge 3$. The right way is to take a big discrete box $V_N$ of side length $N$ and define the two-dimensional DGFF on $V_N$ as a centered Gaussian process with covariance given by the Green's function on $V_N$. Then one can study the connectivity properties of the level-set on $V_N$ as $N$ goes to infinity. However, as shown in \cite{MR3800790}, for any level $h\in\mathbb R$, the level-set above $h$ crosses a macroscopic annulus in $V_N$ with non-vanishing probability as $N$ goes to infinity, which suggests that in some sense there is no non-trivial phase transition for the two-dimensional level-set percolation. Furthermore, the chemical distance (intrinsic distance) between two boundaries of a macroscopic annulus is bounded from above by $N(\log N)^{1/4}$ \cite{MR3800790,MR4112719}. Roughly speaking, the chemical distance has dimension $1$. This inspires us to think that once we truncate the DGFF from two sides to get the so-called two-sided level-set in this paper, whether the chemical distance will be of dimension strictly larger than $1$. Our main result below answers this affirmatively in the sense that if there exists macroscopic (nearest-neighbor) path inside the two-sided level-set, its length must be greater than $N^{1+\varepsilon}$ for some $\varepsilon>0$. Although at present we are not able to show that there could exist some macroscopic path inside the two-sided level-set (This is not obvious, see Question~\ref{que:q1} below), our result gives some expected fractal structure for the two-sided level-set, which is drastically different from the (one-sided) level-set. Next, we introduce our model and then state our main result. For each positive integer $N$, let {$V_{N}=[-N/2,N/2]^2\cap \mathbb Z^2$}. Denote by $\{ \eta^{V_{2N}}(v): v\in V_{2N} \}$ the discrete Gaussian free field (DGFF) on $V_{2N}$ with Dirichlet boundary conditions, which is a mean-zero Gaussian process, vanishing on the boundary, with covariance given by \[ \mathbb E \eta^{V_{2N}} (u) \eta^{V_{2N}} (v) = G_{2N} (u,v) \quad \text{ for } u,v \in V_{2N}, \] where $G_{2N}(u,v)$ is the Green's function of the two-dimensional simple random walk in $V_{2N}$. As usual, we restrict ourselves to consider the DGFF $\eta^{V_{2N}}$ on $V_N$ to avoid boundary issues (This can be made more general, see Remark~\ref{rem:boundary-issue}). Suppose $\lambda > 0$. Let \begin{equation} {\Lambda_{N,\lambda}} : = \{ v\in V_N: |\eta^{V_{2N}}| \le \lambda \}. \end{equation} We say that a vertex $v\in V_N$ is $\lambda$-open if $\left|\eta^{V_{2N}}(v)\right|\le\lambda$, and interpret $\Lambda_{N, \lambda}$ as the ``two-sided'' level set. Let \begin{equation} \label{Eq.defnPke} {\mathcal{P}_{N}^{\kappa, \epsilon} }= \left\{P: P \text { is a path in } V_{N},\|P\| \geq \kappa N, \text { and }|P| \leq N^{1+\epsilon}\right\}, \end{equation} where $\|P\|=\|x-y\|$ if $P$ is a path from $x$ to $y$ and $|P|$ is the length of $P$. We say that $P$ is $\lambda$-open if so is every vertex in $P$. Our main result is the following theorem. \begin{thm}\label{thm:1.1} For each $\lambda>0$, there exists $\epsilon=\epsilon(\lambda)>0$ such that for every $\kappa\in (0,{1})$, \begin{equation}\label{eq:complement-event} \lim _{N \rightarrow \infty}\mathbb{P}\big( P \text { is } \lambda\text{-open for some } P \in {\mathcal{P}_{N}^{\kappa, \epsilon} }\big)=0. \end{equation} \end{thm} \begin{rem}\label{rem:lambda_0}\label{rem:boundary-issue} The choice of working with $V_{2N}$ is for convenience; the above theorem holds with { $V_{2N}$ and $V_{N}$ being respectively replaced with $V_{N}$ and $V_{\delta N}$} for any fixed $0<\delta<1$. \end{rem} \begin{rem} Theorem~\ref{thm:1.1} also holds for the Gaussian free field on metric graphs since percolation on the metric graph is dominated by the percolation on the integer lattice. \end{rem} \begin{rem} Note that it suffices to show \eqref{eq:complement-event} holds for large $\lambda$ since the event is increasing in $\lambda$. In fact, we also obtain some quantitive results, namely, it is able to take $\varepsilon(\lambda)=e^{-a\lambda^2}$ for some absolute constant $a>0$ (see \eqref{eq:eps-lambd}), and the probability in \eqref{eq:complement-event} decays faster than $N^{-c\lambda^{-2}}$ for some absolute constant $c>0$ (see \eqref{eq:decay-rate}). \end{rem} \begin{rem} Furthermore, our method is still effective if $\lambda$ depends on $N$. For example, taking $\lambda=\lambda_N=\sqrt{(2a)^{-1}\log\log N}$, then Theorem~\ref{thm:1.1} shows that for any $\lambda_N$-open path with macroscopic distance, its length should be at least $N(\log N)^{1/2}$. On the other hand, it has been shown in \cite[Theorem 2]{MR1880237} that the maximum of the DGFF is at most $2\sqrt{\frac{2}{\pi}}\log N$ with probability tending to $1$, see \cite{MR4043225} for more about the level-set at heights proportional to the absolute maximum. By symmetry, we see that if we take $\lambda_N=2\sqrt{\frac{2}{\pi}}\log N$, then all points are $\lambda_N$-open with overwhelming probability. Our result stimulates the tempting question of determining the borderline with respect to $\lambda_N$ for linear growth of the chemical distance. \end{rem} \begin{ques}\label{que:q1} Does there exist a large constant $\lambda>0$ such that with non-vanishing probability there exists a $\lambda$-open path $P$ in $V_N$ with $\|P\| \geq \kappa N$? \end{ques} Let $p(\lambda, N)$ denote the probability of the above event. Here, we use non-vanishing to mean that $\inf_{N}p(\lambda, N)>0$. It is readily to see that if $\lambda$ is sufficiently small such that $\rho:=\mathbb P(|Z(0,4)|\le\lambda)<\frac14$ where $Z(0,4)$ is a Gaussian random variable with mean $0$ and variance $4$, then $p(\lambda, N)\le N^2(4\rho)^{\kappa N}$ which vanishes exponentially fast in $N$. However, to show it is not the case for large constant $\lambda$ is quite non-trivial. To the best of our knowledge, this question has not been answered yet. We expect that there exists a non-trivial ``phase transition'' for the two-sided level-set percolation. We would like to mention that recently the authors in \cite{ding2020crossing} show that the probability for (one-sided) level-set crossing a rectangle is bounded away from $0$ and $1$. Another closely related work to this direction is \cite{MR3163210}, in which the ``two-sided'' level-set in $\mathbb Z^d, d\ge 3$ is defined as the random set of points whose absolute value is larger than $h$ (note that this is contrary to our convention for the two-sided level-set), and the associated critical value is proved to be finite for all $d\ge 3$. \begin{ques} Is there a way to take a scaling limit of the two-sided level-set? \end{ques} This might be reminiscent of the Schramm-Sheffield contour line of DGFF in \cite{MR2486487}, which is shown to converge in distribution to $\mathrm{SLE}_4$, as well as the bounded-type thin local sets (BTLS) constructed in \cite{MR3936643} and the first passage set (FPS) introduced in \cite{MR4091511}. Dynkin’s isomorphism enables us to relate the absolute value of the DGFF to the occupation field generated by the random walk loop soup (see \cite{MR2815763,MR3502602}), so all the above questions can be understood in terms of the loop-soup percolation with respect to the occupation field (see \cite{MR2979861,MR3547746,MR3941462} for more related works about the loop-soup). We hope to find some appealing connections between the two-sided level-set and the objects we mentioned. \subsection{Background}\label{subsec:background} The two-dimensional Gaussian free field (GFF) is an important object in statistical physics and the theory of random surfaces \cite{MR2322706}. As {the analog of the Brownian motion with two-dimensional time parameter}, it demonstrates fractal structures in many aspects \cite{MR2486487,MR3947326,MR4019914,MR4076090}. From the perspective of the level-set percolation of the DGFF, we will focus on the chemical distance, which plays a crucial role in {the study of} the fractal structure of clusters in the theory of percolation. Roughly speaking, the chemical distance is the graph distance on the induced (random) subgraph in some probability models. For instance, the chemical distance for classic percolation models can be defined as the length of the shortest path inside open clusters \cite{MR750568,havlin1985chemical}. However, estimating the chemical distance is quite difficult, which often requires subtle analysis of the structure of the shortest path. Especially, for the two-dimensional critical Bernoulli percolation model, physicists expect that there exists an exponent $d_{\mathrm{min}}$ such that \begin{equation} \ell\sim r^{d_{\mathrm{min}}}, \end{equation} where $r$ and $\ell$ are respectively the Euclidean distance and the chemical distance between two vertices $x$ and $y$, and only the case $r < \infty$ is considered. However, a rigorous way to clarify the equivalence above, i.e., the precise meaning of ``$\sim$", remains to be an open problem \cite{MR2334202,MR3698744}. It is expected in the physics community that $d_{\mathrm{min}}$ is universal in the sense that it does not depend on the choice of vertices and the type of lattice. In \cite{MR1712629}, Aizenman and Burchard show that $\ell\ge r^{\eta}$ for some $\eta>1$, which implies that $d_{\mathrm{min}}>1$. {Furthermore}, upper bounds on the chemical distance can be obtained by comparing the shortest horizontal crossing with the lowest crossing \cite{MR3698744,damron2017strict}. Specifically, for the critical Bernoulli bond percolation on the edges of a box of side length $n$, the expectation of chemical distance between the left and right sides of the box is $O(n^{2-\delta}\pi_3(n))$ for some $\delta>0$, where $\pi_3(n)$ is the three-arm probability to distance $n$ \cite{damron2017strict}. Additionally, in the subcritical and supercritical cases of Bernoulli percolation of dimension $d\ge2$, chemical distance is comparable to Euclidean distance \cite{MR762034,MR1404543,MR1068308,MR2319709}. In the critical case in high dimensions, it is shown that macroscopic connecting paths have dimension $2$ \cite{MR2551766,MR2748397,MR3224297}. We next turn to some correlated percolation models, which have been intensively studied recently \cite{biskup2004scaling,MR2915665,MR3692311,MR3800790}. In the supercritical case for a general class of percolation models on $\mathbb Z^d$ {($d\ge3$)}, with long-range correlations (e.g., the random interlacements, the vacant set of random interlacements, the level sets of the GFF), the chemical distance behaves linearly as in the case of Bernoulli percolation; {see \cite{MR3390739} for details. However, the methods developed in three dimensions and higher are invalid in two dimensions, since the two-dimensional DGFF is log-correlated. Thus it becomes quite complicated when we consider the level-set percolation of the DGFF in two dimensions. Recently, {it is shown in \cite{MR3800790} that for level-set percolation of the two-dimensional DGFF, the associated chemical distance between two boundaries of a macroscopic annulus is {$O(Ne^{(\log N)^{\alpha}})$} for any $\alpha>1/2$ with positive probability. Later, the order is improved to $O(N(\log N)^{1/4})$ with high probability, on metric graphs, given connectivity \cite{MR4112719}.} Note that these two results imply that the undetermined chemical distance exponent for level sets of the two-dimensional DGFF is expected to be $1$ in any phase. In this paper, we investigate the two-sided level-set cluster of the DGFF. By Theorem~\ref{thm:1.1}, two vertices $x$ and $y$ has chemical distance $\ge N^{1+\epsilon}$, provided that they are connected. Then it will indicate the fractal structure of the two-sided level-set clusters, contrasting to the afore-mentioned case of no fractality of the level-set clusters. \subsection{Notation conventions}\label{subsec:notations} For the sake of the reader, we list some notations here. For $x=(x_1,x_2), y=(y_1,y_2)\in\mathbb R^2$, let \[ \|x-y\|=\sqrt{|x_1-y_1|^2+|x_2-y_2|^2} \ \text{ and } \ |x-y|_{\infty}=|x_1-y_1|\wedge|x_2-y_2| \] Let $d(x,B)=\inf_{y\in B}\|x-y\|$ and $d(B_1,B_2)=\inf_{x\in B_1}d(x,B_2)$. Similarly, we define $ d_{\infty}(x,B)=\inf_{y\in B}|x-y|_{\infty}$ and $d_{\infty}(B_1,B_2)=\inf_{x\in B_1}d_{\infty}(x,B_2)$. For $x\in\mathbb R^2$ and $\ell>0$, let \[ B(x,\ell)=\{ y\in \mathbb R^2:\|x-y\|\le \ell \} \ \text{ and } \ B_{\infty}(x,\ell)=\{ y\in \mathbb R^2: |x-y|_\infty \le \ell \}. \] Denote \[ V_{\ell}(x)=B_{\infty}(x,\ell/2) \cap\mathbb Z^2. \] For $a\in\mathbb R$, let $\lfloor a\rfloor$ be the greatest integer that is at most $a$. For $r\ge 1$, let $[r]=\{1,\cdots,\lfloor r\rfloor\}$. Throughout this paper, let $C_1,C_2,\cdots>0$ be universal constants. Let $K=2^k$ be large but fixed in terms of $N$ and to be chosen later, where $k$ is a positive integer. Recall $\kappa\in (0,1)$ as stated in Theorem \ref{thm:1.1}. Let $m\in\mathbb Z_+$ be such that \[ K^{m+1} \leq \kappa N<K^{m+2}. \] Note that $m\rightarrow\infty$ since $N\rightarrow\infty$ and $K, \kappa$ are fixed. Suppose $B$ is a box in $\mathbb R^2$ and $B\cap\mathbb Z^2\neq\emptyset$, we denote the lower left corner of $B\cap\mathbb Z^2$ by $z_B$. For each path $P$ in $\mathbb Z^2$, denote by $x_P$ and $y_P$ the starting and ending vertices of $P$, respectively. Denote $\|P\|=\|x_P-y_P\|$. \subsection{Outline of the proof} The general proof strategy we employ in this paper is multi-scale analysis, which is a classic and powerful method in the percolation theory; see for instance \cite{MR1378847,MR1624084,MR3417515,MR3947326}. In order to apply it to prove \eqref{eq:complement-event}, it requires us to combine a contour argument analogous to \cite[Proposition 4]{MR3800790}, which plays an initial role, with the induction analysis analogous to \cite[Lemma 4.4]{MR3947326}. The former is quite similar to \cite{MR3800790}, while the latter is hard in this paper. The main difficulty lies in planning a proper induction strategy and tackling the fluctuation of the harmonic functions in all scales. Section \ref{sec:2-pre} is devoted to preliminaries, for the sake of the reader. We will list basic results about the DGFF, and show some facts required in later proofs. We will also review the tree structure of a path constructed in \cite{MR3947326}. Roughly speaking, a path $P$ in scale $K^j$, i.e. $\| P \|$ is comparable to $K^j$, is associated with a tree $\mathcal{T}_P$ of depth $j$. Nodes at level $r$ in $\mathcal T_P$ are identified as disjointed sub-paths of $P$ in scale $K^{j-r}$, and the parent/child relation of nodes corresponds to path/sub-path relation. Tame paths are those looking like straight lines, and untamed ones are those looking like curves (see Definition \ref{def:tame}). Then, the fact is that untamed nodes are rare in $\mathcal{T}_P$ for all $P \in \mathcal{P}^{\kappa, \epsilon} $ \cite[Proposition 3.6]{MR3947326}, where \[ \mathcal{P}^{\kappa, \epsilon} = \left\{P: P \text { is a path in } V_{N},\|P\| \geq \kappa N, \text { and }|P| \leq N^{1+\epsilon}\right\} \] is defined in \eqref{Eq.defnPke}, and we drop the subscript $N$ for brevity in the context below. Therefore, it remains to show that it is unlikely that tame nodes are all $\lambda$-open, which is actually the essential ingredient of Theorem~\ref{thm:1.1}. In Section \ref{sec:3-open-path}, we will deal with the contour argument. Concretely, we will show that the probability of there existing a tame and open path started in a fixed box decays stretched-exponentially in $K$ (see Theorem \ref{thm:r=0}). To carry this out, note that the existence of a tame and open path implies that a parallelogram $D$ with aspect ratio $O(K)$ has an open crossing. Next, we cut $D$ into $O(\sqrt K)$ sections uniformly, and extract a sub-parallelograms $D_i$ with aspect ratio $O(1)$ from the middle of each section (see Figure \ref{fig:LDP}). Then, exponential decay follows from the following two facts. One is that with positive probability, a parallelogram with aspect ratio $O(1)$ has no open crossings (Lemma~\ref{lem:para-crossing}). The other is that the Gaussian values in different $D_i$'s are roughly independent. In Section \ref{sec:Multi-scale analysis}, we will deal with the induction analysis. Recall that $P$ of scale $K^j$ corresponds to a tree $\mathcal T_P$ of depth $j$. Thus the ratio of tame and open leaves in $\mathcal T_P$ is the average of those in $\mathcal T_{P^{(i)}}$'s, where $P^{(i)}$'s are the children of $P$ and are of number at least $K$. Since $K$ is chosen large, one can apply a large deviation analysis (see Theorem \ref{thm:r=1} and Theorem \ref{thm:xi-bound}). However, we will encounter some technicalities during the proof. Concretely, we need to control the fluctuation of harmonic functions at all scales in an efficient way, so that one can translate the open property into a demand on the GFF at every sub-scale. To this goal, for each level $0\le r\le j$, corresponding to scale $K^{j-r}$, we choose the threshhold $\varepsilon_r$ (see~\eqref{eq:epsilon}) to be large enough to make sure that the harmonic functions at scale $K^{j-r}$ exceed $\varepsilon_r$ with probability decaying sufficiently fast (see Lemmas~\ref{lem:E0}, \ref{lem:E-1} and \ref{lem:E-r}), but on the other hand, $\varepsilon_r$ is not too large in the sense that $\varepsilon_r$'s is a summable sequence. The balance of these two parts ensures that our strategy works. \section{Preliminaries}\label{sec:2-pre} There are some basic facts about the two-dimensional discrete Gaussian free field which will be used intensively throughout this paper. For completeness, we will introduce them in Section \ref{subsec:DGFF}. In Section \ref{subsec:tree-structure}, we will collect the results we need from \cite{MR3947326}, including the tree structure of a path (Proposition \ref{prop:tree}) and the upper bound on the total flow through untamed nodes in the associated tree (Lemma \ref{lem:untame-flow}). \subsection{Properties of two-dimensional DGFF}\label{subsec:DGFF} In this section, we give a rigorous definition for the DGFF and review some standard estimates about the DGFF. Let $B\subseteq \mathbb Z^2$ be finite and non-empty. Denote by $\{ \eta^B(v): v\in B \}$ the DGFF on $B$ with Dirichlet boundary conditions. It is a mean-zero Gaussian process that vanishes on the boundary $\partial B=\{u\in B:\|u-v\|=1 \text{ for some } v\in B^c\}$, with covariance given by \[ \mathbb E \eta^B(u)\eta^B(v)=G_B(u,v) \quad \text{ for } u,v\in B, \] where $G_B(u,v)$ is the Green's function associated with a simple random walk in $B$, i.e., the expected number of visits to $v$ before reaching $\partial B$ for a discrete simple random walk started at $u$. Without loss of generality, we always assume $\eta^B|_{B^c}=0$. To eliminate boundary issues, we will need to consider vertices that have at least an appropriate distance from the boundary. For this purpose, fix $\chi=\frac{1}{10}$, and if $B\subseteq\mathbb Z^2$ is a box of side length $L$, define the box $B^{\chi}:=\{z\in B:d_{\infty}(z,\partial B)>\chi L\}.$ The next lemma says that the DGFF is log-correlated, which can be found in \cite[Eqaution (4)]{MR3947326}. \begin{lem}\label{lem:log-corr} Suppose that $B\subseteq\mathbb Z^2$ is a box of side length $L$. There is a universal constant $C_1>0$ such that \[ \left|\mathbb E\left(\eta^B(u)\eta^B(v)\right)-\frac{2}{\pi}\log\frac{L}{|u-v|_{\infty}\vee 1}\right|\le C_1 \quad \text{for all } u,v \in B^{\chi}. \] \end{lem} The next lemma is the well-known Markov property of the DGFF. A version can be found in \cite[Section 2.2]{MR3947326}. \begin{lem}\label{lem:DMP} Let $D$ be a finite subset of $\mathbb Z^2$, and $B\subseteq D$. Let $\eta^D$ be the DGFF on $D$, $H^B$ be the conditional expectation of $\eta^D$ given $\eta^D|_{B^c\cup\partial B}$. Then \[ \eta^B:=\eta^D-H^B \] is a version of the DGFF on B, and it is independent of $H^B$. In other words, $\eta^D=\eta^B\oplus H^B$ is an orthogonal decomposition. \end{lem} For the next lemma, we quote a version suited to our needs, which follows straightforwardly from the version in \cite[Lemma 3.10]{MR3433630}. \begin{lem}\label{lem:H^2} In addition to the assumptions in Lemma \ref{lem:DMP}, we further assume that $B$ is a box of side length $L$. Then \begin{equation} \mathbb E\left( H^B(x)-H^B(y) \right)^2\le C_2\frac{|y-x|_{\infty}}{L}\quad \text{ for all } x,y \in B^{\chi}, \end{equation} where $C_2>0$ is a universal constant. \end{lem} Next, we estimate the difference of harmonic functions. It will be intensively used for the rest parts of this paper. \begin{lem}\label{lem:fluct-H} In addition to the assumptions in Lemma \ref{lem:H^2}, we further assume that $U$ is a box of side length $\ell$ in $B^{\chi}$. There exists a universal constant $C_3>0$ such that if $\varepsilon\ge C_3\sqrt{\frac{\ell}{L}}$, then for all $z\in U$, \[ \mathbb P\Big(\left|H^B(x)-H^B(z)\right|\ge \varepsilon \text{ for some } x\in U \Big)\le 4\exp\left\{-\frac{\varepsilon^2 L}{8C_2\ell}\right\}. \] \end{lem} We need the following two lemmas to prove Lemma \ref{lem:fluct-H}. \begin{lem}[{Dudley’s inequality, \cite[Lemma 4.1]{MR1088478}}]\label{lem:dudley} Let $U\subseteq \mathbb Z^2$ be a box of side length $\ell$ and $\{G_w: w\in U\}$ be a mean zero Gaussian field satisfying \[ \mathbb{E}\left(G_{z}-G_{w}\right)^{2} \leq|z-w|_{\infty} / \ell \quad \text { for all } z, w \in U. \] Then $\mathbb E\max_{w\in U}G_w\le C_4$, where $C_4>0$ is a universal constant. \end{lem} \begin{lem}[{Borell–Tsirelson inequality, \cite[Lemma 7.1]{MR3184689}}]\label{lem:borell} Let $\{ G_z: z\in X\}$ be a Gaussian field on a finite index set $X$. Set $\sigma^2=\max_{z\in X}\mathrm{Var}(G_z)$. Then \[ \mathbb{P}\left(\left|\max _{z \in X} G_{z}-\mathbb{E} \max _{z \in X} G_{z}\right| \geq a\right) \leq 2 e^{-\frac{a^{2}}{2 \sigma^{2}}} \quad \text { for all } a>0. \] \end{lem} \begin{proof}[Proof of Lemma \ref{lem:fluct-H}] Note that $U\subseteq B^{\chi}\subseteq B\subseteq D$. Fix $z\in U$. For $x\in U$, let $G_x=H^B(x)-H^B(z)$. By Lemma \ref{lem:H^2}, for $x,y\in U$, \begin{gather} \mathbb{E}\left(G_{x}-G_{y}\right)^{2} \leq C_2\frac{|y-x|_{\infty}}{L} \le \frac{C_2\ell}{L}\cdot\frac{|y-x|_{\infty}}{\ell}, \label{eq:Gx-Gy}\\ \mathrm{Var}(G_x)\le C_2\frac{|x-z|_{\infty}}{L}\le \frac{C_2\ell}{L}. \label{eq:Gx} \end{gather} By \eqref{eq:Gx-Gy} and Lemma \ref{lem:dudley}, we have $\mathbb E\max_{x\in U} G_x\le \frac{C_3}{2}\sqrt{\frac{\ell}{L}}$, where $C_3=2C_4\sqrt{C_2}$. Combining the symmetry of Gaussian distribution, we have \begin{align*} &\mathbb P\Big(\left|H^B(x)-H^B(z)\right|\ge \varepsilon \text{ for some } x\in U \Big)\\ \le&2\mathbb P\Big(G_x\ge \varepsilon \text{ for some } x\in U \Big) \le 2\mathbb P\left( \max _{x \in U} G_{x}-\mathbb{E} \max _{x \in U} G_x\ge \varepsilon-\frac{C_3}{2}\sqrt{\frac{\ell}{L}} \right). \end{align*} Noting that $\varepsilon\ge C_3\sqrt{\frac{\ell}{L}}$ and applying Lemma \ref{lem:borell}, we obtain \begin{align*} &\mathbb P\Big(\left|H^B(x)-H^B(z)\right|\ge \varepsilon \text{ for some } x\in U \Big)\\ \le&2\mathbb P\left( \max _{x \in U} G_{x}-\mathbb{E} \max _{x \in U} G_x\ge \frac{\varepsilon}{2}\right) \le 4 \exp\left\{-\frac{\varepsilon^{2}}{8 \sigma^{2}}\right\}, \end{align*} where $\sigma^2=\max_{x\in U}\mathrm{Var}(G_x)\le\frac{C_2\ell}{L}$ by \eqref{eq:Gx}. This concludes the lemma. \end{proof} We will need the following standard estimates on simple random walks. We refer the reader to \cite[Lemma 1]{MR3800790} for a similar derivation. \begin{lem}\label{lem:green's} Let $\ell_1>0$, $\ell_2\ge\ell_1+2$ and $z\in \mathbb Z^2$. Suppose $V_{\ell_1}(z)\subseteq D\subseteq V_{\ell_2}(z)$. Then for all $u\in\partial V_{\ell_1}(z)$, \[ \sum_{v\in \partial V_{\ell_1}(z)} G_{D}(u,v)\le\sum_{v\in \partial V_{\ell_1}(z)} G_{V_{\ell_2}(z)}(u,v)\le 2(\ell_2-\ell_1). \] \end{lem} \subsection{The tree structure associated with a path}\label{subsec:tree-structure} We now briefly recall some facts from \cite[Section 3]{MR3947326} in this section. For an integer $r\ge 1$, let \[ \mathcal{B D}_{r}=\left\{\left[a r-\frac{1}{2},(a+1) r-\frac{1}{2}\right] \times\left[b r-\frac{1}{2},(b+1) r-\frac{1}{2}\right]: a, b \in \mathbb{Z}\right\}. \] Note that $(B\cap\mathbb Z^2)$'s partition $\mathbb Z^2$, where $B$ is taken over $\mathcal{B D}_{r}$. Define the sets of paths \[ {\mathcal{S} \mathcal{L}_{0}:= \mathbb{Z}^{2}}, \ \mathcal{S} \mathcal{L}_{j}:=\left\{P: 1 \leq \frac{1}{K^{j}}\|P\| \leq 1+\frac{1}{K}, P \subseteq B\left(x_{P},\|P\|\right)\right\} \quad \text { for all } j \geq 1, \] recalling $x_P$ and $y_P$ are the two ends of $P$. If $P\in \mathcal{SL}_j$, $P$ is said to be in scale $K^j$. \begin{defi}\label{def:tame} For $j\ge0$ and each $P\in \mathcal{SL}_{j+1}$, let \centerline{$E(P):=\left\{z \in \mathbb{R}^{2}:\left\|x_{P}-z\right\|+\left\|y_{P}-z\right\| \leq\left(1+\frac{2}{K^{2}}\right)\|P\|\right\},$} \centerline{and $\tilde{E}(P)=\left\{z \in \mathbb{R}^{2}: d(z, E(P)) \leq 4 K^{j}\right\}$.} \noindent A path $P$ is said to be tame if $P\subseteq \tilde{E}(P)$ and untamed otherwise. \end{defi} Note that only when a path is in scale $K^j$ with $j\ge 1$ could we say it is tame or untamed. \begin{prop}[{\cite[Proposition 3.1]{MR3947326}}]\label{prop:tree} Suppose that $j\in[m-2]$ and $P\in\mathcal{SL}_{j+1}$. Then, there exists $\ell\in[K^j,(1+\frac1K)K^j]$, a positive integer $d$, and disjoint child-paths $P^{(i)}$of $P$ for $i\in [d]$ such that the following hold. \begin{itemize} \item[(a)] $d\ge K$. \item[(b)] Each box in $\mathcal{BD}_{K^j}$ is visited by at most $12$ sub-paths of the form $P^{(i)}, i \in[d]$. \item[(c)] $P^{(i)} \in \mathcal{S} \mathcal{L}_{j}$ for each $i \in[d]$. \end{itemize} Furthermore, $d\ge \frac12\|P\|$ for $j=0$; and one can extract $d_0$ disjoint sub-paths in $\mathcal{S} \mathcal{L}_{m-1}$ from $P$ with $\|P\| \geq \kappa N$ such that (b) holds with $j=m-1$, where $ d_{0}:=\left\lfloor\frac{\kappa N}{K^{m-1}}\right\rfloor \geq K. $ \end{prop} Fix $P\in\mathcal{SL}_{j}$. The tree $\mathcal{T}_P$ associated with $P$ is constructed as follows. The nodes of $\mathcal{T}_P$ correspond to a family of sub-paths of $P$, where the parent/child relation in $\mathcal{T}_P$ corresponds to path/sub-path relation in the plane by Proposition \ref{prop:tree}. In particular, the root denoted by $\rho$ corresponds to $P$ and the leaves denoted by $\mathcal{L}$ correspond to vertices on $P$. Denote the level of a node $u$ by $L(u)$ with $L(\rho)=0$ and identify $u$ as a sub-path in $\mathcal{SL}_{j-L(u)}$ with $d_u$ children. Each node as a path enjoys the properties in Proposition \ref{prop:tree}. Especially, $P$ with $\|P\| \geq \kappa N$ is associated with a tree $\mathcal{T}_P$ of depth $m$. Let $\theta_P$ be the unit uniform flow on $\mathcal{T}_P$ from $\rho$ to $\mathcal{L}$, with $\theta_{P}(\rho)=1$ and $\theta_{P}(v)=\frac{1}{d_{u}} \theta_{P}(u)$ if $v$ is a child of $u$. For $\delta\in(0,1)$, let \begin{equation}\label{eq:kap-delta-K} \mathcal{P}^{\kappa, \delta, K}:=\left\{P: P \text { is a path in } V_{N},\|P\| \geq \kappa N \text { and }|P| \leq N^{1+\frac{\delta}{K^{2} k}}\right\}. \end{equation} \begin{lem}[{\cite[Proposition 3.6]{MR3947326}}]\label{lem:untame-flow} For each $P\in\mathcal{P}^{\kappa, \delta, K}$, \[ \sum_{u: 1 \leq L(u) \leq m-1} \theta_{P}(u) 1_{ \{u \text{ is untamed}\}} \leq 2 \delta m. \] \end{lem} At the end of this section, we give some definitions that are similar to those in \cite{MR3947326}. Define \[ \mathcal{P}_{j}\left(B\right) :=\left\{P \in \mathcal{S} \mathcal{L}_{j} : x_{P} \in B \right\}, \] \[ T_j(B):=\left\{ P\in\mathcal{P}_{j}(B): P\text{ is tame} \right\}, \] \[ \mathrm{END}_{j} :=\big\{B: B \in \mathcal{B} \mathcal{D}_{K^{j-2}}\text{ and }B\cap V_{N} \neq \emptyset\big\}. \] For $B, B' \in \mathrm{END}_j$, define $T_j(B,B'):=\{ P\in T_j(B): y_P\in B' \}$. \section{Tame paths are unlikely to be open}\label{sec:3-open-path} In this section, we will use a contour argument to show that the probability of there existing a tame and open path started in a fixed box decays stretched- exponentially in $K$ (see Theorem \ref{thm:r=0}). To this goal, we start by showing that, with positive probability, a parallelogram with aspect ratio $O(1)$ has no open crossings (Lemma \ref{lem:para-crossing} ) in Section \ref{subsec:good-para}. Then in Section \ref{subsec:proof-r=0}, we give the proof of Theorem \ref{thm:r=0} by using Lemma \ref{lem:para-crossing} and estimates about harmonic functions in Lemma \ref{lem:fluct-H}. Recall that we call a vertex $x\in V_N$ is $\lambda$-open if $\big|\eta^{V_{2N}}(x)\big|\le \lambda$. Next, we extend this definition a bit. For $V\subseteq V_{2N}$ and $\alpha\in\mathbb R$, we say a vertex $x\in V$ is $(V, \lambda, \alpha)$-open if $|\eta^V(x)+\alpha|\le \lambda$. A path $P\subseteq V$ is said to be $(V, \lambda, \alpha)$-open if so is every vertex in $P$. For brevity, we will occasionally use open to mean $\lambda$-open or $(V, \lambda, \alpha)$-open according to the context. \begin{thm}\label{thm:r=0} For any $\lambda_0>0$, let $\lambda\ge\lambda_0$. There exists $c=c(\lambda_0)>0$ such that the following holds for all $K\ge K_0(\lambda):=e^{c\lambda^2}$. Suppose that $j\in [m-1]$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\alpha\in\mathbb R$. Then, \begin{equation} \mathbb P\big(P \text{ is } (V, \lambda,\alpha)\text{-open for some } P\in T_j(B)\big) \le e^{-0.01\sqrt{K}}. \end{equation} \end{thm} \subsection{Good parallelograms}\label{subsec:good-para} In this section, we consider a closed parallelogram $D$ with corners $(a,b)$, $(a+l,b+h)$, $(a+l,b+h+w)$ and $(a,b+w)$, where $(a,b)\in\mathbb R^2$, $l\ge w\ge 10$ (here $10$ is a somewhat arbitrary choice), and $l\ge h\ge 0$. Especially, we say $D$ is \emph{good} if $a,l\in\mathbb Z$ and $l=16 w$. We call $l, w$, $\theta=\arctan \frac{h}{l}$, and \[ v_0=\left(\left\lfloor\frac{a+h+l-7w\sin^2\theta}{2}\right\rfloor , \left\lfloor \frac{b+h-l+7w\sin\theta\cos\theta}{2} \right\rfloor\right) \] respectively the length, width, angle and anchor of $D$. Note that $\theta\in [0,\frac{\pi}{4}]$ and $v_0\in\mathbb Z^2$. By crossing of good $D$ we mean a path in $D$ connecting the left and right sides of $D$ (see Figure \ref{fig:para}). Let $V$ be a finite set in $\mathbb Z^2$ and $D\cap\mathbb Z^2\subseteq V$, then let $\mathcal{A}(D,V,\lambda,\alpha)$ be the event that there exists a $(V,\lambda,\alpha)$-open crossing of $D$. The reasoning of the following lemma is analogous to that of \cite[Proposition 4]{MR3800790}. \begin{figure} \caption{$D$ and its crossing.} \label{fig:para} \end{figure} \begin{lem}\label{lem:para-crossing} For any $\lambda_0>0$, let $\lambda\ge\lambda_0$. There exists $c'=c'(\lambda_0)>0$ such that if \begin{equation}\label{eq:L-w} L/w\ge e^{c'\lambda^2}, \end{equation} then for any good parallelogram $D$ with width $w$ and anchor $v_0$, and any $\alpha\in\mathbb{R}$, we have \begin{equation}\label{eq:A} \mathbb P\Big( \mathcal{A}\big(D,V_L(v_0),\lambda,\alpha\big) \Big)\le \frac78. \end{equation} \end{lem} \begin{proof} Rotate $D$ around $v_0$ counterclockwise by $i\pi/2$ and denote it by $D_i$, noting $D_0=D$. Since $D$ is good, by our appropriate choice of the anchor $v_0$, $\cup_{i=0}^{3}D_i$ forms an annulus $R$ centered at $v_0$ in $V_{4l}(v_0)$, surrounding $V_{2w}(v_0)$ (see Firgure \ref{fig:rotation}). Let $\mathfrak{C}$ be the collection of all contours in $R$. Here, by contour we mean a path with two endpoints coinciding. We consider a natural partial order on $\mathfrak{C}$: $\mathbf C_1\preceq \mathbf C_2$ if $\mathbf C_1^*\subseteq \mathbf C_2^*$, where $\mathbf C^*$ is the collection of vertices that are surrounded by $\mathbf C$. \begin{figure} \caption{$R$ is the (green) annulus. (Red) curves are open crossings of $D_i$'s.} \label{fig:rotation} \end{figure} Denote $V:=V_L(v_0)$ and $\mathcal{A}_i:=\mathcal{A}\big(D_i,V,\lambda,\alpha\big)$, where correspondingly by crossing of $D_i$ for odd $i$, we mean a path in $D_i$ connecting the top an bottom sides of $D_i$. On the event $\cap_{i=0}^3 \mathcal{A}_i$, we can find at least one $(V,\lambda,\alpha)$-open contour in $R$. Let $\mathscr{C}$ be the random subset of $\mathfrak{C}$ consisting of all open contours in $R$. Then, the partial order above generates a well-defined unique maximum contour on $\mathscr{C}$, which is denoted by $\mathcal C$. To the goal, it remains to show \begin{equation}\label{eq:C*} \mathbb P\big( \mathscr{C}\neq\emptyset\big)\le \frac12. \end{equation} Assuming \eqref{eq:C*} holds, noting $\cap_{i=0}^3 \mathcal{A}_i\subseteq\{ \mathscr{C}\neq\emptyset \}$ and $\mathbb P\left( \mathcal{A}_i\right)=\mathbb P\left( \mathcal{A}_0 \right) $ by rotation invariance, one has $4\big(1-\mathbb P(\mathcal{A}_0)\big)\ge 1-\mathbb P\big(\cap_{i=0}^3 \mathcal{A}_i\big)\ge1/2$, completing the proof. Next, we will prove \eqref{eq:C*}. Denote \[ X:=\frac{1}{|\partial V_{2w}(v_0)|}\sum_{u\in\partial V_{2w}(v_0)}\left(\eta^{V}(u)+\alpha\right). \] By Lemma \ref{lem:log-corr}, if \eqref{eq:L-w} is satisfied for sufficiently large $c'$, then \[ \mathbb E\eta^{V}(u)\eta^{V}(v)\ge \frac{2}{\pi}\log\left(\frac{L}{2w}\right)-C_1\ge\frac{1}{\pi}\log\left(\frac{L}{2w}\right) \ \text{ for all } u,v\in \partial V_{2w}(v_0). \] It follows that \begin{equation}\label{eq:var-X} \mathrm{Var}(X)\ge \frac{1}{\pi}\log\left(\frac{L}{2w}\right). \end{equation} For a deterministic contour $\mathbf C\in\mathfrak{C}$, let $\hat{\mathbf {C}}=(V\backslash\mathbf C^*)\cup\mathbf C$ be the set of points outside $\mathbf C$ but within $V$. Denote $ \mathcal{F}_{\hat{\mathbf C}}:=\sigma\big\{ \eta^{V}(x): x\in \hat{\mathbf C}\big\} $ and $Y:=X-\mathbb E\big(X|\mathcal{F}_{\hat{\mathbf C}}\big)$. Then, \begin{equation}\label{eq:Var-Y} \mathrm{Var}(Y)=\frac{1}{|\partial V_{2w}(v_0)|^2}\sum_{u,v\in\partial V_{2w}(v_0)}G_{\mathbf C^*}(u,v)\le 16, \end{equation} where we have used $ \sum_{v\in\partial V_{2w}(v_0)}G_{\mathbf C^*}(u,v)\le 2(4l-2w) $ by setting $D=\mathbf C^*, \ell_1=2w,\ell_2=4l$ in Lemma \ref{lem:green's}. Note that for each $u\in\partial V_{2w}(v_0)$, \begin{equation}\label{eq:S-tau} \mathbb E\big(\eta^{V}(u)|\mathcal{F}_{\hat{\mathbf C}}\big)=\sum_{v\in \mathbf C}\mathbb P^u(S_{\tau}=v)\cdot\eta^{V}(v), \end{equation} where $\{S_n\}$ is a simple random walk on $\mathbb Z^2$ started from $u$, and $\tau$ is the first time it hits $\mathbf C$. By the definition that $\mathcal{C}$ is the outermost open contour in $\mathscr{C}$, one has $\{ \mathcal{C}=\mathbf C \}\in \mathcal{F}_{\hat{\mathbf C}}$. On the event $\{\mathcal{C}=\mathbf C\}$, we have $|\eta^{V}(v)+\alpha| \le \lambda$ for all $v\in \mathbf C$. Combined with \eqref{eq:S-tau}, it gives that \[ \Big|\mathbb E\left(\eta^{V}(u)+\alpha\big|\mathcal{F}_{\hat{\mathbf C}}\right)\Big| \le \sum_{v\in \mathbf C}\mathbb P^u(S_{\tau}=v)\cdot \big|\eta^{V}(v)+\alpha\big| \le \lambda \ \text{ for all } u\in \partial V_{2w}(v_0), \] implying $ \big|\mathbb E\big(X|\mathcal{F}_{\hat{\mathbf C}}\big)\big| \le \lambda. $ Consequently, $|Y|\le\lambda$ implies $|X|=\big|\mathbb E\big(X|\mathcal{F}_{\hat{\mathbf C}}\big)+Y\big|\le 2\lambda$. Noting that $Y$ and $\mathcal{F}_{\hat{\mathbf C}}$ are independent, \begin{equation*} \mathbb P\big(|X|\le 2\lambda\big| \mathcal{C}=\mathbf C\big) \ge\mathbb P\big( |Y|\le\lambda\big|\mathcal{C}=\mathbf C \big) =\mathbb P\big(|Y|\le\lambda\big). \end{equation*} It follows that \begin{equation}\label{eq:cal-C} \mathbb P\big(\mathscr C\neq\emptyset\big) =\sum_{\mathbf C\in\mathfrak{C}}\mathbb P\big(\mathcal C=\mathbf C\big) =\sum_{\mathbf C\in\mathfrak{C}}\frac{\mathbb P(|X|\le 2\lambda, \mathcal{C}=\mathbf C)}{\mathbb P(|X|\le 2\lambda | \mathcal{C}=\mathbf C)}\le \frac{\mathbb P(|X|\le 2\lambda)}{\mathbb P(|Y|\le \lambda)}. \end{equation} Let $\phi_{\sigma^2}$ be the probability density function of a centered Gaussian random variable with variance $\sigma^2$. Set $\sigma_1^2:=\frac{1}{\pi}\log\left(\frac{L}{2w}\right)$. By \eqref{eq:var-X}, \eqref{eq:Var-Y} and \eqref{eq:cal-C}, \[ \mathbb P\big(\mathscr C\neq\emptyset\big)\le \frac{\phi_{\sigma_1^2}(0)\cdot4\lambda}{\phi_{16}(\lambda_0)\cdot2\lambda_0}. \] Choose large $c'=c'(\lambda_0)$ such that $\sigma_1$ is large enough to make sure the right hand side above is less than $\frac12$. This completes the proof of the lemma. \end{proof} \subsection{Proof of Theorem \ref{thm:r=0}}\label{subsec:proof-r=0} To prove Theorem \ref{thm:r=0}, it suffices to prove the following proposition. \begin{prop}\label{prop:BB'} Let $K\ge K_0(\lambda)$. For all $B, B' \in \mathrm{END}_j$ such that $T_j(B,B')\neq\emptyset$, \begin{equation} \mathbb P\Big(P \text{ is } (V,\lambda,\alpha)\text{-open for some } P\in T_j(B,B')\Big) \le e^{-0.015\sqrt{K}}. \end{equation} \end{prop} \begin{proof}[Proof of Theorem \ref{thm:r=0}, assuming Proposition \ref{prop:BB'}] Note that for $B\in\mathrm{END}_j$, one can find at most $K^5$ boxes $B'$'s in $\mathrm{END}_j$ such that $T_j(B,B')\neq\emptyset$. By a union bound, for $K\ge K_0(\lambda)$, \begin{equation*} \mathbb P\Big(P \text{ is } (V,\lambda,\alpha)\text{-open for some } P\in T_j(B)\Big) \le K^5e^{-0.015\sqrt{K}}\le e^{-0.01\sqrt{K}}. \end{equation*} This completes the proof. \end{proof} We formulate ingredients to prove Proposition \ref{prop:BB'} in the remaining context of this section. In what follows, we will always assume that $B, B' \in \mathrm{END}_j$ and $T_j(B,B')\neq\emptyset$. Let $(x,y)$ and $(x',y')$ be lower-left corners of $B$ and $B'$, respectively. Without loss of generality, suppose that $x'-x\ge y'-y\ge 0$. Then, it is not hard to show the following geometric facts hold for $K\ge 2^{32}$ (see Figure \ref{fig:LDP}). \begin{enumerate} \item[(G1)] One can find a parallelogram $D$ with width $w=20K^{j-1}$ and length $K^j/4$ such that every path in $T_j(B,B')$ contains a crossing of $D$, recalling Definition \ref{def:tame} for the tame path and noting $B, B' \in \mathrm{END}_j$; \item[(G2)] One can extract good $D_i$'s from $D$ for $i\in[\sqrt{K}/8]$ with width $w=20K^{j-1}$ and length $l=16w$ such that $D_i\subseteq V_{4l}(v_i)\subseteq V_i$ for each $i$, where $v_i$ is the anchor of $D_i$, $L=K^{j-1/2}$, $V_i:=V_L(v_i)$, and $V_i$'s are disjoint. \item[(G3)] Let $U=\cup_i V_i$. Then $U\subseteq V_{4K^j}(z_{B})\subseteq V$, where $V$ is the set in the statement of Theorem \ref{thm:r=0}. \end{enumerate} Set $\mathcal{F}_\partial:=\{ \eta^{V}(z):z\in (V\backslash U)\cup\partial U\}$. Let $H_{\partial}(z):=\mathbb E\left(\eta^{V}(z)\big|\mathcal{F}_\partial\right)$ and $ \eta^{V_i}(z):=\eta^{V}(z)-H_{\partial}(z) \text{ for all } z\in V_i. $ By Markov property (Lemma \ref{lem:DMP}), we know that $ \eta^{V_i}=\{\eta^{V_i}(z):z\in V_i \} $ is a DGFF on $V_i$ for each $i\in [\sqrt{K}/8]$, and $\eta^{V_i}$'s are mutually independent by (G2), and they are independent of $H_{\partial}$. Set $ \varepsilon_0=100\sqrt{C_2}, $ where $C_2$ is defined in Lemma \ref{lem:H^2}. Denote \begin{equation} \mathcal{E}_0=\Big\{ \big|H_{\partial}(z)-H_{\partial}(v_{i})\big|\ge \varepsilon_0 \text{ for some } i\in [\sqrt{K}/8] \text{ and }z\in D_i \Big\}. \end{equation} Set $ C_5=(2\vee C_4)^{32}, $ where $C_4$ is defined in Lemma \ref{lem:dudley}. \begin{lem}\label{lem:E0} Let $K\ge C_5$. Then, $\mathbb P\big(\mathcal{E}_0\big)\le e^{-0.5\sqrt{K}}$. \end{lem} \begin{proof} Recall that $w=20K^{j-1}$, $l=16w$, $L=K^{j-1/2}$, and $C_3=2C_4\sqrt{C_2}$. For $K\ge C_5$, we have $C_3\sqrt{\frac{4l}{L}}\le\varepsilon_0$. Setting $\ell=4l$ and $\varepsilon=\varepsilon_0$ in Lemma \ref{lem:fluct-H}, we have \begin{align*} &\mathbb P\Big(\big|H_{\partial}(z)-H_{\partial}(v_{i})\big|\ge \varepsilon_0 \text{ for some } z\in D_i \Big)\\ \le&\mathbb P\Big(\big|H_{\partial}(z)-H_{\partial}(v_{i})\big|\ge \varepsilon_0 \text{ for some } z\in V_{4l}(v_i)\Big) \le 4\exp\left\{-\frac{\varepsilon_0^2 L}{32C_2 l} \right\}\le e^{-0.9\sqrt{K}}, \end{align*} where we have used (G2) in the first inequality, and $K\ge C_5\ge 2^{32}$ in the last inequality. By a union bound, $ \mathbb P\big(\mathcal{E}_0\big)\le \frac18 \sqrt{K}e^{-0.9\sqrt{K}}\le e^{-0.5\sqrt{K}}. $ \end{proof} \begin{figure} \caption{$D$ is the (red) parallelogram, with aspect ratio $O(K)$. The parallelograms with (black) shading are $D_i$'s, with aspect ratio $O(1)$. The (blue) squares are $V_i$'s.} \label{fig:LDP} \end{figure} \begin{proof}[Proof of Proposition \ref{prop:BB'}] Let $w=20K^{j-1}$ and $L=K^{j-1/2}$ as above. Let $c=c(\lambda_0)$ be a constant such that \begin{equation}\label{eq:K_0} K_0(\lambda):=e^{c\lambda^2}\ge 400e^{2c'(\lambda+\varepsilon_0)^2}\vee C_5. \end{equation} For $K\ge K_0(\lambda)$, we have $ L/w\ge e^{c'(\lambda+\varepsilon_0)^2}. $ Then by (G2) and Lemma \ref{lem:para-crossing}, for each $\alpha$, \begin{equation}\label{eq:first-term} \mathbb P\big( \mathcal A\left(D_i,V_i,\lambda+\varepsilon_0,\alpha\right) \big)\le \frac78 \quad \text{ for all } i, \end{equation} recalling that $\mathcal A\left(D_i,V_i,\lambda+\varepsilon_0,\alpha\right)$ is the event that there exists a $\left(V_i,\lambda+\varepsilon_0,\alpha\right)$-open crossing of $D_i$ in $V_i$. Note that for $z\in D_i$, \[ \eta^{V}(z)=\eta^{V_i}(z)+H_{\partial}(z) =\big(\eta^{V_i}(z)+H_{\partial}(v_i)\big)+\big(H_{\partial}(z)-H_{\partial}(v_i)\big). \] By the triangle inequality, if $|H_{\partial}(z)-H_{\partial}(v_i)|\le\varepsilon_0$ for all $z\in D_i$, then $\left|\eta^{V}(z)+\alpha\right|\le\lambda$ implies $ \big|\eta^{V_i}(z)+\alpha+H_{\partial}(v_i)\big|\le\lambda+\varepsilon_0. $ Thus, on the event $\mathcal{E}_0^c$, \[ \mathcal A\subseteq \bigcap_{i\in [\sqrt{K}/8]}\mathcal A\big(D_i,V,\lambda,\alpha\big) \subseteq \bigcap_{i\in [\sqrt{K}/8]}\mathcal A_i, \] where $\mathcal A:=\mathcal A\big(D,V,\lambda,\alpha\big)$, $\mathcal A_i:=\mathcal A\big(D_i,V_i,\lambda+\varepsilon_0,\alpha+H_{\partial}(v_{i})\big)$, and $H_{\partial}(v_{i})$ is regarded as a constant with respect to the DGFF $\eta^{V_i}$ for independence. Therefore, \begin{equation}\label{eq:mathcal-A} \mathbb P\big( \mathcal A \big)\le \mathbb P\Big( \bigcap_{i}\mathcal A_i \Big)+\mathbb P\big( \mathcal{E}_0 \big), \end{equation} where the intersection is over $i\in [\sqrt{K}/8]$. By \eqref{eq:first-term}, \begin{equation}\label{eq:intersection} \mathbb P\Big( \bigcap_{i}\mathcal A_i \Big) \le \mathbb E\Big(\mathbb P\Big(\bigcap_i\mathcal A_i\Big| \mathcal{F}_\partial\Big)\Big) \le \mathbb E \prod_i \mathbb P\big(\mathcal A_i\big| \mathcal{F}_\partial\big) \le \left(\frac78\right)^{\lfloor\sqrt{K}/8\rfloor}, \end{equation} where we have used the conditional independence of $\mathcal A_i$ given $\mathcal{F}_\partial$. Combining (G1), \eqref{eq:mathcal-A}, \eqref{eq:intersection} and Lemma \ref{lem:E0}, for $K\ge K_0(\lambda)\ge C_5\ge 2^{32}$, \[ \mathbb P\Big(P \text{ is } (V,\lambda,\alpha)\text{-open for some } P\in T_j(B,B')\Big) \le \left(\frac78\right)^{\lfloor\sqrt{K}/8\rfloor}+e^{-0.5\sqrt{K}} \le e^{-0.015\sqrt{K}}. \] This completes the proof. \end{proof} \section{Multi-scale analysis on the hierarchical structure of the path}\label{sec:Multi-scale analysis} In this section, we will prove Theorem \ref{thm:1.1}. It suffices to prove the theorem for $\lambda\ge\lambda_0$ with $\lambda_0>0$ fixed, see Remark~\ref{rem:lambda_0}. Note that if there is a $\lambda$-open path $P$ in $\mathcal{P}^{\kappa,\delta,K}$, then all nodes in $\mathcal{T}_P$ are $\lambda$-open. We will prove that this event has probability tending to $0$ as the depth of the tree $m$ tends to infinity, by showing that tame and open nodes are rare (Theorem \ref{thm:xi-bound} below). Note that untamed nodes are rare by Lemma \ref{lem:untame-flow}. Let $j\in[m-1], B\in \mathrm{END}_j$. For each $P\in\mathcal P_j(B)$, recall that $\mathcal T_P$ is a tree of depth $j$, associated with $P$. Each node $u$ of $\mathcal T_P$ is identified with a sub-path of $P$, which is also denoted by $u$ to lighten notation. Let $\mathcal T_{P,r}$ be the collection of nodes of level $r$. Note that the root has level $0$. For each $u\in \mathcal T_{P,r}$, there is a unique starting box $B_u\in \mathrm{END}_{j-r}$ containing the starting point of $u$. Let $\mathscr A$ be the collection of real functions defined on $\cup_{j\in[m-1]}\mathrm{END}_{j}$, i.e., on all end-boxes. We always assume that $\bar\alpha$ is a real function in $\mathscr A$. Note that for any $P$, $\bar\alpha$ induces a function on $\mathcal T_P$ by setting $\bar\alpha_u:=\bar\alpha\big(B_u\big)$ for each $u\in \mathcal T_{P}$. Let $\theta_P$ be the unit uniform flow on $\mathcal T_P$ from $\rho$ to $\mathcal{L}$ (the definition is just before \eqref{eq:kap-delta-K}), where $\rho$ is the root and $\mathcal{L}$ is the set of leaves. For $\lambda>0, V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, define \begin{gather} Y_{P,r,\lambda,\bar\alpha}:=\sum_{u\in \mathcal T_{P,r}}\theta_P(u)1_{\{u \text{ is tame and } (V,\lambda,\bar\alpha_u)\text{-open}\}}, \label{eq:Y-Pr}\\ \xi_{r,\lambda,\bar\alpha,j,B}:=\max\big\{Y_{P,r,\lambda,\bar\alpha}: P\in \mathcal P_j(B)\big\}.\label{eq:xi-Pr} \end{gather} Recall $\lambda\ge\lambda_0$ and $K_0(\lambda)=e^{c\lambda^2}$ as a function of $\lambda$ for some $c=c(\lambda_0)>0$. Noting that \[ \xi_{0,\lambda,\bar\alpha,j,B}=1_{\left\{P \text{ is } (V,\lambda,\bar\alpha_{\rho})\text{-open for some } P\in \mathcal T_j(B) \right\}}, \] the next corollary restates Theorem \ref{thm:r=0}. \begin{cor}\label{cor:r=0} Suppose $K\ge K_0(\lambda)$, $j\in[m-1]$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$. Then, \begin{equation} \mathbb P(\xi_{0,\lambda,\bar\alpha,j,B}>0)\le e^{-0.01\sqrt{K}}. \end{equation} \end{cor} As for $r=1$, we have a similar result. Set $ \varepsilon_{1}=8\sqrt{C_2}$, $ K_1(\lambda)=K_0(\lambda+\varepsilon_{1}). $ We will prove the following theorem in Section \ref{subsec:4.1}. \begin{thm}\label{thm:r=1} Suppose $K\ge K_1(\lambda)$, $j\in[2,m-1]\cap\mathbb Z$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$. Then, \[ \mathbb P(\xi_{1,\lambda,\bar\alpha,j,B}>\delta) \le e^{-K^{1/8}} \quad \text{ for all } \delta\ge\delta_1:=\frac12. \] \end{thm} Before generalizing Theorem \ref{thm:r=1}, let us set our conventions for constants. Set \begin{equation}\label{eq:beta-c_r} \beta=2^{-9}\ \text{ and }\ c_r=(\beta K)^r. \end{equation} Define $\{\delta_r : r\ge0\}$ to be \begin{gather} \delta_0=0, \quad \delta_1=\frac12; \quad \delta_{r+1}=\delta_r+\Delta_r \ \text{ for all } r\ge 1,\label{eq:delta}\\ \text{ where } \Delta_1=\frac{9\log K}{\beta K^{1/8}}; \quad \Delta_{r+1}=\frac{\log(1+2c_{r})+9\beta^{-1}\log K}{c_{r}}\ \text{ for all } \ r\ge 1.\label{eq:Delta} \end{gather} Set \begin{gather} \varepsilon_0=100\sqrt{C_2}, \quad \varepsilon_1=8\sqrt{C_2}; \quad \varepsilon_{r+1}=4\sqrt{C_2}\beta^{r/2}\ \text{ for all } \ r\ge 1, \label{eq:epsilon}\\ K_{r+1}(\lambda)=K_r(\lambda+\varepsilon_{r+1})=K_0\left(\lambda+\sum_{i=1}^{r+1}\varepsilon_i\right) \ \text{ for all } r\ge 0.\label{eq:Kr} \end{gather} We will prove the following theorem by induction on admissible pair $(r,j)$ in Section \ref{subsec:4.2}. \begin{thm}\label{thm:xi-bound} The following holds for any pair $(r,j)$ satisfying $r\in[2,m-2]\cap\mathbb Z$ and $j\in[r+1,m-1]\cap\mathbb Z$. For all $K\ge K_r(\lambda)$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$, we have \[ \mathbb{P}(\xi_{r,\lambda,\bar\alpha,j,B}>\delta) \le 2e^{-c_{r-1}(\delta-\delta_r)} \quad \text{for all } \delta\ge\delta_r. \] \end{thm} In other words, by choosing $\delta_r$ as the threshold for total flows through level $r$, the overflows above $\delta_r$ will have an exponential decay uniformly in other parameters. Furthermore, as in Lemma \ref{lem:E0}, we will use $\{\varepsilon_r : r\ge 1\}$ to bound the fluctuation of harmonic functions at different levels in Lemma \ref{lem:E-1} ($r=1$) and Lemma \ref{lem:E-r} ($r\ge2$), respectively. \subsection{Proof of Theorem \ref{thm:r=1}}\label{subsec:4.1} We assume $j\in[2,m-1]\cap\mathbb Z$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$ in this section. Define $ \mathcal{P}_{j, d}(B):=\big\{P \in \mathcal{P}_{j}\left(B\right) : d_{P}=d\big\}. $ Denote the child-paths of $P$ by $\{P^{(i)}\}_{i\in[d]}$ if $P\in\mathcal{P}_{j, d}(B)$. Note that $d\ge K$ always holds by (a) of Proposition \ref{prop:tree}. Define \[ \mathrm{END}_{j-1, d} :=\left\{\left\{B_i\right\}_{i \in[d]} \subseteq \mathrm{END}_{j-1} :\left|\left\{i : B_{i} \subseteq \tilde B\right\}\right| \leq 12 \text { for each } \tilde B \in \mathcal{B} \mathcal{D}_{K^{j-1}}\right\}, \] and for each sequence $\mathcal{S} :=\left\{ B_i \right\}_{i \in[d]}\in\mathrm{END}_{j-1, d}$, define \[ \mathcal{P}_{j, \mathcal{S}}(B) :=\big\{P \in \mathcal{P}_{j, d}(B) : P^{(i)} \in \mathcal{P}_{j-1}\left(B_i\right) \text { for all } i \in[d]\big\}. \] Furthermore, define \begin{equation*} \mathrm{END}_{j-1, d}(B):=\big\{\mathcal S\in \mathrm{END}_{j-1,d}: \mathcal{P}_{j, \mathcal{S}}(B)\neq\emptyset \big\}. \end{equation*} For the remainder of this paper, we always assume that $\mathcal{S} :=\left\{ B_i \right\}_{i \in[d]}\in\mathrm{END}_{j-1, d}(B)$ and $d\ge K$. Denote for brevity \begin{equation}\label{eq:z-V} z_i:=z_{B_{i}} \ \text{ and } \ V_i:=V_{K^{j-7/8}}(z_i) \ \text { for all } i \in[d]. \end{equation} Note that $V_i\subseteq V_{4K^j}(z_{B})\subseteq V$ for all $i\in[d]$. Let $H_i$ be the conditional expectation of $\eta^V$ given $\eta^V\big|_{ V_i^c\cup\partial V_i }$. By Lemma \ref{lem:DMP}, $\eta^{V_i}:=\eta^V-H_i$ is a DGFF on $V_i$ for each $i\in[d]$. Recall $\varepsilon_{1}=8\sqrt{C_2}$ and define \begin{equation}\label{eq:E_S} \mathcal{E}_{\mathcal S}=\bigcup_{i\in[d]}\Big\{ \big|H_i(x)-H_i(z_i)\big|\ge\varepsilon_{1} \text{ for some } x\in V_{4K^{j-1}}(z_i)\Big\}. \end{equation} For all $P\in\mathcal{P}_{j, \mathcal{S}}(B)$, noting that $P^{(i)}\subseteq V_{4K^{j-1}}(z_i)$ for all $i\in [d]$, and $\eta^{V}(x)=\left(\eta^{V_i}(x)+H_i(z_i)\right)+\big( H_i(x)-H_i(z_i) \big)$ for all $x\in P^{(i)}$; then on the event $\mathcal{E}_{\mathcal S}^c$, by the triangle inequality, $P^{(i)} $ is $(V,\lambda,\alpha_i)$-open implies that it is $\big(V_i,\lambda+\varepsilon_{1},\alpha_i+H_i(z_i)\big)$-open, where $\alpha_i=\bar\alpha(B_i)$ and $H_i(z_i)$ is regarded as a deterministic number with respect to the field $\eta^{V_i}$. Therefore, \[ Y_{P, 1,\lambda, \bar\alpha}\le \frac1d \sum_{i=1}^{d} Y'_{i,0,\lambda+\varepsilon_{1}} \] for all $P\in\mathcal{P}_{j, \mathcal{S}}(B)$ on the event $\mathcal{E}_{\mathcal S}^c$, where $Y'_{i,0,\lambda+\varepsilon_{1}}$ is the indicator function of the event that $P^{(i)}$ is tame and $\left(V_i,\lambda+\varepsilon_{1},\alpha_i+H_i(z_i)\right)$-open. This implies that \begin{equation*} \zeta_{1,\mathcal{S}}\le \frac1d \sum_{i=1}^{d}\xi'_{i} \ \text{ on the event } \mathcal{E}_{\mathcal S}^c, \end{equation*} where $ \zeta_{1,\mathcal{S}} :=\max \big\{Y_{P, 1,\lambda, \bar\alpha}: P \in \mathcal{P}_{j, \mathcal{S}}(B)\big\}, $ and \begin{equation}\label{eq:xi'0} \xi'_{i}:=1_{ \big\{\text{ there exists a } (V_i,\lambda+\varepsilon_{1},\alpha_i+H_i(z_i))\text{-open path in } T_{j-1}\left(B_i\right) \big\} }. \end{equation} It follows that for all $\delta>0$ and $\mathcal S\in\mathrm{END}_{j-1, d}(B)$, \begin{equation}\label{eq:1S-delta} \mathbb P\big(\{\zeta_{1,\mathcal{S}} >\delta\}\cap\mathcal{E}_{\mathcal S}^c\big)\le \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right). \end{equation} Based on an argument similar to \cite[Lemma 4.4]{MR3947326}, we obtain the following lemma. \begin{lem}\label{lem:average-level0} Let $K\ge K_1(\lambda)$. For each $\mathcal{S} :=\left\{ B_i \right\}_{i \in[d]}\in\mathrm{END}_{j-1, d}(B)$, we have \[ \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right)\le e^{ -10^{-4}K^{1/4}\delta d } \quad \text{ for all } \delta\ge\delta_{1}=\frac12. \] \end{lem} \begin{proof} Let $\beta_K=(48K^{1/4})^{-1}$. We will classify $B_i$'s into $\beta_K^{-1}$ groups in the following procedure, such that $V_i$'s in each group are disjoint. Note that if $d_{\infty}(B_{i},B_{i'})\ge 1.5K^{j-7/8}$, then $d_{\infty}(V_i,V_{i'})\ge K^{j-1}$. First, we classify $\mathcal{BD}_{K^{j-1}}$ into $4K^{1/4}=\left( 2K^{j-7/8}/K^{j-1} \right)^2$ families $\tilde{\mathcal G}_s, s\in [4K^{1/4}]$, where $\tilde{\mathcal G}_1$ consists of boxes respectively containing $(2aK^{j-7/8},2bK^{j-7/8})$, $a,b\in\mathbb Z$ and other $\tilde{\mathcal G}_s$'s are its shifts. Let \[ \mathcal{G}_{s}:=\big\{B_{i}: i \in[d],\text{ and } B_{i} \subseteq \tilde B \text{ for some } \tilde B \in \tilde{\mathcal G}_s\big\}. \] Then, by (b) of Proposition \ref{prop:tree}, we can classify each $\mathcal{G}_{s}$ into $12$ groups $\mathcal{G}_{s,t}, t\in [12]$, such that for each $s,t$, a box in $\tilde{\mathcal G_s}$ contains at most one $B_{i}$ in $\mathcal G_{s,t}$. Thus, $V_i$'s in each group $\mathcal G_{s,t}$ are disjoint. Let $V_{s,t}=\cup_i V_i$ be the union of $V_i$'s with $i$ such that $B_{i}\in \mathcal{G}_{s,t}$. Define the $\sigma$-field generated by the information outside $V_{s,t}$ by \[ \mathcal F_{s,t}:=\big\{ \eta^{V} (x): x\in (V \backslash V_{s,t})\cup\partial V_{s,t} \big\}. \] Then, conditioned on $\mathcal F_{s,t}$, $\xi'_i$'s in each group $\mathcal{G}_{s,t}$ are mutually independent. Denote \[W_{s,t}:=\prod_{B_{i}\in \mathcal{G}_{s,t}}e^{a\beta_K(\xi'_i-\delta)}, \] where $\delta\ge\delta_1=\frac12$ and $a$ is a positive number to be set. Then, we have \begin{equation}\label{eq:W-st} \mathbb{E}W_{s,t}^{1/\beta_K} =\mathbb{E}\prod_{B_{i}\in \mathcal{G}_{s,t}}e^{a(\xi'_i-\delta)} =\mathbb{E}\prod_{B_{i}\in \mathcal{G}_{s,t}}\mathbb{E}\left(e^{a(\xi'_i-\delta)} \big| \mathcal F_{s,t}\right). \end{equation} Next, we will estimate $\mathbb{E}\left(e^{a(\xi'_i-\delta)} | \mathcal F_{s,t}\right)$. Since $K\ge K_1(\lambda)= K_0(\lambda+\varepsilon_{1})$, by Corollary \ref{cor:r=0}, $\xi'_i$ is a Bernoulli random variable with $ \mathbb{P}(\xi'_i=1| \mathcal F_{s,t})\le e^{-0.01\sqrt{K}}=:g(K). $ Consequently, \[ \mathbb{E}\left(e^{a(\xi'_i-\delta)} \big| \mathcal F_{s,t}\right)\le e^{a (1-\delta)}g(K)+e^{-a\delta}. \] Set $a=\log\left(\frac{\delta}{1-\delta}g(K)^{-1} \right)$ to optimize the above bound, noting $a\ge\log\left( \frac{\delta_1}{1-\delta_1}g(K)^{-1} \right)=0.01\sqrt{K}>0$. It follows that \begin{equation}\label{eq:mathod-1} \mathbb{E}\left(e^{a(\xi'_i-\delta)} \big| \mathcal F_{s,t}\right)\le f(\delta)g(K)^{\delta}\le 2g(K)^{\delta}, \end{equation} where $f(\delta):=\left( \frac{\delta}{1-\delta} \right)^{1-\delta}+\left( \frac{\delta}{1-\delta} \right)^{-\delta}\le 2$. Combined with \eqref{eq:W-st}, this yields \begin{equation}\label{eq:W-st-2} \mathbb{E}W_{s,t}^{1/\beta_K}\le \prod_{B_{i}\in \mathcal{G}_{s,t}} \left(2g(K)^{\delta} \right). \end{equation} By the Cauchy-Schwarz inequality, \begin{equation}\label{eq:C-S} \mathbb E e^{a\beta_K\sum_i(\xi'_i-\delta)} =\mathbb{E}\prod_{s=1}^{4K^{1/4}}\prod_{t=1}^{12}W_{s,t} \le \prod_{s=1}^{4K^{1/4}}\prod_{t=1}^{12}\left(\mathbb{E}W_{s,t}^{1/\beta_K}\right)^{\beta_K}. \end{equation} Combining \eqref{eq:W-st-2} and \eqref{eq:C-S}, we obtain \begin{align} \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right) &\le \mathbb E e^{a\beta_K\sum_i(\xi'_i-\delta)} \le\prod_{s=1}^{4K^{1/4}}\prod_{t=1}^{12}\prod_{B_i\in \mathcal{G}_{s,t}}\left(2g(K)^{\delta}\right)^{\beta_K}\nonumber\\ &\le\left(2g(K)^{\delta}\right)^{\beta_Kd} \le e^{ -10^{-4}K^{1/4}\delta d },\label{eq:method-2} \end{align} where the last inequality follows from $d\ge K\ge 2^{32}$ and $\delta\ge \frac12$. \end{proof} Recall \eqref{eq:E_S} for the definition of $\mathcal{E}_{\mathcal S}$ and define the event \begin{equation}\label{eq:E1} \mathcal{E}_1:= \bigcup_{d\ge K}\bigcup_{\mathcal S\in\mathrm{END}_{j-1, d}(B)}\mathcal{E}_{\mathcal S}. \end{equation} To prove Theorem \ref{thm:r=1}, we need to estimate $\mathbb P\big(\mathcal{E}_1 \big)$ in addition. The argument is quite similar to Lemma \ref{lem:E0}. \begin{lem}\label{lem:E-1} Let $K\ge C_5$. Then $ \mathbb P\big(\mathcal{E}_1 \big) \le e^{-1.5K^{1/8} }. $ \end{lem} \begin{proof} There are at most $K^7$ boxes in $\mathcal{BD}_{K^{j-3}}$ intersecting with some paths in $\mathcal{P}_j(B)$. Denote them by $B_t$'s. For each $B_t$, denote for brevity \[ z_t=z_{B_t}\ \text{ and } \ V_t=V_{K^{j-7/8}}(z_t). \] Let $H_t$ be the conditional expectation of $\eta^V$ given $\eta^V\big|_{ V_t^c\cup\partial V_t }$. Setting $\ell=4K^{j-1}, L=K^{j-7/8}$ in Lemma \ref{lem:fluct-H}, recalling $C_5=(2\vee C_4)^{32}$, for $K\ge C_5$, we have $ C_3\sqrt{\frac{\ell}{L}}=C_3\sqrt{\frac{4}{K^{1/8}}}\le\varepsilon_1, $ and for all $t$, \[ \mathbb P\Big( \big|H_t(x)-H_t(z_t)\big|\ge\varepsilon_{1} \text{ for some } x\in V_{4K^{j-1}}(z_t) \Big)\le 4e^{ -2K^{1/8}}. \] Note that $\mathcal{E}_1 $ implies that the fluctuation of $H_t$ in $V_{4K^{j-1}}(z_t)$ is greater than $\varepsilon_{1}$ for some $t$. Thus, we obtain $ \mathbb P\left(\mathcal{E}_1 \right) \le 4K^7e^{ -2K^{1/8}} \le e^{ -1.5K^{1/8}}, $ completing the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:r=1}] It can be seen from the definition that \begin{equation}\label{eq:xi1-delta} \mathbb P\big(\xi_{1}>\delta\big) \le \sum_{d=K}^\infty\sum_{\mathcal S\in\mathrm{END}_{j-1,d}(B)} \mathbb P\big(\{\zeta_{1,\mathcal S}>\delta \}\cap\mathcal{E}_{\mathcal S}^c\big)+\mathbb P\big(\mathcal{E}_1\big), \end{equation} Note that there are at most $K^7$ boxes in $\mathcal{BD}_{K^{j-3}}$ intersecting with some path in $\mathcal{P}_j(B)$. Therefore, there are at most $K^{7d}$ sequences in $\mathrm{END}_{j-1,d}(B)$. Combined with \eqref{eq:1S-delta}, Lemma \ref{lem:average-level0} and Lemma \ref{lem:E-1}, and using a union bound, this yields \[ \mathbb P(\xi_{1}>\delta) \le\sum_{d=K}^{\infty}K^{7d}e^{ -10^{-4}K^{1/4}\delta d }+e^{ -1.5K^{1/8}} \le e^{ -K^{1/8}}, \] where in the last inequality we have used $\sum_{d=K}^{\infty}K^{7d}e^{ -10^{-4}K^{1/4}\delta d } \le e^{-K}$ for $\delta\ge\frac12$ and $K\ge 2^{32}$. This completes the proof of the theorem. \end{proof} \subsection{Proof of Theorem \ref{thm:xi-bound}}\label{subsec:4.2} Assume $r\in[2,m-2]\cap\mathbb Z$, $j\in[r+1,m-1]\cap\mathbb Z$, $B\in \mathrm{END}_j,V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$ in this section. The reasoning of the proof of Theorem \ref{thm:xi-bound} is similar to that of Theorem \ref{thm:r=1}. Recall $\mathcal{S} :=\left\{B_{i}\right\}_{i \in[d]}\in\mathrm{END}_{j-1, d}(B)$. Compared with \eqref{eq:z-V}, here we set \[ z_i:=z_{B_{i}} \ \text{ and } \ V_i:=V_{4K^{j-1}}(z_i) \ \text { for all } i \in[d]. \] Noting that $V_i\subseteq V$ for all $i$, let $H_i$ be the conditional expectation of $\eta^V$ given $\eta^V\big|_{ V_i^c\cup\partial V_i }$. By Lemma \ref{lem:DMP}, $\eta^{V_i}:=\eta^V-H_i$ is a GFF on $V_i$ for all $i$. Recall $\beta=2^{-9}$ set in \eqref{eq:beta-c_r} and $\varepsilon_{r+1}=4\sqrt{C_3}\beta^{r/2}$ set in \eqref{eq:epsilon}. Analogous to \eqref{eq:E_S} and \eqref{eq:E1}, we define the events \begin{equation}\label{eq:Ei} \mathcal{E}_{r+1,i}:=\left\{\begin{array}{c} \text { there exists a box } B'\in \mathrm{END}_{j-r-1}\text{ such that } B'\subseteq V_{3K^{j-1}}(z_i) \\ \text{ and }\big|H_i(x)-H_i(z_{B'})\big|\ge\varepsilon_{r+1} \text{ for some } x\in V_{4K^{j-r-1}}(z_{B'}) \end{array}\right\}, \end{equation} \[ \mathcal{E}_{r+1,\mathcal S}:=\bigcup_{i\in [d]}\mathcal{E}_{r+1,i}, \ \text{ and } \ \mathcal{E}_{r+1}:= \bigcup_{d\ge K}\bigcup_{\mathcal S\in\mathrm{END}_{j-1, d}(B)}\mathcal{E}_{r+1,\mathcal S} \ \text{ for all } r\ge 1. \] For $P\in \mathcal{P}_{j, \mathcal{S}}(B), i\in [d], u\in \mathcal{T}_{P^{(i)},r}$, we have $B_u\in \mathrm{END}_{j-r-1}, B_u\subseteq V_{3K^{j-1}}(z_i)$ and $u\subseteq V_{4K^{j-r-1}}(z_{u})$ with $z_u:=z_{B_u}$. Noting that $\eta^{V}(x)=\left(\eta^{V_i}(x)+H_i(z_u)\right)+\left( H_i(x)-H_i(z_u) \right)$ for all $x\in u$, on the event $\mathcal{E}_{r+1,i}^c$, by the triangle inequality, $u$ is $(V,\lambda,\bar\alpha_u)$-open implies that it is $(V_i, \lambda+\varepsilon_{r+1}, \bar\alpha_u+H_i(z_u))$-open. Thus, by an analogous reasoning of \eqref{eq:1S-delta}, for all $\delta>0$ and $\mathcal S\in\mathrm{END}_{j-1, d}(B)$, \begin{equation}\label{eq:r+1S-delta} \mathbb P\left(\{\zeta_{r+1,\mathcal{S}} >\delta\}\cap\mathcal{E}_{r+1,\mathcal S}^c\right)\le \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_{i,r,\lambda+\varepsilon_{r+1},j-1}>\delta\right), \end{equation} where $ \zeta_{r+1,\mathcal{S}} :=\max \big\{Y_{P, r+1,\lambda, \bar\alpha}: P \in \mathcal{P}_{j, \mathcal{S}}(B)\big\}, $ and \begin{equation}\label{eq:xi'_i} \xi'_{i,r,\lambda+\varepsilon_{r+1},j-1}:=\max_{P' \in \mathcal{P}_{j-1}(B_i)}\sum_{u\in \mathcal T_{P',r}}\theta_{P'}(u)1_{\big\{ u \text{ is tame and } (V_i,\lambda+\varepsilon_{r+1},\bar\alpha_u+H_i(z_{u}))\text{-open}\big\}}. \end{equation} In addition, the following lemma is analogous to Lemma \ref{lem:E-1}. \begin{lem}\label{lem:E-r} Let $K\ge C_5$. Then $ \mathbb P\left(\mathcal{E}_{r+1} \right) \le e^{-c_r}, $ where $c_r=(\beta K)^r$ is set in \eqref{eq:beta-c_r}. \end{lem} \begin{proof} There are at most $K^7$ boxes in $\mathrm{END}_{j-1}$ intersecting with some paths in $\mathcal{P}_j(B)$. Denote them by $B_t$'s. For each $B_t$, denote $z_t:=z_{B_t}, V_t:=V_{4K^{j-1}}(z_t)$, and by $H_t$ the conditional expectation of $\eta^V$ given $\eta^V\big|_{ V_t^c\cup\partial V_t }$. Let $\mathcal{E}_{r+1,t}$ be the event as in \eqref{eq:Ei} with $t$ in place of $i$. Note that $\mathcal{E}_{r+1}$ implies $\mathcal{E}_{r+1,t}$ for some $t$. It suffices to estimate the probability of $\mathcal{E}_{r+1,t}$ for all $t$. Setting $\ell=4K^{j-r-1}$ and $L=4K^{j-1}$ in Lemma \ref{lem:fluct-H}, for $K\ge C_5$, we have $ C_3\sqrt{\frac{\ell}{L}}=C_3 K^{-\frac r2}\le \varepsilon_{r+1}, $ and for any box $B'$ in $\mathrm{END}_{j-r-1}$ and $B'\subseteq V_{3K^{j-1}}(z_t)$, we have $V_{4K^{j-r-1}}(z_{B'})\subseteq V_t^{\chi}$, therefore \[ \mathbb P\Big( \big|H_t(x)-H_t(z_t)\big|\ge\varepsilon_{r+1} \text{ for some } x\in V_{4K^{j-r-1}}(z_{B'}) \Big)\le 4\exp\left\{-\frac{\varepsilon_{r+1}^2}{8C_3}K^r \right\}\le 4e^{-2c_r}. \] Note that there are at most $\left( \frac{3K^{j-1}}{K^{j-r-3}} \right)^2\le 9K^{2r+4}$ boxes $B'$'s in $\mathrm{END}_{j-r-1}$ such that $B'\subseteq V_{3K^{j-1}}(z_t)$. By a union bound, $ \mathbb P\left(\mathcal{E}_{r+1} \right) \le K^{7}\cdot 9K^{2r+4} \cdot 4e^{-2c_r} \le e^{-c_r}, $ completing the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:xi-bound}] We will apply induction on $r$, similar to the proof of \cite[Lemma 4.4]{MR3947326}. To this end, we will prove that the following hold for all $r\in[2,m-2]\cap\mathbb Z$ and $j\in [r+1,m-1]\cap\mathbb Z$. (i) Suppose $K\ge K_r(\lambda)$, $B\in \mathrm{END}_j$, $V_{4K^j}(z_{B})\subseteq V\subseteq V_{2N}$, and $\bar\alpha\in\mathscr A$. Then, \[ \mathbb{P}(\xi_{r,\lambda,\bar\alpha,j,B}>\delta) \le 2 e^{-c_{r-1}(\delta-\delta_r)} \quad \text{for all } \delta\ge\delta_r. \] (ii) Suppose $K\ge K_{r+1}(\lambda)$, $B\in \mathrm{END}_{j+1}$, $d\ge K$, $\left\{B_{i}\right\}_{i \in[d]} \in \mathrm{END}_{j, d}(B)$, and denote $\xi'_i:=\xi'_{i,r,\lambda+\varepsilon_{r+1},j}, i\in [d]$, defined in \eqref{eq:xi'_i}. Then, \[ P\left(\frac1d \sum_{i=1}^{d}\xi'_{i}>\delta\right) \le \Big(K^{-9} e^{-\beta c_{r-1}(\delta-\delta_{r+1})}\Big)^d \text{ for all } \delta\ge\delta_{r+1}. \] In Step 1, we will show that (i) implies (ii). In Step 2, we will show (i) for $r+1$ and all $j \in[r+2, m-1] \cap \mathbb{Z}$, provided that (ii) holds for all \(j \in[r+1, m-1] \cap \mathbb{Z}\). In Step 3, we will show (i) holds for $r=2$ and $j\in [3,m-1]\cap\mathbb Z$. \textbf{Step 1.} Suppose (i) holds. We will prove (ii). We can classify $\{B_{i}\}_{i\in [d]}$ into $432(\le 2^9=\beta^{-1})$ groups $\mathcal G_t$'s such that $V_i$'s in each group are disjoint, where $V_i=V_{4K^{j-1}}(z_i)$. Let $V_t$ be the union of $V_i$'s with $i$ such that $B_i\in\mathcal G_t$. Define $ \mathcal F_{t}:=\big\{ \eta^{V} (x): x\in (V \backslash V_{t})\cup\partial V_{t} \big\}. $ Conditioned on $\mathcal F_{t}$, $\xi'_i$'s in each group $\mathcal G_t$ are mutually independent. Next, we will estimate $\mathbb{E}\left(e^{a(\xi'_i-\delta)} \big| \mathcal F_{t}\right)$, where $\delta\ge\delta_{r+1}$ and $a>0$. For each $i\in [d]$, we apply (i) to $\lambda+\varepsilon_{r+1}$, and have for $K\ge K_{r+1}(\lambda):=K _r(\lambda+\varepsilon_{r+1})$, \begin{equation}\label{eq:xi'i} \mathbb P\left(\xi'_i>\delta\big| \mathcal F_{t}\right)\le 2e^{-c_{r-1}(\delta-\delta_r)} \quad \text{ for all } \delta\ge\delta_r. \end{equation} Note that $0\le\xi'_i\le1$. It follows that for each $a>0$, \begin{small} \begin{equation*} \mathbb E \left(e^{a\xi'_i}\Big| \mathcal F_{t}\right) \le e^{a\delta_r}+\int_{\delta_r}^{1} \mathbb P\left(\xi'_i>z\big| \mathcal F_{t}\right)ae^{az} dz\\ =e^{a\delta_r}\left( 1+2a\int_{0}^{1-\delta_r}e^{(a-c_{r-1})z}dz \right). \end{equation*} \end{small} Take $a=c_{r-1}$, then $\mathbb E \left(e^{a\xi'_i}\big| \mathcal F_{t}\right)\le (1+2c_{r-1})e^{c_{r-1}\delta_r}$. Hence for all $\delta\ge\delta_r$, \[ \mathbb E \left(e^{a\left(\xi'_i-\delta\right)}\Big| \mathcal F_{t}\right)\le (1+2c_{r-1})e^{-c_{r-1}(\delta-\delta_r)}. \] Using the same argument from \eqref{eq:mathod-1} to \eqref{eq:method-2} as in Lemma \ref{lem:average-level0}, we obtain \begin{equation}\label{eq:4.3-1} P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right) \le \Big( (1+2c_{r-1})e^{-c_{r-1}(\delta-\delta_r)} \Big)^{\beta d}. \end{equation} Recall $\Delta_r=\delta_{r+1}-\delta_r$ from \eqref{eq:Delta}. We get $ \Big( (1+2c_{r-1})e^{-c_{r-1}(\delta_{r+1}-\delta_r)} \Big)^{\beta}\le K^{-9}. $ Combined with \eqref{eq:4.3-1}, this implies (ii). \textbf{Step 2.} Assuming that (ii) holds for all \(j \in[r+1, m-1] \cap \mathbb{Z}\), we will show (i) for $r+1$ and all $j \in[r+2, m-1] \cap \mathbb{Z}$. Similar to \eqref{eq:xi1-delta}, we have \begin{equation}\label{eq:xi-r+1-delta} \mathbb P\big(\xi_{r+1}>\delta\big) \le \sum_{d=K}^\infty\sum_{\mathcal S\in\mathrm{END}_{j-1,d}(B)} \mathbb P\big(\{\zeta_{r+1,\mathcal S}>\delta \}\cap\mathcal{E}_{r+1,\mathcal S}^c\big)+\mathbb P\big(\mathcal{E}_{r+1}\big). \end{equation} Note that $r+2\le j\le m-1$ implies $r+1\le j-1\le m-1$, then for $K\ge K_{r+1}(\lambda)$, we apply (ii) to $j-1$, and have \[ \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_{i,r,\lambda+\varepsilon_{r+1},j-1}>\delta\right)\le \Big(K^{-9} e^{-\beta c_{r-1}(\delta-\delta_{r+1})}\Big)^d \ \text{ for all } \delta\ge\delta_{r+1}. \] Combined with \eqref{eq:r+1S-delta}, this gives that for each $\mathcal S\in\mathrm{END}_{j-1,d}(B)$, \begin{equation} \mathbb P\big(\{\zeta_{r+1,\mathcal S}>\delta \}\cap\mathcal{E}_{r+1,\mathcal S}^c\big) \le \Big(K^{-9} e^{-\beta c_{r-1}(\delta-\delta_{r+1})}\Big)^d \ \text{ for all } \delta\ge\delta_{r+1}. \end{equation} Note that there are at most $K^{7d}$ sequences in $\mathrm{END}_{j-1,d}(B)$. By a union bound, the first term on the right hand side of \eqref{eq:xi-r+1-delta} is less than \begin{equation}\label{eq:3/2} \sum_{d=K}^{\infty}\Big(K^{-2}e^{-\beta c_{r-1}(\delta-\delta_{r+1})}\Big)^d \le \frac32e^{-c_{r}(\delta-\delta_{r+1})}, \end{equation} since $K^{-2}e^{-\beta c_{r-1}(\delta-\delta_{r+1})}\le K^{-2}\le\frac13$ for $\delta\ge \delta_{r+1}$. Moreover, note that $\delta-\delta_{r+1}\le\frac12$ since $\delta\le1$ and $\delta_{r+1}\ge\frac12$, then by Lemma \ref{lem:E-r}, \begin{equation}\label{eq:1/2} P\big(\mathcal{E}_{r+1}\big)\le e^{-c_r}\le \frac12e^{-c_{r}/2}\le \frac12e^{-c_{r}(\delta-\delta_{r+1})}. \end{equation} Plugging \eqref{eq:3/2} and \eqref{eq:1/2} into \eqref{eq:xi-r+1-delta}, we obtain $ \mathbb P(\xi_{r+1}>\delta) \le 2e^{-c_{r}(\delta-\delta_{r+1})}. $ That is, (i) holds for $r+1$ and all $j \in[r+2, m-1] \cap \mathbb{Z}$. \textbf{Step 3.} We will show that (i) holds for $r=2$ and $j\in [3,m-1]\cap\mathbb Z$. This follows lines in Step 1. We write $\xi'_i=\xi'_{i,1,\lambda+\varepsilon_2,j-1}, i\in [d]$ for brevity. Note that $j-1\in [2,m-1]\cap\mathbb Z$. Then applying Theorem \ref{thm:r=1} to $\lambda+\varepsilon_2$ and $K\ge K_2(\lambda)=K_1(\lambda+\varepsilon_2)$, we obtain \[ \mathbb P\left(\xi'_i>\delta\big| \mathcal F_{t}\right) \le \exp\{ -K^{1/8} \} \ \text{ for all } \delta\ge\delta_{1}, \] playing the role of \eqref{eq:xi'i}. Consequently, for $a=K^{1/8}$, \begin{align*} & \mathbb E \left(e^{a\xi'_i}\Big| \mathcal F_{t}\right) =\int_{0}^{1} \mathbb P\left(\xi'_i>z\big| \mathcal F_{t}\right)ae^{az} dz\\ \le & e^{a\delta_1}-1+\int_{\delta_r}^{1} e^{-K^{1/8}}ae^{az} dz \le e^{a\delta_1}-1+e^{a-K^{1/8}}=e^{a\delta_1}. \end{align*} Hence, it holds that \[ \mathbb E \left(e^{a\left(\xi'_i-\delta\right)}\Big| \mathcal F_{t}\right) \le e^{-K^{1/8}(\delta-\delta_1)}. \] Then, we have \[ \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right) \le \left( e^{-\beta K^{1/8}(\delta-\delta_1)}\right)^d, \] as the counterpart of \eqref{eq:4.3-1}. Recall that $\delta_2-\delta_1=\Delta_1=\frac{9\log K}{\beta K^{1/8}}$, thus for $\delta\ge\delta_2$, \[ \mathbb P\left(\frac1d \sum_{i=1}^{d}\xi'_i>\delta\right) \le \left( K^{-9}e^{-\beta K^{1/8}(\delta-\delta_2)}\right)^d. \] Consequently, \[ \mathbb{P}(\xi_{2}>\delta) \le \sum_{d=K}^{\infty}\Big(K^{-2}e^{-\beta K^{1/8}(\delta-\delta_{2})}\Big)^d+e^{-c_1}, \] as the counterpart of \eqref{eq:xi-r+1-delta}. With estimates similar to \eqref{eq:3/2} and \eqref{eq:1/2} for $r=2$, we conclude that for $K\ge K_2(\lambda)$, \[ \mathbb{P}(\xi_{2}>\delta) \le 2 e^{-c_{1}(\delta-\delta_2)} \] for all $\delta\ge\delta_2,$ completing the proof. \end{proof} \subsection{Proof of Theorem \ref{thm:1.1} } Recall that $v\in V_{2N}$ is $\lambda$-open if $\left| \eta^{V_{2N}}(v) \right|\le\lambda$, i.e., $(V_{2N},\lambda,0)$-open. Define $ \tilde Y_{P,r,\lambda}:=\sum_{u\in \mathcal T_{P,r}}\theta_P(u)1_{\{u \text{ is tame and } \lambda\text{-open}\}}, $ and \[ \tilde \xi_{r,\lambda,j,B}:=\max\big\{\tilde Y_{P,r,\lambda}: P\in \mathcal P_j(B)\big\}\ \text{ for all } B\in\mathrm{END}_j. \] For $j\in [3,m-1]\cap\mathbb Z$ and $2\le r\le j-1$, let $K\ge K_{r}(\lambda)$. Applying Theorem \ref{thm:xi-bound} to $V=V_{2N}$, $\bar\alpha\equiv 0$ and $B\in\mathrm{END}_j$, we get \begin{equation}\label{eq:tilde-xi-bound} \mathbb{P}\left(\tilde\xi_{r,\lambda,j,B}>\delta\right) \le 2 e^{-c_{r-1}(\delta-\delta_r)} \ \text{ for all } \delta\ge\delta_{r+1}, \end{equation} recalling \eqref{eq:beta-c_r}, \eqref{eq:delta} for the definition of $c_r, \delta_r$. Recall $\mathcal{P}^{\kappa, \delta, K}$ from \eqref{eq:kap-delta-K}. For each $P\in\mathcal{P}^{\kappa, \delta, K}$, let $\big\{P^{(i)}:i\in[d_0]\big\}$ be the child-paths of $P$ in $\mathcal{SL}_{m-1}$ from Proposition \ref{prop:tree}. Recall that $L(u)$ is the depth of $u$ with $L(\rho)=0$. For a sub-path $u$ of $P^{(i)}$ in $\mathcal T_P$, denote by $L_i(u):=L(u)-1$ the level of $u$ in $\mathcal T_{P^{(i)}}$. By Lemma \ref{lem:untame-flow}, \begin{equation*} \sum_{i=1}^{d_0}\sum_{u: 0 \leq L_i(u) \leq m-2}\frac{1}{d_0} \theta_{P^{(i)}}(u) 1_{\{u \text{ is untamed}\}}= \sum_{u: 1 \leq L(u) \leq m-1} \theta_{P}(u) 1_{\{u \text{ is untamed}\}}\leq 2 \delta m. \end{equation*} This implies that there is at least one child-path $P^{(i_0)}$ such that \begin{equation}\label{eq:thm1-untamed} \sum_{u: 0 \leq L_{i_0}(u) \leq m-2} \theta_{P^{(i_0)}}(u) 1_{\{u \text{ is untamed}\}}\leq 2 \delta m. \end{equation} Thus if there is a $\lambda$-open path $P$ in $\mathcal{P}^{\kappa, \delta, K}$, there would exist a $\lambda$-open path $\tilde P$ in $\mathcal P_{m-1}(B)$ for some $B\in\mathrm{END}_{m-1}$ such that \eqref{eq:thm1-untamed} holds with $P^{(i_0)}$ replaced with $\tilde P$ and $L_{i_0}(u)$ replaced with $\tilde L(u)$, the depth of $u$ in $\mathcal T_{\tilde P}$. Note that if $\tilde P$ is $\lambda$-open, then all the sub-paths are $\lambda$-open, which leads to \begin{align*} m-1 &=\sum_{u: 0 \leq\tilde L(u) \leq m-2} \theta_{\tilde P}(u) 1_{\{u \text{ is } \lambda\text{-open}\}}\\ &=\sum_{r=0}^{m-2} \tilde Y_{\tilde P,r,\lambda}+\sum_{u: 0 \leq \tilde L(u) \leq m-2} \theta_{\tilde P}(u) 1_{\{u \text{ is untamed}\}} \le \sum_{r=0}^{m-2}\tilde \xi_{r,\lambda,m-1,B}+2\delta m. \end{align*} By the above inequality, in order to prove that for some $\delta>0, K>0$, \begin{equation}\label{eq:P-kap-delta-K} \lim _{N \rightarrow \infty}\mathbb{P}\big( P \text { is } \lambda\text{-open for some } P \in \mathcal{P}^{\kappa, \delta, K} \big)=0, \end{equation} it is sufficient to show that there exists $\delta>0, K(\lambda,\delta)\in (0,\infty)$ such that for $K\ge K(\lambda,\delta)$, \begin{equation}\label{eq:sum-r} \lim_{m\rightarrow\infty}\mathbb P\left( \sum_{r=0}^{m-2}\tilde \xi_{r,\lambda,m-1,B}\ge m-1-2\delta m \text{ for some } B\in\mathrm{END}_{m-1} \right)=0. \end{equation} Recall that $\{\varepsilon_r: r\ge 0\}$ is set in \eqref{eq:epsilon} and $K_r(\lambda)=K_0\left(\lambda+\sum_{i=1}^{r}\varepsilon_i\right)$ is defined in \eqref{eq:Kr}. Noting that $\sum_{i=1}^{\infty}\varepsilon_i<\infty$ and $K_0(\cdot)$ is a increasing function, one has $K_{\infty}(\lambda):=K_0\left(\lambda+\sum_{i=1}^{\infty}\varepsilon_i\right)<\infty$. Recall \eqref{eq:Delta} for the definition of $\Delta_r$. There exists $K(\lambda,\delta)\ge K_{\infty}(\lambda)$ such that the following inequality holds for all $K\ge K(\lambda,\delta)$, \[ \sum_{r=1}^{\infty}\Delta_r=\frac{9\log K}{\beta K^{1/8}}+\sum_{r=1}^{\infty}\frac{\log(1+2c_{r})+9\beta^{-1}\log K}{c_{r}}\le \delta. \] Consequently, for $K\ge K(\lambda,\delta)$, we have $\delta_r\le \frac12+\delta$ for all $r\ge 0$. As $m\ge \delta^{-1}$, \begin{equation}\label{eq:sum-r-} \sum_{r=0}^{m-2}\left(\delta_r+\frac{1}{2^{r+2} } \left(1-8\delta\right)m\right) < m-1-2\delta m, \end{equation} where we set $\delta\in(0,\frac18)$. Since $\kappa N<K^{m+2}$, there are at most $(K^6/\kappa)^2$ boxes in $\mathrm{END}_{m-1}$. By a union bound and \eqref{eq:sum-r-}, for $K\ge K(\lambda,\delta)$ and $m\ge \delta^{-1}$, \begin{equation}\label{eq:sum-r-2} \mathbb P\left( \sum_{r=0}^{m-2}\tilde \xi_{r,\lambda,m-1,B}\ge m-1-2\delta m \text{ for some } B\in\mathrm{END}_{m-1} \right) \le \frac{K^{12}}{\kappa^2}\sum_{r=0}^{m-2}p_{m,r}, \end{equation} where $ p_{m,r}=\mathbb P\left(\tilde \xi_{r,\lambda,m-1,B}>\delta_r+\frac{1}{2^{r+2} } \left(1-8\delta\right)m\right). $ As $m\ge \frac{8}{1-8\delta}$, for $r=0,1$, we have $\frac{1}{2^{r+2} } \left(1-8\delta\right)m\ge 1$, implying $p_{m,r}=0$. Applying \eqref{eq:tilde-xi-bound} to $j=m-1$, then for all $2\le r\le m-2$, \[ p_{m,r}\le 2\exp\left\{ -\frac18\left(1-8\delta\right)\left(\frac{\beta K}{2}\right)^{r-1}m \right\}. \] Furthermore, $\beta K/2\ge 2^{22}$ implies that $\frac18\left(\frac{\beta K}{2}\right)^{r-1}\ge r$ for all $r\ge 2$. Thus, \begin{equation}\label{eq:decay-rate} \sum_{r=0}^{m-2}p_{m,r}\le 2\sum_{r=2}^{m-2}e^{ -\left(1-8\delta\right)mr}\le \frac{2}{e^{\left(1-8\delta\right)m}-1}, \end{equation} which converges to $0$ as $m\rightarrow\infty$. Combined with \eqref{eq:sum-r-2}, this implies \eqref{eq:sum-r}. Especially, set $\delta=\frac{1}{16}$. Then for $\lambda\ge\lambda_0$, there exists $K(\lambda)=e^{b\lambda^2}\ge K(\lambda,1/16)$ for some $b=b(\lambda_0)>0$ such that for $\delta=\frac{1}{16}$ and $K\ge K(\lambda)$, \eqref{eq:P-kap-delta-K} holds. Let \begin{equation}\label{eq:eps-lambd} \epsilon(\lambda)=\frac{1}{16K(\lambda)^2k(\lambda)}, \end{equation} then $\mathcal{P}_{N}^{\kappa,\epsilon(\lambda)}=\mathcal{P}^{\kappa, 1/16, K(\lambda)}$ and \eqref{eq:P-kap-delta-K} implies \eqref{eq:complement-event}. We conclude the proof of Theorem \ref{thm:1.1}. \noindent {\bf Acknowledgments:} This work is supported by NSF of China 11771027. We would like to thank Jian Ding for his suggestions and helpful discussions. \end{document}

\begin{document} \title[Asymptotic formulas for the gamma function]{Asymptotic formulas for the gamma function constructed by bivariate means} \author{Zhen-Hang Yang} \address{Power Supply Service Center, ZPEPC Electric Power Research Institute, Hangzhou, Zhejiang, China, 310007} \email{[email protected]} \date{July 19, 2014} \subjclass[2010]{Primary 33B15, 26E60; Secondary 26D15, 11B83} \keywords{Stirling's formula, gamma function, mean, inqueality, polygamma function} \thanks{This paper is in final form and no version of it will be submitted for publication elsewhere.} \begin{abstract} Let $K,M,N$ denote three bivariate means. In the paper, the author prove the asymptotic formulas for the gamma function have the form of \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+1-\theta \right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-N\left( x+\sigma ,x+1-\sigma \right) } \end{equation*} or \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma \right) ^{K\left( x+\epsilon ,x+1-\epsilon \right) }e^{-M\left( x+\theta ,x+\sigma \right) } \end{equation*} as $x\rightarrow \infty $, where $\epsilon ,\theta ,\sigma $ are fixed real numbers. This idea can be extended to the psi and polygamma functions. As examples, some new asymptotic formulas for the gamma function are presented. \end{abstract} \maketitle \section{Introduction} The Stirling's formula \begin{equation} n!\thicksim \sqrt{2\pi n}n^{n}e^{-n}:=s_{n} \label{S} \end{equation} has important applications in statistical physics, probability theory and and number theory. Due to its practical importance, it has attracted much interest of many mathematicians and have motivated a large number of research papers concerning various generalizations and improvements. Burnside's formula \cite{Burnside-MM-46-1917} \begin{equation} n!\thicksim \sqrt{2\pi }\left( \frac{n+1/2}{e}\right) ^{n+1/2}:=b_{n} \label{B} \end{equation} slight improves (\ref{S}). Gosper \cite{Gosper-PNAS-75-1978} replaced $\sqrt{ 2\pi n}$ by $\sqrt{2\pi \left( n+1/6\right) }$ in (\ref{S}) to get \begin{equation} n!\thicksim \sqrt{2\pi \left( n+\tfrac{1}{6}\right) }\left( \frac{n}{e} \right) ^{n}:=g_{n}, \label{G} \end{equation} which is better than (\ref{S}) and (\ref{B}). In the recent paper \cite {Batir-P-27(1)-2008}, N. Batir obtained an asymptotic formula similar to ( \ref{G}): \begin{equation} n!\thicksim \frac{n^{n+1}e^{-n}\sqrt{2\pi }}{\sqrt{n-1/6}}:=b_{n}^{\prime }, \label{Batir1} \end{equation} which is stronger than (\ref{S}) and (\ref{B}). A more accurate approximation for the factorial function \begin{equation} n!\thicksim \sqrt{2\pi }\left( \frac{n^{2}+n+1/6}{e^{2}}\right) ^{n/2+1/4}:=m_{n} \label{M} \end{equation} was presented in \cite{Mortici-CMI-19(1)-2010} by Mortici. The classical Euler's gamma function $\Gamma $ may be defined by \begin{equation} \Gamma \left( x\right) =\int_{0}^{\infty }t^{x-1}e^{-t}dt \label{Gamma} \end{equation} for $x>0$, and its logarithmic derivative $\psi \left( x\right) =\Gamma ^{\prime }\left( x\right) /\Gamma \left( x\right) $ is known as the psi or digamma function, while $\psi ^{\prime }$, $\psi ^{\prime \prime }$, ... are called polygamma functions (see \cite{Anderson-PAMS-125(11)-1997}). The gamma function is closely related to the Stirling's formula, since $ \Gamma (n+1)=n!$ for all $n\in \mathbb{N}$. This inspires some authors to also pay attention to find better approximations for the gamma function. For example, Ramanujan's \cite[P. 339]{Ramanujan-SB-1988} double inequality for the gamma function: \begin{equation} \sqrt{\pi }\left( \tfrac{x}{e}\right) ^{x}\left( 8x^{3}+4x^{2}+x+\tfrac{1}{ 100}\right) ^{1/6}<\Gamma \left( x+1\right) <\sqrt{\pi }\left( \tfrac{x}{e} \right) ^{x}\left( 8x^{3}+4x^{2}+x+\tfrac{1}{30}\right) ^{1/6} \label{R} \end{equation} for $x\geq 1$. Batir \cite{Batir-AM-91-2008} showed that for $x>0$, \begin{eqnarray} &&\sqrt{2}e^{4/9}\left( \frac{x}{e}\right) ^{x}\sqrt{x+\frac{1}{2}}\exp \left( -\tfrac{1}{6\left( x+3/8\right) }\right) \label{Batir2} \\ &<&\Gamma \left( x+1\right) <\sqrt{2\pi }\left( \frac{x}{e}\right) ^{x}\sqrt{ x+\frac{1}{2}}\exp \left( -\tfrac{1}{6\left( x+3/8\right) }\right) . \notag \end{eqnarray} Mortici \cite{Mortici-AM-93-2009-1} proved that for $x\geq 0$, \begin{eqnarray} \sqrt{2\pi e}e^{-\omega }\left( \frac{x+\omega }{e}\right) ^{x+1/2} &<&\Gamma \left( x+1\right) \leq \alpha \sqrt{2\pi e}e^{-\omega }\left( \frac{x+\omega }{e}\right) ^{x+1/2}, \label{Ml} \\ \beta \sqrt{2\pi e}e^{-\varsigma }\left( \frac{x+\varsigma }{e}\right) ^{x+1/2} &<&\Gamma \left( x+1\right) \leq \sqrt{2\pi e}e^{-\varsigma }\left( \frac{x+\varsigma }{e}\right) ^{x+1/2} \label{Mr} \end{eqnarray} where $\omega =\left( 3-\sqrt{3}\right) /6$, $\alpha =1.072042464...$ and $ \varsigma =\left( 3+\sqrt{3}\right) /6$, $\beta =0.988503589...$. More results involving the asymptotic formulas for the factorial or gamma functions can consult \cite{Shi-JCAM-195-2006}, \cite{Guo-JIPAM-9(1)-2008}, \cite{Mortici-MMN-11(1)-2010}, \cite{Mortici-CMA-61-2011}, \cite {Zhao-PMD-80(3-4)-2012}, \cite{Mortici-MCM-57-2013}, \cite{Qi-JCAM-268-2014} , \cite{Qi-JCAM-268-2014}, \cite{Lu-RJ-35(1)-2014} and the references cited therein). Mortici \cite{Mortici-BTUB-iii-3(52)-2010} presented an idea that by replacing an under-approximation and an upper-approximation of the factorial function by one of their geometric mean to improve certain approximation formula of the factorial. In fact, by observing and analyzing these asymptotic formulas for factorial or gamma function, we find out that they have the common form of \begin{equation} \ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +P_{1}\left( x\right) \ln P_{2}\left( x\right) -P_{3}\left( x\right) +P_{4}\left( x\right) , \label{g-form} \end{equation} where $P_{1}\left( x\right) ,P_{2}\left( x\right) $ and $P_{3}\left( x\right) $ are all means of $x$ and $\left( x+1\right) $, while $P_{4}\left( x\right) $ satisfies $P_{4}\left( \infty \right) =0$. For example, (\ref{S} )--(\ref{M}) can be written as \begin{eqnarray*} &&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n, \\ &&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln \left( n+\frac{1}{2}\right) -\left( n+\frac{1}{2}\right) , \\ &&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n+ \frac{1}{2}\ln \left( 1+\tfrac{1}{6n}\right) , \\ &&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln n-n- \frac{1}{2}\ln \left( 1-\tfrac{1}{6n}\right) , \\ &&\ln n!\thicksim \frac{1}{2}\ln 2\pi +\left( n+\frac{1}{2}\right) \ln \sqrt{ \frac{n^{2}+4n\left( n+1\right) +\left( n+1\right) ^{2}}{6}}-\left( n+\frac{1 }{2}\right) . \end{eqnarray*} Inequalities (\ref{R})--(\ref{Mr}) imply that \begin{eqnarray*} &&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{ 1}{2}\right) \ln x-x+\frac{1}{6}\ln \left( 1+\frac{1}{2x}+\frac{1}{8x^{2}}+ \frac{1}{240x^{3}}\right) , \\ &&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{ 1}{2}\right) \ln x-x+\frac{1}{2}\ln \left( 1+\frac{1}{2x}\right) -\tfrac{1}{ 6\left( x+3/8\right) }, \\ &&\ln \Gamma \left( x+1\right) \thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{ 1}{2}\right) \ln \left( \left( 1-a\right) x+a\left( x+1\right) \right) -\left( \left( 1-a\right) x+a\left( x+1\right) \right) , \end{eqnarray*} where $a=\omega =(3-\sqrt{3})/6$, $\varsigma =(3+\sqrt{3})/6$. The aim of this paper is to prove the validity of the form (\ref{g-form}) which offers such a new way to construct asymptotic formulas for Euler gamma function in terms of bivariate means. Our main results are included in Section 2. Some new examples are presented in the last section. \section{Main results} Before stating and proving our main results, we recall some knowledge on means. Let $I$ be an interval on $\mathbb{R}$. A bivariate real valued function $M:I^{2}\rightarrow \mathbb{R}$ is said to be a bivariate mean if \begin{equation*} \min \left( a,b\right) \leq M\left( a,b\right) \leq \max \left( a,b\right) \end{equation*} for all $a,b\in I$. Clearly, each bivariate mean $M$ is reflexive, that is, \begin{equation*} M\left( a,a\right) =a \end{equation*} for any $a\in I$. $M$ is symmetric if \begin{equation*} M\left( a,b\right) =M\left( b,a\right) \end{equation*} for all $a,b\in I$, and $M$ is said to be homogeneous (of degree one) if \begin{equation} M\left( ta,tb\right) =tM\left( a,b\right) \label{M-h} \end{equation} for any $a,b\in I$ and $t>0$. The lemma is crucial to prove our results. \begin{lemma}[{\protect\cite[Thoerem 1, 2, 3]{Toader.MIA.5.2002}}] \label{Lemma M}If $M:I^{2}\rightarrow \mathbb{R}$ is a differentiable mean, then for $c\in I$, \begin{equation*} M_{a}^{\prime }\left( c,c\right) ,M_{b}^{\prime }\left( c,c\right) \in \left( 0,1\right) \text{ \ and \ }M_{a}^{\prime }\left( c,c\right) +M_{b}^{\prime }\left( c,c\right) =1\text{.} \end{equation*} In particular, if $M$ is symmetric, then \begin{equation*} M_{a}^{\prime }\left( c,c\right) =M_{b}^{\prime }\left( c,c\right) =1/2. \end{equation*} \end{lemma} Now we are in a position to state and prove main results. \begin{theorem} \label{MT-p2><p3}Let $M:\left( 0,\infty \right) \times \left( 0,\infty \right) \rightarrow \left( 0,\infty \right) $ and $N:\left( -\infty ,\infty \right) \times \left( -\infty ,\infty \right) \rightarrow \left( -\infty ,\infty \right) $ be two symmetric, homogeneous and differentiable means and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $ \lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\theta ,\theta ^{\ast },\sigma ,\sigma ^{\ast }$ with $\theta +\theta ^{\ast }=\sigma +\sigma ^{\ast }=1$ such that $x>-\min \left( 1,\theta ,\theta ^{\ast }\right) $, we have \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\theta ^{\ast }\right) ^{x+1/2}e^{-N\left( x+\sigma ,x+\sigma ^{\ast }\right) }e^{r\left( x\right) }\text{, as }x\rightarrow \infty . \end{equation*} \end{theorem} \begin{proof} Since $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the desired result is equivalent to \begin{equation*} \lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{ 2\pi }-\left( x+\frac{1}{2}\right) \ln M\left( x+\theta ,x+\theta ^{\ast }\right) +N\left( x+\sigma ,x+\sigma ^{\ast }\right) \right) =0. \end{equation*} Due to $\lim_{x\rightarrow \infty }r\left( x\right) =0$ and the known relation \begin{equation*} \lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\left( x+ \frac{1}{2}\right) \ln \left( x+\frac{1}{2}\right) +\left( x+\frac{1}{2} \right) \right) =\frac{1}{2}\ln 2\pi , \end{equation*} it suffices to prove that \begin{eqnarray*} D_{1} &:&=\lim_{x\rightarrow \infty }\left( x+\frac{1}{2}\right) \ln \frac{ M\left( x+\theta ,x+\theta ^{\ast }\right) }{x+1/2}=0, \\ D_{2} &:&=\lim_{x\rightarrow \infty }\left( N\left( x+\sigma ,x+\sigma ^{\ast }\right) -\left( x+\frac{1}{2}\right) \right) =0. \end{eqnarray*} Letting $x=1/t$, using the homogeneity of $M$, that is, (\ref{M-h}), and utilizing L'Hospital rule give \begin{eqnarray*} D_{1} &=&\lim_{t\rightarrow 0^{+}}\frac{1+t/2}{t}\ln \frac{M\left( 1+\theta t,1+\theta ^{\ast }t\right) }{1+t/2} \\ &=&\lim_{t\rightarrow 0^{+}}\frac{\ln M\left( 1+\theta t,1+\theta ^{\ast }t\right) -\ln \left( 1+t/2\right) }{t} \\ &=&\lim_{t\rightarrow 0^{+}}\left( \frac{\theta M_{x}\left( 1+\theta t,1+\theta ^{\ast }t\right) +\theta ^{\ast }M_{y}\left( 1+\theta t,1+\theta ^{\ast }t\right) }{M\left( 1+\theta t,1+\theta ^{\ast }t\right) }-\frac{1}{ 2+t}\right) \\ &=&\frac{\theta M_{x}\left( 1,1\right) +\theta ^{\ast }M_{y}\left( 1,1\right) }{M\left( 1,1\right) }-\frac{1}{2}=0, \end{eqnarray*} where the last equality holds due to Lemma \ref{Lemma M}. Similarly, we have \begin{eqnarray*} D_{2} &=&\lim_{x\rightarrow \infty }\left( N\left( x+\sigma ,x+\sigma ^{\ast }\right) -\left( x+\frac{1}{2}\right) \right) \\ &&\overset{1/x=t}{=\!=\!=}\lim_{t\rightarrow 0^{+}}\frac{N\left( 1+\sigma t,1+\sigma ^{\ast }t\right) -\left( 1+t/2\right) }{t} \\ &=&\lim_{t\rightarrow 0^{+}}\left( \sigma N_{x}\left( 1+\sigma t,1+\sigma ^{\ast }t\right) +\sigma ^{\ast }N_{y}\left( 1+\sigma t,1+\sigma ^{\ast }t\right) -\frac{1}{2}\right) \\ &=&\frac{\sigma +\sigma ^{\ast }}{2}-\frac{1}{2}=0, \end{eqnarray*} which proves the desired result. \end{proof} \begin{theorem} \label{MT-p2=p3}Let $M:\left( 0,\infty \right) \times \left( 0,\infty \right) \rightarrow \left( 0,\infty \right) $ be a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\theta ,\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $, we have \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma \right) ^{x+1/2}e^{-M\left( x+\theta ,x+\sigma \right) }e^{r\left( x\right) } \text{, as }x\rightarrow \infty . \end{equation*} \end{theorem} \begin{proof} Since $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the desired result is equivalent to \begin{equation*} \lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{ 2\pi }-\left( x+\frac{1}{2}\right) \ln M\left( x+\theta ,x+\sigma \right) +M\left( x+\theta ,x+\sigma \right) \right) =0. \end{equation*} Similarly, it suffices to prove that \begin{eqnarray*} D_{3} &:&=\lim_{x\rightarrow \infty }\left( \left( x+\frac{1}{2}\right) \ln \frac{M\left( x+\theta ,x+\sigma \right) }{x+1/2}-\left( M\left( x+\theta ,x+\sigma \right) -\left( x+\frac{1}{2}\right) \right) \right) \\ &=&\lim_{x\rightarrow \infty }\left( \left( M\left( x+\theta ,x+\sigma \right) -\left( x+\frac{1}{2}\right) \right) \times \left( \frac{1}{L\left( y,1\right) }-1\right) \right) =0, \end{eqnarray*} where $L\left( a,b\right) $ is the logarithmic mean of positive $a$ and $b$, $y=M\left( x+\theta ,x+\sigma \right) /\left( x+1/2\right) $. Now we first show that \begin{equation*} D_{4}:=M\left( x+\theta ,x+\sigma \right) -\left( x+\frac{1}{2}\right) \end{equation*} is bounded. In fact, by the property of mean we see that \begin{equation*} x+\min \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right) <D_{4}<x+\max \left( \theta ,\sigma \right) -\left( x+\frac{1}{2}\right) \end{equation*} that is, \begin{equation*} \min \left( \theta ,\sigma \right) -\frac{1}{2}<D_{4}<\max \left( \theta ,\sigma \right) -\frac{1}{2}. \end{equation*} It remains to prove that \begin{equation*} \lim_{x\rightarrow \infty }D_{5}:=\lim_{x\rightarrow \infty }\left( \frac{1}{ L\left( y,1\right) }-1\right) =0. \end{equation*} Since \begin{equation*} \frac{x+\min \left( \theta ,\sigma \right) }{x+1/2}<y=\frac{M\left( x+\theta ,x+\sigma \right) }{x+1/2}<\frac{x+\max \left( \theta ,\sigma \right) }{x+1/2 }, \end{equation*} so we have $\lim_{x\rightarrow \infty }y=1$. This together with \begin{equation*} \min \left( y,1\right) \leq L\left( y,1\right) \leq \max \left( y,1\right) \end{equation*} yields $\lim_{x\rightarrow \infty }L\left( y,1\right) =1$, and therefore, $ \lim_{x\rightarrow \infty }D_{5}=0$. This completes the proof. \end{proof} \begin{theorem} \label{MT-p2=p3=x+1/2}Let $K:\left( -\infty ,\infty \right) \times \left( -\infty ,\infty \right) \rightarrow \left( -\infty ,\infty \right) $ be a symmetric, homogeneous and twice differentiable mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\epsilon ,\epsilon ^{\ast }$ with $\epsilon +\epsilon ^{\ast }=1$, we have \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }\left( x+\frac{1}{2}\right) ^{K(x+\epsilon ,x+\epsilon ^{\ast })}e^{-\left( x+1/2\right) }e^{r\left( x\right) }\text{, as }x\rightarrow \infty \end{equation*} \end{theorem} \begin{proof} Due to $\lim_{x\rightarrow \infty }r\left( x\right) =0$, the result in question is equivalent to \begin{equation*} \lim_{x\rightarrow \infty }\left( \ln \Gamma \left( x+1\right) -\ln \sqrt{ 2\pi }-K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) \ln \left( x+\frac{1}{2 }\right) +\left( x+\frac{1}{2}\right) \right) =0. \end{equation*} Clearly, we only need to prove that \begin{equation*} D_{6}:=\lim_{x\rightarrow \infty }\left( K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) -\left( x+\frac{1}{2}\right) \right) \ln \left( x+\frac{1}{2} \right) =0. \end{equation*} By the homogeneity of $K$, we get \begin{eqnarray*} &&D_{6}\!\overset{1/x=t}{=\!=\!=}\lim_{t\rightarrow 0^{+}}\frac{K\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) -\left( 1+t/2\right) }{t}\left( \ln \left( 1+\frac{t}{2}\right) -\ln t\right) \\ &=&\lim_{t\rightarrow 0^{+}}\frac{K\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) -\left( 1+t/2\right) }{t^{2}}\lim_{t\rightarrow 0^{+}}\left( t\ln \left( 1+\frac{t}{2}\right) -t\ln t\right) =0, \end{eqnarray*} where the first limit, by L'Hospital's rule, is equal to \begin{eqnarray*} &&\lim_{t\rightarrow 0^{+}}\frac{\epsilon K_{x}\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) +\epsilon ^{\ast }K_{y}\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) -1/2}{2t} \\ &=&\lim_{t\rightarrow 0^{+}}\frac{\epsilon ^{2}K_{xx}\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) +2\epsilon \epsilon ^{\ast }K_{xy}\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) +\epsilon ^{\ast }K_{yy}\left( 1+\epsilon t,1+\epsilon ^{\ast }t\right) }{2} \\ &=&\frac{\epsilon ^{2}K_{xx}\left( 1,1\right) +2\epsilon \epsilon ^{\ast }K_{xy}\left( 1,1\right) +\epsilon ^{\ast }K_{yy}\left( 1,1\right) }{2}=- \frac{\left( 2\epsilon -1\right) ^{2}}{2}K_{xy}\left( 1,1\right) , \end{eqnarray*} while the second one is clearly equal to zero. The proof ends. \end{proof} By the above three theorems, the following assertion is immediate. \begin{corollary} \label{MCg-form1}Suppose that (i) the function $K:\mathbb{R}^{2}\rightarrow \mathbb{R}$ is a symmetric, homogeneous and twice differentiable mean; (ii) the functions $M:\left( 0,\infty \right) \times \left( 0,\infty \right) \rightarrow \left( 0,\infty \right) $ and $N:\mathbb{R}^{2}\rightarrow \mathbb{R}$ are two symmetric, homogeneous, and differentiable means; (iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty ,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\theta ^{\ast },\sigma ,\sigma ^{\ast }$ with $\epsilon +\epsilon ^{\ast }=\theta +\theta ^{\ast }=\sigma +\sigma ^{\ast }=1$ such that $x>-\min \left( 1,\theta ,\theta ^{\ast }\right) $, we have \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\theta ^{\ast }\right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) }e^{-N\left( x+\sigma ,x+\sigma ^{\ast }\right) }e^{r\left( x\right) },\text{ as }x\rightarrow \infty . \end{equation*} \end{corollary} \begin{corollary} \label{MCg-form2}Suppose that (i) the function $K:\left( -\infty ,\infty \right) ^{2}\rightarrow \left( -\infty ,\infty \right) $ is a symmetric, homogeneous and twice differentiable mean; (ii) the functions $M,N:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ are two means; (iii) the function $r:\left( 0,\infty \right) \rightarrow \left( -\infty ,\infty \right) $ satisfies $\lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\epsilon ,\epsilon ^{\ast },\theta ,\sigma $ with $\epsilon +\epsilon ^{\ast }=1$ such that $x>-\min \left( 1,\theta ,\sigma \right) $, we have \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi }M\left( x+\theta ,x+\sigma \right) ^{K\left( x+\epsilon ,x+\epsilon ^{\ast }\right) }e^{-M\left( x+\theta ,x+\sigma \right) }e^{r\left( x\right) },\text{ as }x\rightarrow \infty . \end{equation*} \end{corollary} Further, it is obvious that our ideas constructing asymptotic formulas for the gamma function in terms of bivariate means can be extended to the psi and polygamma functions. \begin{theorem} Let $M:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ be a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $ \lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\theta $, $\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $, the asymptotic formula for the psi function \begin{equation*} \psi \left( x+1\right) \thicksim \ln M\left( x+\theta ,x+\sigma \right) +r\left( x\right) \end{equation*} holds as $x\rightarrow \infty $. \end{theorem} \begin{proof} It suffices to prove \begin{equation*} \lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln M\left( x+\theta ,x+\sigma \right) \right) =0. \end{equation*} Since $M$ is a mean, we have $x+\min \left( \theta ,\sigma \right) \leq M\left( x+\theta ,x+\sigma \right) \leq x+\max \left( \theta ,\sigma \right) $, and so \begin{equation*} \psi \left( x+1\right) -\ln \left( x+\max \left( \theta ,\sigma \right) \right) <\psi \left( x+1\right) -\ln M\left( x+\theta ,x+\sigma \right) <\psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma \right) \right) , \end{equation*} which yields the inquired result due to \begin{equation*} \lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln \left( x+\max \left( \theta ,\sigma \right) \right) \right) =\lim_{x\rightarrow \infty }\left( \psi \left( x+1\right) -\ln \left( x+\min \left( \theta ,\sigma \right) \right) \right) =0. \end{equation*} \end{proof} \begin{theorem} Let $M:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) $ be a mean and let $r$ be defined on $\left( 0,\infty \right) $ satisfying $ \lim_{x\rightarrow \infty }r\left( x\right) =0$. Then for fixed real numbers $\theta ,\sigma $ such that $x>-\min \left( 1,\theta ,\sigma \right) $, the asymptotic formula for the polygamma function \begin{equation*} \psi ^{(n)}\left( x+1\right) \thicksim \frac{\left( -1\right) ^{n-1}\left( n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma \right) }+r\left( x\right) \end{equation*} holds as $x\rightarrow \infty $. \end{theorem} \begin{proof} It suffices to show \begin{equation*} \lim_{x\rightarrow \infty }\left( \left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left( n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma \right) }\right) =0. \end{equation*} For this purpose, we utilize a known double inequality that for $k\in \mathbb{N}$ \begin{equation*} \frac{(k-1)!}{x^{k}}+\frac{k!}{2x^{k+1}}<\left( -1\right) ^{k+1}\psi ^{(k)}\left( x\right) <\frac{(k-1)!}{x^{k}}+\frac{k!}{x^{k+1}} \end{equation*} holds on $(0,\infty )$ proved by Guo and Qi in \cite[Lemma 3] {Guo-BKMS-47(1)-2010} to get \begin{equation*} \frac{k!}{2x^{k+1}}<\left( -1\right) ^{k+1}\psi ^{(k)}\left( x\right) -\frac{ (k-1)!}{x^{k}}<\frac{k!}{x^{k+1}}. \end{equation*} This implies that \begin{equation} \lim_{x\rightarrow \infty }\left( \left( -1\right) ^{k-1}\psi ^{(k)}\left( x\right) -\frac{(k-1)!}{x^{k}}\right) =0. \label{GQ} \end{equation} On the other hand, without loss of generality, we assume that $\theta \leq \sigma $. By the property of mean, we see that \begin{equation*} x+\theta \leq M\left( x+\theta ,x+\sigma \right) \leq x+\sigma , \end{equation*} and so \begin{eqnarray*} \left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left( n-1\right) !}{\left( x+\theta \right) ^{n}} &<&\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left( n-1\right) !}{M^{n}\left( x+\theta ,x+\sigma \right) } \\ &<&\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{1}{\left( x+\sigma \right) ^{n}}. \end{eqnarray*} Then, by (\ref{GQ}), for $a=\theta ,\sigma $, we get \begin{eqnarray*} &&\left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{\left( n-1\right) !}{\left( x+a\right) ^{n}} \\ &=&\left( \left( -1\right) ^{n-1}\psi ^{(n)}\left( x+1\right) -\frac{(n-1)!}{ \left( x+1\right) ^{n}}\right) +\left( \frac{(n-1)!}{\left( x+1\right) ^{n}}- \frac{\left( n-1\right) !}{\left( x+a\right) ^{n}}\right) \\ &\rightarrow &0+0=0\text{, as }x\rightarrow \infty , \end{eqnarray*} which gives the desired result. Thus we complete the proof. \end{proof} \section{Examples} In this section, we will list some examples to illustrate applications of Theorems \ref{MT-p2><p3} and \ref{MT-p2=p3}. To this end, we first recall the arithmetic mean $A$, geometric mean $G$, and identric (exponential) mean $I$ of two positive numbers $a$ and $b$ defined by \begin{eqnarray*} A\left( a,b\right) &=&\frac{a+b}{2}\text{, \ \ \ }G\left( a,b\right) =\sqrt{ ab}, \\ \mathcal{I}\left( a,b\right) &=&\left( b^{b}/a^{a}\right) ^{1/\left( b-a\right) }/e\text{ if }a\neq b\text{ and }I\left( a,a\right) =a, \end{eqnarray*} (see \cite{Stolarsky-MM-48-1975}, \cite{Yang-MPT-4-1987}). Clearly, these means are symmetric and homogeneous. Another possible mean is defined by \begin{equation} H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) =\frac{ \sum_{k=0}^{n}p_{k}a^{k}b^{n-k}}{\sum_{k=0}^{n-1}q_{k}a^{k}b^{n-1-k}}, \label{H^n,n-1} \end{equation} where \begin{equation} \sum_{k=0}^{n}p_{k}=\sum_{k=0}^{n-1}q_{k}=1. \label{pk-qk1} \end{equation} It is clear that $H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ is homogeneous and satisfies $H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,a\right) =a$. When $p_{k}=p_{n-k}$ and $q_{k}=q_{n-1-k}$, we denote $ H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ by $S_{^{p_{k};q_{k}}}^{n,n-1} \left( a,b\right) $, which can be expressed as \begin{equation} S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) =\frac{\sum_{k=0}^{[n/2]}p_{k} \left( ab\right) ^{k}\left( a^{n-2k}+b^{n-2k}\right) }{\sum_{k=0}^{[\left( n-1\right) /2]}q_{k}\left( ab\right) ^{k}\left( a^{n-1-2k}+b^{n-1-2k}\right) }, \label{S^n,n-1} \end{equation} where $p_{k}$ and $q_{k}$ satisfy \begin{equation} \sum_{k=0}^{[n/2]}\left( 2p_{k}\right) =\sum_{k=0}^{[\left( n-1\right) /2]}\left( 2q_{k}\right) =1, \label{pk-qk2} \end{equation} $[x]$ denotes the integer part of real number $x$. Evidently, $ S_{^{p_{k};q_{k}}}^{n,n-1}$ is symmetric and homogeneous, and $ S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,a\right) =a$. But $ H_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ and $ S_{^{p_{k};q_{k}}}^{n,n-1}\left( a,b\right) $ are not always means of $a$ and $b$. For instance, when $p=2/3$, \begin{equation*} S_{^{p;1/2}}^{2,1}\left( a,b\right) =\frac{pa^{2}+pb^{2}+\left( 1-2p\right) ab}{\left( a+b\right) /2}=\frac{2}{3}\frac{2a^{2}+2b^{2}-ab}{a+b}>\max (a,b) \end{equation*} in the case of $\max (a,b)>4\min \left( a,b\right) $. Indeed, it is easy to prove that $S_{^{p;1/2}}^{2,1}\left( a,b\right) $ is a mean if and only if $ p\in \lbrack 0,1/2]$. Secondly, we recall the so-called completely monotone functions. A function $ f$ is said to be completely monotonic on an interval $I$ , if $f$ has derivatives of all orders on $I$ and satisfies \begin{equation} (-1)^{n}f^{(n)}(x)\geq 0\text{ for all }x\in I\text{ and }n=0,1,2,.... \label{cm} \end{equation} If the inequality (\ref{cm}) is strict, then $f$ is said to be strictly completely monotonic on $I$. It is known (Bernstein's Theorem) that $f$ is completely monotonic on $(0,\infty )$ if and only if \begin{equation*} f(x)=\int_{0}^{\infty }e^{-xt}d\mu \left( t\right) , \end{equation*} where $ \mu $ is a nonnegative measure on $[0,\infty )$ such that the integral converges for all $x>0$, see \cite[p. 161]{Widder-PUPP-1941}. \begin{example} Let \begin{eqnarray*} K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2}, \\ M\left( a,b\right) &=&A^{2/3}\left( a,b\right) G^{1/3}\left( a,b\right) =\left( \frac{a+b}{2}\right) ^{2/3}\left( \sqrt{ab}\right) ^{1/3} \end{eqnarray*} and $\theta =\sigma =0$ in Theorem \ref{MT-p2><p3}. Then we can obtain an asymptotic formulas for the gamma function as follows. \begin{eqnarray*} \ln \Gamma (x+1) &\thicksim &\frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2} \right) \ln \left( \left( x+\frac{1}{2}\right) ^{2/3}\left( \sqrt{x\left( x+1\right) }\right) ^{1/3}\right) -\left( x+\frac{1}{2}\right) \\ &=&\frac{1}{2}\ln 2\pi +\frac{2}{3}\left( x+\frac{1}{2}\right) \ln \left( x+ \frac{1}{2}\right) +\frac{1}{6}\left( x+\frac{1}{2}\right) \ln x \\ &&+\frac{1}{6}\left( x+\frac{1}{2}\right) \ln \left( x+1\right) -\left( x+ \frac{1}{2}\right) ,\text{ as }x\rightarrow \infty . \end{eqnarray*} \end{example} Further, we can prove \begin{proposition} For $x>0$, the function \begin{eqnarray*} f_{1}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\frac{2}{3}\left( x+\frac{1 }{2}\right) \ln \left( x+\frac{1}{2}\right) -\frac{1}{6}\left( x+\frac{1}{2} \right) \ln x \\ &&-\frac{1}{6}\left( x+\frac{1}{2}\right) \ln \left( x+1\right) +\left( x+ \frac{1}{2}\right) \end{eqnarray*} is a completely monotone function. \end{proposition} \begin{proof} Differentiating and utilizing the relations \begin{equation} \psi (x)=\int_{0}^{\infty }\left( \frac{e^{-t}}{t}-\frac{e^{-xt}}{1-e^{-t}} \right) dt\text{ \ and \ }\ln x=\int_{0}^{\infty }\frac{e^{-t}-e^{-xt}}{t}dt \label{psi-ln} \end{equation} yield \begin{eqnarray*} f_{1}^{\prime }(x) &=&\psi \left( x+1\right) -\frac{1}{6}\ln \left( x+1\right) -\frac{1}{6}\ln x-\frac{2}{3}\ln \left( x+\frac{1}{2}\right) + \frac{1}{12\left( x+1\right) }-\frac{1}{12x} \\ &=&\int_{0}^{\infty }\left( \frac{e^{-t}}{t}-\frac{e^{-\left( x+1\right) t}}{ 1-e^{-t}}\right) dt-\int_{0}^{\infty }\frac{e^{-t}-e^{-xt}}{6t} dt-\int_{0}^{\infty }\frac{e^{-t}-e^{-\left( x+1\right) t}}{6t}dt \\ &&-\int_{0}^{\infty }\frac{2\left( e^{-t}-e^{-\left( x+1/2\right) t}\right) }{3t}dt+\frac{1}{12}\int_{0}^{\infty }e^{-\left( x+1\right) t}dt-\frac{1}{12} \int_{0}^{\infty }e^{-xt}dt \\ &=&\int_{0}^{\infty }e^{-xt}\left( \frac{1}{6t}+\frac{e^{-t}}{6t}+\frac{ 2e^{-t/2}}{3t}-\frac{e^{-t/2}}{1-e^{-t}}+\frac{1}{12}\left( e^{-t}-1\right) \right) dt \\ &=&\int_{0}^{\infty }e^{-xt}e^{-t/2}\left( \frac{\cosh \left( t/2\right) }{3t }+\frac{2}{3t}-\frac{1}{2\sinh \left( t/2\right) }-\frac{1}{6}\sinh \frac{t}{ 2}\right) dt \\ &:&=\int_{0}^{\infty }e^{-xt}e^{-t/2}u\left( \frac{t}{2}\right) dt, \end{eqnarray*} where \begin{equation*} u\left( t\right) =\frac{\cosh t}{6t}+\frac{1}{3t}-\frac{1}{2\sinh t}-\frac{1 }{6}\sinh t. \end{equation*} Factoring and expanding in power series lead to \begin{eqnarray*} u\left( t\right) &=&-\frac{t\cosh 2t-\sinh 2t-4\sinh t+5t}{12t\sinh t} \\ &=&-\frac{\sum_{n=1}^{\infty }\frac{2^{2n-2}t^{2n-1}}{\left( 2n-2\right) !} -\sum_{n=1}^{\infty }\frac{2^{2n-1}t^{2n-1}}{\left( 2n-1\right) !} -4\sum_{n=1}^{\infty }\frac{t^{2n-1}}{\left( 2n-1\right) !}+5t}{12t\sinh \left( t/2\right) } \\ &=&-\frac{\sum_{n=3}^{\infty }\frac{\left( 2n-3\right) 2^{2n-2}-4}{\left( 2n-1\right) !}t^{2n-1}}{12t\sinh t}<0 \end{eqnarray*} for $t>0$. This reveals that $-f_{1}^{\prime }$ is a completely monotone function, which together with $f_{1}(x)>\lim_{x\rightarrow \infty }f_{1}(x)=0 $ leads us to the desired result. \end{proof} Using the decreasing property of $f_{1}$ on $\left( 0,\infty \right) $ and notice that \begin{equation*} f_{1}(1)=\ln \frac{2^{3/4}e^{3/2}}{3\sqrt{2\pi }}\text{ \ and \ } f_{1}(\infty )=0 \end{equation*} we immediately get \begin{corollary} For $n\in \mathbb{N}$, it is true that \begin{equation*} \sqrt{2\pi }\left( \frac{(n+1/2)^{4}n\left( n+1\right) }{e^{6}}\right) ^{\left( n+1/2\right) /6}<n!<\frac{2^{3/4}e^{3/2}}{3}\left( \frac{ (n+1/2)^{4}n\left( n+1\right) }{e^{6}}\right) ^{\left( n+1/2\right) /6}, \end{equation*} with the optimal constants $\sqrt{2\pi }\approx 2.5066$ and $ 2^{3/4}e^{3/2}/3\approx 2.5124$. \end{corollary} \begin{example} Let \begin{eqnarray*} K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2}, \\ M\left( a,b\right) &=&\mathcal{I}\left( a,b\right) =\left( b^{b}/a^{a}\right) ^{1/\left( b-a\right) }/e\text{ if }a\neq b\text{ and } I\left( a,a\right) =a \end{eqnarray*} and $\theta =0$ in Theorem \ref{MT-p2><p3}. Then we get the asymptotic formulas: \begin{equation*} \ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \left( (x+1)\ln (x+1)-x\ln x-1\right) -\left( x+\frac{1}{2}\right) , \end{equation*} as $x\rightarrow \infty $. \end{example} And, we have \begin{proposition} For $x>0$, the function \begin{equation*} f_{2}(x)=\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+\frac{1}{2}\right) \left( (x+1)\ln (x+1)-x\ln x-1\right) +x+\frac{1}{2} \end{equation*} is a completely monotone function. \end{proposition} \begin{proof} Differentiation gives \begin{eqnarray*} f_{2}^{\prime }(x) &=&\psi \left( x+1\right) -\left( 2x+\frac{3}{2}\right) \ln \left( x+1\right) +\left( 2x+\frac{1}{2}\right) \ln x+2, \\ f_{2}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -2\ln \left( x+1\right) +2\ln x+\frac{1}{2\left( x+1\right) }+\frac{1}{2x}. \end{eqnarray*} Application of the relations (\ref{psi-ln}), $f_{2}^{\prime \prime }(x)$ can be expressed as \begin{eqnarray*} f_{2}^{\prime \prime }(x) &=&\int_{0}^{\infty }t\frac{e^{-\left( x+1\right) t}}{1-e^{-t}}dt-2\int_{0}^{\infty }\frac{e^{-xt}-e^{-\left( x+1\right) t}}{t} dt+\frac{1}{2}\int_{0}^{\infty }\left( e^{-\left( x+1\right) t}+e^{-xt}\right) dt \\ &=&\int_{0}^{\infty }e^{-xt}\left( \frac{te^{-t}}{1-e^{-t}}-2\frac{1-e^{-t}}{ t}+\frac{1}{2}\left( e^{-t}+1\right) \right) dt \\ &=&\int_{0}^{\infty }e^{-xt}e^{-t/2}\left( \frac{t}{2\sinh \left( t/2\right) }-4\frac{\sinh \left( t/2\right) }{t}+\cosh \frac{t}{2}\right) dt \\ &:&=\int_{0}^{\infty }e^{-xt}e^{-t/2}v\left( \tfrac{t}{2}\right) dt, \end{eqnarray*} where \begin{equation*} v\left( t\right) =\frac{t}{\sinh t}-2\frac{\sinh t}{t}+\cosh t. \end{equation*} Employing hyperbolic version of Wilker inequality proved in \cite {Zhu-MIA-10(4)-2007} (also see \cite{Zhu-AAA-485842-2009}, \cite {Yang-JIA-2014-166}) \begin{equation*} \left( \frac{t}{\sinh t}\right) ^{2}+\frac{t}{\tanh t}>2, \end{equation*} we get \begin{equation*} \frac{\sinh t}{t}v\left( t\right) =\left( \frac{t}{\sinh t}\right) ^{2}+ \frac{t}{\tanh t}-2>0, \end{equation*} and so $f_{2}^{\prime \prime }(x)$ is complete monotone for $x>0$. Hence, $ f_{2}^{\prime }(x)<\lim_{x\rightarrow \infty }f_{2}^{\prime }(x)=0$, and then, $f_{2}(x)>\lim_{x\rightarrow \infty }f_{2}(x)=0$, which indicate that $ f_{2}$ is complete monotone for $x>0$. This completes the proof. \end{proof} The decreasing property of $f_{2}$ on $\left( 0,\infty \right) $ and the facts that \begin{equation*} f_{2}\left( 0^{+}\right) =\ln \frac{e}{\sqrt{2\pi }}\text{, \ }f_{2}\left( 1\right) =\ln \frac{e^{3}}{8}\text{, \ }f_{2}\left( \infty \right) =0 \end{equation*} give the following \begin{corollary} For $x>0$, the sharp double inequality \begin{equation*} \sqrt{2\pi }e^{-2x-1}\frac{(x+1)^{(x+1)\left( x+1/2\right) }}{x^{x\left( x+1/2\right) }}<\Gamma (x+1)<e^{-2x}\frac{(x+1)^{(x+1)\left( x+1/2\right) }}{ x^{x\left( x+1/2\right) }} \end{equation*} holds. For $n\in \mathbb{N}$, it holds that \begin{equation*} \sqrt{2\pi }e^{-2n-1}\frac{(n+1)^{(n+1)\left( n+1/2\right) }}{n^{n\left( n+1/2\right) }}<n!<\frac{e^{3}}{8}e^{-2n-1}\frac{(n+1)^{(n+1)\left( n+1/2\right) }}{n^{n\left( n+1/2\right) }} \end{equation*} with the best constants $\sqrt{2\pi }\approx 2.5066$ and $e^{3}/8\approx 2.5107$. \end{corollary} \begin{example} \label{E-M3,2}Let \begin{eqnarray*} K\left( a,b\right) &=&N\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2}, \\ M\left( a,b\right) &=&M_{^{p;q}}^{3,2}\left( a,b\right) =\frac{ pa^{3}+pb^{3}+\left( 1/2-p\right) a^{2}b+\left( 1/2-p\right) ab^{2}}{ qa^{2}+qb^{2}+(1-2q)ab} \\ &=&\frac{a+b}{2}\frac{2pa^{2}+2pb^{2}+\left( 1-4p\right) ab}{ qa^{2}+qb^{2}+\left( 1-2q\right) ab} \end{eqnarray*} and $\theta =0$ in Theorem \ref{MT-p2><p3}, where $p$ and $q$ are parameters to be determined. Then, we have \begin{eqnarray*} K\left( x,x+1\right) &=&N\left( x,x+1\right) =x+\frac{1}{2}, \\ M\left( x,x+1\right) &=&S_{^{p;q}}^{3,2}\left( x,x+1\right) =\left( x+1/2\right) \frac{x^{2}+x+2p}{x^{2}+x+q}. \end{eqnarray*} Straightforward computations give \begin{eqnarray*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left( x+1/2\right) \ln M_{p;q}^{3,2}\left( x,x+1\right) +x+1/2}{x^{-1}} &=&q-2p- \frac{1}{24}, \\ \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left( x+1/2\right) \ln M_{p;2p+1/24}^{3,2}\left( x,x+1\right) +x+1/2}{x^{-3}} &=&- \frac{160}{1920}\left( p-\frac{23}{160}\right) , \end{eqnarray*} and solving the equation set \begin{equation*} q-2p-\frac{1}{24}=0\text{ and }-\frac{160}{1920}\left( p-\frac{23}{160} \right) =0 \end{equation*} leads to \begin{equation*} p=\frac{23}{160},q=\frac{79}{240}. \end{equation*} And then, \begin{equation*} M\left( x,x+1\right) =\left( x+\frac{1}{2}\right) \frac{x^{2}+x+\frac{23}{80} }{x^{2}+x+\frac{79}{240}}. \end{equation*} It is easy to check that $S_{^{p;q}}^{3,2}\left( a,b\right) $ is a symmetric and homogeneous mean of positive numbers $a$ and $b$ for $p=23/160$, $ q=79/240$. Hence, by Theorem \ref{MT-p2><p3}, we have the optimal asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{x^{2}+x+79/240} -\left( x+\frac{1}{2}\right) , \end{equation*} as $x\rightarrow \infty $, and \begin{equation*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\ln \sqrt{2\pi }-\left( x+1/2\right) \ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{ x^{2}+x+79/240}+x+1/2}{x^{-5}}=-\tfrac{18\,029}{29\,030\,400}. \end{equation*} \end{example} Also, this asymptotic formula have a well property. \begin{proposition} For $x>-1/2$, the function $f_{3}$ defined by \begin{equation} f_{3}\left( x\right) =\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+\frac{1 }{2}\right) \ln \tfrac{\left( x+1/2\right) \left( x^{2}+x+23/80\right) }{ x^{2}+x+79/240}+\left( x+\frac{1}{2}\right) . \label{f3} \end{equation} is increasing and concave. \end{proposition} \begin{proof} Differentiation gives \begin{eqnarray*} f_{3}^{\prime }\left( x\right) &=&\psi \left( x+1\right) +\ln \left( x^{2}+x+ \frac{79}{240}\right) -\ln \left( x^{2}+x+\frac{23}{80}\right) \\ &&-\ln \left( x+\frac{1}{2}\right) -2\frac{\left( x+1/2\right) ^{2}}{ x^{2}+x+23/80}+2\frac{\left( x+1/2\right) ^{2}}{x^{2}+x+79/240}, \end{eqnarray*} \begin{eqnarray*} f_{3}^{\prime \prime }\left( x\right) &=&\psi ^{\prime }\left( x+1\right) +6 \frac{x+1/2}{x^{2}+x+79/240}-6\frac{x+1/2}{x^{2}+x+23/80} \\ &&-\frac{1}{x+1/2}+4\frac{\left( x+1/2\right) ^{3}}{\left( x^{2}+x+23/80\right) ^{2}}-4\frac{\left( x+1/2\right) ^{3}}{\left( x^{2}+x+79/240\right) ^{2}}. \end{eqnarray*} Denote by $x+1/2=t$ and make use of recursive relation \begin{equation} \psi ^{\left( n\right) }(x+1)-\psi ^{\left( n\right) }(x)=\left( -1\right) ^{n}\frac{n!}{x^{n+1}} \label{psi-rel.} \end{equation} yield \begin{eqnarray*} &&f_{3}^{\prime \prime }(t+\frac{1}{2})-f_{3}^{\prime \prime }(t-\frac{1}{2}) \\ &=&-\tfrac{1}{\left( t+1/2\right) ^{2}}+6\tfrac{t+1}{\left( t+1\right) ^{2}+19/240}-6\tfrac{t+1}{\left( t+1\right) ^{2}+3/80}-\frac{1}{\left( t+1\right) }+4\tfrac{\left( t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+3/80\right) ^{2}} \\ &&-4\tfrac{\left( t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+19/240\right) ^{2}}-\left( 6\tfrac{t}{t^{2}+19/240}-6\tfrac{t}{ t^{2}+3/80}-\frac{1}{t}+4\tfrac{t^{3}}{\left( t^{2}+3/80\right) ^{2}}-4 \tfrac{t^{3}}{\left( t^{2}+19/240\right) ^{2}}\right) \\ &=&\frac{f_{31}\left( t\right) }{t\left( t+1\right) \left( t+\frac{1}{2} \right) ^{2}\left( t^{2}+2t+83/80\right) ^{2}\left( t^{2}+3/80\right) ^{2}\left( t^{2}+2t+259/240\right) ^{2}\left( t^{2}+19/240\right) ^{2}}, \end{eqnarray*} where \begin{eqnarray*} f_{31}\left( t\right) &=&\tfrac{18\,029}{138\,240}t^{12}+\tfrac{18\,029}{ 23\,040}t^{11}+\tfrac{83\,674\,657}{41\,472\,000}t^{10}+\tfrac{24\,178\,957}{ 8294\,400}t^{9}+\tfrac{34\,366\,211\,867}{13\,271\,040\,000}t^{8}+\tfrac{ 4894\,651\,067}{3317\,760\,000}t^{7} \\ &&+\tfrac{74\,296\,657\,243}{132\,710\,400\,000}t^{6}+\tfrac{ 20\,147\,292\,749}{132\,710\,400\,000}t^{5}+\tfrac{297\,092\,035\,417}{ 9437\,184\,000\,000}t^{4}+\tfrac{66\,777\,391\,051}{14\,155\,776\,000\,000} t^{3} \\ &&+\tfrac{295\,012\,866\,563}{566\,231\,040\,000\,000}t^{2}+\tfrac{ 3972\,595\,981}{188\,743\,680\,000\,000}t+\tfrac{166\,825\,684\,249}{ 60\,397\,977\,600\,000\,000} \\ &>&0\text{ for }t=x+1/2>0\text{.} \end{eqnarray*} This shows that $f_{3}^{\prime \prime }(t+\frac{1}{2})-f_{3}^{\prime \prime }(t-\frac{1}{2})>0$, that is, $f_{3}^{\prime \prime }(x+1)-f_{3}^{\prime \prime }(x)>0$, and so \begin{equation*} f_{3}^{\prime \prime }(x)<f_{3}^{\prime \prime }(x+1)<f_{3}^{\prime \prime }(x+2)<...<f_{3}^{\prime \prime }(\infty )=0. \end{equation*} It reveals that shows $f_{3}$ is concave on $\left( -1/2,\infty \right) $, and we conclude that, $f_{3}^{\prime }(x)>\lim_{x\rightarrow \infty }f_{3}^{\prime }(x)=0$, which proves the desired result. \end{proof} As a consequence of the above proposition, we have \begin{corollary} For $x>0$, the double inequality \begin{equation*} \sqrt{\tfrac{158e}{69}}\left( \tfrac{x+1/2}{e}\tfrac{x^{2}+x+23/80}{ x^{2}+x+79/240}\right) ^{x+1/2}<\Gamma (x+1)<\sqrt{2\pi }\left( \tfrac{x+1/2 }{e}\tfrac{x^{2}+x+23/80}{x^{2}+x+79/240}\right) ^{x+1/2} \end{equation*} holds true, where $\sqrt{158e/69}\approx 2.4949$ and and $\sqrt{2\pi } \approx 2.5066$ are the best. For $n\in \mathbb{N}$, it is true that \begin{equation*} \left( \tfrac{1118e}{1647}\right) ^{3/2}\left( \tfrac{n+1/2}{e}\tfrac{ n^{2}+n+23/80}{n^{2}+n+79/240}\right) ^{n+1/2}<n!<\sqrt{2\pi }\left( \tfrac{ n+1/2}{e}\tfrac{n^{2}+n+23/80}{n^{2}+n+79/240}\right) ^{n+1/2} \end{equation*} holds true with the best constants $\left( 1118e/1647\right) ^{3/2}\approx 2.5065$ and $\sqrt{2\pi }\approx 2.5066$. \end{corollary} \begin{example} \label{E-N3,2}Let \begin{eqnarray*} K\left( a,b\right) &=&M\left( a,b\right) =A\left( a,b\right) =\frac{a+b}{2}, \\ N\left( a,b\right) &=&S_{^{p;q}}^{3,2}\left( a,b\right) =\frac{ pa^{3}+pb^{3}+\left( 1/2-p\right) ab^{2}+\left( 1/2-p\right) a^{2}b}{ qa^{2}+qb^{2}+\left( 1-2q\right) ab} \\ &=&\frac{a+b}{2}\frac{2pa^{2}+2pb^{2}+\left( 1-4p\right) ab}{ qa^{2}+qb^{2}+\left( 1-2q\right) ab} \end{eqnarray*} and $\sigma =0$ in Theorem \ref{MT-p2><p3}, where $p$ and $q$ are parameters to be determined. Direct computations give \begin{eqnarray*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{ x^{2}+x+2p}{x^{2}+x+q}}{x^{-1}} &=&2p-q-\frac{1}{24}, \\ \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{ x^{2}+x+2p}{x^{2}+x+2p-1/24}}{x^{-3}} &=&\frac{7}{480}-\frac{1}{12}p. \end{eqnarray*} Solving the simultaneous equations \begin{eqnarray*} 2p-q-\frac{1}{24} &=&0, \\ \frac{7}{480}-\frac{1}{12}p &=&0 \end{eqnarray*} leads to $p=7/40$, $q=37/120$. And then, \begin{equation*} N\left( x,x+1\right) =\left( x+1/2\right) \frac{x^{2}+x+7/20}{x^{2}+x+37/120} . \end{equation*} An easy verification shows that $S_{^{p;q}}^{3,2}\left( a,b\right) $ is a symmetric and homogeneous mean of positive numbers $a$ and $b$ for $p=7/40$, $q=37/120$. Hence, by Theorem \ref{MT-p2><p3} we get the best asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\thicksim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln \left( x+\frac{1}{2}\right) -\left( x+\frac{1}{2}\right) \frac{ x^{2}+x+7/20}{x^{2}+x+37/120}, \end{equation*} as $x\rightarrow \infty $. And we have \begin{equation*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln \left( x+1/2\right) +\left( x+1/2\right) \frac{ x^{2}+x+7/20}{x^{2}+x+37/120}}{x^{-5}}=-\frac{1517}{2419\,200}. \end{equation*} \end{example} Now we prove the following assertion related to this asymptotic formula. \begin{proposition} Let the function $f_{4}$ be defined on $\left( -1/2,\infty \right) $ by \begin{equation*} f_{4}(x)=\ln \Gamma (x+1)-\tfrac{1}{2}\ln 2\pi -\left( x+\tfrac{1}{2}\right) \ln (x+\tfrac{1}{2})+\left( x+\tfrac{1}{2}\right) \frac{x^{2}+x+7/20}{ x^{2}+x+37/120}. \end{equation*} Then $f_{4}$ is increasing and convex on $\left( -1/2,\infty \right) $. \end{proposition} \begin{proof} Differentiation gives \begin{eqnarray*} f_{4}^{\prime }(x) &=&\psi \left( x+1\right) -\ln \left( x+\frac{1}{2} \right) +\frac{1}{24}\frac{1}{x^{2}+x+37/120}-\frac{1}{12}\frac{\left( x+1/2\right) ^{2}}{\left( x^{2}+x+37/120\right) ^{2}}, \\ f_{4}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -\frac{1}{x+1/2 }-\frac{1}{4}\frac{x+1/2}{\left( x^{2}+x+37/120\right) ^{2}}+\frac{1}{3} \frac{\left( x+\frac{1}{2}\right) ^{3}}{\left( x^{2}+x+37/120\right) ^{3}}. \end{eqnarray*} Denote by $x+1/2=t$ and make use of recursive relation (\ref{psi-rel.}) yield \begin{eqnarray*} &&f_{4}^{\prime \prime }(t+\frac{1}{2})-f_{4}^{\prime \prime }(t-\frac{1}{2}) \\ &=&-\tfrac{1}{\left( t+1/2\right) ^{2}}-\frac{1}{t+1}-\frac{1}{4}\frac{t+1}{ \left( \left( t+1\right) ^{2}+7/120\right) ^{2}}+\frac{1}{3}\frac{\left( t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+7/120\right) ^{3}} \\ &&-\left( -\frac{1}{t}-\frac{1}{4}\frac{t}{\left( t^{2}+7/120\right) ^{2}}+ \frac{1}{3}\frac{t^{3}}{\left( t^{2}+7/120\right) ^{3}}\right) \\ &=&\frac{f_{41}\left( t\right) }{t\left( t+1\right) \left( t+1/2\right) ^{2}\left( t^{2}+7/120\right) ^{3}\left( t^{2}+2t+127/120\right) ^{3}}, \end{eqnarray*} where \begin{eqnarray*} f_{41}\left( t\right) &=&\frac{1517}{11\,520}t^{8}+\frac{1517}{2880}t^{7}+ \frac{161\,087}{192\,000}t^{6}+\frac{387\,883}{576\,000}t^{5}+\frac{ 39\,563\,149}{138\,240\,000}t^{4} \\ &&+\frac{4462\,549}{69\,120\,000}t^{3}+\frac{67\,788\,161}{8294\,400\,000} t^{2}+\frac{2794\,421}{8294\,400\,000}t+\frac{702\,595\,369}{ 11\,943\,936\,000\,000} \\ &>&0\text{ for }t=x+1/2>0. \end{eqnarray*} This implies that $f_{4}^{\prime \prime }(t+\frac{1}{2})-f_{4}^{\prime \prime }(t-\frac{1}{2})>0$, that is, $f_{4}^{\prime \prime }(x+1)-f_{4}^{\prime \prime }(x)>0$, and so \begin{equation*} f_{4}^{\prime \prime }(x)<f_{4}^{\prime \prime }(x+1)<f_{4}^{\prime \prime }(x+2)<...<f_{4}^{\prime \prime }(\infty )=0. \end{equation*} It reveals that shows $f_{4}$ is concave on $\left( -1/2,\infty \right) $, and therefore, $f_{4}^{\prime }(x)>\lim_{x\rightarrow \infty }f_{4}^{\prime }(x)=0$, which proves the desired result. \end{proof} By the increasing property of $f_{4}$ on $\left( -1/2,\infty \right) $ and the facts \begin{equation*} f_{4}\left( 0\right) =\ln \frac{e^{21/37}}{\sqrt{\pi }}\text{, \ } f_{4}\left( 1\right) =\ln \frac{2e^{423/277}}{3\sqrt{3\pi }}\text{, \ } f_{4}\left( \infty \right) =0, \end{equation*} we have \begin{corollary} For $x>0$, the double inequality \begin{equation*} e^{21/37}\sqrt{2}\left( \tfrac{x+1/2}{\exp \left( \frac{x^{2}+x+7/20}{ x^{2}+x+37/120}\right) }\right) ^{x+1/2}<\Gamma (x+1)<\sqrt{2\pi }\left( \tfrac{x+1/2}{\exp \left( \frac{x^{2}+x+7/20}{x^{2}+x+37/120}\right) } \right) ^{x+1/2} \end{equation*} holds, where $e^{21/37}\sqrt{2}\approx 2.4946$ and $\sqrt{2\pi }\approx 2.5066$ are the best. For $n\in \mathbb{N}$, the double inequality \begin{equation*} e^{423/277}\tfrac{2\sqrt{2}}{3\sqrt{3}}(\tfrac{n+1/2}{e})^{n+1/2}\exp \left( -\tfrac{1}{24}\tfrac{n+1/2}{n^{2}+n+37/120}\right) <n!<\sqrt{2\pi }(\tfrac{ n+1/2}{e})^{n+1/2}\exp \left( -\tfrac{1}{24}\tfrac{n+1/2}{n^{2}+n+37/120} \right) \end{equation*} holds true with the best constants $2\sqrt{2}e^{423/277}/\left( 3\sqrt{3} \right) \approx 2.5065$ and $\sqrt{2\pi }\approx 2.5066$. \end{corollary} \begin{example} \label{E-N4,3}Let \begin{eqnarray*} K\left( a,b\right) &=&M\left( a,b\right) =A\left( a,b\right) =x+1/2, \\ N\left( a,b\right) &=&S_{^{p,q;r}}^{4,3}\left( a,b\right) =\frac{ pa^{4}+pb^{4}+qa^{3}b+qab^{3}+\left( 1-2p-2q\right) a^{2}b^{2}}{ ra^{3}+rb^{3}+\left( 1/2-r\right) a^{2}b+\left( 1/2-r\right) ab^{2}} \end{eqnarray*} and $\sigma =0$ in Theorem \ref{MT-p2><p3}. In a similar way, we can determine that the best parameters satisfy \begin{equation*} r=2p+\frac{1}{2}q-\frac{7}{48}\text{, \ }p=\frac{21}{40}-\frac{7}{4}q\text{, \ }q=\frac{7303}{35\,280}, \end{equation*} which imply \begin{equation*} p=\frac{3281}{20\,160},q=\frac{7303}{35\,280};r=\frac{111}{392}. \end{equation*} Then, \begin{equation} N\left( x,x+1\right) =x+\tfrac{1}{2}+\tfrac{1517}{44\,640}\tfrac{1}{x+1/2}+ \tfrac{343}{44\,640}\tfrac{x+1/2}{x^{2}+x+111/196}:=N_{4/3}\left( x,x+1\right) , \label{N4/3} \end{equation} In this case, we easily check that $S_{^{p,q;r}}^{4,3}\left( a,b\right) $ is a mean of $a$ and $b$. Consequently, from Theorem \ref{MT-p2><p3} the following best asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+1/2\right) \ln (x+1/2)-N_{4/3}\left( x,x+1\right) \end{equation*} holds true as $x\rightarrow \infty $. And, we have \begin{equation*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln \left( x+1/2\right) +N_{4/3}\left( x,x+1\right) }{ x^{-7}}=\tfrac{10\,981}{31\,610\,880}. \end{equation*} \end{example} We now present the monotonicity and convexity involving this asymptotic formula. \begin{proposition} Let $f_{5}$ defined on $\left( -1/2,\infty \right) $ by \begin{equation*} f_{5}(x)=\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln (x+1/2)+N_{4/3}\left( x,x+1\right) , \end{equation*} where $N_{4/3}\left( x,x+1\right) $ is defined\ by (\ref{N4/3}). Then $f_{5}$ is decreasing and convex on $\left( -1/2,\infty \right) $. \end{proposition} \begin{proof} Differentiation gives \begin{eqnarray*} f_{5}^{\prime }(x) &=&\psi \left( x+1\right) -\ln \left( x+\frac{1}{2} \right) -\frac{1517}{44\,640\left( x+1/2\right) ^{2}} \\ &&+\frac{343}{44\,640\left( x^{2}+x+111/196\right) }-\frac{343}{22\,320} \frac{\left( x+1/2\right) ^{2}}{\left( x^{2}+x+111/196\right) ^{2}}, \end{eqnarray*} \begin{eqnarray*} f_{5}^{\prime \prime }(x) &=&\psi ^{\prime }\left( x+1\right) -\frac{1}{x+1/2 }+\frac{1517}{22\,320\left( x+1/2\right) ^{3}} \\ &&-\frac{343}{7440}\frac{x+1/2}{\left( x^{2}+x+111/196\right) ^{2}}+\frac{343 }{5580}\frac{\left( x+1/2\right) ^{3}}{\left( x^{2}+x+111/196\right) ^{3}}. \end{eqnarray*} Denote by $x+1/2=t$ and make use of recursive relation (\ref{psi-rel.}) yield \begin{eqnarray*} &&f_{5}^{\prime \prime }(t+\frac{1}{2})-f_{5}^{\prime \prime }(t-\frac{1}{2}) \\ &=&-\tfrac{1}{\left( t+1/2\right) ^{2}}-\tfrac{1}{\left( t+1\right) }+\tfrac{ 1517}{22\,320\left( t+1\right) ^{3}}-\tfrac{343}{7440}\tfrac{t+1}{\left( \left( t+1\right) ^{2}+31/98\right) ^{2}}+\tfrac{343}{5580}\tfrac{\left( t+1\right) ^{3}}{\left( \left( t+1\right) ^{2}+31/98\right) ^{3}} \\ &&-\left( -\tfrac{1}{t}+\tfrac{1517}{22\,320t^{3}}-\tfrac{343}{7440}\tfrac{t }{\left( t^{2}+31/98\right) ^{2}}+\tfrac{343}{5580}\tfrac{t^{3}}{\left( t^{2}+31/98\right) ^{3}}\right) \\ &=&-\frac{f_{51}\left( t\right) }{80\left( t+1/2\right) ^{2}t^{3}\left( t+1\right) ^{3}\left( t^{2}+2t+129/98\right) ^{3}\left( t^{2}+31/98\right) ^{3}}, \end{eqnarray*} where \begin{eqnarray*} f_{51}\left( t\right) &=&\tfrac{10\,981}{784}t^{10}+\tfrac{54\,905}{784} t^{9}+\tfrac{21\,028\,039}{134\,456}t^{8}+\tfrac{27\,614\,911}{134\,456} t^{7}+\tfrac{294\,820\,517}{1647\,086}t^{6}+\tfrac{739\,744\,471}{6588\,344} t^{5}+ \\ &&\tfrac{138\,266\,105\,451}{2582\,630\,848}t^{4}+\tfrac{25\,165\,604\,049}{ 1291\,315\,424}t^{3}+\tfrac{2726\,271\,884\,261}{506\,195\,646\,208}t^{2}+ \tfrac{574\,150\,150\,569}{506\,195\,646\,208}t+\tfrac{347\,724\,739\,077}{ 3543\,369\,523\,456} \\ &>&0\text{ for }t=x+1/2>0\text{.} \end{eqnarray*} This implies that $f_{5}^{\prime \prime }(t+\frac{1}{2})-f_{5}^{\prime \prime }(t-\frac{1}{2})<0$, that is, $f_{5}^{\prime \prime }(x+1)-f_{5}^{\prime \prime }(x)<0$, and so \begin{equation*} f_{5}^{\prime \prime }(x)>f_{5}^{\prime \prime }(x+1)>f_{5}^{\prime \prime }(x+2)>...>f_{5}^{\prime \prime }(\infty )=0. \end{equation*} It reveals that shows $f_{5}$ is convex on $\left( -1/2,\infty \right) $, and therefore, $f_{5}^{\prime }(x)<\lim_{x\rightarrow \infty }f_{5}^{\prime }(x)=0$, which proves the desired statement. \end{proof} Employing the decreasing property of $f_{5}$ on $\left( -1/2,\infty \right) $ , we obtain \begin{corollary} For $x>0$, the double inequality \begin{eqnarray*} &&\sqrt{2\pi }\left( \tfrac{x+1/2}{e}\right) ^{x+1/2}\exp \left( -\tfrac{1517 }{44\,640}\tfrac{1}{x+1/2}-\tfrac{343}{44\,640}\tfrac{x+1/2}{x^{2}+x+111/196} \right) \\ &<&\Gamma (x+1)<e^{2987/39960}\sqrt{2e}\left( \tfrac{x+1/2}{e}\right) ^{x+1/2}\exp \left( -\tfrac{1517}{44\,640}\tfrac{1}{x+1/2}-\tfrac{343}{ 44\,640}\tfrac{x+1/2}{x^{2}+x+111/196}\right) \end{eqnarray*} holds, where $\sqrt{2\pi }\approx 2.5066$ and $e^{2987/39960}\sqrt{2e} \approx 2.5126$ are the best constants. For $n\in \mathbb{N}$, it holds that \begin{eqnarray*} &&\sqrt{2\pi }\left( \tfrac{n+1/2}{e}\right) ^{n+1/2}\exp \left( -\tfrac{1517 }{44\,640}\tfrac{1}{n+1/2}-\tfrac{343}{44\,640}\tfrac{n+1/2}{n^{2}+n+111/196} \right) \\ &<&n!<\frac{2\sqrt{6}}{9}\exp \left( \tfrac{829\,607}{543\,240}\right) \left( \tfrac{n+1/2}{e}\right) ^{n+1/2}\exp \left( -\tfrac{1517}{44\,640} \tfrac{1}{n+1/2}-\tfrac{343}{44\,640}\tfrac{n+1/2}{n^{2}+n+111/196}\right) \end{eqnarray*} with the best constants $\sqrt{2\pi }\approx 2.5066$ and $2\sqrt{6}\exp \left( \tfrac{829\,607}{543\,240}\right) /9\approx 2.5067$. \end{corollary} Lastly, we give an application example of Theorem \ref{MT-p2=p3}. \begin{example} let \begin{equation*} M\left( a,b\right) =H_{p,q;r}^{2,1}\left( a,b\right) =\frac{ pb^{2}+qa^{2}+(1-p-q)ab}{rb+(1-r)a} \end{equation*} and $\theta =0,\sigma =1$ in Theorem \ref{MT-p2=p3}. Then by the same method previously, we can derive two best arrays \begin{eqnarray*} \left( p_{1},q_{1},r_{1}\right) &=&\left( \frac{129-59\sqrt{3}}{360},\frac{ 129+59\sqrt{3}}{360},\frac{90-29\sqrt{3}}{180}\right) , \\ \left( p_{2},q_{2},r_{2}\right) &=&\left( \frac{129+59\sqrt{3}}{360},\frac{ 129-59\sqrt{3}}{360},\frac{90+29\sqrt{3}}{180}\right) . \end{eqnarray*} Then, \begin{eqnarray} H_{p_{1},q_{1};r_{1}}^{2,1}\left( x,x+1\right) &=&\frac{x^{2}+\frac{180-59 \sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}} :=M_{1}\left( x,x+1\right) , \label{M1} \\ H_{p_{2},q_{2};r_{2}}^{2,1}\left( x,x+1\right) &=&\frac{x^{2}+\frac{180+59 \sqrt{3}}{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}} :=M_{2}\left( x,x+1\right) \label{M2} \end{eqnarray} It is easy to check that $M\left( a,b\right) $ are means of $a$ and $b$ for $ \left( p,q,r\right) =\left( p_{1},q_{1},r_{1}\right) $ and $\left( p_{2},q_{2},r_{2}\right) $. Thus, application of Theorem \ref{MT-p2=p3} implies that both the following two asymptotic formulas \begin{equation*} \ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+1/2\right) \ln M_{i}\left( x,x+1\right) -M_{i}\left( x,x+1\right) \text{, }i=1,2 \end{equation*} are valid as $x\rightarrow \infty $. And, we have \begin{eqnarray*} \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln M_{1}\left( x,x+1\right) +M_{1}\left( x,x+1\right) }{x^{-4}} &=&-\tfrac{1481\sqrt{3}}{2332\,800}, \\ \lim_{x\rightarrow \infty }\tfrac{\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln M_{2}\left( x,x+1\right) +M_{2}\left( x,x+1\right) }{x^{-4}} &=&\tfrac{1481\sqrt{3}}{2332\,800}. \end{eqnarray*} \end{example} The above two asymptotic formulas also have well properties. \begin{proposition} Let $f_{6},f_{7}$ be defined on $\left( 0,\infty \right) $ by \begin{eqnarray*} f_{6}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln M_{1}\left( x,x+1\right) +M_{1}\left( x,x+1\right) , \\ f_{7}(x) &=&\ln \Gamma (x+1)-\frac{1}{2}\ln 2\pi -\left( x+1/2\right) \ln M_{2}\left( x,x+1\right) +M_{2}\left( x,x+1\right) , \end{eqnarray*} where $M_{1}$ and $M_{2}$ are defined\ by (\ref{M1}) and (\ref{M2}), respectively. Then $f_{6}\ $is increasing and concave on $\left( 0,\infty \right) $, while $f_{7}$ is decreasing and convex on $\left( 0,\infty \right) $. \end{proposition} \begin{proof} Differentiation gives \begin{eqnarray*} f_{6}^{\prime }\left( x\right) &=&\psi (x+1)-\ln \frac{x^{2}+\frac{180-59 \sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}- \frac{\left( x+\frac{1}{2}\right) \left( 2x+\frac{180-59\sqrt{3}}{180} \right) }{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}} \\ &&+\frac{x+\frac{1}{2}}{x+\frac{90-29\sqrt{3}}{180}}+\frac{2x+\frac{180-59 \sqrt{3}}{180}}{x+\frac{90-29\sqrt{3}}{180}}-\frac{x^{2}+\frac{180-59\sqrt{3} }{180}x+\frac{129-59\sqrt{3}}{360}}{\left( x+\frac{90-29\sqrt{3}}{180} \right) ^{2}}, \end{eqnarray*} \begin{eqnarray*} f_{6}^{\prime \prime }\left( x\right) &=&\psi ^{\prime }(x+1)-\frac{2x+\frac{ 180-59\sqrt{3}}{180}}{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3} }{360}}+\frac{1}{x+\frac{90-29\sqrt{3}}{180}} \\ &&+\frac{59\sqrt{3}}{180}\frac{x^{2}+\frac{59-26\sqrt{3}}{59}x+\frac{43}{120} -\frac{13\sqrt{3}}{59}}{\left( x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59 \sqrt{3}}{360}\right) ^{2}} \\ &&-\frac{7\sqrt{3}}{45}\frac{1}{\left( x+\frac{90-29\sqrt{3}}{180}\right) ^{2}}-\frac{\sqrt{3}}{180}\frac{x-\frac{629\sqrt{3}-90}{180}}{\left( x+\frac{ 90-29\sqrt{3}}{180}\right) ^{3}}. \end{eqnarray*} Employing the recursive relation (\ref{psi-rel.}) and factoring reveal that \begin{equation*} f_{6}^{\prime \prime }\left( x+1\right) -f_{6}^{\prime \prime }\left( x\right) =\frac{1481\sqrt{3}}{19\,440}\frac{f_{61}\left( x\right) }{ f_{62}\left( x\right) }, \end{equation*} where \begin{eqnarray*} f_{61}\left( x\right) &=&x^{9}+\left( 9-\tfrac{337\,153}{266\,580}\sqrt{3} \right) x^{8}+\left( \tfrac{991\,207\,423}{26\,658\,000}-\tfrac{674\,306}{ 66\,645}\sqrt{3}\right) x^{7} \\ &&+\left( \tfrac{2459\,907\,961}{26\,658\,000}-\tfrac{169\,081\,132\,727}{ 4798\,440\,000}\sqrt{3}\right) x^{6}+\left( \tfrac{4335\,292\,090\,469}{ 28\,790\,640\,000}-\tfrac{55\,797\,724\,727}{799\,740\,000}\sqrt{3}\right) x^{5} \\ &&+\left( \tfrac{956\,621\,902\,709}{5758\,128\,000}-\tfrac{ 148\,442\,768\,304\,491}{1727\,438\,400\,000}\sqrt{3}\right) x^{4} \\ &&+\left( \tfrac{229\,288\,958\,388\,788\,929}{1865\,633\,472\,000\,000}- \tfrac{29\,135\,013\,047\,291}{431\,859\,600\,000}\sqrt{3}\right) x^{3} \\ &&+\left( \tfrac{36\,305\,075\,316\,164\,929}{621\,877\,824\,000\,000}- \tfrac{55\,416\,459\,045\,055\,111\,861}{1679\,070\,124\,800\,000\,000}\sqrt{ 3}\right) x^{2} \\ &&+\left( \tfrac{179\,958\,708\,278\,174\,628\,611}{11\,193\,800\,832\,000 \,000\,000}-\tfrac{7731\,435\,289\,282\,423\,861}{839\,535\,062\,400\,000 \,000}\sqrt{3}\right) x \\ &&+\left( \tfrac{21\,826\,051\,463\,638\,680\,611}{11\,193\,800\,832\,000 \,000\,000}-\tfrac{5586\,677\,417\,732\,710\,687}{4975\,022\,592\,000\,000 \,000}\sqrt{3}\right) , \end{eqnarray*} \begin{eqnarray*} f_{62}\left( x\right) &=&\left( x+1\right) ^{2}\left( x^{2}+\tfrac{180-59 \sqrt{3}}{180}x+\tfrac{129-59\sqrt{3}}{360}\right) ^{2}\left( x^{2}+\tfrac{ 540-59\sqrt{3}}{180}x+\tfrac{283-59\sqrt{3}}{120}\right) ^{2} \\ &&\times \left( x+\tfrac{270-29\sqrt{3}}{180}\right) ^{3}\left( x+\tfrac{ 90-29\sqrt{3}}{180}\right) ^{3}. \end{eqnarray*} By direct verifications we see that all coefficients of $f_{61}$ and $f_{62}$ are positive, so $f_{61}\left( x\right) $, $f_{62}\left( x\right) >0$ for $ x>0$. Therefore, we get $f_{6}^{\prime \prime }\left( x+1\right) -f_{6}^{\prime \prime }\left( x\right) >0$, which yields \begin{equation*} f_{6}^{\prime \prime }(x)<f_{6}^{\prime \prime }(x+1)<f_{6}^{\prime \prime }(x+2)<...<f_{6}^{\prime \prime }(\infty )=0. \end{equation*} It shows that $f_{6}$ is concave on $\left( 0,\infty \right) $, and therefore, $f_{6}^{\prime }(x)>\lim_{x\rightarrow \infty }f_{6}^{\prime }(x)=0$, which proves the monotonicity and concavity of $f_{6}$. In the same way, we can prove the monotonicity and convexity of $f_{7}$ on $ \left( 0,\infty \right) $, whose details are omitted. \end{proof} As direct consequences of previous proposition, we have \begin{corollary} For $x>0$, the double inequality \begin{eqnarray*} &&\delta _{0}\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+ \frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}\right) ^{x+1/2}\exp \left( -\tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{ x+\frac{90-29\sqrt{3}}{180}}\right) \\ &<&\Gamma (x+1)<\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+ \frac{129-59\sqrt{3}}{360}}{x+\frac{90-29\sqrt{3}}{180}}\right) ^{x+1/2}\exp \left( -\tfrac{x^{2}+\frac{180-59\sqrt{3}}{180}x+\frac{129-59\sqrt{3}}{360}}{ x+\frac{90-29\sqrt{3}}{180}}\right) \end{eqnarray*} holds, where $\delta _{0}=\exp f_{6}\left( 0\right) \approx 0.96259$ and $1$ are the best constants. For $n\in \mathbb{N}$, it holds that \begin{eqnarray*} &&\delta _{2}\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+ \frac{129-59\sqrt{3}}{360}}{n+\frac{90-29\sqrt{3}}{180}}\right) ^{n+1/2}\exp \left( -\tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{129-59\sqrt{3}}{360}}{ n+\frac{90-29\sqrt{3}}{180}}\right) \\ &<&n!<\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{ 129-59\sqrt{3}}{360}}{n+\frac{90-29\sqrt{3}}{180}}\right) ^{n+1/2}\exp \left( -\tfrac{n^{2}+\frac{180-59\sqrt{3}}{180}n+\frac{129-59\sqrt{3}}{360}}{ n+\frac{90-29\sqrt{3}}{180}}\right) \end{eqnarray*} with the best constants $\delta _{1}=\exp f_{6}\left( 1\right) \approx 0.99965$ and $1$. \end{corollary} \begin{corollary} For $x>0$, the double inequality \begin{eqnarray*} &&\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59 \sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right) ^{x+1/2}\exp \left( - \tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{ 90+29\sqrt{3}}{180}}\right) \\ &<&\Gamma (x+1)<\tau _{0}\sqrt{2\pi }\left( \tfrac{x^{2}+\frac{180+59\sqrt{3} }{180}x+\frac{129+59\sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right) ^{x+1/2}\exp \left( -\tfrac{x^{2}+\frac{180+59\sqrt{3}}{180}x+\frac{129+59 \sqrt{3}}{360}}{x+\frac{90+29\sqrt{3}}{180}}\right) \end{eqnarray*} holds, where $\tau _{0}=\exp f_{7}\left( 0\right) \approx 1.0020$ and $1$ are the best constants. For $n\in \mathbb{N}$, it holds that \begin{eqnarray*} &&\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59 \sqrt{3}}{360}}{n+\frac{90+29\sqrt{3}}{180}}\right) ^{n+1/2}\exp \left( - \tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59\sqrt{3}}{360}}{n+\frac{ 90+29\sqrt{3}}{180}}\right) \\ &<&n!<\tau _{1}\sqrt{2\pi }\left( \tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+ \frac{129+59\sqrt{3}}{360}}{n+\frac{90+29\sqrt{3}}{180}}\right) ^{n+1/2}\exp \left( -\tfrac{n^{2}+\frac{180+59\sqrt{3}}{180}n+\frac{129+59\sqrt{3}}{360}}{ n+\frac{90+29\sqrt{3}}{180}}\right) \end{eqnarray*} with the best constants $\delta _{1}=\exp f_{7}\left( 1\right) \approx 1.0001 $ and $1$. \end{corollary} \section{Open problems} Inspired by Examples \ref{E-M3,2}--\ref{E-N4,3}, we propose the following problems. \begin{problem} Let $S_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{S^n,n-1} ). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln S_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) -\left( x+\frac{1}{2}\right) :=F_{1}\left( x\right) \end{equation*} holds as $x\rightarrow \infty $ with \begin{equation*} \lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{1}\left( x\right) }{ x^{-2n+1}}=c_{1}\neq 0,\pm \infty . \end{equation*} \end{problem} \begin{problem} Let $S_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{S^n,n-1} ). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln \left( x+\frac{1}{2}\right) -S_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) :=F_{2}\left( x\right) \end{equation*} holds as $x\rightarrow \infty $ with \begin{equation*} \lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{2}\left( x\right) }{ x^{-2n+1}}=c_{2}\neq 0,\pm \infty . \end{equation*} \end{problem} \begin{problem} Let $H_{p_{k};q_{k}}^{n,n-1}\left( a,b\right) $ be defined by (\ref{H^n,n-1} ). Finding $p_{k}$ and $q_{k}$ such that the asymptotic formula for the gamma function \begin{equation*} \ln \Gamma (x+1)\sim \frac{1}{2}\ln 2\pi +\left( x+\frac{1}{2}\right) \ln H_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) -H_{p_{k};q_{k}}^{n,n-1}\left( x,x+1\right) :=F_{3}\left( x\right) \end{equation*} holds as $x\rightarrow \infty $ with \begin{equation*} \lim_{x\rightarrow \infty }\frac{\ln \Gamma (x+1)-F_{1}\left( x\right) }{ x^{-2n}}=c_{3}\neq 0,\pm \infty . \end{equation*} \end{problem} \end{document}

\begin{document} \setlength{\abovedisplayskip}{8pt} \setlength{\belowdisplayskip}{8pt} \setlength{\abovedisplayshortskip}{4pt} \setlength{\belowdisplayshortskip}{8pt} \begin{titlepage} \title{{\bf{\Huge Long-Term Factorization of Affine Pricing Kernels}} \thanks{This paper is based on research supported by the grant CMMI-1536503 from the National Science Foundation.}} \author{Likuan Qin\thanks{[email protected]} } \author{Vadim Linetsky\thanks{[email protected]} } \affil{\emph{Department of Industrial Engineering and Management Sciences}\\ \emph{McCormick School of Engineering and Applied Sciences}\\ \emph{Northwestern University}} \date{} \end{titlepage} \maketitle \begin{abstract} This paper constructs and studies the long-term factorization of affine pricing kernels into discounting at the rate of return on the long bond and the martingale component that accomplishes the change of probability measure to the long forward measure. The principal eigenfunction of the affine pricing kernel germane to the long-term factorization is an exponential-affine function of the state vector with the coefficient vector identified with the fixed point of the Riccati ODE. The long bond volatility and the volatility of the martingale component are explicitly identified in terms of this fixed point. A range of examples from the asset pricing literature is provided to illustrate the theory. \end{abstract} \section{Introduction} The stochastic discount factor (SDF) is a fundamental object in arbitrage-free asset pricing models. It assigns today's prices to risky future payoffs at alternative investment horizons. It accomplishes this by simultaneously discounting the future and adjusting for risk. A familiar representation of the SDF is a factorization into discounting at the risk-free interest rate and a martingale component adjusting for risk. This martingale accomplishes the change of probabilities to the risk-neutral probability measure. More recently \citet{alvarez_2005using}, \citet{hansen_2008consumption}, \citet{hansen_2009} and \citet{hansen_2012} introduce and study an alternative {\em long-term factorization} of the SDF. The {\em transitory component} in the long-term factorization discounts at the rate of return on the pure discount bond of asymptotically long maturity (the {\em long bond}). The {\em permanent component} is a martingale that accomplishes a change of probabilities to the {\em long forward measure}. \citet{linetsky_2014long} study the long-term factorization and the long forward measure in the general semimartingale setting. The long-term factorization of the SDF is particularly convenient in applications to the pricing of long-lived assets and to theoretical and empirical investigations of the term structure of the risk-return trade-off. In addition to the references above, the growing literature on the long-term factorization and its applications includes \citet{hansen_2012pricing}, \citet{hansen_2013}, \citet{borovicka_2014mis}, \citet{borovivcka2011risk}, \citet{borovivcka2016term}, \citet{bakshi_2012}, \citet{bakshia2015recovery}, \citet{christensen2014nonparametric}, \citet{christensen_2013estimating}, \citet{linetsky_2014_cont}, \citet{linetsky2016bond}, \citet{backus2015term}, \citet{filipovic2016linear}, \citet{filipovic2016relation}. Empirical investigations in this literature show that the martingale component in the long-term factorization is highly volatile and economically significant (see, in particular, \citet{bakshi_2012} for results based on pricing kernel bounds, \citet{christensen2014nonparametric} for results based on structural asset pricing models connecting to the macro-economic fundamentals, and \citet{linetsky2016bond} for results based on explicit parameterizations of the pricing kernel, where, in particular, the relationship among the measures ${\mathbb P}$, ${\mathbb Q}$ and ${\mathbb L}$ is empirically investigated). The focus of the present paper is on the analysis of long-term factorization in affine diffusion models, both from the perspective of providing a user's guide to constructing long-term factorization in affine asset pricing models, as well as employing affine models as a convenient laboratory to illustrate the theory of the long-term factorization. Affine diffusions are work-horse models in continuous-time finance due to their analytical and computational tractability (\citet{vasicek_1977equilibrium}, \citet{cox_1985_2}, \citet{duffie_1996}, \citet{duffie_2000}, \citet{dai_2000}, \citet{duffie_2003}). In this paper we show that the principal eigenfunction of \citet{hansen_2009} that determines the long-term factorization, if it exists, is necessarily in the exponential-affine form in affine models, with the coefficient vector in the exponential identified with the fixed point of the corresponding Riccati ODE. This allows us to give a fully explicit treatment and illustrate dynamics of the long bond, the martingale component and the long-forward measure in affine models. In particular, we explicitly verify that when the Riccati ODE associated with the affine pricing kernel possesses a fixed point, the affine model satisfies the sufficient condition in Theorem 3.1 of \citet{linetsky_2014long} so that the long-term limit exists. In Section \ref{Brownian} we review and summarize the long-term factorization in Brownian motion-based models. In Section \ref{exist_affine} we present general results on the long-term factorization of affine pricing kernels. The main results are given in Theorem \ref{affine_long}, where the market price of Brownian risk is explicitly decomposed into the market price of risk under the long forward measure identified with the volatility of the long bond and the remaining market price of risk determining the martingale component accomplishing the change of probabilities from the data-generating to the long forward measure. The latter component is determined by the fixed point of the Riccati ODE. In Section \ref{examples} we study a range of examples of affine pricing kernels from the asset pricing literature. \section{Long-Term Factorization in Brownian Environments} \label{Brownian} We work on a complete filtered probability space $(\Omega,{\mathscr F},({\mathscr F}_{t})_{t\geq 0},{\mathbb P})$. We assume that all uncertainty in the economy is generated by an $n$-dimensional Brownian motion $W_t^{\mathbb{P}}$ and that $(\mathscr{F}_t)_{t\geq 0}$ is the (completed) filtration generated by $W_t^{\mathbb{P}}$. We assume absence of arbitrage and market frictions, so that there exists a strictly positive pricing kernel process in the form of an It\^{o} semimartingale. More precisely, we assume that the pricing kernel follows an It\^{o} process ($\cdot$ denotes vector dot product) $$ dS_t=-r_tS_tdt-S_t \lambda_t \cdot dW_t^{\mathbb{P}} $$ with $\int_0^t |r_s|ds<\infty$ and the market price of Brownian risk vector $\lambda_t$ such that the process $$ M_t^0=e^{-\int_0^t \lambda_s\cdot dW_s^{\mathbb{P}}-\frac{1}{2}\int_0^t \|\lambda_s\|^2 ds} $$ is a martingale (Novikov's condition ${\mathbb E}^{\mathbb P}[e^{\frac{1}{2}\int_0^t \|\lambda_s\|^2 ds}]<\infty$ for each $t>0$ suffices). Under these assumptions the pricing kernel has the risk-neutral factorization \begin{equation} S_t=\frac{1}{A_t}M_t^0=e^{-\int_0^t r_s ds}M_t^0 \eel{rnf} into discounting at the risk-free short rate $r_t$ determining the risk-free asset (money market account) $A_t=e^{\int_0^t r_s ds}$ and the exponential martingale $M^0_t$ with the market price of Brownian risk $\lambda_t$ determining its volatility. We also assume that ${\mathbb E}^{\mathbb{P}}[S_T/S_t]<\infty$ for all $T>t\geq 0$. The integrability of the SDF $S_T/S_t$ for any two dates $T>t$ ensures that that zero-coupon bond price processes $$P_t^T:=\mathbb{E}_t^\mathbb{P}[S_T/S_t], \quad t\in [0,T]$$ are well defined for all maturity dates $T>0$ ($\mathbb{E}_t[\cdot]=\mathbb{E}[\cdot|{\mathscr F}_{t}]$). Since for each $T$ the $T$-maturity zero coupon bond price process $P_t^T$ can be written as $P_t^T=M_t^TP_0^T/S_t$, where $M_t^T=S_t P_t^T/P_0^T= \mathbb{E}_t^\mathbb{P}[S_T]/\mathbb{E}_0^\mathbb{P}[S_T]$ is a positive martingale on $t\in [0,T]$, we can apply the Martingale Representation Theorem to claim that $$dM_t^T=-M_t^T \lambda_t^T \cdot dW^{\mathbb P}_t$$ with some $\lambda_t^T$, and further claim that the bond price process has the representation $$ dP_t^T=(r_t+ \sigma^T_t\cdot \lambda_t) P_t^Tdt+P_t^T \sigma^T_t \cdot dW_t^\mathbb{P} $$ with the volatility process $\sigma^T_t=\lambda_t-\lambda_t^T$. Following \citet{linetsky_2014long}, for each fixed $T>0$ we define a self-financing trading strategy that rolls over investments in $T$-maturity zero-coupon bonds as follows. Fix $T$ and consider a self-financing roll-over strategy that starts at time zero by investing one unit of account in $1/P_{0}^T$ units of the $T$-maturity zero-coupon bond. At time $T$ the bond matures, and the value of the strategy is $1/P_{0}^T$ units of account. We roll the proceeds over by re-investing into $1/(P_{0}^T P_{T}^{2T})$ units of the zero-coupon bond with maturity $2T$. We continue with the roll-over strategy, at each time $kT$ re-investing the proceeds into the bond $P_{kT}^{(k+1)T}$. We denote the valuation process of this self-financing strategy $B_t^T$: \[ B_t^T = \left(\prod_{i=0}^k P_{iT}^{(i+1)T}\right)^{-1} P_{t}^{(k+1)T},\quad t\in [kT,(k+1)T),\quad k=0,1,\ldots. \] For each $T>0$, the process $B_t^T$ is defined for all $t\geq 0$. The process $S_t B_t^T$ extends the martingale $M_t^T$ to all $t\geq 0$. It thus defines the $T$-{\em forward measure} ${\mathbb Q}^T|_{{\mathscr F}_{t}}=M_t^T {\mathbb P}|_{{\mathscr F}_{t}}$ on ${\mathscr F}_{t}$ for each $t\geq 0$, where $T$ now has the meaning of the length of the compounding interval. Under the $T$-forward measure ${\mathbb Q}^T$ extended to all ${\mathscr F}_{t}$, the roll-over strategy $(B_t^T)_{t\geq 0}$ with the compounding interval $T$ serves as the numeraire asset. Following \citet{linetsky_2014long}, we continue to call the measure extended to all ${\mathscr F}_{t}$ for $t\geq 0$ the $T$-forward measure and use the same notation, as it reduces to the standard definition of the forward measure on ${\mathscr F}_{T}$. Since the roll-over strategy $(B^T_t)_{t\geq 0}$ and the positive martingale $M_t^T=S_t B_t^T$ are defined for all $t\geq 0$, we can write the $T$-{\em forward factorization} of the pricing kernel for all $t\geq 0$: \begin{equation} S_t = \frac{1}{B_t^T}M_t^T. \eel{Tfactorization} We now recall the definitions of the {\em long bond} and the {\em long forward measure} from \citet{linetsky_2014long}. \begin{definition}{\bf (Long Bond)} \label{def_longbond} If the wealth processes $(B^T_t)_{t\geq 0}$ of the roll-over strategies in $T$-maturity bonds converge to a strictly positive semimartingale $(B_t^\infty)_{t\geq 0}$ uniformly on compacts in probability as $T\rightarrow \infty$, i.e. for all $t>0$ and $K>0$ \[ \lim_{T\rightarrow \infty} {\mathbb P}(\sup_{s\leq t}|B_s^T-B_s^\infty|>K)=0, \] we call the limit the {\em long bond}. \end{definition} \begin{definition}{\bf (Long Forward Measure)} \label{def_longforward} If there exists a measure $\mathbb{Q}^\infty$ equivalent to $\mathbb{P}$ on each ${\mathscr F}_t$ such that the $T$-forward measures converge strongly to ${\mathbb Q}^\infty$ on each ${\mathscr F}_t$, i.e. \[ \lim_{T\rightarrow \infty}{\mathbb Q}^T(A)={\mathbb Q}^\infty(A) \] for each $A\in {\mathscr F}_t$ and each $t\geq 0$, we call the limit the {\em long forward measure} and denote it ${\mathbb L}$. \end{definition} The following theorem, proved in \citet{linetsky_2014long}, gives a sufficient condition that ensures convergence to the long bond in the semimartingale topology which is stronger than the ucp convergence in Definition 1 and convergence of $T$-forward measures to the long forward measure in total variation, which is stronger than the strong convergence in Definition 2 (we refer to \citet{linetsky_2014long} and the on-line appendix for proofs and details). \begin{theorem}{\bf (Long Term Factorization and the Long Forward Measure)} \label{implication_L1} Suppose that for each $t>0$ the ratio of the ${\mathscr F}_t$-conditional expectation of the pricing kernel $S_T$ to its unconditional expectation converges to a positive limit in $L^1$ as $T\rightarrow \infty$ (under ${\mathbb P}$), i.e. for each $t>0$ there exists an almost surely positive ${\mathscr F}_t$-measurable random variable which we denote $M_t^\infty$ such that \begin{equation} \frac{{\mathbb E}^{\mathbb P}_t[S_T]}{{\mathbb E}^{\mathbb P}[S_T]} \xrightarrow{\rm L^1} M_t^\infty\quad \text{as} \quad T\rightarrow \infty. \eel{PKL1} Then the following results hold:\\ (i) The collection of random variables $(M_t^\infty)_{t\geq0}$ is a positive ${\mathbb P}$-martingale, and the family of martingales $(M_t^T)_{t\geq 0}$ converges to the martingale $(M_t^\infty)_{t\geq0}$ in the semimartingale topology.\\ (ii) The long bond valuation process $(B_t^\infty)_{t\geq0}$ exists, and the roll-over strategies $(B_t^T)_{t\geq 0}$ converge to the long bond $(B_t^\infty)_{t\geq 0}$ in the semimartingale topology.\\ (iii) The pricing kernel possesses the long-term factorization \begin{equation} S_t=\frac{1}{B_t^\infty}M_t^\infty. \eel{ltf} (iv) $T$-forward measures ${\mathbb Q}^T$ converge to the long forward measure ${\mathbb L}$ in total variation on each ${\mathscr F}_t$, and ${\mathbb L}$ is equivalent to ${\mathbb P}$ on ${\mathscr F}_t$ with the Radon-Nikodym derivative $M_t^\infty$. \end{theorem} The process $B_t^\infty$ has the interpretation of the gross return earned starting from time zero up to time $t$ on holding the zero-coupon bond of asymptotically long maturity. The long bond is the numeraire asset under the long forward measure $\mathbb{L}$ since the pricing kernel becomes $1/B_t^\infty$ under $\mathbb{L}$. The long-term factorization of the pricing kernel \eqref{ltf} decomposes it into discounting at the rate of return on the long bond and a martingale component encoding a further risk adjustment. Suppose the condition \eqref{PKL1} in Theorem \ref{implication_L1} holds in the Brownian setting of this paper. Then the long bond valuation process is an It\^{o} semimartingale with the representation $$ dB_t^\infty=(r_t+ \sigma^\infty_t \cdot \lambda_t) B_t^\infty dt + B_t^\infty \sigma^\infty_t \cdot dW_t^\mathbb{P} $$ with some volatility process $\sigma^\infty_t$ such that the process $M_t^\infty=S_t B_t^\infty$ satisfying $$ dM_t^\infty=-M_t^\infty \lambda_t^\infty \cdot dW_t^\mathbb{P} $$ with $\lambda_t^\infty=\lambda_t - \sigma^\infty_t$ is a martingale (the permanent component in the long-term factorization). Thus, the long-term factorization Eq.\eqref{ltf} in the Brownian setting yields a decomposition of the market price of Brownian risk \begin{equation} \lambda_t=\sigma^\infty_t + \lambda_t^\infty \eel{mprdecomposition} into the volatility of the long bond $\sigma_t^\infty$ and the volatility $\lambda_t^\infty$ of the martingale $M_t^\infty$. The change of probability measure from the data-generating measure ${\mathbb P}$ to the long forward measure ${\mathbb L}$ is accomplished via Girsanov's theorem with the ${\mathbb L}$-Brownian motion $W_t^\mathbb{L}=W_t^\mathbb{P}+ \int_0^t \lambda_s^\infty ds.$ \section{Long Term Factorization of Affine Pricing Kernels} \label{exist_affine} We assume that the underlying economy is described by a Markov process $X$. We further assume $X$ is an affine diffusion and the pricing kernel $S$ is exponential affine in $X$ and the time integral of $X$. Affine diffusion models are widely used in continuous-time finance due to their analytical tractability (\citet{vasicek_1977equilibrium}, \citet{cox_1985_2}, \citet{duffie_1996}, \citet{duffie_2000}, \citet{dai_2000}, \citet{duffie_2003}). We start with a brief summary of some of the key facts about affine diffusions. We refer the reader to \citet{filipovic_2009} for details, proofs and references to the literature on affine diffusion. The process we work with solves the following SDE on the state space $E=\mathbb{R}_+^m\times\mathbb{R}^n$ for some $m,n\geq 0$ with $m+n=d$, where $\mathbb{R}_+^m=\big\{x\in \mathbb{R}^m : x_i\geq 0$ for $i=1,...,m\big\}$: \begin{equation} d X_t=b(X_t)dt+\sigma(X_t)d W^{\mathbb{P}}_t,\quad X_0=x, \eel{affinesde} where $W^{\mathbb P}$ is a $d$-dimensional standard Brownian motion and the diffusion matrix $\alpha(x)=\sigma(x)\sigma(x)^\dagger$ (here $^\dagger$ denotes matrix transpose to differentiate it from superscript $^T$) and the drift vector $b(x)$ are both affine in $x$: \begin{equation} \alpha(x)=a+\displaystyle{\sum_{i=1}^d}x_i\alpha_i,\quad b(x)=b+\displaystyle{\sum_{i=1}^d}x_i\beta_i=b+Bx \end{equation} for some $d\times d$-matrices $a$ and $\alpha_i$ and $d$-dimensional vectors $b$ and $\beta_i$, where we denote by $B=(\beta_1,...,\beta_d)$ the $d\times d$-matrix with $i$-th column vector $\beta_i$, $1\leq i\leq d$. The first $m$ coordinates of $X$ are CIR-type and are non-negative, while the last $n$ coordinates are OU-type. Define the index sets $\emph{I}=\{1,...,m\}$ and $\emph{J}=\{m+1,...,m+n\}$. For any vector $\mu$ and matrix $\nu$, and index sets $\emph{M},\emph{N}\in \{I,J\}$, we denote by $\mu_\emph{M}=(\mu_i)_{i\in \emph{M}},$ $\nu_{\emph{M}\emph{N}}=(\nu_{ij})_{i\in \emph{M},j\in \emph{N}}$ the respective sub-vector and sub-matrix. To ensure the process stays in the domain $E={\mathbb R}_+^m\times {\mathbb R}^n$, we need the following assumption (cf. \citet{filipovic_2009}) \begin{assumption}{\bf (Admissibility)}\\ (1) $a_{JJ}$ and $\alpha_{i,JJ}$ are symmetric positive semi-definite for all $i=1,2,...,m$,\\ (2) $a_{II}=0,$ $a_{IJ}=a_{JI}^\dagger=0$,\\ (3) $\alpha_j=0$ for $j\in J$,\\ (4) $\alpha_{i,kl}=\alpha_{i,lk}=0$ for $k\in I\backslash \{i\}$ for all $1\leq k,l\leq d,$\\ (5) $b_I\geq 0$, $B_{IJ}=0$, and $B_{II}$ has non-negative off-diagonal elements. \label{admi_and_nonde} \end{assumption} The condition $b_I\geq 0$ on the constant term in the drift of the CIR-type components ensures that the process stays in the state space $E$. Making a stronger assumption $b_I>0$ ensures that the process instantaneously reflects from the boundary $\partial E$ and re-enters the interior of the state space ${\rm int}E=\mathbb{R}_{++}^m\times\mathbb{R}^n,$ where $\mathbb{R}_{++}^m=\big\{x\in \mathbb{R}^m : x_i> 0$ for $i=1,...,m\big\}$. For any parameters satisfying Assumption \ref{admi_and_nonde}, there exists a unique strong solution of the SDE \eqref{affinesde} (cf. Theorem 8.1 of \citet{filipovic_2009}). Denote by ${\mathbb P}_x$ the law of the solution $X^x$ of the SDE \eqref{affinesde} for $x\in E$, ${\mathbb P}_x(X_t\in A):={\mathbb P}(X^x_t\in A)$. Then $P_t(x,A)={\mathbb P}_x(X_t\in A)$ defined for all $t\geq 0$, Borel subsets $A$ of $E$, and $x\in E$ defines a Markov transition semigroup $(P_t)_{t\geq 0}$ on the Banach space of Borel measurable bounded functions on $E$ by $P_tf(x):=\int_E f(y)P_t(x,dy)$. As shown in \citet{duffie_2003}, this semigroup is {\em Feller}, i.e., it leaves the space of continuous functions vanishing at infinity invariant. Thus, the Markov process $((X_t)_{t\geq 0},({\mathbb P}_x)_{x\in E})$ is a {\em Feller process} on $E$. It has continuous paths in $E$ and has the strong Markov property (cf. \citet{yamada_1971}, Corollary 2, p.162). Thus, it is a Borel right process (in fact, a Hunt process). We make the following assumption about the pricing kernel. \begin{assumption}{\bf (Affine Pricing Kernel)}\label{assumption_affine_PK} We assume that the pricing kernel is exponential-affine in $X$ and its time integral: \begin{equation} S_t=e^{-\gamma t-u^\dagger (X_t-X_0)-\int_0^t \delta^\dagger X_s ds}, \eel{affine_pk} where $\gamma$ is a scalar and $u$ and $\delta$ are $d$-vectors and $^\dagger$ denotes matrix transpose. \end{assumption} The pricing kernel in this form is a positive multiplicative functional of the Markov process $X$. The associated pricing operator ${\mathscr P}_t$ is defined by $$ {\mathscr P}_tf(x)={\mathbb E}^{\mathbb P}_x[S_t f(X_t)] $$ for a payoff $f$ of the Markov state. We refer the reader to Qin and Linetsky (2016a) for a detailed treatment of Markovian pricing operators. The pricing kernel in the form \eqref{affine_pk} is called affine due to the following key result that shows that the term structure of pure discount bond yields is affine in the state vector $X$ (cf. \citet{filipovic_2009} Theorem 4.1). \begin{proposition} \label{affine_ZCB} Let $T_0>0$. The following statements are equivalent: \\ (i) ${\mathbb E}^{\mathbb P}[S_{T_0}]<\infty$ for all fixed initial states $X_0=x\in {\mathbb R}_+^m\times {\mathbb R}^n$. \\ (ii) There exists a unique solution $(\Phi(\cdot),\Psi(\cdot)):[0,T_0]\rightarrow {\mathbb R}\times {\mathbb R}^d$ of the following Riccati system of equations up to time $T_0$: \begin{equation} \begin{split} &\Phi^\prime(t)=-\frac{1}{2}\Psi_J(t)^\dagger a_{JJ}\Psi_J(t)+b^\dagger\Psi(t)+\gamma, \quad \Phi(0)=0,\\ &\Psi_i^\prime(t)=-\frac{1}{2}\Psi(t)^\dagger \alpha_{i}\Psi(t)+\beta_i^\dagger\Psi(t)+\delta_i,\quad i\in\emph{I},\\ &\Psi_J^\prime(t)=B_{JJ}^\dagger\Psi_J(t)+\delta_J,\quad \Psi(0)=u.\\ \end{split} \eel{riccati_d} In either case, the pure discount bond valuation processes (with unit payoffs) are exponential-affine in $X$: \begin{equation} P_t^T=\mathbb{E}^{\mathbb P}_t[ S_T/S_t]= ({\mathscr P}_{T-t}1)(x)= P(T-t,X_t)=e^{-\Phi(T-t)-(\Psi(T-t)-u)^\dagger X^x_t} \eel{representation} for all $0\leq t\leq T\leq t+T_0$ and the SDE initial condition $x\in {\mathbb R}_+^m\times {\mathbb R}^n$. \end{proposition} Since in this paper our standing assumption is that ${\mathbb E}^{\mathbb P}[S_t]<\infty$ for all $t$, in this case the Riccati ODE system has solutions $\Psi(t)$ and $\Phi(t)$ for all $t$, and the bond pricing function entering the expression \eqref{representation} for the zero-coupon bond process \begin{equation} P(t,x)=({\mathscr P}_t1)(x)=e^{-\Phi(t)-(\Psi(t)-u)^\dagger x} \eel{bondfunction} is defined for all $t\geq 0$ and $x\in E$. We next show that an affine pricing kernel always possesses the risk-neutral factorization with the affine short rate function. \begin{theorem}{\bf (Risk-Neutral Factorization of Affine Pricing Kernels)}\label{RN_affine} Suppose $X$ satisfies Assumption \ref{admi_and_nonde} and the pricing kernel satisfies Assumption \ref{assumption_affine_PK} together with the assumption that ${\mathbb E}^{\mathbb P}_x[S_t]<\infty$ for all $t\geq 0$ and every fixed initial state $X_0=x\in {\mathbb R}_+^m\times {\mathbb R}^n$.\\ (i) Then the pricing kernel admits the risk-neutral factorization $$S_t=e^{-\int_0^t r(X_s)ds}M^0_t$$ with the affine short rate \begin{equation} r(x)=g+h^\dagger x,\, \eel{affineshortr} with \begin{equation} g=\gamma-\frac{1}{2}u_J^\dagger a_{JJ} u_J+ b^\dagger u, \, h_i=\delta_i-\frac{1}{2}u^\dagger\alpha_i u+\beta_i^\dagger u,\, i\in I, \, h_J=\delta_J+B_{JJ}^\dagger u_J \eel{gh} and the martingale $$M^0_t=e^{-\int_0^t\lambda_s^\dagger dW_s^{\mathbb P}-\frac{1}{2}\int_0^t \|\lambda_s\|^2ds}$$ with the market price of Brownian risk (column $d$-vector) \begin{equation} \lambda_t = \sigma(X_t)^\dagger u, \eel{mprvaffine} where $\sigma(x)$ is the volatility matrix of the state variable $X$ in the SDE \eqref{affinesde} and $$\|\lambda_t\|^2=\lambda_t^\dagger\lambda_t=u^\dagger \alpha(X_t)u.$$ \\ (ii) Under the risk-neutral measure ${\mathbb Q}$ defined by the martingale $M$, the dynamics of $X$ reads \begin{equation} d X_t=(b(X_t)-\alpha(X_t)u) dt+\sigma(X_t)d W^{\mathbb{Q}}_t, \eel{affineq} where $W^{\mathbb Q}_t=W_t^{\mathbb P} +\int_0^t \lambda_s ds$ is the standard Brownian motion under ${\mathbb Q}$. \end{theorem} \begin{proof} (i) Define a process $M_t^0:=S_te^{\int_0^t r(X_s)ds}$. It is also in the form of Eq.\eqref{affine_pk} with $\gamma$ replaced by $\gamma-g$ and $\delta$ replaced by $\delta-h$. Thus, Proposition \ref{affine_ZCB} also holds if we replace $S_t$ with $M_t^0$, replace $\gamma$ with $\gamma-g$ and replace $\delta$ with $\delta-h$, i.e. $\mathbb{E}_t^\mathbb{P}[M_T/M_t]=e^{-\Phi(T-t)-(\Psi(T-t)-u)^\dagger X^x_t},$ where \begin{equation} \begin{split} &\Phi^\prime(t)=-\frac{1}{2}\Psi_J(t)^\dagger a_{JJ}\Psi_J(t)+b^\dagger\Psi(t)+\gamma-g, \quad \Phi(0)=0,\\ &\Psi_i^\prime(t)=-\frac{1}{2}\Psi(t)^\dagger \alpha_{i}\Psi(t)+\beta_i^\dagger\Psi(t)+\delta_i-h_i,\quad i\in\emph{I},\\ &\Psi_J^\prime(t)=B_{JJ}^\dagger\Psi_J(t)+\delta_J-h_J,\quad \Psi(0)=u.\\ \end{split} \end{equation} With the choice of $g$ and $h$ in Eq.\eqref{gh}, the solution to the above ODE is $\Phi(t)=0$ and $\Psi(0)=u$, which implies $\mathbb{E}_t^\mathbb{P}[M_T/M_t]=1$. This shows that $M_t^0$ is a martingale. Furthermore, using the SDE for the affine state $X$, we can cast $M_t^0$ in the exponential martingale form $e^{-\int_0^t \lambda_s^\dagger dW_s^{\mathbb{P}}-\frac{1}{2}\int_0^t \|\lambda_s\|^2ds}$. with $\lambda_t$ given in \eqref{mprvaffine}. \noindent(ii) The SDE for $X$ under $\mathbb{Q}$ follows from Girsanov's Theorem. $\Box$ \end{proof} We next turn to the long term factorization of the affine pricing kernel. \begin{theorem}{\bf (Long Term Factorization of Affine Pricing Kernels)} \label{affine_long} Suppose the solution $\Psi(t)$ of the Riccati ODE \eqref{riccati_d} converges to a fixed point $v\in {\mathbb R}^d$: \begin{equation} \lim_{t\rightarrow\infty}\Psi(t)=v. \eel{psi_converge} Then the following results hold.\\ (i) Condition Eq.\eqref{PKL1} is satisfied and, hence, all results in Theorem \ref{implication_L1} hold. \\ (ii) The long bond is given by \begin{equation} B_t^\infty=e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}, \eel{long_bond_affine} where \begin{equation} \pi(x)=e^{(u-v)^\dagger x} \eel{affineeigen} is the positive exponential-affine eigenfunction of the pricing operator ${\mathscr P}_t$ $$ {\mathscr P}_t \pi(x)=e^{-\lambda t}\pi(x) $$ with the eigenvalue $e^{-\lambda t}$ with \begin{equation} \lambda=\gamma-\frac{1}{2}v_J^\dagger a_{JJ}v_J+ b^\dagger v \eel{affineeigenv} interpreted as the limiting long-term zero-coupon yield: \begin{equation} \lim_{t\rightarrow \infty}\frac{-\ln P(t,x)}{t}=\lambda \eel{asymptyield} for all $x$.\\ (iii) The long bond has the ${\mathbb P}$-measure dynamics: \begin{equation} dB_t^\infty = (r(X_t)+(\sigma^\infty_t)^\dagger \lambda_t )B_t^\infty dt + B_t^\infty (\sigma_t^\infty)^\dagger dW_t^{\mathbb P}, \end{equation} where the (column vector) volatility of the long bond is given by: \begin{equation} \sigma_t^\infty=\sigma(X_t)^\dagger(u-v). \eel{sigmainf} (iv) The martingale component in the long-term factorization of the PK $M^\infty_t=S_t B_t^\infty$ can be written in the form \begin{equation} M^\infty_t=e^{-\int_0^t (\lambda_s^\infty)^\dagger dW_s^{\mathbb P}-\frac{1}{2}\int_0^t \|\lambda_s^\infty\|^2 ds}, \eel{minfty} where \begin{equation} \lambda^\infty_t=\lambda_t - \sigma_t^\infty=\sigma(X_t)^\dagger v. \eel{gammainfty} (v) The long-term decomposition of the market price of Brownian risk is given by: \begin{equation} \lambda_t = \sigma_t^\infty + \lambda^\infty_t, \end{equation} where $\sigma_t^\infty$ is the volatility of the long bond \eqref{sigmainf} and $\lambda_t^\infty$ given in \eqref{gammainfty} defines the martingale \eqref{minfty}.\\ (vi) Under the long forward measure $\mathbb{L}$ the state vector $X_t$ solves the following SDE \begin{equation} dX_t=(b(X_t)-\alpha(X_t)v)dt+\sigma(X_t)dW_t^\mathbb{L}, \eel{affinel} where $W_t^\mathbb{L}=W_t^\mathbb{P}+\int_0^t \lambda_s^\infty ds$ is the d-dimensional Brownian motion under $\mathbb{L}$, and the long bond has the ${\mathbb L}$-measure dynamics: \begin{equation} dB_t^\infty = (r(X_t)+\|\sigma_s^\infty\|^2 )B_t^\infty dt + B_t^\infty (\sigma_t^\infty)^\dagger dW_t^{\mathbb L}. \end{equation} \end{theorem} \begin{proof} Since the solution of the Riccati ODE $\Psi(t)$ converges to a constant as $t\rightarrow \infty$, the right hand side of Eq.\eqref{riccati_d} also converges to a constant. This implies that $\Psi'(t)$ also converges to a constant. This constant must vanish, otherwise $\Psi(t)$ cannot converge to a constant. Thus, the right hand side of Eq.\eqref{riccati_d} also converges to zero. All these imply that $\Psi(t)=v$ is a stationary solution of the Riccati equation Eq.\eqref{riccati_d}. Applying Proposition \ref{affine_ZCB} to the affine kernel of the form $1/B_t^\infty$, where $B_t^\infty$ is the process defined in \eqref{long_bond_affine}, it then follows that $\pi(x)$ defined in Eq.\eqref{affineeigen} is an eigenfunction of the pricing operator with the eigenvalue \eqref{affineeigenv}. We can then verify that $$M_t^\infty := S_t e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}$$ is a martingale (with $M_0^\infty=1$). We can use it to define a new probability measure $$\mathbb{Q}^\pi|_{\mathscr{F}_t}:=M_t^\infty\mathbb{P}|_{\mathscr{F}_t}$$ associated with the eigenfunction $\pi(x)$. The dynamics of $X_t$ under $\mathbb{Q}^\pi$ follows from Girsanov's Theorem. We stress that $\pi(x)$ is the eigenfunction of the pricing semigroup operator, rather than merely an eigenfunction of the generator. It is generally possible for an eigenfunction of the generator to fail to be an eigenfunction of the semigroup. That case will lead to a mere local martingale. In our case, $\pi(x)$ is an eigenfunction of the semigroup by construction, and the process $M_t^\infty$ is a martingale, rather than a mere local martingale. We now show that the condition \eqref{PKL1} holds under our assumptions in Theorem 3.2. We first re-write it under the probability measure $\mathbb{Q}^{\pi}$: \begin{equation} \lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{Q}^\pi}\left[\left|\frac{P_t^T}{P_0^TB_t^\infty}-1\right|\right]=0. \eel{BHS_L1} We will now verify that this indeed holds under our assumptions. First observe that by Eq.\eqref{representation}: \begin{equation} \frac{P_t^T}{P_0^TB_t^\infty}=e^{-\lambda t-(\Phi(T-t)-\phi(T))-(\Psi(T-t)-v)^\dagger (X_t- X_0)}. \end{equation} Since $\lim_{T\rightarrow\infty}\Psi(T)=v$ and $\lim_{T\rightarrow\infty}\Phi^\prime(T)=\lambda$, we have that $$\lim_{T\rightarrow\infty}\frac{P_t^T}{P_0^TB_t^\infty}=1$$ almost surely. Next, we show $L^1$ convergence. First, we observe that for any $\epsilon>0$ there exists $T_0$ such that for all $T>T_0$ $$|\Psi_i(T-t)-v_i|\leq\epsilon$$ for all $i\in I$ and $$e^{-\lambda t-(\Phi(T-t)-\phi(T))+(\Psi(T)-v)^\dagger X_0}\leq 1+\epsilon.$$ Thus, $$\left|\frac{P_t^T}{P_0^TB_t^\infty}-1\right|\leq1+\left|\frac{P_t^T}{P_0^TB_t^\infty}\right|\leq1+(1+\epsilon)\sum_{k_i=\pm\epsilon}e^{k^\dagger X_t}.$$ Since $X_t$ remains affine under $\mathbb{Q}^\pi$, by Theorem 4.1 of \citet{filipovic_2009} there exists $\epsilon>0$ such that $e^{k^\dagger X_t}$ is integrable under $\mathbb{Q}^\pi$ for all vectors $k$ such that $k_i=\pm\epsilon$. Thus, by the Dominated Convergence Theorem, Eq.\eqref{BHS_L1} holds. This proves (i) and (ii) (Eq.\eqref{asymptyield} follows from Eq.\eqref{bondfunction} and the fact $\Phi'(t)\rightarrow\lambda$ as $t\rightarrow\infty$). (iii) follows from Eq.\eqref{long_bond_affine} and Ito's formula. To prove (iv), we note that by Theorem \ref{implication_L1} $M_t^\infty$ is a martingale. By It\^{o}'s formula, its volatility is $-\lambda_t^\infty$. This proves (iv). Part (v) follows from Eq.\eqref{gammainfty}. To prove (vi), first note that Eq.\eqref{minfty} and Girsanov's theorem implies that $W_t^\mathbb{L}=W_t^\mathbb{P}+\int_0^t\lambda_s^\infty ds$ is an $\mathbb{L}$-Brownian motion. The dynamics of $X_t$ and $B_t^\infty$ under $\mathbb{L}$ then follows. $\Box$ \end{proof} The economic meaning of Theorem 3 is that the existence of a fixed point $v$ of the solution to the Riccati equation is sufficient for existence of the long term limit. The fixed point $v$ itself identifies the volatility of the long bond in Eq.(17) and the long-term zero-coupon yield in Eq.(16) via the principal eigenvalue (15). We note that the condition in Theorem 3.2 of \citet{linetsky_2014long} is automatically satisfied in affine models. Indeed, from Eq.\eqref{bondfunction} when the Riccati equation has a fixed point $v$, from Theorem 3.2 in this paper we have $$ \lim_{T\rightarrow \infty}\frac{P(T-t,x)}{P(T,x)}=e^{\lambda t}, $$ and we can write $P(t,x)=e^{-\lambda t}L_x(t)$, where $L_x(t)=e^{\lambda t}P(t,x)$ is a slowly varying function of time $t$ for each $x$. By Eq.\eqref{asymptyield}, the eigenvalue $\lambda$ is identified with the asymptotic long-term zero-coupon yield. We note that since $\Psi(t)=v$ is a stationary solution of the Riccati ODE \eqref{riccati_d}, the vector $v$ satisfies the following {\em quadratic vector equation}: $$\frac{1}{2}v^\dagger \alpha_{i}v+\beta_i^\dagger v-\delta_i=0,\quad i\in\emph{I},\quad B_{JJ}^\dagger v_J-\delta_J=0.$$ However, in general this quadratic vector equation may have multiple solutions leading to multiple exponential-affine eigenfunctions. In order to determine the solution that defines the long-term factorization, if it exists, it is essential to verify that $v$ is the limiting solution of the Riccatti ODE, i.e. that Eq.\eqref{psi_converge} holds. In this regard, we recall that \citet{linetsky_2014_cont} identified the unique {\em recurrent eigenfunction} $\pi_R$ of an affine pricing kernel with the {\em minimal} solution of the quadratic vector equation (see Appendix F in the on-line e-companion to \citet{linetsky_2014_cont}). We recall that, for a Markovian pricing kernel $S$ (see \citet{hansen_2009} and \citet{linetsky_2014_cont}), we can associate a martingale $$M^\pi_t=S_t e^{\lambda t}\frac{\pi(X_t)}{\pi(X_0)}$$ with {\em any} positive eigenfunction $\pi(x)$. In general, positive eigenfunctions are not unique. \citet{linetsky_2014_cont} proved uniqueness of a recurrent eigenfunction $\pi_R$ defined as such a positive eigenfunction of the pricing kernel $S$, i.e. $${\mathbb E}_x^{\mathbb P}[S_t \pi(X_t)]=e^{-\lambda t}\pi(x)$$ for some $\lambda$, that, under the locally equivalent probability measure (eigen-measure) ${\mathbb Q}^{\pi_R}$ defined by using the associated martingale $M_t^{\pi_R}$ as the Radon-Nikodym derivative, the Markov state process $X$ is recurrent. However, in general, without additional assumptions, the recurrent eigenfunction $\pi_R$ associated with the minimal solution to the quadratic vector equation may or may not coincide with the eigenfunction $\pi_L$ germane to the long-term limit and, thus, the long forward measure may or may not coincide with the recurrent eigenmeasure (the fixed point $v$ of the Riccati ODE may or may not be the minimal solution of the quadratic vector equation). Under additional exponential ergodicity assumptions the fixed point of the Riccati ODE is necessarily the minimal solution of the quadratic vector equation and $\pi_R=\pi_L$. If the exponential ergodicity assumption is not satisfied, they may differ, or one may exist, while the other does not exist. We refer the reader to \citet{linetsky_2014_cont} and \citet{linetsky_2014long} for the exponential ergodicity assumption. Analytical tractability of affine models allows us to provide fully explicit examples to illustrate these theoretical possibilities. In the next section we give a range of examples. \section{Examples} \label{examples} \subsection{Cox-Ingersoll-Ross Model} \label{example_cir} Suppose the state follows a CIR diffusion (\citet{cox_1985_2}): \begin{equation} dX_t=(a -\kappa_{\mathbb P} X_t)dt+\sigma\sqrt{X_t}dW^{\mathbb{P}}_t, \eel{cir} where $a>0$, $\sigma>0$, $\kappa_{\mathbb P}\in {\mathbb R}$, and $W^{\mathbb{P}}$ is a one-dimensional standard Brownian motion (in this case $m=d=1$ and $n=0$). Consider the CIR pricing kernel in the form \eqref{affine_pk}. The short rate is given by \eqref{affineshortr} with $g=\gamma+au$ and $h=\delta-u\kappa_{\mathbb P}-u^2\sigma^2/2$. For simplicity we choose $\gamma=-au$ and $\delta=1+u\kappa_{\mathbb P}+u^2\sigma^2/2,$ so that the short rate can be identified with the state variable, $r_t=X_t$. The market price of Brownian risk is $\lambda_t=\sigma u \sqrt{X_t}$. Under ${\mathbb Q}$ the short rate follows the process \eqref{affineq}, which is again a CIR diffusion, but with a different rate of mean reversion: \begin{equation} \kappa_{\mathbb Q}=\kappa_{\mathbb P}+\sigma^2u. \end{equation} The fixed point $v$ of the Riccati ODE $$\Psi'(t)=-\frac{1}{2}\sigma^2\Psi^2(t)-\kappa_{\mathbb P}\Psi(t)+\delta$$ with the initial condition $\Psi(0)=u$ can be readily determined. Since $-\frac{1}{2}u^2\sigma^2-u\kappa_{\mathbb P}+\delta=1>0$, we know that $\Psi(0)=u$ is between the two roots of the quadratic equation $-\frac{1}{2}\sigma^2 x^2-\kappa_{\mathbb P} x+\delta=0$. This immediately implies that $\Psi(t)$ converges to the larger root, i.e. \begin{equation} \lim_{t\rightarrow\infty}\Psi(t)=\frac{\sqrt{\kappa_{\mathbb P}^2+2\sigma^2\delta}-\kappa_{\mathbb P}}{\sigma^2}=\frac{\sqrt{\kappa_{\mathbb Q}^2+2\sigma^2}-\kappa_{\mathbb P}}{\sigma^2}=\frac{\kappa_{\mathbb L}-\kappa_{\mathbb P}}{\sigma^2}=:v, \end{equation} where we introduce the following notation: $$ \kappa_{\mathbb L}=\sqrt{\kappa_{\mathbb Q}^2+2\sigma^2}. $$ Thus, the long bond in the CIR model is given by \begin{equation} B_t^\infty=e^{\lambda t-\frac{\kappa_{\mathbb L}-\kappa_{\mathbb Q}}{\sigma^2}(X_t-X_0)} \eel{cirlongbond} with \begin{equation} \lambda=\frac{a(\kappa_{\mathbb L}-\kappa_{\mathbb Q})}{\sigma^2} \eel{cireigenvalue} and the long bond volatility $$ \sigma_t^\infty=-\frac{\kappa_{\mathbb L}-\kappa_{\mathbb Q}}{\sigma} \sqrt{X_t}. $$ Under the long forward measure the state follows the process \eqref{affinel}, which is again a CIR diffusion, but with the different rate of mean reversion $\kappa_{\mathbb L}>\kappa_{\mathbb Q}$. The fixed point $v$ is proportional to the difference between the rate of mean reversion under the long forward measure ${\mathbb L}$ and the data generating measure ${\mathbb P}$. It defines the market price of risk under ${\mathbb L}$ via $\lambda_t^\infty = v\sigma \sqrt{X_t}$. We note that if one selects $u=(-\kappa_\mathbb{P}\pm\sqrt{\kappa_\mathbb{P}^2-2\sigma^2})/\sigma^2$ in the specification of the pricing kernel, then $v=0$ and $\lambda_t^\infty=0$, so the margingale component in the long term factorization is degenerate, and the pricing kernel is in the transition independent form. In this case, $\kappa_{\mathbb P}=\kappa_{\mathbb L}$ so that the data-generating measure coincides with the long-forward measure. This is the condition of Ross' recovery theorem (see \citet{linetsky_2014_cont} for more details). Since the closed form solution for the CIR zero-coupon bond pricing function is available (\citet{cox_1985_2}), these results can also be recovered by directly calculating the limit $$\lim_{T\rightarrow \infty}\frac{P(T-t,y)}{P(T,x)}=e^{\lambda t}\frac{\pi(y)}{\pi(x)}$$ with the eigenvalue $\lambda$ given by Eq.\eqref{cireigenvalue} and the eigenfunction $\pi(x)=e^{-\frac{\kappa_{\mathbb L}-\kappa_{\mathbb Q}}{\sigma^2}x}$. \begin{remark} \citet{borovicka_2014mis} in their Example 4 on p.2513 also consider an exponential-affine pricing kernel driven by a single CIR factor. However, their specification of the PK is in a special form such that $h=0$ in Eq.(4) for the short rate (which corresponds to the choice $\delta=u\kappa_{\mathbb P}+u^2\sigma^2/2$ in our parameterization). Thus, all dependence on the CIR factor is contained in the martingale component in the risk-neutral factorization of their PK, with the short rate being constant. In this special case the long bond is deterministic and the long forward measure is simply equal to the risk-neutral measure since the short rate is independent of the state variable. In this special case the pricing operator has two distinct positive eigenfunctions. One of the eigenfunctions is constant. This eigenfunction defines the risk-neutral measure, which coincides with the long forward measure in this case due to independence of the short rate and the eigenfunction of the state variable. The second eigenfunction (Eq.(19) in \citet{borovicka_2014mis}) defines a probability measure, which is distinct from the risk-neutral measure and, hence, distinct from the long forward measure as well. Depending on the specific parameter values of the CIR process, either one of the two eigenfunctions may serve as the recurrent eigenfunction. The eigenmeasure associated with the other eigenfunction will not be recurrent, as the CIR process will have a non-mean reverting drift under that measure. \end{remark} \subsection{CIR Model with Absorption at Zero: ${\mathbb L}$ Exists, ${\mathbb Q}^{\pi_R}$ Does Not Exist} \label{example_absorb} We next consider a degenerate CIR model \eqref{cir} with $a=0$, $\sigma>0$, and $\kappa\in {\mathbb R}$. When $a$ vanishes, the diffusion has an absorbing boundary at zero, i.e. there is a positive probability to reach zero in finite time and, once reached, the process stays at zero with probability one for all subsequent times. Consider a pricing kernel in the form of Eq.\eqref{affine_pk}. The short rate is given by \eqref{affineshortr} with $g=\gamma$ and $h=\delta-u\kappa_\mathbb{P}-\frac{1}{2}u^2\sigma^2$. We assume $\gamma=0$ and $\delta=1+u\kappa_\mathbb{P}+\frac{1}{2}u^2\sigma^2>0$, so that short rate $r_t$ takes values in $\mathbb{R}_+$. The market price of Brownian risk is $\lambda_t=\sigma u \sqrt{X_t}$, and under ${\mathbb Q}$ the short rate follows the process \eqref{affineq}, which is again a CIR diffusion with an absorbing boundary at zero, but with a different rate of mean reversion $\kappa_\mathbb{Q}=\kappa_\mathbb{P}+\sigma^2 u$. It is clear that under any locally equivalent measure, zero remains absorbing and thus no recurrent eigenfunction exists. Nevertheless, we can proceed in the same way as in our analysis of the CIR model to show that $$B_t^\infty=e^{-\frac{\kappa_\mathbb{L}-\kappa_\mathbb{Q}}{\sigma^2}(X_t-X_0)}$$ with $\kappa_\mathbb{L}=\sqrt{\kappa_\mathbb{Q}^2+2\sigma^2}$ is the long bond and $X_t$ solves the CIR SDE \eqref{cir} with $a=0$ and mean-reverting rate $\kappa_\mathbb{L}$ under ${\mathbb L}$. In fact, the treatment of the long bond and the long forward measure is exactly the same as in the non-degenerate example with $a>0$, even though this case is transient with absorption at zero. The eigenvalue degenerates in this case, $\lambda=0$, and the asymptotic long-term zero-coupon yield vanishes, corresponding to the eventual absorption of the short rate at zero. \subsection{Vasicek Model} \label{example_ou} Our next example is the \citet{vasicek_1977equilibrium} model with the state variable following the OU diffusion: \[ dX_t=\kappa(\theta_\mathbb{P}-X_t)dt+\sigma dW^{\mathbb{P}}_t \] with $\kappa>0$, $\sigma>0$ (in this case $m=0$, $n=d=1$). Consider the pricing kernel in the form \eqref{affine_pk}. The short rate is given by \eqref{affineshortr} with $g=\gamma+u\kappa\theta_\mathbb{P}-\frac{1}{2}u^2\sigma^2$ and $h=\delta-u\kappa$. For simplicity we choose $\gamma=-u\kappa\theta_\mathbb{P}+\frac{1}{2}u^2\sigma^2$ and $\delta=1+u\kappa,$ so that the short rate is identified with the state variable, $r_t=X_t$. The market price of Brownian risk is constant in this case, $\lambda_t=\sigma u$. Under ${\mathbb Q}$ the short rate follows the process \eqref{affineq}, which in this case is again the OU diffusion, but with a different long run mean $$\theta_\mathbb{Q}=\theta_\mathbb{P}-\frac{\sigma^2 u}{\kappa}$$ (the rate of mean reversion $\kappa$ remains the same). The explicit solution to the ODE $\Psi'(t)=-\kappa\Psi(t)+\delta$ with the initial condition $\Psi(0)=u$ is $$\Psi(t)=-(\frac{\delta}{\kappa}+u)e^{-\kappa t}+\frac{\delta}{\kappa},$$ and the limit yields the fixed point $\lim_{t\rightarrow\infty}\Psi(t)=\frac{\delta}{\kappa}=:v.$ Thus, the long bond in the Vasicek model is given by $$B_t^\infty=e^{\lambda t-\frac{1}{\kappa}(X_t-X_0)}$$ with the long-term yield $$ \lambda=\theta_\mathbb{Q}-\frac{\sigma^2}{2\kappa^2} $$ and the long bond volatility $$\sigma_t^\infty=-\frac{\sigma}{\kappa}.$$ Under the long forward measure the short rate follows the process \eqref{affinel}, which is again the OU diffusion, but with a different long run mean $$\theta_\mathbb{L}=\theta_\mathbb{Q}-\frac{\sigma^2}{\kappa^2}$$ (the rate of mean reversion remains the same). \subsection{Non-mean-reverting Gaussian Model: $\mathbb{Q}^{\pi_R}$ Exists, $\mathbb{L}$ Does not Exist}\label{L_no_exist} Suppose $X_t$ is a Gaussian diffusion with affine drift and constant volatility \begin{equation} dX_t=\kappa(\theta-X_t)dt+\sigma dW^{\mathbb{P}}_t, \end{equation} but now with $\kappa<0$, so that the process is not mean-reverting. Consider a risk-neutral pricing kernel that discounts at the rate $r_t=X_t$, i.e. $S_t=e^{-\int_0^t X_s ds}$. Then the pure discount bond price is given by $P_t^T=P(X_t,T-t)$ with \begin{equation} P(x,t)=A(t)e^{-x B(t)}, \end{equation} \begin{equation} B(t)=\frac{1-e^{-\kappa t}}{\kappa},\enskip A(t)=\exp\Big\{(\theta-\frac{\sigma^2}{2\kappa^2})(B(t)-t)-\frac{\sigma^2}{4\kappa}B^2(t)\Big\}. \eel{bp_ou} It is easy to see that the ratio $P(y,T-t)/P(x,T)$ does not have a finite limit as $T\rightarrow \infty$ and, hence, $P_t^T/P_0^T$ does not converge as $T\rightarrow \infty$. Thus, the long bond and the long forward measure $\mathbb{L}$ do not exist in this case. However, the recurrent eigenfunction $\pi_R$ and the recurrent eigen-measure $\mathbb{Q}^{\pi_R}$ do exist in this case and are explicitly given in Section 6.1.3 of \citet{linetsky_2014_cont}. Under $\mathbb{Q}^{\pi_R}$, $X_t$ is the OU process with mean reversion (since $\kappa<0$): \begin{equation} dX_t=(\sigma^2/\kappa-\kappa\theta+\kappa X_t)dt+\sigma dW_t^{\mathbb{Q}^{\pi_R}}. \end{equation} \subsection{ Breeden Model} Our next example is a special case of \citet{breeden_1979intertemporal} consumption CAPM considered in Example 3.8 of \citet{hansen_2009}. There are two independent factors, a stochastic volatility factor $X_t^v$ evolving according to the CIR process \begin{equation} dX_t^v=\kappa_v(\theta_v-X_t^v)dt+\sigma_v\sqrt{X_t^v} dW_t^{v,\mathbb{P}} \end{equation} and a mean-reverting growth rate factor $X_t^g$ evolving according to the OU process \[ dX_t^g=\kappa_g(\theta_g-X_t^g)dt+\sigma_g dW_t^{g,\mathbb{P}}. \] Here it is assumed that $\kappa_v,\kappa_g>0$, $\theta_v,\theta_g>0$, $\sigma_g>0$, $\sigma_v<0$ (so that a positive increment to $W^v$ reduces volatility), and $2\kappa_v\theta_v\geq \sigma_v^2$ (so that volatility stays strictly positive). Suppose that equilibrium consumption evolves according to \begin{equation} dc_t=X_t^g dt+\sqrt{X_t^v} dW_t^{v,\mathbb{P}}+\sigma_c dW_t^{g,\mathbb{P}}, \end{equation} where $c_t$ is the logarithm of consumption $C_t$. Thus, $X^g$ models predictability in the growth rate and $X^v$ models predictability in volatility. Suppose also that the representative consumer's preferences are given by \begin{equation} \mathbb{E}\left[\int_0^\infty e^{-b t}\frac{C_t^{1-a}-1}{1-a}dt\right] \end{equation} for $a,b>0$. Then the implied pricing kernel $S_t$ is \begin{equation} S_t=e^{-bt}C_t^{-a}=\exp\left(-a\int_0^t X_s^g ds-b t-a\int_0^t \sqrt{X_s^v} dW_s^{v,\mathbb{P}}-a\int_0^t \sigma_c dW_t^{g,\mathbb{P}}\right). \end{equation} Using the SDEs for $X^g$ and $X^v$ it can be cast in the affine form \eqref{affine_pk}: \begin{equation} \begin{array}{ll} S_t & =\exp\left( -\gamma t-\frac{a}{\sigma_v}(X_t^v-X_0^v)-\frac{a\sigma_c}{\sigma_g}(X_t^g-X_0^g)\right. \\ & \left.\quad-\frac{a\kappa_v}{\sigma_v}\int_0^t X_s^v ds-(a+\frac{a\sigma_c\kappa_g}{\sigma_g})\int_0^t X_s^gds\right),\\ \end{array} \end{equation} where $\gamma=b-\frac{a\kappa_v\theta_v}{\sigma_v}-\frac{a\sigma_c\kappa_g\theta_g}{\sigma_g}$. \begin{proposition} If $\kappa_g>0$ (mean-reverting growth rate) and $\kappa_v+\sqrt{\kappa_v^2+2a\kappa_v\sigma_v}+a\sigma_v>0$, Eq.\eqref{psi_converge} holds and, thus, Theorem \ref{affine_long} applies. The long bond is given by \begin{equation} B_t^\infty=\exp\left(\lambda t+(\frac{a}{\sigma_v}-v_1)(X_t^v-X_0^v)+(\frac{a\sigma_c}{\sigma_g}-v_2)(X_t^g-X_0^g)\right), \end{equation} where $\lambda=\gamma-\frac{1}{2}\sigma_g^2v_2^2+\kappa_v\theta_v v_1+\kappa_g\theta_g v_2$, $v_1=(\sqrt{\kappa_v^2+2a\kappa_v\sigma_v}-\kappa_v)/\sigma_v^2$, $v_2=a(1/\kappa_g+\sigma_c/\sigma_g)$, and the state variables have the following dynamics under ${\mathbb L}$: \begin{equation} dX_t^v=\left(\kappa_v\theta_v-\sqrt{\kappa_v^2+2a\kappa_v\sigma_v}X_t^v\right)dt+\sigma_v\sqrt{X_t^v}dW_t^{v,\mathbb{L}}, \end{equation} \begin{equation} dX_t^g=\kappa_g\left(\theta_g-\frac{a\sigma_g^2}{\kappa_g^2}-\frac{a\sigma_c\sigma_g}{\kappa_g}-X_t^g\right)dt+\sigma_g dW_t^{g,\mathbb{L}}. \end{equation} \end{proposition} \begin{proof} In this model Eq.\eqref{riccati_d} reduces to \begin{equation} \begin{split} &\Phi^\prime(t)=-\frac{1}{2}\sigma_g^2\Psi_2(t)^2 +\kappa_v\theta_v\Psi_1(t)+\kappa_g\theta_g\Psi_2(t)+\gamma, \quad \Phi(0)=0,\\ &\Psi_1^\prime(t)=-\frac{1}{2}\sigma_v^2\Psi_1(t)^2 -\kappa_v\Psi_1(t)+\frac{a\kappa_v}{\sigma_v},\quad \Psi_1(0)=\frac{a}{\sigma_v},\\ &\Psi_2^\prime(t)=-\kappa_g\Psi_2(t)+a+\frac{a\sigma_c\kappa_g}{\sigma_g},\quad \Psi_2(0)=\frac{a\sigma_c}{\sigma_g}.\\ \end{split} \end{equation} In this special case $\Psi_1(t)$ and $\Psi_2(t)$ are separated and thus can be analyzed independently. It is easy to see that if $\kappa_g>0$ then $\Psi_2(t)$ converges to $v_2$. When $\kappa_v+\sqrt{\kappa_v^2+2a\kappa_v\sigma_v}+a\sigma_v>0$, $\frac{a}{\sigma_v}$ is greater than the smaller root of the second order equation $-\frac{1}{2}\sigma_v^2\Psi_1(t)^2 -\kappa_v\Psi_1(t)+\frac{a\kappa_v}{\sigma_v}$, which implies that $\Psi_1(t)$ converges to the larger root of the second-order equation for $v_1$. The eigenvalue and the dynamics of the state variable can be computed accordingly. $\Box$. \end{proof} The proof essentially combines the proofs in Examples \ref{example_cir} and \ref{example_ou}. Similar to these examples, we observe that the rate of mean reversion of the volatility factor is higher under the long forward measure, $\sqrt{\kappa_v^2+2a\kappa_v\sigma_v}>\kappa_v$, while the rate of mean reversion of the growth rate remains the same, but its long run level is lower under ${\mathbb L}$. \subsection{ \citet{borovicka_2014mis} Continuous-Time Long-Run Risks Model} Our next example is a continuous-time version of the long-run risks model of \citet{bansal_2004risks} studied by \citet{borovicka_2014mis}. It features growth rate predictability and stochastic volatility in the aggregate consumption and recursive preferences. The model is calibrated to the consumption dynamics in \citet{bansal_2004risks}. The two-dimensional state modeling growth rate predictability and stochastic volatility follows the affine dynamics: {\small\begin{equation} d\begin{bmatrix} X^1_t\\ X^2_t\\ \end{bmatrix} =\left( \begin{bmatrix} 0.013\\ 0 \end{bmatrix}+ \begin{bmatrix} -0.013&0\\ 0&-0.021\\ \end{bmatrix} \begin{bmatrix} X^1_t\\X^2_t \end{bmatrix}\right)dt+\sqrt{X_t^1} \begin{bmatrix} -0.038&0\\ 0&0.00034 \\ \end{bmatrix} d \begin{bmatrix} W^{1,\mathbb{P}}_t\\ W^{2,\mathbb{P}}_t\\ \end{bmatrix}, \end{equation}} where $W^{i,\mathbb{P}}_t,$ $i=1,2,$ are two independent Brownian motions. Here $X^1_t$ is the stochastic volatility factor following a CIR process and $X^2_t$ is an OU-type mean-reverting growth rate factor with stochastic volatility. The aggregate consumption process $C_t$ in this model evolves according to \begin{equation} d\log C_t=0.0015dt+X^2_tdt+\sqrt{X^1_t} 0.0078 dW^{3,\mathbb{P}}_t, \end{equation} where $W^{3,\mathbb{P}}$ is a third independent Brownian motion modeling direct shocks to consumption. Numerical parameters are from \citet{borovicka_2014mis} and are calibrated to monthly frequency (here time is measured in months). The representative agent in this model is endowed with recursive homothetic preferences and a unitary elasticity of substitution. \citet{borovicka_2014mis} solve for the pricing kernel: \[ d\log S_t=-0.0035dt-0.0118X^1_tdt-X^2_t dt-\sqrt{X^1_t}\Big[0.0298\quad0.1330\quad0.0780\Big]dW^{\mathbb{P}}_t, \] where the three-dimensional Brownian motion $W^{\mathbb{P}}_t=(W^{i,\mathbb{P}}_t)_{i=1,2,3}$ is viewed as a column vector. We now cast this model specification in the {\em three-dimensional} affine form of Assumption \ref{assumption_affine_PK}. To this end, we introduce a third factor $X^3_t=\log S_t$. We can then write the pricing kernel in the exponential affine form $S_t=e^{X_t^3}$, where the state vector $(X^1_t,X^2_t,X^3_t)$ follows a three-dimensional affine diffusion driven by a three-dimensional Brownian motion: \begin{equation} dX_t =\left( b+ B X_t\right)dt+\sqrt{X_t^1} \rho dW^{\mathbb{P}}_t, \end{equation} where the numerical values for entries of the three-dimensional vector $b$ and $3\times 3$-matrices $B$ and $\rho$ are given above. We can now directly apply our general results for affine pricing kernels. First, by Theorem \ref{RN_affine}, the short rate is $r(X_t)=0.0035-0.00057798 X^1_t+X^2_t$ and depends only on the factors $X^1$ and $X^2$ and is independent of $X^3$. The risk-neutral (${\mathbb Q}$-measure) dynamics is given by: \begin{equation} d\begin{bmatrix} X^1_t\\ X^2_t\\ X^3_t\\ \end{bmatrix} =\left( \begin{bmatrix} 0.013\\0\\-0.0035 \end{bmatrix}+ \begin{bmatrix} -0.0119&0&0\\ -0.00004522&-0.021&0\\ 0.0129&-1&0\\ \end{bmatrix} \begin{bmatrix} X^1_t\\X^2_t\\X^3_t \end{bmatrix}\right)dt+\sqrt{X_t^1}\rho dW^{\mathbb{Q}}_t, \end{equation} where \begin{equation} \rho=\begin{bmatrix} -0.038&0&0\\0&0.00034&0\\-0.0298&-0.1330&-0.0780\\ \end{bmatrix}. \end{equation} The vector $\Psi(t)=(\Psi_1(t),\Psi_2(t),\Psi_3(t))^\dagger$ solves the ODE (here $\alpha:=\rho\rho^\dagger$): $$ \Psi_1^\prime(t)=-\frac{1}{2}\Psi(t)^\dagger \alpha\Psi(t)+B_{11}\Psi_1(t)+B_{21}\Psi_2(t)+B_{31}\Psi_3(t), $$ $$ \Psi_2^\prime(t)=B_{22}\Psi_2(t)+B_{32}\Psi_3(t),\quad \Psi_3^\prime(t)=0 $$ with $\Phi(0)=\Psi_1(0)=\Psi_2(0)=0, \Psi_3(0)=-1$. It is immediate that $$\Psi_3(t)\equiv-1\quad \text{and}\quad \Psi_2(t)=\frac{B_{32}}{B_{22}}(1-e^{B_{22}t})$$ and, since $B_{22}<0$, $$\lim_{t\rightarrow\infty}\Psi_2(t)=B_{32}/B_{22}=47.6191:=v_2.$$ To see $\Psi_1(t)$ convergence, notice that we can write $-\frac{1}{2}\Psi(t)^\dagger \alpha\Psi(t)+B_{11}\Psi_1(t)+B_{21}\Psi_2(t)+B_{31}\Psi_3(t)=c_1(\Psi_1(t))^2 + c_2 \Psi_1(t) + c_3 (\Psi_2(t))^2 + c_4 \Psi_2(t) + c_5$, where $c_1, c_2, c_3, c_4, c_5<0$. Since $\Psi_1(0)=\Psi_2(0)=0$, we have $\Psi_1'(0)<0$. Since $\Psi_2(t)>0$ and it is easy to see that $\Psi_1(t)<0$. Since $\Psi_2(t)<v_2$, we have $c_1(\Psi_1(t))^2 + c_2 \Psi_1(t) + c_3 (\Psi_2(t))^2 + c_4 \Psi_2(t) + c_5>c_1(\Psi_1(t))^2 + c_2 \Psi_1(t) + c_3 v_2^2+c_4 v_2+c_5$. We can check that $c_1(\Psi_1(t))^2 + c_2 \Psi_1(t) + c_3 v_2^2+c_4 v_2+c_5=0$ has two negative roots. Denote the larger root $v_1$, we see that $\Psi_1(t)>v_1$. Combining these facts, we see that $\Psi_1(t)$ converges to $v_1$. The exact value of $v_1$ has to be determined numerically. The numerical solution yields $$v_1=\lim_{t\rightarrow\infty}\Psi_1(t)=-0.2449.$$ In Figure \ref{phipsi}, we plot the functions $\Psi_1(t)$ and $\Psi_2(t)$, as well as the gross return $B_t^{t+T}$ on the $T$-bond over the period $[0,t]$ as a function of $T$. In this numerical example we take $t=12$ months, so we are looking at the one-year holding period return, and assume that the initial state $X_0$ and the state $X_t$ are both equal to the stationary mean under $\mathbb{P}$. We observe that in this model specification $\Psi(t)$ and $B_t^{t+T}$ are already very close to the fixed point for $t$ around 30 years (360 months). \begin{figure} \caption{Plot of $\Psi_1(t)$, $\Psi_2(t)$ and $B_t^{t+T}$. Time is measured in months.} \label{phipsi} \end{figure} By Theorem \ref{affine_long}, the eigenfunction determining the long bond is $\pi(x)=e^{-v_1 x^1-v_2 x^2},$ corresponding to the eigenvalue (note this is not annualized yield since time unit is in month) $$\lambda=b_1v_1+b_2v_2-b_3=0.0003163,$$ the long bond is given by $$B_t^\infty=e^{\lambda t - v_1(X_t^1-X_0^1)-v_2(X_t^2-X_0^2)},$$ the martingale component is given by $$M_t^\infty=e^{\lambda t - v_1(X_t^1-X_0^1)-v_2(X_t^2-X_0^2)+X_t^3},$$ and the state vector $(X^1_t, X_t^2, X^3_t)$ has the following dynamics under the long forward measure ${\mathbb L}$: \begin{equation} d\begin{bmatrix} X^1_t\\ X^2_t\\ X^3_t\\ \end{bmatrix} =\left( \begin{bmatrix} 0.013\\0\\-0.0035 \end{bmatrix}+ \begin{bmatrix} -0.0115&0&0\\ -0.00005074&-0.021&0\\ 0.0153&-1&0\\ \end{bmatrix} \begin{bmatrix} X^1_t\\X^2_t\\X^3_t \end{bmatrix}\right)dt+\sqrt{X_t^1}\rho dW^{\mathbb{L}}_t. \end{equation} As already observed by \citet{borovicka_2014mis}, in this model the state dynamics under the long forward measure ${\mathbb L}$ is close to the state dynamics under the risk-neutral measure ${\mathbb Q}$ and is substantially distinct from the dynamics under the data-generating measure ${\mathbb P}$ due to the volatile martingale component $M_t^\infty$. However, our approach to the analysis of this model is different from the analysis of \citet{borovicka_2014mis}. We cast it as a three-factor affine model and directly apply our Theorem \ref{affine_long} for affine models that is, in turn, a consequence of our Theorem \ref{implication_L1} for semimartingale models. We only need to determine the fixed point \eqref{psi_converge} of the Riccati equation. Existence of the long bond, the long term factorization of the pricing kernel, and the long forward measure then immediately follow from Theorem \ref{affine_long}, without any need to verify ergodicity. In fact, the three-factor affine process $(X^1_t,X_t^2,X_t^3)$ is not ergodic, and not even recurrent, as is immediately seen from the dynamics of $X^3$. In contrast, the approach in \citet{borovicka_2014mis} relies on the two-dimensional mean-reverting affine diffusion $(X^1_t,X_t^2)$. Namely, since the Perron-Frobenius theory of \citet{hansen_2009} requires ergodicity to single out the principal eigenfunction and ascertain its relevance to the long-term factorization, \citet{borovicka_2014mis} implicitly split the pricing kernel into the product of two sub-kernels, a multiplicative functional of the two-dimensional Markov process $(X^1_t,X_t^2)$ and the additional factor in the form $e^{-\int_0^t 0.0780\sqrt{X_s^1}dW_s^{3,{\mathbb P}}}$. The Perron-Frobenius theory of \citet{hansen_2009} is then applied to the multiplicative functional of the two-dimensional Markov process $(X^1_t,X_t^2)$. In contrast, in our approach we do not require ergodicity and work directly with the non-ergodic three-dimensional process and verify that the Riccati ODE possesses a fixed point, which is already sufficient for existence of the long-term factorization in affine models by Theorem \ref{affine_long}. \section{Conclusion} This paper constructs and studies the long-term factorization of affine pricing kernels into discounting at the rate of return on the long bond and the martingale component that accomplishes the change of probability measure to the long forward measure. It is shown that the principal eigenfunction of the affine pricing kernel germane to the long-term factorization is an exponential-affine function of the state vector with the coefficient vector identified with the fixed point of the Riccati ODE. The long bond volatility and the volatility of the martingale component are explicitly identified in terms of this fixed point. When analyzing a given affine model, a research needs to establish whether the Riccati ODE possesses a fixed point. If the fixed point is determined, the long-term factorization then follows. It is shown how the long-term factorization plays out in a variety of asset pricing models, including single factor CIR and Vasicek models, a two-factor version of Breeden's CCAPM, and the three-factor long-run risks model studied in \citet{borovicka_2014mis}. \end{document}

\begin{document} \title{Hochschild cohomology of the algebra of conformal endomorphisms} \section{Introduction} The notion of a conformal (Lie) algebra emerged in \cite{KacValgBeginners} as a tool in the theory of vertex algebras which goes back to mathematical physics \cite{BPZ1983} and representation theory (see, e.g., \cite{Borch}). From the algebraic point of view, the structure of a vertex algebra is a breed of two structures: a differential left-symmetric algebra and a Lie conformal algebra \cite{BK-Field2002}. The structure theory of (finite) Lie conformal algebra was developed in \cite{DK1998}, irreducible representations of simple and indecomposable semisimple finite Lie conformal algebras were described in \cite{ChengKac}. Given a finite conformal module $M$ over a Lie conformal algebra $L$, the representation of $L$ on $M$ is a homomorphism from $L$ to the Lie conformal algebra of conformal endomorphisms $\mathrm{gc}\,(M)$, see \cite[Ch.~2]{KacValgBeginners}. The latter is an analogue of the ``ordinary'' Lie algebra $\mathrm{gl}\,(V)$ of a linear space $V$ in the category of conformal algebras. As in ordinary algebras, $\mathrm{gc}\,(M)$ is the commutator algebra of an {\em associative} conformal algebra $\mathop {\fam 0 Cend} \nolimits (M)$. Thus the study of associative conformal algebras (and $\mathop {\fam 0 Cend} \nolimits (M)$, in particular) is essential for representation theory of Lie conformal algebras and, as a corollary, for vertex algebras theory. A systematic study of $\mathop {\fam 0 Cend} \nolimits (M)$ was performed in \cite{BKL2003}, its simple subalgebras were described in \cite{Kol2006Adv}. The most interesting case is when $M$ is a free $H$-module of rank $k$, then $\mathop {\fam 0 Cend} \nolimits (M)$ is denoted $\mathop {\fam 0 Cend} \nolimits_k$. This system plays the same role in the theory of conformal algebras as the matrix algebra $M_k(\Bbbk )$ does in the ordinary algebra. The homological studies for conformal algebras starts from the paper \cite{BKV}. Conceptually, to define (co)chains, (co)cycles, and (co)boundaries for a particular class of algebras over a field $\Bbbk $, one needs to know what a multilinear mapping is, how to combine such mappings, and how symmetric groups act on multilinear mappings. All these notions have their analogues in the category of modules over cocommutative bialgebras, that is, these are pseudo-tensor categories \cite{BDK}. In particular, the definition of Hochschild cohomologies for an associative algebra in the pseudo-tensor category over the polynomial bialgebra $H=\Bbbk [\partial ]$, where $\partial $ is a primitive element, coincides with the definition of Hochschild cohomology of associative conformal algebras in \cite{BKV}. It is well-known since \cite{Hoch1943} that for the associative algebra $\mathop {\fam 0 End} \nolimits (V)$ of linear transformations of a finite-dimensional space $V$ all $n$th Hochschild cohomology groups are trivial for $n\ge 1$. The problem of description of conformal Hochschild cohomologies of $\mathop {\fam 0 Cend} \nolimits (M)$ for a finite $H$-module $M$ was stated in \cite{BKV}. In \cite{Dolg2009}, it was shown that the second Hochschild cohomology group of $C=\mathop {\fam 0 Cend} \nolimits (M)$ is trivial for all conformal bimodules over~$C$, which was a partial solution of the problem from \cite{BKV}. The purpose of this paper is to complete solving this problem and prove that all $n$th Hochschild cohomology groups of $\mathop {\fam 0 Cend} \nolimits (M)$ for $n\ge 2$ with coefficients in all conformal bimodules over $\mathop {\fam 0 Cend} \nolimits (M)$. Note that the classical argument (see \cite{Hoch1943}) based on the isomorphism $\mathrm H^n(A,M)\simeq \mathrm H^{n-1}(A, \mathrm{Hom}\,(A,M))$ does not work for conformal algebras since the analogue of $\mathrm{Hom}$ denoted $\mathrm{Chom}$ (see \cite{KacValgBeginners}) does not carry a structure of conformal bimodule due to locality issues. As shown in \cite{BKV}, the calculation of conformal Hochschild cohomology $\mathrm H^\bullet (C,M)$ of an associative conformal algebra $C$ with coefficients in a conformal bimodule $M$ over $C$ is based on the ordinary Hochschild cohomology $\mathrm H^\bullet (\mathcal A_+(C), M)$, where $\mathcal A_+(C)$ is the positive part of the coefficient algebra of~$C$. For $C=\mathop {\fam 0 Cend} \nolimits_k$, the positive part $\mathcal A_+(\mathop {\fam 0 Cend} \nolimits_k)$ of its coefficient algebra is isomorphic to the matrix algebra over the first Weyl algebra $W_1$, i.e., the unital associative algebra generated by two elements $p$, $q$ such that $qp-pq=1$. The series of Weyl algebras (and, in particular, the first one) is under intensive study in various areas of mathematics. Homological properties of these algebras were considered, for example, in \cite{GHL, Rine, Hart}. For instance, the global dimension of the Weyl algebra $W_n$, $n\ge 1$, essentially depends on the characteristic of the base field. One of the by-products of this paper is an explicit computation of the 3rd Hochschild cohomology group of the first Weyl algebra by means of the Anick resolution. We apply the Morse matching method to transform a bar-resolution of the first Weyl algebra into its Anick resolution and calculate explicitly $\mathrm H^3(W_1, M)$ for an arbitrary $W_1$-bimodule~$M$. As a result, we solve a problem stated in \cite{BKV} on the computation of Hochschild cohomologies of the conformal algebra $\mathop {\fam 0 Cend} \nolimits_k$: we prove $\mathrm H^n(\mathop {\fam 0 Cend} \nolimits_k, M)=0$ for all $n\ge 2$ and for all conformal bimodules $M$ over $\mathop {\fam 0 Cend} \nolimits_k$. \section{Morse matching method for constructing the Anick resolution}\label{sec:MorseMatching} The idea of D. Anick on the construction of a relatively small free resolution for an augmented algebra has shown its effectiveness in a series of applications \cite{AK2020,A2021,A2022-Cn, A2022, Akl, lopatkin}. Let us briefly observe the main points of this construction and its application to the computation of Hoch\-schild cohomologies of associative algebras. Suppose $\Lambda $ is a unital associative algebra equipped with a homomorphism $\varepsilon: \Lambda \to \Bbbk $, $\varepsilon(1)=1$ (augmentation). Denote by $A$ the cokernel $\Lambda/\Bbbk $ of the inverse embedding $\eta : \Bbbk \to \Lambda $ and consider the two-sided bar resolution of free $\Lambda$-bimodules \[ 0\leftarrow \Bbbk \leftarrow \mathrm{B}_{0} \leftarrow \mathrm{B}_1 \leftarrow \dots \leftarrow \mathrm{B}_n \leftarrow \mathrm{B}_{n+1} \leftarrow \dots , \] where $\mathrm{B}_0 = \Lambda\otimes \Lambda $, $\mathrm{B}_n = \Lambda \otimes A^{\otimes n}\otimes \Lambda $ for $n\ge 1$. We will denote a tensor $a_1\otimes \dots \otimes a_n \in A^{\otimes n}$ as $[a_1|\ldots |a_n]$ and omit the tensor product signs between $\Lambda $ and $A^{\otimes n}$. The arrows $d_{n+1}: \mathrm{B}_{n+1} \to \mathrm{B}_n$ are $\Lambda $-bimodule homomorphisms given by \begin{equation}\label{eq:Bar-Differential} d_{n+1}[a_1|\ldots |a_{n+1}] = a_1[a_2|\ldots |a_{n+1}] +\sum\limits_{i=1}^n (-1)^i[a_1| \ldots |a_ia_{i+1}|\ldots |a_{n+1}] + (-1)^{n+1} [a_1|\ldots |a_n] a_{n+1}, \end{equation} for $n>0$, and \[ d_1: [a]\mapsto a\otimes 1 - 1\otimes a, \quad d_0: a\otimes b \mapsto \varepsilon(ab). \] If $M$ is an arbitrary unital $\Lambda $-bimodule then \[ \Hom_{\Lambda{-}\Lambda} (\mathrm B_n, M) \simeq \Hom (A^{\otimes n}, M) \] as linear spaces, and for every $\varphi \in \Hom_{\Lambda{-}\Lambda} (\mathrm B_n, M)$ the composition $\varphi d_{n+1}: \mathrm B_{n+1} \to M$ corresponds to the $\Bbbk $-linear map $\Delta^n (\varphi ): A^{\otimes (n+1)}\to M$ which is given by the Hochschild differential formula. Therefore, if we start with an associative algebra $A$, join an exterior identity to get $\Lambda = A\otimes \Bbbk 1$ with $\varepsilon(A)=0$, then the cochain complex \[ \big ( \Hom_{\Lambda{-}\Lambda} (\mathrm B_\bullet , M), \Delta^\bullet \big ) \] coincides with Hochschild complex $\mathrm {C}^\bullet (A,M)$. The bar resolution $(\mathrm B_\bullet , d_\bullet)$ is easy to construct but it is too large for particular computations. Therefore, it is reasonable to replace $(\mathrm B_\bullet , d_\bullet)$ with a smaller but homotopy equivalent resolution, e.g., the {\em Anick resolution} $(\mathrm A_\bullet, \delta_\bullet)$, \[ 0\leftarrow \Bbbk \leftarrow \mathrm A_0 \leftarrow \mathrm A_1 \leftarrow \dots \leftarrow \mathrm A_n \leftarrow \mathrm A_{n+1} \leftarrow \dots, \quad \delta_{n+1}: \mathrm A_{n+1}\to \mathrm A_n. \] Then, given an $A$-bimodule (hence, a unital $\Lambda $-bimodule), the cohomologies of the complex \[ \big ( \Hom_{\Lambda{-}\Lambda} (\mathrm A_\bullet, M), \Delta^\bullet \big ), \quad \Delta^{n}\varphi = \varphi \delta_{n+1}, \ \varphi \in \Hom_{\Lambda{-}\Lambda} (\mathrm A_n, M), \] coincide with the Hochschild cohomologies $\mathrm H^\bullet (A,M)$. Suppose $X$ is a set of generators of the algebra $A$. Denote by $X^*$ the set of nonempty words in $X$, and let $\Bbbk \langle X\rangle $ stand for the linear span of $X^*$, this is the free associative algebra generated by~$X$. Let $\Sigma \subset \Bbbk \langle X\rangle $ be a Gr\"obner--Shirshov basis of $A$ relative to an appropriate monomial order (e.g., deg-lex order). We will denote by $V = \overline \Sigma $ the set of principal parts of relations from~$\Sigma $ (called {\em obstructions}). Recall that $\mathrm A_0 = \mathrm B_0 = \Lambda \otimes \Lambda $, $\mathrm A_n = \Lambda \otimes \Bbbk V^{(n-1)}\otimes \Lambda $, where $V^{(k)}$ stands for the set of {\em Anick $k$-chains}. By definition (see \cite{Anick1983}), $V^{(0)}=\{[x] \mid x\in X\}$, $V^{(1)} = \{[x|s] \mid x\in X, s\in X^*, xs\in V\}$, and for $k\ge 2$ the set $V^{(k)}$ is constructed on the words in $X^*$ obtained by consecutive ``hooking'' of the words from $\overline \Sigma $. This definition becomes transparent in the case when the defining relations $\Sigma $ contain at most quadratic monomials, so that all words in $V $ are of length two. For $n \ge 1$, an Anick $n$-chain is a word $v=[x_{0}|\ldots |x_{n}] \in X^{*}$ such that $x_ix_{i+1}\in V$ for $i=0,\ldots,n-1$. \begin{example}\label{exmp:UnivEnvelope} Let $\mathfrak g$ be a Lie algebra over $\Bbbk $ with a linearly ordered basis $X$. Denote $[x,y]\in \Bbbk X$, $x,y\in X$, the Lie product in $\mathfrak g$. Set $\Sigma = \{ xy-yx-[x,y] \mid x,y\in X, x>y \}$, $A=\Bbbk \langle X\rangle /(\Sigma )$. Then $\Lambda = A\oplus \Bbbk 1 $ is exactly the universal enveloping associative algebra $U(\mathfrak g)$. Then $V^{(k)} = \{[x_0|x_1|\ldots |x_k] \mid x_0>x_1>\dots >x_k, x_i\in X \}$. The elements of $V^{(k)}$ are in obvious one-to-one correspondence with the basis of $\wedge^{k+1}\mathfrak g$. \end{example} The Anick differentials $\delta_{n+1}:\mathrm A_{n+1}\to \mathrm A_n$ were computed in \cite{Anick1983} by means of a complicated inductive procedure. In order to make this computation easier, in \cite{JollWelker} and, independently, in \cite{Skoldb}, it was proposed to use algebraic discrete Morse theory developed in \cite{formancell, formanguide} to construct a smaller complex (of free modules) which is homotopy equivalent to a given one. In particular, given a bar resolution of an augmented algebra $\Lambda $, the resulting complex is the Anick resolution. The Morse matching method for computing the Anick resolution \cite{JollWelker}, \cite{Skoldb} is also described in \cite{lopatkin, A2022}. In a few words, the problem is to choose a set of edges in the weighted directed graph describing the structure of the bar resolution. Then one has to transform the graph by means of inverting the matched edges. Inverting means not only switch of direction, but also replacing the weight $c$ of the matched edge with $-c{-1}$. In the resulting graph, the non-matched vertices (critical cells) are exactly the Anick chains. Finally, in order to calcuate the Anick differential $\delta_{n+1}$ on a chain $w$ from $V^{(n)}$ one has to track all paths from $w$ to vertices from $V^{(n-1)}$. The weight of each path is equal to the product of the weights of all its edges. \begin{example}\label{exmp:Heisen-3} Let $\mathfrak g = H_3$ be the Heisenberg Lie algebra. The universal enveloping algebra $U(H_3)$ is generated by the elements $x,y,z$, relative to the following relations: \[ xy=yx+z,\quad xz=zx,\quad yz=zy. \] Assume $x>y>z$. Then the Anick $n$-chains are: \[ V^{(1)}=\{[x|y],[x|z],[y|z]\}, \quad V^{(2)}=\{[x|y|z]\}, \quad V^{(n)}=\emptyset,\ n\ge 3. \] In order to compute $\delta_3[x|y|z]$, consider a fragment of the bar resolution graph and choose a Morse matching (dashed edges on Fig.~\ref{Fig1}). Tracking the paths and collecting similar terms lead to the following answer: \[ \delta_3[x|y|z]=x[y|z]-[y|z]x+[x|z]y -y[x|z]+z[x|y]-[x|y]z. \] \begin{figure} \caption{Calculating the Anick differential of $[x|y|z]$ for $U(H_3)$} \label{Fig1} \end{figure} \end{example} \begin{remark} The differential in Example~\ref{exmp:Heisen-3} corresponds to ``two-sided'' resolution. The restriction to the left module case (i.e., when multiplication by $x,y,z$ from the right is zero) leads us exactly to the Chevalley--Eilenberg differential for the Lie algebra $H_3$. This is a general observation: given a Lie algebra $\mathfrak g$, the ``left'' Anick resolution for $U(\mathfrak g)$ coincides with the Chevalley--Eilenberg resolution for~$\mathfrak g$. \end{remark} When applied to the settings of Example~\ref{exmp:UnivEnvelope}, the Anick differential for $U(\mathfrak g)$ coincides with the Chevalley--Eilenberg differential for the Lie algebra~$\mathfrak g$. \section{Conformal endomorphisms and the 1st Weyl algebra} From now on, $\Bbbk $ is a field of characteristic zero, $H=\Bbbk [\partial ]$ is the polynomial algebra in one variable. Suppose $V$ and $M$ are two $H$-modules. A {\em conformal homomorphism} \cite{KacValgBeginners} from $V$ to $M$ is a $\Bbbk $-linear map \[ \varphi _\lambda : V\to M[\lambda ]=\Bbbk [\partial,\lambda ]\otimes _H M \] such that \[ \varphi_\lambda (f(\partial ) v) = f(\partial+\lambda )\varphi_\lambda (v) \] for all $v\in V$, $f=f(\partial) \in H$. If $M=V$ then the space of all conformal homomorphisms from $V$ to $M$ is denoted $\mathop {\fam 0 Cend} \nolimits (V)$. This is also an $H$-module: \[ (\partial \varphi)_\lambda = -\lambda \varphi_\lambda , \] and if $V$ is a finitely generated $H$-module then $\mathop {\fam 0 Cend} \nolimits (V)$ is an {\em associative conformal algebra} \cite{KacValgBeginners}: for every $\varphi,\psi \in \mathop {\fam 0 Cend} \nolimits(V)$ we have \[ (\varphi \oo\lambda \psi ) \in \mathop {\fam 0 Cend} \nolimits (V) \] defined by the rule \[ (\varphi \oo\lambda \psi )_\mu = \varphi_\lambda \psi _{\mu+\lambda }. \] If $V$ is a free $H$-module of rank $k\in \mathbb N$ then $\mathop {\fam 0 Cend} \nolimits(V)$ is denoted $\mathop {\fam 0 Cend} \nolimits_k$. Up to an isomorphism (see \cite{BKL2003, Kol2006Adv}), one may identify $\mathop {\fam 0 Cend} \nolimits_k$ with the space of all $(k\times k)$-matrices over the polynomial ring $\Bbbk [\partial , x]$ equipped with the operation \[ f(\partial, x)\oo\lambda g(\partial , x) = f(-\lambda , x)g(\partial+\lambda , x+\lambda ), \] $f,g\in \Bbbk [\partial, x]$. For matrices, the operation $(\cdot\oo\lambda \cdot)$ is extended by the ordinary row-column rule. Let $H$ act from the right on the Lawrent polynomials $\Bbbk [t,t^{-1}]$ in such a way that $\partial = -d/dt$. For every conformal algebra $C$ in the sense of \cite{KacValgBeginners}, one may define the {\em coefficient algebra} $\mathcal A(C)$ as the linear space $\Bbbk[t,t^{-1}]\otimes _H C$ equipped with the multiplication \begin{equation}\label{eq:CoeffProd} a(n)b(m) = \sum\limits_{s\ge 0} \binom{n}{s} (a\oo{s} b)(n+m-s) \end{equation} where $t^n\otimes _H a = a(n)$ for $a\in C$, $n\in \mathbb Z$, and $(a\oo s b) $ stands for the coefficient at $\lambda^s/s!$ of $(a\oo\lambda b)$, $a,b\in C$. For polynomials from $\mathop {\fam 0 Cend} \nolimits_1$, for example, we have \[ f(x)\oo{s} g(x) = f(x)\dfrac{d^s}{dx^s} g(x) \] by the Taylor formula. The subspace of $\mathcal A(C)$ spanned by all $a(n)$, $n\ge 0$, $a\in C$, is a subalgebra of $\mathcal A(C)$ denoted $\mathcal A_+(C)$. For instance, $\mathcal A(\mathop {\fam 0 Cend} \nolimits_1) = \Bbbk [t,t^{-1},x]$ as a linear space, the isomorphism identifies $t^n\otimes _H x^m$, $n\in \mathbb Z$, $m\in \mathbb Z_+$, with $x^m t^n \in \Bbbk [t,t^{-1},x]$. The product of two such monomials is calculated via \eqref{eq:CoeffProd}. For example, \[ t^n \cdot xt^m = (1\oo{0} x)t^{n+m} + n (1\oo{1} x) t^{n+m-1} = x t^{n+m} + n t^{n+m-1}, \] so $tx = xt +1$, $t^{-1}x = xt^{-1} - t^{-2}$, etc. Hence, $\mathcal A(\mathop {\fam 0 Cend} \nolimits_1)$ is isomorphic to the localization of the first Weyl algebra $W_1 = \Bbbk \langle p,q\mid qp-pq=1\rangle $ relative to the multiplicative set $\{q^s\mid s\ge 0\}$. The positive part $\mathcal A_+(\mathop {\fam 0 Cend} \nolimits_1)$ is isomorphic to the Weyl algebra itself, so $\mathcal A_+(\mathop {\fam 0 Cend} \nolimits_k) \simeq M_k(W_1)$. Let $C$ be an associative conformal algebra, and let $M$ be a conformal bimodule over~$C$. Then $M$ is a bimodule over the ordinary associative algebra $A=\mathcal A_+(C)$, the action is given by \[ a(n)\cdot u = a\oo{n} u,\quad u \cdot a(n) = \{u\oo{n} a\} = \sum\limits_{s\ge 0} (-1)^{n+s} \dfrac{1}{s!} \partial^s (u\oo{n+s} a), \] for $u\in M$, $a\in C$, $n\in \mathbb Z_+$. The {\em basic Hochschild complex} \cite{BKV} of $C$ with coefficients in~$M$ is isomorphic to the Hochschild complex of $A=\mathcal A_+(C)$ with coefficients in the same bimodule~$M$. There is a linear map \[ D_n : \C^n( A,M) \to \C^n( A ,M) \] given by \[ (D_n f)(a_1(m_1),\ldots, a_n(m_n) ) = \partial f(a_1(n_1),\ldots, a_n(m_n)) + \sum\limits_{i=1}^n m_i f(a_1(n_1),\ldots, a_i(m_i-1), \ldots, a_n(m_n)), \] for $f\in \C^n( A ,M)$. The maps $D_n$ are induced by the derivation $\partial: a(m)\mapsto -ma(m-1)$ on the algebra $A$. Since $D_{n+1}\Delta^n = \Delta^n D_n$, the image $D_\bullet \C^\bullet (A,M)$ is a subcomplex of $\C^\bullet (A,M)$, and the quotient \begin{equation}\label{eq:RestrictedComplex} \overline{\C}^\bullet(A,M) = \C^\bullet (A,M) / D_\bullet \C^\bullet (A,M) \end{equation} is isomorphic to the {\em reduced Hochschild complex} of the conformal algebra $C$ (see \cite[Theorem 6.1, Corollary 6.1]{BKV}). \begin{proposition}\label{prop:MainTool} If $C$ is an associative conformal algebra, $A = \mathcal A_+(C)$, $M$ is a conformal bimodule over $C$, and $\mathrm H^q(A,M)=0$ for all $q\ge 3$, then $\mathrm H^q(\overline{\C}^\bullet (A,M)) = 0$ for all $q\ge 3$. \end{proposition} \begin{proof} The short exact sequence \[ 0\to D_\bullet \C^\bullet (A,M) \to \C^\bullet (A,M) \to \overline{\C}^\bullet(A,M) \to 0 \] gives rise to the long exact sequence of cohomologies \[ \begin{aligned} \dots \to{}& \mathrm H^q (D_\bullet \C^\bullet (A,M)) \to \mathrm H^q (\C^\bullet (A,M)) \to \mathrm H^q (\overline{\C}^\bullet (A,M)) \\ \to{}& \mathrm H^{q+1} (D_\bullet \C^\bullet (A,M)) \to \mathrm H^{q+1} (\C^\bullet (A,M)) \to \mathrm H^{q+1} (\overline{\C}^\bullet (A,M)) \to \dots \end{aligned} \] By \cite[Proposition 2.1]{BKV}, the complexes $\C^\bullet =\C^\bullet (A,M)$ and $D_\bullet \C^\bullet $ are isomorphic in positive degrees. Hence, under the conditions of the statement, $\mathrm H^q (\overline{\C}^\bullet (A,M))$, $q\ge 3$, is clamped between zeros, thus it is zero itself. \end{proof} \section{Two-sided Anick resolution for the first Weyl algebra} In this section, we apply the Morse matching method described in Section \ref{sec:MorseMatching} to compute the 3rd Hochschild cohomology of the first Weyl algebra with coefficients in an arbitrary bimodule. The Weyl algebra $W_1$ is generated by the elements $q,p,e$, relative to the following relations: \[ qp=pq+e,\quad pe=p,\quad qe=q,\quad eq=q,\quad ep=p,\quad ee=e. \] Assume $q>p>e$. Then the sets of Anick $n$-chains for $n=1,2,3$ are easy to find: \[ \begin{aligned} V^{(1)}= {} & \{ [q|p],[q|e],[p|e],[e|q],[e|p],[e|e]\}, \\ V^{(2)}={} & \{ [q|p|e],[e|q|p],[q|e|p],[p|e|q],[q|e|e],[p|e|e],[e|e|q],[e|e|p], [e|q|e],\\ & [e|p|e],[q|e|q],[p|e|p],[e|e|e]\}, \\ V^{(3)}= {}& \{ [q|p|e|e],[e|q|p|e],[q|e|p|e],[p|e|q|e],[q|e|e|e],[p|e|e|e],\\ & [e|q|e|e], [e|p|e|e],[q|e|q|e],[p|e|p|e],[e|e|e|e],[e|e|q|p],[e|q|e|p],\\ & [e|p|e|q],[e|e|e|q],[e|e|e|p], [e|e|q|e], [e|e|p|e],[e|q|e|q],\\ & [e|p|e|p],[q|e|e|p],[p|e|e|q],[q|e|e|q],[p|e|e|p], [q|p|e|q], [q|p|e|p]\}. \end{aligned} \] In order to compute $\mathrm H^3(W_1, M)$ for an arbitrary $W_1$-bimodule $M$ we need to know the Anick differentials on $V^{(2)}$ and $V^{(3)}$. For example, consider a fragment of the graph constructed from the bar resolution of $\Lambda = W_1\oplus \Bbbk 1$ with the vertex $[q|p|e]$ with a matched edge $[p|q|e]\to [pq|e]$, see Fig.~\ref{Fig2}\,a. Note that $[p|q]$ is not an Anick chain thus should not be a critical cell. Indeed, the vertex $[p|q]$ belongs to another matched edge $[p|q]\to [pq]$ which also appears in the bar resolution graph, see Fig.~\ref{Fig2}\,b. In a similar way, construct a fragment with the vertex $[e|q|p]$ on Fig.~\ref{Fig3}\,a: all ending vertices of this fragment are either Anick chains or $[p|q]$ which is already matched. Note that the vertices $[e|p|q]$ and $[q|p|e]$ belong to matched edges. As a final example, consider the fragment with $[e|q|p|e]$ (Fig.~\ref{Fig3}\,b): all ending vertices of this graph are either Anick chains or already matched ones. In the sequel, we will often omit symbols $|$ in the elements of $V^{(n)}$. \begin{figure} \caption{Calculating the Anick differential of $[q|p|e]$ and $[q|p]$} \label{Fig2} \end{figure} \begin{figure} \caption{Calculating the Anick differential of $[e|q|p]$ and $[e|q|p|e]$} \label{Fig3} \end{figure} In the same way, one may compute Anick differentials on the other chains from $V^{(2)}$ and $V^{(3)}$. As a resul, we get the following statements. \begin{lemma}\label{lem:DiffV2} The mapping $\delta _3: \mathrm A_3\to \mathrm A_2$ is defined by \[ \begin{aligned} \delta_3[qpe]= {}&q[pe]-p[qe]-[ee]+[qp]-[qp]e, \\ \delta_3[eqp]={}&e[qp]-[qp]+[ep]q+[ee]-[eq]p,\\ \delta_3[qep]={}& q[ep]-[qe]p,\\ \delta_3[peq]={}& p[eq]-[pe]q,\\ \delta_3[qee]={}& q[ee]-[qe]e,\\ \delta_3[pee]={}& p[ee]-[pe]e,\\ \delta_3[eeq]={}& e[eq]-[ee]q,\\ \delta_3[eep]={}& e[ep]-[ee]q,\\ \delta_3[eqe]={}& e[qe]-[qe]+[eq]-[eq]e,\\ \delta_3[epe] ={}& e[pe]-[pe]+[pe]-[ep]e,\\ \delta_3[qeq]={}& q[eq]-[qe]q,\\ \delta_3[pep]={}& p[ep]-[pe]p,\\ \delta_3[eee] ={}& e[ee]-[ee]e. \end{aligned} \] \end{lemma} \begin{lemma}\label{lem:DiffV3} The mapping $\delta _4: \mathrm A_4\to \mathrm A_3$ is defined by \[ \begin{aligned} \delta_4[qpee]={}&q[pee]-p[qee]-[eee]+[qpe]e, \\ \delta_4[qeep] ={}& q[eep]-[qep]+[qee]p,\\ \delta_4[peeq]={}& p[eeq]-[peq]+[pee]q,\\ \delta_4[qeee] ={}& q[eee]-[qee]+[qee]e,\\ \delta_4[peee]={}& p[eee]-[pee]+[pee]e,\\ \delta_4[eeeq]={}& e[eeq]-[eeq]+[eee]q,\\ \delta_4[eeep]={}& e[eep]-[eep]+[eee]p,\\ \delta_4[eeqe]={}& e[eqe]-[eeq]+[eeq]e,\\ \delta_4[eepe]={}& e[epe]-[eep]+[eep]e,\\ \delta_4[qeeq]={}& q[eeq]-[qeq]+[qee]q,\\ \delta_4[peep]={}& p[eep]-[pep]+[pee]p,\\ \delta_4[eeqp]={}& e[eqp]-[eep]q-[eee]+[eeq]p,\\ \delta_4[eqpe]={}& e[qpe]-[qpe]+[eee]-[eqp]+[eqp]e,\\ \delta_4[qepe]={}& q[epe]-[qep]+[qep]e,\\ \delta_4[peqe]={}& p[eqe]-[peq]+[peq]e,\\ \delta_4[eqee]={}& e[qee]-[qee]+[eqe]e,\\ \delta_4[epee]={}& e[pee]-[pee]+[epe]e,\\ \delta_4[qeqe]={}& q[eqe]-[qeq]+[qeq]e,\\ \delta_4[pepe]={}& p[epe]-[pep]+[pep]e,\\ \delta_4[eeee]={}& e[eee]-[eee]+[eee]e,\\ \delta_4[eqep]={}& e[qep]-[qep]+[eqe]p,\\ \delta_4[epeq]={}& e[peq]-[peq]+[epe]q,\\ \delta_4[epep]={}& e[pep]-[pep]+[epe]p,\\ \delta_4[eqeq]={}& e[qeq]-[qeq]+[eqe]q,\\ \delta_4[qpeq]={}& q[peq]-[eeq]-p[qeq]+[qpe]q,\\ \delta_4[qpep]={}& q[pep]-[eep]-p[qep]+[qpe]p. \end{aligned} \] \end{lemma} \begin{theorem}\label{thm:WeylCohomology} For an arbitrary $W_1$-bimodule $M$, the Hochchild cohomology group $\mathrm H^3(W_1,M)$ is trivial. \end{theorem} \begin{proof} It is enough to find the respective cohomology group of the complex $\Hom_{\Lambda{-}\Lambda} (\mathrm A_\bullet , M)$, where $\Lambda = W_1\oplus \Bbbk 1$, as above. Note that an arbitrary bimodule $M$ over $W_1$ is a direct sum of four components: \[ M = M_{1,1}\oplus M_{0,1}\oplus M_{1,0}\oplus M_{0,0}, \] where the identity element $e\in W_1$ act on $M_{i,j}$ in such a way that $eu= iu$, $ue = ju$, for $u\in M_{i,j}$, $i,j\in \{0,1\}$. Hence, we may consider cohomologies with coefficients on the summands $M_{i,j}$ separately. First, assume $M=M_{1,1}$, i.e., $eu=ue=u$ for all $u\in M$. Suppose $\varphi : \mathrm A_3 \to M$ is a cocycle, i.e., $\Delta^3(\varphi)=\varphi \delta_{4}=0$. Apply $\varphi $ to all relations in Lemma~\ref{lem:DiffV3}: since zero emerges in all right-hand sides, we get the following relations on the values of $\varphi $ on the basis of $\mathrm A_3$ as of a free $\Lambda $-bimodule: \begin{equation}\label{eq:CocycleRelations} \begin{aligned} \varphi[qpe]={}& -q\varphi[pee]+p\varphi[qee]+\varphi[eee], \\ \varphi[qep]={}& q\varphi[eep]+\varphi[qee]p,\\ \varphi[peq] ={}&p\varphi[eeq]+\varphi[pee]q,\\ q\varphi[eee]={}& p\varphi[eee] = \varphi[eee]q = \varphi[eee]p = \varphi[eqe] = \varphi[epe]=0,\\ \varphi[qeq]&=q\varphi[eeq]+\varphi[qee]q,\\ \varphi[pep]&=p\varphi[eep]+\varphi[pee]p,\\ \varphi[eqp]&=\varphi[eep]q+\varphi[eee]-\varphi[eeq]p. \end{aligned} \end{equation} As a corollary, \[ \varphi[eee] =e\varphi[eee]=q(p\varphi[eee])-p(q\varphi[eee])=0. \] Hence, $\varphi $ is completely determined by its values \[ \varphi[eeq],\ \varphi[eep],\ \varphi[qee],\ \varphi[pee]. \] Let us define $\psi\in \Hom_{\Lambda{-}\Lambda }(\mathrm A_2,M)$ in such a way that \[ \psi[eq]=\varphi[eeq], \ \psi[ep]=\varphi[eep], \ \psi[qe]=-\varphi[qee], \ \psi[pe]=-\varphi[pee], \] and $ \psi[ee]=\psi[qp] = 0$. Then $\Delta^2(\psi ) = \psi \delta_3$ is a coboundary, and \[ \begin{aligned} (\psi \delta_3)[eeq]&=e\psi[eq]-\psi[ee]q=\varphi[eeq]+0=\varphi[eeq],\\ (\psi \delta_3)[eep]&=e\psi[ep]-\psi[ee]p=\varphi[eep]+0=\varphi[eep], \\ (\psi \delta_3)[qee]&=q\psi[ee]-\psi[qe]e=0+\varphi[qee]=\varphi[qee],\\ (\psi \delta_3)[pee]&=p\psi[ee]-\psi[pe]e=0+\varphi[pee]=\varphi[pee]. \end{aligned} \] Hence, $\Delta^2(\psi )= \varphi $, i.e., every 3-cocycle is a coboundary, so $\mathrm H^3(W_1,M)=0$ for every bimodule $M$ over~$W_1$. Next, assume $M= M_{1,0}$, i.e., $eu=u$ and $ue=0$ for all $u\in M$. It follows from \ref{eq:CocycleRelations} that \begin{equation}\label{eq:CocycleRelations left} \begin{gathered} q\varphi[pee] -p\varphi[qee]-\varphi[eee]=0, \quad \varphi[qep]= q\varphi[eep],\quad \varphi[peq] = p\varphi[eeq],\\ \varphi[qee]= q[eee],\quad \varphi[pee]= p[eee],\quad \varphi[eqe]= [eeq],\quad \varphi[epe]= [eep],\\ \varphi[qeq]= q\varphi[eeq],\quad \varphi[pep]= p\varphi[eep],\quad \varphi[eqp]=\varphi[eee]. \end{gathered} \end{equation} Therefore, $\varphi $ is completely determined by its values $\varphi[eeq]$, $\varphi[eep]$, $\varphi[eee]$, $\varphi[qpe]$. Let us define $\psi\in \Hom_{\Lambda }(\mathrm A_2,M)$ in such a way that \begin{gather*} \psi[eq]=\varphi[eeq], \ \psi[ep]=\varphi[eep], \ \psi[qe]=\varphi[qee],\\ \ \psi[pe]=\varphi[pee], \ \psi[ee]=\varphi[eee], \ \psi[qp]=\varphi[qpe]. \end{gather*} Then $\Delta^2(\psi ) = \psi \delta_3$ is a coboundary, and \[ \begin{aligned} (\psi \delta_3)[eeq]&=e\psi[eq]=\psi[eq]=\varphi[eeq],\\ (\psi \delta_3)[eep]&=e\psi[ep]=\psi[ep]=\varphi[eep], \\ (\psi \delta_3)[eee]&=e\psi[ee]=\varphi[eee]=\varphi[eee],\\ (\psi \delta_3)[qpe]&=q\psi[pe]-p\psi[qe]-\psi[ee]+\psi[qp]\\ &=q\varphi[pee]-p\varphi[qee]-\varphi[eee]+\varphi[qpe]\\ &=0+\varphi[qpe]=\varphi[qpe]. \end{aligned} \] Hence, $\Delta^2(\psi )= \varphi $, i.e., every 3-cocycle is a coboundary, so $\mathrm H^3(W_1,M)=0$. The cases of right-unital ($M_{0,1}$) and trivial ($M_{0,0}$) modules are completely analogous. \end{proof} Since for every associative algebra $A$ and for every $A$-bimodule $M$ we have $\mathrm H^{n+1}(A,M) = \mathrm H^{n}(A, \Hom(A,M))$, all higher cohomologies (for $n\ge 3$) also vanish. \begin{corollary} For every $n\ge 3$ we have $\mathrm H^n(W_1,M)=0$. \end{corollary} { The Hochschild cohomology is invariant under Morita equivalence of algebras, and it is known that an algebra $A$ is Morita equivalent to the algebra of matrices $M_n(A)$ \cite{Keller}, \cite[Chapter~7]{Lam}, \cite[Chapter~1]{Loday} so $\mathrm H^n(M_k(W_1),M)=\mathrm H^n(W_1,M)=0$. } As a corollary, we obtain the following description of conformal Hochschild cohomologies of the associative conformal algebra $\mathop {\fam 0 Cend} \nolimits_k$. \begin{theorem} Let $M$ be a conformal bimodule over $\mathop {\fam 0 Cend} \nolimits_k$, $k\ge 1$. Then $\mathrm H^n( \mathop {\fam 0 Cend} \nolimits_k,M )=0$ for $n\ge 2$. \end{theorem} \begin{proof} Proposition~\ref{prop:MainTool} immediately implies $\mathrm H^n( \mathop {\fam 0 Cend} \nolimits_k,M )=0$ for $n\ge 3$. For $n=2$, the result was obtained in \cite{Dolg2009}. \end{proof} \subsection*{Acknowledgments} The work was supported by Russian Science Foundation, project 23-21-00504. \input biblio \end{document}

\begin{document} \selectlanguage{english} \begin{center}\begin{large}\textbf{A Principal--Agent Model of Trading Under Market Impact\footnote{The research leading to these results has received funding from the ERC (grant agreement 249415-RMAC), from the Swiss Finance Institute project {\sl Systemic Risk and Dynamic Contract Theory}, as well as the SFB 649 {\sl Economic Risk}, and it is gratefully acknowledged.} \\ -Crossing networks interacting with dealer markets-}\end{large} Jana Bielagk\footnote{Department of Mathematics, Humboldt-University Berlin, Unter den Linden 6, 10099 Berlin, Germany. \\ \hspace{1cm} [email protected]}, Ulrich Horst\footnote{Department of Mathematics, Humboldt-University Berlin, Unter den Linden 6, 10099 Berlin, Germany. \\[email protected]} \& Santiago Moreno--Bromberg\footnote{ Center for Finance and Insurance, Department of Banking and Finance, University of Zurich, Plattenstr. 14, 8032 Zurich, Switzerland. [email protected]} \end{center} \begin{abstract} We use a principal--agent model to analyze the structure of a book--driven \textit{dealer market} when the dealer faces competition from a crossing network or dark pool. The agents are privately informed about their \textit{types} (e.g. their portfolios), which is something that the dealer must take into account when engaging his counterparties. Instead of trading with the dealer, the agents may chose to trade in a \textit{crossing network}. We show that the presence of such a network results in more types being serviced by the dealer and that, under certain conditions and due to reduced adverse selection effects, the book's \textit{spread} shrinks. We allow for the pricing on the dealer market to determine the structure of the crossing network and show that the same conditions that lead to a reduction of the spread imply the existence of an equilibrium book/crossing network pair. \noindent\textit{AMS Classification}: 49K30; 65K10; 91A13; 91B24. \noindent\textit{Keywords}: Asymmetric information; crossing networks; dealer markets; non--linear pricing; principal--agent games. \end{abstract} \section{\large{Introduction}} Recently, the analysis of optimal trading under market impact has received considerable attention. Starting with the contribution of~\cite{AlmgrenChriss00}, the existence of optimal trading strategies under illiquidity has been established by many authors, including ~\cite{Forsyth2012}, \cite{GatheralSchied11}, \cite{KratzSchoeneborn13} and~\cite{SchiedSchoenebornTehranchi10}, just to name a few. The literature on trading under illiquidity typically assumes that block trading takes place under some (exogenous) pricing schedule, which describes the liquidity available for trading at different price levels. This article studies the impact of a CN on a DM within the scope of principal--agent models under hidden information (adverse selection). This asymmetric--information approach is a significant departure from the settings of the articles mentioned above. Specifically, we consider a one--period model where block trading is modeled via a risk--neutral dealer or market--maker who provides liquidity to a heterogeneous (in terms of idiosyncratic characteristics or ``types'') group of privately--informed investors or traders. Extending the seminal work on asset pricing under asymmetric information in~\cite{BMR}, we assume that each investor has an outside option that provides him with a type--dependent reservation utility that the dealer may not be able to match without making a loss. We allow the dealer to abstain from trading with investors whose outside options would be too costly to match. The fact that the dealer may choose between excluding agents, matching their outside options (which in some cases yields him strictly positive profits) or offering them contracts that result in utilities that strictly dominate their reservation ones, implies that a rich structure (in terms of the partition of the type space) may emerge in equilibrium. For instance, in Example~\ref{RichStructure} we analyze a scenario where the type space is partitioned into two intervals where the agents' outside options are matched, one where they are excluded and three where they earn positive rents. In more mundane terms, within a portfolio--liquidation framework, we may think of traders who need to unwind portfolios whose sizes are private information and who can either trade in a DM or a CN, the latter providing some of them with trading options that the dealer may be unable improve upon without suffering losses. To the best of our knowledge, such adverse--selection models have thus far only been considered by~\cite{BJ:03} and~\cite{Page}. The latter analyzes, in quite a general setting where the set of consumer types is a Polish space and the contract space an arbitrary compact metric space, the problem of a monopolist who faces both an adverse-selection problem (as in the work at hand) as well as a moral-hazard one relative to contract performance. \cite{BJ:03}, on the other hand, only studies the adverse-selection problem in a finite-dimensional setting. This allows him to find a quasi-explicit representation of the optimal contract using Lagrange-multiplier techniques. He identifies conditions for the optimal contract to be separating, to be non--stochastic and to induce full participation. Furthermore, he also discusses the nature of the solution when bunching occurs. He does not, however, analyze the case where the dealer's choices may have an impact on the structure of the reservation--utility function, which in turn would influence his decisions. Our study of such a feedback loop is novel and it is a crucial component in our analysis of the interactions between DMs and CNs, which is typically not unidirectional. To account for the fact that many off--exchange venues settle trades at prices taken from primary venues, we state sufficient conditions for the existence of an equilibrium pricing schedule. By this we mean that there exists a pricing schedule in the DM such that, if trades in the CN are settled at the best bid and ask prices from the DM, then the dealer's optimal pricing schedule is precisely that schedule. In order to study the impact of a type--dependent outside option, we first analyze the benchmark case where the said option is trivial, i.e. all traders may abstain from engaging the dealer and in turn earn (or lose) nothing. In such a setting the dealer is able to match the traders' outside options by offering ``nothing in exchange for nothing'', which is costless. This analysis follows~\cite{BMR}. Next we look at the general case where the traders' reservation utilities are type dependent and the dealer need not be able to match them without incurring losses. It is well known that asymmetric information results, in equilibrium, in some traders being kept to their reservation utilities. This is due to the adverse--selection costs. Intuitively, these costs increase with the profitability of trading with high--type traders (e.g. investors with large portfolios). This suggests that when mostly high--type traders benefit from the outside option in terms of the latter strictly dominating what the dealer would have offered them in the benchmark case, then more low--type traders will be serviced in equilibrium. As a consequence of the reduced adverse--selection costs, more investors engage in trading, either in the DM or the CN. Our analysis further suggests that the presence of the CN is welfare improving even for investors for whom trading in the CN is not beneficial. We also provide sufficient conditions that guarantee that the competition from the CN results in a narrower spread in the DM. Overall, we propose a benchmark model of optimal block trading of privately--informed traders with an endogenous pricing schedule, analyze the impact of a CN on pricing schedules in DMs and prove an existence result of equilibria of best bid and ask prices in our trading game. \subsection*{Related literature} \cite{HorstNaujokat13} and~\cite{KratzSchoeneborn13} were the first to allow orders to be simultaneously submitted both to a dealer market (DM) and to an off--exchange venue such as a crossing network (CN) or a dark pool (DP). These are alternative trading facilities that allow investors to reduce their market impact by submitting liquidity that is shielded from the public view. The downside is that trade execution is uncertain: trades take place only when the matching liquidity is or becomes available. In such a case, trades are typically settled at prices prevailing in an associated primary venue, which significantly reduces the cost of large trades if settled in a CN or in a DP. The aforementioned articles on optimal, simultaneous trading in DMs and CNs do not allow for an impact of off--exchange trading on the dynamics of the associated DM. Equilibrium models analyzing the impact of alternative trading venues on DMs and trading behavior have been extensively analyzed in the financial--economics literature; see, e.g.~\cite{Glosten} and~\cite{PS} and the references therein. To simplify the analysis of market impact, this literature typically assumes that the market participants trade only a single unit of the stock. For instance, in their seminal work, \cite{HM} derives conditions for the viability of the alternative trading institutions in a modeling framework where a random number of informed and liquidity traders, each buying or selling a single unit, chooses between a DM and a CN. In their model, dealers receive multiple single--unit orders and cannot distinguish between the informed and the liquidity orders. Hence, their bid--ask spread corresponds to each order's market impact. \cite{DDH} consider the allocation of order flow between a CN and a DM when trading in both markets takes place at exogenously given prices. They show that small differences in the traders' preferences generate a unique equilibrium, in which patient traders use the CN whereas impatient traders submit orders directly to the DM. Due to the fact that prices are exogenous, the equilibrium market share of the CN is fully determined by the price differential between the markets, together with the distribution of the traders' liquidity preferences. In contrast with the two preceding works, where interactions between DMs and CNs are studied, \cite{Buti} take an alternative approach and analyze a dynamic model with single--unit traders who may place market or limit orders in a limit--order book (LOB). Alternatively, should they have access to it, the agents may place an immediate--or--cancel order in a dark pool (DP). Agents differ in their valuation of the asset and their access to the DP. The authors find that, whenever the LOB is illiquid, the presence of a DP leads to widening spreads and to a decline in the book's depth; thus, to a deterioration of market quality and welfare. This, in spite of the fact that, on average, trade volume increases. These negative effects are generally decreasing in the depth of the LOB. The take--home message offered is that, when studying interacting LOBs and DPs, there is a trade--off between trade and volume creation on the one hand, and book depth and spread on the other one. In terms of the aforementioned effects of the presence of the CN, whereas increases in the number of participating agents and welfare are generic, the narrowing of the spread does not seem to be so. For instance in \cite{Buti}, the presence of a DP results in a migration of liquidity and hence an increasing spread --- an effect that cannot appear in our setting where all traders are liquidity takers. Contrastingly, \cite{Buti_Data}, provide empirical evidence that high DP activity is associated with narrower spreads, but no causality is concluded. In~\cite{Zhu2014}, asymmetric information divides agents into informed and (uninformed) liquidity traders. When a CN complements an existing DM, the spread widens because the liquidity traders move to the CN, whereas the informed ones, who tend to be on one side of the market, prefer the DM. In our setting, agent heterogeneity corresponds to different endowments or preferences, but there is no distinction at the level of access to information. Hence, the spread originates due to the adverse--selection problem faced by the dealer. The remainder of this article is structured as follows. Our model and main results are presented in Section \ref{sec:Model}. Existence of a solution to the dealer's optimization problem is established in Section \ref{sec:ExistenceSol}. Section \ref{sec:ImpactSpread} studies the impact of a CN on the spread. Section \ref{sec:ExistenceEqui} establishes our result regarding the existence of equilibrium price schedules. A specific application to a portfolio--liquidation problem with dark--pool trading is analyzed in Section \ref{sec:DPtrading} and Section~\ref{sec:Conclusions} concludes. \section{\large{Model and main results}}\label{sec:Model} \noindent We consider a quote--driven market for an asset, in which a risk--neutral \textit{dealer} engages a group of privately--informed \textit{traders}\footnote{Our dealer is called the {\sc principal} in the contract--theory jargon and the traders are usually referred to as the {\sc agents}.}. The dealer market (DM for short) is described by a pricing schedule $T:\mathbb{R}\to\mathbb{R}.$ In other words, $q$ units of the asset are offered to be traded, on a take--it--or--leave--it basis, for the amount $T(q)$. For $q\in\mathbb{R},$ we refer to the pair $\big(q, T(q)\big)$ as a \textit{contract}. We assume that $T(0)=0$ and that $T$ is absolutely continuous. Thus, we may write \begin{equation*} T(q) = \int_0^q t(s)ds,\quad q\geq 0, \end{equation*} and analogously for negative values of $q.$ Here $t(s)$ is the marginal price at which the $s$--th unit is traded. As we shall see below, pricing schedules are, in general, not differentiable at zero. Hence, for a particular schedule $T$ the \textit{spread} is \begin{equation*} \mathcal{S}(T) := |T'(0_+) - T'(0_-)|=|t(0_+) - t(0_-)|, \end{equation*} where $t(0_-)$ and $t(0_+)$ are the \textit{best--bid} and \textit{best--ask} prices, respectively. We denote by $C:\mathbb{R}\to\mathbb{R}$ the dealer's inventory or risk costs associated with a position $q$, e.g. the impact costs of unwinding a portfolio of size $q$ in a limit order book. We assume that the mapping $q\mapsto C(q)$ is strictly convex, coercive and that it satisfies $C(0) = 0.$ The traders' idiosyncratic characteristics are represented by the index $\theta$ that runs over a closed interval $\Theta:=[\underline{\theta}, \overline{\theta}],$ called the set of \textit{types}. We assume that zero belongs to the interior of $\Theta.$ Saying that a trader's type is $\theta$ means that if he trades $q$ shares for $T(q)$ dollars his utility is $u(\theta, q) - T(q),$ where \begin{equation*} u(\theta, q):=\theta \psi_1(q) + \psi_2(q) \end{equation*} and $\psi_1,\psi_2:\mathbb{R}\to\mathbb{R}$ are smooth functions that satisfy $\psi_1(0)=\psi_2(0)=0,$ $\psi_1$ is strictly increasing and $C(q)-\Psi_2(q)\geq 0$ holds for all $q\in\mathbb{R}$. Thus far, with our choice of preferences the traders enjoy a type--independent \textit{reservation utility} of zero, should they decide to abstain from trading in the DM. Such an action is commonly referred to agents choosing their \textit{outside option}. As $C(0) = 0$, providing $\big(0, T(0)\big)$ is costless to the dealer and, since $\big(0, T(0)\big)$ yields all agents their reservation utility, in the absence of any other trading opportunity, we may equate the contract $(0,0)$ to the traders' outside option. Besides participating in the DM, each trader has the possibility to submit an order to a \textit{crossing network} (CN). The latter is an alternative trading venue where trades take place at fixed bid/ask prices $\pi:=\big(\pi_-, \pi_+\big)$, but where execution might not be guaranteed.\footnote{In other words, the crossing network presents agents with possibly better prices at the cost of an uncertain execution. CN trading often benefits agents who intend to unwind large positions, which might result in a price impact.} The possibility of trading in the crossing network modifies the traders' outside option to the extent that now they may choose between abstaining from all trading and earning zero or participating in the CN if the corresponding expected utility is non--negative. For a specific $\pi,$ the quantity $u_0(\theta; \pi)\geq 0$ represents the expected utility of the $\theta$--type investor who decides to take his (now extended) outside option. In the sequel we indulge in a slight abuse of the language and also refer to $u_0(\cdot; \pi)$ as the agents' outside option(s). Following \cite{DDH,HM} we focus on the case where a trader chooses exclusively between his outside option and trading in the DM, i.e. we do not allow for simultaneous participation in the DM and the CN. Initially we take $\pi$ as given, but later we analyze the case where it is endogenously determined through the interaction between the DM and the CN via the feedback of the spread in the former into the pricing in the latter. We work under the following assumption:\footnote{Once an assumption has been made, we consider it to be standing for the remainder of the paper.} \begin{Assumption}\label{ass:cost of access} There is a fixed cost $\kappa>0$ of accessing the outside option such that, for all $\pi\in\mathbb{R}^2,$ the function $u_0(\cdot; \pi)$ can be written as $ u_0(\cdot; \pi) =\max\big\{\widetilde{u}_0(\cdot; \pi) - \kappa, 0\big\}, $ where $\widetilde{u}_0(0; \pi) = 0.$ \end{Assumption} Trading over the DM is anonymous; the dealer is unable to determine a trader's type before he engages the latter. The only ex--ante information the dealer has is the distribution of the individual types over $\Theta,$ which is described by a density $f:\Theta\to\mathbb{R}_+.$ In the sequel we specify the traders' and the dealer's optimization problems and analyze the impact of the CN on the DM, especially on its spread. \subsection{\large{The traders' problem}}\label{ssec:Agents} Until further notice we consider $\pi$ to be fixed. The problem of a trader of type $\theta$ is to determine, for a given pricing schedule $T,$ \begin{equation*} q_m(\theta) := \text{argmax}\Big\{u(\theta, q) - T(q)\Big\} \end{equation*} and then choose, for $q_m\in q_m(\theta),$ between his \textit{indirect--utility} $v(\theta):=u\big(\theta, q_m\big) - T\big(q_m\big)$ from trading in the DM and his outside option $u_0(\theta; \pi).$ As the supremum of affine functions, the indirect utility function is convex. The choice of a pricing schedule $T$ induces a partition of the type space. We say that a trader of type $\theta$ \textit{participates} in the DM if $ v(\theta)\geq u_0(\theta; \pi), $ assuming that ties are broken in the dealer's favor. Conversely, we say that a trader of type $\theta$ \textit{is excluded} from trading in the DM if $ v(\theta) < u_0(\theta; \pi). $ For a given schedule $T,$ we denote the set of excluded types by $\Theta_e(T;\pi).$ Observe that, in the absence of a CN, there is no loss of generality in assuming that all traders participate. We say that a trader of type $\theta$ is \textit{fully serviced} if he earns strictly positive profits from interacting with the dealer. \subsection{\large{The dealer's problem}}\label{ssec:BenchmarkProblemMM} The \textit{Revelation Principle} (see, e.g. Meyerson~\cite{Meyerson:91}) says that, when studying Nash--equilibrium outcomes in adverse-selection games such as ours, there is no loss of generality in focusing on direct--revelation mechanisms, i.e. those mechanisms where the set of types indexes the contracts. Furthermore, from the \textit{Taxation Principle} (see, e.g. Rochet~\cite{R:85}) there is also no loss of generality in writing $\tau(\theta)$ instead of $T(q(\theta)),$ where $\tau:\Theta\to\mathbb{R}$ is an absolutely continuous function. From this point on we shall, therefore, study our principal--agent game through books of the form $\big\{\big(q(\theta), \tau(\theta)\big),\theta\in\Theta\big\}$ and drop $T$ from the specification of the indirect--utility functions. We also write $\Theta_e(q,\tau;\pi)$ instead of $\Theta_e(T;\pi)$ for the set of excluded types. At the onset, a trader of type $\theta$ could misrepresent his type by choosing a contract $\big(q(\widetilde{\theta}), \tau(\widetilde{\theta})\big),$ with $\widetilde{\theta}\neq \theta.$ The dealer strives to avoid this situation, since he wants to exploit the information contained in the density of types. This requires that he offers \textit{incentive--compatible} books, i.e. those that satisfy \begin{equation*} \max_{\widetilde{\theta}\in\Theta}\big\{u\big(\theta, q(\widetilde{\theta})\big) - \tau(\widetilde{\theta})\big\} = u\big(\theta, q(\theta)\big) - \tau(\theta). \end{equation*} In the presence of an incentive--compatible book, the contract that yields a trader of type $\theta$ his indirect utility is precisely the one the dealer has designed for him. Since the dealer is risk neutral, his goal is to maximize his expected income from engaging the traders. Taking into account the impact of the CN on the traders' optimal actions, his problem is to devise $(q^*, \tau^*)$ so as to solve the problem \begin{equation*} \begin{array}{cc} \mathcal{P}(\pi) := & \left\{ \begin{array}{l} \sup_{(q, \tau)} \int_{\Theta_e^c(q, \tau;\pi)}\Big(\tau(\theta) - C\big(q(\theta)\big)\Big)f(\theta)d\theta\\ \text{s.t. }\\ (q(\theta), \tau(\theta))\in\text{argmax}_{\widetilde{\theta}\in\Theta}\big\{u\big(\theta, q(\widetilde{\theta})\big)-\tau(\widetilde{\theta})\big\},\\ \tau \text{ is absolutely continuous}. \end{array} \right. \end{array} \end{equation*} Due to the \textit{Envelope Theorem}, if a contract $\big\{(q(\theta), \tau(\theta)\big),\theta\in\Theta\big\}$ is incentive compatible, then $\psi_1(q(\theta))$ belongs to the subdifferential $\partial v(\theta)$. Since for almost all $\theta\in\Theta$ it holds that $\partial v(\theta)=v'(\theta)$ and $\psi_1$ is strictly increasing, we have for almost all $\theta \in \Theta$ that \begin{equation}\label{eq:QualGrad} q(\theta) = \psi_1^{-1}\big(v'(\theta)\big). \end{equation} Therefore, starting from a convex indirect--utility function we can recover, for almost all types, the quantities in the incentive--compatible book that generated it. Furthermore, the indirect utility function may be written as \begin{equation}\label{eq:IndUt} \begin{split} v(\theta) & = \theta \psi_1\big(\psi_1^{-1}\big(v'(\theta)\big)\big) + \psi_2\big(\psi_1^{-1}\big(v'(\theta)\big)\big) -\tau(\theta) \\ & = \theta\,v'(\theta) + \psi\big(v'(\theta)\big) - \tau(\theta), \end{split} \end{equation} where $\psi:=\psi_2\circ \psi_1^{-1}.$ It follows from Eqs.~\eqref{eq:QualGrad} and~\eqref{eq:IndUt} that the traders' indirect utility function contains all the information about the quantities and the pricing schedule, which allows us to write $\Theta_e^c(v;\pi)$ instead of $\Theta_e^c(q, \tau;\pi).$ In particular, introducing the functions \begin{equation*} \widetilde{K}(q):=C\big(\psi_1^{-1}(q)\big) - \psi_2\big(\psi_1^{-1}(q)\big) \quad\text{and}\quad i(\theta, v, q):=\theta\cdot q - v - \widetilde{K}(q) \end{equation*} and denoting by $\mathcal{C}$ the cone of all real--valued convex functions over $\Theta$, we can restate the dealer's problem as \begin{equation*} \mathcal{P}(\pi) =\sup_{v\in\mathcal{C}} \int_{\Theta_e^c(v;\pi)} i\big(\theta,v(\theta),v'(\theta)\big)f(\theta) d\theta. \end{equation*} We prove in Theorem~\ref{thm:Main1} below that, under suitable assumptions, Problem $\mathcal{P}(\pi)$ admits a solution. The latter is, in fact, quasi--unique in the sense that on the set of participating types the solution is indeed unique. However, agents are excluded by offering their types any incentive--compatible, indirect--utility function that lies below $u_0.$ In other words, there is no uniqueness on the set of excluded types. From the agents' point of view there is no ambiguity: they either trade with the specialist or they take their outside option. The non--uniqueness is also a non--issue for the specialist, since it it only appears in subdomains of the type space that he does not access. With this in mind, in the sequel we denote by $v(\cdot;\pi)$ ``the'' solution to Problem $\mathcal{P}(\pi)$. \begin{Assumption}\label{ass:qc} The functions $\psi_1, \psi_2$ and $C$ are such that $\widetilde{K}$ is strictly convex, coercive, continuously differentiable and it satisfies $\widetilde{K}'(0)=0.$ \end{Assumption} Determining the set of types who do participate but who earn zero profits is essential to our analysis, since it is precisely at the \textit{boundary types} where $t(0_-)$ and $t(0_+)$ are determined. We prove in Lemma~\ref{lemma:tradingSB} that, by virtue of Assumption~\ref{ass:cost of access}, these limits are always well defined. For any $v\in\mathcal{C}$, we shall refer to \begin{equation*} \Theta_0(v):=\big\{\theta\in\Theta\,|\,v(\theta)=0\big\} \end{equation*} as the set of \textit{reserved traders}. Whenever we refer to the reserved set corresponding to the solution $v(\cdot;\pi)$ to $\mathcal{P}(\pi)$ we write $\Theta_0(\pi).$ We prove in Proposition~\ref{lm:ZeroatZero} that there is no loss of generality in assuming that any feasible $v\in\mathcal{C}$ satisfies $v(0)=0;$ thus, $\Theta_0(v)\neq\emptyset.$ \begin{remark}\label{rmk:wellpossed} A well defined spread requires $\Theta_0(\pi)$ to be a proper interval $[\underline{\theta}_0(\pi), \overline{\theta}_0(\pi)],$ which will follow from Assumption~\ref{ass:cost of access}, and that there exists $\epsilon>0$ such that $(\underline{\theta}_0(\pi)-\epsilon, \underline{\theta}_0(\pi))$ and $(\overline{\theta}_0(\pi), \overline{\theta}_0(\pi)+\epsilon)$ belong to the set of fully--serviced traders. The existence of such an $\epsilon$ is proved in Lemma~\ref{lemma:tradingSB}. Economically, this conditions means that the CN is not beneficial for low--type traders. We shall encounter several instances where the proofs of our results concern conditions on points to the left of $\underline{\theta}_0(\pi)$ or to the right of $\overline{\theta}_0(\pi)$ that are analogous. So as to streamline the said proofs, whenever we find ourselves in one of these ``either--or'' situations, we deal only with the positive case. \end{remark} \noindent We are now ready to state the first main result of this paper, whose proof is given in Section~\ref{sec:ExistenceSol} below. \begin{theorem} \label{thm:Main1} Problem $\mathcal{P}(\pi)$ admits a solution, which is unique on the set of participating types. \end{theorem} Our second main result concerns the effect of the CN on the spread and the set of participating traders if, disregarding negative expected unwinding costs, the dealer can match the CN. \begin{Assumption}\label{ass:matching} There exists an incentive compatible book $\big\{(q_c(\theta), \tau_c(\theta) \big), \theta\in\Theta\big\}$ such that for almost all $\theta\in\Theta$ it holds that $ u\big(\theta, q_c(\theta) \big) - \tau_c(\theta)= u_0(\theta; \pi). $ \end{Assumption} Assumption~\ref{ass:matching} implies that $u_0(\cdot; \pi)$ is also a convex function. The case where $u_0(\cdot; \pi)$ is concave is somewhat simpler, since it boils down to exclusion without matching. \noindent The following theorem analyzes the impact of the CN on the DM and the traders' welfare. \begin{theorem} \label{thm:Main2} For a given price $\pi=(\pi_-, \pi_+)$ let $\mathcal{S}_m$ and $\mathcal{S}_o$ be the spreads with and without the presence of the crossing network and $v_o$ and $v(\cdot;\pi)$ the corresponding indirect--utility functions, respectively. In the presence of the crossing network \begin{enumerate} \item less types are reserved, i.e. $\Theta_0(v_o)\supseteq \Theta_0(\pi).$ Furthermore, the inclusion is strict if there exists $\theta\in\Theta$ such that $u_0(\theta;\pi)>v_o(\theta);$ \item if the types are uniformly distributed ($f\equiv(\overline{\theta}-\underline{\theta})^{-1}$) the spread narrows, i.e. $\mathcal{S}_o \geq \mathcal{S}_m;$ \item the typewise welfare increases, i.e. $v_o(\theta)\leq v(\theta;\pi)$ for all $\theta\in\Theta.$ \end{enumerate} \end{theorem} In the sequel we use the subindexes $``m"$ and $``o"$ to distinguish structures or quantities with and without a CN, respectively. \subsection{\large{Equilibrium}}\label{ssec:Equilibrium} It is natural to assume that pricing in the DM has an impact on the pricing schedule $\pi.$ For example, the CN could be a \textit{dark pool}, where trading takes place at the best--bid and best--ask prices of the primary market. We analyze such an example, within a portfolio--liquidation framework, in Section~\ref{sec:DPtrading}. The pecuniary interaction between the DM and the CN, however, is not unidirectional if the dealer anticipates the effect that his choice of book structure has on the CN. Our main focus is the impact of the CN on the spread in the DM. Specifically, if we denote by $t(0;\pi):=\big(t(0_-;\pi), t(0_+;\pi)\big)$ the best bid--ask prices in the DM for a given CN price schedule $\pi,$ then we call $\pi^*$ an \textit{equilibrium price} if $\pi^* = t(0;\pi^*). $ We make the following natural assumption on the impact of $\pi$ on the traders' outside option. \begin{Assumption}\label{ass:monotone} Let $\pi_1\le \pi_2,$ where ``$\leq$'' is the lexicographic order in $\mathbb{R}^2,$ then for all $\theta\in\Theta$ it holds that $ u_0(\theta;\pi_1)\geq u_0(\theta;\pi_2). $ Furthermore, we assume that there exists $\big(\underline{\pi}_-, \overline{\pi}_+\big)\in\mathbb{R}^2$ such that $u_0(\cdot;\pi)\leq 0$ for all $(\pi_-,\pi_+)$ such that ${\pi}_-\leq\underline{\pi}_-$ and $\overline{\pi}_+\leq{\pi}_+.$ \end{Assumption} \noindent The following is our main result on the existence of an equilibrium price. \begin{theorem} \label{thm:Main3} If types are uniformly distributed, then the mapping $\pi\mapsto t(0; \pi)$ has a fixed point. \end{theorem} Summarizing, we have that the dealer can correctly anticipate the movements in prices in the CN when he designs the optimal pricing schedule for the DM. Furthermore, the presence of the CN is beneficial in terms of liquidity, market participation and the traders' welfare. \begin{remark} The uniformity of the distribution of types in Theorems~\ref{thm:Main2} and~\ref{thm:Main3} can be relaxed, which is something we postpone to the corresponding proofs, where the required notation is introduced. \end{remark} \section{\large{Existence of a solution to Problem $\mathcal{P}(\pi)$}}\label{sec:ExistenceSol} In this section we prove the existence of a solution to the dealer's problem in the presence of a CN. Even though, strictly speaking, this result is a particular case of Theorem 4.4 in~\cite{Page}, for the reader's convenience we present a proof in our simpler setting. Some of the arguments are somewhat standard, but we include them for completeness. The first important result that we require is that the dealer's optimal choices will lead to him never losing money on types that participate. \begin{proposition}\label{prop:PosProf} If $(q^*, \tau^*):\Theta\to\mathbb{R}^2$ is an optimal allocation, then for all participating types it holds that $\tau^*(\theta) - C\big(q^*(\theta)\big)\geq 0.$ \end{proposition} \noindent\begin{Proof} Assume the contrary, i.e. that the set \begin{equation*} \widetilde{\Theta}:=\big\{\theta\,|\, v(\theta;\pi)\ge u_0(\theta;\pi), \tau^*(\theta)<C\big(q^*(\theta)\big)\big\}, \end{equation*} where $v(\theta;\pi)=u\big(\theta, q^*(\theta)\big) - \tau^*(\theta)$ has positive measure. Define a new pricing schedule via \begin{equation*} \widetilde{\tau}(\theta):=\max\big\{\tau^*(\theta), C\big(q^*(\theta)\big)\big\}. \end{equation*} The incentives for types in $\widetilde{\Theta}^c$ do not change, since their prices remain unchanged, whereas prices for others have increased. Profits corresponding to trading with types in $\widetilde{\Theta}$ increase to zero. As a consequence the dealer's welfare strictly increases, which violates the optimality of $(q^*, \tau^*).$ \end{Proof} A consequence of Proposition~\ref{prop:PosProf} is that, together with Assumption~\ref{ass:qc}, it allows us to restrict the feasible set of the dealer's problem to a compact one. We prove this in several steps, \begin{lemma}\label{lm:ZeroatZero} If $v:\Theta\to\mathbb{R}$ is a non--negative, convex function that solves $\mathcal{P},$ then $v(0)=0.$ \end{lemma} \noindent\begin{Proof} Assume that $v\in\mathcal{C}$ solves $\mathcal{P}$ and $v(0)>0.$ This implies that $\psi_2\big(q(0)\big) - \tau(0) \geq 0.$ Since, from Assumption~\ref{ass:cost of access}, a trader of type $\theta=0$ has no access to a profitable outside option, then he participates. From Proposition~\ref{prop:PosProf} it must then hold that $\tau(0)\geq C\big(q(0)\big)$ which in turn implies that $\psi_2\big(q(0)\big)\geq C\big(q(0)\big).$ This relation, however, can only hold for $q(0)=0,$ which implies that $\tau(0)=v(0)=0.$ \end{Proof} \begin{lemma}\label{lm:BoundedQ} There exists $\overline{q}\geq 0$ such that if $v$ is feasible, then $|\partial v|\leq\overline{q}.$ \end{lemma} \noindent\begin{Proof} From Assumption~\ref{ass:qc} and the compactness of $\Theta$ we have that the mapping $ q\mapsto i(\theta, v, q) $ tends to $-\infty$ as $|q|\to\infty$ uniformly on $\Theta$ for $v\geq 0.$ From Proposition~\ref{prop:PosProf} $i\big(\theta, v(\theta), v'(\theta)\big)$ must be non--negative for all participating types, which concludes the proof. \end{Proof} From Lemmas~\ref{lm:ZeroatZero} and~\ref{lm:BoundedQ} we have that the quantity $\max_{\theta\in\Theta}\big\{u_0(\theta;\pi)\big\} + \overline{q}\|\Theta\|$ is an upper bound for any feasible choice of $v,$ which yields the following \begin{corollary}\label{cor:UnifBound} The feasible set $\mathcal{A}\subset\mathcal{C}$ of Problem $\mathcal{P}$ is uniformly bounded and uniformly equicontinuous. \end{corollary} \noindent\begin{Proof} A uniform bound is $\max_{\theta\in\Theta}\big\{u_0(\theta;\pi)\big\} + \overline{q}\|\Theta\|.$ Lemma~\ref{lm:BoundedQ} guarantees that for any $v\in\mathcal{A}$ it holds that $|\partial v|\leq\overline{q}.$ In other words, $\mathcal{A}$ is composed of convex functions whose subdifferentials are uniformly bounded, hence $\mathcal{A}$ is uniformly equicontinuous. \end{Proof} Notice that, when it comes to determining quantities and prices for trader types who do participate, Proposition~\ref{prop:PosProf} results in the dealer having to solve the problem \begin{equation*} \begin{array}{cc} \widetilde{\mathcal{P}}(\pi):= & \left\{ \begin{array}{ll} \sup_{v\in\mathcal{A}} \int_{\Theta}\big(i\big(\theta, v(\theta), v'(\theta)\big)\big)_+ f(\theta)d\theta\\ \text{s.t. } v(\theta)\geq u_0(\theta;\pi)\text{ for all } \theta\in\Theta. \end{array} \right. \end{array} \end{equation*} The last auxiliary result that we need is the following proposition, whose proof is a direct consequence of Fatou's Lemma, together with Lemmas~\ref{lm:ZeroatZero} and~\ref{lm:BoundedQ}. \begin{proposition}\label{prop:USC} The mapping $v\mapsto\int_{\Theta}\big(i(\theta, v(\theta), v'(\theta))\big)_+ f(\theta)d\theta$ is upper semi--continuous in $\mathcal{A}$ with respect to uniform convergence. \end{proposition} \noindent We are now ready to prove our first main result: \noindent\textbf{Proof of Theorem \ref{thm:Main1}:} Assume that $\mathcal{A}\bigcap\big\{v\in\mathcal{C} | v(\cdot)\geq u_0(\cdot ; \pi)\big\}$ is non--empty and consider a maximizing sequence $\big\{\widetilde{v}_n\big\}_{n\in\mathbb{N}}$ of Problem $\widetilde{\mathcal{P}}(\pi).$ From Corollary~\ref{cor:UnifBound} we have that, passing to a subsequence if necessary, there exists $\widetilde{v}\in\mathcal{A}$ such that $\widetilde{v}_n\to\widetilde{v}$ uniformly. A direct application of Proposition~\ref{prop:USC} yields that $\widetilde{v}$ is a solution to $\widetilde{\mathcal{P}}(\pi).$ To finalize the proof we must construct from $\widetilde{v}$ a solution to Problem $\mathcal{P}(\pi).$ To this end, let us define the sets \begin{equation*} \Theta_-:=\big\{\theta\in\Theta | i\big(\theta, \widetilde{v}(\theta), \widetilde{v}'(\theta)\big)<0\big\}\quad\text{and}\quad \Theta_+:=\Theta_-^c. \end{equation*} It is well known that if a sequence of convex functions converges uniformly (to a convex function), then there is also uniform convergence of the derivatives wherever they exist, which is almost everywhere. This fact, together with the continuity of the mappings $\theta\mapsto\widetilde{v}(\theta)$ and $(\theta,v, q)\mapsto i(\theta,v, q),$ implies that $\Theta_-$ is the union of a disjoint set of open intervals: \begin{equation*} \Theta_-=\bigcup_{i=1}^{\infty}(a_i, b_i). \end{equation*} Define, for each $i\geq 1,$ \begin{equation*} \widetilde{v}_{a,i}:=\inf\big\{q | q\in\partial \widetilde{v}(a_i)\big\}\quad\text{and}\quad \widetilde{v}_{b,i}:=\sup\big\{q | q\in\partial \widetilde{v}(b_i)\big\} \end{equation*} and consider the support lines to $\text{graph}\{\widetilde{v}\}$ at $a_i$ and $b_i$ given by \begin{equation*} l_i(\theta)=\widetilde{v}(a_i)+\widetilde{v}_{a,i}(\theta-a_i)\quad\text{and}\quad L_i(\theta)=\widetilde{v}(b_i)+\widetilde{v}_{b,i}(\theta-b_i), \end{equation*} respectively. Let $c_i\in(a_i, b_i)$ be, for each $i\geq 1,$ the unique solution to the equation $l_i(\theta)=L_i(\theta)$ and define on $(a_i, b_i)=:\Theta_i$ \begin{equation*} \begin{array}{cc} v_i^*(\theta):= & \left\{ \begin{array}{ll} l_i(\theta) & \theta\leq c_i;\\ L_i(\theta) & \theta> c_i. \end{array} \right. \end{array} \end{equation*} Finally define \begin{equation*} \begin{array}{cc} v^*(\theta):= & \left\{ \begin{array}{ll} \widetilde{v}(\theta) & \theta\in\Theta_+;\\ v_i^*(\theta) & \theta\in\Theta_i, i\in\mathbb{N}, \end{array} \right. \end{array} \end{equation*} then $v^*$ is a solution to Problem $\mathcal{P}(\pi)$ and $\Theta_e(v^*)=\Theta_-,$ which concludes the proof. \begin{remark}\label{rem:QuasiUniqueness} If the specialist can profitably match all agents' outside option, then the quasi--uniqueness of a solution to Problem $\mathcal{P}(\pi)$ is in fact uniqueness and it follows directly from Assumption~\ref{ass:qc}. Indeed, in such a case \begin{equation*} \big(i(\theta, v(\theta), v'(\theta))\big)_+=\big(i(\theta, v(\theta), v'(\theta))\big) \end{equation*} nd problem $\widetilde{\mathcal{P}}(\pi)$ is one of maximizing a strictly concave, coercive functional over a convex set that is closed with respect to uniform convergence. In the general case, we construct the quasi--unique solution in Section~\ref{sec:ModelCN}. Assumption~\ref{ass:qc} remains crucial, since it guarantees that the maximization problems through which we define the optimal quantities have unique maximizers. \end{remark} \section{\large{The impact of a crossing network}}\label{sec:ImpactSpread} {In this section we look at the impact that a CN has on the spread, on participation and on the traders' welfare. In order to do so, we provide a characterization of the solution to Problem $\mathcal{P}(\pi).$ It should be noted that, given the restriction of candidate solutions to $\mathcal{C},$ we cannot simply make use of the Euler--Lagrange equations to solve the variational problem, since the said equations are only satisfied when the constraints do not bind.} \subsection{\large{A benchmark without a CN}}\label{sec:ModelBench} We first analyze the benchmark case where the traders do not have access to a CN. The corresponding dealer's problem is denoted by $\mathcal{P}_o.$ Recall that all trader types have a zero reservation utility, which the dealer is able to match costlessly by offering the contract $(0, 0).$ The point of making this normalization is to simplify the constraints in the dealer's optimization problem. This will not be possible in the presence of a CN since, even if the dealer were able to match the utility that investors enjoy if they trade in the CN, this would be in general not costless. We take a Lagrange--multiplier approach to provide a characterization of the solution to Problem $\mathcal{P}_o.$ To this end, let us introduce the following definition: \begin{equation*} I[v] := \int_{\Theta}i\big(\theta, v(\theta), v'(\theta)\big)f(\theta)d\theta. \end{equation*} Let $BV_+(\Theta)$ be the space of non--negative functions of bounded variation $\gamma:\Theta\to\mathbb{R}_+,$ which we place in duality with $C(\Theta, \mathbb{R}),$ the space of real--valued, continuous functions on $\Theta,$ via the standard pairing \begin{equation*} \langle v, \gamma\rangle :=\int_{\Theta} v(\theta)d\gamma(\theta) \end{equation*} for $v\in C(\Theta, \mathbb{R}),$ where $d\gamma$ is the distributional derivative of $\gamma.$ Furthermore, it follows from Pontryagin's Maximum Principle and the fact that $f$ is a probability density function that there is no loss of generality in assuming that $\gamma$ is absolutely continuous and that $\gamma(\overline{\theta}) = 1.$ The Lagrangian for the dealer's problem is \begin{equation*}\label{eq:Lagrangian} \mathcal{L}(v, \gamma) := I[v]+\langle v, \gamma\rangle,\quad v\in\mathcal{C}, \end{equation*} with corresponding Karush--Kuhn--Tucker conditions \begin{equation}\label{eq:KT} \langle v, \gamma\rangle = 0\quad\text{and}\quad d\gamma(\theta)=0\Rightarrow v(\theta)>0. \end{equation} The next result is the formalization of the \textit{vox populi} saying that ``quality does not jump''. Regularity properties of the solutions to variational problems subject to convexity constraints were studied by Carlier and Lachand--Robert in~\cite{CarLach}, and their methodology can be directly adapted to prove the following result. \begin{proposition}\label{prop:NoJumps1} If $v\in\mathcal{C}$ is a stationary point of $\mathcal{L}(v, \gamma),$ then $v\in C^1(\Theta).$ \end{proposition} The fact that, at the optimum, the mapping $\theta\mapsto v'(\theta)$ is continuous, implies that $q$ is also a continuous function of the types. This will prove to be extremely useful, specially in the presence of a crossing network. If we integrate by parts, then $\mathcal{L}(v, \gamma)$ can be transformed into \begin{equation*} \Sigma(q, \gamma):=\int_{\Theta}\bigg(\Big(\theta+\frac{F(\theta) -\gamma(\theta)}{f(\theta)}\Big)\psi_1\big(q(\theta)\big) - \widetilde{C}\big(q(\theta)\big)\bigg)f(\theta)d\theta, \end{equation*} where $q(\theta) =\psi_1^{-1}\big(v'(\theta)\big),$ as described above, and $\widetilde{C}(q):=C(q)-\Psi_2(q).$ The idea now is to maximize the mapping \begin{equation*} q \mapsto \sigma(\theta, q, \Gamma) := \Big(\theta +\frac{F(\theta) -\Gamma}{f(\theta)}\Big)\psi_1(q) - \widetilde{C}\big(q\big) \end{equation*} pointwise, for a given fixed $\Gamma$ (in the sequel we use $\Gamma$ whenever we are dealing with an arbitrary but fixed value of $\gamma$). From Assumption~\ref{ass:qc} it follows that we can write down the unique maximizer as \begin{equation*} l(\theta, \Gamma):=K^{-1}\Big(\frac{F(\theta) + \theta\,f(\theta) -\Gamma}{f(\theta)}\Big), \end{equation*} where $K(q):=\widetilde{C}'(q)/\Psi_1'(q).$ For each $\theta\in\Theta$ and $\Gamma\in [0, 1],$ the quantity $l(\theta, \Gamma)$ is a candidate for the optimal $q(\theta)$ and convexity (or incentive compatibility) is verified if the mapping $\theta\mapsto l(\theta, \Gamma)$ is increasing. The crux is then to determine the Lagrange multiplier $\gamma.$ In the sequel we denote $\Theta_o:=\Theta_0(v_o^*),$ where $v_o^*$ solves Problem $\mathcal{P}_o.$ In other words, if $\theta\in\Theta_o$, then $q(\theta)=T(\theta)=v(\theta)=0.$ From Lemma~\ref{lm:ZeroatZero} we have that, unless $v(\underline{\theta})=0,$ the quantity $q(\underline{\theta})<0$ and the complementary--slackness condition imply that $\gamma(\theta)=0$ for $\theta\in[\underline{\theta}, \widetilde{\theta})$ for some $\widetilde{\theta}>\underline{\theta}.$ The left endpoint $\underline{\theta}_0$ of $\Theta_o$ is then determined by solving the equation \begin{equation*} K^{-1}\Big(\theta + \frac{F(\theta)}{f(\theta)}\Big)=0. \end{equation*} Furthermore, since $v$ must be convex, once $v(\hat{\theta})>0$ then $v(\theta)>0$ for all $\theta>\hat{\theta}.$ This implies that the right endpoint $\overline{\theta}_0$ of $\Theta_o$ is determined by solving the equation \begin{equation*} K^{-1}\Big(\theta-\frac{1 - F(\theta)}{f(\theta)}\Big)=0. \end{equation*} The quantities $F(\theta)/f(\theta)$ and $(1 - F(\theta))/f(\theta)$ are know as the \textit{hazard rates}, and sufficient conditions for the mapping $\theta\mapsto l(\theta, \Gamma)$ to be non--decreasing are \begin{equation*} \frac{d}{d\theta}\left(\frac{F(\theta)}{f(\theta)}\right)\geq 0 \geq \frac{d}{d\theta}\Big(\frac{1 - F(\theta)}{f(\theta)}\Big), \end{equation*} see, e.g. Biais et al.~\cite{BMR} for a discussion on this condition. Let us assume that we have determined $\Theta_o$. What remains is then to connect the participation constraint with the spread. Differentiating Eq.~\eqref{eq:IndUt} and noting that $v'(\theta)=\psi_1(q(\theta))$ we have that \begin{equation*} \tau'(\theta) = q'(\theta)\big(\theta\psi_1'(q(\theta)) + \psi_2'(q(\theta))\big). \end{equation*} Observe that $\tau'(\underline{\theta}_0)$ and $\tau'(\overline{\theta}_0)$ are in fact $T'(0_-)$ and $T'(0_+),$ since by construction $q(\underline{\theta}_0) = q(\overline{\theta}_0)=0.$ If we define $\phi_1:= \psi_1'(0)$ and $\phi_2:=\psi_2'(0)$, then we have that the spread is given by the expressions \begin{equation}\label{eq:Spread} t(0_-) = q'(\underline{\theta}_{0}-)\big(\underline{\theta}_0\phi_1 + \phi_2\big)\quad\text{and}\quad t(0_+) = q'(\overline{\theta}_{0}+)\big(\overline{\theta}_0\phi_1 + \phi_2\big). \end{equation} Our objective in Section~\ref{sec:ModelCN} is to compare the values above to those obtained in the presence of a crossing network. Before we proceed we present two examples so as to illustrate the use of the methodology described hitherto. The first revisits Mussa \& Rosen \cite{MR:78}. The second is slightly more advanced. We shall use it below to illustrate the complex structure of equilibrium pricing schedules and utilities in the presence of CNs. \begin{example} \label{MussaRosen} Let us assume that $\Theta = [-r,r]$ for some $r>0$, that types are uniformly distributed and that \begin{equation*} u(\theta,q) = \theta q. \end{equation*} We also set $C(q)=0.5\,q^2.$ By direct computation we find that $\underline{\theta}_0 = -\frac{r}{2}$ and $\overline{\theta}_0 = \frac{r}{2}.$ Since a trader of type $\theta \in \Theta_o$ is brought down to reservation utility and hence trades $q(\theta)=0,$ the expression \begin{equation*} q(\theta) = \theta + \frac{F(\theta) - \gamma(\theta)}{f(\theta)}=2\theta+r-2r\gamma(\theta) \end{equation*} implies that the Lagrange multiplier is \begin{equation*} \gamma(\theta) = \left\{ \begin{array}{ll} 0 & \theta < \underline{\theta}_0 \\ \frac{1}{2} + \frac{\theta}{r} & \theta \in \Theta_o \\ 1 & \theta > \overline{\theta}_0 \end{array} \right. . \end{equation*} In particular, $q'(\underline{\theta}_{0}-) = q'(\overline{\theta}_{0}+) = 2$ and hence $t(0_-) = -r$ and $t(0_+) = r$. Thus, the spread increases linearly in the highest/lowest type. \end{example} \begin{example} \label{ex1} Let us assume that the distribution of types over $\Theta=[-1,1]$ is given by $f(\theta)=(2\theta+3)/4$ for $\theta \in [-1,0)$ and $f(\theta)=(3-2\theta)/4$ for $\theta \in [0,1]$; that $C(q)=0.5\,q^2$ and that $u(\theta, q)-\tau = \theta\cdot q + 0.25\,q^2-\tau.$ It is straightforward to show that the conditions on the Hazard rates are satisfied and that \begin{equation*} K^{-1}\Big(\theta+\frac{F(\theta)}{f(\theta)}\Big)={2}\Big[\frac{3\theta^2+6\theta+2}{2\theta+3}\Big] \quad\text{and}\quad K^{-1}\Big(\theta-\frac{1 - F(\theta)}{f(\theta)}\Big)={2}\Big[\frac{3\theta^2-6\theta+2}{2\theta-3}\Big]. \end{equation*} Furthermore, $\Theta_o\approx\big[-0.423, 0.423\big].$ For the spread, we have that $t(0_-)=q'(\underline{\theta}_0)\underline{\theta}_0\approx -1.359$ and $t(0_+)=q'(\overline{\theta}_0)\overline{\theta}_0\approx 1.359.$ In order to obtain $v$ we integrate $q$ (since $\Psi_1(q)=q$) and take into account that $v\equiv 0$ over $\Theta_o.$ We plot graph$\{v_o\}$ in Figure~\ref{fig:Benchmark}, as well as the per--type profits of the dealer. \begin{figure} \caption{An example without a crossing network} \label{fig:IndUt} \label{fig:SpecProf} \label{fig:Benchmark} \end{figure} \end{example} \subsection{\large{Introducing a crossing network}}\label{sec:ModelCN} Let us now analyze the dealer's problem when the market participants have access to a CN that yields a trader of type $\theta$ the expected utility $u_0(\theta; \pi).$ In this setting it is no longer without loss of generality to assume that all traders participate in the DM, given that enforcing participation (which can be done thanks to Assumption~\ref{ass:matching}) may result in losses to the dealer. The latter may, as a consequence, choose to abstain from trading with a set of types $\Theta_e(v)$ by offering an incentive--compatible book whose corresponding indirect--utility function lies strictly under $u_0(\theta; \pi)$ for $\theta\in\Theta_e(v)$. The resulting problem for the dealer would be \begin{equation*} \mathcal{P}(\pi) = \sup_{v\in\mathcal{C}} \int_{\Theta}\Big(\theta\,v'(t) - v(t) - \widetilde{K}\big(v'(\theta)\big)\Big){\mbox{\rm{1}\hspace{-0.09in}\rm{1}\hspace{0.00in}}}_{\{\Theta_e^c(v)\}}(\theta)f(\theta)d\theta. \end{equation*} Dealing with the presence of the zero--one indicator function ${\mbox{\rm{1}\hspace{-0.09in}\rm{1}\hspace{0.00in}}}_{\{\Theta_e^c\}}$ is quite cumbersome (see, e.g. Horst \& Moreno--Bromberg~\cite{HoMo2}), since its domain of definition may change with different book choices. In contrast to the setting studied in~\cite{HoMo2}, however, here the CN is passive. This lack of non--cooperative--games component allows for an alternative way to proceed. To this end, we make use of the following \textit{accounting trick}, which was introduced by Jullien~\cite{BJ:03}: Let us assume that the dealer had access to a fictitious market such that the unwinding costs from trading in it, denoted in the sequel by $C_c,$ satisfy $C_c(q(\theta)) = \tau(\theta)$ for almost all $\theta\in\Theta.$ In this way, we may again assume that the dealer trades with all market participants, but now his costs of unwinding are given by the function $\mathbb{C}:\mathbb{R}\to\mathbb{R}$ defined as \begin{equation*} \mathbb{C}(q):=\min\big\{C(q), C_c(q)\big\},\quad q\in\mathbb{R}. \end{equation*} In terms of incentives, nothing is distorted by introducing the cost function $\mathbb{C},$ but we must identify the points where there is switching from using $C$ to using $C_c$ and vice versa. These switching points will determine the regions of market segmentation. If we define, for any traded quantity $q,$ the function $\widetilde{\mathbb{C}}(q) := \mathbb{C}\big(q\big) - \psi_2\big(q\big),$ then we may re--use the machinery from Section~\ref{sec:ModelBench} with minor modifications;\footnote{Observe that Assumptions~\ref{ass:cost of access} and~\ref{ass:matching} imply that $\widetilde{\mathbb{C}}$ satisfies Assumption~\ref{ass:qc}.} namely, denoting by $\mathbb{I}$ the energy corresponding to the cost function $\mathbb{C},$ we may write the Lagrangian of the dealer's problem as \begin{equation*}\label{eq:LagrangianCN} \mathbb{L}(v, \gamma) := \mathbb{I}[v]+\langle v - u_0(\cdot; \pi), \gamma\rangle, \end{equation*} with the corresponding complementary--slackness conditions. From here we may proceed as in Section~\ref{sec:ModelBench} to find the quantities that the dealer will choose to offer. Strictly speaking we should find the pointwise maximizer in $q$ of the expression \begin{equation}\label{eq:VirtualSurplus} \Big(\theta +\frac{F(\theta) -\Gamma}{f(\theta)}\Big)\Psi_1(q) - \mathbb{K}(q), \end{equation} where $\mathbb{K}(q):=\widetilde{\mathbb{C}}(q)-\Psi_2(q).$ This may fortunately be avoided, given that whenever $\mathbb{C}(q)=C_c(q)$, the participation constraint binds and $q(\theta)=q_c(\theta).$ Before proceeding to the proof of Theorem~\ref{thm:Main2}, we study the mechanism used by the dealer to choose between excluding types, matching the CN and trading with them while offering strictly positive rents. Whenever the participation constraint does not bind, the dealer selects the quantity to be chosen via the pointwise maximization of the mapping $q\mapsto \sigma(\theta, q, \Gamma).$ What makes the current problem trickier than the case without a CN is that now we must pay more attention to the evolution of the multiplier $\gamma.$ If we compare $l(\theta, 0)$ and $l(\theta, 1)$ to $q_c(\theta)$ we may pinpoint the set where the participation constraint may bind. Observe that $\big\{l(\theta, 1),\theta\in\Theta\big\}$ and $\big\{l(\theta, 0), \theta\in\Theta\big\}$ are the sets of the lowest and highest quantities the dealer may offer in an individually--rational way. Hence, as long as $l(\theta, 1)\leq q_c(\theta) \leq l(\theta, 0)$ there is the possibility of \textit{profitable matching}. There might be instances where the participation constraint is binding for some type $\theta\in\Theta$, i.e. $\big(q(\theta), \tau(\theta)\big)=\big(q_c(\theta), \tau_c(\theta)\big),$ and $\tau_c(\theta) - C\big(q_c(\theta)\big)<0.$ In such cases $\mathbb{C}\big(q_c(\theta)\big)=C_c\big(q_c(\theta)\big)$ and $\theta\in\Theta_e(v)$ for the corresponding indirect utility function, and we say there is \textit{exclusion}. \begin{remark}\label{rem:QuasiUniquenessReprise} It is at this point that the quasi--uniqueness mentioned in Remark~\ref{rem:QuasiUniqueness} can be addressed. The principal's problem $\mathbb{P}(\pi)$ using the cost function $\mathbb{C}$ results in the condition \begin{equation*} \big(i(\theta, v(\theta), v'(\theta))\big)_+=\big(i(\theta, v(\theta), v'(\theta))\big) \end{equation*} being trivially satisfied. As a consequence, problem $\mathbb{P}(\pi)$ admits a unique solution. The latter coincides, by construction, with the solution to $\mathcal{P}(\pi)$ whenever $\mathbb{C}(q(\theta))=C(q(\theta))$. The caveat is that the solution to problem $\mathbb{P}(\pi)$ is blind towards what is offered to excluded types, since here their outside option is costlessly matched (they are effectively reserved). Constructing incentive compatible contracts for the excluded types is, thanks to the convexity of the indirect utility function, relatively simple. For instance if an interval of types $(\theta_1, \theta_2)$ were excluded (but $\theta_1$ and $\theta_2$ participated) one could consider any two supporting lines to $graph\{v(\cdot;\pi)$ at $(\theta_1, v(\theta_1;\pi))$ and $\theta_2, v(\theta_2;\pi)$. From the resulting indirect--utility function on $(\theta_1, \theta_2)$ one could extract the corresponding quantities and prices. The resulting global convexity of the indirect--utility function offered by the principal would imply that all incentives would remain unchanged. Whether the principal would suffer losses from the contracts offered to types on $(\theta_1, \theta_2)$ would be irrelevant, since the corresponding agents do not participate. \end{remark} As mentioned above, here it is not necessary to determine $\gamma(\theta)$ in order to do likewise with $q(\theta).$ On the other hand, however, if we interpret $\gamma$ as the shadow cost of satisfying the participation constraint, we may wish to identify the multiplier so as to have a measure of the impact of the CN on the dealer's profits. The following result, which deals with points where there is switching between matching and fully servicing, extends Proposition~\ref{prop:NoJumps1}. \begin{proposition}\label{prop:NoJumps2} For $\pi\in\mathbb{R}^2$ given, let $\widetilde{\theta}\in\Theta$ be such that there exists $\epsilon>0$ such that $v(\theta;\pi)=u_0(\theta;\pi)$ on $(\widetilde{\theta}-\epsilon, \widetilde{\theta}]$ and $v(\theta;\pi)>u_0(\theta;\pi)$ on $(\widetilde{\theta}, \widetilde{\theta}+\epsilon]$. Furthermore, assume that \begin{equation*} \int_{\widetilde{\theta}-\epsilon}^{\widetilde{\theta}}\big(\tau(\theta)-C(q(\theta))\big)f(\theta)d\theta>0, \end{equation*} where $\big\{\big( q(\theta), \tau(\theta) \big), \theta \in \Theta \big\}$ implements $v(\cdot;\pi).$ In other words, there is profitable matching on $(\widetilde{\theta}-\epsilon, \widetilde{\theta}]$ and the dealer fully services types on $(\widetilde{\theta}, \widetilde{\theta}+\epsilon].$ Then $\partial v(\widetilde{\theta};\pi)$ is a singleton. The result also holds if the order of the matching and full-servicing intervals is switched. \end{proposition} \noindent The rationale behind Proposition~\ref{prop:NoJumps2} is that, as long as the dealer is able to match the traders' outside option without incurring in a loss, it is possible to normalize the latter to zero and directly apply Proposition~\ref{prop:NoJumps1}. This is, naturally, not the case when matching $u_0$ results in losses. We put Proposition~\ref{prop:NoJumps2} to work in Example~\ref{RichStructure}. Before moving on, we present below a modification to Example~\ref{ex1} that shows how even agents without access to a non--trivial outside option benefit from the presence of the CN and that the optimal Lagrange multiplier need not be continuous. \begin{example}\label{ex2} Let $f,$ $\Theta,$ $C$ and $u$ be as in Example \ref{ex1} and assume that the CN offers the traders the following expected profits: \begin{equation*} u_0(\theta; (3.2, 3.2))=\begin{cases} -0.975\theta - 0.52, \text{ if } \theta\leq -\frac{8}{15};\\ 0.975\theta - 0.52, \text{ if } \theta\geq \frac{8}{15};\\ \text{convex and negative for } \theta\in(-\frac{8}{15}, \frac{8}{15}). \end{cases} \end{equation*} Matching this outside option would require the dealer to offer the contracts $(\pm 0.975, 0.52)$. This is profitable, hence the indirect utility never lies below $u_0$. To illustrate this, we have plotted the indirect--utility function in Figure~\ref{fig:IndUtNoEx}. It strictly dominates the one plotted in Figure~\ref{fig:IndUt} for all types who earn positive profits. The smooth pasting condition ($l(\theta,\gamma(\theta))=q_c(\theta)$ where $v$ touches $u_0$, i.e. in $\pm 0.675$) determines the optimal Lagrange multiplier, namely $\gamma(-1)=0$ and $\gamma \equiv 0.030$ on $(-1,-0.389]$. For positive types we obtain symmetrically $\gamma(1)=1$ and $\gamma \equiv 0.970$ on $[0.389,1)$. The new spread, given by $\big(t(0_-), t(0_+)\big)=(-1.282, 1.282)$, is strictly smaller than in the case without a CN. \begin{figure} \caption{An example without exclusion} \label{fig:IndUtNoEx} \label{fig:LMNoEx} \label{fig:NoExclusion} \end{figure} \end{example} The following result will prove to be essential for the results in Section~\ref{sec:ExistenceEqui}. It guarantees,by virtue of Assumption~\ref{ass:cost of access}, our notion of the spread is well defined in the presence of a CN and could be loosely summarized by saying that the first (in terms of moving away from $\theta=0$) types to earn positive utility trade in the DM. \begin{lemma}\label{lemma:tradingSB} There exists $\epsilon=\epsilon(\pi)$ such that the types that belong to $(\underline{\theta}_0(\pi)-\epsilon,\underline{\theta}_0(\pi) )\cup (\overline{\theta}_0(\pi), \overline{\theta}_0(\pi)+\epsilon)$ are fully serviced. \end{lemma} \noindent\begin{Proof} Let us denote by $\hat{\theta}$ the positive solution to the equation $u_0(\theta;\pi)=0.$ If there exists $\eta>0$ such that types on $(\hat{\theta}, \hat{\theta}+\eta)$ can be matched profitably, then the result follows either because $\overline{\theta}_0(\pi)<\hat{\theta}$ or because $\overline{\theta}_0(\pi)=\hat{\theta}$ and the types on $(\hat{\theta}, \hat{\theta}+\epsilon),$ for some $0<\epsilon\leq \eta,$ are fully serviced. Let us now assume that such an $\eta$ does not exist, we claim then that $\overline{\theta}_0(\pi)<\hat{\theta}$ must hold. Proceeding by the way of contradiction, let us assume that $\overline{\theta}_0(\pi)=\hat{\theta}$ (which is equivalent to $\overline{\theta}_0(\pi)\geq\hat{\theta}$) and that there exists $\delta>0$ such that $(\hat{\theta}, \hat{\theta}+\delta)\subset\Theta_e(\pi).$ This configuration can be improved upon as follows: let $a>0$ be such that $\hat{\theta}-a>0.$ By construction $l(\hat{\theta}-a, \gamma(\hat{\theta}-a))=0.$ Let us fix $\gamma(\theta)\equiv \gamma(\hat{\theta}-a)=:\Gamma(a)$ for $\theta\in(\hat{\theta}-a, \theta_a),$ where $\theta_a$ the solution to $v_a(\theta)=u_0(\theta;\pi)$ on $(\hat{\theta}-a,\overline{\theta}]$ if it exists or $\theta_a=\overline{\theta}$ otherwise, given that we denote by $v_a$ the indirect--utility function corresponding to $\Gamma(a)$. In particular $\theta_a>\hat{\theta}$ and $l(\theta,\Gamma(a))>0$ for $\theta \in (\hat{\theta}-a, \theta_a)$. We now have that types $\theta\in(\hat{\theta}-a, \theta_a)$ are fully serviced. By Assumption~\ref{ass:cost of access}, $v'_a(\hat{\theta}-a)=0<u_0'(\hat{\theta};\pi);$ therefore, there exists $a_1>0$ such that for all $a\leq a_1$ it holds that $\theta_a < \hat{\theta}+\delta.$ If we could show that there exists $a\leq a_1$ such that the principal could offer types in $(\hat{\theta}-a, \theta_a)$ the quantities $q_a(\theta)=l(\theta, \Gamma(a))$ at a profit, we would contradict the optimality of $\overline{\theta}_0(\pi)$ and the proof would be finalized, since incentives above $\theta_a$ would not be distorted and the principal's profits would strictly increase. In order to do so, observe that the principal's typewise profit when offering $q_a(\theta)$ is \begin{equation*} P(\theta):=\theta\Psi_1(q_a(\theta))+\Psi_2(q_a(\theta))-v_a(\theta)-C\big(q_a(\theta)\big). \end{equation*} In particular, $P(\hat{\theta}-a)=0$ and \begin{align*} P'(\hat{\theta}-a)& =\Psi_1(q_a(\hat{\theta}-a))+(\hat{\theta}-a)\Psi_1'(q_a(\hat{\theta}-a))q_a'(\hat{\theta}-a)+v_a'(\hat{\theta}-a)-\widetilde{C}'\big(q_a(\hat{\theta}-a)\big)q_a'(\hat{\theta}-a)\\ & =\Psi_1(0)+(\hat{\theta}-a)\Psi_1'(0)q_a'(\hat{\theta}-a)+v_a'(\hat{\theta}-a)-\widetilde{C}'\big(0\big)q_a'(\hat{\theta}-a)\\ & = (\hat{\theta}-a)\Psi_1'(0)q_a'(\hat{\theta}-a). \end{align*} The step from the second to the third equality follows, because by construction $v_a'(\hat{\theta}-a)=0;$ by assumption $\Psi_1(0)=0$ and, from Assumption~\ref{ass:qc}, $\widetilde{C}'\big(0\big)=0.$ Furthermore, since $\Psi_1$ is strictly increasing and $q_a'(\hat{\theta}-a)>0,$ then $P'(\hat{\theta}-a)>0.$ Therefore, there exists $b>0$ such that $P(\theta)> 0$ if $\theta\in(\hat{\theta}-a, \hat{\theta}-a+b).$ As a consequence, if $a<a_1$ is small enough, then $P(\theta)>0$ for $\theta\in(\hat{\theta}-a, \theta_a),$ as required. \end{Proof} \noindent We are now in the position to present the proof of our second main result. \noindent\textbf{{Proof of Theorem~\ref{thm:Main2}.}} (1) Observe that if $\pi$ is such that $\big(\underline{\theta}_0(\pi), \overline{\theta}_0(\pi)\big)=\Theta_0(\pi)\subset \Theta_o,$ then the result follows immediately from Lemma~\ref{lemma:tradingSB}. If we revert the inclusion, two situations are possible, since the addition of the CN--constraint to Problem $\mathcal{P}_o$ may or may not bind for some types. The latter case being trivial, let us look at the case where there is a point $\theta_a>\overline{\theta}_0$ on which it holds that $v_o(\theta_a)=u_0(\theta_b;\pi)$ and such that $v_o(\theta)>u_0(\theta;\pi)$ for $\theta<\theta_a$ and vice versa for $\theta>\theta_a.$ The Lagrange multiplier $\gamma_m$ is active on $(\theta_a, \overline{\theta}],$ which implies that $\gamma_m(\theta_a)<1.$ We know from~\cite{BJ:03}, p. 9, that for all $\theta$ such that $l(\theta, \Gamma)>0,$ the latter is decreasing in $\Gamma.$ As a consequence, the root of the equation \begin{equation*} K^{-1}\Big(\theta+\frac{F(\theta)-\gamma_m(\theta_a)}{f(\theta)}\Big) = 0 \end{equation*} is strictly smaller than that of $l(\theta,1)=0,$ which yields the desired result. \noindent (2) Let us denote by $t_o(0_-)$ and $t_o(0_+)$ the best bid and ask prices without the presence of a CN and by $t_m(0_-)$ and $t_m(0_+)$ the corresponding marginal prices with one; thus, \begin{equation*} t_o(0_-)=q_o'(\underline{\theta}_{0,o-})\big(\underline{\theta}_{0,o}\phi_1 + \phi_2\big)\text{ and } t_o(0_+) = q_o'(\overline{\theta}_{0,o+})\big(\overline{\theta}_{0,o}\phi_1 + \phi_2\big) \end{equation*} and \begin{equation*} t_m(0_-)=q_m'(\underline{\theta}_{0,m-})\big(\underline{\theta}_{0,m}\phi_1 + \phi_2\big)\text{ and } t_m(0_+) = q_m'(\overline{\theta}_{0,m+})\big(\overline{\theta}_{0,m}\phi_1 + \phi_2\big). \end{equation*} From Part (1) we know that $\underline{\theta}_{0,o}\leq\underline{\theta}_{0,m}$ (both negative) and $\overline{\theta}_{0,m}\leq\overline{\theta}_{0,o}$ (both positive) and, since $\phi_1$ and $\phi_2$ do not depend on the presence of the CN, all we have left to do is show that \begin{equation*} q_m'(\underline{\theta}_{0,m-})\leq q_o'(\underline{\theta}_{0,o-})\text{ and }q_m'(\underline{\theta}_{0,m+})\leq q_o'(\underline{\theta}_{0,o+}). \end{equation*} Using the well--known relation $(f^{-1})'(a)=1/f'(f^{-1}(a))$ we have that \begin{align*} q_m'(\underline{\theta}_{0,m-})& = \frac{1}{K'\Big(K^{-1}\big(\underline{\theta}_{0,m} - \frac{\gamma(\underline{\theta}_{0,m-})-F(\underline{\theta}_{0,m})}{f(\underline{\theta}_{0,m})}\big)\Big)}\frac{d}{d\theta}\big(\theta-\frac{\gamma(\theta)-F(\theta)}{f(\theta)}\big)\Big|_{\theta=\underline{\theta}_{0,m-}}\\ & = \frac{1}{K'\big(q_m(\underline{\theta}_{0,m})\big)}\frac{d}{d\theta}\big(\theta-\frac{\gamma(\theta)-F(\theta)}{f(\theta)}\big)\Big|_{\theta=\underline{\theta}_{0,m-}}\\ & = \frac{1}{K'(0)}\Big(1-\frac{d}{d\theta}\big(\frac{\gamma(\underline{\theta}_{0,m-})-F(\theta)}{f(\theta)}\big)\Big)\Big|_{\theta=\underline{\theta}_{0,m}}, \end{align*} where we have used the fact that $\gamma$ is constant on $(\underline{\theta}_{0,m}-\delta,\underline{\theta}_{0,m})$ for some $\delta>0.$ We may proceed analogously for the other three quantities. We have to show that \begin{align}\begin{split}\label{eq:condMonot} \frac{1}{K'(0)}\frac{d}{d\theta}\Big(\frac{\gamma(\underline{\theta}_{0,m-})-F(\theta)}{f(\theta)}\Big)\Big|_{\theta=\underline{\theta}_{0,m}} & \geq\frac{1}{K'(0)}\frac{d}{d\theta}\Big(\frac{-F(\theta)}{f(\theta)}\Big)\Big|_{\theta=\underline{\theta}_{0,o}}\\ \frac{1}{K'(0)}\frac{d}{d\theta}\Big(\frac{\gamma(\overline{\theta}_{0,m+})-F(\theta)}{f(\theta)}\Big)\Big|_{\theta=\overline{\theta}_{0,m}} & \geq\frac{1}{K'(0)}\frac{d}{d\theta}\Big(\frac{1-F(\theta)}{f(\theta)}\Big)\Big|_{\theta=\overline{\theta}_{0,o}}, \end{split} \end{align} which hold with equality under the assumption that $f\equiv(\overline{\theta}-\underline{\theta})^{-1}.$ \noindent (3) If follows from Part (1) that, if $\theta$ participates in the presence of the CN, then $q_o(\theta)\leq q_m(\theta).$ Assume now that the inequality $v_o(\theta) > v(\theta; \pi)$ holds for all $\theta$ in a non--empty interval $(\theta_1, \theta_2)$ and $v_o(\theta_1) = v(\theta_1; \pi)$ and $v_o(\theta_2) = v(\theta_2; \pi).$ By the convexity of $v_o$ and $v(\cdot; \pi),$ this would imply the existence of $\theta_3\in(\theta_1, \theta_2)$ such that $v_o'(\theta) > v'(\theta; \pi)$ holds almost surely in $(\theta_1, \theta_3).$ However $v_o'(\theta)=\psi_1(q_o(\theta)),$ $v'(\theta; \pi)=\psi_1(q_m(\theta))$ and $\psi_1$ is strictly increasing; hence, this would imply that $q_o(\theta)>q_m(\theta)$ for almost all $\theta\in(\theta_1, \theta_3),$ which is a contradiction. $\Box$ We finalize this section with two examples that showcase the results obtained thus far. Example~\ref{MussaRosenEx} showcases that, in the simple case where the outside option is such that the dealer will (only) exclude all high--enough (in absolute value) types, then the results of Theorem~\ref{thm:Main2} follow trivially. \begin{example}\label{MussaRosenEx} Let us revisit Example~\ref{MussaRosen} with an extremely steep outside option that will warrant exclusion, namely, for $r_0<r$ let \begin{equation*} u_0(\theta)=\begin{cases} \infty,\text{ if }\, \theta\in [-r,-r_0)\bigcup (r_0, r];\\ 0,\,\,\text{ otherwise}. \end{cases} \end{equation*} Recall that, for a given value $\Gamma$ of the Lagrange multiplier, the corresponding quantity is \begin{equation*} q(\theta;\Gamma):=2\theta+r-2r\Gamma. \end{equation*} In Example~\ref{MussaRosen} the participation constraint does not bind for high types. In particular, $\gamma\equiv 0$ on $[-r,\underline{\theta}_0)$ and to find the left--hand endpoint of the reserved set we set $\Gamma=0$ and solve $2\theta+r=0.$ In the current setting, the participation constraint must bind for $\theta<-r_0$ and the multiplier will be constant on $(-r_0, \underline{\theta}_0(\Gamma)),$ where \begin{equation*} \underline{\theta}_0(\Gamma):=-\frac{r}{2}\big[1-2\Gamma\big]. \end{equation*} By construction, the choice of $\Gamma$ will bear no weight on the trader types that will be serviced to the left of $\theta=-r_0,$ but only on how many additional low types benefit from the presence of the outside option. By integrating $q(\theta;\Gamma)$ and noting that the corresponding indirect--utility function $v(\cdot;\Gamma)$ must satisfy $v(\underline{\theta}_0(\Gamma);\Gamma)=0,$ we have, for $\theta\in [-r_0,\underline{\theta}_0(\Gamma)]$ \begin{equation*} v(\theta;\Gamma)=\theta^2+\theta r\big[1-2\Gamma\big]+\frac{r^2}{4}\big[1-2\Gamma\big]^2. \end{equation*} Since the indirect--utility function also satisfies $v(\theta;\Gamma)=\theta q(\theta;\Gamma)-\tau(\theta;\Gamma),$ we have that the dealer market on $[-r_0,\underline{\theta}_0(\Gamma)]$ is described by the quantity--price pairs $\big(q(\theta;\Gamma), \theta^2-\frac{r^2}{4}\big[1-2\Gamma\big]^2\big).$ As a consequence, the per--type profit is \begin{equation*} \Pi(\theta;\Gamma):=-\theta^2-\frac{3}{4}r^2\big[1-2\Gamma\big]^2-2\theta r\big[1-2\Gamma\big], \end{equation*} where the third term on the right--hand side is positive and dominates the first two. Finally, we have that each choice of $\Gamma$ will result in the dealer obtaining the aggregate profits from negative types \begin{equation*} P(\Gamma):=\frac{1}{2r}\int_{-r_0}^{\underline{\theta}_0(\Gamma)}\Pi(\theta;\Gamma)d\theta. \end{equation*} The mapping $\Gamma\mapsto P(\Gamma)$ is strictly concave and the first--order conditions yield that it is maximized at $\Gamma=(r-r_0)/(2r).$ As a result $\underline{\theta}_0(\Gamma)=-r_0/2$ and $v(\theta;\Gamma)=\theta^2+r_0\theta+r_0^2/4,$ which correspond to the boundary of the reserved set and the indirect--utility function for negative trader types in the problem without a CN on $[-r_0, r_0].$ \end{example} \begin{example}\label{RichStructure} We stay with the basic setup of Examples \ref{ex1} and \ref{ex2}, but now assume that $u_0(\theta; \pi)=\Big(\frac{1- \pi_+}{3}\theta^{6/5}-0.001\Big)_+$ for $\theta\geq 0$ and $u_0(\theta; \pi)\equiv 0$ otherwise. For any type $\theta$ such that $u_0(\theta)>0$ it holds that \begin{equation*} \big(q_c(\theta), \tau_c(\theta)\big) =\Big(\frac{2}{5}(1-\pi_+)\theta^{1/5},\frac{2}{5}(1-\pi_+)\theta^{6/5} + \frac{1}{25}(1-\pi_+)^2\theta^{2/5} - \big(\frac13(1-\pi_+)\theta^{6/5}-0.001\big)_+\Big). \end{equation*} We assume $\pi=(0,1/2).$ The first thing to notice is that the dealer's per-type profit for offering $(q_c(\theta),\tau_c(\theta))$, i.e. $\tau_c(\theta)-C(q_c(\theta))=\theta^{6/5}/30 - \theta^{2/5}/100+0.001$, is negative for types $\theta\in(0.0035, 0.1667)$. On the other hand, the inequality $u_0(\theta; 1/2)\geq 0$ only holds for $\theta\geq 0.014.$ Combining both arguments we see that $\Theta_e(\pi)\subset(0.014, 0.1667).$ Next we observe that the inequality \begin{equation*} l(\theta,1) = K^{-1}\Big(\theta-\frac{1 - F(\theta)}{f(\theta)}\Big)\geq \frac{\sqrt[5]{\theta}}{5} \end{equation*} holds for all $\theta\in [0.4761, 1].$ Hence profitable matching may occur on the interval $(0.1667, 0.4761),$ over which $q(\theta)=q_c(\theta)$ and $\mathbb{C}\big(q(\theta)\big)=C\big(q(\theta)\big).$ Furthermore, Proposition~\ref{prop:NoJumps2} implies that the corresponding indirect utility function will be differentiable at $\theta= 0.4761.$ In order to obtain $v(\theta; \pi)$ for $\theta\in [0.4761, 1],$ we integrate $l(\cdot, 1)$ and determine the corresponding integration constant $c$ by equating \begin{equation*} 2\int_0^{0.4761}\left(\frac{3\theta^2-6\theta+2}{2\theta-3}\right)d\theta + c = \frac{1}{6}(0.4761)^{6/5} - 0.001. \end{equation*} We know from the example without a CN that $\gamma(t)=0$ for $\theta\in[-1, -0.423).$ On $[-0.423, 0)$ the multiplier must satisfy \begin{equation*} K^{-1}\Big(\theta-\frac{\gamma(\theta) - F(\theta)}{f(\theta)}\Big)=0, \end{equation*} which results in $\gamma(\theta)=(3\theta^2+6\theta+2)/4$ on the said interval. What remains to be determined is $\overline{\theta}_0$ and $\gamma(\overline{\theta}_0).$ To this end, we define the family of functions $v(\cdot;\Gamma)$ such that $v'(\theta;\Gamma)=l(\theta,\Gamma)$ whenever this quantity is positive and $v(\theta;\Gamma)=0$ for $\theta\in[0,\theta(\Gamma)],$ where $\theta(\Gamma)$ is the solution to the equation $l(\theta,\Gamma)=0.$ Since $\gamma(0)=0.5,$ we have that $\Gamma>0.5.$\footnote{Pasting when passing from servicing to excluding need not be smooth.} In fact, $\Gamma=\gamma(\overline{\theta}_0)=0.5105,$ $\overline{\theta}_0=0.007$ and the intersection of $v(\cdot;\Gamma)$ and $u_0(\cdot; 1/2)$ occurs at $\theta= 0.0159.$ Summarizing, the types on $[-1,-0.423)\cup(0.007, 0.0159]\cup(0.1667, 1]$ are fully serviced, those on $[-0.423, 0.007]$ are reserved and the ones that lie on $(0.0159, 0.1667)$ are excluded. The left--hand side of the spread is the same as in the example without a CN, whereas the right--hand side is $t(0_+)=0.0281.$ This is significantly smaller than in Example \ref{ex1}. Determining $\gamma(\theta)$ on $(0, 0.007]$ is relatively simple, as we again must solve $l(\theta,\gamma(\theta))=0,$ which results in $\gamma(\theta)=(-3\theta^2+6\theta+2)/4$. Finally, in order to determine $\gamma$ on $\Theta_e(\pi)$ we must rewrite the virtual surplus using $\mathbb{C}(q(\theta))=\tau_c(\theta),$ which results in ${\mathbb{C}(q)=(5^5/6)q^6-(1/4)q^2+0.001}.$ The pointwise maximization of the resulting virtual surplus must equal $q_c(\theta)=\sqrt[5]{\theta}/5.$ After some lengthy arithmetic that we choose to spare the reader from, we obtain \begin{equation*} \gamma(\theta)=F(\theta)-f(\theta)\bigg[5^5 q_c(\theta)^5-\theta\bigg] = F(\theta)\quad\text{for }\theta\in\Theta_e(\pi). \end{equation*} Finally, in the profitable--matching region we solve $l(\theta,\gamma(\theta))=\sqrt[5]{\theta}/5$ so as to find the multiplier, which yields \begin{equation*} \gamma(\theta)=F(\theta)-f(\theta)\bigg[\frac{1}{10}\theta^{1/5}-\theta\bigg]\quad\text{for }\theta\in [0.1667,0.4761). \end{equation*} or \begin{equation*} \gamma(\theta)=\frac{1}{10}\theta^{1/5}\cdot \frac{2\theta-3}{4} - \frac{3\theta^2-6\theta-2}{4} \quad\text{for }\theta\in [0.1667,0.4761). \end{equation*} Observe that, in contrast with Example~\ref{ex2}, here $\gamma(\theta)=1$ for types that are strictly smaller than one. This means that the rightmost types do not profit from introduction of CN via changes in the quantities they are offered, but rather from changes in the corresponding prices. Intuitively speaking this has to do with how steep the outside option is for large types and, as a consequence, whether or not it will be matched over a non-trivial interval. We present in Figure~\ref{fig:IndUtOpt} the indirect utilities for positive types (the ones for negative ones being the same as in Figure~\ref{fig:IndUt}). The values of $\gamma$ have been plotted in Figure~\ref{fig:OptMult}. In Figure~\ref{fig:WithExZoom} we provide a magnification around small values of $\theta$ so as to highlight the switching between reservation, full servicing and exclusion. Observe the jump of the Lagrange multiplier at the boundary between fully--serviced and excluded types (Figure~\ref{fig:OptMultZoom}) and between excluded and matched ones (Figure~\ref{fig:OptMult}). \begin{figure} \caption{An example with exclusion} \label{fig:IndUtOpt} \label{fig:OptMult} \label{fig:WithEx} \end{figure} \begin{figure} \caption{An example with exclusion (magnified)} \label{fig:IndUtOptZoom} \label{fig:OptMultZoom} \label{fig:WithExZoom} \end{figure} \end{example} \noindent We shall revisit this example in the upcoming section, where we look into the existence of equilibrium prices in the CN. \section{\large{An equilibrium price in the crossing network}}\label{sec:ExistenceEqui} {In this section we prove the existence of an equilibrium price $\pi^*.$ We first observe that, from Assumption~\ref{ass:monotone}, there is no loss of generality in assuming that $\pi^*$ belongs to some closed and bounded subset of $\mathbb{R}^2,$ which we denote by $\Pi.$ As a consequence we have that $t(0;\cdot):\Pi\to\Pi.$ The restriction of possible equilibrium prices to $\Pi,$ together with Assumptions~\ref{ass:cost of access} and~\ref{ass:monotone}, yields the next result.} \begin{lemma}\label{lm:BondedParticipation} There exists a non--empty interval $[\epsilon_1, \epsilon_2]\subset\Theta$ such that \begin{enumerate} \item $0\in (\epsilon_1, \epsilon_2);$ \item $u_0(\theta; \pi) = 0$ for all $\theta\in[\epsilon_1, \epsilon_2]$ and all $\pi\in\Pi.$ \end{enumerate} \end{lemma} In the sequel we make use of the results obtained in Section~\ref{sec:ModelCN} to show that the mapping $\pi\mapsto t(0; \pi)$ has the required monotonicity properties so as to use the following result (see, e.g. Aliprantis \& Border~\cite{AB}): \begin{theorem}\label{thm:Tarski}(Tarski's Fixed Point Theorem) Let $(X, \leq)$ be a non--empty, complete lattice. If $f:X\to X$ is order preserving, then the set of fixed points of $f$ is also a non--empty, complete lattice. \end{theorem} \noindent We are now ready to give the proof of our third main result. \noindent\textbf{Proof of Theorem \ref{thm:Main3}.} Lemmas~\ref{lemma:tradingSB} and ~\ref{lm:BondedParticipation} guarantee that we have a well--defined spread; thus, we may decompose the analysis of the mapping $\pi\mapsto t(0; \pi)$ into that of the mappings $\pi_-\mapsto t(0_-; \pi_-)$ and $\pi_+\mapsto t(0_+; \pi_+).$ In other words, for a given price $\pi,$ the dealer's optimal response to $u_0(\cdot;\pi)$ is, modulo a normalization of $\gamma,$ equivalent to the combination of his actions towards negative and positive types separately. We shall concentrate on the existence of a fixed point of the mapping $\pi_+\mapsto t(0_+; \pi_+).$ From Assumption~\ref{ass:monotone} we have that if $\pi_{1+}<\pi_{2+}$, then $u_0(\theta;\pi_{1+})>u_0(\theta;\pi_{2+})$ for all $\theta>0.$ If for $i=1,2$ it holds that $u_0(\theta;\pi_{i+})<v_o(\theta)$ for all $\theta>0$, then $v(\theta;\pi_{1+})=v(\theta;\pi_{2+})$ on the same domain and $t(0_+; \pi_{1+})=t(0_+; \pi_{2+}).$ Next assume that $u_0(\theta;\pi_{i+})\geq v_o(\theta)$ on a subset $\Theta_i$ of $(0, \overline{\theta}],$ for $i=1,2.$ Given that $u_0(\theta;\pi_{1+})>u_0(\theta;\pi_{2+})$ for all $\theta>0,$ then $\overline{\theta}(\pi_1)<\overline{\theta}(\pi_2)$ and the first point $\widetilde{\theta}_1$ such that $v(\theta;\pi_{1+})=u_0(\theta;\pi_{1+})$ holds satisfies $\widetilde{\theta}_1<\widetilde{\theta}_2,$ where the latter is the analogous to $\widetilde{\theta}_1$ in the presence of $u_0(\theta;\pi_{2+}).$ The existence of $\widetilde{\theta}_1$ and $\widetilde{\theta}_2$ is guaranteed by the fact that in both cases the indirect--utility functions intersect the corresponding outside options. Arguing as in the proof of Theorem~\ref{thm:Main2}, Part (2), this also implies that $\overline{\theta}_0(\pi_{1})<\overline{\theta}_0(\pi_{2});$ hence $t(0_+; \pi_{1+})<t(0_+; \pi_{2+}).$ In other words, the mapping $\pi_{+}\mapsto t(0_+; \pi_{+})$ is order--preserving and, using Tarski's Fixed Point Theorem, we may conclude it has a fixed point. $\Box$ \begin{remark} The requirement of uniformly distributed types can be relaxed to the extent that if $f$ and $K$ are such that Conditions~\eqref{eq:condMonot} are satisfied, then the required monotonicity properties still apply. Unfortunately, these conditions cannot be verified ex--ante, since they include the end points of the set of reserved traders. \end{remark} \begin{example} Let us go back to our example with exclusion, but introduce the feedback loop between the DM and the CN through the iteration $\pi_{i+1}= t(0;\pi_i).$ We initialize the recursion by setting $\pi_0=(0, 1/2)$ and $\kappa=0.001$, which are the parameters in the aforementioned example. \begin{figure} \caption{ The indirect--utility functions corresponding to the iteration $\pi_{i+1}=t(0; \pi_{i})$.} \label{fig:EquilPriceZoomIn} \end{figure} We observe a very swift convergence. Indeed, it takes only four iterations to reach $\|v(\cdot;\pi_i)-v(\cdot; \pi_{i+1})\|_{\infty}\leq 10^{-5}$ and the indirect--utility functions in the third and fourth iteration are almost indistinguishable. The equilibrium price is $\pi^*=(0, 0.015)$. We present in Figure~\ref{fig:EquilPriceZoomIn} the plots of the first four iterates. It is evident that each iteration results in a smaller set of reserved traders and in a higher indirect utility for all types. The spreads, the right endpoints of the reserved regions, the Lagrange multipliers at the right endpoint of the reserved regions and the exclusion regions are provided in Table~\ref{table1}. It is interesting to observe that, as the spread decreases to its equilibrium level, the number of trader types that are reserved decreases and the sets of excluded types grow (in terms of inclusions). This last fact obeys the fact that, when the traders have a more attractive outside option, it is harder for the dealer to match it profitably. \begin{table*}[ht!] \small \centering \caption{The numbers of the feedback loop}\label{table1} \begin{tabular}{@{}ccc ccc cccc cccc @{}}\toprule $\pi_+$ & $\Theta_o$ & $\Gamma$& $\Theta_e(\pi_+)$ \\ \midrule $ 1/2$ & [-0.423,0.0070] & 0.5105 & [0.0159, 0.1667]\\ $0.0281$ & [-0.423,0.0040] & 0.5061 & [0.0083, 0.4872]\\ $0.0161$ & [-0.423,0.0040] & 0.5060 & [0.0082, 0.4954]\\ $0.0158$ & [-0.423,0.0040] & 0.5060 & [0.0082, 0.4955]\\ \bottomrule \\ \end{tabular} \end{table*} \end{example} \section{\large{Portfolio liquidation and dark--pool trading}}\label{sec:DPtrading} {In this section we present an application of our methodology to portfolio liquidation. We assume that the market participants' aim is to liquidate their current holdings on some traded asset. The sizes of the traders' portfolios are heterogeneous and saying that a trader's type is $\theta$ means that he holds $\theta$ shares of the asset prior to trading.} We set $\Theta=[-1, 1]$ and $f\equiv 1/2.$ If a trader of type $\theta$ trades $q$ shares for $\tau$ dollars, his utility is \begin{equation*} \hat{u}(\theta, q)- \tau:=-\alpha(\theta-q)^2-\tau, \end{equation*} where $0<\alpha$ denotes the traders' (homogeneous) sensitivity towards inventory holdings. Notice that $-\alpha\theta^2$ is the type--dependent reservation utility of a trader of type $\theta.$ If we ``normalize" the said utility to zero, we may write \begin{equation*} u(\theta ,q) - \tau = 2\alpha\theta q -\alpha q^2 -\tau. \end{equation*} In this example the crossing network takes the form of a \textit{dark pool} (DP). Choosing to trade in the latter entails two kinds of costs for the traders: On the one hand, there is a direct, fixed cost $\kappa>0$ of engaging in dark--pool trading. On the other hand, execution in the DP is not guaranteed. We denote by $p\in [0, 1]$ the probability that an order is executed where we assume for simplicity that the probability of order execution is independent of the order size. Pricing in the DP is linear. Namely, for a given execution price $\pi$, the utility that a trader of type $\theta$ extracts from submitting an order of $q$ shares to be traded in the DP is \begin{equation*} p\big[(2\theta\alpha - \pi)q - \alpha q^2\big] - \kappa, \end{equation*} where again we have normalized reservation utilities to zero. The problem of optimal submission to the DP for a $\theta$--type trader is \begin{equation*} \max_q \Big\{p\big[(2\theta\alpha - \pi)q - \alpha q^2\big]\Big\}, \end{equation*} which yields the optimal submission level \begin{equation*} q_d(\theta):= \theta-\frac{\pi}{2\alpha}. \end{equation*} We obtain that opting for the DP results in a trader of type $\theta$ enjoying the expected utility \begin{equation*} u_0(\theta;\pi) = \alpha p\left(\theta-\frac{\pi}{2\alpha}\right)^2 - \kappa. \end{equation*} We assume that $p\pi^2<4\alpha\kappa$ so as to keep the DP unattractive for small types. We assume that the dealer's costs/profits of unwinding a portfolio of size $q$ are $C(q)=\epsilon\,q+\beta q^2$ where $\beta>0$ and $\epsilon$ is non--negative. Observe that, since $u_0(\cdot;\pi)$ does not satisfy Assumption~\ref{ass:cost of access}, some restrictions must be imposed on the problem's parameters so as to still have Lemma~\ref{lm:BondedParticipation}. Namely, it must hold that \begin{equation}\label{eq:RestParamDP} \pi<2\sqrt{\frac{\alpha\kappa}{p}}. \end{equation} Condition~\eqref{eq:RestParamDP} imposes a hard upper bound on possible equilibrium DP prices. It should be noted that Assumption~\ref{ass:monotone} is not satisfied by $u_0(\cdot;\pi),$ which, together with the way in which we shall define the pricing feedback loop from the DM to the DP, implies that our equilibrium result does not apply ``as is'' to the current setting. \subsection{\large{The dealer market without a dark pool}} In the absence of a dark pool, the dealer's optimal choices of quantities are, for negative types \begin{equation*} l(\theta, 0)=\frac{\alpha}{\alpha+\beta}\big(2\theta+1\big)-\frac{\epsilon}{2(\alpha+\beta)} \end{equation*} and for positive types \begin{equation*} l(\theta, 1)=\frac{\alpha}{\alpha+\beta}\big(2\theta-1\big)-\frac{\epsilon}{2(\alpha+\beta)}, \end{equation*} where the boundary of $\Theta_0$ is given by \begin{equation*} \underline{\theta}_0=\frac{1}{2}\Big(\frac{\epsilon}{2\alpha}-1\Big)\quad\text{and}\quad \overline{\theta}_0=\frac{1}{2}\Big(\frac{\epsilon}{2\alpha}+1\Big). \end{equation*} In order to guarantee that $\Theta_0\subset[-1,1]$ the condition $\epsilon<2\alpha$ must be imposed on the corresponding parameters. From the relation $v'(\theta)=\Psi_1\big(q(\theta)\big)$ we have that the indirect--utility function is \begin{equation*} v(\theta)=\begin{cases} \frac{2\alpha^2}{\alpha+\beta}\theta^2 +\frac{\alpha}{\alpha+\beta}\big(2\alpha-\epsilon\big)\theta+c_1, & \theta\leq\underline{\theta}_0;\\ \frac{2\alpha^2}{\alpha+\beta}\theta^2 -\frac{\alpha}{\alpha+\beta}\big(2\alpha+\epsilon\big)\theta+c_2, & \theta\geq\overline{\theta}_0, \end{cases} \end{equation*} where \begin{equation*} c_1=\frac{2\alpha^2}{4(\alpha+\beta)}\Big(\frac{\epsilon}{2\alpha}+1\Big)^2 +\frac{\alpha(2\alpha+\epsilon)}{2(\alpha+\beta)}\Big(\frac{\epsilon}{2\alpha}+1\Big)\text{ and } c_2=\frac{2\alpha^2}{4(\alpha+\beta)}\Big(\frac{\epsilon}{2\alpha}-1\Big)^2 -\frac{\alpha(2\alpha+\epsilon)}{2(\alpha+\beta)}\Big(\frac{\epsilon}{2\alpha}-1\Big). \end{equation*} When it comes to the spread, observe that $q'\equiv \frac{2\alpha}{\alpha+\beta},$ $\psi_1\equiv 2\alpha$ and $\psi_2\equiv0,$ which yields \begin{equation*} [t(0_-), t(0_+)]=\frac{4\alpha^2}{\alpha+\beta}[\underline{\theta}_0,\overline{\theta}_0]. \end{equation*} Below we analyze how the spread changes with the introduction of the DP. \subsection{\large{The impact of a dark pool}} We first take an exogenous execution price $\pi$ and determine, for each $\theta\in\Theta,$ what is the quantity--price pair $\big(q_c(\theta;\pi), \tau_c(\theta;\pi)\big)$ that the dealer must offer so as to match a DP with execution price $\pi.$ Using the relation $q_c(\theta;\pi)=u_0'(\theta;\pi)$ we obtain \begin{align}\begin{split} q_c(\theta;\pi)=& \ 2\alpha p\left(\theta-\frac{\pi}{2\alpha}\right) \text{ and}\\ \tau_c(\theta;\pi)=& \ \kappa+4\alpha^2 p(\theta -\alpha p)\left(\theta-\frac{\pi}{2\alpha}\right)-\alpha p\left(\theta-\frac{\pi}{2\alpha}\right)^2. \end{split}\end{align} From the Envelope Theorem and the structure of $u(\theta,q)$ we have that the traders' indirect utility function satisfies \begin{equation}\label{eq:QualEnvel} \frac{v'(\theta)}{2\alpha}=l\big(\theta,\gamma(\theta)\big). \end{equation} In order to determine the spread in the presence of the DP we must determine $\underline{\theta}_{0, m}$ and $\overline{\theta}_{0, m}$ together with $\gamma\big(\underline{\theta}_{0, m}\big)$ and $\gamma\big(\overline{\theta}_{0, m}\big).$ For an arbitrary $\Gamma\in [0, 1]$ we have \begin{equation*} l(\theta,\Gamma)=\frac{\alpha}{\alpha+\beta}\big[2\theta+1-2\Gamma\big]-\frac{\epsilon}{2(\alpha+\beta)}. \end{equation*} Indexed by $\Gamma,$ the candidates for $\underline{\theta}_{0, m}$ are then given by \begin{equation*} \underline{\theta}_{0, m}(\Gamma)=\frac{1}{2}\Big(\frac{\epsilon}{2\alpha}+2\Gamma-1\Big). \end{equation*} Since it must hold that $\underline{\theta}_{0, m}(\Gamma)\leq 0$, then $\Gamma\leq 0.5(1-\epsilon/2\alpha).$ Integrating Expression~\eqref{eq:QualEnvel} we have that, on the interval $[\widetilde{\theta}_{m}(\Gamma), \underline{\theta}_{0, m}(\Gamma)],$ the traders' indirect utility is given by \begin{equation}\label{eq:IndUtGamma} v(\theta;\Gamma)=\frac{2\alpha^2}{\alpha+\beta}\theta^2+2\alpha\Big[\frac{\alpha}{\alpha+\beta}(1-2\Gamma)-\frac{\epsilon}{2(\alpha+\beta)}\Big]\theta+c_{1,m}, \end{equation} where $\widetilde{\theta}_{m}(\Gamma)$ is the first intersection to the left of $\underline{\theta}_{0, m}(\Gamma)$ of $v(\cdot; \Gamma)$ and $u_0(\cdot;\pi)$ and $c_{1,m}$ is determined by the equation \begin{equation*} v\big(\underline{\theta}_{0, m}(\Gamma);\Gamma\big)=0. \end{equation*} Unless the inequality $\Gamma\leq 0.5(1-\epsilon/2\alpha)$ is tight, in which case the types below $\widetilde{\theta}_{m}(\Gamma)$ are excluded, Proposition~\ref{prop:NoJumps2} implies that $\Gamma$ must be chosen so as to satisfy the smooth--pasting condition $u_0'\big(\widetilde{\theta}_{m}(\Gamma);\pi\big)=v'\big(\widetilde{\theta}_{m}(\Gamma);\pi\big),$ which is equivalent to \begin{equation*} \widetilde{\theta}_{m}(\Gamma)=\Big[\frac{2\alpha}{\alpha+\beta}-p\Big]^{-1} \Big[\frac{\epsilon}{2(\alpha+\beta)}-\frac{\alpha}{\alpha+\beta}(1-2\Gamma)-\frac{p\pi}{2\alpha}\Big]. \end{equation*} Observe that, besides the requirement $\Gamma\geq 0.5(1-\epsilon/2\alpha),$ the strategy to determine $\overline{\theta}_{0, m}$ is exactly the same as for $\underline{\theta}_{0, m}.$ Summarizing, from Eq.~\eqref{eq:IndUtGamma} we observe that, if $\Gamma_-$ and $\Gamma_+$ correspond to the optimal choices for the negative and positive endpoints of $\Theta_{0}(\pi),$ then \begin{equation*} q'\big(\underline{\theta}_{0, m}(\Gamma_-)\big)=\frac{1}{2\alpha}v''\big(\underline{\theta}_{0, m}(\Gamma_-);\Gamma_-\big)= \frac{1}{2\alpha}v''\big(\overline{\theta}_{0, m}(\Gamma_+);\Gamma_+\big)=q'\big(\overline{\theta}_{0, m}(\Gamma_+)\big)=\frac{2\alpha}{\alpha+\beta}. \end{equation*} The spread is then \begin{equation*} [t_m(0_-), t_m(0_+)]=\frac{4\alpha^2}{\alpha+\beta}[\underline{\theta}_{0, m}(\Gamma_-),\overline{\theta}_{0, m}(\Gamma_+)]\subset \frac{4\alpha^2}{\alpha+\beta}[\underline{\theta}_0,\overline{\theta}_0], \end{equation*} i.e. the presence of a dark pool strictly narrows the spread in the dealer's market. \subsection{\large{An equilibrium price}} A standard (but not unique) way in which dark--pool prices are generated is by computing the average of some publicly available best--bid and best--ask prices. In the case of the US, this is usually the mid--quote of the National Best Bid and Offer (NBBO). Borrowing from this idea we define the price--iteration in the DP as follows: \begin{equation*} \pi_{i+1}=\frac{1}{2}\big(t_i(0_+)-t_i(0_-)\big),\quad i\in\mathbb{N}, \end{equation*} where $\{t_i(0_-), t_i(0_+)\}$ are the best bid and ask prices in the DM in the presence of a DP with execution price $\pi_i.$ We know from the previous section that the sequence $\{\pi_i, i\in\mathbb{N}\}\subset ((4\alpha^2)/(\alpha+\beta))[\underline{\theta}_0,\overline{\theta}_0];$ hence, by the Bolzano--Weierstrass Theorem it has at least one convergent subsequence. The limit of each of the said subsequences will be an equilibrium price. The (possible) non--uniqueness of these prices is due to the fact that by virtue of its definition, the sequence of dark--pool prices need not be monotonic. The problem of non-uniqueness of equilibria in models of competing DMs and CNs has been observed before. We refer to \cite{DDH} for a detailed discussion. \section{\large{Conclusions}}\label{sec:Conclusions} We have presented an adverse--selection model to study the structure of the limit--order book of a dealer who provides liquidity to traders of unknown preferences. Furthermore, we have established a link between the traders' indirect--utility function and the bid--ask spread in the DM. Making use of the aforementioned link, we have studied how the presence of a type--dependent outside option impacts the spread of the DM, as well as the set of trader types who participate in the DM and their welfare. In particular, we have shown, in a portfolio--liquidation setting, that the presence of a dark pool results in a shrinkage of the spread in the DM. Finally, we have established that, under certain conditions, the feedback loop introduced by the impact that the spread has on the structure of the outside option leads to an equilibrium price. \end{document} \end{document}

\begin{document} \begin{CJK*}{GBK}{song} \CJKindent \title{Differential inequality of the second derivative that leads to normality} \author{Qiaoyu Chen, Shahar Nevo, Xuecheng Pang } \date{} \maketitle \vskip 3mm \SHOUYEJIAOZHU{2010 Mathematical subject classification 30A10,~30D45.} \SHOUYEJIAOZHU{ keywords and phrases, Normal family, Differential inequality.} \centerline{{\CJKfamily{hei} Abstract}} \noindent Let~$\mathcal{F}$~ be a family of functions meromorphic in a domain ~D.~ If~$\{\df{|f^{''}|}{1+|f|^{3}}:f\in \mathcal{F} \}$~is locally uniformly bounded away from zero,~then~$\mathcal{F}$~ is normal. \noindent I.~Introduction. \\ Recently,~progress was occurred concerning the study of the connection between differential inequalities and normality. ~A natural point of departure for this subject is the well-known theorem due to F.Marty.\\ \CJKfamily{hei}{Marty's Theorem }~[8, P.75]\quad A family ~$\mathcal{F}$~of functions meromorphic in a domain ~D~ is normal~ if and only if~~$\{f^{\#}:~f \in \mathcal{F}\}$~is locally uniformly bounded in ~D~.\\ Following Marty's Theorem,~L.~Royden proved the followiing generalization.\\ \CJKfamily{hei}{Theorem R}[7]\quad Let ~$\mathcal{F}$~ be a family of functions meromorphic in a domain ~D, ~with the property that for each compact set ~$K\subset D$~, ~there is a positive increasing function ~$h_K$~, ~such that ~$|f'(z)|\leq h_K(|f(z)|)$~for all ~$f\in \mathcal{F} $~ ~and ~$z\in K$.~Then~$\mathcal{F}$~is normal in~D.\\ This result was significantly extended further in various directions, ~see ~$[3],[9]~and~[11]$. ~S.Y.Li and H.Xie established a different kind of generalization ~of Marty's Theorem that involves higher derivatives. \\ \CJKfamily{hei}{Theorem LX} ~$[4]$\quad Let~$\mathcal{F}$~ be a family of functions meromorphic in a domain ~D,~such that each $f\in \mathcal{F} $~has zeros only of multiplicities $\geq k~,k\in N$.~Then~$\mathcal{F}$~is normal in D if and only if the family $$\left\{\df {|f^{(k)}(z)|}{1+|f(z)|^{k+1}}:f\in \mathcal{F}\right\}$$ is locally uniformly bounded in D.\\ In~$[6]$,~the second and the third authors gave a counterexample to the validity of Theorem LX,~without the condition on the multiplicities of zeros for ~the case ~$k=2$.\\ Concerning differential inequalities ~with the reversed sign of the inequality,~J.~Grahl,~and the ~second author proved the ~following result,~that may be ~considered as a counterpart to ~Marty's Theorem.\\ \CJKfamily{hei}{Theorem GN }\quad $[1]$\quad Let ~$\mathcal{F}$~ be a family of functions meromorphic in D,~and~$c>0$~.~If ~$f^{\#}(z)>c$~ for every~$f\in \mathcal{F} $~and~$z\in D$,~then~$\mathcal{F}$~is normal in ~D.\\ N.Steinmetz ~$[10]$,~gave a shorter proof of Theorem GN,~using the Schwarzian derivative and some Well-known facts on linear differential equations.\\ Then in~$[5]$,~X.J.Liu together ~with the second and third ~authors generalized Theorem GN ~and proved the following result.\\ \CJKfamily{hei}{Theorem LNP}\quad Let~$1\leq\alpha< \infty$~ and~$c>0$.~Let ~~$\mathcal{F}$~ be the family of all meroforphic functions ~$f $ ~in~D, ~such that $${\df {|f'(z)|}{1+|f(z)|^{\alpha}}>C}$$ ~for every ~$ z\in D$.\\~ Then the following hold:\\ (1) If ~$\alpha>1$,~then~$\mathcal{F}$~is normal ~in ~D.~\\ (2) If~$\alpha=1$,~then~$\mathcal{F}$~is quasi-normal in ~D~ but not necessarily normal. \\ Observe that (2) of the theorem is a differential inequalities that distinguish between quasi-normality to normality. \\ In this paper, we continue to ~study differential inequality ~with the reversed sign ~$(``\geq ")$~ ~and prove the following theorem.\\ \CJKfamily{hei}{Theorem 1.}\quad Let ~D~be a domain in ~$\mathbb{C}$~and let ~$c>0$.~Then the family ~~$\mathcal{F}$~ of all functions ~f~meromorphic in ~D~,~such that ~$$\df {|f^{''}(z)|}{1+|f(z)|^{3}}> C$$~for ~every ~$z\in D$~is normal. \\ Observe that the above differential ~inequality is the reversed inequality to that of Theorem LX in the case ~$k=2$.~\\ Let us set some notation. \\ For ~$z_0 \in C $~and ~$r>0$.~$\Delta(z_0, r) = \{z: |z - z_0| < r\}$,~$\overline{\Delta}(z_0, r) = \{z: |z - z_0| \leq r\}$. We write~$f_n(z)\overset\chi \Rightarrow f(z)$~on ~$D$~to indicate that the sequence $\{f_n(z)\}$ converges to ~$f(z)$ in the spherical metric, uniformly on compact subsets of ~$D$, and $f_n(z)\Rightarrow f(z)$~on ~$D$ if the convergence is also in the Euclidean metric.\\ II\quad Proof of Theorem 1.\\ Since ~$|f''| > c$~ for every~$f \in\mathcal{F},$~it ~follows that~$\{f'':f \in\mathcal{F}\}$~is normal in ~D~.~Let ~$\{f_n\}^{\infty}_{n=1}$~be a sequence of functions from $\mathcal{F}$.~Without loss of ~generality,~we can assume that ~$f^{''}_n(z)\overset\chi \Rightarrow H$ in D~.~Let us separate into two cases.\\ Case 1.\quad$f_n,n\geq 1$~are holomorphic ~functions~in ~D~.\\ Case 1.1\quad H~is holomorphic function~in~D~.\\ Since normality is a local property. ~It is enough to prove that ~$\{f_n\}$~is normal at each point of ~D~. Let ~$z_0\in D$~without loss of generality, ~we can assume that~$z_0=0$~.By the assumption on ~H~, ~there exist some~$r >0,~M >C$,~such that~$|f^{''}_n(z)|\leq M$~ ~for every~$z\in \Delta(0,r)$~if ~$n$~ is large ~enough.~We then get for large enough ~$n$~ and~$z\in \Delta(0,r)$~that ~$1+|f_n(z)|^3 \leq \frac {2 M}{C}$ ~and we deduce ~that ~$\{f_n\}^{\infty}_{n=1}$~ is normal at ~$z=0,$~ ~as required.\\ Case 1.2 \quad$H \equiv\infty$~in~D.\\ Again,~let $z_0 \in D$ and assume that $z_0 =0.$~ Let $r >0$ be such that $\overline{\Delta}(0, r)\subset D$. Without loss of generality,~we can assume that ~$|f^{''}_n(z)|> 1$~ ~for every~$z\in\Delta(0,r),~n\in\mathbb{N}$.~Then~$\log |f_n^{''}|$~is a positive harmonic function in $\Delta(0,r)$.\\ From Harnack's inequality we then get that\\ (1)~$$|f_n^{''}(z)|\leq|f_n^{''}(0)|^{\frac{1+|z|}{1-|z|}} $$~for every $z\in\Delta(0,r),~n\in\mathbb{N}.$\\ Let us fix some~$0<\rho<\displaystyle{\frac{r}{2}}$.~Then \\(2) $$\frac{r+\rho}{r-\rho}<3.$$\\ For every $n\geq 1,$ let ~$z_n\in\{z:|z|=\rho\}$~be such that $$ |f_n(z_n)|=\max\limits_{|z|\leq\rho}|f_n(z)|=M(\rho,f_n)$$ By Cauchy's Inequality ,~we get that\\ $$|f^{''}_n(0)|\leq \df{2}{\rho^2}M(\rho,f_n)=\df{2}{\rho^2}|f_n(z_n)|.$$ Hence,~by (1),~we get $$C\leq \df{|f^{''}_n(z_n)|}{1+|f_n(z_n)|^3}\leq \df{|f^{''}_n(z_n)|}{|f_n(z_n)|^3}\leq \df{|f^{''}_n(0)|^{\df{r+\rho}{r-\rho}}}{|f_n(z_n)|^3} \leq \left(\df{2}{\rho^2}\right)^{\df{r+\rho}{r-\rho}}\left|f_n(z_n)\right|^{\df{r+\rho}{r-\rho}-\displaystyle{3}},$$ Thus, by (2) $$M(\rho,f_n)=|f(z_n)|\leq \left(\df{1}{C}\left(\df{2}{\rho^2}\right)^{\df{r+\rho} {r-\rho}}\right)^{\df{1}{3-\df{r+\rho}{r-\rho}}},$$ which means that $\{f_n\}$ is locally uniformly bounded in $\Delta(0,\rho)$ and thus $\{f_n\}$ is normal at $z=0$.\\ Case 2 \quad $f_n$ are meromorphic functions with pole in $D$.\\ By Case 1 we have to prove normality only at point $z_0$,~where $H(z_0)=\infty$.~Such points exist if $H$ is a meromorphic function with poles in $D$ or if $H\equiv \infty$. So let $z_0$ be such that $H(z_0)\equiv \infty$.~Without loss of generality , we can assume that $z_0=0$. After moving to a subsequence, that without loss of generality will also be denoted by $\{f_n\}_1^{\infty}$,~we can assume that there is a sequence $\zeta_n\rightarrow 0$~such that ~$f_n(\zeta_n)=\infty$.~For if it was not the case,then for some $\delta >0$ ~and large enough ~$n$~,$f_n$~would be holomorphic in $\Delta(0,\delta)$,and then we would get the asserted normality by case (1).\\ Also we can assume the existence of \\ (3)\quad\quad a sequence~$\eta_n\rightarrow 0$~such that $f_n(\eta_n)=0$.\\ Indeed,~since ~$H(z_0)=\infty$~there exists some ~$\delta> 0$~such that for large enough ~$n$~$\min\limits_{z\in \Delta (0,\delta)}|f_n^{''}|>1$.\\ Combining it with ~$f_n\neq 0$~ in some neighbourhood of ~$z=0$~gives the normality at~$z=0$~by Gu's Criterion [2]. \\ We can also assume that ~$\{f_n^{'}\}$~ is not normal at~$z=0$.~Indeed, if~$\{f_n^{'}\}$~would be normal at ~$z=0$,~then by Marty's theorem there exist ~$r_1>0$~and ~$M>0$~such that for large enough ~$n$,~ $\df{|f_n^{''}(z)|}{1+|f_n^{'}(z)|^2}<M $ for ~$z\in \Delta(0,r_1)$.~ Since ~$H(0)=\infty $,there exists some ~$r_2\leq r_1$~such that for large enough ~$n,$~$|f_n^{''}(z)|\geq 2M$~for ~$z\in\Delta(0,r_2)$.\\ We thus have for large enough ~$n$~and ~$z\in \Delta(0,r_2)$,~ $1+|f_n^{'}(z)|^2>\df{|f_n^{''}(z)|}{M}\geq 2$~and thus ~$|f^{'}_n(z)|\geq 1$.We then get $$\df{|f^{'}_n(z)|^2}{|f^{''}_n(z)|}=\df{|f^{'}_n(z)|^2}{1+|f^{'}_n(z)|^2}\cdot \df{1+|f^{'}_n(z)|^2}{|f^{''}_n(z)|}\geq \df{1^2}{1+1^2}\cdot \df{1}{M}=\df{1}{2M}.$$ Hence We have for large enough ~$n$~and ~ $z\in \Delta(0,r_2)$\\ $(4)\quad\quad\displaystyle{\df{|f^{'}_n(z)|^2}{1+|f_n(z)|^3}=\df{|f^{'}_n(z)|^2}{|f^{''}_n(z)|}\cdot \df{|f^{''}_n(z)|}{1+|f_n(z)|^3}>\df{1}{2M}\cdot C}.$ \\ Now,for every ~$x\geq 0,\df{\sqrt{1+x^2}}{1+x}\geq \df{1}{\sqrt{2}}$,~and by taking square root of (4),~we get $$\df{|f^{'}_n(z)|}{1+|f_n(z)|^\frac{3}{2}}=\df{|f^{'}_n(z)|}{\sqrt{1+|f_n(z)|^3}}\cdot \df{\sqrt{1+|f_n(z)|^3}}{1+|f_n(z)|^\frac{3}{2}}>\sqrt{\df{C}{2M}}\cdot\df{1}{\sqrt{2}} .$$ \\ By (1) of Theorem LNP, ~with ~$\alpha=\frac{3}{2}>1$,~we deduce that~$\{f_n\}$ ~ is normal in $\Delta(0,r_2)$ and~we are done. \\ Thus we can assume that~$\{f'_n\}$ ~is not normal at ~$z=0$. \\ Similarly to (3) we can assume~that there is a sequence ~$s_n\to 0$~such that~$f^{'}_n(s_n)=0$.\\ We claim that we can assume that~$\{\df{f^{'}_n}{f^{''}_n}\}_{n=1}^{\infty}$~is not normal at ~$z=0$. \\ Otherwise,~after moving to a~subsequence that will also be denoted by~$\{\df{f^{'}_n}{f^{''}_n}\}_{n=1}^{\infty}$ we have~$\df{f^{'}_n}{f^{''}_n}\Rightarrow H_1$ in $\Delta(0,r)$ , for some~$r > 0$.~Since ~$f^{''}_n \neq 0$~and $\df{f^{'}_n}{f^{''}_n}(\zeta_n)=0$~ then ~$H_1$~must be holomorphic function in~$\Delta(0,r).$ Differentiation then gives\\ (5)\quad\quad $1-\df{f^{'}_n f^{''}_n }{(f^{''}_n)^2}\Rightarrow H^{'}_1$~in $\Delta(0,r).$\\ At ~$z=s_n$~ the left hand of~(5)~is equal to 1.~on the other hand ~in some small neighbourhood of~$z=\zeta_n$,~We have $f_n(z)=\frac{A_n}{z-z_n}+\hat{f}_n(z),~$ where $A_n\neq 0$ is a constant,~and $\hat{f}_n(z)$ is analytic. ~Here we used that according to the ~assumption of Theorem ~$1$~,~all ~poles of ~$f_n$~must be simple. \\ Hence we have ~$f^{'}_n(z)=\df{-A_n}{(z-\zeta_n)^2}+\hat{f}_n^{'}(z), f^{''}_n(z)=\df{2A_n}{(z-\zeta_n)^3}+\hat{f}_n^{''}(z), f_n^{(3)}(z)=\df{-6A_n}{(z-z_n)^4}+\hat{f}_n^{(3)}(z)$.~ Then the left hand of (5) get at~$z=\zeta_n.$~The value ~$1-\df{6}{4}=-\df{1}{2}\neq 1$, a contradiction .\\ Claim ~ there exist ~$r>0$~and ~$k>0$~such that for large enough ~$n$, $|\df{f_n}{f_n^{''}}(z)|,~~\left|\df{f_n^2}{f_n^{''}}(z)\right|\leq K$~for ~$z\in \Delta(0,r)$.\\ Proof of Claim \quad Since ~$H(0)=\infty $,there exist ~$r>0$~and ~$M>0$~such that $\overline{\Delta}(0,r)\subset D$~and such that for large enough $n$, $|f_n^{''}(z)|>M$ for ~$z\in \Delta (0,r)$.\\ Now ,if $|f_n(z)|\leq |f_n^{''}(z)|^{\frac{1}{3}}$ then \\ (6) \quad\quad $|\df{f_n}{f_n^{''}}(z)|\leq \df{|f_n^{''}(z)|^{\frac{1}{3}}}{|f_n^{''}(z)|}\leq \df{1}{M^{\frac{2}{3}}}$\\ and\\ (7)\quad\quad$|\df{f_n^2}{f_n^{''}}(z)|\leq \df{|f_n^{''}(z)|^{\frac{2}{3}}}{|f_n^{''}(z)|}\leq \frac{1}{M^{\frac{1}{3}}}$.\\ If on the other hand $|f_n(z)|\geq |f_n^{''}(z)|^{\frac{1}{3}}$,~then since $\df{x}{1+x^3}\leq \df{2^{\frac{2}{3}}}{3}$~for ~$x\geq 0$,~we get \\ (8)\quad\quad $|\df{f_n}{f_n^{''}}(z)|=\df{1+|f_n(z)|^3}{|f_n^{''}(z)|}\cdot \df{|f_n(z)|}{1+|f_n(z)|^3}\leq \df{1}{C}\cdot \df{2^{\frac{2}{3}}}{3}$.\\ Also We have $\df{x^2}{1+x^3}\leq \frac{2^{\frac{2}{3}}}{3}$~for ~$x\geq 0$ and thus\\ (9)\quad\quad$|\df{f_n^2}{f_n^{''}}(z)|=\df{1+|f_n(z)|^3}{|f_n^{''}(z)|}\cdot \df{|f_n^2(z)|}{1+|f_n(z)|^3}\leq \df{1}{C}\cdot \df{2^{\frac{2}{3}}}{3}$.\\ The claim then follows by taking $k=\max{\{\frac{1}{M^{\frac{2}{3}}},\frac{1}{M^{\frac{1}{3}}},\frac{1}{C}\cdot\frac{2^{\frac{2}{3}}}{3}\}}$ and consider (6),(7),(8)and (9).\\ From the claim we deduce that $\{\df{f_n}{f_n^{''}}\}_{=1}^\infty$ and $\{\df{f_n^2}{f_n^{''}}\}_{=1}^\infty$ are normal in $\Delta(0,r)$,~so after moving to a subsequence,~that also will be denote by $\{f_n\}_{n=1}^\infty $,~we get that \\ (10)\quad\quad$\df{f_n}{f_n^{''}}\rightarrow H_1$ in $\Delta(0,r)$\\ and\\ (11)\quad\quad$\df{f_n^2}{f_n^{''}}\rightarrow H_2$ in $\Delta(0,r)$\\ From the claim it follows that $H_1$ and $H_2$ are holomorphic in $\Delta(0,r)$.\\ Differentiating (10) and (11) gives respectively \\ (12)\quad\quad$\df{f_n^{'}}{f_n^{''}}-\df{f_n^{(3)}}{f_n^{''}}\cdot f_n \Rightarrow H_1^{'}$ in $\Delta(0,r)$\\ and \\ (13)\quad\quad$2f_n\cdot\df{f_n^{'}}{f_n^{''}}-f_n^2\cdot\df{f_n^{(3)}}{{f^{''}_n}^2}\Rightarrow H_2^{'}$ in $\Delta(0,r)$.\\ Since $\{f_n''\}_{n=1}^\infty$ is normal, there exists some $k_1>0$ such that $ \frac{|f_n^{(3)}(z)|}{1+|f_n''(z)|}\leq k_1 $ for every $n\geq1$ and for every $z\in\Delta(0,r)$. Since in addition for large enough $n$, $|f_n''(z)|>M$, then \begin{eqnarray*} \frac{|f_n^{(3)}(z)|}{|f_n^{''}(z)|^2}&=&\frac{|f_n^{(3)}(z)|}{1+|f_n^{''}(z)|^2}\frac{1+|f_n^{''}(z)|^2}{|f_n^{''}(z)|^2}\\ &\leq&k_1(1+\frac{1}{M^2}):=k_2. \end{eqnarray*} \\ Thus~\\ (14)\quad\quad$ \frac{|f_n^{''}(z)|^2}{|f_n^{(3)}(z)|}\geq\frac{1}{k^2} $~for large enough n. \\ Now since we assume that $\{\frac{f_n'}{f_n''}\}$ is not normal at $z=0$, then after moving to a subsequence, that also will be denoted by $\{f_n\}_{n=1}^\infty$, we get that there exists a sequence of points $t_n\rightarrow0$, such that $$ \frac{f_n'}{f_n''}(t_n):=M_n\rightarrow\infty, \quad M_n\in \mathbb{C}. $$ Substituting $z=t_ n$ in $(12)$ gives\\ (15) $$ M_n-\frac{f_n^{(3)}\cdot f_n}{f_n''^2}(t_n):=\varepsilon_n\rightarrow H_1'(0). $$ Hence $$ f_n(t_n)=(M_n-\varepsilon_n)\frac{f_n^{''2}}{f_n^{(3)}}(t_n) $$ From $(15)$ we get,~ by substituting $z=t_n$ in $(13)$ $$ 2(M_n-\varepsilon_n)\frac{f_n^{''2}}{f_n^{(3)}}(t_n)M_n-(M_n-\varepsilon_n)^2\left(\frac{f_n^{''2}}{f_n^{(3)}}(t_n)\right)^2\frac{f_n^{(3)}}{f_n^{''2}}(t_n):=\delta_n\rightarrow H_2'(0). $$ \\ From this we get after simplifying $$ (M_n^2-\varepsilon_n^2)\frac{f_n^{''2}}{f_n^{(3)}}(t_n)=\delta_n. $$ But by (14) the left hand above tends to $\infty$~ as $n\rightarrow\infty$, while the right hand is bounded, a contradiction.\\ This completes the proof of Theorem 1. QIAOYU CHEN, DEPARTMENT OF MATHEMATICS, EAST CHINA NORMAL UNIVERSITY,SHANG HAI 200241,P.R.CHINA\\ E-mail address: [email protected] SHAHAR NEVO, DEPARTMENT OF MATHEMATICS, BAR-ILAN UNIVERSITY, 52900 RAMAT-GAN, ISRAEL \\ E-mail address: [email protected] XUECHENG PANG, DEPARTMENT OF MATHEMATICS, EAST CHINA NORMAL UNIVERSITY,SHANG HAI 200241,P.R.CHINA\\ E-mail address: [email protected] \end{CJK*} \end{document}

\begin{document} \title{On Change of Variable Formulas for non-anticipative functionals } \author{M. Mania$^{1)}$ and R. Tevzadze$^{2)}$} \date{~} \maketitle \begin{center} $^{1)}$ A. Razmadze Mathematical Institute of Tbilisi State University, 6 Tamarashvili Str., Tbilisi 0177; and Georgian-American University, 8 Aleksidze Str., Tbilisi 0193, Georgia, \newline(e-mail: [email protected]) \\ $^{2)}$ Georgian-American University, 8 Aleksidze Str., Tbilisi 0193, Georgia, Georgian Technical Univercity, 77 Kostava str., 0175, Institute of Cybernetics, 5 Euli str., 0186, Tbilisi, Georgia \newline(e-mail: [email protected]) \end{center} \begin{abstract} {\bf Abstract.} For non-anticipative functionals, differentiable in Chitashvili's sense, the It\^o formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established. \end{abstract} \noindent {\it 2010 Mathematics Subject Classification. 90A09, 60H30, 90C39} \ \noindent {\it Keywords}: The It\^o formula, semimartingales, non-anticipative functionals, functional derivatives \section{Introduction} The classical It\^o \cite{ito} formula shows that for a sufficiently smooth function\\ $(f(t,x), t\ge0, x\in R)$ the transformed process $f(t,X_t)$ is a semimartingale for any semimartingale $X$ and provides a decomposition of the process $f(t,X_t)$ as a sum of stochastic integral relative to $X$ and a process of finite variation. This formula is applicable to functions of the current value of semimartingales, but in many applications, such as statistics of random processes, stochastic optimal control or mathematical finance, uncertainty affects through the whole history of the process and it is necessary to consider functionals of entire path of a semimartingale. In 2009 Dupire \cite{Dupire} proposed a method to extend the It\^o formula for non-anticipative functionals using naturally defined pathwise time and space derivatives. The space derivative measures the sensitivity of a functional $f:D([0,T], R)\to R$ to a variation in the endpoint of a path $\omega\in D([0,T], R)$ and is defined as a limit $$ \partial_\omega f(t,\omega)=\lim_{h\to 0}\frac{f(t,\omega+hI_{[t,T]})-f(t,\omega)}{h}, $$ if this limit exists, where $D([0,T])$ is the space of RCLL ( right continuous with left limits) functions. Similarly is defined the second order space derivative $\partial_{\omega\omega}f:= \partial_{\omega}(f_{\omega}).$ The definition of the time derivative is based on the flat extension of a path $\omega$ up to time $t+h$ and is defined as a limit $$ \partial_t f(t,\omega)=\lim_{h\to 0+}\frac{f(t+h,\omega^t)-f(t,\omega)}{h}, $$ whenever this limit exists, where $\omega^t=\omega(.\wedge t)$ is the path of $\omega$ stopped at time $t$. If a continuous non-anticipative functional $f$ is from $C^{1,2}$ , i.e., if $\partial_t f, \partial_\omega f$, $\partial_{\omega\omega}f$ exist and are continuous with respect to the metric $d_\infty$ (defined in section 2) and $X$ is a continuous semimartingale, Dupire \cite{Dupire} proved that the process $f(t,X)$ is also a semimartingale and $$ f(t, X)=f(0, X)+\int_0^t\partial_t f(s,X)ds+\int_0^t\partial_\omega f(s,X)dX_s $$ \begin{equation}\label{itoc} +\frac{1}{2}\int_0^t\partial_{\omega\omega}f(s, X)d\langle X\rangle_s. \end{equation} For the special case of $f(t,X_t)$ these derivatives coincide with the usual space and time derivatives and the above formula reduces to the standard It\^o formula. Erlier related works are the works by Ahn \cite{ahn} and Tevzadze \cite{T2}, where It\^o's formula was derived in very particular cases of functionals that assume the knowledge of the whole path without path dependent dynamics. Further works extending this theory and corresponding references one can see in \cite{CF1}, \cite{CF2}, \cite{LScS},\cite{O}. Motivated by applications in stochastic optimal control, before Dupire's work, Chitashvili (1983) defined differentiability of non-anticipative functionals in a different way and proved the corresponding It\^o formula for continuous semimartingales. His definition is based on "hypothetical" change of variable formula for continuous functions of finite variation. We formulate Chitashvili's definition of differentiability and present his change of variable formula in a simplified form and for one-dimensional case. Let $C_{[0,T]}$ be the space of continuous functions on $[0,T]$ equipped with the uniform norm. Let $f(t,\omega)$ be non-anticipative continuous mapping of $C_{[0,T]}$ into $C_{[0,T]}$ and denote by ${\cal V}_{[0,T]}$ the space of functions of finite variation on $[0,T]$. A continuous non-anticipative functional $f$ is differentiable if there exist continuous functionals $f^0$ and $f^1$ such that for all $\omega\in C_{[0,T]}\cap {\cal V}_{[0,T]}$ \begin{equation}\label{chd} f(t,\omega)=f(0,\omega)+\int_0^tf^0(s,\omega)ds+\int_0^tf^1(s,\omega)d\omega_s. \end{equation} A functional $f$ is two times differentiable if $f^1$ is differentiable, i.e., if there exist continuous functionals $f^{0,1}$ and $f^{1,1}$ satisfying \begin{equation}\label{chd2} f^1(t,\omega)=f^1(0,\omega)+\int_0^tf^{1,0}(s,\omega)ds+\int_0^tf^{1,1}(s,\omega)d\omega_s. \end{equation} for all $\omega\in C_{[0,T]}\cap {\cal V}_{[0,T]}$. Here functionals $f^0, f^1$ and $f^{1,1}$ play the role of time, space and the second order space derivatives respectively. It was proved by Chitashvili \cite{Ch} that if the functional $f$ is two times differentiable then the process $f(t,X)$ is a semimartingale for any continuous semimartingale $X$ and is represented as $$ f(t, X)=f(0, X)+\int_0^tf^0(s,X)ds+\int_0^tf^1(s,X)dX_s $$ \begin{equation}\label{itoc} +\frac{1}{2}\int_0^tf^{1,1}(s, X)d\langle X\rangle_s. \end{equation} The idea of the proof of change of variable formula (\ref{itoc}) for semimartingales is to use the change of variable formula for functions of finite variations, first for the function $f$ and then for its derivative $f^1$, before approximating a continuous semimartingale $X$ by processes of finite variation. In the paper Ren et al \cite{RTZ} a wider class of $C^{1,2}$ functionals was proposed, which is based on the Ito formula itself. We formulate this definition in equivalent form and in one-dimensional case. The function $f$ belongs to $C^{1,2}_{RTZ}$, if $f$ is a continuous non-anticipative functional on $[0,T]\times C_{[0,T]}$ and there exist continuous non-anticipative functionals $\alpha, z, \gamma$, such that \begin{equation}\label{itoc2} f(t, X)=f(0, X)+\int_0^t\alpha(s,X)ds+\int_0^tz(s,X)dX_s +\frac{1}{2}\int_0^t\gamma(s, X)d\langle X\rangle_s \end{equation} for any continuous semimartingale $X$. The functionals $\alpha, z$ and $\gamma$ also play the role of time, first and second order space derivatives respectively. Since any process of finite variation is a semimartingale and any deterministic semimartingale is a function of finite variation, it follows from $f\in C^{1,2}_{RTZ}$ that $f$ is differentiable in the Chitashvili sense and \begin{equation}\label{ChT} \alpha=f^0,\;\;\;z=f^1. \end{equation} Becides, any $C^{1,2}$ process in the Dupire or Chitashvili sense is in $C^{1,2}_{RTZ}$, which is a consequence of the functional It\^o formula proved in \cite{Dupire} and \cite{Ch} respectively. Although, the definition of the class $C^{1,2}_{RTZ}$ does not require that $\gamma$ be (in some sense) the derivative of $z$, but if $f\in C^{1,2}$ in the Chitashvili sense, then beside equality (\ref{ChT}) we also have that $\gamma=f^{1,1}$ (i.e., $\gamma=z^1$). Our goal is to extend the formula (\ref{itoc}) for RCLL (or cadlag in French terminology) semimartingales and to establish how Dupire's, Chitashvili's and other derivatives are related. Since the bumped path used in the definition of Dupire's vertical derivative is not continuous even if $\omega$ is continuous, to compare derivatives defined by (\ref{chd}) with Dupire's derivatives, one should extend Chitashvili's definition to RCLL processes, or to modify Dupire's derivative in such a way that perturbation of continuous paths remain continuous. The direct extension of Chitashvili's definition of differentiability for RCLL functions is following: A continuous functional $f$ is differentiable, if there exist continuous functionals $f^0$ and $f^1$ (continuous with respect to the metric $d_\infty$ defined by (\ref{rho})) such that $ f(\cdot,\omega)\in {\cal V}_{[0,T]}$ for all $\omega\in {\cal V}_{[0,T]}$ and \begin{equation}\label{xvii} f(t,\omega)=f(0,\omega)+\int_0^tf^0(s,\omega)ds+\int_0^tf^1(s-,\omega)d\omega_s \end{equation} $$ +\sum_{s\le t}\big[f(s,\omega)-f(s-,\omega)-f^1(s-,\omega)\Delta\omega_s\big], $$ for all $(t,\omega)\in [0,T]\times {\cal V}_{[0,T]}$. In order to compare Dupire's derivatives with Chitashvili's derivatives, we introduce another type of vertical derivative where, unlike to Dupire's derivative $\partial_\omega f$, the path deformation of continuous paths are also continuous. We say that a non-anticipative functional $f(t,\omega)$ is vertically differentiable and denote this differential by $D_\omega f(t,\omega)$, if the limit \begin{equation} D_\omega f(t,\omega):=\lim_{h\to0, h>0}\frac{f(t+h,\omega^{t}+\chi_{t,h})-f(t+h,\omega^{t})}{h}, \end{equation} exists for all $(t,\omega)\in [0,T]\times {D}_{[0,T]}$, where $$ \chi_{t,h}(s)=(s-t)1_{(t,t+h]}(s)+h1_{(t+h,T]}(s). $$ Let $f(t,\omega)$ be differentiable in the sense of (\ref{xvii}). Then, as proved in Proposition 1, \begin{equation} f^0(t,\omega)=\partial_t f(t,\omega)\;\;\;\;\text{and}\;\;\;\; f^1(t,\omega)=D_\omega f(t,\omega). \end{equation} for all $(t,\omega)\in [0,T]\times {D}_{[0,T]}$. Thus, $f^0$ coincides with Dupire's time derivative, but $f^1$ is equal to $D_\omega f$ which is different from Dupire's vertical derivative in general. The simplest counterexample is $f(t,\omega)=\omega_t-\omega_{t-}$. It is evident that in this case $\partial_\omega f=1$ and $D_\omega f=0$. In general, if $g(t,\omega):=f(t-,\omega)$ then $D_\omega g(t,\omega)=D_\omega f(t,\omega)$ and $\partial_\omega g(t,\omega)=0$ if corresponding derivatives of $f$ exist. However, under stronger conditions, e.g. if $f\in C^{1,1}$ in the Dupire sense, then $D_\omega f$ exists and $D_\omega f=f^1=\partial_\omega f.$ The paper is organized as follows: In section 2 we extend Citashvili's change of variable formula for RCLL semimartingales and give an application of this formula on the convergence of ordinary integrals to the stochastic integrals. In section 3 we establish relations between different type of derivatives for non-anticipative functionals. \section{The It\^o formula according to Chitashvili for cadlag semimartingales} Let $\Omega:= D([0,T], R)$ be the set of c\`{a}dl\`{a}g paths. Denote by $\omega$ the elements of $\Omega$, by $\omega_t$ the value of $\omega$ at time $t$ and let $\omega^t=\omega(\cdot\wedge t)$ be the path of $\omega$ stopped at $t$. Let $B$ be the canonical process defined by $B_t(\omega)=\omega_t$, $\mathbb{F}=(F_t,t\in[0,T])$ the corresponding filtration and let $\Lambda:= [0,T]\times\Omega$. The functional $f:[0,T]\times D[0,T]\to R$ is non-anticipative if $$ f(t,\omega)=f(t,\omega^t) $$ for all $\omega\in D[0,T]$, i.e., the process $f(t,\omega)$ depends only on the path of $\omega$ up to time $t$ and is $\mathbb{F}$- adapted. Following Dupire, we define semi-norms on $\Omega$ and a pseudo-metric on $\Lambda$ as follows: for any $(t, \omega), ( t', \omega') \in\Lambda$, \begin{eqnarray} \label{rho} \|\omega\|_{t}&:=& \sup_{0\le s\le t} | \omega_s|,\nonumber\\[-8pt]\\[-8pt] d_\infty\bigl((t, \omega),\bigl( t', \omega'\bigr) \bigr)&:=& \bigl|t-t'\bigr| + \sup_{0\le s\le T} \bigl|\omega_{t\wedge s} - \omega'_{t'\wedge s}\bigr|.\nonumber \end{eqnarray} Then $(\Omega, \|\cdot\|_{T})$ is a Banach space and $(\Lambda, d_\infty)$ is a complete pseudo-metric space. Let ${\cal V}={\cal V}[0,T]$ be the set of finite variation paths from $\Omega$. Note that, if $f\in C(\Lambda)$, then from $\Delta \omega_t=0$ follows $f(t,\omega)-f(t-,\omega)=0$, since $d_\infty((t_n,\omega),(t,\omega))\to 0$ when $t_n\uparrow t$. Hence $f(t,\omega)-f(t-,\omega)\neq 0$ means $\Delta \omega_t\neq 0$. Note that any functional $f:[0,T]\times\Omega\to R$ continuous with respect to $d_\infty$ is non-anticipative. In this paper we consider only $d_\infty$-continuous, and hence non-anticipative, functionals. {\bf {Definition 1.}} We say that a continuous functional $f\in C([0,T]\times \Omega)$ is differentiable , if there exist $f^0\in C([0,T]\times \Omega)$ and $f^1\in C([0,T]\times \Omega)$ such that for all $\omega\in {\cal V}$ the process $ f(t,\omega)$ is of finite variation and \begin{equation}\label{xv} f(t,\omega)=f(0,\omega)+\int_0^tf^0(s,\omega)ds+\int_0^tf^1(s-,\omega)d\omega_s \end{equation} $$ +\sum_{s\le t}\big[f(s,\omega)-f(s-,\omega)-f^1(s-,\omega)\Delta\omega_s\big], $$ for all $(t,\omega)\in [0,T]\times\cal V$. A functional $f$ is two times differentiable if $f^1$ is differentiable, i.e., if there exist $f^{0,1}\in C([0,T]\times \Omega)$ and $f^{1,1}\in C([0,T]\times \Omega)$ such that for all $(t,\omega)\in [0,T]\times\cal V$ \begin{equation}\label{two} f^1(t,\omega)=f^1(0,\omega)+\int_0^tf^{1,0}(s,\omega)ds+\int_0^tf^{1,1}(s-,\omega)d\omega_s + V^1(t,\omega), \end{equation} where $$ V^1(t,\omega)=\sum_{s\le t}\big(f^1(s,\omega)-f^1(s-,\omega)-f^{1, 1}(s-,\omega)\Delta\omega_s\big). $$ Now we give a generalization of Theorem 2 from Chitashvili \cite{Ch} for general cadlag (RCLL) semimartingales. \begin{thr} Let $f$ be two times differentiable in the sense of Definition 1 and assume that for some $K>0$ \begin{equation}\label{v} |f(t,\omega)-f(t-,\omega)-f^1(t-,\omega)\Delta\omega_t|\le K(\Delta\omega_t)^2,\;\; \forall\omega\in\cal V. \end{equation} Then for any semimartingale $X$ the process $f(t,X)$ is a semimartingale and $$ f(t, X)=f(0, X)+\int_0^tf^0(s,X)ds+\int_0^tf^1(s-,X)dX_s $$ \begin{equation}\label{ito} +\frac{1}{2}\int_0^tf^{1,1}(s, X)d\langle X^c\rangle_s+\sum_{s\le t}\big[f(s,X)-f(s-,X)-f^1(s-,X)\Delta X_s\big]. \end{equation} \end{thr} {\it Proof.} Let first assume that $X$ is a semimartingale with the decomposition \begin{equation}\label{dec0} X_t=A_t+M_t, t\in[0,T], \end{equation} where $M$ is a continuous local martingale and $A$ is a process of finite variation having only finite number of jumps, i.e., the jumps of $A$ are exhausted by graphs of finite number of stopping times $(\tau_i, 1\le i\le l, l<\infty)$. Let $X_t^n= A_t+M^n_t$ and \begin{equation} M^n_t= n\int_0^tM_s\exp(-n(\langle M\rangle_t-\langle M\rangle_s)d\langle M\rangle_s. \end{equation} It is proved in \cite{Ch} that \begin{equation}\label{mc} \sup_{s\le t}|M^n_s-M_t|\to 0, \;\;\;as\;\;n\to\infty,\;\;\; a.s. \end{equation} Since $X^n$ is of bounded variation, $f$ is differentiable and $\Delta X^n_t=\Delta A_t=\Delta X_t$, it follows from (\ref{xv}) that $$ f(t,X ^n)=f(0, X)+\int_0^tf^0(s,X^n)ds $$ $$ +\int_0^tf^1(s-,X^n)dX_s +\int_0^tf^1(s-,X^n)d(M^n_s-M_s) $$ \begin{equation}\label{itod} +\sum_{s\le t}\big(f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X_s\big). \end{equation} Since $X$ admits finite number of jumps, by continuity of $f$ and $f^1$, \begin{equation}\label{jumpb} \sum_{s\le t}\big(f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X_s\big)\to \end{equation} $$ \to\sum_{s\le t}\big(f(s,X)-f(s-,X)-f^1(s-,X)\Delta X_s\big) $$ The continuity of $f, f^0, f^1$ and relation (\ref{mc}) imly that \begin{equation}\label{fxn1a} f(t,X^n)\to f(t,X),\;\;\;as\;\;n\to\infty,\;\;\; a.s., \end{equation} \begin{equation}\label{fxn22a} \int_0^tf^0(s,X^n)ds\to\int_0^tf^0(s,X)ds\;\;\;as\;\;n\to\infty,\;\;\; a.s.. \end{equation} by the dominated convergence theorem and \begin{equation}\label{fxn4a} \int_0^tf^1(s-,X^n)dX_s\to\int_0^tf^1(s-,X)dX_s\;\;\;as\;\;n\to\infty,\;\;\; a.s.. \end{equation} by the dominated convergence theorem for stochastic integrals. Here we may use the dominated convergence theorem, since by continuity of $f^i ( i=0,1)$ the process $\sup_{n, s\le t}|f^i(s-, X^n)|$ is locally bounded (see Lemma A1). Let us show now that \begin{equation}\label{fx12aa} \int_0^tf^1(s-,X^n)d(M^n_s-M_s)\to\frac{1}{2}\int_0^tf^{1,1}(s,X)d\langle M\rangle_s. \end{equation} Integration by parts and (\ref{two}) give $$ \int_0^tf^1(s,X^n)d(M^n_s-M_s)=(M^n_t-M_t)f^1(t,X^n)- $$ $$ -\int_0^t(M^n_s-M_s)f^{1,0}(s,X^n)ds-\int_0^t(M^n_s-M_s)f^{1,1}(s-,X^n)dA_s $$ $$ -\int_0^t(M^n_s-M_s)f^{1,1}(s-,X^n)dX^n_s-\int_0^t(M^n_s-M^c_s)dV^1(s, X^n)= $$ \begin{equation}\label{i5} =I^1_t(n)+I^2_t(n)+I^3_t(n)+I^4_t(n) +I_t^5(n). \end{equation} $I^1_t(n)\to 0$ (as $n\to\infty$, a.s.) by continuity of $f^1$ and (\ref{mc}). $I^2_t(n)$ and $I_t^3(n)$ tend to zero (as $n\to\infty$, a.s.) by continuity of $f^{1,0}$ and $f^{1,1}$, relation (\ref{mc}) and by the dominated convergence theorem (using the same arguments as in (\ref{fxn22a})-(\ref{fxn4a})). Moreover, since $A$ admits finite number of jumps at $(\tau_i, 1\le i\le l)$ \begin{equation}\label{jump2} I_t(5)=\sum_{s\le t}(M^n_s-M_s)\big(f^1(s,X^n)-f^1(s-,X^n)-f^{1,1}(s-,X^n)\Delta A_s\big) \end{equation} $$ =\sum_{i\le l}(M^n_{\tau_i}-M_{{\tau_i}})\big(f^1(\tau_i,X^n)-f^1(\tau_i-,X^n)-f^{1,1}(\tau_i-,X^n)\Delta A_{\tau_i}\big) $$ $$ \le \sup_{s\le t}|M^n_s-M_s|\big(2l\sup_{n, s\le t}|f^1(s,X^n)|+\sup_{n, s\le t}|f^{1,1}(s,X^n)|\sum_{i\le l}|\Delta A_{\tau_i}|\big)\to 0, $$ as $n\to\infty$, since the continuity of $f^1, f^{1,1}$, relation (\ref{mc}) and Lemma A1 imply that $\sup_{n, s\le t}|f^1(s,X^n)|+\sup_{n, s\le t}|f^{1,1}(s,X^n)|<\infty$ (a.s.) Let us consider now the term $$ I_t^4(n)=\int_0^t(M_s-M^n_s)f^{1,1}(s,X^n)dM^n_s $$ Let $$ K^n_t=\int_0^t(M_s-M^n_s)dM^n_s. $$ Using the formula of integration by parts we have $$ K^n_t=-\frac{1}{2}(M_t^n)^2+M_t M_t^n-\int_0^tM_s^ndM_s $$ and it follows from (\ref{mc}), the dominated convergence theorem and equality $M_t^2=2\int_0^tM_sdM_s+\langle M\rangle_t$, that \begin{equation}\label{kn} sup_{s\le t}|K^n_s-\frac{1}{2}\langle M\rangle_s|\to 0, \;\;\;as\;\;n\to\infty,\;\;\; a.s. \end{equation} From definition of $M^n$, using the formula of integration by parts, it follows that $M^n$ admits representation $$ M^n_t=n\int_0^t(M_s-M^n_s)d\langle M\rangle_s. $$ Therefore $$ K^n_t=n\int_0^t(M_s-M^n_s)^2d\langle M\rangle_s. $$ This implies that $K^n$ is a sequence of increasing processes, which is stochastically bounded by (\ref{kn}) (i.e. satisfies the condition UT from (\cite{JMP}) and by theorem 6.2 of(\cite{JMP}) (it follows also from lemma 12 of \cite{CF1}) $$ \int_0^t(M_s-M^n_s)f^{1,1}(s,X^n)dM^n_s= $$ $$ =\int_0^tf^{1,1}(s,X^n)dK^n_s\to\frac{1}{2}\int_0^tf^{1,1}(s,X)d\langle M\rangle_s,\;\;\;n\to\infty, $$ which (together with (\ref{i5})) implies the convergence (\ref{fx12aa}). Therefore, the formula (\ref{ito}) for the process $X$ with decomposition (\ref{dec0}) follows by passage to the limit in (\ref{itod}) using relations (\ref{jumpb})-(\ref{fx12aa}). Note that in this cased the condition (\ref{v}) is not needed. Let consider now the general case. Any semimartingale $X$ admits a decomposition $X_t=A_t+M_t$, where $A$ is a process of finite variation and $M$ is a locally square integrable martingale (such decomposition is not unique, but the continuous martingale parts coincide for all such decompositions of $X$, which is sufficient for our goals) see \cite{J}. Let $M_t=M_t^c+M^d_t$, where $M^c$ and $M^d$ are continuous and purely discontinuous martingale parts of $M$ respectively. Let $A_t=A_t^c+A_t^d$ be the decomposition of $A$, where $A^c$ and $A^d$ are continuous and purely discontinuous processes of finite variations respectively. Note that $A^d$ is the sum of its jumps, whereas $M^d$ is the sum of compensated jumps of $M$. So, we shall use the decomposition \begin{equation}\label{dec1} X_t=A_t^c+A_t^d+M_t^c+M_t^d \end{equation} for $X$ and using localization arguments, without loss of generality, one can assume that $M^c$ and $M^d$ are square integrable martingales. Let $M^d_t(n)$ be the compensated sum of jumps of $M$ of amplitude greater than $1/n$, which is a martingale of finite variation and is expressed as a difference \begin{equation}\label{jump} M^d_t(n)=B^n_t-\widetilde{ B_t^n}, \end{equation} where $B^n_t=\sum_{s\le t}\Delta M_sI_{(|\Delta M_s|\ge 1/n)}$ and $\widetilde{B^n}$ is the dual predictable projection of $B^n$. It can be expressed also as compensated stochastic integral (see \cite{DM}) $$ M^d_t(n)=\int_0^tI_{(|\Delta M_s|>\frac{1}{n})}{}_{\overset{\bullet}C}dM_s, $$ where by $H{}_{\overset{\bullet}C}Y$ we denote the compensated stochastic integral. Since $$ M^d_t(n)-M_t^d=\int_0^tI_{(0<|\Delta M_s|\le\frac{1}{n})}{}_{\overset{\bullet}C}dM_s, $$ it follows from Doob's inequality and from \cite{DM} (theorem 33, Ch.VIII) that $$ E\sup_{s\le t}|M_s^d(n)-M_s^d|^2\le const E[M^d(n)-M^d]_t= const E[I_{(0<|\Delta M|\le\frac{1}{n})}{}_{\overset{\bullet}C}M] $$ $$ \le const E\int_0^tI_{(0<|\Delta M_s|\le\frac{1}{n})}d[M]_s\to 0, \;\;\;as\;\;n\to\infty $$ by dominated convergence theorem, since $E[M^d]_T<\infty$. Hence \begin{equation}\label{md} \sup_{s\le t}|M^d_s(n)-M^d_s|\to 0, \;\;\;as\;\;n\to\infty,\;\;\; a.s. \end{equation} for some subsequence, for which we preserve the same notation. Let $$ A_t^d(n)=\sum_{s\le t}I_{(|\Delta A_s|>\frac{1}{n})}\Delta A_s=\int_0^tI_{(|\Delta A_s|>\frac{1}{n})}dA_s. $$ Since $$ |A^d_t-A_t^d(n)|\le\int_0^tI_{(0<|\Delta A_s|\le\frac{1}{n})}|dA_s| $$ we have that \begin{equation}\label{ad} \sup_{s\le t}|A^d_s(n)-A^d_t|\to 0, \;\;\;as\;\;n\to\infty,\;\;\; a.s. \end{equation} Let $$ X^n_t= A^c_t+A_t^d(n)+M_t^d(n)+M_t^c. $$ Relations (\ref{md}) and (\ref{ad}) imply that \begin{equation}\label{x} \sup_{s\le t}|X_s(n)-X_s|\to 0, \;\;\;as\;\;n\to\infty,\;\;\; a.s., \end{equation} Thus, $X^n$ is a sum of continuous local martingale $M^c$ and a process of finite variation $A^c_t+A_t^d(n)+M_t^d(n)$ which admits only finite number of jumps for every $n\ge 1$. Therefore, as it is already proved, $$ f(t,X^n)=f(0,X^n)+\int_0^tf^0(s,X^n)ds+\int_0^tf^1(s-,X^n)dX_s $$ $$ +\int_0^tf^1(s-,X^n)d(M_s^n(d)-M_s^d)+\int_0^tf^1(s-,X^n)d(A_s^n(d)-A_s^d) $$ $$ +\frac{1}{2}\int_0^tf^{1,1}(s, X)d\langle X^c\rangle_s $$ \begin{equation}\label{fxnv} +\sum_{s\le t}\big(f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X^n_s\big). \end{equation} By continuity of $f, f^0$ and $f^1$ \begin{equation}\label{fxn1} f(t,X^n)\to f(t,X),\;\;\;as\;\;n\to\infty,\;\;\; a.s., \end{equation} \begin{equation}\label{fxn22} \int_0^tf^0(s,X^n)ds\to\int_0^tf^0(s,X)ds\;\;\;as\;\;n\to\infty,\;\;\; a.s.. \end{equation} by the dominated convergence theorem and \begin{equation}\label{fxn4} \int_0^tf^1(s-,X^n)dX_s\to\int_0^tf^1(s-,X)dX_s\;\;\;as\;\;n\to\infty,\;\;\; a.s.. \end{equation} by the dominated convergence theorem for stochastic integrals (using the same arguments as in (\ref{fxn22a})- (\ref{fxn4a})). By properties of compensated stochastic integrals $$ \int_0^tf^1(s-,X^n)d(M^d_s(n)-M^d_s)=\int_0^tf^1(s-,X^n)I_{(0<|\Delta M_s|\le\frac{1}{n})}{}_{\overset{\bullet}C}dM_s $$ and using theorem 33, Ch. VIII from \cite{DM} $$ E\big(\int_0^tf^1(s-,X^n)I_{(0<|\Delta M_s|\le\frac{1}{n})}{}_{\overset{\bullet}C}dM_s\big)^2 $$ \begin{equation}\label{fx} \le const E\int_0^t(f^1(s-,X^n))^2I_{(0<|\Delta M_s|\le\frac{1}{n})}d[M^d]_s\to 0\;\;\;as\;\;n\to\infty \end{equation} by dominated convergence theorem, since $\sup_{n, s\le t}(f^1(s,X^n))^2$ is locally bounded (by Lemma A1 from appendix), $I_{(0<|\Delta M_s|\le\frac{1}{n})}\to 0$ and $E[M^d]_T<\infty$. Similarly, $\int_0^tf^1(s-,X^n)d(A_s^n(d)-A_s^d)$ also tends to zero, since \begin{equation}\label{fxan} \int_0^tf^1(s-,X^n)d(A_s^n(d)-A_s^d)\le \int_0^t|f^1(s-,X^n)|I_{(0<|\Delta A_s|\le\frac{1}{n})}|dA_s|\to 0. \end{equation} From (\ref{jump}) $$ \Delta M^n_s(d)=\Delta M_sI_{(|\Delta M_s|\ge 1/n)} - \big( \Delta MI_{(|\Delta M|\ge 1/n)}\big)_s^p, $$ where $Y^p$ is the usual projection of $Y$. Here we used the fact that the jump of the dual projection of $B^n$ is the usual projection of the jump, i.e. $\Delta\widetilde{B^n_t}=(\Delta B^n)_t^p$. Therefore, using condition (\ref{v}) we have that $$ |(f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X^n_s|\le const. (\Delta X^n_s)^2 $$ $$ = const. \big(\Delta A_sI_{(|\Delta A_s|\ge 1/n)}+\Delta M_sI_{(|\Delta M_s|\ge 1/n)} - ( \Delta MI_{(|\Delta M|\ge 1/n)})_s^p\big)^2 $$ \begin{equation}\label{jump2} \le 3 const.\big( (\Delta A_s)^2+(\Delta M_s)^2+ E( (\Delta M_s)^2/F_{s-})\big). \end{equation} Since, it follows from (\ref{x}) and continuity of $f$ and $f^1$, that $$ f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X^n_s\to f(s,X)-f(s-,X)-f^1(s-,X)\Delta X_s $$ and $$ \sum_{s\le t}\big ( (\Delta A_s)^2+(\Delta M_s)^2+ E( (\Delta M_s)^2/F_{s-})\big ) < \infty, $$ the dominated convergence theorem implies that $$ \sum_{s\le t}\big(f(s,X^n)-f(s-,X^n)-f^1(s-,X^n)\Delta X^n_s\big) $$ \begin{equation}\label{jump3} \to\sum_{s\le t}\big(f(s,X)-f(s-,X)-f^1(s-,X)\Delta X_s\big), \;\;\;as\;\;\;n\to\infty. \end{equation} Therefore, passing to the limit in (\ref{fxnv}) it follows from (\ref{fxn1})-(\ref{jump3}) that (\ref{ito}) holds.\qed Now we give one application of the change of variable formula (\ref{ito}) to the convergence of stochastic integrals. If $g(t,x), t\ge0, x\in R)$ is a function of two variables admitting continuous partial derivatives $\partial g(t,x)/\partial t$, $\partial g(t,x)/\partial x$ and $V^n$ is a sequence of processes of finite variations converging to the Wiener process, then it was proved by Wong and Zakai \cite{WZ} that the sequence of ordinary integrals $\int_0^tg(s,V^n_s)dV^n_s$ converges to the Stratanovich stochastic integral. The following assertion generalizes this result for non-anticipative functionals $g(t,\omega)$. {\bf Corollary}. Assume that $f(t,\omega)$ is differentiable in the sense of Definition 1 and there is a continuous on $[0,T]\times D([0,T])$ functional $F(t,\omega)$ such that \begin{equation}\label{str0} F(t,\omega)=\int_0^tf(s-,\omega)d\omega_s \end{equation} For all $\omega\in{\cal V}_{[0,T]}$. Let $X$ be a cadlag semimartingale and let $(V^n,n\ge1)$ be a sequence of processes of finite variation converging to $X$ uniformly on $[0, T]$. Then \begin{equation}\label{str} \lim_{n\to\infty}\int_0^tf(s-, V^n)dV^n_s= \int_0^tf(s-,X)dX_s +\frac{1}{2}\int_0^tf^{1}(s, X)d\langle X^c\rangle_s. \end{equation} Proof: By continuity of $F$ and (\ref{str0}) \begin{equation}\label{str1} \lim_{n\to\infty}\int_0^tf(s-, V^n)dV^n_s=\lim_{n\to\infty}F(t,V^n)=F(t,X). \end{equation} It is evident that $$ F^1(t,\omega)=f(t,\omega),\;\; F^0(t,\omega)=0\;\;\text{and}\;\;\;F(t,\omega)-F(t-,\omega)-F^1(t-,\omega)\Delta\omega_t=0, $$ Thus, $F$ is two times differentiable in the sense of definition 1 and condition (\ref{v}) is automatically satisfied. Therefore, by the It\^o formula (\ref{ito}) $$ F(t,X)=\int_0^tf(s-,X)dX_s +\frac{1}{2}\int_0^tf^{1}(s, X)d\langle X^c\rangle_s, $$ which, together with (\ref{str1}) implies the convergence (\ref{str}). \section{The relations between various definitions of functional derivatives} Following Dupire \cite{Dupire} we define time and space derivatives, called also horizontal and vertical derivatives of the non-anticipative functionals. {\bf Definition 2}. A non-anticipative functional $f(t,\omega)$ is said to be horizontally differentiable at $(t,\omega)\in\Lambda$ if the limit \begin{equation} \label{hatpat} \partial_t f(t,\omega):= \lim_{h\to0, h>0} \frac {1}{h} \bigl[f (t+h,\omega^t )-f (t, \omega) \bigr],\qquad t<T, \end{equation} exists. If $ \partial_t f(t,\omega)$ exists for all $(t,\omega)\in\Lambda$, then the non-anticipating functional $\partial f_t$ is called the horizontal derivative of $f$. A non-anticipative functional $f(t,\omega)$ is vertically differentiable at $(t,\omega)\in\Lambda$ if \begin{equation} \label{hatpax} \partial_{\omega} f(t,\omega):= \lim_{h\to0}\frac {1}{h} \bigl[ f(t,\omega+ h 1_{[t,T]}) - f(t,\omega) \bigr], \end{equation} exists. If $f$ is vertically differentiable at all $(t,\omega)\in\Lambda$ then the map $\partial _\omega f :\Lambda\to R$ defines a non-anticipative map, called the vertical derivative of $f$. Similarly one can define \begin{equation} \partial_{\omega\omega}f:= \partial_{\omega }(\partial f_{\omega}),\qquad. \end{equation} Define $C^{1,k}([0, T )\times \Omega)$ as the set of functionals $f$, which are \begin{itemize} \item horizontally differentiable with $\partial_t f$ continuous at fixed times, \item $k$ times vertically differentiable with continuous $\partial_{\omega}^k f$. \end{itemize} The following assertion follows from the generalized It\^o formula for cadlag semimartingales proved in \cite{CF1} (see also \cite{LScS}). \begin{thr} Let $f\in C^{1,1}([0,T]\times \Omega)$. Then for all $(t,\omega)\in [0,T]\times \cal V$ \begin{eqnarray*} f(t,\omega)=f(0,\omega)+\int_0^t\partial_t f(s,\omega)ds+\int_0^t\partial_\omega f(s-,\omega)d\omega_s\\ +\sum_{s\le t}(f(s,\omega)-f(s-,\omega)-\partial_{\omega}f(s-,\omega)\Delta\omega_s) \end{eqnarray*} and $f(t,\omega)\in\cal V$ for all $\omega\in\cal V$. \end{thr} {\bf Corollary}. If $f\in C^{1,1}([0,T]\times \Omega)$, then $f$ is differentiable in the sense of Definition 1 and $$ \partial_tf=f^0,\;\;\;\;\partial _\omega f= f^1. $$ In order to compare Dupire's derivatives with Chitashvili's derivative (the derivative in the sense of Definition 1), we introduce another type of vertical derivative where, unlike to Dupire's derivative $\partial_\omega f$, the path deformation of continuous paths remain continuous. {\bf Definition 3}. We say that a non-anticipative functional $f(t,\omega)$ is vertically differentiable and denote this differential by $D_\omega f(t,\omega)$, if the limit \begin{equation} D_\omega f(t,\omega):=\lim_{h\to0, h>0}\frac{f(t+h,\omega^{t}+\chi_{t,h})-f(t+h,\omega^{t})}{h}, \end{equation} exists for all $(t,\omega)\in [0,T]\times \Omega$, where $$ \chi_{t,h}(s)=(s-t)1_{(t,t+h]}(s)+h1_{(t+h,T]}(s). $$ The second order derivative is defined similarly $$ D_{\omega\omega}f=D_\omega(D_\omega f). $$ Note that, if $f(t,\omega)=g(\omega_t)$ for any $\omega\in D[0,T]$, where $g=(g(x), x\in R)$ is a differentiable function, then $D_\omega f(t,\omega)$ (so as $\partial _\omega f(t,\omega)$) coincides with $g'(\omega_t)$. \begin{prop}\label{11} Let $f\in C([0,T]\times \Omega)$ be differentiable in the sense of Definition 1, i.e., there exist $f^0, f^1\in C([0,T]\times \Omega)$, such that for all $(t,\omega)\in [0,T]\times\cal V$ \begin{equation}\label{xv22} f(t,\omega)=f(0,\omega)+\int_0^tf^0(s,\omega)ds+\int_0^tf^1(s-,\omega)d\omega_s + V(t,\omega), \end{equation} where $$ V(t,\omega):=\sum_{s\le t}\big[f(s,\omega)-f(s-,\omega)-f^1(s-,\omega)\Delta\omega_s\big] $$ is of finite variation for all $\omega\in {\cal V}$. Then for all $(t,\omega)\in [0,T]\times D([0,T])$ \begin{equation} f^0(t,\omega)=\partial_t f(t,\omega)\;\;\;\;\text{and}\;\;\;\; f^1(t,\omega)=D_\omega f(t,\omega). \end{equation} \end{prop} {\it Proof.} Since $\omega^t$ is constant on $[t,T]$ and $f(t,\omega^t)=f(t,\omega)$, if $s\le t$, from (\ref{xv22}) we have that for any $ \omega\in{\cal V}$ \begin{equation}\label{xv23} f(t+h,\omega^t)=f(0,\omega)+\int_0^tf^0(s,\omega)ds+\int_t^{t+h}f^0(s,\omega^t)ds+ \end{equation} $$ + \int_0^tf^1(s-,\omega)d\omega_s + V(t,\omega) $$ and \begin{equation}\label{xv24} f(t+h,\omega^t+\chi_{t,h})=f(0,\omega)+\int_0^tf^0(s,\omega)ds+ \int_0^tf^1(s-,\omega)d\omega_s+ \end{equation} $$ +\int_t^{t+h}f^0(s,\omega^t+\chi_{t,h})ds+ \int_t^{t+h}f^1(s,\omega^t+\chi_{t,h})ds+ V(t,\omega). $$ Therefore $$ \partial_t f(t,\omega)=\lim_{h\to0}\frac{f(t+h,\omega^{t})-f(t,\omega)}{h}= $$ $$ =\lim_{h\to0}\frac{1}{h}\int_{t}^{t+h}f^0(s,\omega^{t})ds= f^0(t,\omega) $$ by continuity of $f^0$. It is evident that $\chi_{t,h}(s)\le h$ and $$\frac{\chi_{t,h}(s)-\chi_{t,0}(s)}{h}=\frac{\chi_{t,h}(s)}{h}\to 1_{[t,T]}(s)\;as\;h\to0+,\;\forall s\in [0,T].$$ Trerefore, relations (\ref{xv24})-(\ref{xv23}) and continuity of $f^1$ and $f^0$ imply that \begin{eqnarray*} D_\omega f(t,\omega)=\lim_{h\to0}\frac{f(t+h,\omega^{t}+\chi_{t,h})-f(t+h,\omega^{t})}{h}=\\ =\lim_{h\to0}\frac{1}{h}\int_{t}^{t+h}\big(f^0(s,\omega^{t}+\chi_{t,h})-f^0(s,\omega^{t})\big)ds\\ +\lim_{h\to0}\frac{1}{h}\int_{t}^{t+h}f^1(s,\omega^{t}+\chi_{t,h})ds= f^1(t,\omega) \end{eqnarray*} for any $\omega\in\cal V([0,T])$ and by continuity of $f^1$ this equality is true for all $\omega\in D([0,T])$. {\bf Remark.} If $f\in C([0,T]\times \Omega)$ is two times differentiable in the sense of Definition 1, then similarly one can show that $$ f^{1,1}(t,\omega)=D_{\omega\omega}f(s,\omega). $$ {\bf Corrolary 1.} Let $f\in C^{1,1}([0,T]\times \Omega)$. Then for all $(t,\omega)\in \Lambda$ \begin{eqnarray*} \partial_\omega f(t,\omega)=f^1(t,\omega)=D_\omega f(t,\omega). \end{eqnarray*} In general $ \partial_\omega f(t,\omega)$ and $D_\omega f(t,\omega)$ are not equal. {\bf Counterexample 1}. Let $g=(g(x),x\in r)$ be a bounded differentiable function and let $f(t,\omega)=g(\omega_t)-g(\omega_{t-})$. Then $\partial_\omega f(t,\omega)=g'(\omega_t)$ and \begin{eqnarray*} D_\omega f(t,\omega)=\lim_{h\to 0+}\frac{f(t+h,\omega^{t}+\chi_{t,h})-f(t+h,\omega^{t})}{h}=0,\; \rm{since} \\varphi(t+h,\omega^{t}+\chi_{t,h})-f(t+h,\omega^{t})=g(\omega_t+h)-g(\omega_t+h)-g(\omega_t)+g(\omega_t)=0. \end{eqnarray*} It is evident that $f\bar\in C^{1,1}(\Lambda)$, since $f\bar\in C(\Lambda)$ and $\partial_t f=\infty.$ The following assertion shows that if $f$ belongs to the class $C^{1,2}(\Lambda)$ of non-anticipative functionals, then $\partial f_\omega (t,\omega)$ and $\partial f_{\omega\omega} (t,\omega)$ are uniquelly determined by the restriction of $f$ to continuous paths. This assertion is proved by Cont and Fournie \cite{CF} (see also \cite{BCC}) in a complicated way. We give a simple proof based on Proposition 1. {\bf Corrolary 2.} Let $f^1$ and $f^2$ belong to $\in C^{1,2}(\Lambda)$ in the Dupire sense and \begin{equation}\label{f1f2} f^1(t,\omega)=f^2(t,\omega)\;\;\;\;\text{for all}\;\;\;(t,\omega)\in [0,T]\times C([0,T]). \end{equation} Then \begin{equation}\label{f12} \partial_\omega f^1(t,\omega)=\partial_\omega f^2(t,\omega),\;\;\;\partial_{\omega\omega} f^1(t,\omega)=\partial_{\omega\omega}f^2(t,\omega) \end{equation} for all $(\omega,t)\in [0,T]\times C([0,T])$. {\it Proof}. By Theorem 2 \begin{equation}\label{bv} f^i(t,\omega)=f^i(0,\omega)+\int_0^t\partial_t f^i(s,\omega)ds+\int_0^t\partial_\omega f^i(s,\omega)d\omega_s\;\;\;i=1,2, \end{equation} for all $\omega\in C([0,T])\cap{\cal V}([0,T])$. It follows from Proposition 1 that $$ \partial_\omega f^i(t,\omega)=D_\omega f^i(t,\omega);\;\;\; i=1,2. $$ Since $\omega^{t}+\chi_{t,h}\in C([0,T])$ if $\omega\in C([0,T])$, by definition of $D_\omega$ and equality (\ref{f1f2}) we have \begin{equation}\label{df12} D_\omega f^1(t,\omega)=D_\omega f^2(t,\omega)\;\;\;\;\text{for all}\;\;\;(t,\omega)\in [0,T]\times C([0,T])), \end{equation} which implies that \begin{equation}\label{f12} \partial_\omega f^1(t,\omega)=\partial_\omega f^2(t,\omega),\;\;\;\;\text{for all}\;\;\;(t,\omega)\in [0,T]\times C([0,T]), \end{equation} It is evident that $\partial_t f^1(t,\omega)=\partial_t f^2(t,\omega)$ for all $(\omega,t)\in [0,T]\times C([0,T])$. Therefore, comparing the It\^o formulas (\ref{itoc}) for $f^1(t,\omega)$ and $f^2(t,\omega)$ we obtain that $$ \int_t^u\partial_{\omega\omega}f^1(s,\omega)d\langle\omega\rangle_s=\int_t^u\partial_{\omega\omega}f^2(s,\omega)d\langle\omega\rangle_s $$ for any continuous semimartinale $\omega$. Dividing both parts of this equality by $\langle\omega\rangle_u-\langle\omega\rangle_t$ and passing to the limit as $u\to t$, we obtain that $\partial_{\omega\omega} f^1(t,\omega)=\partial_{\omega\omega}f^2(t,\omega)$ for any continuous semimartingale and by continuity of $\partial_{\omega\omega} f^1(t,\omega)$ and $\partial_{\omega\omega}f^2(t,\omega)$ this equality will be true for all $\omega\in C([0,T])$. \begin{prop}\label{33} Let $f\in C([0,T]\times \Omega)$ be differentiable in the sense of Definition 1 and \begin{equation}\label{xv25} \left|f(t,\omega)-f(t-,\omega)-\Delta\omega_tf^1(t-,\omega)\right|\le K|\Delta\omega_t|^2 \end{equation} for some $K>0$. Then $$ f^0(t,\omega)=\partial_t f(t,\omega), \;\;\;\;\forall(t,\omega)\in \Lambda, $$ $$ f^1(t,\omega)=\partial_\omega f(t,\omega),\;\;\;\;\forall \omega\in C[0,T] $$ (or for all $\omega$ continuous at $t$). \end{prop} {\it Proof.} For $\omega\in D[0,T]$ let $\tilde\omega=\omega_s$ if $s<t$ and $\tilde\omega=\omega_{s-}+h$, if $s\ge t$, i.e. $\tilde\omega=\omega^{t-}+h1_{[t,T]}$, hence $\Delta\tilde\omega_s=h$. Therefore, using condition (\ref{xv25}) for $\tilde\omega$ we have \begin{eqnarray*} \left|\frac{f(t,\omega^{t-}+h1_{[t,T]})-f(t-,\omega)}{h}-f^1(t-,\omega)\right|\le K|h|,\;\forall h. \end{eqnarray*} It follows from here that $$ \lim_{h\to0}\frac{f(t,\omega^{t-}+h1_{[t,T]})-f(t-,\omega)}{h}=f^1(t-,\omega), $$ which implies that $f^1(t,\omega)=\partial_\omega f(t,\omega)$ if $\omega$ is continuous at $t$. Equality $f^1(t,\omega)=\partial_tf(t,\omega), \forall(t,\omega)\in \Lambda$ is proved in Proposition 1.\qed Now we introduce definition of space derivatives which can be calculated pathwise along the differentiable paths and using such derivatives in Theorem 3 below a change of variables formula for functions of finite variations is proved, which gives sufficient conditions for the existence of derivatives in the Chitashvili sense. {\bf Definition 4}. We say that a non-anticipative functional $f(t,\omega)$ is differentiable, if the limits $f_t\;f_\omega\in C(\Lambda)$ exist, where \begin{eqnarray*} f_t(t,\omega)=\lim_{h\to0, h>0}\frac{f(t+h,\omega^{t})-f(t,\omega)}{h}, \;\;\;\; \forall(t,\omega)\in [0,T]\times D[0,T]\\ f_{\omega}(t,\omega)=\lim_{h\to0, h>0}\frac{f(t+h,\omega)-f(t+h,\omega^{t})}{\omega_{t+h}-\omega_t},\;\;\;\;\forall(t,\omega)\in [0,T]\times C^1[0,T].\\ \end{eqnarray*} \begin{prop}\label{22} Let $f$ be differentiable in the sense of definition 4. Then $\forall(t,\omega)\in [0,T]\times C^1[0,T]$ \beq\label{itt} f(t,\omega)-f(0,\omega)=\int_0^tf_t(s,\omega)ds+\int_0^tf_\omega(s,\omega)d\omega_s. \eeq \end{prop} {\it Proof}. We have \begin{eqnarray*} \lim_{h\to0, h>0}\frac{f(t+h,\omega)-f(t,\omega)}{h}\\ = \lim_{h\to0+}\frac{f(t+h,\omega)-f(t+h,\omega^{t})}{{\omega_{t+h}-\omega_t}}\times \frac{{\omega_{t+h}-\omega_t}}{h} \\ +\lim_{h\to0+}\frac{f(t+h,\omega^{t})-f(t,\omega)}{h} =\omega'(t)f_\omega(t,\omega)+f_t(t,\omega),\\ \forall(t,\omega)\in [0,T]\times C^1[0,T]. \end{eqnarray*} Hence right derivative of \begin{eqnarray*} f(t,\omega)-f(0,\omega)-\int_0^tf_t(s,\omega)ds-\int_0^tf_\omega(s,\omega)\omega'_sds \end{eqnarray*} is zero for each $\omega\in C^1$. By the Lemma A2 of appendix formula (\ref{itt}) is satisfied. \begin{thr} Let $f\in C(\Lambda)$ and $f_t,f_\omega\in C(\Lambda)$ are derivatives in the sense of definition 4. Assume also that for any $\omega\in\cal V$ $$ \sum_{s\le t}|f(s,\omega)-f(s-,\omega)|<\infty. $$ Then \begin{eqnarray*} f(t,\omega)=f(0,\omega)+\int_0^tf_t(s,\omega)ds+\int_0^tf_\omega(s,\omega)d\omega_s^c\\ +\sum_{s\le t}(f(s,\omega)-f(s-,\omega)). \end{eqnarray*} \end{thr} {\it Proof}. For $\omega\in V$ we have $\omega=\omega^c+\omega^d,\;\omega^d=\sum_{s\le t}\Delta\omega_s,\;\omega^c\in C$. Set $$\omega^{d,n}=\sum_{s\le t,|\Delta\omega_s|>\frac1n}\Delta\omega_s,\;\omega^n=\omega^c+\omega^{d,n}.$$ It is evident that as $n\to 0$ $$ |\omega^n-\omega|_T=|\omega^d-\omega^{d,n}|_T=\max_t|\int_0^t1_{(|\Delta\omega_s|\le\frac1n)}d\omega_s^d\le\int_0^T1_{(|\Delta\omega_s|\le\frac1n)}dvar_s(\omega^d)\to 0. $$ We know that discontinuity points of $f$ are also discontinuity points of $\omega$. Let $\{t_1<...<t_k\}=\{s:|\Delta\omega_s|>\frac1n\}\cup\{0,T\}$. Denote by $\omega^{\varepsilon}\in C'$ a differentiable approximation of $\omega^c$, such that $var_T(\omega^\varepsilon-\omega^c)<\varepsilon$ and let $\omega^{n,\varepsilon}=\omega^\varepsilon+\omega^{d,n}$. Then by Proposition \ref{22} \begin{eqnarray*} f(t,\omega^{n,\varepsilon})-f(t_i,\omega^{n,\varepsilon})-\int_{t_i}^tf_t(s,\omega^{n,\varepsilon})ds-\int_{t_i}^tf_\omega^{n,\varepsilon}(s,\omega^{n,\varepsilon})\omega^{'\varepsilon}_sds=0,\;t\in[t_i,t_{i+1}) \end{eqnarray*} and \begin{eqnarray*} f(T,\omega^{n,\varepsilon})-f(0,\omega^{n,\varepsilon})=\sum_{i\ge 1} \big(f(t_{i},\omega^{n,\varepsilon})-f(t_{i-1},\omega^{n,\varepsilon})\big)\\ =\sum \big (f(t_{i}-,\omega^{n,\varepsilon})-f(t_{i-1},\omega^{n,\varepsilon})\big )+\sum \big (f(t_i,\omega^{n,\varepsilon})-f(t_i-,\omega^{n,\varepsilon})\big )\\ =\sum\int_{t_{i-1}}^{t_i}f_t(s,\omega^{n,\varepsilon})ds+\sum\int_{t_{i-1}}^{t_i}f_\omega(s,\omega^{n,\varepsilon})\omega^{'\varepsilon}_sds+\sum \big (f(t_i,\omega^{n,\varepsilon})-f(t_i-,\omega^{n,\varepsilon})\big )\\ =\sum\int_{t_{i-1}}^{t_i}f_t(s,\omega^{n,\varepsilon})ds+\sum\int_{t_{i-1}}^{t_i}f_\omega(s,\omega^{n,\varepsilon})d\omega_s^{n,\varepsilon}\\ -\sum f_\omega(t_i-,\omega^{n,\varepsilon})\Delta\omega^{n,\varepsilon}_{t_i}+\sum \big (f(t_i,\omega^{n,\varepsilon})-f(t_i-,\omega^{n,\varepsilon})\big )\\ =\int_0^Tf_s(s,\omega^{n,\varepsilon})ds+\int_0^Tf_\omega(s,\omega^{n,\varepsilon})d\omega^{n,\varepsilon}_s\\ +\sum\big ( f(t_i,\omega^{n,\varepsilon})-f(t_i-,\omega^{n,\varepsilon})-f_\omega(t_i-,\omega^{n,\varepsilon})\Delta\omega^{n,\varepsilon}_{t_i}\big )\\ =\int_0^Tf_t(s,\omega^{n,\varepsilon})ds+\int_0^Tf_\omega(s,\omega^{n,\varepsilon})d\omega_s^\varepsilon+\sum\big ( f(t_i,\omega^{n,\varepsilon})-f(t_i-,\omega^{n,\varepsilon})\big ). \end{eqnarray*} Since $f(t,\omega^{n,\varepsilon})$ admits finite number of jumps and $\sup_\varepsilon var_T\omega^\varepsilon<\infty$, passing to the limit as $\varepsilon\to0$ we get \begin{eqnarray*} f(T,\omega^{n})-f(0,\omega^{n})\\ =\int_0^Tf_t(s,\omega^{n})ds+\int_0^Tf_\omega(s,\omega^{n})d\omega_s^c+\sum\big ( f(t_i,\omega^n)-f(t_i-,\omega^n)\big ). \end{eqnarray*} By the continuity of functionals $f,\;f_t,\;f_\omega$ and Lemma A1 from the appendix $$f(t,\omega^n)\to f(t,\omega),\;\int_0^tf_t(s,\omega^n)ds\to \int_0^tf_t(s,\omega)ds, $$ $$\int_0^tf_t(s,\omega^n)d\omega_s^c\to \int_0^tf_t(s,\omega)d\omega_s^c,\;as\;n\to\infty.$$ It remains to show convergence of the sum. Since\\ $f^d(t,\omega)=\sum_{s\le t}f(s,\omega)-f(s-,\omega)$ is of finite variation \begin{eqnarray*} f^d(t,\omega)=\sum \big ( f(t_i,\omega^{n})-f(t_i-,\omega^{n})\big )-\sum \big (f(t_i,\omega)-f(t_i-,\omega)\big )\\ =\sum_{s\le t} (f(s,\omega)-f(s-,\omega))1_{(|\Delta\omega_s|\le\frac1n)}\\ =\int_0^t1_{(|\Delta\omega_s|\le\frac1n)}df^d(s,\omega)\to 0,\; as\;n\to\infty, \end{eqnarray*} by the dominated convergence theorem. {\bf Corollary.} If $f$ satisfies conditions of Theorem 3 then $f$ is differentiable in the sense of Definition 1. \section{Appendix} The following lemma is a modification of lemma 6 of \cite{LScS}. {\bf Lemma A1}. Let $X_n,X\in \Omega$ be a sequence of paths, such that $||X_n-X||_T\to 0$ as $n\to\infty$. Let $f\in C(\Lambda)$. Then $$\sup_{t\le T}|f(t,X_n)-f(t,X)|\overset{n\to\infty}\to 0.$$ {\it Proof}. If not then $\exists \varepsilon > 0$, a sequence of integers $n_k, k=1,...$, and a sequence $s_k\in [0, T ]$ such that \beq\label{uni} |f(s_k, X_{n_k})-f(s_k, X)| \ge \varepsilon \eeq By moving to a subsequence we can assume without loss of generality that either $s_k \to s^*,\;s_k\ge s^*$ or $s_k \to s^*,\;s_k< s^*$ for some $s^*\in[0, T]$. In the first case by continuity assumption we get \begin{eqnarray*} |f(s_k, X_{n_k})-f(s_k, X)|\le |f(s_k, X_{n_k})-f(s^*, X)|\\+|f(s_k, X)-f(s^*, X)| \to 0, \end{eqnarray*} since $d_\infty((s_k,X_{n_k}),(s^*, X))\to 0,\;d_\infty((s_k,X),(s^*, X))\to 0$ . In the second case we have \begin{eqnarray*} |f(s_k, X_{n_k})-f(s_k, X)| \le |f(s_k, X_{n_k})-f(s^*, X^{s^*-})|\\ +|f(s_k, X)-f(s^*, X^{s^*-})|\to 0, \end{eqnarray*} since $d_\infty((s_k, X_{n_k}),(s^*, X^{s^*-}))\to 0,\;d_\infty((s_k, X),(s^*, X^{s^*-}))\to 0$ . This contradicts (\ref{uni}). We shall need also the following assertion {\bf Lemma A2}. Let $f$ be a real-valued, continuous function, defined on an arbitrary interval $I$ of the real line. If $f$ is right (or left) differentiable at every point $a \in I$, which is not the supremum (infimum) of the interval, and if this right (left) derivative is always zero, then $f$ is a constant. {\it Proof}. For a proof by contradiction, assume there exist $a < b$ in $I$ such that $f(a) \neq f(b)$. Then \begin{eqnarray*} \varepsilon :={\frac {|f(b)-f(a)|}{2(b-a)}}>0. \end{eqnarray*} Define $c$ as the infimum of all those $x$ in the interval $(a,b]$ for which the difference quotient of $f$ exceeds $\varepsilon$ in absolute value, i.e. \begin{eqnarray*} c=\inf\{\,x\in (a,b]\mid |f(x)-f(a)|>\varepsilon (x-a)\,\}. \end{eqnarray*} Due to the continuity of $f$, it follows that $c < b$ and $|f(c)-–f(a)|=\varepsilon(c–-a)$. At $c$ the right derivative of $f$ is zero by assumption, hence there exists $d$ in the interval $(c,b]$ with $|f(x)–-f(c)|\le\varepsilon(x–-c)$ for all $x \in (c,d]$. Hence, by the triangle inequality, \begin{eqnarray*} |f(x)-f(a)|\leq |f(x)-f(c)|+|f(c)-f(a)|\leq \varepsilon (x-a) \end{eqnarray*} for all $x$ in $[c,d)$, which contradicts the definition of $c$. \end{document}

\begin{document} \title{On $C^*$-algebras associated to actions of discrete subgroups of $\SL(2,\mathbb{R})$ on the punctured plane} \author{Jacopo Bassi} \maketitle \begin{abstract} \noindent Dynamical conditions that guarantee stability for discrete transformation group $C^*$-algebras are determined. The results are applied to the case of some discrete subgroups of $\SL(2,\mathbb{R})$ acting on the punctured plane by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^*$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $\SL(2,\mathbb{R})$. \end{abstract} \section{Introduction} Transformation group $C^*$-algebras represent a tool for the construction of examples of structure and classification theorems for $C^*$-algebras and provide a way to interpret dynamical properties on the $C^*$-algebraic level. Typical examples are the $C^*$-algebras associated to minimal homeomorphisms on infinite compact metric spaces with finite covering dimension (\cite{rieffel-ir,giordano-putnam-skau,toms-winter}), or more generally, free minimal actions of countable residually finite groups with asymptotically finite-dimensional box space on compact metric spaces with finite covering dimension (\cite{swz}). In these cases the structure is that of an $ASH$-algebra and classification is provided by the Elliott invariant. Moving to the non-unital setting, such classification results are still available in the case the $C^*$-algebra is stable and contains projections, assuming a suitable Rokhlin type property for the action. In these situations the resulting transformation group $C^*$-algebra is a stabilized $ASH$-algebra. Examples come from free and minimal actions of the real numbers on compact metric spaces admitting compact transversals (\cite{hsww}), where stability is reminiscent of freeness and the transversal produces a projection in the crossed product. On the other hand, stable simple $\mathcal{Z}$-stable projectionless $C^*$-algebras admit a description of the isomorphism classes of hereditary $C^*$-subalgebras and countable generated Hilbert $C^*$-modules in terms of Cuntz equivalence of positive elements (\cite{zstable_projless}). Dynamical conditions which ensure stability of a transformation group $C^*$-algebra were given in \cite{green}, where it is proved that $C^*$-algebras arising from actions that are free and wandering on compacts are trivial fields of compact operators. For the case of more general $C^*$-algebras, other characterizations of stability are contained in \cite{rordam-fp,brttw}.\\ The present paper focuses on transformation group $C^*$-algebras associated to the action of discrete subgroups os $\SL(2,\mathbb{R})$ on the punctured plane, by means of matrix multiplication of vectors. Ergodic properties of such dynamical systems have been investigated in several places and the duality with the horocycle flow on the corresponding homogeneous spaces for $\SL(2,\mathbb{R})$ has been successfully employed in \cite{furstenberg,ledrappier, mau_weiss}. The study of such dynamical systems and their generalizations has a number of interesting applications, as observed in \cite{goro_weiss}, as the quantitative Oppenheim conjecture, quantitative estimates of the denseness of certain projections associated to irreducible lattices and strengthenings of distribution results concerning actions of lattices by automorphisms. The first part of this work focuses on the study of the distribution of orbits of compact sets on the punctured plane under the action of discrete subgroups of $\SL(2,\mathbb{R})$ containing two hyperbolic elements with different axes. Rather than studying the asymptotics of the distribution of such orbits under an increasing family of finite subsets in the lattice, as in \cite{nogueira2002}, \cite{nogueira2010} and \cite{guilloux}, we consider the possibility to find, at every step, an element in the group that \textit{squeezes} enough the image of the compact set under the action of any element in the finite subset. This property of the action resembles the fact that such discrete subgroups of $\SL(2,\mathbb{R})$ actually contain an abundance of hyperbolic elements and represents a weaker version of the wandering on compacts assumption considered in \cite{green}. This dynamical condition guarantees the existence of invertible approximants for the elements in the crossed product $C^*$-algebra. By appealing to \cite{rordam-fp}, we show that in the case of actions that are contractive in a suitable sense, this property is enough in order to ensure stability of the crossed product $C^*$-algebra. The "dual" approach is used in the last part to find properties of the crossed product arising from an action of a cocompact subgroup of $\SL(2,\mathbb{R})$ on $\mathbb{R}^2 \backslash \{0\}$ by establishing a $*$-isomorphism between this $C^*$-algebra and the $C^*$-algebra associated to the horocycle flow on the corresponding homogeneous space for $\SL(2,\mathbb{R})$.\\ \subsection{Notation} If $G$ is a locally compact group and $A$ is a $C^*$-algebra, by an action of $G$ on $A$ we mean a continuous group homomorphism from $G$ to the group $\Aut (A)$ of $*$-automorphisms of $A$, endowed with the topology of pointwise convergence. If $X$ is a locally compact Hausdorff space, by an action of $G$ on $X$ we mean a continuous map $G \times X \rightarrow X$ that is associative and such that the identity of the group leaves every point of the space fixed. If a locally compact group $G$ acts on a locally compact Hausdorff space $X$ by means of an action $\alpha : G \times X \rightarrow X$, we denote by $C_0 (X) \rtimes G$ the associated (full) transformation group $C^*$-algebra, that is the full crossed product $C^*$-algebra relative to the action $\hat{\alpha}_g (f)=f \circ g^{-1}$ for $g \in G$, $f \in C_0 (X)$. Similarly $C_0 (X) \rtimes_r G$ is the reduced transformation group $C^*$-algebra, that is the reduced crossed product relative to the same action. If $X$ and $Y$ are two Hilbert modules over a $C^*$-algebra, we write $X\Subset Y$ to mean that $X$ is compactly contained in $Y$ in the sense of \cite{cuntz_hm} Section 1. If $F \subset S$ is an inclusion of sets, we write $F\Subset S$ to mean that $F$ has finite cardinality. If $X$ is a topological space and $S\subset X$ a subset, we denote by $S^\circ$ its interior. \section{Weak stable rank $1$} The concept of stable rank for $C^*$-algebras was introduced by Rieffel in \cite{rieffel} as a noncommutative analogue of the covering dimension of a space and the case of stable rank $1$ is of particular interest (see for example \cite{cuntz_hm} and \cite{open_proj}). Conditions under which a transformation group $C^*$-algebra has stable rank $1$ have been given in \cite{poon} for actions of the integers; for actions of other groups with finite Rokhlin dimension on compact spaces such conditions can be obtained by combining the results in \cite{hwz}, \cite{szabo} or \cite{swz} and \cite{rordam-sr}, under some other assumptions, as for example, the existence of an invariant measure. If $A$ is a $C^*$-algebra, it is said to have stable rank $1$ if every element in its minimal unitization $\tilde{A}$ can be approximated by invertible elements in $\tilde{A}$. We will consider a more restrictive (non-stable) approximation property, which was used in a crucial way in \cite{brttw}. The following definition was given by Hannes Thiel during a lecture about the Cuntz semigroup in the Winter semester 2016/2017 at the University of M{\"u}nster. \begin{defn} \label{defn2.0} Let $A$ be a $C^*$-algebra. Then $A$ has \textit{weak stable rank $1$}, $\wsr (A)=1$, if $A \subset \overline{GL(\tilde{A})}$. \end{defn} Another variation of the concept of stable rank $1$ is the following \begin{defn}[\cite{zstable_projless} Definition 3.1] Let $A$ be a $C^*$-algebra. Then $A$ has \textit{almost stable rank $1$}, $\asr (A)=1$, if $\wsr (B) =1$ for every hereditary $C^*$-subalgebra $B \subset A$. \end{defn} A $C^*$-algebra $A$ is said to be stable if $A\otimes \mathbb{K} \simeq A$, where $\mathbb{K}$ denotes the $C^*$-algebra of compact operators on a separable Hilbert space. Stable $C^*$-algebras always have weak stable rank $1$ by Lemma 4.3.2 of \cite{brttw} and their multiplier algebra is properly infinite by \cite{rordam-fp} Lemma 3.4. The connection between stability and stable rank in the $\sigma$-unital case was already investigated in \cite{rordam-fp} Proposition 3.5 and Proposition 3.6. For our purpose, we need the following slight variation of the results contained in \cite{rordam-fp}: \begin{thm} \label{thm2.0} Let $A$ be a $\sigma$-unital $C^*$-algebra. The following are equivalent \begin{itemize} \item[(i)] $\wsr (A)=1$ and $M(A)$ is properly infinite; \item[(ii)] $A$ is stable. \end{itemize} If $A$ is simple, they are equivalent to \begin{itemize} \item[(iii)] $\wsr(A)=1$ and $M(A)$ is infinite. \end{itemize} \end{thm} \proof The proof of Lemma 3.2 of \cite{rordam-fp} applies under the hypothesis of weak stable rank $1$, hence if $\wsr (A) =1$ and $M(A)$ is properly infinite, then $A$ is stable by the considerations in the proof of \cite{rordam-fp} Proposition 3.6. As already observed, for any stable $C^*$-algebra $A$, $\wsr(A)=1$ and its multiplier algebra is properly infinite. In the simple case the result follows by an application of Lemma 3.3 of \cite{rordam-fp} and the proof is complete.\\ In order to obtain stability for a transformation group $C^*$-algebra, we introduce a certain dynamical condition and observe that it guarantees weak stable rank $1$; this is the content of the rest of this section. We will deduce infiniteness properties for the multiplier algebra by adapting the results contained in \cite{sth} to the locally compact case in the next section. \begin{defn} \label{defn2.1} Let $G$ be a discrete group acting on a locally compact Hausdorff space $X$. The action is said to be \textit{squeezing} if for every $F\Subset G$ and every $C \subset X$ compact there exists $\gamma \in G$ such that \[ \gamma g \gamma h C \cap \gamma g C \cap C =\emptyset \nonumber \] for all $g,h \in F$. \end{defn} Note that Definition \ref{defn2.1} only makes sense for actions on locally compact non-compact spaces, since the space itself is globally fixed by any homeomorphism. \begin{pro} \label{prop2.1} Let $G$ be a discrete group acting on a locally compact Hausdorff space $X$ by means of a squeezing action. Then $\wsr (C_0 (X) \rtimes G) =1$. \end{pro} \proof Every element in $C_0 (X) \rtimes G$ can be approximated by elements in $C_c (G, C_c (X))$, hence it is enough to prove that any element in $C_c (G, C_c (X))$ is the limit of invertible elements in $(C_0 (X) \rtimes G )^\sim$. Let $F \Subset G$ and $z=\sum_{g \in F} z_g u_g$ be such that $z_g \in C_c (X)$ for every $g \in F$. Define $C := \bigcup_{g \in F} \supp (z_g)$ and let $K \subset X$ be a compact subset such that $C \subsetneq K^\circ$. There exists a continuous function $f : X \rightarrow [0,1]$ such that $\supp (f) \subset K$, $f|_{C} =1$; furthermore, since the action is squeezing, there is a group element $\gamma \in G$ such that \[ \gamma g \gamma h K \cap \gamma g K \cap K = \emptyset \nonumber \qquad \mbox{ for all }\quad g,h \in F \cup F^{-1} \cup \{ e\}. \] From our choice of $f$, it follows that we can write $z = f z = (f u_{\gamma})(u_{\gamma^{-1}} z)$. Computing the third power of $u_{\gamma^{-1}} z$ we obtain \[ (u_{\gamma^{-1}} z)^3 = \sum_{g,g' , g'' \in F} (z_g \circ \gamma) (z_{g'} \circ (\gamma^{-1} g \gamma^{-1})^{-1}) (z_{g''} \circ (\gamma^{-1} g \gamma^{-1} g' \gamma^{-1})^{-1}) u_{\gamma^{-1} g \gamma^{-1} g' \gamma^{-1} g''}. \nonumber \] For every $s \in G$ and $\phi \in C_c (X)$ we have $\supp (\phi \circ s^{-1}) = s \supp (\phi)$ and so from our choice of $K$ and $\gamma$, we see that \[ \begin{split} &\supp (z_g \circ \gamma) \cap \supp (z_{g'} \circ (\gamma^{-1} g \gamma^{-1})^{-1}) \cap \supp (z_{g''} \circ (\gamma^{-1} g \gamma^{-1} g' \gamma^{-1})^{-1}) \\ &\subset \gamma^{-1} (K \cap g \gamma^{-1} K \cap g \gamma^{-1} g' \gamma^{-1} K) = \emptyset, \end{split} \nonumber \] since $K \cap g \gamma^{-1} K \cap g \gamma^{-1} g' \gamma^{-1} K =\emptyset$ if and only if $\gamma(g')^{-1} \gamma g^{-1} K \cap \gamma (g')^{-1} K \cap K = \emptyset$. Hence \[ (u_\gamma^{-1} z)^3 = 0 \nonumber \] and $u_\gamma z$ is nilpotent. In the same way we obtain \[ (f u _{\gamma})^3 = f (f\circ \gamma^{-1}) (f\circ \gamma^{-2}) u_{\gamma^{3}}=0 \nonumber \] since $\gamma^2 K \cap \gamma K \cap K = \emptyset$. Hence $z$ is a product of nilpotent elements, thus is the limit of invertible elements in $(C_0 (X) \rtimes G)^\sim$ (cfr. \cite{rordam-uhf} 4.1) and the claim follows. \begin{rem} \label{oss2.0} Natural variations of Definition \ref{defn2.1} lead to the same result of Proposition \ref{prop2.1}. The reason why we chose this form is that it fits in the discussion of Section 4. \end{rem} \begin{rem} \label{oss2.1} Proposition \ref{prop2.1} applies to the reduced crossed product as well. \end{rem} \section{Contractive and paradoxical actions} In the last section we determined a condition on an action of a discrete group that guarantees weak stable rank $1$ for the transformation group $C^*$-algebra. In view of Theorem \ref{thm2.0} this section is devoted to find conditions that guarantee infiniteness properties for the multiplier algebra of the crossed product $C^*$-algebra.\\ If $A$ is any $C^*$-algebra and $G$ a discrete group acting on it, then $A\rtimes G$ is isomorphic to an ideal in $M(A) \rtimes G$, where the action of $G$ on $M(A)$ is the extension of the action on $A$. Then there is a unital $*$-homomorphism $\phi : M(A) \rtimes G \rightarrow M(A \rtimes G)$; if we identify $M(A\rtimes G)$ with the $C^*$-algebra of double centralizers on $A\rtimes G$ and $A\rtimes G$ with its isomorphic image in $M(A) \rtimes G$, $\phi (x) y = xy$ for any $x$ in $M(A) \rtimes G$ and $y$ in $A\rtimes G$. The same results apply to the reduced crossed product as well. This will be the framework for the following considerations.\\ In virtue of the above discussion, all the results we state in the rest of this section concerning full transformation group $C^*$-algebras hold true for the reduced transformation group $C^*$-algebras as well. The same applies to the results contained in the next section, where in order to prove the analogue of Proposition \ref{prop123} for the reduced crossed product, one can use the extension of the surjective $*$-homomorphism from the full crossed product to the reduced crossed product to the multiplier algebras. The concept of contractive action (see below) was already considered in \cite{sth} page 22 and has to be compared with the more restrictive Definition 2.1 of \cite{delaroche}. \begin{defn} \label{defn3.1} Let $G$ be a discrete group acting on a locally compact Hausdorff space $X$. The action is said to be \textit{contractive} if there exist an open set $U \subset X$ and an element $t \in G$ such that $t \overline{U} \subsetneq U$. In this case $(U,t)$ is called a \textit{contractive pair} and $U$ a \textit{contractive set}. \end{defn} The notion of scaling element was introduced in \cite{blackadar-cuntz} and was used to characterize stable algebraically simple $C^*$-algebras. \begin{defn}[\cite{blackadar-cuntz} Definition 1.1] \label{defn3.3} Let $A$ be a $C^*$-algebra and $x$ an element in $A$. $x$ is called a \textit{scaling element} if $x^* x (xx^*) = xx^*$ and $x^* x \neq xx^*$. \end{defn} \begin{pro} \label{prop3.1} Let $G$ be a discrete group acting on a locally compact Hausdorff space $X$. Consider the following properties: \begin{itemize} \item[(i)] The action of $G$ on $X$ is contractive. \item[(ii)] There exists a scaling elementary tensor in $C_c (X, C_b (X))$. \end{itemize} Then $(ii) \Rightarrow (i)$. If $X$ is normal, then $(i) \Rightarrow (ii)$. \end{pro} \proof $(ii) \Rightarrow (i)$: Let $x=u_t f$ be a scaling elementary tensor in $C_c (G, C_b (X))$ and $U$ the interior of $\supp(f)$. Since $x^* x = |f|^2$ and $xx^* = | f \circ t^{-1} |^2$, the condition $x^* x xx^* = xx^*$ implies $|f| |_{t\overline{U}} =1$; in particular $t\overline{U} \subset U$. Suppose that $t\overline{U} =U$. Then \[ |f| |_{U^c} =0, \quad |f||_U = |f||_{t\overline{U}} = 1|_{t\overline{U}} \nonumber \] and \[ |f\circ t^{-1} ||_{U^c} = |f\circ t^{-1} ||_{(t\overline{U})^c} =| f\circ t^{-1} ||_{t (U)^c}=0. \nonumber \] Since $G$ acts by homeomorphisms, $U$ is a clopen set and $t^{-1} U = t \overline{U}$, which entails \[ |f\circ t^{-1}||_U =1|_U = 1|_{t \overline{U}}. \nonumber \] This would imply $|f|= |f\circ t^{-1}|$ and $x^* x=xx^*$. Hence $t\overline{U} \subsetneq U$.\\ Suppose now that $X$ is normal and let $(U,t)$ be a contractive pair. Take $\xi \in U \backslash (t\overline{U})$. By Urysohn Lemma (normality) there exists a continuous function $f : X \rightarrow [0,1]$ that is $0$ on $U^c$ and $1$ on $\{ \xi \} \cup (t\overline{U})$. The element $x := u_t f \in C_c (G, C_b (X))$ satisfies $x^* x = f^2$, $xx^* = (f\circ t^{-1} )^2$ and $x^* x (xx^*)=xx^*$. Since $\supp (f\circ t^{-1} ) \subsetneq \supp (f)$, we have $x^* x \neq xx^*$, completing the proof. \begin{cor} \label{cor3.3.1} Let $G$ be a group acting on a locally compact normal Hausdorff space by means of a contractive action. Then $M(C_0 (X) \rtimes G)$ is infinite. \end{cor} \proof Let $x$ be as in Proposition \ref{prop3.1}, we want to show that $\phi(x^*x) \neq \phi (xx^*)$ ($\phi$ is defined at the beginning of the section). For take $\xi \in U \backslash (t\overline{U})$ and let $f \in C_c (X)$ be such that $f(\xi)=1$. Then $(x^*x f)(\xi) \neq 0$ and $(xx^* f)(\xi)=0$ and so $\phi (x^* x) \neq \phi (xx^*)$. As shown in \cite{blackadar-cuntz} Theorem 3.1 the element $\phi(x)+(1-\phi(x^* x))^{1/2}$ is a nontrivial isometry and the claim follows.\\ A variation of the concept of contractive action is the following (see \cite{sth} Lemma 2.3.2) and is a particular case of Definition 2.3.6 of \cite{sth}. \begin{defn} \label{defn3.33} Let $X$ be a locally compact Hausdorff space and $G$ a discrete group acting on it. We say that the action is \textit{paradoxical} if there are positive natural numbers $n$, $m$, group elements $t_1 ,..., t_{n+m}$ and non-empty open sets $U_1 ,..., U_{n+m}$ such that $\bigcup_{i=1}^n U_i = \bigcup_{i=n+1}^{n+m} U_i = X$, $\bigcup_{i=1}^{n+m} t_i (U_i) \subsetneq X$ and $t_i U_i \cap t_j U_j = \emptyset$ for every $i\neq j$. \end{defn} Adapting the ideas (and methods) of \cite{sth} Lemma 2.3.7 to the locally compact case, we have the following \begin{pro} \label{prop3.2} Let $G$ be a discrete group acting on a locally compact normal Hausdorff space $X$. If the action is paradoxical, then $M(C_0 (X) \rtimes G)$ is properly infinite. \end{pro} \proof Let $n$, $m$, $t_1 ,..., t_{n+m}$ and $U_1 ,..., U_{n+m}$ be as in Definition \ref{defn3.33}. Taking unions and relabeling we can suppose $t_i \neq t_j$ for $i\neq j$. Let $F:= \{ t_1 ,..., t_n \}$, $F' := \{ t_{n+1} ,..., t_{n+m}\}$.\\ Since $X$ is normal we can take a partition of unity $\{\phi_t\}_{t \in F}$ subordinated to $\{U_i\}_{i=1}^n$ and a partition of unity $\{ \psi_{s}\}_{s \in F'}$ subordinated to $\{U_i\}_{i=n+1}^{n+m}$. Consider the extension of the action of $G$ to $C_b (X)$ and the associated crossed product $C^*$-algebra $C_b (X) \rtimes G$.\\ Define $x:= \sum_{t \in F} u_t \phi_t^{1/2}$ and $y:= \sum_{t' \in F'} u_{s} \psi_{s}^{1/2}$. Then \[ x^* x = y^* y = 1. \nonumber \] Note now that \[ x^*y = \sum_{t \in F, s \in F'} \phi_t^{1/2} (\psi_s^{1/2} \circ s^{-1} t ) u_{t^{-1} s} =0 \nonumber \] and so $xx^* \perp yy^*$.\\ Let $\phi :C_b (X) \rtimes G \rightarrow M(C_0 (X) \rtimes G)$ be as at the beginning of this section. Take a positive function $f \in C_c(X)$ that takes the value $1$ on a point $\xi \in (\bigcup_{1\leq i \leq n} t_i U_i)^c$. Then \[ xx^* f = \sum_{t, t' \in F} (\phi_t^{1/2} \circ t^{-1}) (\phi_{t'}^{1/2} \circ t^{-1}) u_{t(t')^{-1}} f \nonumber \] entails $0=(xx^* f)(\xi) \neq f (\xi) =1$. Hence $xx^* f \neq f$ and $\phi (xx^*)\neq \phi (1)=1$. The same applies to $yy^*$ and so $1 \in M(C_0 (X) \rtimes G)$ is properly infinite, as claimed.\\ If a discrete group $G$ acts on a locally compact Hausdorff space $X$, the action is said to be \textit{topologically free} if for every $F \Subset G$ the set $\bigcap_{t \in F \backslash \{e\}} \{ x \in X \; | \; tx \neq x \}$ is dense in $X$ (\cite{archbold-spielberg} Definition 1). Combining Proposition \ref{prop3.2} with the results of Section $2$ we obtain \begin{thm} \label{thm3.1} Let $G$ be a discrete group acting on a locally compact metric space by means of an action that is paradoxical and squeezing. Then $C_0 (X) \rtimes G$ is stable. If the action is topologically free, minimal, squeezing and contractive, then $C_0 (X) \rtimes_r G$ is stable. \end{thm} \proof Since $X$ is second countable, $C_0 (X) \rtimes G$ is separable, hence $\sigma$-unital. The result follows from Theorem \ref{thm2.0}, Proposition \ref{prop3.2} and Proposition \ref{prop2.1}. If the action is topologically free and minimal, then $C_0 (X)\rtimes_r G$ is simple by \cite{archbold-spielberg}; hence Theorem \ref{thm2.0} applies also in this situation. \section{The case of discrete subgroups of $\SL(2,\mathbb{R})$} A Fuchsian group $\Gamma$ is a discrete subgroup of $\PSL(2,\mathbb{R})$ (\cite{katok} Definition 2.2) and as such it acts on the hyperbolic plane $\mathbb{H}$ and on its boundary $\partial \mathbb{H} =\mathbb{R}\cup \{\infty\} \simeq \mathbb{R} \mathbb{P}^1$ by means of M{\"o}bius transformations. Let $G$ be a discrete subgroup of $\SL(2,\mathbb{R})$ acting on $\mathbb{R}^2 \backslash \{0\}$ by means of matrix multiplication of vectors. The quotient map $\pi : \mathbb{R}^2 \backslash \{0\} \rightarrow \mathbb{R} \mathbb{P}^1$ induces an action of $G$ on $\mathbb{R}\mathbb{P}^1$, which factors through the action of the corresponding Fuchsian group $p (G)$, where $p: \SL(2,\mathbb{R}) \rightarrow \PSL(2,\mathbb{R})$ is the quotient by the normal subgroup $\{-1,+1\}$ of $\SL(2,\mathbb{R})$.\\ If $\gamma$ is a hyperbolic element (\cite{katok} 2.1) in $\PSL(2,\mathbb{R})$ or $\SL(2,\mathbb{R})$ acting on $\mathbb{RP}^1$, we denote by $\gamma^{- (+)}$ its repelling (attracting) fixed point. For a subset of $\SL(2,\mathbb{R})$ or $\PSL(2,\mathbb{R})$ consisting of hyperbolic transformations, we say that its elements have different axes if the fixed-point sets for the action of the elements on $\mathbb{R} \mathbb{P}^1$ are pairwise disjoint. Note that in both $\SL(2,\mathbb{R})$ and $\PSL(2,\mathbb{R})$, discreteness of a subgroup $G$ implies that whenever two hyperbolic elements in $G$ have a common axis, then both their axes coincide. \begin{lem} \label{lem5} Let $\Gamma$ be a Fuchsian group containing two hyperbolic elements with different axes. Then for every $F\Subset \Gamma$ there exists a hyperbolic element $\gamma \in \Gamma$ such that \[ g \gamma^+ \neq \gamma^- \qquad \forall g \in F. \nonumber \] The same is true if $\Gamma$ is a group generated by a hyperbolic element. \end{lem} \proof Let $F \Subset \Gamma$. If $\Gamma$ contains two hyperbolic elements with different axes, then it contains infinitely many, hence we can take $\eta$, $\delta$ hyperbolic with different axes and such that the fixed points of $\eta$ are not fixed by any elements in $F$. Suppose that $F$ is such that for every $n \in \mathbb{N}$ there is a $g \in F$ with $g \eta^n \delta^+ =g(\eta^n \delta \eta^{-n})^+ = (\eta^n \delta \eta^{-n})^- = \eta^n \delta^-$. Then, passing to a subsequence \[ \exists g \in F \quad \mbox{ s.t. } \quad g\eta^{n_k} \delta^+ = \eta^{n_k} \delta^-. \nonumber \] Both $\eta^{n_k} \delta^+$ and $\eta^{n_k} \delta^-$ converge to $\eta^+$ and so $\eta^+$ is fixed by $g$, a contradiction. \begin{pro} \label{prop4.1} Let $G$ be a discrete subgroup of $\SL(2,\mathbb{R})$ such that $p(G)$ is a Fuchsian group containing two hyperbolic elements with different axes or a Fuchsian group generated by a hyperbolic element. Then the action of $G$ on $\mathbb{R}^2 \backslash \{0\}$ is squeezing. \end{pro} \proof Let $F \Subset G$ and let be given an orthonormal basis $\{e_1 , e_2\}$ for $\mathbb{R}^2$. By Lemma \ref{lem5} there is a $\gamma \in p (G)$ such that $p(g) \gamma^+ \neq \gamma^-$ for every $g \in G$. Let $h \in G$ be such that $p(h) = \gamma$. Hence $h$ is hyperbolic and is conjugated in $\SL(2,\mathbb{R})$ to a diagonal matrix: \[ h = u^{-1} \Lambda u =u^{-1} \left( \begin{array}{cc} \lambda & 0 \\ 0 & \lambda^{-1} \end{array}\right)u, \qquad |\lambda| >1. \nonumber \] Let $g'$ and $g$ be elements in $G$ and suppose that the upper-left diagonal entry of the matrix $ug'u^{-1}$ vanishes: $(ug'u^{-1})_{1,1} =0$. This means that $\langle e_1, ug'u^{-1}e_1\rangle =0$, or equivalently, $ug'u^{-1} e_1 \in \mathbb{R} e_2$; hence, since the image of $u^{-1} e_1$ under the quotient map $\pi : \mathbb{R}^2 \backslash \{0\} \rightarrow \mathbb{R} \mathbb{P}^1$ is $\gamma^+$ and the image of $u^{-1} e_2$ under the same map is $\gamma^-$, looking at the action of $p(G)$ on $\mathbb{R} \mathbb{P}^1$ we obtain $p (g') \gamma^+ = \gamma^-$, contradicting the assumption. Hence $(ug'u^{-1})_{1,1}\neq 0$. Define $g_u := ugu^{-1}$, $g'_u := ug' u^{-1}$ and compute for $n \in \mathbb{N}$ \[ \Lambda^n g_u= \left(\begin{array}{cc} \lambda^n (g_{u})_{1,1} & \lambda^n (g_u)_{1,2} \\ \lambda^{-n} (g_{u})_{2,1} & \lambda^{-n} (g_{u})_{2,2} \end{array}\right), \nonumber \] and \[ \Lambda^n g'_u \Lambda^n g_u = \left( \begin{array}{cc} \lambda^{2n} (g'_u)_{1,1} (g_{u})_{1,1} + (g'_u)_{1,2} (g_u)_{2,1} & \lambda^{2n} (g'_u)_{1,1} (g_u)_{1,2} + (g'_u)_{1,2} (g_u)_{2,2} \\ (g'_u)_{2,1} (g_u)_{1,1} + \lambda^{-2n} (g'_u)_{2,2} (g_u)_{2,1} & (g'_u)_{2,1} (g_u)_{1,2} + \lambda^{-2n} (g'_u)_{2,2} (g_u)_{2,2} \end{array}\right). \nonumber \] Let $C \subset \mathbb{R}^2 \backslash \{ 0 \}$ be a compact subset; take real positive numbers $r_1$ and $r_2$ such that the compact crown $C_{r_1 , r_2} = \{ z \in \mathbb{R}^2 | r_1 \leq \| z \| \leq r_2\}$ contains $uC$. We want to show that there exists $n >0$ such that \[ \Lambda^n g'_u \Lambda^n g_u C_{r_1 , r_2} \cap \Lambda^n g'_u C_{r_1 , r_2} \cap C_{r_1,r_2} = \emptyset, \qquad \forall g,g' \in F. \nonumber \] Let $(x,y)^t \in \mathbb{R}^2$ be such that $\Lambda^n g'_u \Lambda^n g_u (x,y)^t$ belongs to $uC$. In particular, this entails \[ \| \Lambda^n g'_u \Lambda^n g_u (x,y)^t \| \leq r_2 \nonumber \] and taking the first coordinate: \[ | \lambda^{2n} (g'_u)_{1,1} [(g_u)_{1,1} x + (g_u)_{1,2} y] + [ (g'_u)_{1,2} (g_u)_{2,2} x + (g'_u)_{1,2} (g_u)_{2,2} y ]| \leq r_2. \nonumber \] Hence \begin{equation} \label{eq4.1} |(g_u)_{1,1} x + (g_u)_{1,2} y| \leq \frac{r_2 + | (g'_u)_{1,2} (g_u)_{2,2} x + (g'_u)_{1,2} (g_u)_{2,2} y |}{\lambda^{2n} | (g'_u)_{1,1}|} \end{equation} for every $(x,y)^t \in (\Lambda^n g'_u \Lambda^n g_u)^{-1} C$. Furthermore, if $(x,y)^t \in \mathbb{R}^2$ is such that $\Lambda^n g_u (x,y)^t $ belongs to $C_{r_1,r_2}$, then \begin{equation} \label{eq4.2} \begin{split} r_1^2 &\leq [\lambda^n (g_u)_{1,1} x + \lambda^n (g_u)_{1,2} y]^2 + [\lambda^{-n} (g_u)_{2,1} x + \lambda^{-n} (g_u)_{2,2} y ]^2\\ & = \lambda^{2n} [(g_u)_{1,1} x + (g_u)_{1,2} y]^2 + \lambda^{-2n}[ (g_u)_{2,1} x + (g_u)_{2,2} y ]^2. \end{split} \end{equation} Combining (\ref{eq4.1}) and (\ref{eq4.2}) we obtain \begin{equation} \label{eq4.3} r_1^2 \leq \frac{[r_2 + | (g'_u)_{1,2} (g_u)_{2,2} x + (g'_u)_{1,2} (g_u)_{2,2} y |]^2}{\lambda^{2n} | (g'_u)_{1,1}|^2} + \lambda^{-2n} [(g_u)_{2,1} x + (g_u)_{2,2} y]^2 \end{equation} for every $(x,y)^t \in (\Lambda^n g'_u \Lambda^n g_u)^{-1} C_{r_1,r_2} \cap (\Lambda^n g_u)^{-1} C_{r_1,r_2}$. If $(x,y)^t$ belongs to $C_{r_1,r_2}$, then there is a constant $M >0$ such that \[ \frac{| (g'_u)_{1,2} (g_u)_{2,2} x + (g'_u)_{1,2} (g_u)_{2,2} y |]^2}{ | (g'_u)_{1,1}|^2} + [(g_u)_{2,1} x + (g_u)_{2,2} y]^2 \leq M \nonumber \] and this constant does not depend on the choice of $g$, $g'$ in $F$. So, by (\ref{eq4.3}), for $n$ large enough \[ (\Lambda^n g'_u \Lambda^n g_u)^{-1} C_{r_1,r_2} \cap (\Lambda^n g_u)^{-1} C_{r_1,r_2} \cap C_{r_1,r_2} = \emptyset, \nonumber \] which entails \[ C_{r_1,r_2} \cap \Lambda^n g'_u C_{r_1,r_2} \cap \Lambda^n g'_u \Lambda^n g_u C_{r_1,r_2}=\emptyset \nonumber \] and so \[ u^{-1}C_{r_1,r_2} \cap h^n g' u^{-1}C_{r_1,r_2} \cap h^n g' h^n g u^{-1}C_{r_1,r_2}=\emptyset. \nonumber \] The result follows since $C \subset u^{-1} C_{r_1 , r_2}$. \\ Hence we have determined a class of discrete subgroups of $\SL(2,\mathbb{R})$ whose action on $\mathbb{R}^2 \backslash \{0\}$ is squeezing. Conditions under which this action is contractive or paradoxical are the content of the following \begin{pro} \label{prop4.2} Let $G$ be a discrete subgroup of $\SL(2,\mathbb{R})$ acting on $\mathbb{R}^2 \backslash \{ 0 \}$ by means of matrix multiplication of vectors. If $G$ contains a hyperbolic element, then the action is contractive. If $G$ contains at least two hyperbolic elements with different axes, then the action is paradoxical. \end{pro} \proof Suppose that $G$ contains a hyperbolic element, then the same is true for its image under the quotient map $p : \SL(2,\mathbb{R}) \rightarrow \PSL(2,\mathbb{R})$. Since the action of $\Gamma = p (G)$ on $\mathbb{R}\mathbb{P}^1$ is by homeomorphisms and every hyperbolic element in $\Gamma$ is conjugated in $\PSL(2,\mathbb{R})$ to a Moebius transformation of the form $z \mapsto \lambda^2 z$ for some $\lambda >1$, it follows that the action of $\Gamma$ on $\mathbb{R}\mathbb{P}^1$ is contractive. Hence there are $U \subset \mathbb{R}\mathbb{P}^1$ and $\gamma \in \Gamma$ such that \begin{equation} \gamma \overline{U} \subsetneq U. \nonumber \end{equation} In the case $G$ contains at least two hyperbolic elements with different axes, then the same is true for $\Gamma$ and as is well known, in this case $\Gamma$ contains a countable subset of hyperbolic elements with different axes. In order to see this, let $\gamma$, $\eta$ be hyperbolic elements in $\Gamma$ with different axes; then the elements in the sequence $\{ \eta^n \gamma \eta^{-n}\}_{n \in \mathbb{N}}$ are hyperbolic transformations with different axes. In particular, for every $n, m \geq 2$ natural numbers there are group elements $\gamma_1 ,..., \gamma_{n+m}$ and contractive open sets $U_1 ,..., U_{n+m}$, where for each $i=1,...,n+m$ $U_i$ contains the attracting fixed point $\gamma_i^+$ of $\gamma_i$, such that \begin{equation} \label{eq2} \bigcup_{i=1}^n U_i = \bigcup_{j=n+1}^{n+m} U_j = \mathbb{R}\mathbb{P}^1, \end{equation} \begin{equation} \label{eq3} \gamma_i U_i \cap \gamma_j U_j =\emptyset \qquad \forall i\neq j. \end{equation} Hence, we just need to observe that the same holds after replacing the sets $U_i$ with $\pi^{-1} (U_i)$ and the elements $\gamma_i$ with some representatives in $G$. Equation (\ref{eq2}) automatically holds for the sets $\pi^{-1} (U_i) \subset \mathbb{R}^2 \backslash \{0\}$. Choose a representative $g_i \in G$ for every $\gamma_i \in \Gamma$; since the action of $G$ on $\mathbb{R}\mathbb{P}^1$ factors through the action of $\Gamma$, equation (\ref{eq3}) can be replaced by \[ g_i U_i \cap g_j U_j =\emptyset \qquad \forall i\neq j. \nonumber \] By equivariance of the quotient map $\pi : \mathbb{R}^2 \backslash \{0\} \rightarrow \mathbb{R} \mathbb{P}^1$ it follows that \[ g_i (\pi^{-1} (U_i)) \cap g_j (\pi^{-1} (U_j)) = \emptyset \qquad \forall i\neq j. \nonumber \] We are left to check that the inverse image of a contractive open set is again a contractive open set. Since the map $\mathbb{R}^2 \backslash \{0\} \rightarrow \mathbb{R}\mathbb{P}^1$ is a quotient by a group action (the group is $\mathbb{R}^\times$), it is open and so the inverse image of the closure of a set is the closure of the inverse image of the same set; hence, if $(U, g)$ is a contractive pair with $U \subset \mathbb{R}\mathbb{P}^1$ and $g \in G$, then \[ g \overline{( \pi^{-1} (U))} = g \pi^{-1} (\overline{U}) = \pi^{-1} (g \overline{U}) \subsetneq \pi^{-1} (U). \nonumber \] The proof is complete. \begin{cor} \label{cor4.1} Let $G$ be a discrete subgroup of $\SL(2,\mathbb{R})$ such that $p (G) \subset \PSL(2,\mathbb{R})$ is a Fuchsian group containing two hyperbolic elements with different axes. The transformation group $C^*$-algebra $C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G$ is stable.\\ If $p(G)$ is generated by a hyperbolic transformation, then $\wsr (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G) =1$ and $M(C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)$ is infinite. \end{cor} \proof Follows from Proposition \ref{prop4.2}, Proposition \ref{prop4.1}, Theorem \ref{thm3.1} and Corollary \ref{cor3.3.1}.\\ Corollary \ref{cor4.1} applies to the case of discrete subgroups of $\SL(2,\mathbb{R})$ associated to Fuchsian groups of the first kind (\cite{katok} 4.5), hence in particular the cocompact ones. Non-lattice subgroups to which Corollary \ref{cor4.1} applies are considered in \cite{semenova}.\\ In Proposition \ref{prop4.2} we deduced paradoxicality for the action of a discrete subgroup $G$ of $\SL(2,\mathbb{R})$ on $\mathbb{R}^2 \backslash \{0\}$ from paradoxicality of the action of the corresponding Fuchsian group on $\mathbb{R} \mathbb{P}^1$ and concluded from this fact that the multiplier algebra of $C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G$ is properly infinite. It follows from \cite{glasner} Example VII.3.6 that if $\Gamma$ is a Fuchsian group of the first kind, then its action on $\mathbb{R} \mathbb{P}^1$ is extremely proximal (see \cite{glasner} page 96 for the definition) and this property represents a stronger form of paradoxicality, hence stronger infiniteness properties for the multiplier algebra of the transformation group $C^*$-algebra are expected in this case. Note that in \cite{boundary} an extremely proximal action is called a strong boundary action.\\ The next Proposition is a consequence of the results contained in \cite{boundary} and \cite{kra}. \begin{lem} \label{lem4.4} Let $G$ and $H$ be locally compact groups and $A$, $B$ be $C^*$-algebras. Suppose $G$ acts on $A$ and $H$ acts on $B$ and that there is an equivariant involutive homomorphism $\phi : C_c (G,A) \rightarrow C_c (H,B)$ which is continuous for the $L^1$-norms. Then there is a $*$-homomorphism $\hat{\phi} : A\rtimes G \rightarrow B\rtimes H$. \end{lem} \proof If $\rho: L^1 (H,B) \rightarrow \mathfrak{H}$ is a nondegenerate $L^1$-continuous involutive representation of $L^1 (H, B)$, then the composition $\rho \circ \phi : C_c (G, A) \rightarrow \mathfrak{H}$ is $L^1$-continuous as well. Hence $\| \phi (f) \| \leq \| f \|$ for every $f \in C_c (G,A)$ by \cite{williams} Corollary 2.46, as claimed. \begin{pro} \label{prop123} Let $G$ be a discrete subgroup of $\SL(2,\mathbb{R})$ such that $p(G) \subset \PSL(2,\mathbb{R})$ is a finitely generated Fuchsian group of the first kind not containing elements of order $2$. Then $M(C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)$ contains a Kirchberg algebra in the UCT class as a unital $C^*$-subalgebra. \end{pro} \proof The quotient map $p : \mathbb{R}^2 \backslash \{0\} \rightarrow \mathbb{R} \mathbb{P}^1$ is surjective and equivariant with respect to the action of $G$, hence it induces a unital $*$-homomorphism $ C(\mathbb{R} \mathbb{P}^1) \rtimes G \rightarrow C_b (\mathbb{R}^2 \backslash \{0\} )\rtimes G$ which can be composed with the unital $*$-homomorphism $C_b (\mathbb{R}^2 \backslash \{0\})\rtimes G \rightarrow M(C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G)$ introduced at the beginning of Section 3 in order to obtain a unital $*$-homomorphism $\phi : C(\mathbb{R} \mathbb{P}^1)\rtimes G \rightarrow M(C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G)$. By \cite{kra}, finitely generated Fuchsian groups of the first kind not admitting elements of order $2$ lift to $\SL(2,\mathbb{R})$. Denote by $\kappa : \Gamma \rightarrow \kappa (\Gamma) \subset G$ a lift. Since the action of $G$ on $\mathbb{R}\mathbb{P}^1$ factors through the action of $\Gamma$, the map \[ \psi_c : C_c (\Gamma , C(\mathbb{R} \mathbb{P}^1)) \rightarrow C_c (\kappa (\Gamma) , C(\mathbb{R}\mathbb{P}^1)) \nonumber \] \[ f \mapsto f \circ \kappa^{-1} \nonumber \] is an involutive homomorphism and it preserves the $L^1$-norm, as well as the inclusion $C_c (\kappa (\Gamma), C(\mathbb{R} \mathbb{P}^1)) \rightarrow C_c (G, C(\mathbb{R} \mathbb{P}^1))$. By Lemma \ref{lem4.4} there is a (unital) $*$-homomorphism $\psi : C(\mathbb{R}\mathbb{P}^1)\rtimes \Gamma \rightarrow C(\mathbb{R}\mathbb{P}^1) \rtimes G$. Hence $\phi \circ \psi: C(\mathbb{R} \mathbb{P}^1 ) \rtimes \Gamma \rightarrow M(C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)$ is a unital $*$-homomorphism. By \cite{boundary} Theorem 5 the $C^*$-algebra $C(\mathbb{R} \mathbb{P}^1 ) \rtimes \Gamma$ is a unital Kirchberg algebra in the UCT class, hence $\psi \circ \phi$ is injective and the result follows. \section{Cocompact subgroups of $\SL(2,\mathbb{R})$} Consider the one-parameter subgroup of $\SL(2,\mathbb{R})$ \[ N:= \{ n(t) \in \SL(2,\mathbb{R}) \; | \; n(t)=\left( \begin{array}{cc} 1 & t \\ 0 & 1 \end{array}\right), \quad t \in \mathbb{R} \}. \nonumber \] Given a discrete subgroup $G$ of $\SL(2,\mathbb{R})$, one can define a flow on the corresponding homogeneous space $G \backslash \SL(2,\mathbb{R})$ by $Gg \mapsto Gg n(-t)$; this is called the \textit{horocycle flow} (\cite{ew} 11.3.1). The stabilizer of the point $(1,0)^t$ in $\mathbb{R}^2 \backslash \{0\}$ for the action of $\SL(2,\mathbb{R})$ is $N$ and so the quotient $\SL(2,\mathbb{R})/N$, endowed with the action of $\SL(2,\mathbb{R})$ given by left multiplication, is isomorphic, as a dynamical system, to $\mathbb{R}^2 \backslash \{0\}$. The interplay between the action of $G$ on $\mathbb{R}^2 \backslash \{0\}$ and the horocycle flow on $G \backslash \SL(2,\mathbb{R})$ is employed in the following \begin{pro} \label{prop2.1} Let $G$ be a discrete cocompact subgroup of $\SL(2,\mathbb{R})$. The transformation group $C^*$-algebra $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is simple, separable, stable, $\mathcal{Z}$-stable, with a unique lower semicontinuous $2$-quasitrace and it has almost stable rank $1$. In particular it satisfies the hypothesis of \cite{io} Theorem 3.5. \end{pro} \proof Since $G$ is countable and $\mathbb{R}^2 \backslash \{0\}$ is a locally compact second countable Hausdorff space, $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is separable.\\ As already observed in the discussion after Corollary \ref{cor4.1}, $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is stable in the case $G$ is cocompact. Since the action of $G$ on $\mathbb{R}^2 \backslash \{0\}$ is free and minimal (\cite{ergtopdyn} Theorem IV.1.9), $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is simple (\cite{archbold-spielberg}). By simplicity, the non-trivial lower semicontinuous traces on $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ are semifinite (\cite{dixmier} 6.1.3) and so, in virtue of \cite{green2} Proposition 25 and Proposition 26, the restriction map sets up a bijection with the lower semicontinuous semifinite $G$-invariant traces on $C_0 (\mathbb{R}^2 \backslash \{0\})$. Every such trace is uniquely given by integration against a $G$-invariant Radon measure. By Furstenberg Theorem (\cite{furstenberg}) there is exactly one such non-trivial measure. Hence $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ admits a unique non-trivial lower semicontinuous trace. Since the action of $G$ on $\mathbb{R}^2 \backslash \{0\}$ is amenable, $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is nuclear (\cite{delaroche2} Theorem 3.4). By exactness it admits a unique non-trivial lower semicontinuous $2$-quasitrace (\cite{kirchberg}).\\ In virtue of Corollary 9.1 and Corollary 6.7 of \cite{hsww} the $C^*$-algebra $C(G \backslash \SL(2,\mathbb{R})) \rtimes N$ is stable; hence it follows from Green's imprimitivity Theorem (\cite{williams} Corollary 4.11) that $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G \simeq C(G \backslash \SL(2,\mathbb{R})) \rtimes N$. By \cite{hsww} Corollary 9.1 and Theorem 3.5, $C(G \backslash \SL(2,\mathbb{R}) )\rtimes N$ has finite nuclear dimension; hence, \cite{tikusis} Corollary 8.7 entails $\mathcal{Z}$-stability.\\ As observed in \cite{noncommgeom} page 129, $C(G\backslash \SL(2,\mathbb{R})) \rtimes N$ is projectionless; hence, \cite{zstable_projless} Corollary 3.2 applies and $\asr (C(G\backslash \SL(2,\mathbb{R})) \rtimes N) =1$.\\ The result follows since the Cuntz semigroup of a stable $\mathcal{Z}$-stable $C^*$-algebra is almost unperforated (\cite{rordam-sr} Theorem 4.5).\\ \begin{rem} The stability of $C_0 (\mathbb{R}^2 \backslash \{0\}) G$ in Proposition \ref{prop2.1} can also be established directly from that of $C(G \backslash \SL(2,\mathbb{R})) \rtimes N$. In fact, the rest of the proof shows that $C(G \backslash \SL(2,\mathbb{R})) \rtimes N$ satisfies the hypothesis of \cite{io} Theorem 3.5. Since $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is a hereditary $C^*$-subalgebra of $C(G \backslash \SL(2,\mathbb{R})) \rtimes N$, it is then enough to prove that the non-trivial lower semicontinuous trace on $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is unbounded. But this follows since it is induced by the Lebesgue measure. \end{rem} As a consequence we obtain the following properties for the $C^*$-algebra associated to the action of a cocompact discrete subgroup of $\SL(2,\mathbb{R})$ on $\mathbb{R}^2 \backslash\{0\}$ \begin{cor} \label{horo_1} Let $G$ be a cocompact discrete subgroup of $\SL(2,\mathbb{R})$, $\tau$ the lower semicontinuous trace associated to the Lebesgue measure $\mu_L$ on $\mathbb{R}^2 \backslash \{0\}$ and $d_\tau$ the corresponding functional on the Cuntz semigroup $Cu (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)$. Then \[ \Ped (C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G)=\{ x \in C(\mathbb{R}^2 \backslash \{0\}) \rtimes G\; : \; d_\tau ([|x|]) < \infty\}. \nonumber \] Every hereditary $C^*$-subalgebra of $C(\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is either algebraically simple or isomorphic to $C(\mathbb{R}^2 \backslash \{0\}) \rtimes G$. \end{cor} \begin{cor} \label{horo_3} Let $G$ be a cocompact discrete subgroup of $\SL(2,\mathbb{R})$. Every countably generated right Hilbert module for $C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G$ is isomorphic to one of the form \[ \overline{f \cdot (C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G)}, \qquad f \in C_0 (\mathbb{R}^2 \backslash \{0\}). \nonumber \] For two such Hilbert modules we have \[ \overline{f \cdot (C_0 (\mathbb{R}^2 \backslash \{0\}) \rtimes G)} \simeq \overline{g \cdot (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)} \qquad \Leftrightarrow \qquad \mu_L (\supp (f)) = \mu_L (\supp (g)) \nonumber \] and there exists a Hilbert module $E$ such that \[ \overline{f \cdot (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)} \simeq E \Subset \overline{g \cdot (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G)} \nonumber \] if and only if $\mu_L (\supp (f)) < \mu_L (\supp (g))$. \end{cor} \proof The Cuntz semigroup of the $C^*$-algebra $C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes G$ is stably finite by \cite{cuntz_t} Proposition 5.2.10, hence by \cite{cuntz_t} Proposition 5.3.16 it does not contain compact elements, since this $C^*$-algebra is projectionless. Hence the countably generated Hilbert modules correspond to soft elements and Cuntz equivalence of soft elements is implemented by the unique (up to scalar multiples) nontrivial functional associated to the unique (up to scalar multiples) lower semicontinuous trace $\tau$. It follows that all the possible values in the range of the dimension function are obtained by Cuntz equivalence classes of elements in $C_0 (\mathbb{R}^2 \backslash \{0\})$ since, for every $f \in C_0 (\mathbb{R}^2 \backslash \{0\})$, we have $d_\tau (f) = \mu_L (\supp (f))$. The result follows from Theorem 3.5 of \cite{io}. \section{Final remarks} It follows from the results in the last section that if $G$ is a cocompact discrete subgroup of $\SL(2,\mathbb{R})$, the Cuntz classes of elements in the transformation group $C^*$-algebra $C_0 (\mathbb{R}^2 \backslash\{0\})\rtimes G$ are generated by continuous functions on the plane. It might be possible that this property can be derived from the dynamics. It can be shown that if we restrict to discrete subgroups of $\SL(2,\mathbb{R})$ which are the inverse images under the quotient map $p: \SL(2,\mathbb{R}) \rightarrow \PSL(2,\mathbb{R})$ of fundamental groups of hyperbolic Riemann surfaces, the construction of the $C^*$-algebra associated to the horocycle flow on the corresponding homogeneous space of $\SL(2,\mathbb{R})$ induces a functor from a category whose objects are hyperbolic Riemann surfaces and the morphisms are finite sheeted holomorphic coverings to the usual category of $C^*$-algebras; this suggests that it might be possible do detect the holomorphic structure at the $C^*$-algebraic level. Observe that, if $\mathcal{M}_g$ is a compact Riemann surface of genus $g$, after identifying $p^{-1} (\pi_1 (\mathcal{M}_g)) \backslash \SL(2,\mathbb{R})$ with the unit tangent bundle $T_1 (\mathcal{M}_g)$, the Thom-Connes isomorphism (\cite{thom-connes}) gives a way to compute the $K$-theory of $C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes p^{-1}(\pi_1 (\mathcal{M}_g)) \simeq C(T_1 (\mathcal{M}_g))\rtimes \mathbb{R}$ and it reads \[ K_0 (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes p^{-1}(\pi_1 (\mathcal{M}_g))) = \mathbb{Z}^{2g+1}, \nonumber \] \[ K_1 (C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes p^{-1}(\pi_1 (\mathcal{M}_g)))= \mathbb{Z}^{2g+1} \oplus \mathbb{Z} /(2g-2) . \nonumber \] Both the order and the scale in $K_0$ are trivial since $C_0 (\mathbb{R}^2 \backslash \{0\})\rtimes p^{-1}(\pi_1 (\mathcal{M}_g))$ is projectionless and stable.\\ Furthermore, by \cite{thom-connes} Corollary 2 the range of the pairing between $K_0$ and the unique trace is determined by the Ruelle-Sullivan current associated to this flow (see \cite{noncommgeom} 5-$\alpha$), which is trivial by \cite{paternain}. Thus the Elliott invariant contains information only about the genus, or equivalently, the homeomorphic class of the Riemann surface. In particular, if the Elliott conjecture holds true for this class of $C^*$-algebras and if it is possible to detect the holomorphic structure at the level of the $C^*$-algebras, this should correspond to something finer than the $C^*$-algebraic structure. \section{Aknowledgements} The author thanks Prof. Wilhelm Winter for the hospitality at the Westf\"alische Wilhelms-Universit\"at of M\"unster and Prof. Roberto Longo for the hospitality at the Universit\`a degli Studi di Roma Tor Vergata for the period of this research. Many thanks go to Prof. Ludwik D\k abrowski who carefully read and gave his important feedback on the parts of this paper that are contained in the author's PhD thesis. The author also thanks the anonymous referee for the valuable comments on a previous version of the manuscript which led to an improved exposition. This research is partially supported by INdAM. \end{document}

\begin{document} \title{ Reduced Free Products of Finite Dimensional $C^*$-Algebras } \par \author{ Nikolay A. Ivanov } \date{\today} \address{\hskip-\parindent Nikolay Ivanov \\ Department of Mathematics \\ Texas A\&M University \\ College Station TX 77843-3368, USA} \email{[email protected]} \begin{abstract} We find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional $C^*$-algebras with specified traces on them. \end{abstract} \maketitle \section{Introduction and Definitions} The notion of reduced free product of a family of $C^*$-algebras with specified states on them was introduced independently by Avitzour (\cite{A82}) and Voiculescu (\cite{V85}). We will recall this notion and some of its properties here. \par \begin{defi} The couple $(A,\phi)$, where $A$ is a unital $C^*$-algebra and $\phi$ a state is called a $C^*$-noncommutative probability space or $C^*$-NCPS. \end{defi} \par \begin{defi} Let $(A,\phi)$ be a $C^*$-NCPS and $\{ A_i | i \in I \}$ be a family of $C^*$-subalgebras of $A$, s.t. $1_A \in A_i$, $\forall i\in I$, where $I$ is an index set. We say that the family $\{ A_i |i \in I \}$ is free if $\phi(a_1...a_n)=0$, whenever $a_j \in A_{i_j} $ with $i_1\neq i_2\neq ... \neq i_n$ and $\phi(a_j)=0$, $\forall j \in \{ 1,...n \}$. A family of subsets $\{ S_i | i \in I \}$ $\subset$ $A$ is $*$-free if $\{ C^*(S_i \cup \{ 1_A \} ) | i \in I \}$ is free. \end{defi} Let $\{ (A_i,\phi_i) | i \in I \}$ be a family of $C^*$-NCPS such that the GNS representations of $A_i$ associated to $\phi_i$ are all faithful. Then there is a unique $C^*$-NCPS $(A,\phi) \overset{def}{=} \underset{i \in I}{*} (A_i,\phi_i)$ with unital embeddings $A_i \hookrightarrow A$, s.t. \\ (1) $\phi|_{A_i}=\phi_i$ \\ (2) the family $\{ A_i | i \in I \}$ is free in $(A,\phi)$ \\ (3) $A$ is the $C^*$-algebra generated by $\underset{i \in I}{\bigcup}A_i$ \\ (4) the GNS representation of $A$ associated to $\phi$ is faithful. \\ And also: \\ (5) If $\phi_i$ are all traces then $\phi$ is a trace too (\cite{V85}). \\ (6) If $\phi_i$ are all faithful then $\phi$ is faithful too (\cite{D98}). \par In the above situation $A$ is called the reduced free product algebra and $\phi$ is called the free product state. Also the construction of the reduced free product is based on defining a free product Hilbert space, which turns out to be $\mathfrak{H}_A$ - the GNS Hilbert space for $A$, associated to $\phi$. \par \begin{example} If $\{ G_i | i \in I \}$ is a family of discrete groups and $C^*_r(G_i)$ are the reduced group $C^*$-algebras, corresponding to the left regular representations of $G_i$ on $l^2(G_i)$ respectively, and if $\tau_i$ are the canonical traces on $C^*_r(G_i)$, $i \in I$, then we have $\underset{i \in I}{*} (C^*_r(G_i), \tau_i)=(C^*_r(\underset{i \in I}{*}G_i), \tau)$, where $\tau$ is the canonical trace on the group $C^*$-algebra $C^*_r(\underset{i \in I}{*} G_i)$. \end{example} Reduced free products satisfy the following property: \begin{lemma}[\cite{DR98}] Let $I$ be an index set and let $(A_i,\phi_i)$ be a $C^*$-NCPS ($i \in I$), where each $\phi_i$ is faithful. Let $(B,\psi)$ be a $C^*$-NCPS with $\psi$ faithful. Let \begin{center} $(A,\phi) = \underset{i\in I}{*} (A_i,\phi_i)$. \end{center} Given unital $*$-homomorphisms, $\pi_i : A_i \rightarrow B$, such that $\psi \circ \pi_i = \phi_i$ and $\{ \pi_i(A_i) \}_{i\in I}$ is free in $(B, \psi)$, there is a $*$-homomorphism, $\pi : A \rightarrow B$ such that $\pi|_{A_i} = \pi$ and $\psi \circ \pi = \phi$. \end{lemma} \par From now on we will be concerned only with $C^*$-algebras equipped with tracial states. \par The study of simplicity and uniqueness of trace for reduced free products of $C^*$-algebras, one can say, started with the paper of Powers \cite{P75}. In this paper Powers proved that the reduced $C^*$-algebra of the free group on two generators $F_2$ is simple and has a unique trace - the canonical one. In \cite{C79} Choi showed the same for the "Choi algebra" $C_r^*(\mathbb{Z}_2 * \mathbb{Z}_3)$ and then Paschke and Salinas in \cite{PS79} generalized the result to the case of $C_r^*(G_1 * G_2)$, where $G_1, G_2$ are discrete groups, such that $G_1$ has at least two and $G_2$ at least three elements. After that Avitzour in \cite{A82} gave a sufficient condition for simplicity and uniqueness of trace for reduced free products of $C^*$-algebras, generalizing the previous results. He proved: \begin{thm}[\cite{A82}] Let \begin{equation*} (\mathfrak{A}, \tau) = (A, \tau_A) * (B, \tau_B), \end{equation*} where $\tau_A$ and $\tau_B$ are traces and $(A,\tau_A)$ and $(B,\tau_B)$ have faithful GNS representations. Suppose that there are unitaries $u,v \in A$ and $w \in B$, such that $\tau_A(u) = \tau_A(v) = \tau_A(u^* v) = 0$ and $\tau_B(w) = 0$. Then $\mathfrak{A}$ is simple and has a unique trace $\tau$. \end{thm} {\em Note:} It is clear that $uw$ satisfies $\tau((uw)^n) = 0$, $\forall n \in \mathbb{Z} \backslash \{ 0 \}$. Unitaries with this property we define below. \section{Statement of the Main Result and Preliminaries} We adopt the following notation: \\ If $A_0$, ... , $A_n$ are unital $C^*$-algebras equipped with traces $\tau_0$, ... , $\tau_n$ respectively, then $A=\underset{\alpha_0}{\overset{p_0}{A_0}} \bigoplus \underset{\alpha_1}{\overset{p_1}{A_1}} \bigoplus ... \bigoplus \underset{\alpha_n}{\overset{p_n}{A_n}}$ will mean that the $C^*$-algebra $A$ is isomorphic to the direct sum of $A_0$, ... , $A_n$, and is such that $A_i$ are supported on the projections $p_i$. Also $A$ comes with a trace (let's call it $\tau$) given by the formula $\tau=\alpha_0\tau_0 + \alpha_1\tau_1 + ... + \alpha_n\tau_n$. Here of course $\alpha_0$, $\alpha_1$, ... , $\alpha_n > 0$ and $\alpha_0 + \alpha_1 + ... + \alpha_n = 1$. \begin{defi} If $(A,\tau)$ is a $C^*$-NCPS and $u\in A$ is a unitary with $\tau(u^n)=0$, $\forall n \in \mathbb{Z} \backslash \{ 0 \}$, then we call $u$ a Haar unitary. \par If $1_A \in B \subset A$ is a unital abelian $C^*$-subalgebra of $A$ we call $B$ a diffuse abelian $C^*$-subalgebra of $A$ if $\tau|_B$ is given by an atomless measure on the spectrum of $B$. We also call $B$ a unital diffuse abelian $C^*$-algebra. \end{defi} From Proposition 4.1(i), Proposition 4.3 of \cite{DHR97} we can conclude the following: \begin{prop} If $(B,\tau)$ is a $C^*$-NCPS with $B$-abelian, then $B$ is diffuse abelian if and only if $B$ contains a Haar unitary. \end{prop} $C^*$-algebras of the form $(\underset{\alpha}{\overset{p}{\mathbb{C}}} \bigoplus \underset{1-\alpha}{\overset{1-p}{\mathbb{C}}})*(\underset{\beta}{\overset{q}{\mathbb{C}}}\bigoplus \underset{1-\beta}{\overset{1-q}{\mathbb{C}}})$ have been described explicitly in \cite{ABH91} (see also \cite{D99LN}): \begin{thm} Let $1 > \alpha \geqq \beta \geqq \frac{1}{2}$ and let \begin{center} $( A,\tau ) = ( \underset{\alpha }{\overset{p}{\mathbb{C}}} \oplus \underset{1-\alpha }{\overset{1-p}{\mathbb{C}}} ) * ( \underset{\beta }{\overset{q}{\mathbb{C}}}\oplus \underset{1-\beta}{\overset{1-q}{\mathbb{C}}} ) $. \end{center} If $\alpha > \beta$ then \begin{equation*} A=\underset{\alpha -\beta }{\overset{p\wedge (1-q)}{\mathbb{C}}}\oplus C([a,b], M_2(\mathbb{C}))\oplus \underset{\alpha + \beta -1}{\overset{p\wedge q}{\mathbb{C}}} , \end{equation*} for some $0 < a < b < 1$. Furthermore, in the above picture \begin{center} $p=1 \oplus \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \oplus 1 ,$ \end{center} \begin{equation*} q=0\oplus \begin{pmatrix} t & \sqrt{t(1-t)} \\ \sqrt{t(1-t)} & 1-t \end{pmatrix} \oplus 1 , \end{equation*} and the faithful trace $\tau$ is given by the indicated weights on the projections $p\wedge (1-q)$ and $p\wedge q$, together with an atomless measure, whose support is $[a,b]$. \par If $\alpha =\beta > \frac{1}{2}$ then \begin{equation*} A=\{\ f:[0,b]\rightarrow M_2(\mathbb{C}) |\ f\ is\ continuous\ and\ f(0)\ is\ diagonal\ \} \oplus \underset{\alpha + \beta -1}{\overset{p\wedge q}{\mathbb{C}}}, \end{equation*} for some $0 < b < 1$. Furthermore, in the above picture \begin{center} $p= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \oplus 1,$ \end{center} \begin{equation*} q= \begin{pmatrix} {t} & {\sqrt{t(1-t)}} \\ {\sqrt{t(1-t)}} & {1-t} \end{pmatrix} \oplus 1, \end{equation*} and the faithful trace $\tau$ is given by the indicated weight on the projection $p\wedge q$, together with an atomless measure on $[0,b]$. \par If $\alpha = \beta = \frac{1}{2}$ then \begin{equation*} A=\{\ f:[0,1]\rightarrow M_2(\mathbb{C}) |\ f\ is\ continuous\ and\ f(0)\ and\ f(1)\ are\ diagonal\ \}. \end{equation*} Furthermore in the above picture \begin{center} $p= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} ,$ \end{center} \begin{equation*} q= \begin {pmatrix} t & \sqrt{t(1-t)} \\ \sqrt{t(1-t)} & 1-t \end{pmatrix} , \end{equation*} and the faithful trace $\tau$ is given by an atomless measure, whose support is $[0,1]$. \end{thm} The question of describing the reduced free product of a finite family of finite dimensional abelian $C^*$-algebras was studied by Dykema in \cite{D99}. He proved the following theorem: \begin{thm}[\cite{D99}] Let \begin{equation*} (\mathfrak{A},\phi )=(\underset{\alpha_0}{\overset{p_0}{A_0}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_n}{\overset{p_n}{\mathbb{C}}})*(\underset{\beta_0}{\overset{q_0}{B_0}} \oplus \underset{\beta_1}{\overset{q_1}{\mathbb{C}}} \oplus ... \oplus \underset{\beta_m}{\overset{q_m}{\mathbb{C}}}), \end{equation*} where $\alpha_0 \geq 0$ and $\beta_0 \geq 0$ and $A_0$ and $B_0$ are equipped with traces $\phi(p_0)^{-1} \phi|_{A_0}$, $\phi(q_0)^{-1} \phi|_{B_0}$ and $A_0$ and $B_0$ have diffuse abelian $C^*$-subalgebras, and where $n \geq 1$, $m \geq 1$ (if $\alpha_0 = 0$ or $\beta_0 = 0$, or both, then, of course, we don't impose any conditions on $A_0$ or $B_0$, or both respectively). Suppose also that $\dim(A) \geq 2$, $\dim(B) \geq 2$, and $\dim(A) + \dim(B) \geq 5$. \par Then \begin{equation*} \mathfrak{A} = \overset{r_0}{\mathfrak{A}_0} \oplus \underset{(i',j)\in L_+}{\bigoplus} \underset{\alpha_i + \beta_i -1}{\overset{p_i \wedge q_j}{\mathbb{C}}}, \end{equation*} where $L_+ = \{ (i,j)| 1 \leq i \leq n$, $1 \leq j \leq m$ and $\alpha_i + \beta_j > 1 \}$, and where $\mathfrak{A}_0$ has a unital, diffuse abelian sublagebra supported on $r_0 p_1$ and another one supported on $r_0 q_1$. \par Let $ L_0 = \{(i,j)| 1 \leq i \leq n$, $1 \leq j \leq m$ and $\alpha_i + \beta_j = 1 \} .$ \par If $L_0$ is empty then $\mathfrak{A}_0$ is simple and $\phi(r_0)^{-1} \phi|_{\mathfrak{A}_{0}}$ is the unique trace on $\mathfrak{A}_0.$ \par If $L_0$ is not empty, then for each $(i,j) \in L_0$ there is a $*$-homomorphism $\pi_{(i,j)}: \mathfrak{A}_0 \rightarrow \mathbb{C}$ such that $\pi_{(i,j)}(r_0 p_i) = 1 = \pi_{(i,j)}(r_0 q_j).$ Then: \\ (1) $\mathfrak{A}_{00} \overset{def}{=} \underset{(i,j)\in L_0}{\bigcap} \ker (\pi_{(i,j)})$ \\ is simple and nonunital, and $\phi(r_0)^{-1} \phi|_{\mathfrak{A}_{00}}$ is the unique trace on $\mathfrak{A}_{00}.$ \\ (2) For each $i\in \{1,...n \}, \ r_0 p_i$ is full in $\mathfrak{A}_0 \cap \underset{i' \neq i}{\underset{(i',j) \in L_0}{\bigcap}} \ker (\pi_{(i',j)}).$ \\ (3) For each $j \in \{ 1, ... , m \}, \ r_0 q_j$ is full in $\mathfrak{A}_{0} \cap \underset{j' \neq j}{\underset{(i,j') \in L_0}{\bigcap}} \ker (\pi_{(i,j')}).$ \end{thm} One can define von Neumann algebra free products, similarly to reduced free products of $C^*$-algebras. We will denote by $\mathbb{M}_n$ the $C^*$-algebra (von Neumann algebra) of $n \times n$ matrices with complex coefficients. \par Dykema studied the case of von Neumann algebra free products of finite dimensional (von Neumann) algebras: \begin{thm}[\cite{D93}] Let \begin{equation*} A = \underset{\alpha_0}{\overset{p_0}{L(F_s)}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{M}_{n_1}}} \oplus ... \oplus \underset{\alpha_k}{\overset{p_k}{\mathbb{M}_{n_k}}} \end{equation*} and \begin{equation*} B = \underset{\beta_0}{\overset{q_0}{L(F_r)}} \oplus \underset{\beta_1}{\overset{q_1}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\beta_l}{\overset{q_l}{\mathbb{M}_{m_l}}}, \end{equation*} where $L(F_s), L(F_r)$ are interpolated free group factors, $\alpha_0, \beta_0 \geq 0$, and where $\dim(A) \geq 2$, $\dim(B) \geq 2$ and $\dim(A) + \dim(B)\geq 5$. Then for the von Neumann algebra free product we have: \begin{equation*} A*B = L(F_t) \oplus \underset{(i,j) \in L_+}{\bigoplus} \underset{\gamma_{ij}}{\overset{f_{ij}}{\mathbb{M}_{N(i,j)}}}, \end{equation*} where $L_+ = \{(i,j) | 1 \leq i \leq k, 1 \leq j \leq l, (\frac{\alpha_i}{n_i^2}) + (\frac{\beta_j}{m_j^2}) > 1 \}$, $N(i,j) = max(n_i, m_j)$, $\gamma_{ij} = N(i,j)^2 \cdot (\frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} - 1)$, and $f_{ij} \leq p_i \wedge q_j$. \end{thm} {\em Note:} $t$ can be determined from the other data, which makes sense only if the interpolated free group factors are all different. We will use only the fact that $L(F_t)$ is a factor. For definitions and properties of interpolated free group factors see \cite{Ra94} and \cite{D94}. \par In this paper we will extend the result of Theorem 2.4 to the case of reduced free products of finite dimensional $C^*$-algebras with specified traces on them. We will prove: \begin{thm} Let \begin{equation*} (\mathfrak{A},\phi )=(\underset{\alpha_0}{\overset{p_0}{A_0}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{M}_{n_1}}} \oplus ... \oplus \underset{\alpha_k}{\overset{p_k}{\mathbb{M}_{n_k}}})*(\underset{\beta_0}{\overset{q_0}{B_0}} \oplus \underset{\beta_1}{\overset{q_1}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\beta_l}{\overset{q_l} {\mathbb{M}_{m_l}}}), \end{equation*} where $\alpha_0, \beta_0 \geq 0$, $\alpha_i > 0$, for $i=1,..,k$ and $\beta_j > 0$, for $j=1,...,l$, and where $\phi(p_0)^{-1} \phi|_{A_0}$ and $\phi(q_0)^{-1} \phi|_{B_0}$ are traces on $A_0$ and $B_0$ respectivelly. Suppose that $\dim(A) \geq 2$, $\dim(B) \geq 2$, $\dim(A) + \dim(B) \geq 5$, and that both $A_0$ and $B_0$ contain unital, diffuse abelian $C^*$-subalgebras (if $\alpha_0 > 0$, respectivelly $\beta_0 > 0$). Then \begin{equation*} \mathfrak{A}= \underset{\gamma}{\overset{f}{\mathfrak{A}_0}} \oplus \underset{(i,j)\in L_+}{\bigoplus} \underset{\gamma_{ij}}{\overset{f_{ij}}{\mathbb{M}_{N(i,j)}}}, \end{equation*} where $L_+ = \{ (i,j)| \frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} > 1 \}$, $N(i,j) = max(n_i,m_j)$, $\gamma_{ij} = N(i,j)^2(\frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} -1)$, $f_{ij} \leq p_i \wedge q_j$. There is a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on $f p_1$ and another one, supported on $f q_1$. \par If $L_0 = \{ (i,j)| \frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} = 1 \},$ is empty, then $\mathfrak{A}_0$ is simple with a unique trace. If $L_0$ is not empty, then $\forall (i,j) \in L_0 ,\ \exists \pi_{(i,j)} : \mathfrak{A}_{0} \rightarrow \mathbb{M}_{N(i,j)}$ a unital $*$-homomorphism, such that $\pi_{(i,j)}(f p_i) = \pi_{(i,j)}(f q_j) = 1$. Then: \\ (1) $\mathfrak{A}_{00} \overset{def}{=} \underset{(i,j) \in L_0}{\bigcap} \ker (\pi_{(i,j)})$ is simple and nonunital, and has a unique trace $\phi(f )^{-1} \phi |_{\mathfrak{A}_{00}}$. \\ (2) For each $i \in \{ 1, ..., k \}$, $f p_i$ is full in $\mathfrak{A}_0 \cap \underset{i' \neq i}{\underset{(i',j) \in L_0}{\bigcap}} \ker(\pi_{(i',j)})$. \\ (3) For each $j \in \{ 1, ..., l \}$, $f q_j$ is full in $\mathfrak{A}_0 \cap \underset{j' \neq j}{\underset{(i,j') \in L_0}{\bigcap}} \ker(\pi_{(i,j')})$. \end{thm} \section{Beginning of the Proof - A Special Case} In order to prove this theorem we will start with a simpler case. We will study first the $C^*$-algebras of the form $(A,\tau) \overset{def}{=} $ $ ( \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_m}{\overset{p_m} {\mathbb{C}}})*(\mathbb{M}_n, tr_n)$ with $0 < \alpha_1 \leq ... \leq \alpha_m$. We chose a set of matrix units for $\mathbb{M}_n$ and denote them by $\{ e_{ij}|i,j \in \{1,...n \} \} $ as usual. Let's take the (trace zero) permutation unitary $$ u \overset{def}{=} \begin{pmatrix} 0 & 1 & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & 1 \\ 1 & 0 & ... & 0 \end{pmatrix} \in \mathbb{M}_n.$$ \\ We see that $\Ad(u)(e_{11}) = u e_{11} u^* = e_{nn}$ and for $2 \leq i \leq n$, $\Ad(u)(e_{ii}) = u e_{ii} u^* = e_{(i-1) (i-1)}$. \par It's clear that $$A = C^*(\{p_1, ..., p_m \}, \{ e_{ii}\}_{i=1}^n, u).$$ Then it is also clear that $$A = C^*(\{ u^ip_1u^{-i} , ... , u^ip_mu^{-i} \}_{i=0}^{n-1}, \{ e_{ij} \}_{i=1}^{n}, u).$$ We want to show that the family $$\{ \{ \mathbb{C} \cdot u^ip_1u^{-i} \oplus ,..., \oplus \mathbb{C} \cdot u^ip_mu^{-i} \}_{i=0}^{n-1},\ \{ \mathbb{C} \cdot e_{11} \oplus ... \oplus \mathbb{C} \cdot e_{nn} \} \}$$ is free. We will prove something more general. We denote $$B \overset{def}{=} C^*( \{ u^kp_1u^{-k}, ... , u^kp_mu^{-k} \}_{k=0}^{n-1}, \{ e_{11}, ... ,e_{nn} \} ).$$ Let $l$ be an integer and $l|n$, $1 < l < n$ (if such $l$ exists). Let $$E \overset{def}{=} C^*( \{ \{ u^kp_1u^{-k}, ... , u^kp_mu^{-k} \}_{k=0}^{l-1}, \{ e_{11} , ... , e_{nn} \}, \{ u^l, u^{2l}, ... , u^{n-l} \} \} ).$$ It's easy to see that $$C^* ( \{ e_{11}, ... , e_{nn} \}, \{ u^l, u^{2l}, ... , u^{n-l} \} )= \underbrace{\mathbb{M}_{ \frac{n}{l} } \oplus ... \oplus \mathbb{M}_{ \frac{n}{l} }}_{l-times} \subset \mathbb{M}_n.$$ We will adopt the following notation from \cite {D99LN}: \par Let $(D, \varphi)$ be a $C^*$-NCPS and $1_D \in D_1, ..., D_k \subset D$ be a family of unital $C^*$-subalgebras of $D$, having a common unit $1_D$. We denote by $D^{\circ} \overset{def}{=} \{ d\in D | \varphi(d)=0 \}$ (analoguously for $D_1$, ..., etc). We denote by $\Lambda^{\circ}(D_1^{\circ}, D_2^{\circ} , ..., D_k^{\circ})$ the set of all words of the form $d_1 d_2 \cdots d_j$ and of nonzero length, where $d_t \in D_{i_t}^{\circ}$, for some $1 \leq i_t \leq k$ and $i_t \neq i_{t+1}$ for any $1 \leq t \leq j-1$. \\ \par We have the following \begin{lemma} If everything is as above, then: (i) The family $\{ \{ u^kp_1u^{-k} , ... , u^kp_mu^{-k} \}_{k=0}^{n-1},$ $\{ e_{11} , ... ,e_{nn} \} \}$ is free in $(A,\tau)$. And more generally if $$\omega \in \Lambda^{\circ}( C^*(p_1, ..., p_m)^{\circ}, ..., C^*(u^{n-1}p_1u^{1-n}, ..., u^{n-1}p_mu^{1-n})^{\circ}, C^*(e_{11}, ..., e_{nn})^{\circ}),$$ then $\tau(\omega u^r)=0$ for all $0 \leq r \leq n-1$. (ii) The family $\{ \{ u^kp_1u^{-k} , ... , uu^kp_mu^{-k} \}_{k=0}^{l-1},$ $\{ e_{11} , ... , e_{nn} , u^l, u^{2l}, ... u^{n-l} \} \}$ is free in $(A,\tau)$. And more generally if $$\omega \in \Lambda^{\circ}(C^*(p_1,..., p_m)^{\circ},..., C^*(u^{l-1}p_1u^{1-l},..., u^{l-1}p_mu^{1-l})^{\circ}, C^*( e_{11}, ..., e_{nn}, u^l,..., u^{n-l})^{\circ}),$$ then $\tau(\omega u^r)=0$ for all $0 \leq r \leq l-1$. \end{lemma} \begin{proof} Each letter $\alpha \in C^*( \{ u^kp_1u^{-k}, ... , u^kp_mu^{-k} \})$ with $\tau(\alpha) = 0$ can be represented as $\alpha = u^k \alpha' u^{-k}$ with $\tau(\alpha') = 0$, and $\alpha' \in C^*( \{ p_1, ..., p_m \} )$. \par Case (i): \\ \par Each $$\omega \in \Lambda^{\circ}( C^*(p_1, ..., p_m)^{\circ}, ..., C^*(u^{n-1}p_1u^{1-n}, ..., u^{n-1}p_mu^{1-n})^{\circ}, C^*(e_{11}, ..., e_{nn})^{\circ})$$ is of one of the four following types: \begin{equation} \omega = \alpha_{11} \alpha_{12} \cdots \alpha_{1i_1} \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1} \alpha_{t1} \cdots \alpha_{ti_t}, \end{equation} \begin{equation} \omega = \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1} \alpha_{t1} \cdots \alpha_{ti_t}, \end{equation} \begin{equation} \omega = \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1}, \end{equation} \begin{equation} \omega = \alpha_{11} \alpha_{12} \cdots \alpha_{1i_1} \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1}, \end{equation} where $\alpha_{ij} \in C^*(u^{k_{ij}}p_1u^{k_{ij}}, ..., u^{k_{ij}}p_mu^{k_{ij}})^{\circ}$ with $0 \leq k_{ij} \leq n-1$, $k_{ij} \neq k_{i(j+1)}$ and $\beta_i \in C^*(e_{11}, ..., e_{nn})^{\circ}$. \\ \par We consider the following two cases: \\ (a) We look at $\alpha_{ji} \alpha_{ji+1}$ with $\alpha_{jc}$ $\in$ $C^*(\{ u^{k_{c}}p_1u^{-k_{c}}, ... , u^{k_{c}}p_mu^{-k_{c}} \} )^{\circ}$ for $c=i, i+1$. We write $\alpha_{jc} = u^{k_{c}} \alpha'_{jc} u^{-k_{c}}$ with $\alpha'_{jc} \in C^*( \{ p_1, ... , p_m \} )^{\circ}$ for $c = i, i+1$. So $\alpha_{ji} \alpha_{ji+1} =$ \\ $u^{k_i} \alpha'_{ji} u^{k_{i+1} - k_i} \alpha'_{ji+1} u^{-k_{i+1}}$. Here $\alpha'_{ji}$ and $\alpha'_{ji+1}$ are free from $u^{k_{i+1} - k_i}$ in $(A,\tau)$ (Notice that we have $k_{i+1} - k_i \neq 0$). \\ (b) We look at $\alpha_{ji_j} \beta_j \alpha_{(j+1) 1}$ with $\beta \in C^*( \{e_{11} , ... , e_{nn} \} )^{\circ},$ \\ $\alpha_{(j+1)1} \in C^*( \{ u^{k_{j+1}}p_1u^{-k_{j+1}} , ... , u^{k_{j+1}} p_m u^ {-k _{j+1}} \} )^{\circ},$ \\ $\alpha_{ji_j} \in C^*( \{u^{k_j} p_1 u^{-k_j} , ... , u^{k_j} p_m u^{-k_j} \} )^{\circ}$. Now we write $\alpha_{ji_j} = u^{k_j} \alpha'_{ji_j} u^{-k_j}$ and $\alpha_{(j+1)1} = u^{k_{j+1}} \alpha'_{(j+1)1} u^{-k_{j+1}}$ with $\alpha'_{ji_j} , \alpha'_{(j+1)1} \in C^*( \{ p_1 , ..., p_m \} )^{\circ}$. We see that $\alpha_{ji_j} \beta_j \alpha_{(j+1)1} =$ $u^{k_j} \alpha'_{ji_j} u^{-k_j} \beta_j u^{k_{j+1}} \alpha'_{(j+1)1} u^{-k_{j+1}}$. If $k_j = k_{j+1}$ then $\tau(u^{-k_j} \beta_j u^{k_{j+1}}) = \tau(u^{k_{j+1}} u^{-k_j} \beta_j) = \tau(\beta_j) = 0$ since $\tau$ is a trace. If $k_j \neq k_{j+1}$ then $\tau(u^{-k_j} \beta_j u^{k_{j+1}}) = \tau(u^{k_{j+1}} u^{-k_j} \beta_j)$ and $u^{k_{j+1} - k_j} \beta_j \in \mathbb{M}_n$ is a linear combination of off-diagonal elements, so $\tau(u^{k_{j+1}} u^{-k_j} \beta_j) = 0$ also. Notice that $\alpha'_{ji_j}$ and $\alpha'_{(j+1)1}$ are free from $u^{-k_j} \beta_j u^{k_{j+1}}$ in $(A,\tau)$. \\ Now we expand all the letters in the word $\omega$ according to the cases (a) and (b). We see that we obtain a word, consisting of letters of zero trace, such that every two consequitive letters come either from $C^*( \{p_1, ..., p_m \} )$ or from $\mathbb{M}_n$. So $\tau(\omega) = 0$. It only remains to look at the case of the word $\omega u^r$ which is the word $\omega$, but ending in $u^r$. There are two principally different cases for $\omega u^r$ from the all four possible choices for $\omega$: \\ In cases (1) and (2) $\alpha_{ti_t} = u^k \alpha'_{ti_t} u^{-k}$ for some $0 \leq k \leq n-1$ with $\alpha'_{ti_t} \in C^*( \{ p_1 , ..., p_m \} )^{\circ}$. So the word will end in $u^k \alpha'_{ti_t} u^{r-k}$. If $r = k$ then $\alpha'_{ti_t}$ will be the last letter with trace zero and everything else will be the same as for $\omega$, so the whole word will have trace $0$. If $k \neq r$ then $\tau(u^{r-k}) = 0$ and $u^{r-k}$ is free from $\alpha'_{ti_t}$ so the word in this case will be of zero trace too. \\ In cases (3) and (4) if $\beta_{t-1} u^{r}$ is the whole word then $\beta_{t-1} u^{r}$ is a linear combination of off-diagonal elements of $\mathbb{M}_n$, and so its trace is $0$. If not then $\alpha_{(t-1)i_{t-1}} = u^k \alpha'_{(t-1)i_{t-1}} u^{-k}$ with $\alpha'_{(t-1)i_{t-1}} \in C^*( \{ p_1 , ... , p_m \} )^{\circ}$. So the word ends in \\ $ u^k \alpha'_{(t-1)i_{t-1}} u^{-k} \beta_{t-1} u^{r} $. Similarly as above we see that $\tau(u^{-k} \beta_{t-1} u^{r}) = 0$ for all values of $k$ and $r$. The rest of the word we treat as above and conclude that it's of zero trace in this case too. \\ So in all cases $\tau( \omega u^r) = 0$ just what we had to show. \\ \par Case (ii): \\ \par As in case (i) $$\omega \in \Lambda^{\circ}(C^*(p_1,..., p_m)^{\circ},..., C^*(u^{l-1}p_1u^{1-l},..., u^{l-1}p_mu^{1-l})^{\circ}, C^*( e_{11},..., e_{nn}, u^l,..., u^{n-l})^{\circ})$$ is of one of the following types: \\ \begin{equation} \omega = \alpha_{11} \alpha_{12} \cdots \alpha_{1i_1} \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1} \alpha_{t1} \cdots \alpha_{ti_t}, \end{equation} \begin{equation} \omega = \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1} \alpha_{t1} \cdots \alpha_{ti_t}, \end{equation} \begin{equation} \omega = \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1}, \end{equation} \begin{equation} \omega = \alpha_{11} \alpha_{12} \cdots \alpha_{1i_1} \beta_1 \alpha_{21} \cdots \alpha_{2i_2} \beta_2 \alpha_{31} \cdots \alpha_{t-1i_{t-1}} \beta_{t-1}, \end{equation} where $\alpha_{ij} \in C^*(u^{k_{ij}}p_1u^{k_{ij}}, ..., u^{k_{ij}}p_mu^{k_{ij}})^{\circ}$ with $0 \leq k_{ij} \leq l-1$ and $k_{ij} \neq k_{(i+1)j}$ and $\beta_i \in C^*(e_{11}, ..., e_{nn}, u^l, u^{2l}, ..., u^{n-l})^{\circ}$. \\ \par Similarly as case (i) we consider two cases: \\ (a) We look at $\alpha_{ji} \alpha_{ji+1}$ with $\alpha_{jc}$ $\in$ $C^*(\{ u^{k_{c}}p_1u^{-k_{c}}, ... , u^{k_{c}}p_mu^{-k_{c}} \} )$, and $0 \leq k_c \leq l-1$ for $c=i, i+1$. We write $\alpha_{jc} = u^{k_c} \alpha'_{jc} u^{-k_c}$ with $\alpha'_{jc} \in C^*( \{ p_1, ... , p_m \} )^{\circ}$ for $c = i, i+1$. It follows $\alpha_{ji} \alpha_{ji+1} =$ $u^{k_i} \alpha'_{ji} u^{k_{i+1} - k_i} \alpha'_{ji+1} u^{-k_{i+1}}$. Here $\alpha'_{ji}$ and $\alpha'_{ji+1}$ are free from $u^{k_{i+1} - k_i}$ in $(A,\tau)$ (and again $k_{i+1} - k_i \neq 0$). \\ (b) We look at $\alpha_{ji_j} \beta_j \alpha_{(j+1) 1}$ with $\beta_j \in C^*( \{e_{11} , ... , e_{nn} \} , \{ u^l, u^{2l}, ..., u^{n-l} \} )^{\circ},$ \\ $\alpha_{(j+1)1} \in C^*( \{ u^{k_{j+1}}p_1u^{-k_{j+1}} , ... , u^{k_{j+1}} p_m u^ {-k _{j+1}} \} )^{\circ},$ \\ $\alpha_{ji_j} \in C^*( \{u^{k_j} p_1 u^{-k_j} , ... , u^{k_j} p_m u^{-k_j} \} )^{\circ}$, where in this case $k_j, k_{j+1} \in \{ 0, ..., l-1 \}$. Again we write $\alpha_{ji_j} = u^{k_j} \alpha'_{ji_j} u^{-k_j}$ and $\alpha_{(j+1)1} = u^{k_{j+1}} \alpha'_{(j+1)1} u^{-k_{j+1}}$ with $\alpha'_{ji_j} , \alpha'_{(j+1)1} \in C^*( \{ p_1 , ... , p_m \} )^{\circ},$. We have $\alpha_{ji_j} \beta_j \alpha_{(j+1)1} =$ $u^{k_j} \alpha'_{ji_j} u^{-k_j} \beta_j u^{k_{j+1}} \alpha'_{(j+1)1} u^{-k_{j+1}}$. \\ We only need to show that $\tau(u^{-k_j} \beta_j u^{k_{j+1}}) = 0$. $\tau(u^{-k_j} \beta_j u^{k_{j+1}}) = \tau(u^{k_{j+1}} u^{-k_j} \beta_j) = \tau(u^{k_{j+1} - k_j} \beta_j)$. The case $ k_{j+1} = k_j$ is clear. Notice that if $ k_{j+1} \neq k_j$ then $0 < k_{j+1} - k_j \leq l-1$. Is it clear that $u^{k_{j+1} - k_j} \cdot \Span ( \{ e_{11}, ..., e_{nn} \}) \subset \mathbb{M}_n$ consists of liner combination of off-diagonal elements. The same is clear for $u^{k_{j+1} - k_j} \cdot \Span( \{ u^l, u^{2l} , ..., u^{n-l} \} ) \subset \mathbb{M}_n $. It's not difficult to see then that $$u^{k_{j+1} - k_j} \cdot \Alg ( \{ e_{11}, ..., e_{nn} \}, \{ u^l, u^{2l}, ..., u^{n-l} \} )$$ will consist of linear span of the union of the off-diagonal entries among $\{ e_{ij} | 1 \leq i,j \leq n \}$ present in $u^{k_{j+1} - k_j} \cdot \Span( \{e_{11}, ..., e_{nn} \})$ and the ones present in \\ $u^{k_{j+1} - k_j} \cdot \Span( \{ u^l, u^{2l}, ..., u^{n-l} \} )$. This shows that $u^{k_{j+1} - k_j} \beta_j$ will be also a linear span of off-diagonal entries in $\mathbb{M}_n$ and will have trace $0$. So $\tau(u^{-k_j} \beta_j u^{k_{j+1}}) = 0$. In this case also $\alpha'_{ji_j}$ and $\alpha'_{(j+1)1}$ are free from $u^{-k_j} \beta_j u^{k_{j+1}}$ in $(A,\tau)$. \\ We expand all the letters of the word $\omega$ and see that it is of trace $0$ similarly as in case (i). For the word $\omega u^r$ with $0 \leq r \leq l-1$ we argue similarly as in case (i). Again there are two principally different cases: \\ In cases (5) and (6) $\alpha_{ti_t} = u^k \alpha'_{ti_t} u^{-k}$ for some $0 \leq k \leq l-1$ with $\alpha'_{ti_t} \in C^*( \{ p_1 , ..., p_m \} )^{\circ}$. So the word will end in $u^k \alpha'_{ti_t} u^{r-k}$. If $r = k$ then $\alpha'_{ti_t}$ will be the last letter with trace zero and everything else will be the same as for $\omega$, so the whole word will have trace $0$. If $k \neq r$ then $\tau(u^{r-k}) = 0$ and $u^{r-k}$ is free from $\alpha'_{ti_t}$ so the word in this case will be of zero trace too. In cases (7) and (8) $\beta_{t-1} u^r$ then this is a linear combination of off-diagonal elements as we showed in case (ii)-(b). If not we write $\alpha_{(t-1)i_{t-1}} = u^k \alpha'_{(t-1)i_{t-1}} u^{-k}$ with $0 \leq k \leq l-1$ and $\alpha'_{(t-1)i_{t-1}} \in C^*( \{ p_1, ..., p_m \} )^{\circ}$. So the word that we are looking at will end in $u^k \alpha'_{(t-1)i_{t-1}} u^{-k} \beta_{t-1} u^{r} $. Since $0 \leq k,r \leq l-1$ similarly as in case (ii)-(b) we see that $\tau(u^{-k} \beta_{t-1} u^{r}) = 0$. We treat the remaining part of the word as above and conclude that in this case the word has trace $0$. \\ \par So in all cases $\tau(\omega u^r) = 0$ just what we had to show. \\ \par This proves the lemma. \end{proof} From properties (5) and (6) of the reduced free product it follows that $\tau$ is a faithful trace. From Lemma 1.4 it follows that $$B = (\mathbb{C} \cdot e_{11} \oplus ... \oplus \mathbb{C} \cdot e_{nn}) * (\underset{k=0}{\overset{n-1}{*}} (\mathbb{C} \cdot u^k p_1 u^{-k} \oplus ... \oplus \mathbb{C} \cdot u^k p_m u^{-k})),$$ $$\cong ( \underset{\frac{1}{n}}{\mathbb{C}} \oplus ... \oplus \underset{\frac{1}{n}}{\mathbb{C}} ) * (\underset{k=0}{\overset{n-1}{*}} (\underset{\alpha_1}{\mathbb{C}} \oplus ... \oplus \underset{\alpha_m}{\mathbb{C}}))$$ and that $$E = C^*( \{ e_{11}, ..., e_{nn}, u_l, u^{2l}, ..., u^{n-l} \} ) * (\underset{k=0}{\overset{l-1}{*}} (\mathbb{C} \cdot u^k p_1 u^{-k} \oplus ... \oplus \mathbb{C} \cdot u^k p_m u^{-k})),$$ $$\cong (\underset{\frac{l}{n}}{\mathbb{M}_{\frac{n}{l}}} \oplus ... \oplus \underset{\frac{l}{n}}{\mathbb{M}_{\frac{n}{l}}}) * (\underset{k=0}{\overset{l-1}{*}} (\underset{\alpha_1}{\mathbb{C}} \oplus ... \oplus \underset{\alpha_m}{\mathbb{C}})).$$ \begin{cor} If everything is as above: (1) For $b \in B$ and $0 < k \leq n-1$ we have $\tau(b u^k) = 0$, so also $\tau(u^k b) = 0$. \par (2) For $e \in E$ and $0 < k \leq l-1$ we have $\tau(e u^k) = 0$, so also $\tau(u^k e) = 0$. \end{cor} For $(B, \tau|_B)$ and $(E, \tau|_E)$ we have that $\mathfrak{H}_B \subset \mathfrak{H}_E \subset \mathfrak{H}_A$. If $a \in A$ we will denote by $\hat{a} \in \mathfrak{H}_A$ the vector in $\mathfrak{H}_A$, corresponding to $a$ by the GNS construction. We will show that \begin{cor} If everything is as above: \\ (1) $u^{k_1} \mathfrak{H}_B \bot u^{k_2} \mathfrak{H}_B$ for $k_1 \neq k_2$, $0 \leq k_1, k_2 \leq n-1$. \par (2) $u^{k_1} \mathfrak{H}_E \bot u^{k_2} \mathfrak{H}_E$ for $k_1 \neq k_2$, $0 \leq k_1, k_2 \leq l-1$. \end{cor} \begin{proof} (1) Take $ b_1, b_2 \in B $. We have $\langle u^{k_1} \hat{b_1} , u^{k_2} \hat{b_2} \rangle = \tau(u^{k_2} b_2 b_1^* u^{-k_1}) = \tau(b_2 b_1^* u^{k_2 - k_1}) = 0,$ by the above Corollary. \\ (2) Similarly take $e_1, e_2 \in E$, so $\langle u^{k_1} \hat{e_1}, u^{k_2} \hat{e_2} \rangle = \tau(u^{k_2} e_2 e_1^* u^{-k_1}) = \tau(e_2 e_1^* u^{k_2 - k_1}) = 0,$ again by the above Corollary. \end{proof} Now $\mathfrak{H}_A$ can be written in the form $\mathfrak{H}_A = \underset{i=0}{\overset{n-1}{\bigoplus}} u^i \mathfrak{H}_B$ as a Hilbert space because of the Corollary above. Denote by $P_i$ the projection $P_i : \mathfrak{H}_A \rightarrow \mathfrak{H}_A$ onto the subspace $u^i \mathfrak{H}_B$. Now it's also true that $A = \underset{i=0}{\overset{n-1}{\bigoplus}} u^i B$ as a Banach space. To see this we notice that $\Span\{u^iB, i=0, ...n-1\}$ is dense in $A$, also that $u^i B,\ 0 \leq i \leq n-1$ are closed in $A$. Now take a sequence $\{ \sum_{i=0}^{n-1} u^i b_{mi} \}_{m=1}^{\infty}$ converging to an element $a \in A$ ($b_{mi} \in B$). Then for each $i$ we have $\{ P_j \sum_{i=0}^{n-1} u^i b_{mi} P_0 \}_{m=1}^{\infty} = \{ P_j u^j b_{mj} P_0 \}_{m=1}^{\infty}$ converges (to $P_j a P_0$), consequently the sequence $\{ b_{mj} \}_{m=1}^{\infty}$ converges to an element $b_j$ in $B$ $\forall 0 \leq j \leq n-1$. So $a = \sum_{i=0}^{n-1} u^i b_i$. Finally the fact that $u^{i_1} B \cap u^{i_2} B = 0$, for $i_1 \neq i_2$ follows easily from $u^{i_1} \mathfrak{H}_B \cap u^{i_2} \mathfrak{H}_B = 0$, for $i_1 \neq i_2$ and the fact that the trace $\tau$ is faithful. We also have $A = \underset{i=0}{\overset{n-1}{\bigoplus}} B u^i$. \par Let $C$ is a $C^*$-algebra and $\Gamma$ is a discrete group with a given action $\alpha : \Gamma \rightarrow Aut(C)$ on $C$. By $C \rtimes \Gamma$ we will denote the reduced crossed product of $C$ by $\Gamma$. It will be clear what group action we take. \par Let's denote by $G$ the multiplicative group, generated by the automorphism $\Ad(u)$ of $B$. Then $G \cong \mathbb{Z}_n$ and by what we proved above $\mathfrak{H}_A \cong L^2(G,\mathfrak{H}_B)$. \begin{lemma} $A \cong B \rtimes G$ \end{lemma} \begin{proof} We have to show that the action of $A$ on $\mathfrak{H}_A$ "agrees" with the crossed product action. Take $a= \underset{k=0}{\overset{n-1}{\sum}} b_k u^k \in A$, $b_k \in B, k=0, 1, ..., n-1$ and take $\xi = \underset{k=0}{\overset{n-1}{\sum}} u^k \hat{b'_k} \in \mathfrak{H}_A$, $b'_k \in B, k=0, 1, ..., n-1$. Then $$a(\xi) = \underset{k=0}{\overset{n-1}{\sum}} \underset{m=0}{\overset{n-1}{\sum}} b_k u^k u^m \hat{b'_m} = \underset{k=0}{\overset{n-1}{\sum}} \underset{m=0}{\overset{n-1}{\sum}} u^{k+m} . (u^{-k-m} b_k u^{k+m} ) \hat{b'_m},$$ $$= \underset{s=0}{\overset{n-1}{\sum}} \underset{k=0}{\overset{n-1}{\sum}} (u^s . \Ad(u^{-s})(b_k) ) (\widehat{b'_{s-k(mod\ n)}}).$$ This shows that the action of $A$ on $\mathfrak{H}_A$ is the crossed product action. \end{proof} To study simplicity in this situation, we can invoke Theorem 4.2 from \cite{O75} and Theorem 6.5 from \cite{OP78}, or with the same success, use the following result from \cite{K81}: \begin{thm}[\cite{K81}] Let $\Gamma$ be a discrete group of automorphisms of $C^*$-algebra $\mathfrak{B}$. If $\mathfrak{B}$ is simple and if each $\gamma$ is outer for the multiplier algebra $M(\mathfrak{B})$ of $\mathfrak{B}$, $\forall \gamma \in \Gamma \backslash \{ 1 \} $, then the reduced crossed product of $\mathfrak{B}$ by $\Gamma$, $\mathfrak{B} \rtimes \Gamma$, is simple. \end{thm} An automorphism $\omega$ of a $C^*$-algebra $\mathfrak{B}$ , contained in a $C^*$-algebra $\mathfrak{A}$ is outer for $\mathfrak{A}$, if there doesn't exist a unitary $w \in \mathfrak{A}$ with the property $\omega = \Ad(w)$. \par A representation $\pi$ of a $C^*$-algebra $\mathfrak{A}$ on a Hilbert space $\mathfrak{H}$ is called non-degenerate if there doesn't exist a vector $\xi \in \mathfrak{H}$, $\xi \neq 0$, such that $\pi(\mathfrak{A}) \xi = 0$. \par The idealizer of a $C^*$-algebra $\mathfrak{A}$ in a $C^*$-algebra $\mathfrak{B}$ ($\mathfrak{A} \subset \mathfrak{B}$) is the largest $C^*$-subalgebra of $\mathfrak{B}$ in which $\mathfrak{A}$ is an ideal. \\ We will not give a definition of multiplier algebra of a $C^*$-algebra. Instead we will give the following property from \cite{APT73}, which we will use (see \cite{APT73} for more details on multiplier algebras): \begin{prop}[\cite{APT73}] Each nondegenerate faithful representation $\pi$ of a $C^*$-algebra $\mathfrak{A}$ extends uniquely to a faithful representation of $M(\mathfrak{A})$, and $\pi(M(\mathfrak{A}))$ is the idealizer of $\pi(\mathfrak{A})$ in its weak closure. \end{prop} Suppose that we have a faithful representation $\pi$of a $C^*$ algebra $\mathfrak{A}$ on a Hilbert space $\mathfrak{H}$. If confusion is impossible we will denote by $\bar{\mathfrak{A}}$ (in $\mathfrak{H}$) the weak closure of $\pi(\mathfrak{A})$ in $\mathbb{B}(\mathfrak{H})$. \par To study uniqueness of trace we invoke a theorem of B$\acute{e}$dos from \cite{B93}. \par Let $\mathfrak{A}$ be a simple, unital $C^*$-algebra with a unique trace $\varphi$ and let $(\pi_{\mathfrak{A}}, \mathfrak{H}_{\mathfrak{A}}, \widehat{1_{\mathfrak{A}}})$ denote the GNS-triple associated to $\varphi$. The trace $\varphi$ is faithful by the simplicity of $\mathfrak{A}$ and $\mathfrak{A}$ is isomorphic to $\pi_{\mathfrak{A}}(\mathfrak{A})$. Let $\alpha \in Aut(\mathfrak{A})$. The trace $\varphi$ is $\alpha$-invariant by the uniqueness of $\varphi$. Then $\alpha$ is implemented on $\mathfrak{H}_{\mathfrak{A}}$ by the unitary operator $U_{\alpha}$ given by $U_{\alpha}(\hat{a}) = \alpha(a) \cdot \widehat{1_{\mathfrak{A}}}$, $a \in \mathfrak{A}$. Then we denote the extension of $\alpha$ to the weak closure $\bar{\mathfrak{A}}$ (in $\mathfrak{H}_{\mathfrak{A}}$) of $\pi_{\mathfrak{A}}(\mathfrak{A})$ on $\mathbb{B}(\mathfrak{H}_{\mathfrak{A}})$ by $\tilde{\alpha} \overset{def}{=} \Ad(U_{\alpha})$. We will say that $\alpha$ is $\varphi$-outer if $\tilde{\alpha}$ is outer for $\bar{\mathfrak{A}}$. \begin{thm}[\cite{B93}] Suppose $\mathfrak{A}$ is a simple unital $C^*$-algebra with a unique trace $\varphi$ and that $\Gamma$ is a discrete group with a representation $\alpha : \Gamma \rightarrow Aut(\mathfrak{A})$, such that $\alpha_{\gamma}$ is $\varphi$-outer $\forall \gamma \in \Gamma \backslash \{ 1 \}$. Then the reduced crossed product $\mathfrak{A} \rtimes \Gamma$ is simple with a unique trace $\tau$ given by $\tau = \varphi \circ E$, where $E$ is the canonical conditional expectation from $\mathfrak{A} \rtimes \Gamma$ onto $\mathfrak{A}$. \end{thm} Let's now return to the $C^*$-algebra $(A,\tau) = ( \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_m}{\overset{p_m}{\mathbb{C}}})*(\mathbb{M}_n, tr_n)$, with $\alpha_1 \leq \alpha_2 \leq ... \leq \alpha_m$. If $B \subset E \subset A$ are as in the beginning of this section, then the representations of $B$, $E$ and $A$ on $\mathfrak{H}_A$ are all nondegenerate. Also we have the following: \begin{lemma} The weak closure of $B$ in $\mathbb{B}(\mathfrak{H}_B)$ and the one in $\mathbb{B}(\mathfrak{H}_A)$ are the same (or $\bar{B}$ (in $\mathfrak{H}_B$) $\cong$ $\bar{B}$ (in $\mathfrak{H}_A$)). Analoguously, $\bar{E}$ (in $\mathfrak{H}_E$) $\cong$ $\bar{E}$ (in $\mathfrak{H}_A$). \end{lemma} \begin{proof} For $b \in B \subset A$ we have $b(u^t h) = u^t (\Ad(u^{-t} b))(h)$ for $h \in \mathfrak{H}_B$ and $0 \leq t \leq n-1$. Taking a weak limit in $\mathbb{B}(\mathfrak{H}_B)$ we obtain the same equation $\forall \bar{b} \in \bar{B}$ (in $\mathfrak{H}_B$): $\bar{b}(u^th) = u^t(\Ad(u^{-t})(\bar{b}))(h)$, which shows, of course, that $\bar{b}$ has a unique extension to $\mathbb{B}(\mathfrak{H}_A)$. Conversely if $\tilde{b} \in \bar{B}$ (in $\mathfrak{H}_A$), then since $\mathfrak{H}_B$ is invariant for $B$ it will be invariant for $\tilde{b}$ also. So the restriction of $\tilde{b}$ to $\mathfrak{H}_B$ is the element we are looking for. \par Analoguously if $e \in E$ and if $h_0 + u^l h_1 + ... + u^{n-l} h_{\frac{n}{l}-1} \in \mathfrak{H}_E$, then for $0 \leq t \leq l-1$ we have $e(u^t(h_0 + u^l h_1 + ... + u^{n-l} h_{\frac{n}{l}-1})) = u^t(\Ad(u^{-t})(e))(h_0 + u^l h_1 + ... + u^{n-l} h_{\frac{n}{l}-1})$. And again for an element $\bar{e} \in \bar{E}$ (in $\mathfrak{H}_E$) we see that $\bar{e}$ has a unique extension to an element of $\bar{E}$ (in $\mathfrak{H}_A$). Conversely an element $\tilde{e} \in \bar{E}$ (in $\mathfrak{H}_A$) has $\mathfrak{H}_E$ as an invariant subspace, so we can restrict it to $\mathfrak{H}_E$ to obtain an element in $\bar{E}$ (in $\mathfrak{H}_E$). \end{proof} We will state the following theorem from \cite{D99}, which we will frequently use: \begin{thm}[\cite{D99}] Let $\mathfrak{A}$ and $\mathfrak{B}$ be unital $C^*$-algebras with traces $\tau_{\mathfrak{A}}$ and $\tau_{\mathfrak{B}}$ respectively, whose GNS representations are faithful. Let \begin{center} $(\mathfrak{C}, \tau) = (\mathfrak{A}, \tau_{\mathfrak{A}}) * (\mathfrak{B}, \tau_{\mathfrak{B}})$. \end{center} Suppose that $\mathfrak{B} \neq \mathbb{C}$ and that $\mathfrak{A}$ has a unital, diffuse abelian $C^*$-subalgebra $\mathfrak{D}$ ($1_{\mathfrak{A}} \in \mathfrak{D} \subseteq \mathfrak{A}$). Then $\mathfrak{C}$ is simple with a unique trace $\tau$. \end{thm} Using repeatedly Theorem 2.4 we see that $$B = (\mathbb{C} \cdot e_{11} \oplus ... \oplus \mathbb{C} \cdot e_{nn}) * (\underset{k=0}{\overset{n-1}{*}} (\mathbb{C} \cdot u^k p_1 u^{-k} \oplus ... \oplus \mathbb{C} \cdot u^k p_m u^{-k})),$$ $$\cong (U \oplus \underset{max \{ n\alpha_m - n + 1,\ 0 \} }{\overset{\tilde{p}}{\mathbb{C}}}) * (\underset{\frac{1}{n}}{\overset{e_{11}}{\mathbb{C}}} \oplus ... \oplus \underset{\frac{1}{n}}{\overset{e_{nn}}{\mathbb{C}}}),$$ where $U$ has a unital, diffuse abelian $C^*$-subalgebra, and where $\tilde{p} = \underset{i=0}{\overset{n-1}{\wedge}} u^i p_m u^{-i}$. \par We will consider the following 3 cases, for $\alpha_1 \leq \alpha_2 \leq ... \leq \alpha_m$: \\ \par (I) $\alpha_m < 1-\frac{1}{n^2}$. \par (II) $\alpha_m = 1-\frac{1}{n^2}$. \par (III) $\alpha_m > 1-\frac{1}{n^2}$. \\ \par We will organize those cases in few lemmas: \par \par (I) \begin{lemma} If $A$ is as above, then for $\alpha_m < 1-\frac{1}{n^2}$ we have that $A$ is simple with a unique trace. \end{lemma} \begin{proof} We consider: \\ (1) $\alpha_m \leq 1-\frac{1}{n}$. \\ Then $B \cong U * (\underset{\frac{1}{n}}{\overset{e_{11}}{\mathbb{C}}} \oplus ... \oplus \underset{\frac{1}{n}}{\overset{e_{nn}}{\mathbb{C}}})$ with $U$ containing a unital, diffuse abelian $C^*$-subalgebra (from Theorem 2.4). From the Theorem 3.9 we see that $B$ is simple with a unique trace. \\ (2) $1-\frac{1}{n} < \alpha_m < 1-\frac{1}{n^2}$. \\ Then $B \cong (U \oplus \underset{n\alpha_m - n + 1}{\overset{\tilde{p}}{\mathbb{C}}}) * (\underset{\frac{1}{n}}{\overset{e_{11}}{\mathbb{C}}} \oplus ... \oplus \underset{\frac{1}{n}}{\overset{e_{nn}}{\mathbb{C}}})$ with $U$ having a unital, diffuse abelian $C^*$-subalgebra. Using Theorem 2.4 one more time we see that $B$ is simple with a unique trace in this case also. \par We know that $A = B \rtimes G$, where $G = \langle \Ad(u) \rangle \cong \mathbb{Z}_n$. Since $B$ is unital then the multiplier algebra $M(B)$ coinsides with $B$. We note also that since $\bar{B}$ (in $\mathfrak{H}_B$) is isomorphic to $\bar{B}$ (in $\mathfrak{H}_A$) to prove that some element of $Aut(B)$ is $\tau_B$-outer it's enough to prove that this automorphism is outer for $\bar{B}$ (in $\mathfrak{H}_A$) (and it will be outer for $M(B) = B$ also). Making these observations and using Theorem 3.5 and Theorem 3.7 we see that if we prove that $\Ad(u^i)$ is outer for $\bar{B}$ (in $\mathfrak{H}_A$), $\forall 0 < i \leq n-1$, then it will follow that $A$ is simple with a unique trace. We will show that $\Ad(u^i)$ is outer for $\bar{B}$ (in $\mathfrak{H}_A$) (we will write just $\bar{*}$ for $\bar{*}$ (in $\mathfrak{H}_A$) and omit writting $\mathfrak{H}_A$ - all the closures will be in $\mathbb{B}(\mathfrak{H}_{\mathfrak{A}})$) for the case $\alpha_m \leq 1-\frac{1}{n^2}$. \par Fix $0 < k \leq n-1$. Since $u^k \mathfrak{H}_B \perp \mathfrak{H}_B$ it follows that $u^k \notin \bar{B}$ (in $\mathfrak{H}_A$). Suppose $\exists w \in \bar{B}$, such that $\Ad(u^k) = \Ad(w)$ on $\bar{B}$. Then $u^k w u^{-k} = w w w^* = w$ and $u^k w^* u^{-k} = w w^* w^* = w^*$ and this implies that $u^k$, $u^{-k}$, $w$ and $w^*$ commute, so it follows $u^k w^*$ commutes with $\overline{C^*(B, u^k)}$, so it belongs to its center. If $k \nmid n$ then $\overline{C^*(B, u^k)} = \bar{A}$ and by Theorem 2.5 $\bar{A}$ (in $\mathfrak{H}_A$)is a factor, so $u^k w^*$ is a multiple of $1_A$, which contradicts the fact $u^k \notin \bar{B}$. If $k=l \mid n$, then $\overline{C^*(B, u^k)} = \bar{E}$ and $\bar{E}$ (in $\mathfrak{H}_A$) $\cong$ $\bar{E}$ (in $\mathfrak{H}_E$) is a factor too (by Theorem 2.5), so this implies again that $u^k w^*$ is a multiple of $1_A = 1_E$, so this is a contradiction again and this proves that $\Ad(u^k)$ are outer for $\bar{B}$, $\forall 0 < k \leq n-1$. This concludes the proof. \end{proof} (III) \begin{lemma} If $A$ is as above, then for $\alpha_m > 1-\frac{1}{n^2}$ we have $A = A_0 \oplus \underset{n^2 \alpha_m - n^2 + 1}{\mathbb{M}_n}$, where $A_0$ is simple with a unique trace. \end{lemma} \begin{proof} In this case $B \cong (U \oplus \underset{n\alpha_m - n + 1}{\overset{\tilde{p}}{\mathbb{C}}}) * (\underset{\frac{1}{n}}{\overset{e_{11}}{\mathbb{C}}} \oplus ... \oplus \underset{\frac{1}{n}}{\overset{e_{nn}}{\mathbb{C}}})$, where $U$ has a unital, diffuse abelian $C^*$-subalgebra. Form Theorem 2.4 we see that $B \cong \overset{\tilde{p}_0}{B_0} \oplus \underset{n \alpha_m - n + \frac{1}{n}}{\overset{e_{11} \wedge \tilde{p}}{\mathbb{C}}} \oplus ... \oplus \underset{n \alpha_m - n + \frac{1}{n}}{\overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}}$ with $\tilde{p}_0 = 1- e_{11} \wedge \tilde{p} - ... - e_{nn} \wedge \tilde{p}$, and $B_0$ being a unital, simple and having a unique trace. It's easy to see that $\Ad(u)$ permutes $\{ e_{ii} | 1 \leq i \leq n \}$ and that $\Ad(u)$ permutes $\{ u^i p_j u^{-i} | 0 \leq i \leq n-1 \}$ for each $1 \leq j \leq m$. But since $\tilde{p} = \underset{i=0}{\overset{n-1}{\wedge}} u^i p_m u^{-i}$ we see that $\Ad(u)(\tilde{p}) = \tilde{p}$. This shows that $\Ad(u)$ permutes $\{ e_{ii} \wedge \tilde{p} | 1 \leq i \leq n \}$. This shows that $\Ad(\tilde{p}_0 u)$ is an automorphism of $B_0$ and that $\Ad((1-\tilde{p}_0) u)$ is an automorphism of $\overset{e_{11} \wedge \tilde{p}}{\mathbb{C}} \oplus ... \oplus \overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}$. If we denote $G_1 = \langle \Ad(\tilde{p}_0 u) \rangle$ and $G_2 = \langle \Ad((1-\tilde{p}_0) u) \rangle$, then we have $A = B_0 \rtimes G_1 \oplus (\overset{e_{11} \wedge \tilde{p}}{\mathbb{C}} \oplus ... \oplus \overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}) \rtimes G_2$. Now it's easy to see that $(\overset{e_{11} \wedge \tilde{p}} {\mathbb{C}} \oplus ... \oplus \overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}) \rtimes G_2 = C^*(\{ e_{11} \wedge \tilde{p}, ..., e_{nn} \wedge \tilde{p} \}, (1-\tilde{p}_0) u) = (1-\tilde{p}_0).C^*( \{ e_{11}, ..., e_{nn} \}, u) \cong \mathbb{M}_n$ (because $\tilde{p}_0$ is a central projection). To study $A_0 \overset{def}{=} B_0 \rtimes G_1$ we have to consider the automorphisms $\Ad(\tilde{p}_0 u)$. From Lemma 3.8 we see that $$\overline{B_0 \oplus \overset{e_{11} \wedge\tilde{p}}{\mathbb{C}} \oplus ... \oplus \overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}}\ (in\ \mathfrak{H}_B) \cong \overline{B_0 \oplus \overset{e_{11} \wedge\tilde{p}}{\mathbb{C}} \oplus ... \oplus \overset{e_{nn} \wedge \tilde{p}}{\mathbb{C}}}\ (in\ \mathfrak{H}_A).$$ This implies $\bar{B}_0$ (in $\mathfrak{H}_{B_0}$) $\cong$ $\bar{B}_0$ (in $\mathfrak{H}_{A_0}$). This is because $\mathfrak{H}_{A_0} = \tilde{p}_0 \mathfrak{H}_A$ and $\mathfrak{H}_{B_0} = \tilde{p}_0 \mathfrak{H}_B$ (which is clear, since $\mathfrak{H}_{A_0}$ and $\mathfrak{H}_{B_0}$ are direct summands in $\mathfrak{H}_A$ and $\mathfrak{H}_B$ respectivelly). For some $l | n$ if we denote $E_0 \overset{def}{=} \tilde{p}_0 E$ then by the same reasoning as above $$E = E_0 \oplus (1-\tilde{p}_0). C^*(\{ e_{11}, ..., e_{nn} \}, u^l) \cong E_0 \oplus (\underbrace{\mathbb{M}_{\frac{n}{l}} \oplus ... \oplus \mathbb{M}_{\frac{n}{l}}}_{l-times}).$$ So we similarly have $\bar{E_0}$ (in $\mathfrak{H}_{E_0}$) $\cong$ $\bar{E_0}$ (in $\mathfrak{H}_{A_0}$). We use Theorem 2.5 and see that $\bar{A} \cong L(F_t) \oplus \mathbb{M}_n$ and that $$\bar{E} \cong L(F_{t'}) \oplus (\underbrace{\mathbb{M}_{\frac{n}{l}} \oplus ... \oplus \mathbb{M}_{\frac{n}{l}}}_{l-times}),$$ for some $1 < t, t' < \infty$. This shows that $\bar{A_0}$ and $\bar{E_0}$ are both factors. Now for $\Ad(\tilde{p_0} u^k)$, $1 \leq k \leq n-1$ we can make the same reasoning as in the case (I) to show that $\Ad(\tilde{p_0} u^k)$ are all outer for $\bar{B_0}$, $\forall 1 \leq k \leq n-1$. Now we use Theorem 3.5 and Theorem 3.7 to finish the proof. Notice that the trace of the support projection of $\mathbb{M}_n$, $e_{11} \wedge \tilde{p} + ... + e_{nn} \wedge \tilde{p}$, is $n^2 \alpha_m - n^2 + 1$. \end{proof} (II) \\ \par We already proved that $\Ad(u^k)$ are outer for $\bar{B}$, $\forall 1 \leq k \leq n-1$. Using Theorem 2.4 we see $B \cong (U \oplus \underset{1-\frac{1}{n}}{\overset{\tilde{p}}{\mathbb{C}}}) * (\underset{\frac{1}{n}}{\overset{e_{11}}{\mathbb{C}}} \oplus ... \oplus \underset{\frac{1}{n}}{\overset{e_{nn}}{\mathbb{C}}})$ with $U$ having a unital, diffuse abelian $C^*$-subalgebra. There are $*$-homomorphisms $\pi_i : B \rightarrow \mathbb{C}$, $1 \leq i \leq n$ with $\pi_i(\tilde{p}) = \pi_i(e_{ii}) = 1$, and such that $B_0 \overset{def}{=} \underset{i=0}{\overset{n-1}{\bigcap}} \ker(\pi_i)$ is simple with a unique trace. Now if $1 \leq k \leq n-1$, then $B_0 \bigcap \Ad(u^k)(B_0) = $ either $0$ or $B_0$, because $B_0$ and $\Ad(u^k)(B_0)$ are simple ideals in $B$. The first possibility is actually impossible, because of dimension reasons, so this shows that $B_0$ is invariant for $\Ad(u^k)$, $1 \leq k \leq n-1$. In other words $\Ad(u^k) \in Aut(B_0)$. Similarly as in Lemma 3.4 it can be shown that $$A_0 \overset{def}{=} C^*(B_0 \oplus B_0 u \oplus ... \oplus B_0 u^{n-1}) \cong B_0 \rtimes \{ \Ad(u^k) | 0 \leq k \leq n-1 \} \subset A.$$ \begin{lemma} We have a short split-exact sequence: \begin{center} $0 \hookrightarrow A_0 \rightarrow A \overset{\curvearrowleft}{\rightarrow} \mathbb{M}_n \rightarrow 0$. \end{center} \end{lemma} \begin{proof} It's clear that we have the short exact sequence \begin{equation*} 0 \rightarrow B_0 \hookrightarrow B \overset{\pi}{\longrightarrow} \underbrace{\mathbb{C} \oplus ... \oplus \mathbb{C}}_{n-times} \rightarrow 0, \end{equation*} where $\pi \overset{def}{=} (\pi_1, ..., \pi_n)$. We think $\pi$ to be a map from $B$ to $diag(\mathbb{M}_n)$, defined by $$\pi(b) = \begin{pmatrix} \pi_1(b) & 0 & ... & 0 \\ 0 & \pi_2(b) & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & \pi_n(b) \end{pmatrix} .$$ Now since $\pi_i(\tilde{p}) = \pi_i(e_{ii}) = 1$ and $\Ad(u)(e_{11}) = u e_{11} u^* = e_{nn}$ and for $2 \leq i \leq n$, $\Ad(u)(e_{ii}) = u e_{ii} u^* = e_{(i-1) (i-1)}$, then $\pi_i \circ \Ad(u)(e_{(i+1) (i+1)}) = \pi_i \circ \Ad(u)(\tilde{p}) = 1$ for $1 \leq i \leq n-1$ and $\pi_n \circ \Ad(u)(e_{1 1}) = \pi_n \circ \Ad(u)(\tilde{p}) = 1$. So since two $*$-homomorphism of a $C^*$-algebra, which coinside on a set of generators of the $C^*$-algebra, are identical, we have $\pi_i \circ \Ad(u) = \pi_{i+1}$ for $1 \leq i \leq n-1$ and $\pi_n \circ \Ad(u) = \pi_1$. Define $\tilde{\pi} : A \rightarrow \mathbb{M}_n$ by $\underset{k=0}{\overset{n-1}{\sum}} b_ku^k \mapsto \underset{k=0}{\overset{n-1}{\sum}} \pi(b_k) W^k$ (with $b_k \in B$), where $W \in \mathbb{M}_n$ is represented by the matrix, which represent $u \in \mathbb{M}_n \subset A$, namely $$W \overset{def}{=} \begin{pmatrix} 0 & 1 & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & 1 \\ 1 & 0 & ... & 0 \end{pmatrix} .$$ We will show that if $b \in B$ and $0 \leq k \leq n-1$, then $\pi(u^k b u^{-k}) = W^k \pi(b) W^{-k}$. For this it's enough to show that $\pi(u b u^{-1}) = W \pi(b) W^{-1}$. For the matrix units $\{ E_{ij} | 1 \leq i,j \leq n \}$ we have as above $W E_{ii} W^* = E_{(i-1) (i-1)}$ for $2 \leq i \leq n-1$ and $W E_{11} W^* = E_{nn}$. So $$W \begin{pmatrix} \pi_1(b) & 0 & ... & 0 \\ 0 & \pi_2(b) & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & \pi_n(b) \end{pmatrix} W^* = \begin{pmatrix} \pi_2(b) & 0 & ... & 0 \\ 0 & \pi_3(b) & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & \pi_1(b) \end{pmatrix} ,$$ $$ = \begin{pmatrix} \pi_1(\Ad(u)(b)) & 0 & ... & 0 \\ 0 & \pi_2(\Ad(u)(b)) & ... & 0 \\ . & . & . & . \\ 0 & 0 & ... & \pi_n(\Ad(u)(b)) \end{pmatrix} = \pi(\Ad(u)(b)),$$ just what we wanted. \par Now for $b \in B$ and $0 \leq k \leq n-1$ we have $$\tilde{\pi}((b u^k)^*) = \tilde{\pi}(u^{-k} b^*) = \tilde{\pi}(u^{-k} b^* u^k u^{-k}) = \pi(u^{-k} b^* u^k) W^{-k} = W^{-k} \pi(b^*) W^k W^{-k} , $$ $$ = W^{-k} \pi(b)^* = (\pi(b) W^k)^* = (\tilde{\pi}(b u^k))^*.$$ Also if $b, b' \in B$ and $0 \leq k, k' \leq n-1$, then $$\tilde{\pi}((b' u^{k'}).(b u^k)) = \tilde{\pi}(b'(u^{k'} b u^{-k'}) u^{k+k'}) = \pi(b'(u^{k'} b u^{-k'})) W^{k+k'},$$ $$= \pi(b') \pi(u^{k'} b u^{-k'}) W^{k+k'} = \pi(b') W^{k'} \pi(b) W^{-k'} W^{k+k'} = \tilde{\pi}(b' u^{k'}) \tilde{\pi}(b u^k).$$ This proves that that $\tilde{\pi}$ is a $*$-homomorphism. Continuity follows from continuity of $\pi$ and the Banach space representation $A = \underset{i=0}{\overset{n-1}{\bigoplus}} Bu^i$. \par Clearly $A_0 = \underset{i=0}{\overset{n-1}{\bigoplus}} B_0 u^i$ as a Banach space. It's also clear by the definition of $\tilde{\pi}$ that $A_0 \subset \ker(\tilde{\pi})$. Since $A_0$ has a Banach space codimension $n^2$ in $A$, and so does $\ker(\tilde{\pi})$, then we must have $A_0 = \ker(\tilde{\pi})$. \par From the construction of the map $\tilde{\pi}$ we see that $\tilde{\pi}(e_{ii}) = E_{ii}$, since $\pi(e_{ii}) = E_{ii}$ and also $\tilde{\pi}(u^k) = W^k$. Since $\{e_{ii} | 1 \leq i \leq n \} \cup \{ W^k | 0 \leq k \leq n-1 \}$ generate $\mathbb{M}_n$, then we have $\tilde{\pi}(e_{ij}) = E_{ij}$, so the inclusion map $s: \mathbb{M}_n \rightarrow A$ given by $E_{ij} \mapsto e_{ij}$ is a right inverse for $\tilde{\pi}$. \end{proof} From this lemma follows that we can write $A = A_0 \oplus \mathbb{M}_n$ as a Banach space. \begin{lemma} If $\eta$ is a trace on $A_0$, then the linear functional on $A$ $\tilde{\eta}$, defined by $\tilde{\eta}(a_0 \oplus M) = \eta(a_0) + tr_n(M)$, where $a_0 \in A_0$ and $M \in \mathbb{M}_n$ is a trace and $\tilde{\eta}$ is the unique extension of $\eta$ to a trace on $A$ (of norm 1). \end{lemma} \begin{proof} The functional $\eta$ can be extended in at most one way to a tracial state on $A$, because of the requirement $\tilde{\eta}(1_A) = 1$, the fact that $\mathbb{M}_n$ sits as a subalgebra in $A$, and the uniqueness on trace on $\mathbb{M}_n$. Since $\tilde{\eta}(1_A) = 1$, to show that $\tilde{\eta}$ is a trace we need to show that $\tilde{\eta}$ is positive and satisfies the trace property. For the trace property: If $x ,y \in A$ then we need to show $\tilde{\eta}(xy) = \tilde{\eta}(yx)$. It is easy to see, that to prove this it's enough to prove that if $a_0 \in A_0$ and $M \in \mathbb{M}_n$, then $\eta(a_0 M) = \eta(M a_0)$. Since $\eta$ is linear and $a_0$ is a linear combination of 4 positive elements we can think, without loss of generality, that $a_0 \geq 0$. Then $a_0 = a_0^{1/2} a_0^{1/2}$ and $M a_0^{1/2}, a_0^{1/2} M \in A_0$, so since $\eta$ is a trace on $A_0$, we have $\eta(M a_0) = \eta((M a_0^{1/2}) a_0^{1/2}) = \eta(a_0^{1/2}(M a_0^{1/2})) = \eta((a_0^{1/2} M) a_0^{1/2}) = \eta(a_0^{1/2}(a_0^{1/2} M)) = \eta(a_0 M).$ This shows that $\tilde{\eta}$ satisfies the trace property. It remains to show positivity. Suppose $a_0 \oplus M \geq 0$. We must show $\eta(a_0 \oplus M) \geq 0$. Write $M = \underset{i=0}{\overset{n}{\sum}} \underset{j=0}{\overset{n}{\sum}} m_{ij} e_{ij}$ and $a_0 = \underset{i=0}{\overset{n}{\sum}} \underset{j=0}{\overset{n}{\sum}} e_{ii} a_0 e_{jj}$ Since $\tilde{\eta}$ is a trace if $i \neq j$, then $\tilde{\eta}(e_{ii} a_0 e_{jj}) = \tilde{\eta}(e_{jj} e_{ii} a_0) = 0$, so this shows that $\tilde{\eta}(a_0 \oplus M) = \underset{i=0}{\overset{n}{\sum}} (\frac{m_{ii}}{n} + \eta(e_{ii} a_0 e_{ii}))$. Clearly $a_0 \oplus M \geq 0$ implies $\forall 1 \leq i \leq n, e_{ii} (a_0 \oplus M) e_{ii} \geq 0$. So to show positivity we only need to show $\forall 1 \leq i \leq n$ $\tilde{\eta}(e_{ii}(a_0 + M)e_{ii}) \geq 0$, given $\forall 1 \leq i \leq n, m_{ii} e_{ii} + e_{ii} a_0 e_{ii} \geq 0$. Suppose that for some $i$, $m_{ii} < 0$. Then it follows that $e_{ii} a_0 e_{ii} \geq -m_{ii} e_{ii}$, so $e_{ii} a_0 e_{ii} \in e_{ii} A_0 e_{ii}$ is invertible, which implies $e_{ii} \in A_0$, that is not true. So this shows that $m_{ii} \geq 0$, and $m_{ii} e_{ii} \geq -e_{ii} a_0 e_{ii}$. If $\{ \epsilon_{\gamma} \}$ is an approximate unit for $A_0$, then positivity of $\eta$ implies $1 = \| \eta \| = \underset{\gamma}{\lim}\ \eta(\epsilon_{\gamma})$. Since $\eta$ is a trace we have $\underset{\gamma}{\lim}\ \eta(\epsilon_{\gamma} e_{ii}) = \frac{1}{n}$. Since $\forall \gamma,\ m_{ii}\epsilon_{\gamma}^{1/2} e_{ii} \epsilon_{\gamma}^{1/2} \geq - \epsilon_{\gamma}^{1/2} e_{ii} a_0 e_{ii} \epsilon_{\gamma}^{1/2}$, then $$tr_n(m_{ii} e_{ii}) = \frac{m_{ii}}{n} = \underset{\gamma}{\lim}\ \eta(m_{ii} e_{ii} \epsilon_{\gamma}) = \underset{\gamma}{\lim}\ \eta(m_{ii} \epsilon_{\gamma}^{1/2} e_{ii} \epsilon_{\gamma}^{1/2}) \geq \underset{\gamma}{\lim}\ \eta(\epsilon_{\gamma}^{1/2} e_{ii} a_0 e_{ii} \epsilon_{\gamma}^{1/2}),$$ $$ = \underset{\gamma}{\lim}\ \eta(e_{ii} a_0 e_{ii} \epsilon_{\gamma}) = \eta(e_{ii} a_0 e_{ii}).$$ This finishes the proof of positivity and the proof of the lemma. \end{proof} \begin{remark} We will show below that $\tau|_{A_0}$ is the unique trace on $A_0$. Since we have $A = A_0 \oplus \mathbb{M}_n$ as a Banach space, then clearly the free product trace $\tau$ on $A$ is given by $\tau(a_0 \oplus M) = \tau|_{A_0}(a_0) + tr_n(M)$, where $a_0 \oplus M \in A_0 \oplus \mathbb{M}_n = A$. All tracial positive linear functionals of norm $\leq 1$ on $A_0$ are of the form $t\tau|_{A_0}$, where $0 \leq t \leq 1$. Then there will be no other traces on $A$ then the family $\lambda_t \overset{def}{=} t \tau|_{A_0} \oplus tr_n$. To show that these are traces indeed, we can use the above lemma (it is still true, no mater that the norm of $t \tau_{A_0}$ can be less than one), or we can represent them as a convex linear combination $\lambda_t = t \tau + (1-t)\mu$ of the free product trace $\tau$ and the trace $\mu$, defined by $\mu(a_0 \oplus M) = tr_n(M) = tr_n(\tilde{\pi}(a_0 \oplus M))$. \end{remark} \begin{lemma} $\bar{B_0}$ (in $\mathfrak{H}_A$) $=$ $\bar{B}$ (in $\mathfrak{H}_A$). \end{lemma} \begin{proof} Let's take $D \overset{def}{=} ( \overset{1-\tilde{p}}{\mathbb{C}} \oplus \overset{\tilde{p}}{\mathbb{C}} ) * ( \overset{e_{11}}{\mathbb{C}} \oplus \overset{e_{22} + ... + e_{nn}}{\mathbb{C}}) \subset B$. Denote $D_0 \overset{def}{=} D\cap B_0$. From Theorem 2.3 follows that $D \cong \{ f: [0,b] \rightarrow \mathbb{M}_2 | f$ is continuous and $f(0)$ - diagonal$\}$ $\oplus \overset{\tilde{p}\wedge (1-e_{11})} {\mathbb{C}}$, where $0 < b < 1$ and $\tau|_D$ is given by an atomless measure $\mu$ on $\{ f: [0,b] \rightarrow \mathbb{M}_2 | f$ is continuous and $f(0)$ - diagonal $\}$, $\tilde{p}$ is represented by $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \oplus 1$, and $e_{11}$ is represented by $\begin{pmatrix} 1-t & \sqrt{t(1-t)} \\ \sqrt{t(1-t)} & t \end{pmatrix} \oplus 0$. A $*$-homomorphism, defined on the generators of a $C^*$-algebra can be extended in at most one way to the whole $C^*$-algebra. This observation, together with $\pi_1(e_{11}) = \pi_1(\tilde{p}) = 1$ and $\pi_i(e_{22} = ... + e_{nn}) = \pi(\tilde{p}) = 1$ implies that $\pi_1|_D(f \oplus c) = f_{11}(0)$ and $\pi_i|_D(f \oplus c) = c$ for $2 \leq i \leq n-1$. This means that $D_0 = \{ f: [0,b] \rightarrow \mathbb{M}_2 | f$ is continuous and $f_{11}(0) = f_{12}(0) = f_{21}(0) = 0 \} \oplus 0$. Now we see $\bar{D_0}$ (in $\mathfrak{H}_D$) $\cong$ $\mathbb{M}_2 \otimes L^{\infty}([0,b], \mu) \oplus 0$, so then $e_{11} \in \bar{D_0}$ (in $\mathfrak{H}_D$). So we can find sequence $\{ \varepsilon_n \}$ of self-adjoined elements (functions) of $D_0$, supported on $e_{11}$, weakly converging to $e_{11}$ on $\mathfrak{H}_D$ and such that $\{ \varepsilon_n^2 \}$ also converges weakly to $e_{11}$ on $\mathfrak{H}_D$. Then take $a_1, a_2 \in A$. in $\mathfrak{H}_A$ we have $\langle \widehat{a_1}, (\varepsilon_n^2 - e_{11})\widehat{a_2} \rangle = \tau( (\varepsilon_n^2 - e_{11}) a_2 a_1^*) = \tau((\varepsilon_n - e_{11}) a_2 a_1^* (\varepsilon_n - e_{11})) \leq 4 \| a_2 a_1^* \| \tau(\varepsilon_n^2 - e_{11})$ (The last inequality is obtained by representing $a_2 a_1^*$ as a linear combination of 4 positive elements and using Cauchy-Bounjakovsky-Schwartz inequality). This shows that $e_{11} \in \bar{D_0}$ (in $\mathfrak{H}_A$) $\subset \bar{B_0}$ (in $\mathfrak{H}_A$). Analoguously $e_{ii} \in \bar{B_0}$ (in $\mathfrak{H}_A$), so this shows $\bar{B_0} = \bar{B}$ (in $\mathfrak{H}_A$). \end{proof} It easily follows now that \begin{cor} $\bar{A_0}$ (in $\mathfrak{H}_A$) $=$ $\bar{A}$ (in $\mathfrak{H}_A$). \end{cor} The representation of $B_0$ on $\mathfrak{H}_A$ is faithful and nondegenerate, and we can use Proposition 3.6, together with Theorem 3.5 and the fact that $\Ad(u^k)$ are outer for $\bar{B} = \bar{B_0}$ to get: \begin{lemma} $A_0 = B_0 \rtimes G$ is simple. \end{lemma} For the uniqueness of trace we need to modify a little the proof Theorem 3.7 (which is Theorem 1 in \cite{B93}, stated for "nontwisted" crossed products). \begin{lemma} $A_0 = B_0 \rtimes G$ has a unique trace, $\tau|_{A_0}$. \end{lemma} \begin{proof} Above we already proved that $\{ \Ad(u^k) | 1 \leq k \leq n-1 \}$ are $\tau|_{B_0}$-outer for $B_0$. \par Suppose that $\eta$ is a trace on $A_0$. We will show that $\tau|_{A_0} = \eta$. We consider the GNS representation of $B$, associated to $\tau|_B$. By repeating the proof of Lemma 3.13 we see that $\bar{B_0}$ (in $\mathfrak{H}_B$) $=$ $\bar{B}$ (in $\mathfrak{H}_B$). The simplicity of $B_0$ allows us to identify $B_0$ with $\pi_{\tau|_B}(B_0)$. We will also identify $B_0$ with it's canonical copy in $A_0$. $A_0$ is generated by $\{ b_0 \in B_0 \} \cup \{ u^k | 0 \leq k \leq n-1 \}$ and $\{ \Ad(u^k) | 0 \leq k \leq n-1 \}$ extend to $\bar{B_0}$ (in $\mathfrak{H}_A$), so also to $\bar{B_0}$ (in $\mathfrak{H}_B$) ( $\cong \bar{B}$ (in $\mathfrak{H}_A$)). Now we can form the von Neumann algebra crossed product $\tilde{A} \overset{def}{=} \bar{B_0} \rtimes \{ \Ad(u^k) | 0 \leq k \leq n-1 \} \cong \bar{B} \rtimes \{ \Ad(u^k) | 0 \leq k \leq n-1 \}$, where the weak closures are in $\mathfrak{H}_B$. Clearly $\tilde{A} \cong \bar{A}$ (in $\mathfrak{H}_A$). Denote by $\widetilde{\tau_{B_0}}$ the extension of $\tau|_{B_0}$ to $\bar{B_0}$ (in $\mathfrak{H}_A$), given by $\widetilde{\tau_{B_0}}(x) = \langle x(\widehat{1_A}), \widehat{1_A} \rangle_{\mathfrak{H}_A}$. By Proposition 3.19 of Chapter V in \cite{T79}, $\widetilde{\tau_{B_0}}$ is a faithful normal trace on $\bar{B_0}$ (in $\mathfrak{H}_A$). Now from the fact that $\bar{B_0}$ (in $\mathfrak{H}_A$) is a factor and using Lemma 1 from \cite{L81} we get that $\widetilde{\tau_{B_0}}$ is unique on $\bar{B_0}$ (in $\mathfrak{H}_A$). By the same argument we have that the extension $\widetilde{\tau_{A_0}}$ of $\tau|_{A_0}$ to $\bar{A_0}$ (in $\mathfrak{H}_{A}$) $\cong$ $\bar{A}$ (in $\mathfrak{H}_A$) is unique, since $\bar{A_0}$ (in $\mathfrak{H}_{A}$) $\cong$ $\bar{A}$ (in $\mathfrak{H}_A$) is a factor. \par We take the unique extension of $\eta$ to $A$. We will call it again $\eta$ for convenience. \\ We denote by $\mathfrak{H}'_{C}$ the GNS Hilbert space for $C$, corresponding to $\eta|_C$ (for $C$ $=$ $A$, $B$, $B_0$, $A_0$). Since $\eta|_{B_0} = \tau|_{B_0}$ it follows that $\bar{B_0}$ (in $\mathfrak{H}'_{B_0}$) $\cong$ $\bar{B}$ (in $\mathfrak{H}'_B$) and of course $\mathfrak{H}'_{B_0} = \mathfrak{H}'_B$. Then similarly as in Lemma 3.12 we get that $\bar{A_0}$ (in $\mathfrak{H}'_{A_0}$) $\cong$ $\bar{A}$ (in $\mathfrak{H}'_{A}$), so $\mathfrak{H}'_{A_0} = \mathfrak{H}'_{A}$ (this can be done, since $\tau|_{B_0} = \eta|_{B_0}$). Now again by Proposition 3.19 of Chapter V in \cite{T79} we have that $\tilde{\eta}(x) \overset{def}{=} \langle x(\widehat{1_A}), \widehat{1_A} \rangle_{\mathfrak{H}'_A}$ ($\widehat{1_A}$ is abuse of notation - in this case it's the element, corresponding to $1_A$ in $\mathfrak{H}'_A$) defines a faithful normal trace on $\overline{\pi'_{A}(A)}$ (in $\mathfrak{H}'_A$). In particular $\tilde{\eta}|_{\overline{\pi'_A(B)}}$ is a faithful normal trace on $\overline{\pi'_A(B)}$ (in $\mathfrak{H}'_{A}$). By uniqueness of $\tau|_{B_0}$ we have $\tau|_{B_0} = \eta|_{B_0}$, so for $b_0 \in B_0$ we have $\tilde{\tau} (b_0) = \tau(b_0) = \eta(b_0) = \langle \pi'_{A}(b_0)(\widehat{1_A}), \widehat{1_A} \rangle_{\mathfrak{H}'_A} = \tilde{\eta}(\pi'_{A}(b_0))$. \par Since $B_0$ is simple, it follows that $\pi'_{A}|_{B_0}$ is a $*$-isomorphism from $B_0$ onto $\pi'_{A}(B_0)$ and from Exercise 7.6.7 in \cite{KR86} it follows that $\pi'_{A}|_{B_0}$ extends to a $*$-isomorphism from $\bar{B_0}$ (in $\mathfrak{H}_A$) $\cong$ $\bar{B}$ (in $\mathfrak{H}_A$) onto $\overline{\pi'_{A}(B_0)}$ (in $\mathfrak{H}'_A$) $\cong$ $\overline{\pi'_{A}(B)}$ (in $\mathfrak{H}'_A$). We will denote this $*$-isomorphism by $\theta$. We set $w \overset{def}{=} \pi'_A(u)$, $\beta \overset{def}{=} \theta \Ad(u) \theta^{-1} \in Aut(\overline{\pi'_A(B)}$ (in $\mathfrak{H}'_A$)). For $b_0 \in B_0$ we have $w \pi'_A(b_0) w^* = \pi'_A(u b_0 u^*) = \pi'_A((\Ad(u))(b_0)) = \beta (\pi'_A(b_0))$. So by weak continuity follows $\beta = \Ad(w)$ on $\overline{\pi'_A(B)}$ (in $\mathfrak{H}'_A$). Since $\bar{B}$ (in $\mathfrak{H}_A$) is a factor and $\{ \Ad(u^k) | 1 \leq k \leq n-1 \}$ are all outer, Kallman's Theorem (Corrolary 1.2 in \cite{Ka69}) gives us that $\{ \Ad(u^k) | 1 \leq k \leq n-1 \}$ act freely on $\bar{B}$ (in $\mathfrak{H}_A$). Namely if $\bar{b} \in \bar{B}$ (in $\mathfrak{H}_A$), and if $\forall \bar{b}' \in \bar{B}$ (in $\mathfrak{H}_A$), $\bar{b} \bar{b}' = \Ad(u^k)(\bar{b}') \bar{b}$, then $\bar{b} = 0$. Then by the above settings it is clear that $\{ \Ad(w^k) | 1 \leq k \leq n-1 \}$ also act freely on $\overline{\pi'_A(B)}$ (in $\mathfrak{H}'_A$). \par Since $\tilde{\eta}$ is a faithful normal trace on $\overline{\pi'_A(A)}$ (in $\mathfrak{H}'_A$), then by Proposition 2.36 of Chapter V in \cite{T79} there exists a faithful conditional expectation $P: \overline{\pi'_A(A)} \rightarrow \overline{\pi'_A(B)}$ (both weak closures are in $\mathfrak{H}'_A$). $\forall x \in \overline{\pi'_A(B)}$ (in $\mathfrak{H}'_A$), and $\forall 1 \leq k \leq n-1$, $\Ad(w^k)(x) w^k = w^k x$. Applying $P$ we get $\Ad(w^k)(x)(P(w^k)) = P(w^k) x$, so by the free action of $\Ad(w^k)$ we get that $P(w^k) = 0$, $\forall 1 \leq k \leq n-1$. It's clear that $\{ \overline{\pi'_A(B)} \} \cup \{ w^k | 1 \leq k \leq n-1 \}$ generates $\overline{\pi'_A(A)}$ (in $\mathfrak{H}'_A$) as a von Neumann algebra. Now we use Proposition 22.2 from \cite{S81}. It gives us a $*$-isomorphism $\Phi : \overline{\pi'_A(A)}$ (in $\mathfrak{H}'_A$) $\rightarrow \bar{B} \rtimes \{ \Ad(u^k) | 1 \leq k \leq n-1 \} \cong \bar{A}$ (last two weak closures are in $\mathfrak{H}_A$) with $\Phi(\theta(x)) = x,$ $x\in \bar{B}$ (in $\mathfrak{H}_A$), $\Phi(w) = u$. So since $\bar{A}$ (in $\mathfrak{H}_A$) is a finite factor, so is $\overline{\pi'_A(A)}$ (in $\mathfrak{H}'_A$), and so it's trace $\tilde{\eta}$ is unique. Hence, $\tilde{\eta} = \tilde{\tau} \circ \Phi$, and so $\forall b \in B$, and $\forall 1 \leq k \leq n-1$ we have $\eta(b u^k) = \tilde{\eta}(\pi'_A(b) \pi'_A(u^k)) = \tilde{\tau}(\Phi(\pi'_A(b)) \Phi(\pi'_A(u^k))) = \tilde{\tau}(\Phi(\theta(b)) \Phi(w^k)) = \tilde{\tau}(b u^k) = \tau(b u^k)$. By continuity and linearity of both traces we get $\eta = \tau$, just what we want. \end{proof} We conclude this section by proving the following \begin{prop} Let \begin{center} $(A,\tau) \overset{def}{=} ( \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_m}{\overset{p_m} {\mathbb{C}}})*(\mathbb{M}_n, tr_n)$, \end{center} where $\alpha_1 \leq \alpha_2 \leq ... \leq \alpha_m$. Then: \par (I) If $\alpha_m < 1-\frac{1}{n^2}$, then $A$ is unital, simple with a unique trace $\tau$. \par (II) If $\alpha_m = 1-\frac{1}{n^2}$, then we have a short exact sequence $0 \rightarrow A_0 \rightarrow A \rightarrow \mathbb{M}_n \rightarrow 0$, where $A$ has no central projections, and $A_0$ is nonunital, simple with a unique trace $\tau|_{A_0}$. \par (III) If $\alpha_m > 1-\frac{1}{n^2}$, then $A = \underset{n^2 - n^2 \alpha_m}{\overset{f}{A_0}} \oplus \underset{n^2 \alpha_m - n^2 + 1}{\overset{1-f}{\mathbb{M}_n}}$, where $1-f \leq p_m$, and where $A_0$ is unital, simple and has a unique trace $(n^2 - n^2 \alpha_m)^{-1} \tau|_{A_0}$. \par Let $f$ means the identity projection for cases (I) and (II). Then in all cases for each of the projections $f p_1, ..., f p_m$ we have a unital, diffuse abelian $C^*$-subalgebra of $A$, supported on it. \par In all the cases $p_m$ is a full projection in $A$. \end{prop} \begin{proof} We have to prove the second part of the proposition, since the first part follows from Lemma 3.10, Lemma 3.11, Lemma 3.12, Lemma 3.17 and Lemma 3.18. From the discussion above we see that in all cases we have $fA = fB \rtimes \{ \Ad(f u^k f) | 0 \leq k \leq n-1 \}$, where $B$ and $\{ \Ad(f u^k ) | 0 \leq k \leq n-1 \}$ are as above. So the existence of the unital, diffuse abelian $C^*$-sublagebras follows from Theorem 2.4, applied to $B$. \par In the case (I) $p_m$ is clearly full, since $A$ is simple. In the case (III) it's easy to see that $p_m \wedge f \neq 0$ and $p_m \geq (1-f)$, so since $A_0$ and $\mathbb{M}_n$ are simple in this case, then $p_m$ is full in $A$. In case (II) it follows from Theorem 2.4 that $p_m$ is full in $B$, and consequently in $A$. \end{proof} \section{ The General Case} In this section we prove the general case of Theorem 2.6, using the result from the previous section (Proposition 3.19). The prove of the general case involves techniques from \cite{D99}. So we will need two technical results from there. \par The first one is Proposition 2.8 in \cite{D99} (see also \cite{D93}): \begin{prop} Let $A = A_1 \oplus A_2$ be a direct sum of unital $C^*$-algebras and let $p = 1 \oplus 0 \in A$. Suppose $\phi_A$ is a state on $A$ with $0 < \alpha \overset{def}{=} \phi_A(p) < 1$. Let $B$ be a unital $C^*$-algebra with a state $\phi_B$ and let $(\mathfrak{A}, \phi) = (A, \phi_A) * (B, \phi_B)$. Let $\mathfrak{A}_1$ be the $C^*$-subalgebra of $\mathfrak{A}$ generated by $(0 \oplus A_2) + \mathbb{C} p \subseteq A$, toghether with $B$. In other words \begin{equation*} (\mathfrak{A}_1, \phi|_{\mathfrak{A}_1}) = (\underset{\alpha}{\overset{p}{\mathbb{C}}} \oplus \underset{1-\alpha}{\overset{1-p}{A_2}}) * (B, \phi_B). \end{equation*} Then $p \mathfrak{A} p$ is generated by $p \mathfrak{A}_1 p$ and $A_1 \oplus 0 \subset A$, which are free in $(p \mathfrak{A} p, \frac{1}{\alpha} \phi|_{p \mathfrak{A} p})$. In other words \begin{equation*} (p \mathfrak{A} p, \frac{1}{\alpha} \phi|_{p \mathfrak{A} p}) \cong (p \mathfrak{A}_1 p,\frac{1}{\alpha} \phi|_{p \mathfrak{A}_1 p}) * (A_1, \frac{1}{\alpha} \phi_A|_{A_1}). \end{equation*} \end{prop} \begin{remark} This proposition was proved for the case of von Neumann algebras in \cite{D93}. It is true also in the case of $C^*$-algebras. \end{remark} The second result is Proposition 2.5 (ii) of \cite{D99}, which is easy and we give its proof also: \begin{prop} Let $A$ be a $C^*$-algebra. Take $h \in A, h \geq 0$, and let $B$ be the hereditary subalgebra $\overline{hAh}$ of $A$ ( $\overline{*}$ means norm closure). Suppose that $B$ is full in $A$. Then if $B$ has a unique trace, then $A$ has at most one tracial state. \end{prop} \begin{proof} It's easy to see that $\Span \{ xhahy | a,x,y \in A \}$ is norm dense in $A$. If $\tau$ is a tracial state on $A$ then $\tau(xhahy) = \tau(h^{1/2} ahyx h^{1/2})$. Since $h^{1/2} ahyx h^{1/2} \in B$, $\tau$ is uniquely determined by $\tau_B$. \end{proof} It is clear that Proposition 3.19 agrees with Theorem 2.6, so it is a special case. \par As a next step we look at a $C^*$-algebra of the form \begin{equation*} (M, \tau) = (\underset{\alpha_0'}{\overset{p_0'}{A_0}} \oplus \underset{\alpha_1'}{\overset{p_1'}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\alpha_k'}{\overset{p_k'}{\mathbb{M}_{m_k}}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * (\mathbb{M}_n, tr_n), \end{equation*} where $A_0$ comes with a specified trace and has a unital, diffuse abelian $C^*$-subalgebra with unit $p'_0$. Also we suppose that $\alpha'_0 \geq 0$, $0 < \alpha_1' \leq ... \leq \alpha_k'$, $0 < \alpha_1 \leq ... \leq \alpha_l$, $m_1, ..., m_k \geq 2$, and either $\alpha'_0 > 0$ or $k \geq 1$, or both. Let's denote $p_0 \overset{def}{=} p_0' + p_1' + ... + p_k'$, $B_0 \overset{def}{=} \underset{\alpha_1'}{\overset{p_1'}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\alpha_k'}{\overset{p_k'}{\mathbb{M}_{m_k}}}$, and $\alpha_0 \overset{def}{=} \alpha'_0 + \alpha'_1 + ... + \alpha'_k = \tau(p_0)$. \\ Let's have a look at the $C^*$-subalgebras $N$ and $N'$ of $M$ given by \begin{equation*} (N, \tau|_N) = (\underset{\alpha_0}{\overset{p_0}{\mathbb{C}}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * (\mathbb{M}_n, tr_n) \end{equation*} and \begin{equation*} (N', \tau|_{N'}) = (\underset{\alpha_0'}{\overset{p_0'}{\mathbb{C}}} \oplus \underset{\alpha_1'}{\overset{p_1'}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_k'}{\overset{p_k'}{\mathbb{C}}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * (\mathbb{M}_n, tr_n). \end{equation*} We studied the $C^*$-algebras, having the form of $N$ and $N'$ in the previous section. A brief description is as follows: \par If $\alpha_0, \alpha_l < 1-\frac{1}{n^2}$, then $N$ is simple with a unique trace and $N'$ is also simple with a unique trace. For each of the projections $p_0', p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $N'$, supported on it. \par If $\alpha_0$, or $\alpha_l$ $= 1-\frac{1}{n^2}$, then $N$ has no central projections, and we have a short exact sequence $0 \rightarrow N_0 \rightarrow N \rightarrow \mathbb{M}_n \rightarrow 0$, with $N_0$ being simple with a unique trace. Moreover $p_0$ or $p_l$ respectivelly is full in $N$. For each of the projections $p_0', p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $N'$, supported on it. \par If $\alpha_0$ or $\alpha_l$ $> 1-\frac{1}{n^2}$, then $N = \overset{q}{N_0} \oplus \mathbb{M}_n$, with $N_0$ being simple and having a unique trace. \par We consider 2 cases: \par (I) case: $\alpha_l \geq \alpha_0$. \par (1) $\alpha_l < 1-\frac{1}{n^2}$. \par In this case $N$ and $N'$ are simple and has unique traces, and $p_0$ is full in $N$ and consequently $1_M = 1_N$ is contained in $\langle p_0 \rangle_N$ - the ideal of $N$, generated by $p_0$. Since $\langle p_0 \rangle_N \subset \langle p_0 \rangle_M$ it follows that $p_0$ is full also in $M$. From Proposition 4.1 we get $p_0 M p_0 \cong (A_0 \oplus B_0) * p_0 N p_0$. Then from Theorem 3.9 follows that $p_0 M p_0$ is simple and has a unique trace. Since $p_0$ is a full projection, Proposition 4.3 tells us that $M$ is simple and $\tau$ is its unique trace. For each of the projections $p_0', p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $M$, supported on it, and comming from $N'$. \par (2) $\alpha_l = 1-\frac{1}{n^2}$. \par In this case it is also true that for each of the projections $p_0', p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $M$, supported on it, and comming from $N'$. It is easy to see that $M$ is the linear span of $p_0 M p_0$, $p_0 M (1- p_0) N (1- p_0)$, $(1 -p_0) N p_0 M p_0$, $(1- p_0) N p_0 M p_0 N (1- p_0)$ and $(1- p_0) N (1- p_0)$. We know that we have a $*$-homomorphism $\pi : N \rightarrow M_n$, such that $\pi(p_l) = 1$. Then it is clear that $\pi(p_0) = 0$, so we can extend $\pi$ to a linear map $\tilde{\pi}$ on $M$, defining it to equal $0$ on $p_0 M p_0$, $p_0 M (1- p_0) N (1- p_0)$, $(1 -p_0) N p_0 M p_0$ and $(1- p_0) N p_0 M p_0 N (1- p_0)$. It is also clear then that $\tilde{\pi}$ will actually be a $*$-homomorphism. Since $\ker(\pi)$ is simple in $N$ and $p_0 \in \ker(\pi)$, then $p_0$ is full in $\ker(\pi) \subset N$, so by the above representation of $M$ as a linear span we see that $p_0$ is full in $\ker(\tilde{\pi})$ also. From Proposition 4.1 follows that $p_0 M p_0 \cong (A_0 \oplus B_0) * (p_0 N p_0)$. Since $p_0 N p_0$ has a unital, diffuse abelian $C^*$-subalgebra with unit $p_0$, it follows from Theorem 3.9 that $p_0 M p_0$ is simple and has a unique trace (to make this conclusion we could use Theorem 1.5 instead). Now since $p_0 M p_0$ is full and hereditary in $\ker(\tilde{\pi})$, from Proposition 4.3 follows that $\ker(\tilde{\pi})$ is simple and has a unique trace. \par (3) $\alpha_l > 1-\frac{1}{n^2}$. \par In this case $N = \underset{n^2 - n^2 \alpha_l}{\overset{q}{N_0}} \oplus \underset{n^2 \alpha_l - n^2 + 1}{\overset{1-q}{\mathbb{M}_n}}$ and also $N' = \underset{n^2 - n^2 \alpha_l}{\overset{q}{N'_0}} \oplus \underset{n^2 \alpha_l - n^2 + 1}{\overset{1-q}{\mathbb{M}_n}}$ with $N_0$ and $N'_0$ being simple with unique traces. For each of the projections $q p_0', q p_1' , ..., q p_k', q p_1, ..., q p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $M$, supported on it, and coming from $N'_0$. \par Since $p_0 \leq q$ we can write $M$ as a linear span of $p_0 M p_0$, $p_0 M p_0 N_0 (1- p_0)$, $(1- p_0) N_0 p_0 M p_0$, $(1- p_0) N_0 p_0 M p_0 N_0 (1- p_0)$, $(1- p_0) N_0 (1- p_0)$ and $\mathbb{M}_n$. So we can write $M = \underset{n^2 - n^2 \alpha_l}{\overset{q}{M_0}} \oplus \underset{n^2 \alpha_l - n^2 + 1}{\overset{1-q}{\mathbb{M}_n}}$, where $M_0 \overset{def}{=} q M q \supset N_0$. We know that $p_0$ is full in $N_0$, so as before we can write $1_{M_0} = 1_{N_0} \in \langle p_0 \rangle_{N_0} \subset \langle p_0 \rangle_{M_0}$, so $\langle p_0 \rangle_{M_0} = M_0$. Because of Proposition 4.1, we can write $p_0 M_0 p_0 \cong (A_0 \oplus B_0) * (p_0 N_0 p_0)$. Since $p_0 N_0 p_0$ has a unital, diffuse abelian $C^*$-subalgebra with unit $p_0$, then from Theorem 3.9 (or from Theorem 1.5) it follows that $p_0 M_0 p_0$ is simple with a unique trace. Since $p_0 M_0 p_0$ is full and hereditary in $M_0$, Proposition 4.3 yields that $M_0$ is simple with a unique trace. \par (II) $\alpha_0$ $>$ $\alpha_l$. \par (1) $\alpha_0 \leq 1- \frac{1}{n^2}$. \par In this case $p_0$ is full in $N$ and also in $N'$, so $1_M = 1_N \in \langle p_0 \rangle_N$, which means $p_0$ is full in $M$ also. $p_0 M p_0$ is a full hereditary $C^*$-subalgebra of $M$ and $p_0 M p_0 \cong (A_0 \oplus B_0) * p_0 N p_0$ by Proposition 4.1. Since $p_0 N p_0$ has a diffuse abelian $C^*$-subalgebra, Theorem 3.9 (or Theorem 1.5) shows that $p_0 M p_0$ is simple with a unique trace and then by Proposition 4.3 follows that the same is true for $M$. For each of the projections $p_0', p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $M$, supported on it, comming from $N'$. \par (2) $\alpha_0$ $> 1-\frac{1}{n^2}$. \\ We have 3 cases: \par (2$'$) $\alpha'_0 > 1-\frac{1}{n^2}$. \par In this case $N \cong \overset{q}{N_0} \oplus \mathbb{M}_n$ and $N' \cong \overset{q'}{N'_0} \oplus \mathbb{M}_n$, where $q \leq q'$, with $N_0$ and $N'_0$ being simple and having unique traces. It is easy to see that $p'_1, ..., p'_k, p_1, ..., p_l \leq q'$, so for each of the projections $p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $N'$, supported on it. So those $C^*$-subalgebras live in $M$ also. We have a unital, diffuse abelian $C^*$-subalgebra of $A_0$, supported on $1_{A_0}$, which yields a unital, diffuse abelian $C^*$-subalgebra on $M$, supported on $p'_0$. It is clear that $p_0$ is full in $N$, so as before, $1_M = 1_N \in \langle p_0 \rangle_N$, so $p_0$ is full in $M$ also, so $p_0 M p_0$ is a full hereditary $C^*$-subalgebra of $M$. From Proposition 4.1 we have $p_0 M p_0 \cong (A_0 \oplus B_0) * ( p_0 N_0 p_0 \oplus \mathbb{M}_n)$. It is easy to see that $\mathbb{M}_n$, for $n \geq 2$ contains two $tr_n$-orthogonal zero-trace unitaries. Since also $p_0 N_0 p_0$ has a unital, diffuse abelian $C^*$-subalgebra, supported on $1_{N_0}$, it is easy to see (using Proposition 2.2) that it also contains two $\tau|{N_0}$-orthogonal, zero-trace unitaries. Then the conditions of Theorem 1.5 are satisfied. This means that $p_0 M p_0$ is simple with a unique trace and Proposition 4.3 implies that $M$ is simple with a unique trace also. \par (2$''$) $\alpha'_k > 1-\frac{1}{n^2}$. \par Let's denote $$N'' = (\underset{\alpha_0'}{\overset{p_0'}{A_0}} \oplus \underset{\alpha_1'}{\overset{p_1'}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\alpha_{k-1}'}{\overset{p_{k-1}'}{\mathbb{M}_{m_{k-1}}}} \oplus \underset{\alpha_{k}'}{\overset{p_k'}{\mathbb{C}}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * (\mathbb{M}_n, tr_n).$$ Then $N''$ satisfies the conditions of case (I,3) and so $N'' \cong \overset{q}{N''_0} \oplus \mathbb{M}_n$. Clearly $p_0', p_1', ..., p_{k-1}', p_1, ..., p_l \leq q$, so for each of the projections $p_0', p_1', ..., p_{k-1}', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $N''_0$, supported on it. Those $C^*$-algebras live in $M$ also. From case (I,3) we have that $p'_k$ is full in $N''$ and as before $1_M = 1_{N''} \in \langle p'_k \rangle_{N''}$ implies that $p'_k$ is full in $M$ also. From Proposition 4.1 follows that $p'_k M p'_k \cong (p'_k N''_0 p'_k \oplus \mathbb{M}_n) * \mathbb{M}_{m_k}$. Since $N''_0$ has a unital, diffuse abelian $C^*$-subalgebra, supported on $q p'_k$, then an argument, similar to the one we made in case (II, 2$"$), allows to apply Theorem 1.5 to get that $p'_k M p'_k$ is simple with a unique trac. By Proposition 4.3 follows that the same is true for $M$. The unital, diffuse abelian $C^*$-subalgebra of $M$, supported on $p'_k$, we can get by applying the note after Theorem 1.5 to $p'_k M p'_k \cong (p'_k N''_0 p'_k \oplus \mathbb{M}_n) * \mathbb{M}_{m_k}$. \par (2$'''$) $\alpha'_0$ and $\alpha'_k$ $\leq 1-\frac{1}{n^2}$. \par In this case $N \cong \overset{q}{N_0} \oplus \mathbb{M}_n$, with $N_0$ being simple and having a unique trace. Moreover $N'$ has no central projections and for each of the projections $p'_0, p_1', ..., p_k', p_1, ..., p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $N'$, supported on it. So those $C^*$-subalgebras live in $M$ also. It is clear that $p_0$ is full in $N$, so as before $1_M = 1_N \in \langle p_0 \rangle_N$, so $p_0$ is full in $M$ also, so $p_0 M p_0$ is a full hereditary $C^*$-subalgebra of $M$. From Proposition 4.1 we have $p_0 M p_0 \cong (A_0 \oplus B_0) * ( p_0 N_0 p_0 \oplus \mathbb{M}_n)$. Since $A_0$ and $p_0 N_0 p_0$ both have unital, diffuse abelian $C^*$-subalgebras, supported on their units, it is easy to see (using Proposition 2.2), that the conditions of Theorem 1.5 are satisfied. This means that $p_0 M p_0$ is simple with a unique trace and Proposition 4.3 yields that $M$ is simple with a unique trace also. \par We summarize the discussion above in the following \begin{prop} Let \begin{equation*} (M,\tau) \overset{def}{=} (\underset{\alpha_0'}{\overset{p_0'}{A_0}} \oplus \underset{\alpha_1'}{\overset{p_1'}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\alpha_k'}{\overset{p_k'}{\mathbb{M}_{m_k}}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * (\mathbb{M}_n, tr_n), \end{equation*} where $n \geq 2$, $\alpha'_0 \geq 0$, $\alpha'_1 \leq \alpha'_2 \leq ... \leq \alpha'_k$, $\alpha_1 \leq ... \leq \alpha_l$, $m_1, ..., m_k \geq 2$, and $\overset{p'_0}{A_0} \oplus 0$ has a unital, diffuse abelian $C^*$-subalgebra, having $p'_0$ as a unit. Then: \par (I) If $\alpha_l < 1-\frac{1}{n^2}$, then $M$ is unital, simple with a unique trace $\tau$. \par (II) If $\alpha_l = 1-\frac{1}{n^2}$, then we have a short exact sequence $0 \rightarrow M_0 \rightarrow M \rightarrow \mathbb{M}_n \rightarrow 0$, where $M$ has no central projections and $M_0$ is nonunital, simple with a unique trace $\tau|_{M_0}$. \par (III) If $\alpha_l > 1-\frac{1}{n^2}$, then $M = \underset{n^2 - n^2 \alpha_l}{\overset{f}{M_0}} \oplus \underset{n^2 \alpha_l - n^2 + 1}{\overset{1-f}{\mathbb{M}_n}}$, where $1-f \leq p_l$, and where $M_0$ is unital, simple and has a unique trace $(n^2 - n^2 \alpha_l)^{-1} \tau|_{M_0}$. \par Let $f$ means the identity projection for cases (I) and (II). Then in all cases for each of the projections $f p_0', f p_1', ..., f p_k', f p_1, ..., f p_l$ we have a unital, diffuse abelian $C^*$-subalgebra of $M$, supported on it. \par In all the cases $p_l$ is a full projection in $M$. \end{prop} To prove Theorem 2.6 we will use Proposition 4.4. First let's check that Proposition 4.4 agrees with the conclusion of Theorem 2.6. We can write $$(M,\tau) \overset{def}{=} (\underset{\alpha_0'}{\overset{p_0'}{A_0}} \oplus \underset{\alpha_1'}{\overset{p_1'} {\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\alpha_k'}{\overset{p_k'}{\mathbb{M}_{m_k}}} \oplus \underset{\alpha_1}{\overset{p_1} {\mathbb{C}}} \oplus ... \oplus \underset{\alpha_l}{\overset{p_l}{\mathbb{C}}}) * \underset{\beta_1}{\overset{q_1}{\mathbb{M}_n}},$$ where $q_1 = 1_M$ and $\beta_1 = 1$. It is easy to see that $L_0 = \{ (l,1) | \frac{\alpha_l}{1^2} + \frac{1}{n^2} = 1 \} = \{ (l,1) | \alpha_l = 1-\frac{1}{n^2} \}$, which is not empty if and only if $\alpha_l = 1-\frac{1}{n^2}$. Also $L_+ = \{ (l,1) | \frac{\alpha_l}{1^2} + \frac{1}{n^2} > 1 \} = \{ (l,1) | \alpha_l > 1-\frac{1}{n^2} \}$, and here $L_+$ is not empty if and only if $\alpha_l > 1-\frac{1}{n^2}$. If both $L_+$ and $L_0$ are empty, then $M$ is simple with a unique trace. If $L_0$ is not empty, then clearly $L_+$ is empty, so we have no central projections and a short exact sequence $0 \rightarrow M_0 \rightarrow M \rightarrow \mathbb{M}_n \rightarrow 0$, with $M_0$ being simple with a unique trace. In this case all nontrivial projections are full in $M$. If $L_+$ is not empty, then clearly $L_0$ is empty and so $M = \underset{n^2 -n^2 \alpha_l}{\overset{q}{M_0}} \oplus \underset{n^2(\frac{\alpha_l}{1^2} + \frac{1}{n^2} - 1)}{\overset{1-q}{\mathbb{M}_n}}$, where $M_0$ is simple with a unique trace. $p_l$ is full in $M$. \\ \par $Proof\ of\ Theorem\ 2.6:$ \\ \par Now to prove Theorem 2.6 we start with \begin{equation*} (\mathfrak{A},\phi )=(\underset{\alpha_0}{\overset{p_0}{A_0}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{M}_{n_1}}} \oplus ... \oplus \underset{\alpha_k}{\overset{p_k}{\mathbb{M}_{n_k}}})*(\underset{\beta_0}{\overset{q_0}{B_0}} \oplus \underset{\beta_1}{\overset{q_1}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\beta_l}{\overset{q_l}{\mathbb{M}_{m_l}}}), \end{equation*} where $A_0$ and $B_0$ have unital, diffuse abelian $C^*$-subalgebras, supported on their units (we allow $\alpha_0 = 0$ or/and $\beta_0 = 0$). The case where $n_1 = ... = n_k = m_1 = ... = m_l = 1$ is treated in Theorem 2.5. The case where $\alpha_0 = 0$, $k = 1$, and $n_k > 1$ was treated in Proposition 4.4. So we can suppose without loss of generality that $n_k \geq 2$ and either $k > 1$ or $\alpha_0 > 0$ or both. To prove that the conclusions of Theorem 2.6 takes place in this case we will use induction on $\card \{ i | n_i \geq 2 \} + \card \{ j | m_j \geq 2 \}$, having Theorem 2.5 ($\card \{ i | n_i \geq 2 \} + \card \{ j | m_j \geq 2 \} = 0$) as first step of the induction. We look at \begin{equation*} (\mathfrak{B},\phi|_\mathfrak{B})=(\underset{\alpha_0}{\overset{p_0}{A_0}} \oplus \underset{\alpha_1}{\overset{p_1}{\mathbb{M}_{n_1}}} \oplus ... \oplus \underset{\alpha_{k-1}}{\overset{p_{k-1}}{\mathbb{M}_{n_{k-1}}}} \oplus \underset{\alpha_k}{\overset{p_k}{\mathbb{C}}}) * (\underset{\beta_0}{\overset{q_0}{B_0}} \oplus \underset{\beta_1}{\overset{q_1}{\mathbb{M}_{m_1}}} \oplus ... \oplus \underset{\beta_l}{\overset{q_l}{\mathbb{M}_{m_l}}}) \subset (\mathfrak{A},\phi). \end{equation*} We suppose that Theorem 2.6 is true for $(\mathfrak{B},\phi|_\mathfrak{B})$ and we will prove it for $(\mathfrak{A},\phi )$. This will be the induction step and will prove Theorem 2.6. \par Denote $L_0^{\mathfrak{A}} \overset{def}{=} \{ (i,j)| \frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} = 1 \}$, $L_0^{\mathfrak{B}} \overset{def}{=} \{ (i,j)| i \leq k-1$ and $\frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} = 1 \} \cup \{ (k,j) | \frac{\alpha_k}{1^2} + \frac{\beta_j}{m_j^2} = 1 \}$ and similarly $L_+^{\mathfrak{A}} \overset{def}{=} \{ (i,j)| \frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} > 1 \}$, and $L_+^{\mathfrak{B}} \overset{def}{=} \{ (i,j)| i \leq k-1$ and $\frac{\alpha_i}{n_i^2} + \frac{\beta_j}{m_j^2} > 1 \} \cup \{ (k,j) | \frac{\alpha_k}{1^2} + \frac{\beta_j}{m_j^2} > 1 \}$. Clearly $L_0^{\mathfrak{A}} \cap \{ 1 \leq i \leq k-1 \} = L_0^{\mathfrak{B}} \cap \{ 1 \leq i \leq k-1 \}$ and similarly $L_+^{\mathfrak{A}} \cap \{ 1 \leq i \leq k-1 \} = L_+^{\mathfrak{B}} \cap \{ 1 \leq i \leq k-1 \}$. Let $N_{\mathfrak{A}}(i,j) = max(n_i, m_j)$ and let $N_{\mathfrak{B}}(i,j) = N_{\mathfrak{A}}(i,j), 1 \leq i \leq k-1$, and $N_{\mathfrak{B}}(k,j) = m_j$. By assumption \begin{equation*} \mathfrak{B}= \underset{\delta}{\overset{g}{\mathfrak{B}_0}} \oplus \underset{(i,j)\in L_+^{\mathfrak{B}}}{\bigoplus} \underset{\delta_{ij}}{\overset{g_{ij}}{\mathbb{M}_{N_{\mathfrak{B}}(i,j)}}}. \end{equation*} We want to show that \begin{equation} \mathfrak{A} = \underset{\gamma}{\overset{f}{\mathfrak{A}_0}} \oplus \underset{(i,j)\in L_+^{\mathfrak{A}}}{\bigoplus} \underset{\gamma_{ij}}{\overset{f_{ij}}{\mathbb{M}_{N_{\mathfrak{A}}(i,j)}}}. \end{equation} We can represent $\mathfrak{A}$ as the span of $p_k \mathfrak{A} p_k$, $p_k \mathfrak{A} p_k \mathfrak{B} (1-p_k)$, $(1-p_k) \mathfrak{B} p_k \mathfrak{A} p_k$, $(1-p_k) \mathfrak{B} p_k \mathfrak{A} p_k \mathfrak{B} (1-p_k)$, and $(1-p_k) \mathfrak{B} (1-p_k)$. From the fact that $g_{kj} \leq p_k$ and $g_{ij} \leq 1-p_k, \forall 1 \leq i \leq k-1$ we see that $p_k \mathfrak{B} (1-p_k) = p_k \mathfrak{B}_0 (1-p_k)$, $(1-p_k) \mathfrak{B} p_k = (1-p_k) \mathfrak{B}_0 p_k$, and $(1-p_k) \mathfrak{B} (1-p_k) = (1-p_k) \mathfrak{B}_0 (1-p_k) \oplus \underset{i \neq k}{\underset{(i,j) \in L_+^{\mathfrak{B}}}{\bigoplus}} \mathbb{M}_{N(i,j)}$. All this tells us that we can represent $\mathfrak{A}$ as the span of $p_k \mathfrak{A} p_k$, $p_k \mathfrak{A} p_k \mathfrak{B}_0 (1-p_k)$, $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A} p_k$, $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A} p_k \mathfrak{B}_0 (1-p_k)$, $ (1-p_k) \mathfrak{B}_0 (1-p_k)$, and $\underset{i \neq k}{\underset{(i,j)\in L_+^{\mathfrak{B}}}{\bigoplus}} \underset{\delta_{ij}}{\overset{g_{ij}}{\mathbb{M}_{N(i,j)}}}$. \par In order to show that $\mathfrak{A}$ has the form (9), we need to look at $p_k \mathfrak{A} p_k$. From Proposition 4.1 we have $$p_k \mathfrak{A} p_k \cong (p_k \mathfrak{B} p_k) * \mathbb{M}_{n_k} \cong (\underset{\frac{\delta}{\alpha_k}}{\overset{g}{p_k \mathfrak{B}_0 p_k}} \oplus \underset{(k,j)\in L_+^{\mathfrak{B}}}{\bigoplus} \underset{\frac{\delta_{kj}}{\alpha_k}}{\overset{g_{kj}}{\mathbb{M}_{N(k,j)}}}) * \mathbb{M}_{n_k}.$$ Since by assumption $p_k \mathfrak{B}_0 p_k$ has a unital, diffuse abelian $C^*$-subalgebra, supported on $1_{p_k \mathfrak{B}_0 p_k}$, we can use Proposition 4.4 to determine the form of $p_k \mathfrak{A} p_k$. \par Thus $p_k \mathfrak{A} p_k$: \par (i) Is simple with a unique trace if whenever for all $1 \leq r \leq l$ with $N(k,r) = 1$ we have $\frac{\delta_{kr}}{\alpha_k} < 1 - \frac{1}{n_k^2}$. \par (ii) Is an extension $0 \rightarrow I \rightarrow p_k \mathfrak{A} p_k \rightarrow \mathbb{M}_{n_k} \rightarrow 0$ if $\exists 1 \leq r \leq l$, with $N(k,r) = 1$, and $\frac{\delta_{kr}}{\alpha_k} = 1 - \frac{1}{n_k^2}$. Moreover $I$ is simple with a unique trace and has no central projections. \par (iii) Has the form $p_k \mathfrak{A} p_k = I \oplus \underset{n_k^2(\frac{\delta_{kr}}{\alpha_k} - 1 + \frac{1}{n_k^2})}{\mathbb{M}_{n_k}}$, where $I$ is unital, simple with a unique trace whenever $\exists 1 \leq r \leq l$ with $N(k,r) = 1$, and $\frac{\delta_{kr}}{\alpha_k} > 1 - \frac{1}{n_k^2}$. \par By assumption $\delta_{ij} = N(i,j)^2 (\frac{\alpha_i}{n_i^2} +\frac{\beta_j}{m_j^2} - 1)$, so when $r$ satisfies the conditions of case (iii) above, then $m_r = 1$ and $n_k^2(\frac{\delta_{kr}}{\alpha_k} - 1 + \frac{1}{n_k^2}) = n_k^2(\frac{\alpha_k + \beta_r - 1}{\alpha_k} + \frac{1}{n_k^2} -1) = \frac{n_k^2}{\alpha_k}(\frac{\alpha_k}{n_k^2} + \frac{\beta_r}{1^2} - 1)$, just what we needed to show. Defining $\mathfrak{A}_0 \overset{def}{=} (1-(\underset{(i,j) \in L_+^{\mathfrak{A}}}{\oplus} f_{ij})) \mathfrak{A} (1-(\underset{(i,j) \in L_+^{\mathfrak{A}}}{\oplus} f_{ij}))$, we see that $\mathfrak{A}$ has the form (9). \par We need to study $\mathfrak{A}_0$ now. Since clearly $g \leq f$, we see that $\mathfrak{A} p_k \mathfrak{B}_0 = \mathfrak{A} p_k g \mathfrak{B}_0 = \mathfrak{A} g p_k \mathfrak{B}_0 = \mathfrak{A}_0 p_k \mathfrak{B}_0$ and similarly $\mathfrak{A} p_k \mathfrak{B}_0 = \mathfrak{A}_0 p_k \mathfrak{B}_0$. From this and from what we proved above follows that: \begin{gather} \mathfrak{A}_0 \text{ is the span of } p_k \mathfrak{A}_0 p_k,\ (1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k, \\ \notag p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k),\ (1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k), \text{ and } (1-p_k) \mathfrak{B}_0 (1-p_k). \end{gather} We need to show that for each of the projections $f p_s$, $0 \leq s \leq k$ and $f q_t$, $1 \leq t \leq l$, we have a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on it. The ones, supported on $f p_s$, $1 \leq s \leq k-1$ come from $(1-p_k) \mathfrak{B}_0 (1-p_k)$ by the induction hypothesis. The one with unit $f p_k$ comes from the representation $p_k \mathfrak{A} p_k \cong (p_k \mathfrak{B} p_k) * \mathbb{M}_{n_k}$ and Proposition 4.4. For $1 \leq s \leq l$ we have \begin{gather} q_s \mathfrak{A} q_s \cong \underset{\frac{\gamma}{\beta_s}}{\overset{f q_s}{{q_s \mathfrak{A}_0 q_s}}} \oplus \underset{1 \leq i \leq k-1}{\underset{(i,s) \in L_+^{\mathfrak{A}}}{\bigoplus}} \underset{\frac{\gamma_{is}}{\beta_s}}{\overset{f_{is}}{\mathbb{M}_{N_{\mathfrak{A}}(i,s)}}} \oplus \underset{\frac{\gamma_{ks}}{\beta_s}}{\overset{f_{ks}}{\mathbb{M}_{N_{\mathfrak{A}}(k,s)}}} \end{gather} and \begin{gather} q_s \mathfrak{B} q_s \cong \underset{\frac{\delta}{\beta_s}}{\overset{g q_s}{q_s \mathfrak{B}_0 q_s}} \oplus \underset{1 \leq i \leq k-1}{\underset{(i,s) \in L_+^{\mathfrak{B}}}{\bigoplus}} \underset{\frac{\delta_{is}}{\beta_s}}{\overset{g_{is}}{\mathbb{M}_{N_{\mathfrak{B}}(i,s)}}} \oplus \underset{\frac{\delta_{ks}}{\beta_s}}{\overset{g_{ks}}{\mathbb{M}_{N_{\mathfrak{B}}(k,s)}}}. \end{gather} From what we showed above follows that for $1 \leq i \leq k-1$ we have $\gamma_{is} = \delta_{is}$ and $f_{is} = g_{is}$. If $(k,s) \notin L_+^{\mathfrak{B}}$, (or $\alpha_k < 1 - \frac{\beta_s}{m_s^2}$), then $(k,s) \notin L_+^{\mathfrak{A}}$ and by (11) and (12) we see that $gq_s = fq_s$ and so in $\mathfrak{A}_0$ we have a unital, diffuse abelian $C^*$-subalgebra with unit $gq_s = fq_s$, which comes from $\mathfrak{B}_0$. If $(k,s) \in L_+^{\mathfrak{B}}$, then $gq_s \lvertneqq fq_s$ and since we have a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on $gq_s$, comming from $\mathfrak{B}_0$, we need only to find a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on $fq_s - gq_s$ and its direct sum with the one supported on $gq_s$ will be a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on $fq_s$. But from the form (11) and (12) it is clear that $fq_s - gq_s \leq g_{ks}$, since from (11) and (12) $(f_{1s} + ... + f_{(k-1)s}) q_s \mathfrak{A} q_s (f_{1s} + ... + f_{(k-1)s}) = (g_{1s} + ... + g_{(k-1)s}) q_s \mathfrak{B} q_s (g_{1s} + ... + g_{(k-1)s})$. It is also clear then that $ fq_s - gq_s = f g_{ks} \leq p_k$, since $gq_s \perp g_{ks}$. We look for this $C^*$-subalgebra in $$p_k \mathfrak{A} p_k = \underset{\frac{\gamma}{\alpha_k}}{\overset{fp_k}{ p_k \mathfrak{A}_0 p_k }} \oplus \underset{(k,j)\in L_+^{\mathfrak{A}}}{\bigoplus} \underset{\frac{\gamma_{kj}}{\alpha_k}}{\overset{f_{kj}} {\mathbb{M}_{N_{\mathfrak{A}}(k,j)}}} \cong (p_k \mathfrak{B} p_k) * \mathbb{M}_{n_k},$$ $$ \cong (\underset{\frac{\delta}{\alpha_k}}{\overset{g}{p_k \mathfrak{B}_0 p_k}} \oplus \underset{(k,j)\in L_+^{\mathfrak{B}}}{\bigoplus} \underset{\frac{\delta_{kj}}{\alpha_k}}{\overset{g_{kj}}{\mathbb{M}_{N_{\mathfrak{B}}(k,j)}}}) * \mathbb{M}_{n_k}.$$ Proposition 4.4 gives us a unital, diffuse abelian $C^*$-subalgebra of $p_k \mathfrak{A}_0 p_k$, supported on $(f p_k) g_{ks} = f g_{ks} = fq_s -gq_s$. This proves that we have a unital, diffuse abelian $C^*$-subalgebra of $\mathfrak{A}_0$, supported on $fq_s$. \par Now we have to study the ideal structure of $\mathfrak{A}_0$, knowing by the induction hypothesis, the form of $\mathfrak{B}$. We will use the "span representation" of $\mathfrak{A}_0$ (10). \par For each $(i,j) \in L_0^{\mathfrak{B}}$ we know the existance of $*$-homomorphisms $\pi_{(i,j)}^{\mathfrak{B}_0} : \mathfrak{B}_0 \rightarrow \mathbb{M}_{N_{\mathfrak{B}}(i,j)}$. For $i \neq k$ we can write those as $\pi_{(i,j)}^{\mathfrak{B}_0} : \mathfrak{B}_0 \rightarrow \mathbb{M}_{N_{\mathfrak{A}}(i,j)}$ and since the support of $\pi_{(i,j)}^{\mathfrak{B}_0}$ is contained in $(1-p_k)$, using (10), we can extend linearly $\pi_{(i,j)}^{\mathfrak{B}_0}$ to $\pi_{(i,j)}^{\mathfrak{A}_0} : \mathfrak{A}_0 \rightarrow \mathbb{M}_{N_{\mathfrak{A}}(i,j)}$, by defining it to be zero on $p_k \mathfrak{A}_0 p_k$, $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k$, $p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k)$, and $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k)$. Clearly $\pi_{(i,j)}^{\mathfrak{A}_0}$ is a $*$-homomorphism also. \par By the induction hypothesis we know that $g p_k$ is full in $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{B}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{B}_0}) \subset \mathfrak{B}_0$ and by (10), and the way we extended $\pi_{(i,j)}^{\mathfrak{B}_0}$, we see that $f p_k$ is full in $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}) \subset \mathfrak{A}_0$. Then $p_k \mathfrak{A}_0 p_k$ is full and hereditary in $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, so by the Rieffel correspondence from \cite{R82}, we have that $p_k \mathfrak{A}_0 p_k$ and $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ have the same ideal structure. \par Above we saw that \begin{gather} p_k \mathfrak{A} p_k = \underset{\frac{\gamma}{\alpha_k}}{\overset{fp_k}{ p_k \mathfrak{A}_0 p_k }} \oplus \underset{(k,j)\in L_+^{\mathfrak{A}}}{\bigoplus} \underset{\frac{\gamma_{kj}}{\alpha_k}}{\overset{f_{kj}} {\mathbb{M}_{N_{\mathfrak{A}}(k,j)}}} \cong (p_k \mathfrak{B} p_k) * \mathbb{M}_{n_k} \cong \\ \notag \cong (\underset{\frac{\delta}{\alpha_k}}{\overset{gp_k}{p_k \mathfrak{B}_0 p_k}} \oplus \underset{(k,j)\in L_+^{\mathfrak{B}}}{\bigoplus} \underset{\frac{\delta_{kj}}{\alpha_k}}{\overset{g_{kj}}{\mathbb{M}_{N_{\mathfrak{B}}(k,j)}}}) * \mathbb{M}_{n_k}. \end{gather} From Proposition 4.4 follows that $p_k \mathfrak{A}_0 p_k$ is not simple if and only if $\exists 1 \leq s \leq m$, such that $(k,s) \in L_+^{\mathfrak{B}}, m_s = 1$ with $\frac{\delta_{ks}}{\alpha_k} = 1-\frac{1}{n_k^2}$, where $\delta_{ks} = \alpha_k + \beta_s -1$. This means that $\frac{\alpha_k + \beta_s -1}{\alpha_k} = 1 - \frac{1}{n_k^2}$, which is equivalent to $\frac{\beta_s}{1^2} + \frac{\alpha_k}{n_k^2} = 1$, so this implies $(k,s) \in L_0^{\mathfrak{A}}$. If this is the case (13), together with Proposition 4.4 gives us a $*$-homomorphism $\pi'_{(k,s)} : p_k \mathfrak{A}_0 p_k \rightarrow \mathbb{M}_{n_k}$, such that $\ker(\pi'_{(k,s)}) \subset p_k \mathfrak{A}_0 p_k$ is simple with a unique trace. Using (10) we extend $\pi'_{(k,s)}$ linearly to a linear map $\pi_{(k,s)}^{\mathfrak{A}_0} : \mathfrak{A}_0 \rightarrow \mathbb{M}_{n_k}$, by defining $\pi_{(k,s)}^{\mathfrak{A}_0}$ to be zero on $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k$, $p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k)$, $(1-p_k) \mathfrak{B}_0 p_k \mathfrak{A}_0 p_k \mathfrak{B}_0 (1-p_k)$, and $(1-p_k) \mathfrak{B}_0 (1-p_k)$. Similarly as before, $\pi_{(k,s)}^{\mathfrak{A}_0}$ turns out to be a $*$-homomorphism. By the Rieffel correspondence of the ideals of $p_k \mathfrak{A}_0 p_k$ and $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, it is easy to see that the simple ideal $\ker(\pi'_{(k,s)}) \subset p_k \mathfrak{A}_0 p_k$ corresponds to the ideal $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}) \subset \underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, so $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ is simple. To see that $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ has a unique trace we notice that from the construction of $\pi_{(i,j)}^{\mathfrak{A}_0}$ we have $\ker(\pi'_{(k,s)}) = p_k \ker(\pi_{(k,s)}^{\mathfrak{A}_0}) p_k = p_k \underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}) p_k$ (the last equality is true because $p_k \mathfrak{A}_0 p_k \subset \ underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$). Now we argue similarly as in the proof of Proposition 4.3, using the fact that $\ker(\pi'_{(k,s)})$ has a unique trace: Suppose that $\rho$ is a trace on $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. It is easy to see that $\Span \{ x p_k a p_k y | x, y, a \in \underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}), a \geq 0 \}$ is dense in $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, since $\ker(\pi'_{(k,s)})$ is full in $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. Then since $p_k a p_k \geq 0$ we have $\rho(x p_k a p_k y) = \rho((p_k a p_k) y x) = \rho((p_k a p_k)^{1/2} y x (p_k a p_k)^{1/2})$ and since $(p_k a p_k)^{1/2} y x (p_k a p_k)^{1/2}$ is supported on $p_k$, it follows that $(p_k a p_k)^{1/2} y x (p_k a p_k)^{1/2} \in p_k \underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}) p_k = \ker(\pi'_{(k,s)})$, so $\rho$ is uniquely determined by $\rho|_{\ker(\pi'_{(k,s)})}$ and hence $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ has a unique trace. \par If $\nexists 1 \leq s \leq m$ with $(k,s) \in L_0^{\mathfrak{A}}$ it follows from what we said above, that $p_k \mathfrak{A}_0 p_k$ is simple with a unique trace. But since $p_k \mathfrak{A}_0 p_k$ is full and hereditary in $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0}) = \underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ it follows that $\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$ is simple with a unique trace in this case too. \par We showed already that $f p_k$ is full in $\underset{i \neq k}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. Now let $1 \leq r \leq k-1$. We need to show that $f p_r$ is full in $\underset{i \neq r}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. From (11) and (12) follows that $f-g \leq p_k$. So $f p_r = g p_r$ for all $1 \leq r \leq k-1$. From the way we constructed $\pi_{(i,j)}^{\mathfrak{A}_0}$ is clear that $f p_r \in \underset{i \neq r}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. It is also true that $f p_r \notin \ker(\pi_{(r,j)}^{\mathfrak{A}_0})$ for any $1 \leq j \leq l$. So the smallest ideal of $\mathfrak{A}_0$, that contains $f p_r$, is $\underset{i \neq r}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, meaning that we must have $\langle f p_r \rangle_{\mathfrak{A}_0} = \underset{i \neq r}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. \par Finally, we need to show that for all $1 \leq s \leq l$ we have that $f q_s$ is full in $\underset{j \neq s}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. Let $(i,j) \in L_0^{\mathfrak{A}}$ with $i \neq k$, $j \neq s$. Since $g q_s \in \ker(\pi_{(i,j)}^{\mathfrak{B}})$ and since $(f-g)q_s \leq p_k$, the way we extended $\pi_{(i,j)}^{\mathfrak{B}}$ to $\pi_{(i,j)}^{\mathfrak{A}}$ shows that $f q_s \in \ker(\pi_{(i,j)}^{\mathfrak{B}})$. Let $(i,s) \in L_0^{\mathfrak{A}}$ and $i \neq k$. Then we know that $g q_s \notin \ker(\pi_{(i,j)}^{\mathfrak{B}})$, which implies $f q_s \notin \ker(\pi_{(i,j)}^{\mathfrak{A}})$. Suppose $(k,s) \in L_0^{\mathfrak{A}}$. Then $m_s = 1$ and (13), Proposition 4.4, and the way we extended $\pi'_{(k,s)}$ to $\pi_{(k,s)}^{\mathfrak{A}_0}$ show, that $f g_{ks} = fq_s - gq_s$ is full in $p_k \mathfrak{A}_0 p_k$, meaning that $fq_s -gq_s$, and consequently $fq_s$, is not contained in $\ker(\pi_{(k,s)}^{\mathfrak{A}_0})$. Finally let $j \neq s$, and suppose $(k,j) \in L_0^{\mathfrak{A}}$. This means that $(k,j) \in L_+^{\mathfrak{B}}$ and also that the trace of $q_j$ is so big, that $(i,s) \notin L_+^{\mathfrak{B}}$ and $(i,s) \notin L_0^{\mathfrak{B}}$ for any $1 \leq i \leq k$. Then (12) shows that $q_s \leq g$. The way we defined $\pi_{(k,j)}^{\mathfrak{A}_0}$ using (13) and Proposition 4.4 shows us that $\mathfrak{B}_0 \subset \ker(\pi_{(k,j)}^{\mathfrak{A}_0})$ in this case. This shows $q_s = g q_s = fq_s \in \ker(\pi_{(k,j)}^{\mathfrak{A}_0})$. All this tells us that the smallest ideal of $\mathfrak{A}_0$, containing $fq_s$, is $\underset{j \neq s}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$, and therefore $\langle fq_s \rangle_{\mathfrak{A}_0} = \underset{j \neq s}{\underset{(i,j) \in L_0^{\mathfrak{A}}}{\bigcap}} \ker(\pi_{(i,j)}^{\mathfrak{A}_0})$. \par This concludes the proof of Theorem 2.6. \qed {\em Acknowledgements.} I would like to thank Ken Dykema, my advisor, for the many helpful conversations I had with him, for the moral support and for reading the first version of this paper. I would also like to thank Ron Douglas and Roger Smith for some discussions. \end{document}

\begin{document} \title{Numerical homogenization of H(curl)-problems} \author{Dietmar Gallistl\footnotemark[2] \and Patrick Henning\footnotemark[3]\and Barbara Verf\"urth\footnotemark[4]} \date{} \maketitle \renewcommand{\arabic{footnote}}{\fnsymbol{footnote}} \footnotetext[2]{Institut f\"ur Angewandte und Numerische Mathematik, Karlsruher Institut f\"ur Technologie, Englerstr. 2, D-76131 Karlsruhe, Germany} \footnotetext[3]{Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsv\"agen 25, SE-100 44 Stockholm, Sweden} \footnotetext[4]{Applied Mathematics, Westf\"alische Wilhelms-Uni\-ver\-si\-t\"at M\"unster, Einsteinstr. 62, D-48149 M\"unster, Germany} \renewcommand{\arabic{footnote}}{\arabic{footnote}} \begin{Abstract} If an elliptic differential operator associated with an $\mathbf{H}(\mathrm{curl})$-problem involves rough (rapidly varying) coefficients, then solutions to the corresponding $\mathbf{H}(\mathrm{curl})$-problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest order N{\'e}d{\'e}lec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, $\mathbf{H}(\mathrm{curl})$-stable, quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh-size) in the $\mathbf{H}(\mathrm{curl})$ norm are obtained provided the right-hand side belongs to $\mathbf{H}(\mathrm{div})$. With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first order corrector, including corresponding quantitative error estimates without the requirement of scale separation. \end{Abstract} \begin{keywords} multiscale method, wave propagation, Maxwell's equations, finite element method, a priori error estimates \end{keywords} \begin{AMS} 35Q61, 65N12, 65N15, 65N30, 78M10 \end{AMS} \section{Introduction} Electromagnetic wave propagation plays an essential role in many physical applications, for instance, in the large field of wave optics. In the last years, multiscale and heterogeneous materials are studied with great interest, e.g., in the context of photonic crystals \cite{JJWM08phc}. These materials can exhibit unusual and astonishing (optical) properties, such as band gaps, perfect transmission or negative refraction \cite{CJJP02negrefraction, EP04negphC, LS15negindex}. These problems are modeled by Maxwell's equations, which involve the curl-operator and the associated Sobolev space $\VH(\curl)$. Additionally, the coefficients in the problems are rapidly oscillating on a fine scale for the context of photonic crystals and metamaterials. The numerical simulation and approximation of the solution is then a challenging task for the following three reasons. 1.\ As with all multiscale problems, a direct treatment with standard methods in infeasible in many cases because it needs grids which resolve all discontinuities or oscillations of the material parameters. 2.\ Solutions to $\VH(\curl)$-problems with discontinuous coefficients in Lip\-schitz domains can have arbitrarily low regularity, see \cite{BGL13regularitymaxwell, CDN99maxwellinterface, Cost90regmaxwellremark}. Hence, standard methods (see e.g., \cite{Monk} for an overview) suffer from bad convergence rates and fine meshes are needed to have a tolerably small error. 3.\ Due to the large kernel of the curl-operator, we cannot expect that the $L^2$-norm is of a lower order as the full $\VH(\curl)$-norm (the energy norm). Thus, it is necessary to consider dual norms or the Helmholtz decomposition to obtain improved a priori error estimates. In order to deal with the rapidly oscillating material parameters, we consider multiscale methods and thereby aim at a feasible numerical simulation. In general, these methods try to decompose the exact solution into a macroscopic contribution (without oscillations), which can be discretized on a coarse mesh, and a fine-scale contribution. Analytical homogenization for locally periodic $\VH(\curl)$-problems shows that there exists such a decomposition, where the macroscopic part is a good approximation in $H^{-1}$ and an additional fine-scale corrector leads to a good approximation in $L^2$ and $\VH(\curl)$, cf.\ \cite{CH15homerrormaxwell, HOV15maxwellHMM, Well2}. Based on these analytical results, multiscale methods are developed, e.g., the Heterogeneous Multiscale Method in \cite{HOV15maxwellHMM, CFS17hmmmaxwell} and asymptotic expansion methods in \cite{CZAL10maxwell}. The question is now in how far such considerations can be extended beyond the (locally) periodic case. The main contribution of this paper is the numerical homogenization of $\VH(\curl)$-elliptic problems -- beyond the periodic case and without assuming scale separation. The main findings can be summarized as follows. We show that the exact solution can indeed be decomposed into a coarse and fine part, using a suitable interpolation operator. The coarse part gives an optimal approximation in the $H^{-1}$-norm, the best we can hope for in this situation. In order to obtain optimal $L^2$ and $\VH(\curl)$ approximations, we have to add a so called fine-scale corrector or corrector Green's operator. This corrector shows exponential decay and can therefore be truncated to local patches of macroscopic elements, so that it can be computed efficiently. This technique of numerical homogenization is known as Localized Orthogonal Decomposition (LOD) and it was originally proposed by M{\aa}lqvist and Peterseim \cite{MP14LOD} to solve elliptic multiscale problems through an orthogonalization procedure with a problem-specific \quotes{multiscale} inner product. The LOD has been extensively studied in the context of Lagrange finite elements \cite{HM14LODbdry, HP13oversampl}, where we particularly refer to the contributions written on wave phenomena \cite{AH17LODwaves, BrG16, bgp2017, GP15scatteringPG, OV16a, P15LODhelmholtz, PeS17}. Aside from Lagrange finite elements, an LOD application in Raviart-Thomas spaces was given in \cite{HHM16LODmixed}. A crucial ingredient for numerical homogenization procedures in the spirit of LODs is the choice of a suitable interpolation operator. As we will see later, in our case we require it to be computable, $\VH(\curl)$-stable, (quasi-)local and that it commutes with the curl-operator. Constructing an operator that enjoys such properties is a very delicate task and a lot of operators have been suggested -- with different backgrounds and applications in mind. The nodal interpolation operator, see e.g.\ \cite[Thm.\ 5.41]{Monk}, and the interpolation operators introduced in \cite{DB05maxwellpintpol} are not well-defined on $\VH(\curl)$ and hence lack the required stability. Various (quasi)-interpolation operators are constructed as composition of smoothing and some (nodal) interpolation, such as \cite{Chr07intpol, CW08intpol, DH14aposteriorimaxwell, EG15intpol, Sch05multilevel,Sch08aposteriori}. For all of them, the kernel of the operator is practically hard or even impossible to compute and they only fulfill the projection \emph{or} the locality property. Finally, we mention the interpolation operator of \cite{EG15intpolbestapprox} which is local and a projection, however, which does not commute with the exterior derivative. A suitable candidate (and to the authors' best knowledge, the only one) that enjoys all required properties was proposed by Falk and Winther in \cite{FalkWinther2014}. This paper thereby also shows the applicability of the Falk-Winther operator. In this context, we mention two results, which may be of own interest: a localized regular decomposition of the interpolation error (in the spirit of \cite{Sch08aposteriori}), and the practicable implementation of the Falk-Winther operator as a matrix. The last point admits the efficient implementation of our numerical scheme and we refer to \cite{EHMP16LODimpl} for general considerations. The paper is organized as follows. Section \ref{sec:setting} introduces the general curl-curl-problem under consideration and briefly mentions its relation to Maxwell's equations. In Section \ref{sec:motivation}, we give a short motivation of our approach from periodic homogenization. Section \ref{sec:intpol} introduces the necessary notation for meshes, finite element spaces, and interpolation operators. We introduce the Corrector Green's Operator in Section \ref{sec:LODideal} and show its approximation properties. We localize the corrector operator in Section \ref{sec:LOD} and present the main apriori error estimates. The proofs of the decay of the correctors are given in Section \ref{sec:decaycorrectors}. Details on the definition of the interpolation operator and its implementation are given in Section \ref{sec:intpolimpl}. The notation $a\lesssim b$ is used for $a\leq Cb$ with a constant $C$ independent of the mesh size $H$ and the oversampling parameter $m$. It will be used in (technical) proofs for simplicity and readability. \section{Model problem} \label{sec:setting} Let $\Omega\subset \mathbb{R}^3$ be an open, bounded, contractible domain with polyhedral Lipschitz boundary. We consider the following so called curl-curl-problem: Find $\Vu:\Omega\to\mathbb{C}^3$ such that \begin{equation} \label{eq:curlcurl} \begin{split} \curl(\mu\curl\Vu)+\kappa\Vu&=\Vf\quad\text{in }\Omega,\\ \Vu\times \Vn&=0\quad\text{on }\partial \Omega \end{split} \end{equation} with the outer unit normal $\Vn$ of $\Omega$. Exact assumptions on the parameters $\mu$ and $\kappa$ and the right-hand side $\Vf$ are given in Assumption~\ref{asspt:sesquiform} below, but we implicitly assume that the above problem is a multiscale problem, i.e.\ the coefficients $\mu$ and $\kappa$ are rapidly varying on a very fine sale. Such curl-curl-problems arise in various formulations and reductions of Maxwell's equations and we shortly give a few examples. In all cases, our coefficient $\mu$ equals $\tilde{\mu}^{-1}$ with the magnetic permeability $\tilde{\mu}$, a material parameter. The right-hand side $\Vf$ is related to (source) current densities. One possible example are Maxwell's equations in a linear conductive medium, subject to Ohm's law, together with the so called time-harmonic ansatz $\hat{\Vpsi}(x,t)=\Vpsi(x)\exp(-i\omega t)$ for all fields. In this case, one obtains the above curl-curl-problem with $\Vu=\VE$, the electric field, and $\kappa=i\omega\sigma-\omega^2\varepsilon$ related to the electric permittivity $\varepsilon$ and the conductivity $\sigma$ of the material. Another example are implicit time-step discretizations of eddy current simulations, where the above curl-curl-problem has to be solved in each time step. In that case $\Vu$ is the vector potential associated with the magnetic field and $\kappa\approx\sigma/\tau$, where $\tau$ is the time-step size. Coefficients with multiscale properties can for instance arise in the context of photonic crystals. Before we define the variational problem associated with our general curl-curl-problem \eqref{eq:curlcurl}, we need to introduce some function spaces. In the following, bold face letters will indicate vector-valued quantities and all functions are complex-valued, unless explicitly mentioned. For any bounded subdomain $G\subset \Omega$, we define the space \[\VH(\curl, G):=\{ \Vv\in L^2(G, \mathbb{C}^3)|\curl\Vv\in L^2(G, \mathbb{C}^3)\}\] with the inner product $(\Vv, \Vw)_{\VH(\curl, G)}:=(\curl\Vv, \curl\Vw)_{L^2(G)}+(\Vv, \Vw)_{L^2(G)}$ with the complex $L^2$-inner product. We will omit the domain $G$ if it is equal to the full domain $\Omega$. The restriction of $\VH(\curl, \Omega)$ to functions with a zero tangential trace is defined as \[\VH_0(\curl, \Omega):=\{\Vv\in \VH(\curl, \Omega)|\hspace{3pt} \Vv\times \Vn \vert_{\partial \Omega} =0\}. \] Similarly, we define the space \[\VH(\Div, G):=\{\Vv\in L^2(G, \mathbb{C}^3)|\Div \Vv\in L^2(G, \mathbb{C})\}\] with corresponding inner product $(\cdot, \cdot)_{\VH(\Div, G)}$. For more details we refer to \cite{Monk}. We make the following assumptions on the data of our problem. \begin{assumption} \label{asspt:sesquiform} Let $\Vf\in \VH(\Div, \Omega)$ and let $\mu\in L^\infty(\Omega, \mathbb{R}^{3 \times 3})$ and $\kappa\in L^\infty(\Omega, \mathbb{C}^{3 \times 3})$. For any open subset $G\subset\Omega$, we define the sesquilinear form $\CB_{G}: \VH(\curl,G)\times \VH(\curl,G)\to \mathbb{C}$ as \begin{equation} \label{eq:sesquiform} \CB_{G}(\Vv, \Vpsi):=(\mu\curl \Vv, \curl\Vpsi)_{L^2(G)} +(\kappa\Vv, \Vpsi)_{L^2(G)}, \end{equation} and set $\CB:=\CB_\Omega$. The form $\CB_{G}$ is obviously continuous, i.e.\ there is $C_B>0$ such that \begin{equation*} |\CB_{G}(\Vv, \Vpsi)|\leq C_B\|\Vv\|_{\VH(\curl,G)}\|\Vpsi\|_{\VH(\curl,G)} \quad\text{for all }\Vv,\Vpsi\in\VH(\curl,G). \end{equation*} We furthermore assume that $\mu$ and $\kappa$ are such that $\CB: \VH_0(\curl)\times \VH_0(\curl)\to \mathbb{C}$ is $\VH_0(\curl)$-elliptic, i.e.\ there is $\alpha>0$ such that \[ |\CB(\Vv, \Vv)|\geq \alpha\|\Vv\|^2_{\VH(\curl)} \quad\text{for all }\Vv\in\VH_0(\curl) . \] \end{assumption} We now give a precise definition of our model problem for this article. Let Assumption \ref{asspt:sesquiform} be fulfilled. We look for $\Vu\in \VH_0(\curl, \Omega)$ such that \begin{equation} \label{eq:problem} \CB(\Vu, \Vpsi)=(\Vf, \Vpsi)_{L^2(\Omega)} \quad\text{for all } \Vpsi\in \VH_0(\curl, \Omega). \end{equation} Existence and uniqueness of a solution to \eqref{eq:problem} follow from the Lax-Milgram-Babu{\v{s}}ka theorem \cite{Bab70fem}. Assumption \ref{asspt:sesquiform} is fulfilled in the following two important examples mentioned at the beginning: (i) a strictly positive real function in the identity term, i.e.\ $\kappa\in L^\infty(\Omega, \mathbb{R})$, as it occurs in the time-step discretization of eddy-current problems; (ii) a complex $\kappa$ with strictly negative real part and strictly positive imaginary part, as it occurs for time-harmonic Maxwell's equations in a conductive medium. Further possibilities of $\mu$ and $\kappa$ yielding an $\VH(\curl)$-elliptic problem are described in \cite{FR05maxwell}. \begin{remark} The assumption of contractibility of $\Omega$ is only required to ensure the existence of local regular decompositions later used in the proof of Lemma \ref{lem:localregulardecomp}. We note that this assumption can be relaxed by assuming that $\Omega$ is simply connected in certain local subdomains formed by unions of tetrahedra (i.e. in patches of the form $\UN(\Omega_P)$, using the notation from Lemma \ref{lem:localregulardecomp}). \end{remark} \section{Motivation of the approach} \label{sec:motivation} For the sake of the argument, let us consider model problem \eqref{eq:curlcurl} for the case that the coefficients $\mu$ and $\kappa$ are replaced by parametrized multiscale coefficients $\mu_{\delta}$ and $\kappa_\delta$, respectively. Here, $0<\delta \ll 1$ is a small parameter that characterizes the roughness of the coefficient or respectively the speed of the variations, i.e.\ the smaller $\delta$, the faster the oscillations of $\mu_{\delta}$ and $\kappa_\delta$. If we discretize this model problem in the lowest order N{\'e}d{\'e}lec finite element space $\mathring{\CN}(\CT_H)$, we have the classical error estimate of the form \begin{align*} \inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu_{\delta} - \mathbf{v}_H \|_{\VH(\curl)} \le C H \left( \| \Vu_{\delta} \|_{H^1(\Omega)} + \| \curl \Vu_{\delta} \|_{H^1(\Omega)} \right). \end{align*} However, if the coefficients $\mu_{\delta}$ and $\kappa_\delta$ are discontinuous the necessary regularity for this estimate is not available, see \cite{Cost90regmaxwellremark, CDN99maxwellinterface, BGL13regularitymaxwell}. On the other hand, if $\mu_{\delta}$ and $\kappa_\delta$ are sufficiently regular but $\delta$ small, then we face the blow-up with $\| \Vu_{\delta} \|_{H^1(\Omega)} + \| \curl \Vu_{\delta} \|_{H^1(\Omega)}\rightarrow \infty$ for $\delta \rightarrow 0$, which makes the estimate useless in practice, unless the mesh size $H$ becomes very small to compensate for the blow-up. This does not change if we replace the $\VH(\curl)$-norm by the $L^2(\Omega)$-norm since both norms are equivalent in our setting. To understand if there exist any meaningful approximations of $\Vu_{\delta}$ in $\mathring{\CN}(\CT_H)$ (even on coarse meshes), we make a short excursus to classical homogenization theory. For that we assume that the coefficients $\mu_{\delta}(x)=\mu(x/\delta)$ and $\kappa_\delta(x)=\kappa(x/\delta)$ are periodically oscillating with period $\delta$. In this case it is known (cf.\ \cite{CFS17hmmmaxwell, HOV15maxwellHMM, Well2}) that the sequence of exact solutions $\Vu_{\delta}$ converges weakly in $\VH_0(\curl)$ to a \emph{homogenized} function $\Vu_{0}$. Since $\Vu_0 \in \VH_0(\curl)$ is $\delta$-independent and slow, it can be well approximated in $\mathring{\CN}(\CT_H)$. Furthermore, there exists a \emph{corrector} $\mathcal{K}_{\delta}(\Vu_0)$ such that \[\Vu_{\delta} \approx (\id + \mathcal{K}_{\delta})\Vu_0 \] is a good approximation in $\VH(\curl)$, i.e.\ the error converges strongly to zero with \[ \| \Vu_{\delta} -( \Vu_0 + \mathcal{K}_{\delta}(\Vu_0)) \|_{\VH(\curl)} \rightarrow 0 \qquad \mbox{for } \delta \rightarrow 0. \] Here the nature of the corrector is revealed by two estimates. In fact, $\mathcal{K}_{\delta}(\Vu_0)$ admits a decomposition into a gradient part and a part with small amplitude (cf. \cite{HOV15maxwellHMM, CH15homerrormaxwell, Well2}) such that \[ \mathcal{K}_{\delta}(\Vu_0) = \Vz_{\delta} + \nabla \theta_{\delta} \] with \begin{align} \label{hom-corrector-est-1} \delta^{-1} \| \Vz_{\delta} \|_{L^2(\Omega)} + \| \Vz_{\delta} \|_{\VH(\curl)} &\le C\| \Vu_0 \|_{\VH(\curl)}\\ \label{hom-corrector-est-2} \text{and}\qquad\delta^{-1} \| \theta_{\delta} \|_{L^2(\Omega)} + \| \nabla \theta_{\delta} \|_{L^2(\Omega)} &\le C \| \Vu_0 \|_{\VH(\curl)}, \end{align} where $C=C(\alpha,C_B)$ only depends on the constants appearing in Assumption \ref{asspt:sesquiform}. First, we immediately see that the estimates imply that $\mathcal{K}_{\delta}(\Vu_0)$ is $\VH(\curl)$-stable in the sense that it holds \begin{align*} \| \mathcal{K}_{\delta}(\Vu_0) \|_{\VH(\curl)} \le C \| \Vu_0 \|_{\VH(\curl)}. \end{align*} Second, and more interestingly, we see that alone from the above properties, we can conclude that $\Vu_0$ \emph{must} be a good approximation of the exact solution in the space $H^{-1}(\Omega,\mathbb{C}^3)$. In fact, using \eqref{hom-corrector-est-1} and \eqref{hom-corrector-est-2} we have for any $\mathbf{v}\in H^1_0(\Omega,\mathbb{C}^3)$ with $\| \mathbf{v} \|_{H^1(\Omega)}=1$ that \begin{align*} \left|\int_{\Omega} \mathcal{K}_{\delta}(\Vu_0) \cdot \mathbf{v} \right|= \left|\int_{\Omega} \Vz_{\delta} \cdot \mathbf{v} - \int_{\Omega} \theta_{\delta} \hspace{2pt} (\nabla \cdot \mathbf{v}) \right| \le \| \Vz_{\delta} \|_{L^2(\Omega)} + \| \theta_{\delta} \|_{L^2(\Omega)} \le C \delta \| \Vu_0 \|_{\VH(\curl)}. \end{align*} Consequently we have strong convergence in $H^{-1}(\Omega)$ with \begin{align*} \| \Vu_{\delta} - \Vu_0 \|_{H^{-1}(\Omega)} \le \| \Vu_{\delta} - ( \Vu_0 + \mathcal{K}_{\delta}(\Vu_0))\|_{H^{-1}(\Omega)} + \| \mathcal{K}_{\delta}(\Vu_0) \|_{H^{-1}(\Omega)} \overset{\delta \rightarrow 0}{\longrightarrow} 0. \end{align*} We conclude two things. Firstly, even though the coarse space $\mathring{\CN}(\CT_H)$ does not contain good $\VH(\curl)$- or $L^2$-approximations, it still contains meaningful approximations in $H^{-1}(\Omega)$. Secondly, the fact that the coarse part $\Vu_0$ is a good $H^{-1}$-approximation of $\Vu_{\delta}$ is an intrinsic conclusion from the properties of the correction $\mathcal{K}_{\delta}(\Vu_0)$. In this paper we are concerned with the question if the above considerations can be transferred to a discrete setting beyond the assumption of periodicity. More precisely, defining a coarse level of resolution through the space $\mathring{\CN}(\CT_H)$, we ask if it is possible to find a coarse function $\Vu_H$ and an (efficiently computable) $\VH(\curl)$-stable operator $\mathcal{K}$, such that \begin{align} \label{motivation:int-estimates} \| \Vu_{\delta} - \Vu_H \|_{H^{-1}(\Omega)} \le C H \qquad \mbox{and} \qquad \| \Vu_{\delta} - (I+\mathcal{K})\Vu_H \|_{\VH(\curl)} \le CH, \end{align} with $C$ being independent of the oscillations in terms of $\delta$. A natural ansatz for the coarse part is $\Vu_H=\pi_H( \Vu_{\delta} )$ for a suitable projection $\pi_H : \VH(\curl) \rightarrow \mathring{\CN}(\CT_H)$. However, from the considerations above we know that $\Vu_H=\pi_H( \Vu_{\delta} )$ can only be a good $H^{-1}$-approximation if the error fulfills a discrete analog to the estimates \eqref{hom-corrector-est-1} and \eqref{hom-corrector-est-2}. Since $\Vu_{\delta} - \pi_H( \Vu_{\delta} )$ is nothing but an interpolation error, we can immediately derive a sufficient condition for our choice of $\pi_H$: we need that, for any $\Vv\in \VH_0(\curl, \Omega)$, there are $\Vz\in \VH^1_0(\Omega)$ and $\theta\in H^1_0(\Omega)$ such that \[\Vv-\pi_H \Vv=\Vz+\nabla \theta\] and \begin{equation} \label{motivation:properties-pi-H} \begin{split} H^{-1}\|\Vz\|_{L^2(\Omega)}+\|\nabla \Vz\|_{\VH(\curl)} &\leq C \|\curl\Vv\|_{L^2(\Omega)},\\ H^{-1}\|\theta\|_{L^2(\Omega)}+\|\nabla \theta\|_{L^2(\Omega)}&\leq C \|\curl\Vv\|_{L^2(\Omega)}. \end{split} \end{equation} This is a sufficient condition for $\pi_H$. Note that the above properties are not fulfilled for e.g. the $L^2$-projection. This resembles the fact that the $L^2$-projection does typically not yield a good $H^{-1}$-approximation in our setting. We conclude this paragraph by summarizing that if we have a projection $\pi_H$ fulfilling \eqref{motivation:properties-pi-H}, then we can define a coarse scale numerically through the space $\mathring{\CN}(\CT_H) = \mbox{im}(\pi_H)$. On the other hand, to ensure that the corrector inherits the desired decomposition with estimates \eqref{motivation:int-estimates}, it needs to be constructed such that it maps into the kernel of the projection operator, i.e. $\mbox{im}(\mathcal{K})\subset\mbox{ker}(\pi_H)$. \section{Mesh and interpolation operator} \label{sec:intpol} In this section we introduce the basic notation for establishing our coarse scale discretization and we will present a projection operator that fulfills the sufficient conditions derived in the previous section. Let $\CT_H$ be a regular partition of $\Omega$ into tetrahedra, such that $\cup\CT_H=\overline{\Omega}$ and any two distinct $T, T'\in \CT_H$ are either disjoint or share a common vertex, edge or face. We assume the partition $\CT_H$ to be shape-regular and quasi-uniform. The global mesh size is defined as $H:=\max\{ \diam(T)|T\in \CT_{H}\}$. $\CT_H$ is a coarse mesh in the sense that it does not resolve the fine-scale oscillations of the parameters. Given any (possibly even not connected) subdomain $G\subset \overline{\Omega}$ define its neighborhood via \[\UN(G):=\Int(\cup\{T\in \CT_{H}|T\cap\overline{G}\neq \emptyset\})\] and for any $m\geq 2$ the patches \[\UN^1(G):=\UN(G)\qquad \text{and}\qquad\UN^m(G):=\UN(\UN^{m-1}(G)),\] see Figure \ref{fig:patch} for an example. The shape regularity implies that there is a uniform bound $C_{\ol, m}$ on the number of elements in the $m$-th order patch \[\max_{T\in \CT_{H}}\operatorname{card}\{K\in \CT_{H}|K\subset\overline{\UN^m(T)}\}\leq C_{\ol, m}\] and the quasi-uniformity implies that $C_{\ol, m}$ depends polynomially on $m$. We abbreviate $C_{\ol}:=C_{\ol, 1}$. \begin{figure} \caption{Triangle $T$ (in black) and its first and second order patches (additional elements for $\UN(T)$ in dark gray and additional elements for $\UN^2(T)$ in light gray).} \label{fig:patch} \end{figure} The space of $\CT_H$-piecewise affine and continuous functions is denoted by $\CS^1(\CT_H)$. We denote the lowest order N{\'e}d{\'e}lec finite element, cf.\ \cite[Section 5.5]{Monk}, by \[ \mathring{\CN}(\CT_H):=\{\Vv\in \VH_0(\curl)|\forall T\in \CT_H: \Vv|_T(\Vx)=\Va_T\times\Vx+\Vb_T \text{ with }\Va_T, \Vb_T\in\mathbb{C}^3\} \] and the space of Raviart--Thomas fields by \[ \mathring{\CR\CT}(\CT_H):=\{\Vv\in \VH_0(\Div)|\forall T\in \CT_H: \Vv|_T(\Vx)=\Va_T\cdot\Vx+\Vb_T \text{ with }\Va_T\in \mathbb{C}, \Vb_T\in\mathbb{C}^3\}. \] As motivated in Section \ref{sec:motivation} we require an $\VH(\curl)$-stable interpolation operator $\pi_H^E:\VH_0(\curl)\to \mathring{\CN}(\CT_H)$ that allows for a decomposition with the estimates such as \eqref{motivation:properties-pi-H}. However, from the view point of numerical homogenization where corrector problems should be localized to small subdomains, we also need that $\pi_H^E$ is local and (as we will see later) that it fits into a commuting diagram with other stable interpolation operators for lowest order $H^1(\Omega)$, $\VH(\Div)$ and $L^2(\Omega)$ elements. As discussed in the introduction, the only suitable candidate is the Falk-Winther interpolation operator $\pi_H^E$ \cite{FalkWinther2014}. We postpone a precise definition of $\pi_H^E$ to Section \ref{sec:intpolimpl} and just summarize its most important properties in the following proposition. \begin{proposition}\label{p:proj-pi-H-E} There exists a projection $\pi_H^E:\VH_0(\curl)\to \mathring{\CN}(\CT_H)$ with the following local stability properties: For all $\Vv\in \VH_0(\curl)$ and all $T\in \CT_H$ it holds that \begin{align} \label{eq:stabilityL2} \|\pi_H^E(\Vv)\|_{L^2(T)}&\leq C_\pi \bigl(\|\Vv\|_{L^2(\UN(T))}+H\|\curl\Vv\|_{L^2(\UN(T))}\bigr),\\* \label{eq:stabilitycurl} \|\curl\pi_H^E(\Vv)\|_{L^2(T)}&\leq C_\pi \|\curl\Vv\|_{L^2(\UN(T))}. \end{align} Furthermore, there exists a projection $\pi_H^F:\VH_0(\Div)\to \mathring{\mathcal{RT}}(\CT_H)$ to the Raviart-Thomas space such that the following commutation property holds \[\curl\pi_H^E(\Vv)=\pi_H^F(\curl \Vv).\] \end{proposition} \begin{proof} See \cite{FalkWinther2014} for a proof, which can be adapted to the present case of homogeneous boundary values. \end{proof} As explained in the motivation in Section \ref{sec:motivation}, we also require that $\pi_H^E$ allows for a regular decomposition in the sense of \eqref{motivation:properties-pi-H}. In general, regular decompositions are an important tool for the study of $\VH(\curl)$-elliptic problems and involve that a vector field $\Vv\in \VH_0(\curl)$ is split -- in a non-unique way -- into a gradient and a (regular) remainder in $\VH^1$, see \cite{Hipt02FEem, PZ02Schwarz}. In contrast to the Helmholtz decomposition, this splitting is not orthogonal with respect to the $L^2$-inner product. If the function $\Vv\in \VH_0(\curl)$ is additionally known to be in the kernel of a suitable quasi-interpolation, a modified decomposition can be derived that is localized and $H$-weighted. In particular, the weighting with $H$ allows for estimates similar as the one stated in \eqref{motivation:properties-pi-H}. The first proof of such a modified decomposition was given by Sch\"oberl \cite{Sch08aposteriori}. In the following we shall use his results and the locality of the Falk-Winther operator to recover a similar decomposition for the projection $\pi_H^E$. More precisely, we have the following lemma which is crucial for our analysis. \begin{lemma} \label{lem:localregulardecomp} Let $\pi_H^E$ denote the projection from Proposition \ref{p:proj-pi-H-E}. For any $\Vv\in \VH_0(\curl, \Omega)$, there are $\Vz\in \VH^1_0(\Omega)$ and $\theta\in H^1_0(\Omega)$ such that \[\Vv-\pi_H^E(\Vv)=\Vz+\nabla \theta\] with the local bounds for every $T\in \CT_H$ \begin{equation} \label{eq:regulardecomp} \begin{split} H^{-1}\|\Vz\|_{L^2(T)}+\|\nabla \Vz\|_{L^2(T)}&\leq C_z\|\curl\Vv\|_{L^2(\UN^3(T))},\\ H^{-1}\|\theta\|_{L^2(T)}+\|\nabla \theta\|_{L^2(T)}&\leq C_\theta\bigl(\|\Vv\|_{L^2(\UN^3(T))}+H\|\curl\Vv\|_{L^2(\UN^3(T))}\bigr), \end{split} \end{equation} where $\nabla \Vz$ stands for the Jacobi matrix of $\Vz$. Here $C_z$ and $C_\theta$ are generic constants that only depend on the regularity of the coarse mesh. \end{lemma} Observe that \eqref{eq:regulardecomp} implies the earlier formulated sufficient condition \eqref{motivation:properties-pi-H}. \begin{proof} Let $\Vv\in \VH_0(\curl, \Omega)$. Denote by $I_H^S:\VH_0(\curl,\Omega)\to \mathring{\CN}(\CT_H)$ the quasi-interpolation operator introduced by Sch\"oberl in \cite{Sch08aposteriori}. It is shown in \cite[Theorem 6]{Sch08aposteriori} that there exists a decomposition \begin{equation} \label{eq:schoeberlstab-p1} \Vv-I_H^S(\Vv) = \sum_{\substack{P \text{ vertex}\\ \text{of }\CT_H}} \Vv_P \end{equation} where, for any vertex $P$, $\Vv_P\in \VH_0(\curl, \Omega_P)$ and $\Omega_P$ the support of the local hat function associated with $P$. Moreover, \cite[Theorem 6]{Sch08aposteriori} provides the stability estimates \begin{equation}\label{eq:schoeberlstab} \| \Vv_P \|_{L^2(\Omega_P)} \lesssim \|\Vv\|_{L^2(\UN(\Omega_P))} \quad\text{and}\quad \|\curl \Vv_P \|_{L^2(\Omega_P)} \lesssim \|\curl \Vv\|_{L^2(\UN(\Omega_P))} \end{equation} for any vertex $P$. With these results we deduce, since $\pi_H^E$ is a projection onto the finite element space, that \begin{align*} \Vv-\pi_H^E(\Vv) =\Vv-I_H^S(\Vv)-\pi_H^E(\Vv-I_H^S\Vv) =\sum_{\substack{P \text{ vertex}\\ \text{of }\CT_H}}(\id-\pi_H^E)(\Vv_P). \end{align*} Due to the locality of $\pi_H^E$, we have $(\id-\pi_H^E)(\Vv_P)\in \VH_0(\curl, \UN(\Omega_P))$. The local stability of $\pi_H^E$, \eqref{eq:stabilityL2} and \eqref{eq:stabilitycurl}, and the stability \eqref{eq:schoeberlstab} imply \begin{align*} \|(\id-\pi_H^E)(\Vv_P)\|_{L^2(\UN(\Omega_P))}&\lesssim \|\Vv\|_{L^2(\UN(\Omega_P))}+H\|\curl\Vv\|_{L^2(\UN(\Omega_P))},\\* \|\curl(\id-\pi_H^E)(\Vv_P)\|_{L^2(\UN(\Omega_P))}&\lesssim \|\curl\Vv\|_{L^2(\UN(\Omega_P))}, \end{align*} We can now apply the regular splitting to $\Vv_P$ (cf.\ \cite{PZ02Schwarz}), i.e.\ there are $\Vz_P\in \VH^1_0(\UN(\Omega_P))$, $\theta_P\in H^1_0(\UN(\Omega_P))$ such that $\Vv_P=\Vz_P+\nabla \theta_P$ and with the estimates \begin{align*} H^{-1}\|\Vz_P\|_{L^2(\UN(\Omega_P))}+\|\nabla \Vz_P\|_{L^2(\UN(\Omega_P))}&\lesssim \|\curl((\id-\pi_H^E)(\Vv_P))\|_{L^2(\UN(\Omega_P))},\\* H^{-1}\|\theta_P\|_{L^2(\UN(\Omega_P))}+\|\nabla \theta_P\|_{L^2(\UN(\Omega_P))}&\lesssim \|(\id-\pi_H^E)(\Vv_P)\|_{L^2(\UN(\Omega_P))}. \end{align*} Set $\Vz=\sum_P\Vz_P$ and $\theta=\sum_P\theta_P$, which is a regular decomposition of $\Vv-\pi_H^E(\Vv)$. The local estimates follows from the foregoing estimates for $\Vv_P$ and the decomposition \eqref{eq:schoeberlstab-p1} which yields \begin{align*} H^{-1}\|\Vz\|_{L^2(T)}+\|\nabla \Vz\|_{L^2(T)}&\leq \sum_{\substack{P \text{ vertex}\\ \text{of } T}} \left( H^{-1}\| \Vz_P \|_{L^2(\Omega_P)}+\|\nabla \Vz_P \|_{L^2(\Omega_P)} \right)\\ &\lesssim \sum_{\substack{P \text{ vertex}\\ \text{of } T}} \|\curl (\id-\pi_H^E)(\Vv_P)\|_{L^2(\UN(\Omega_P))} \lesssim \|\curl\Vv\|_{L^2(\UN^3(T))}. \end{align*} The local estimate for $\theta$ follows analogously. \end{proof} \section{The Corrector Green's Operator} \label{sec:LODideal} In this section we introduce an ideal \emph{Corrector Green's Operator} that allows us to derive a decomposition of the exact solution into a coarse part (which is a good approximation in $H^{-1}(\Omega,\mathbb{C}^3)$) and two different corrector contributions. For simplicity, we let from now on $\mathcal{L} : \VH_0(\curl) \rightarrow \VH_0(\curl)^{\prime}$ denote the differential operator associated with the sesquilinear form $\CB(\cdot,\cdot)$, i.e. $\mathcal{L}(v)(w)=\CB(v,w)$. Using the Falk-Winter interpolation operator $\pi_H^E$ for the N{\'e}d{\'e}lec elements, we split the space $\VH_0(\curl)$ into the finite, low-dimensional coarse space $\mathring{\CN}(\CT_H)=\mbox{im}(\pi_H^E)$ and a corrector space given as the kernel of $\pi_H^E$, i.e.\ we set $\VW:=\ker (\pi_H^E)\subset \VH_0(\curl)$. This yields the direct sum splitting $\VH_0(\curl)=\mathring{\CN}(\CT_H)\oplus\VW$. Note that $\VW$ is closed since it is the kernel of a continuous (i.e. $\VH(\curl)$-stable) operator. With this the ideal Corrector Green's Operator is defined as follows. \begin{definition}[Corrector Green's Operator] For $\mathbf{F} \in \VH_0(\curl)^\prime$, we define the Corrector Green's Operator \begin{align} \label{cor-greens-op} \mathcal{G}: \VH_0(\curl)^{\prime} \rightarrow \VW \hspace{40pt} \mbox{by} \hspace{40pt} \CB(\mathcal{G}(\mathbf{F}) , \Vw )=\mathbf{F}(\Vw)\qquad \mbox{for all } \Vw\in \VW. \end{align} It is well-defined by the Lax-Milgram-Babu{\v{s}}ka theorem, which is applicable since $\CB(\cdot,\cdot)$ is $\VH_0(\curl)$-elliptic and since $\VW$ is a closed subspace of $\VH_0(\curl)$. \end{definition} Using the Corrector Green's Operator we obtain the following decomposition of the exact solution. \begin{lemma}[Ideal decomposition] \label{lemma:ideal-decompos} The exact solution $\Vu\in\VH_0(\curl)$ to \eqref{eq:problem} and $\Vu_H:=\pi_H^E(\Vu)$ admit the decomposition \[ \Vu = \Vu_H - (\mathcal{G} \circ \mathcal{L})(\Vu_H) + \mathcal{G}(\Vf). \] \end{lemma} \begin{proof} Since $\VH_0(\curl)=\mathring{\CN}(\CT_H)\oplus\VW$, we can write $\Vu$ uniquely as \[ \Vu = \pi_H^E(\Vu) + (\id - \pi_H^E)(\Vu) = \Vu_H + (\id - \pi_H^E)(\Vu), \] where $(\id - \pi_H^E)(\Vu) \in \VW$ by the projection property of $\pi_H^E$. Using the differential equation for test functions $\Vw\in \VW$ yields that \begin{align*} \CB( (\id - \pi_H^E)(\Vu) , \Vw )= - \CB( \Vu_H , \Vw ) + (\Vf, \Vw)_{L^2(\Omega)} = - \CB( (\mathcal{G} \circ \mathcal{L})(\Vu_H) , \Vw ) + \CB( \mathcal{G}(\Vf) , \Vw ). \end{align*} Since this holds for all $\Vw\in \VW$ and since $\mathcal{G}(\Vf) - (\mathcal{G} \circ \mathcal{L})(\Vu_H) \in \VW$, we conclude that \[ (\id - \pi_H^E)(\Vu) = \mathcal{G}(\Vf) - (\mathcal{G} \circ \mathcal{L})(\Vu_H), \] which finishes the proof. \end{proof} The Corrector Green's Operator has the following approximation and stability properties, which reveal that its contributions is always negligible in the $\VH(\Div)^\prime$-norm and negligible in the $\VH(\curl)$-norm if applied to a function in $\VH(\Div)$. \begin{lemma}[Ideal corrector estimates] \label{lemma:corrector-props} Any $\mathbf{F} \in \VH_0(\curl)^{\prime}$ satisfies \begin{align} \label{green-est-Hcurl-1} H \| \mathcal{G}(\mathbf{F}) \|_{\VH(\curl)} + \| \mathcal{G}(\mathbf{F}) \|_{\VH(\Div)^{\prime}} \le C H \alpha^{-1} \| \mathbf{F} \|_{\VH_0(\curl)^{\prime}}. \end{align} If $\mathbf{F} = \mathbf{f} \in \VH(\Div)$ we even have \begin{align} \label{green-est-Hdiv-1} H \| \mathcal{G}(\mathbf{f}) \|_{\VH(\curl)} + \| \mathcal{G}(\mathbf{f}) \|_{\VH(\Div)^{\prime}} \le C H^2 \alpha^{-1} \| \mathbf{f} \|_{\VH(\Div)}. \end{align} Here, the constant $C$ does only depend on the maximum number of neighbors of a coarse element and the generic constants appearing in Lemma \ref{lem:localregulardecomp}. \end{lemma} Note that this result is still valid if we replace the $\VH(\Div)^{\prime}$-norm by the $H^{-1}(\Omega,\mathbb{C}^3)$-norm. \begin{proof} The stability estimate $\| \mathcal{G}(\mathbf{F}) \|_{\VH(\curl)} \le \alpha^{-1} \| \mathbf{F} \|_{\VH_0(\curl)^{\prime}}$ is obvious. Next, with $\mathcal{G}(\mathbf{F})\in\VW$ and Lemma \ref{lem:localregulardecomp} we have \begin{equation}\label{green-est-Hdiv-1-proof} \begin{aligned} \| \mathcal{G}(\mathbf{F}) \|_{\VH(\Div)^{\prime}} &= \underset{\| \mathbf{v} \|_{\VH(\Div)}=1}{\sup_{\mathbf{v}\in \VH(\Div)}} \left|\int_{\Omega} \Vz \cdot \mathbf{v} - \int_{\Omega} \theta (\nabla \cdot \mathbf{v}) \right| \\ & \le ( \| \Vz \|_{L^2(\Omega)}^2 + \| \theta \|_{L^2(\Omega)}^2 )^{1/2} \le C H \| \mathcal{G}(\mathbf{F}) \|_{\VH(\curl)} \le C H \alpha^{-1} \| \mathbf{F} \|_{\VH_0(\curl)^{\prime}}, \end{aligned} \end{equation} which proves \eqref{green-est-Hcurl-1}. Note that this estimate exploited $\theta \in H^{1}_0(\Omega)$, which is why we do not require the function $\mathbf{v}$ to have a vanishing normal trace. Let us now consider the case that $\mathbf{F} = \mathbf{f} \in \VH(\Div)$. We have by \eqref{green-est-Hdiv-1-proof} that \begin{align*} \alpha \| \mathcal{G}( \mathbf{f} ) \|_{\VH(\curl)}^2 \le \| \mathcal{G}( \mathbf{f}) \|_{\VH(\Div)^{\prime}} \| \mathbf{f} \|_{\VH(\Div)} \le C H \| \mathcal{G}(\mathbf{f}) \|_{\VH(\curl)} \| \mathbf{f} \|_{\VH(\Div)}. \end{align*} We conclude $\| \mathcal{G}( \mathbf{f} ) \|_{\VH(\curl)} \le C H \alpha^{-1} \| \mathbf{f} \|_{\VH(\Div)}$. Finally, we can use this estimate again in \eqref{green-est-Hdiv-1-proof} to obtain \begin{align*} \| \mathcal{G}(\Vf) \|_{\VH(\Div)^{\prime}} \le C H \| \mathcal{G}(\Vf) \|_{\VH(\curl)} \le C H^2 \alpha^{-1} \| \mathbf{f} \|_{\VH(\Div)}. \end{align*} This finishes the proof. \end{proof} An immediate conclusion of Lemmas \ref{lemma:ideal-decompos} and \ref{lemma:corrector-props} is the following. \begin{conclusion} \label{conclusion-ideal-corr-est} Let $\Vu$ denote the exact solution to \eqref{eq:curlcurl} for $ \mathbf{f} \in \VH(\Div)$. Then with the coarse part $\Vu_H:=\pi_H^E(\Vu)$ and corrector operator $\mathcal{K} := - \mathcal{G} \circ \mathcal{L}$ it holds \begin{align*} H^{-1}\| \Vu - (\id + \mathcal{K})\Vu_H \|_{\VH(\Div)^{\prime}} + \| \Vu - (\id + \mathcal{K})\Vu_H \|_{\VH(\curl)} + \| \Vu - \Vu_H \|_{\VH(\Div)^{\prime}} \le C H \| \mathbf{f} \|_{\VH(\Div)} . \end{align*} Here, $C$ only depends on $\alpha$, the mesh regularity and on the constants appearing in Lemma \ref{lem:localregulardecomp}. \end{conclusion} \begin{proof} The estimates for $\Vu - (\id + \mathcal{K})\Vu_H =\mathcal{G}(\Vf)$ directly follow from \eqref{green-est-Hdiv-1}. For the estimate of $\Vu - \Vu_H =\mathcal{K}\Vu_H + \mathcal{G} \Vf$, observe that \eqref{green-est-Hcurl-1} and Proposition~\ref{p:proj-pi-H-E} imply \begin{equation*} \| \mathcal{K}\Vu_H \|_{\VH(\Div)^{\prime}} \lesssim H \| \CL\Vu_H \|_{\VH_0(\curl)^{\prime}} \lesssim H \| \Vu_H \|_{\VH(\curl)} = H \| \pi_H^E \Vu \|_{\VH(\curl)} \lesssim H \| \Vu \|_{\VH(\curl)} . \end{equation*} Thus, the proof follows from the stability of the problem and the the triangle inequality. \end{proof} It only remains to derive an equation that characterizes $(\id + \mathcal{K})\Vu_H$ as the unique solution of a variational problem. This is done in the following theorem. \begin{theorem} We consider the setting of Conclusion \ref{conclusion-ideal-corr-est}. Then $\Vu_H=\pi_H^E(\Vu) \in \mathring{\CN}(\CT_H)$ is characterized as the unique solution to \begin{align} \label{ideal-lod} \CB( \hspace{2pt} (\id + \mathcal{K})\Vu_H , (\id + \mathcal{K}^{\ast})\Vv_H \hspace{1pt} ) = ( \Vf, (\id + \mathcal{K}^{\ast})\Vv_H )_{L^2(\Omega)} \qquad \mbox{for all } \Vv_H \in \mathring{\CN}(\CT_H). \end{align} Here, $\mathcal{K}^{\ast}$ is the adjoint operator to $\mathcal{K}$. The sesquilinear form $\CB( \hspace{1pt} (\id + \mathcal{K})\hspace{3pt}\cdot \hspace{2pt}, (\id + \mathcal{K}^{\ast})\hspace{2pt}\cdot \hspace{2pt} )$ is $\VH(\curl)$-elliptic on $\mathring{\CN}(\CT_H)$. \end{theorem} Observe that we have the simplification $\mathcal{K}^{\ast}=\mathcal{K}$ if the differential operator $\mathcal{L}$ is self-adjoint as it is typically the case for $\VH(\curl)$-problems. \begin{proof} Since Lemma \ref{lemma:ideal-decompos} guarantees $\Vu = \Vu_H - (\mathcal{G} \circ \mathcal{L})(\Vu_H) + \mathcal{G}(\Vf)$, the weak formulation \eqref{eq:problem} yields \begin{align*} \CB( \Vu_H - (\mathcal{G} \circ \mathcal{L})(\Vu_H) + \mathcal{G}(\Vf) , \Vv_H ) = ( \Vf, \Vv_H )_{L^2(\Omega)} \qquad \mbox{for all } \Vv_H \in \mathring{\CN}(\CT_H). \end{align*} We observe that by definition of $\mathcal{G}$ we have \begin{align*} \CB( \mathcal{G}(\Vf) , \Vv_H ) = ( \Vf , (\mathcal{G} \circ \mathcal{L})^{\ast}\Vv_H )_{L^2(\Omega)} \end{align*} and \begin{align*} \CB( \Vu_H - (\mathcal{G} \circ \mathcal{L})(\Vu_H) , (\mathcal{G} \circ \mathcal{L})^{\ast}\Vv_H ) = 0. \end{align*} Combining the three equations shows that $(\id + \mathcal{K})\Vu_H$ is a solution to \eqref{ideal-lod}. The uniqueness follows from the following norm equivalence \begin{align*} \| \Vu_H \|_{\VH(\curl)} = \| \pi_H^E((\id + \mathcal{K})\Vu_H) \|_{\VH(\curl)} \le C \| (\id + \mathcal{K})\Vu_H \|_{\VH(\curl)} \le C \| \Vu_H \|_{\VH(\curl)}. \end{align*} This is also the reason why the $\VH(\curl)$-ellipticity of $\CB( \cdot, \cdot)$ implies the $\VH(\curl)$-ellipticity of $\CB( \hspace{1pt} (\id + \mathcal{K})\hspace{3pt}\cdot \hspace{2pt}, (\id + \mathcal{K}^{\ast})\hspace{2pt}\cdot \hspace{2pt} )$ on $\mathring{\CN}(\CT_H)$. \end{proof} \textbf{Numerical homogenization}. Let us summarize the most important findings and relate them to (numerical) homogenization. We defined a \emph{homogenization scale} through the coarse FE space $\mathring{\CN}(\CT_H)$. We proved that there exists a numerically homogenized function $\Vu_H \in \mathring{\CN}(\CT_H)$ which approximates the exact solution well in $\VH(\Div)^{\prime}$ with \begin{align*} \| \Vu - \Vu_H \|_{\VH(\Div)^{\prime}} \le C H \| \mathbf{f} \|_{\VH(\Div)}. \end{align*} From the periodic homogenization theory (cf. Section \ref{sec:motivation}) we know that this is the best we can expect and that $\Vu_H$ is typically not a good $L^2$-approximation due to the large kernel of the curl-operator. Furthermore, we showed the existence of an $\VH(\curl)$-stable corrector operator $\mathcal{K}: \mathring{\CN}(\CT_H) \rightarrow \VW$ that corrects the homogenized solution in such a way that the approximation is also accurate in $\VH(\curl)$ with \begin{align*} \| \Vu - (\id + \mathcal{K})\Vu_H \|_{\VH(\curl)} \le C H \| \mathbf{f} \|_{\VH(\Div)}. \end{align*} Since $\mathcal{K} = - \mathcal{G} \circ \mathcal{L}$, we know that we can characterize $\mathcal{K} (\Vv_H) \in \VW$ as the unique solution to the (ideal) corrector problem \begin{align} \label{ideal-corrector-problem} \CB( \mathcal{K} (\Vv_H) , \Vw )=- \CB( \Vv_H , \Vw ) \qquad \mbox{for all } \Vw\in \VW. \end{align} The above result shows that $(\id + \mathcal{K})\Vu_H$ approximates the analytical solution with linear rate without any assumptions on the regularity of the problem or the structure of the coefficients that define $\CB(\cdot,\cdot)$. Also it does not assume that the mesh resolves the possible fine-scale features of the coefficient. On the other hand, the ideal corrector problem \eqref{ideal-corrector-problem} is global, which significantly limits its practical usability in terms of real computations. However, as we will see next, the corrector Green's function associated with problem \eqref{cor-greens-op} shows an exponential decay measured in units of $H$. This property will allow us to split the global corrector problem \eqref{ideal-corrector-problem} into several smaller problems on subdomains, similar to how we encounter it in classical homogenization theory. We show the exponential decay of the corrector Green's function indirectly through the properties of its corresponding Green's operator $\mathcal{G}$. The localization is established in Section \ref{sec:LOD}, whereas we prove the decay in Section \ref{sec:decaycorrectors}. \section{Quasi-local numerical homogenization} \label{sec:LOD} In this section we describe how the ideal corrector $\mathcal{K}$ can be approximated by a sum of local correctors, without destroying the overall approximation order. This is of central importance for an efficient computability. Furthermore, it also reveals that the new corrector is a quasi-local operator, which is in line with homogenization theory. We start with quantifying the decay properties of the Corrector Green's Operator in Section \ref{subsec:idealapprox}. In Section \ref{subsec:LODlocal} we apply the result to our numerical homogenization setting and state the error estimates for the \quotes{localized} corrector operator. We close with a few remarks on a fully discrete realization of the localized corrector operator in Section \ref{subsec:discreteLOD}. \subsection{Exponential decay and localized corrector} \label{subsec:idealapprox} The property that $\mathcal{K}$ can be approximated by local correctors is directly linked to the decay properties of the Green's function associated with problem \eqref{cor-greens-op}. These decay properties can be quantified explicitly by measuring distances between points in units of the coarse mesh size $H$. We have the following result, which states -- loosely speaking -- in which distance from the support of a source term $\mathbf{F}$, becomes the $\VH(\curl)$-norm of $\mathcal{G}(\mathbf{F})$ negligibly small. For that, recall the definition of the element patches from the beginning of Section \ref{sec:intpol}, where $\UN^m(T)$ denotes the patch that consists of a coarse element $T \in \CT_H$ and $m$ layers of coarse elements around it. A proof of the following proposition is given in Section \ref{sec:decaycorrectors}. \begin{proposition} \label{prop:decaycorrector1} Let $T\in \CT_H$ denote a coarse element and $m\in \mathbb{N}$ a number of layers. Furthermore, let $\mathbf{F}_T \in \VH_0(\curl)^{\prime}$ denote a local source functional in the sense that $\mathbf{F}_T(\Vv)=0$ for all $\Vv \in \VH_0(\curl)$ with $\supp(\Vv) \subset \Omega \setminus T$. Then there exists $0<\tilde{\beta}<1$, independent of $H$, $T$, $m$ and $\mathbf{F}_T$, such that \begin{equation} \label{eq:decaycorrector1} \| \mathcal{G}(\mathbf{F}_T) \|_{\VH(\curl, \Omega\setminus \UN^m(T))}\lesssim \tilde{\beta}^m\| \mathbf{F}_T \|_{\VH_0(\curl)^{\prime}}. \end{equation} \end{proposition} In order to use this result to approximate $\mathcal{K}(\Vv_H) = - (\mathcal{G} \circ \mathcal{L})\Vv_H$ (which has a nonlocal argument), we introduce, for any $T\in\CT_H$, localized differential operators $\CL_T:\VH(\curl,T)\to\VH(\curl,\Omega)'$ with \[\langle \mathcal{L}_T(\Vu), \Vv \rangle := \CB_T(\Vu, \Vv ),\] where $\CB_T(\cdot, \cdot )$ denotes the restriction of $\CB(\cdot, \cdot )$ to the element $T$. By linearity of $\mathcal{G}$ we have that \[\mathcal{G} \circ \mathcal{L} = \sum_{T \in \CT_H} \mathcal{G} \circ \mathcal{L}_T\] and consequently we can write \[ \mathcal{K}( \Vv_H ) = \sum_{T \in \CT_H} \mathcal{G}( \mathbf{F}_T ), \qquad \mbox{with } \mathbf{F}_T:= - \mathcal{L}_T(\Vv_H). \] Obviously, $\mathcal{G}( \mathbf{F}_T )$ fits into the setting of Proposition \ref{prop:decaycorrector1}. This suggests to truncate the individual computations of $\mathcal{G}( \mathbf{F}_T )$ to a small patch $\UN^m(T)$ and then collect the results to construct a global approximation for the corrector. Typically, $m$ is referred to as \emph{oversampling parameter}. The strategy is detailed in the following definition. \begin{definition}[Localized Corrector Approximation] \label{de:loc-correctors} For an element $T\in \CT_H$ we define the element patch $\Omega_T:=\UN^m(T)$ of order $m\in \mathbb{N}$. Let $\mathbf{F} \in \VH_0(\curl)^{\prime}$ be the sum of local functionals with $\mathbf{F} =\sum_{T\in \CT_H} \mathbf{F}_T$, where $\mathbf{F}_T \in \VH_0(\curl)^{\prime}$ is as in Proposition \ref{prop:decaycorrector1}. Furthermore, let $\VW(\Omega_T)\subset \VW$ denote the space of functions from $\VW$ that vanish outside $\Omega_T$, i.e. \[\VW(\Omega_T)=\{\Vw\in\VW|\Vw=0 \text{ \textrm{outside} }\Omega_T\}.\] We call $\mathcal{G}_{T,m}( \mathbf{F}_T ) \in \VW(\Omega_T)$ the \emph{localized corrector} if it solves \begin{equation} \label{eq:correctorlocal} \CB( \mathcal{G}_{T,m}( \mathbf{F}_T ) , \Vw )=\mathbf{F}_T(\Vw)\qquad \mbox{for all } \Vw\in \VW(\Omega_T). \end{equation} With this, the global corrector approximation is given by \begin{align*} \mathcal{G}_{m}(\mathbf{F}) := \sum_{T\in \CT_H} \mathcal{G}_{T,m}( \mathbf{F}_T ). \end{align*} \end{definition} Observe that problem \eqref{eq:correctorlocal} is only formulated on the patch $\Omega_T$ and that it admits a unique solution by the Lax-Milgram-Babu{\v{s}}ka theorem. Based on decay properties stated in Proposition \ref{prop:decaycorrector1}, we can derive the following error estimate for the difference between the exact corrector $\mathcal{G}(\mathbf{F})$ and its approximation $\mathcal{G}_{m}(\mathbf{F})$ obtained by an $m$th level truncation. The proof of the following result is again postponed to Section \ref{sec:decaycorrectors}. \begin{theorem} \label{thm:errorcorrectors} We consider the setting of Definition \ref{de:loc-correctors} with ideal Green's Corrector $\mathcal{G}(\mathbf{F})$ and its $m$th level truncated approximation $\mathcal{G}_{m}(\mathbf{F})$. Then there exist constants $C_{d}>0$ and $0<\beta<1$ (both independent of $H$ and $m$) such that \begin{align} \label{eq:errorcorrector} \| \mathcal{G}(\mathbf{F}) - \mathcal{G}_{m}(\mathbf{F}) \|_{\VH(\curl)}&\leq C_{d} \sqrt{C_{\ol,m}}\,\beta^m \left( \sum_{T\in \CT_H} \| \mathbf{F}_T \|_{\VH_0(\curl)^{\prime}}^2 \right)^{1/2} \end{align} and \begin{align} \label{eq:errorcorrector-2} \| \mathcal{G}(\mathbf{F}) - \mathcal{G}_{m}(\mathbf{F}) \|_{\VH(\Div)^{\prime}}&\leq C_{d} \sqrt{C_{\ol,m}}\, \beta^m H \left( \sum_{T\in \CT_H} \| \mathbf{F}_T \|_{\VH_0(\curl)^{\prime}}^2 \right)^{1/2}. \end{align} \end{theorem} As a direct conclusion from Theorem \ref{thm:errorcorrectors} we obtain the main result of this paper that we present in the next subsection. \subsection{The quasi-local corrector and homogenization} \label{subsec:LODlocal} Following the above motivation we split the ideal corrector $\mathcal{K}(\Vv_H) =- (\mathcal{G} \circ \mathcal{L})\Vv_H$ into a sum of quasi-local contributions of the form $\sum_{T \in \CT_H} (\mathcal{G} \circ \mathcal{L}_T)\Vv_H$. Applying Theorem \ref{thm:errorcorrectors}, we obtain the following result. \begin{conclusion} \label{conclusion-main-result} Let $\mathcal{K}_m := - \sum_{T \in \CT_H} (\mathcal{G}_{T,m} \circ \mathcal{L}_T): \mathring{\CN}(\CT_H) \rightarrow \VW$ denote the localized corrector operator obtained by truncation of $m$th order. Then it holds \begin{align} \label{conclusion-main-result-est} \inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu - (\id + \mathcal{K}_m)\mathbf{v}_H \|_{\VH(\curl)} \le C \left( H + \sqrt{C_{\ol,m}} \beta^m \right) \| \Vf \|_{\VH(\Div)}. \end{align} \end{conclusion} Note that even though the ideal corrector $\mathcal{K}$ is a non-local operator, we can approximate it by a quasi-local corrector $\mathcal{K}_m$. Here, the quasi-locality is seen by the fact that, if $\mathcal{K}$ is applied to a function $\Vv_H$ with local support, the image $\mathcal{K}(\Vv_H)$ will typically still have a global support in $\Omega$. On the other hand, if $\mathcal{K}_m$ is applied to a locally supported function, the support will only increase by a layer with thickness of order $mH$. \begin{proof}[Proof of Conclusion \ref{conclusion-main-result}] With $\mathcal{K}_m = - \sum_{T \in \CT_H} (\mathcal{G}_{T,m} \circ \mathcal{L}_T)$ we apply Conclusion~\ref{conclusion-ideal-corr-est} and Theorem \ref{thm:errorcorrectors} to obtain \begin{equation*} \begin{aligned} & \inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu - (\id + \mathcal{K}_m)\mathbf{v}_H \|_{\VH(\curl)} \le \| \Vu - (\id + \mathcal{K})\mathbf{u}_H \|_{\VH(\curl)} + \|(\mathcal{K} - \mathcal{K}_m)\mathbf{u}_H \|_{\VH(\curl)}\\ & \qquad\qquad\qquad\qquad \le C H \| \Vf \|_{\VH(\Div)} + C \sqrt{C_{\ol,m}}\, \beta^m \left( \sum_{T\in \CT_H} \| \mathcal{L}_T(\mathbf{u}_H) \|_{\VH_0(\curl)^{\prime}}^2 \right)^{1/2}, \end{aligned} \end{equation*} where we observe with $\| \mathcal{L}_T(\mathbf{v}_H) \|_{\VH_0(\curl)^{\prime}} \le C \| \mathbf{v}_H \|_{\VH(\curl,T)}$ that \begin{align*} \sum_{T\in \CT_H} \| \mathcal{L}_T(\mathbf{u}_H) \|_{\VH_0(\curl)^{\prime}}^2 \le C \| \mathbf{u}_H \|_{\VH(\curl)}^2 = C \| \pi_H^E(\Vu) \|_{\VH(\curl)}^2 \le C \| \Vu \|_{\VH(\curl)}^2 \le C \| \Vf \|_{\VH(\Div)}^2. \end{align*} \end{proof} Conclusion \ref{conclusion-main-result} has immediate implications from the computational point of view. First, we observe that $\mathcal{K}_m$ can be computed by solving local decoupled problems. Considering a basis $\{ \boldsymbol{\Phi}_k | \hspace{3pt} 1 \le k \le N \}$ of $\mathring{\CN}(\CT_H)$, we require to determine $\mathcal{K}_m(\boldsymbol{\Phi}_k)$. For that, we consider all $T \in \CT_H$ with $T \subset \supp(\boldsymbol{\Phi}_k)$ and solve for $\mathcal{K}_{T,m}(\boldsymbol{\Phi}_k) \in \VW(\hspace{1pt}\UN^m(T)\hspace{1pt})$ with \begin{align} \label{loc-corrector-problems} \CB_{\UN^m(T)}( \mathcal{K}_{T,m}(\boldsymbol{\Phi}_k), \Vw ) = - \CB_{T}( \boldsymbol{\Phi}_k , \Vw ) \qquad \mbox{for all } \Vw \in \VW(\hspace{1pt}\UN^m(T)\hspace{1pt}). \end{align} The global corrector approximation is now given by \[ \mathcal{K}_m(\boldsymbol{\Phi}_k) = \underset{ T \subset \supp(\boldsymbol{\Phi}_k) }{\sum_{ T \in \CT_H }} \mathcal{K}_{T,m}(\boldsymbol{\Phi}_k). \] Next, we observe that selecting the localization parameter $m$ such that \[ m\gtrsim \lvert \log H\rvert \big/ \lvert \log \beta\rvert, \] we have with Conclusion \ref{conclusion-main-result} that \begin{align} \label{curl-est-m-logH}\inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu - (\id + \mathcal{K}_m)\mathbf{v}_H \|_{\VH(\curl)} \le C H \| \Vf \|_{\VH(\Div)}, \end{align} which is of the same order as for the ideal corrector $\mathcal{K}$. Consequently, we can consider the Galerkin finite element method, where we seek $\Vu_{H,m} \in \mathring{\CN}(\CT_H)$ such that \begin{align*} \CB( (\id + \mathcal{K}_m)\Vu_{H,m} , (\id + \mathcal{K}_m)\mathbf{v}_H ) = (\mathbf{f} , (\id + \mathcal{K}_m)\mathbf{v}_H )_{L^2(\Omega)} \qquad \mbox{for all } \Vv_{H,m} \in \mathring{\CN}(\CT_H). \end{align*} Since a Galerkin method yields the $\VH(\curl)$-quasi-best approximation of $\Vu$ in the space \linebreak[4]$(\id + \mathcal{K}_m)\mathring{\CN}(\CT_H)$ we have with \eqref{curl-est-m-logH} that \begin{align*} \| \Vu - (\id + \mathcal{K}_m)\Vu_{H,m} \|_{\VH(\curl)} \le C H \| \Vf \|_{\VH(\Div)} \end{align*} and we have with \eqref{green-est-Hcurl-1}, \eqref{eq:errorcorrector-2} and the $\VH(\curl)$-stability of $\pi_H^E$ that \begin{align*} \| \Vu - \Vu_{H,m} \|_{\VH(\Div)^{\prime}} \le C H \| \Vf \|_{\VH(\Div)}. \end{align*} This result is a homogenization result in the sense that it yields a coarse function $\Vu_{H,m}$ that approximates the exact solution in $\VH(\Div)^{\prime}$. Furthermore, it yields an appropriate (quasi-local) corrector $\mathcal{K}_m(\Vu_{H,m})$ that is required for an accurate approximation in $\VH(\curl)$. \begin{remark}[Refined estimates] With a more careful proof, the constants in the estimate of Conclusion \ref{conclusion-main-result} can be specified as \begin{eqnarray*} \label{eq:errorLOD-refined} \lefteqn{\inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu - (\id + \mathcal{K}_m)\mathbf{v}_H \|_{\VH(\curl)}}\\ \nonumber&\leq& \alpha^{-1}(1+H)\bigl(H\max\{C_z, C_\theta\} \sqrt{C_{\ol,3}}+C_d C_\pi C_B^2\sqrt{C_{\ol,m}C_{\ol}}\, \beta^m\bigr)\|\Vf\|_{\VH(\Div)}, \end{eqnarray*} where $\alpha$ and $C_B$ are as in Assumption \ref{asspt:sesquiform}, $C_{d}$ is the constant appearing in the decay estimate \eqref{eq:errorcorrector}, $C_\pi$ is as in Proposition \ref{p:proj-pi-H-E}, $C_z$ and $C_\theta$ are from \eqref{eq:regulardecomp} and $C_{\ol,m}$ as detailed at the beginning of Section \ref{sec:intpol}. Note that if $m$ is large enough so that $\UN^m(T)=\Omega$ for all $T \in \CT_H$, we have as a refinement of Conclusion \ref{conclusion-ideal-corr-est} that \begin{eqnarray*} \inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu - (\id + \mathcal{K})\mathbf{v}_H \|_{\VH(\curl)} \leq \alpha^{-1}(1+H)\bigl(H\max\{C_z, C_\theta\} \sqrt{C_{\ol,3}} \bigr)\|\Vf\|_{\VH(\Div)}. \end{eqnarray*} \end{remark} \subsection{A fully discrete localized multiscale method} \label{subsec:discreteLOD} The procedure described in the previous section is still not yet \quotes{ready to use} for a practical computation as the local corrector problems \eqref{loc-corrector-problems} involve the infinite dimensional spaces $\VW(\Omega_T)$. Hence, we require an additional fine scale discretization of the corrector problems (just like the cell problems in periodic homogenization theory can typically not be solved analytically). For a fully discrete formulation, we introduce a second shape-regular partition $\CT_h$ of $\Omega$ into tetrahedra. This partition may be non-uniform and is assumed to be obtained from $\CT_H$ by at least one global refinement. It is a fine discretization in the sense that $h<H$ and that $\CT_h$ resolves all fine-scale features of the coefficients. Let $\mathring{\CN}(\CT_h)\subset\VH_0(\curl)$ denote the space of N{\'e}d{\'e}lec elements with respect to the partition $\CT_h$. We then introduce the space \[\VW_h(\Omega_T):=\VW(\Omega_T)\cap\mathring{\CN}(\CT_h)=\{\Vv_h\in\mathring{\CN}(\CT_h)|\Vv_h=0\text{ outside }\Omega_T, \pi_H^E(\Vv_h)=0\}\] and discretize the corrector problem \eqref{loc-corrector-problems} with this new space. The corresponding correctors are denoted by $\mathcal{K}_{T,m,h}$ and $\mathcal{K}_{m,h}$. With this modification we can prove analogously to the error estimate \eqref{conclusion-main-result-est} that it holds \begin{align} \label{conclusion-main-result-est-h} \inf_{\mathbf{v}_H \in \mathring{\CN}(\CT_H)} \| \Vu_h - (\id + \mathcal{K}_{m,h})\mathbf{v}_H \|_{\VH(\curl)} \le C \left( H + \sqrt{C_{\ol,m}} \tilde{\beta}^m \right) \| \Vf \|_{\VH(\Div)}, \end{align} where $\Vu_h$ is the Galerkin approximation of $\Vu$ in the discrete fine space $\mathring{\CN}(\CT_h)$. If $\CT_h$ is fine enough, we can assume that $\Vu_h$ is a good $\VH(\curl)$-approximation to the true solution $\Vu$. Consequently, it is justified to formulate a fully discrete (localized) multiscale method by seeking $\Vu_{H,h,m}^{\ms}:=(\id+\mathcal{K}_{m,h})\Vu_H$ with $\Vu_H\in \mathring{\CN}(\CT_H)$ such that \begin{equation} \label{eq:discreteLOD} \CB(\Vu_{H,h,m}^{\ms}, (\id+\mathcal{K}_{m,h})\Vv_H)=(\Vf, (\id+\mathcal{K}_{m,h})\Vv_H)_{L^2(\Omega)}\qquad\mbox{for all } \Vv_H\in\mathring{\CN}(\CT_H). \end{equation} As before, we can conclude from \eqref{conclusion-main-result-est-h} together with the choice $m\gtrsim \lvert \log H\rvert/\lvert\log \beta\rvert$, that it holds \begin{align*} \| \Vu_h - \Vu_{H,h,m}^{\ms} \|_{\VH(\curl)} + \| \Vu_h - \pi_H^E \Vu_{H,h,m}^{\ms} \|_{\VH(\Div)^{\prime}} \le C H \| \Vf \|_{\VH(\Div)}. \end{align*} Thus, the additional fine-scale discretization does not affect the overall error estimates and we therefore concentrate in the proofs (for simplicity) on the semi-discrete case as detailed in Sections \ref{subsec:idealapprox} and \ref{subsec:LODlocal}. Compared to the fully-discrete case, only some small modifications are needed in the proofs for the decay of the correctors. These modifications are outlined at the end of Section \ref{sec:decaycorrectors}. Note that $\Vu_h$ is not needed in the practical implementation of the method. \section{Proof of the decay for the Corrector Green's Operator} \label{sec:decaycorrectors} In this section, we prove Proposition \ref{prop:decaycorrector1} and Theorem \ref{thm:errorcorrectors}. Since the latter one is based on the first result, we start with proving the exponential decay of the Green's function associated with $\mathcal{G}$. Recall that we quantified the decay indirectly through estimates of the form \begin{equation*} \| \mathcal{G}(\mathbf{F}_T) \|_{\VH(\curl, \Omega\setminus \UN^m(T))}\lesssim \tilde{\beta}^m\| \mathbf{F}_T \|_{\VH_0(\curl)^{\prime}}, \end{equation*} where $\mathbf{F}_T$ is a $T$-local functional and $0<\tilde{\beta}<1$. \begin{proof}[Proof of Proposition \ref{prop:decaycorrector1}] Let $\eta\in \CS^1(\CT_H)\subset H^1(\Omega)$ be a scalar-valued, piece-wise linear and globally continuous cut-off function with \begin{equation*} \eta=0\qquad \text{in}\quad \UN^{m-6}(T)\qquad \qquad\qquad \eta=1\qquad \text{in}\quad \Omega\setminus\UN^{m-5}(T). \end{equation*} Denote $\CR=\supp(\nabla \eta)$ and $\Vphi:=\mathcal{G}(\mathbf{F}_T) \in \VW$. In the following we use $\UN^k(\CR)=\UN^{m-5+k}(T)\setminus \UN^{m-6-k}(T)$. Note that $\|\nabla \eta\|_{L^\infty(\CR)}\sim H^{-1}$. Furthermore, let $\Vphi=\Vphi-\pi_H^E\Vphi=\Vz+\nabla \theta$ be the splitting from Lemma \ref{lem:localregulardecomp}. We obtain with $\eta\leq 1$, the coercivity, and the product rule that \begin{align*} \alpha\|\Vphi\|^2_{\VH(\curl, \Omega\setminus\UN^m(T))}&\leq \bigl|(\mu\curl\Vphi, \eta\curl\Vphi)_{L^2(\Omega)}+(\kappa\Vphi, \eta\Vphi)_{L^2(\Omega)}\bigr|\\ &=\bigl|(\mu\curl\Vphi, \eta\curl\Vz)_{L^2(\Omega)}+(\kappa\Vphi, \eta\nabla\theta+\eta\Vz)_{L^2(\Omega)}\bigr|\\ &\leq M_1 +M_2+M_3+M_4+M_5, \end{align*} where \begin{align*} & M_1:=\Bigl|\bigl(\mu\curl\Vphi, \curl(\id-\pi_H^E)(\eta\Vz)\bigr)_{L^2(\Omega)} &&\hspace{-23pt}+\enspace \bigl(\kappa \Vphi, (\id-\pi_H^E) (\eta\Vz+\nabla(\eta\theta))\bigr)_{L^2(\Omega)}\Bigr|, \\ &M_2:=\Bigl|\bigl(\mu\curl\Vphi, \curl\pi_H^ E(\eta\Vz)\bigr)_{L^2(\Omega)}\Bigr|, && M_3:=\Bigl|\bigl(\kappa \Vphi,\pi_H^E(\eta\Vz+\nabla(\eta\Vphi))\bigr)_{L^2(\Omega)}\Bigr|, \\ & M_4:=\Bigl|\bigl(\mu\curl \Vphi, \nabla \eta\times \Vz\bigr)_{L^2(\Omega)}\Bigr|, && M_5:=\Bigl|\bigl(\kappa\Vphi, \theta\nabla \eta\bigr)_{L^2(\Omega)}\Bigr|. \end{align*} We used the product rule $\curl(\eta\Vz)=\nabla\eta\times \Vz+\eta\curl\Vz$ here. We now estimate the five terms separately. Let $\Vw:=(\id-\pi_H^E)(\eta\Vz+\nabla(\eta\theta))$ and note that (i) $\curl\Vw=\curl(\id-\pi_H^E)(\eta\Vz)$, (ii) $\Vw\in \VW$, (iii) $\supp\Vw\subset\Omega\setminus T$. Using the definition of the Corrector Green's Operator in \eqref{cor-greens-op} and the fact that $\mathbf{F}_T(\Vw)=0$ yields $M_1=0$. For $M_2$, note that the commuting property of the projections $\pi^E$ and $\pi^F$ implies $\curl\pi_H^E(\Vz)=\pi_H^F(\curl \Vz)=\pi_H^F(\curl\Vphi)=\curl\pi_H^E\Vphi=0$ because $\Vphi\in \VW$. Using the stability of $\pi_H^E$ \eqref{eq:stabilitycurl} and Lemma \ref{lem:localregulardecomp}, we can estimate $M_2$ as \begin{align*} M_2&\lesssim \|\curl\Vphi\|_{L^2(\UN(\CR))}\|\curl\pi_H^E(\eta\Vz)\|_{L^2(\UN(\CR))}\lesssim \|\curl\Vphi\|_{L^2(\UN(\CR))}\|\curl(\eta\Vz)\|_{L^2(\UN^2(\CR))}\\ &\lesssim \|\curl\Vphi\|_{L^2(\UN(\CR))}\Bigl(\|\nabla\eta\|_{L^\infty(\CR)}\|\Vz\|_{L^2(\CR)} +\|\eta\|_{L^\infty(\UN^2(\CR))}\|\curl\Vz\|_{L^2(\UN^{m-3}(T)\setminus \UN^{m-6}(T)))}\Bigr)\\ &\lesssim \|\curl\Vphi\|_{L^2(\UN(\CR))}\|\curl\Vphi\|_{L^2(\UN^{m}(T)\setminus \UN^{m-9}(T))}. \end{align*} In a similar manner, we obtain for $M_3$ that \begin{align*} M_3&\lesssim\|\Vphi\|_{L^2(\UN(\CR))}\Bigl(\|\eta \Vz\|_{L^2(\UN^2(\CR))}+\|\nabla(\eta\theta)\|_{L^2(\UN^2(\CR))}+H\|\curl(\eta\Vz)\|_{L^2(\UN^2(\CR))}\Bigr)\\ &\lesssim \|\Vphi\|_{L^2(\UN(\CR))}\Bigl(\|\Vphi\|_{L^2(\UN^{m}(T)\setminus \UN^{m-9}(T))}+H\|\curl\Vphi\|_{L^2(\UN^{m}(T)\setminus \UN^{m-9}(T))}\Bigr). \end{align*} Simply using Lemma \ref{lem:localregulardecomp}, we deduce for $M_4$ and $M_5$ \begin{align*} M_4&\lesssim \|\curl\Vphi\|_{L^2(\CR)}\|\curl\Vphi\|_{L^2(\UN^3(\CR))}, \\ M_5&\lesssim \|\Vphi\|_{L^2(\CR)} (\|\Vphi\|_{L^2(\UN^3(\CR))} + H\|\curl \Vphi\|_{L^2(\UN^3(\CR))}). \end{align*} All in all, this gives \begin{equation*} \|\Vphi\|^2_{\VH(\curl, \Omega\setminus \UN^m(T))}\leq \tilde{C} \|\Vphi\|^2_{\VH(\curl, \UN^{m}(T)\setminus \UN^{m-9}(T) )} \end{equation*} for some $\tilde{C}>0$. Moreover, it holds that \begin{equation*} \|\Vphi\|^2_{\VH(\curl, \Omega\setminus \UN^m(T))} = \|\Vphi\|^2_{\VH(\curl, \Omega\setminus \UN^{m-9}(T))} - \|\Vphi\|^2_{\VH(\curl, \UN^m(T)\setminus \UN^{m-9}(T))}. \end{equation*} Thus, we obtain finally with $\tilde{\beta}_{\mathrm{pre}}:=(1+\tilde{C}^{-1})^{-1}<1$, a re-iteration of the above argument, and Lemma~\ref{lemma:corrector-props} that \begin{equation*} \|\Vphi\|^2_{\VH(\curl, \Omega\setminus \UN^m(T))}\lesssim \tilde{\beta}_{\mathrm{pre}}^{\lfloor m/9\rfloor}\|\Vphi\|^2_{\VH(\curl)}\lesssim \tilde{\beta}_{\mathrm{pre}}^{\lfloor m/9\rfloor}\| \mathbf{F}_T \|^2_{\VH_0(\curl)^\prime}. \end{equation*} Algebraic manipulations give the assertion. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:errorcorrectors}] We start by proving the following local estimate \begin{align} \label{eq:errorcorrectorlocal} \| \mathcal{G}( \mathbf{F}_T )-\mathcal{G}_{T,m}( \mathbf{F}_T ) \|_{\VH(\curl)}&\leq C_1 \tilde{\beta}^m \| \mathbf{F}_T \|_{\VH_0(\curl)^{\prime}} \end{align} for some constant $C_1>0$ and $0<\tilde{\beta}<1$. Let $\eta\in \CS^1(\CT_H)$ be a piece-wise linear and globally continuous cut-off function with \begin{align*} \eta=0 \qquad \text{in} \quad \Omega\setminus \UN^{m-1}(T)\qquad\qquad\qquad\eta=1\qquad\text{in}\quad \UN^{m-2}(T). \end{align*} Due to C{\'e}a's Lemma we have \begin{align*} \| \mathcal{G}( \mathbf{F}_T ) - \mathcal{G}_{T,m}( \mathbf{F}_T ) \|_{\VH(\curl)}\lesssim \inf_{\Vw_{T,m}\in \VW(\Omega_T)}\| \mathcal{G}( \mathbf{F}_T ) -\Vw_{T,m}\|_{\VH(\curl)}. \end{align*} We use the splitting of Lemma \ref{lem:localregulardecomp} and write $\mathcal{G}( \mathbf{F}_T )=(\id-\pi_H^E)(\mathcal{G}( \mathbf{F}_T ))=\Vz+\nabla\theta$. Then we choose $\Vw_{T,m}=(\id-\pi_H^E)(\eta\Vz+\nabla(\eta\theta))\in \VW(\Omega_T)$ and derive with the stability of $\pi_H^E$ and \eqref{eq:regulardecomp} \begin{align*} \|\mathcal{G}( \mathbf{F}_T )-\mathcal{G}_{T,m}( \mathbf{F}_T )\|_{\VH(\curl)}&\lesssim \|(\id-\pi_H^E)(\mathcal{G}( \mathbf{F}_T )-\eta\Vz - \nabla(\eta\theta))\|_{\VH(\curl)}\\ &=\|(\id-\pi_H^E)((1-\eta)\Vz+\nabla((1-\eta)\theta))\|_{\VH(\curl)}\\ &\lesssim \|(1-\eta)\Vz\|_{L^2(\Omega\setminus\{\eta=1\})}+\|\nabla((1-\eta)\theta)\|_{L^2(\Omega\setminus\{\eta=1\})}\\* &\quad+(1+H)\|\curl((1-\eta)\Vz)\|_{L^2(\Omega\setminus\{\eta=1\})}\\ &\lesssim (1+H)\,\| \mathcal{G}( \mathbf{F}_T )\|_{\VH(\curl, \UN^3(\Omega\setminus\{\eta=1\}))}. \end{align*} Combination with Proposition \ref{prop:decaycorrector1} gives estimate \eqref{eq:errorcorrectorlocal}. To prove the main estimate of Theorem \ref{thm:errorcorrectors}, i.e.\ estimate \eqref{eq:errorcorrector}, we define, for a given simplex $T\in\CT_H$, the piece-wise linear, globally continuous cut-off function $\eta_T\in \CS^1(\CT_H)$ via \begin{align*} \eta_T=0\qquad \text{in}\quad \UN^{m+1}(T)\qquad\qquad\qquad \eta_T=1\qquad\text{in}\quad \Omega\setminus\UN^{m+2}(T). \end{align*} Denote $\Vw:=(\mathcal{G}-\mathcal{G}_m)(\mathbf{F})=\sum_{T \in \CT_H} \Vw_T$ with $\Vw_T:=(\mathcal{G}-\mathcal{G}_{T,m})(\mathbf{F}_T)$ and split $\Vw$ according to Lemma \ref{lem:localregulardecomp} as $\Vw=\Vw-\pi_H^E(\Vw)=\Vz+\nabla\theta$. Due to the ellipticity of $\CB$ and its sesquilinearity, we have \begin{align*} \alpha\|\Vw\|^2_{\VH(\curl)} \leq \Bigl|\sum_{T\in\CT_H}\CB(\Vw_T,\Vw)\Bigr|\leq \sum_{T\in \CT_H}|\CB(\Vw_T,\Vz+\nabla\theta )| \leq \sum_{T\in \CT_H} (A_T + B_T) \end{align*} where, for any $T\in\CT_H$, we abbreviate \begin{equation*} A_T:=|\CB(\Vw_T,(1-\eta_T)\Vz+\nabla((1-\eta_T)\theta))| \quad\text{and}\quad B_T:=|\CB(\Vw_T,\eta_T\Vz+\nabla(\eta_T\theta))| . \end{equation*} For the term $A_T$, we derive by using the properties of the cut-off function and the regular decomposition \eqref{eq:regulardecomp} \begin{align*} A_T&\lesssim\|\Vw_T\|_{\VH(\curl)}\|(1-\eta_T)\Vz+\nabla((1-\eta_T)\theta)\|_{\VH(\curl, \{\eta_T\neq 1\})}\\ &\leq \|\Vw_T\|_{\VH(\curl)}\,(1+H)\,\|\Vw\|_{\VH(\curl, \UN^3(\{\eta_T\neq 1\}))}. \end{align*} The term $B_T$ can be split as \begin{align*} B_T\leq |\CB(\Vw_T,(\id-\pi_H^E)(\eta_T\Vz+\nabla(\eta_T\theta)))|+|\CB(\Vw_T,\pi_H^E(\eta_T\Vz+\nabla(\eta_T\theta)))|. \end{align*} Denoting $\Vphi:=(\id-\pi_H^E)(\eta_T\Vz+\nabla(\eta_T\theta))$, we observe $\Vphi\in \VW$ and $\supp\Vphi\subset\Omega\setminus \UN^m(T)$. Because $\Vphi\in\VW$ with support outside $T$, we have $\CB(\mathcal{G}(\mathbf{F}_T),\Vphi)=\mathbf{F}_T(\Vphi)=0$. Since $\Vphi$ has support outside $\UN^m(T)=\Omega_T$, but $\mathcal{G}_{T,m}(\mathbf{F}_T)\in \VW(\Omega_T)$, we also have $\CB(\mathcal{G}_{T,m}(\mathbf{F}_T),\Vphi)=0$. All in all, this means $\CB(\Vw_T , \Vphi )=0$. Using the stability of $\pi_H^E$ \eqref{eq:stabilityL2}, \eqref{eq:stabilitycurl} and the regular decomposition \eqref{eq:regulardecomp}, we obtain \begin{align*} B_T &\leq |\CB(\Vw_T,\pi_H^E(\eta_T\Vz+\nabla(\eta_T\theta)))|\\* &\lesssim\|\Vw_T\|_{\VH(\curl)}\bigl(\|\eta_T\Vz+\nabla(\eta_T\theta)\|_{L^2(\UN^2(\{\eta_T\neq 1\}))}+(1+H)\|\curl(\eta_T\Vz)\|_{L^2(\UN^2(\{\eta_T\neq 1\}))}\bigr)\\ &\lesssim \|\Vw_T\|_{\VH(\curl)}(1+H)\,\|\Vw\|_{\VH(\curl, \UN^5(\{\eta_T\neq 1\}))}. \end{align*} Combining the estimates for $A_T$ and $B_T$ and observing that $\{\eta_T\neq 1\} =\UN^{m+2}(T)$, we deduce \begin{align*} \alpha\|\Vw\|_{\VH(\curl)}^2&\lesssim \sum_{T\in\CT_H}\|\Vw_T\|_{\VH(\curl)}\,\|\Vw\|_{\VH(\curl, \UN^{m+7}(T))} \lesssim \sqrt{C_{\ol, m}}\, \|\Vw\|_{\VH(\curl)}\sqrt{\sum_{T\in\CT_H}\|\Vw_T\|_{\VH(\curl)}^2}. \end{align*} Combination with estimate \eqref{eq:errorcorrectorlocal} finishes the proof of \eqref{eq:errorcorrector}. Finally, estimate \eqref{eq:errorcorrector-2} follows with \begin{align*} \|\Vw\|_{\VH(\Div)^{\prime}} \leq C H \|\Vw\|_{\VH(\curl)}. \end{align*} \end{proof} \textbf{Changes for the fully discrete localized method.}\hspace{2pt} Let us briefly consider the fully-discrete setting described in Section \ref{subsec:discreteLOD}. Here we note that, up to a modification of the constants, Theorem \ref{thm:errorcorrectors} also holds for the difference $(\mathcal{G}_h - \mathcal{G}_{h,m})(\mathbf{F})$, where $\mathcal{G}_h(\mathbf{F})$ is the Galerkin approximation of $\mathcal{G}(\mathbf{F})$ in the discrete space $\VW_h:=\{\Vv_h\in\mathring{\CN}(\CT_h)|\pi_H^E(\Vv_h)=0\}$ and where $\mathcal{G}_{h,m}(\mathbf{F})$ is defined analogously to $\mathcal{G}_{h,m}(\mathbf{F})$ but where $\VW_h(\Omega_T):=\{ \Vw_h \in \VW_h| \hspace{3pt} \Vw_h \equiv 0 \mbox{ in } \Omega \setminus \Omega_T \}$ replaces $\VW(\Omega_T)$ in the local problems. Again, the central observation is a decay result similar to Proposition \ref{prop:decaycorrector1}, but now for $\mathcal{G}_{h}(\mathbf{F}_T)$. A few modifications to the proof have to be made, though: The product of the cut-off function $\eta$ and the regular decomposition $\Vz+\nabla\theta$ does not lie in $\mathring{\CN}(\CT_h)$. Therefore, an additional interpolation operator into $\mathring{\CN}(\CT_H)$ has to be applied. Here it is tempting to just use the nodal interpolation operator and its stability on piece-wise polynomials, since $\eta \hspace{2pt} \mathcal{G}_{h}(\mathbf{F}_T)$ is a piece-wise (quadratic) polynomial. However, the regular decomposition employed is no longer piece-wise polynomial and we hence have to use the Falk-Winther operator $\pi_h^E$ onto the fine space $\mathring{\CN}(\CT_h)$ here. This means that we have the following modified terms in the proof of Proposition \ref{prop:decaycorrector1}: \begin{align*} \tilde{M}_1&:=\Bigl|\bigl(\mu\curl\Vphi, \curl(\id-\pi_H^E)\pi_h^E(\eta\Vz)\bigr)_{L^2(\Omega)} &&\hspace{-17pt}+\enspace \bigl(\kappa \Vphi, (\id-\pi_H^E)\pi_h^E(\eta\Vz+\nabla(\eta\theta))\bigr)_{L^2(\Omega)}\Bigr|, \\ \tilde{M}_2&:=\Bigl|\bigl(\mu\curl\Vphi, \curl\pi_H^ E\pi_h^E(\eta\Vz)\bigr)_{L^2(\Omega)}\Bigr|, && \tilde{M}_3:=\Bigl|\bigl(\kappa \Vphi,\pi_H^E\pi_h^E(\eta\Vz+\nabla(\eta\Vz))\bigr)_{L^2(\Omega)}\Bigr|. \end{align*} They can be treated similarly to $M_1$, $M_2$ and $M_3$, using in addition the stability of $\pi_h^E$. Note that the additional interpolation operator $\pi_h^E$ will enlarge the patches slightly, so that we should define $\eta$ via \begin{align*} \eta=0\qquad \text{in}\quad \UN^{m-8}(T)\qquad\qquad\qquad\eta=1\qquad\text{in}\quad \Omega\setminus \UN^{m-7}(T). \end{align*} The terms $M_4$ and $M_5$ remain unchanged, and we moreover get the terms \begin{align*} \tilde{M}_6:=\Bigl|\bigl(\mu\curl\Vphi, \curl(\id-\pi_h^E)(\eta\Vz)\bigr)_{L^2(\Omega)}\Bigr|, \qquad \tilde{M}_7:=\Bigl|\bigl(\kappa \Vphi, (\id-\pi_h^E)(\eta\Vz+\nabla(\eta\theta))\bigr)_{L^2(\Omega)}\Bigr|. \end{align*} These can be estimated simply using the stability of $\pi_h^E$, the properties of $\eta$ and the regular decomposition \eqref{eq:regulardecomp}. \section{Falk--Winther interpolation} \label{sec:intpolimpl} This section briefly describes the construction of the bounded local cochain projection of \cite{FalkWinther2014} for the present case of $\VH(\curl)$-problems in three space dimensions. The two-dimensional case is thoroughly described in the gentle introductory paper \cite{FalkWinther2015}. After giving the definition of the operator, we describe how it can be represented as a matrix. This is important because the interpolation operator is part of the algorithm and not a mere theoretical tool and therefore required in a practical realization. \subsection{Definition of the operator} Let $\Delta_0$ denote the set of vertices of $\CT_H$ and let $\mathring{\Delta}_0:=\Delta_0\cap\Omega$ denote the interior vertices. Let $\Delta_1$ denote the set of edges and let $\mathring{\Delta}_1$ denote the interior edges, i.e., the elements of $\Delta_1$ that are not a subset of $\partial\Omega$. The space $\mathring{\CN}(\CT_H)$ is spanned by the well-known edge-oriented basis $(\Vpsi_E)_{E\in\mathring{\Delta}_1}$ defined for any $E\in\mathring{\Delta}_1$ through the property \begin{equation*} \fint_E \Vpsi_E\cdot \Vt_E\,ds = 1 \quad\text{and}\quad \fint_{E'} \Vpsi_E\cdot \Vt_E\,ds = 0 \quad\text{for all }E'\in\mathring{\Delta}_1\setminus\{E\}. \end{equation*} Here $\Vt_E$ denotes the unit tangent to the edge $E$ with a globally fixed sign. Any vertex $z\in\Delta_0$ possesses a nodal patch (sometimes also called macroelement) \begin{equation*} \omega_z:=\Int\Big(\bigcup\{T\in\CT_H : z\in T\}\Big). \end{equation*} For any edge $E\in\Delta_1$ shared by two vertices $z_1,z_2\in\Delta_0$ such that $E=\operatorname{conv}\{z_1,z_2\}$, the extended edge patch reads \begin{equation*} \omega_E^{\mathit{ext}} := \omega_{z_1}\cup\omega_{z_2}. \end{equation*} The restriction of the mesh $\CT_H$ to $\omega_E^{\mathit{ext}}$ is denoted by $\CT_H(\omega_E^{\mathit{ext}})$. Let $\CS^1(\CT_H(\omega_E^{\mathit{ext}}))$ denote the (scalar-valued) first-order Lagrange finite element space with respect to $\CT_H(\omega_E^{\mathit{ext}})$ and let $\CN(\CT_H(\omega_E^{\mathit{ext}}))$ denote the lowest-order N\'ed\'elec finite element space over $\CT_H(\omega_E^{\mathit{ext}})$. The operator \[ Q^1_E: \VH(\curl, \omega_E^{\mathit{ext}}) \to \CN(\CT_H(\omega_E^{\mathit{ext}})) \] is defined for any $\Vu\in \VH(\curl, \omega_E^{\mathit{ext}})$ via \begin{equation*} \begin{aligned} (\Vu-Q^1_E \Vu, \nabla \tau) &= 0 \quad &&\text{for all } \tau\in \CS^1(\CT_H(\omega_E^{\mathit{ext}})) \\ (\curl (\Vu-Q^1_E \Vu),\curl \Vv) &=0 &&\text{for all } \Vv\in \CN(\CT_H(\omega_E^{\mathit{ext}})). \end{aligned} \end{equation*} Given any vertex $y\in\Delta_0$, define the piecewise constant function $z^0_y$ by \begin{equation*} z^0_y = \begin{cases} (\operatorname{meas}(\omega_y))^{-1} &\text{in } \omega_y \\ 0 &\text{in } \Omega\setminus\omega_y \end{cases} \end{equation*} Given any edge $E\in\Delta_1$ shared by vertices $y_1,y_2\in\Delta_0$ such that $E=\operatorname{conv}\{y_1,y_2\}$, define \begin{equation*} (\delta z^0)_E := z^0_{y_2} - z^0_{y_1} . \end{equation*} Let $E\in\Delta_1$ and denote by $\mathring{\mathcal{RT}}(\CT_H(\omega_E^{\mathit{ext}}))$ the lowest-order Raviart--Thomas space with respect to $\CT_H(\omega_E^{\mathit{ext}})$ with vanishing normal trace on the boundary $\partial (\omega_E^{\mathit{ext}})$. Let for any $E\in\Delta_1$ the field $\Vz_E^1\in\mathring{\mathcal{RT}}(\CT_H(\omega_E^{\mathit{ext}}))$ be defined by \begin{equation*} \begin{aligned} \Div \Vz_E^1 &=-(\delta z^0)_E \quad && \\ (\Vz_E^1,\curl\Vtau) &= 0 &&\text{for all } \Vtau\in\mathring{\CN}(\CT_H(\omega_E^{\mathit{ext}})) \end{aligned} \end{equation*} where $\mathring{\CN}(\CT_H(\omega_E^{\mathit{ext}}))$ denotes the N\'ed\'elec finite element functions over $\CT_H(\omega_E^{\mathit{ext}})$ with vanishing tangential trace on the boundary $\partial(\omega_E^{\mathit{ext}})$. The operator $M^1:L^2(\Omega;\mathbb{C}^3)\to\mathring{\CN}(\CT_H)$ maps any $\Vu\in L^2(\Omega;\mathbb{C}^3)$ to \begin{equation*} M^1\Vu := \sum_{E\in\mathring{\Delta}_1} (\operatorname{length}(E))^{-1} \int_{\omega_E^{\mathit{ext}}} \Vu\cdot \Vz_E^1\,dx\, \Vpsi_E. \end{equation*} The operator \[ Q^1_{y,-} : \VH(\curl,\omega_E^{\mathit{ext}}) \to \CS^1(\CT_H(\omega_E^{\mathit{ext}})) \] is the solution operator of a local discrete Neumann problem. For any $\Vu\in \VH(\curl, \omega_E^{\mathit{ext}})$, the function $ Q^1_{y,-} \Vu $ solves \begin{equation*} \begin{aligned} (\Vu-\nabla Q^1_{y,-} \Vu,\nabla v) &= 0 \quad&&\text{for all } v\in \CS^1(\CT_H(\omega_E^{\mathit{ext}})) \\ \int_{\omega_E^{\mathit{ext}}} Q^1_{y,-} \Vu\,dx & = 0. && \end{aligned} \end{equation*} Define now the operator $S^1:\VH_0(\curl,\Omega)\to \mathring{\CN}(\CT_H)$ via \begin{equation}\label{e:S1def1} S^1 \Vu := M^1 \Vu + \sum_{y\in\mathring{\Delta}_0} (Q^1_{y,-}\Vu)(y)\nabla \lambda_y . \end{equation} The second sum on the right-hand side can be rewritten in terms of the basis functions $\Vpsi_E$. The inclusion $\nabla \mathring{\CS}^1(\CT_H)\subseteq \mathring{\CN}(\CT_H)$ follows from the principles of finite element exterior calculus \cite{ArnoldFalkWinther2006,ArnoldFalkWinther2010}. Given an interior vertex $y\in\mathring{\Delta}_0$, the expansion in terms of the basis $(\Vpsi_E)_{E\in\mathring{\Delta}_1}$ reads \begin{equation*} \nabla\lambda_z = \sum_{E\in\mathring{\Delta}_1} \fint_E \nabla\lambda_z\cdot \Vt_E\,ds\,\Vpsi_E = \sum_{E\in\Delta_1(z)} \frac{\operatorname{sign}(\Vt_E\cdot\nabla\lambda_z)}{\operatorname{length}(E)} \Vpsi_E \end{equation*} where $\Delta_1(z)\subseteq\mathring{\Delta}_1$ is the set of all edges that contain $z$. Thus, $S^1$ from \eqref{e:S1def1} can be rewritten as \begin{equation}\label{e:S1def2} S^1 \Vu := M^1 \Vu + \sum_{E\in\mathring{\Delta}_1} (\operatorname{length}(E))^{-1} \big((Q^1_{y_2(E),-}\Vu)(y_2(E)) - (Q^1_{y_1(E),-}\Vu)(y_1(E))\big) \Vpsi_E \end{equation} where $y_1(E)$ and $y_2(E)$ denote the endpoints of $E$ (with the orientation convention $\Vt_E = (y_2(E)-y_1(E))/\operatorname{length}(E)$). Finally, the Falk-Winter interpolation operator $\pi_H^E:\VH_0(\curl, \Omega)\to\mathring{\CN}(\CT_H)$ is defined as \begin{equation}\label{e:R1def} \pi_H^E \Vu := S^1 \Vu + \sum_{E\in\mathring{\Delta}_1} \fint_E \big((\id-S^1)Q^1_E \Vu\big)\cdot \Vt_E\,ds \,\Vpsi_E . \end{equation} \subsection{Algorithmic aspects} Given a mesh $\CT_H$ and a refinement $\CT_h$, the linear projection $\pi_H : \mathring{\CN}(\CT_h)\to \mathring{\CN}(\CT_H)$ can be represented by a matrix $\mathsf{P}\in\mathbb R^{\dim \mathring{\CN}(\CT_H)\times\dim \mathring{\CN}(\CT_h)}$. This subsection briefly sketches the assembling of that matrix. The procedure involves the solution of local discrete problems on the macroelements. It is important to note that these problems are of small size because the mesh $\CT_h$ is a refinement of $\CT_H$. Given an interior edge $E\in\mathring{\Delta}_1^H$ of $\CT_H$ and an interior edge $e\in\mathring{\Delta}_1^h$ of $\CT_h$, the interpolation $\pi_H \Vpsi_e$ has an expansion \begin{equation*} \pi_H \Vpsi_e= \sum_{E'\in\mathring{\Delta}_1^H} c_{E'} \Vpsi_{E'} \end{equation*} for real coefficients $(c_{E'})_{E'\in\mathring{\Delta}_1^H}$. The coefficient $c_E$ is zero whenever $e$ is not contained in the closure of the extended edge patch $\overline{\omega}_E^{\mathit{ext}}$. The assembling can therefore be organized in a loop over all interior edges in $\mathring{\Delta}_1^H$. Given a global numbering of the edges in $\mathring{\Delta}_1^H$, each edge $E\in\mathring{\Delta}_1^H$ is equipped with a unique index $I_H(E)\in\{1,\dots,\operatorname{card}(\mathring{\Delta}_1^H)\}$. Similarly, the numbering of edges in $\mathring{\Delta}_1^h$ is denoted by $I_h$. The matrix $\mathsf{P}=\mathsf{P_1}+\mathsf{P_2}$ will be composed as the sum of matrices $\mathsf{P_1}$, $\mathsf{P_2}$ that represent the two summands on the right-hand side of \eqref{e:R1def}. Those will be assembled in loops over the interior edges. Matrices $\mathsf{P_1}$, $\mathsf{P_2}$ are initialized as empty sparse matrices. \subsubsection{Operator $\mathsf{P_1}$} \noindent \textbf{for} $E\in\mathring{\Delta}_1^H$ \textbf{do} Let the interior edges in $\mathring{\Delta}_1^h$ that lie inside $\overline{\omega}_E^{\mathit{ext}}$ be denoted with $\{e_1,e_2,\dots,e_N\}$ for some $N\in\mathbb N$. The entries $\mathsf{P}_1(I_H(E),[I_h(e_1)\dots I_h(e_N)])$ of the matrix $\mathsf{P}_1$ are now determined as follows. Compute $\Vz^1_E \in \mathring{\mathcal{RT}}(\CT_H({\omega}_E^{\mathit{ext}}))$. The matrix $\mathsf{M}_E\in\mathbb R^{1\times N}$ defined via \[ \mathsf{M}_E := (\operatorname{length}(E))^{-1} \left[ \int_{{\omega}_E^{\mathit{ext}}} \Vz^1_E\cdot\Vpsi_{e_j}\,dx \right]_{j=1}^N \] represents the map of the basis functions on the fine mesh to the coefficient of $M^1$ contributing to $\Vpsi_E$ on the coarse mesh. Denote by $\mathsf{A}_{y_j(E)}$ and $\mathsf{B}_{y_j(E)}$ ($j=1,2$) the stiffness and right-hand side matrix representing the system for the operator $Q_{y_j(E),-}$ \begin{align*} \mathsf{A}_{y_j(E)} &:= \left[ \int_{\omega_{y_j(E)}} \nabla \phi_y \cdot\nabla \phi_z\,dx \right]_{y,z\in\Delta_0(\CT_H(\omega_{y_j(E)}))}, \\ \mathsf{B}_{y_j(E)} &:= \left[ \int_{\omega_{y_j(E)}} \nabla \phi_y \cdot\Vpsi_{e_j}\,dx \right]_{\substack{y\in\Delta_0(\CT_H(\omega_{y_j(E)}))\\ j=1,\dots,N}}. \end{align*} After enhancing the system to $\tilde{\mathsf{A}}_{y_j(E)}$ and $\tilde{\mathsf{B}}_{y_j(E)}$ (with a Lagrange multiplier accounting for the mean constraint), it is uniquely solvable. Set $\tilde{\mathsf{Q}}_{y_j(E)} = \tilde{\mathsf{A}}_{y_j(E)}^{-1}\tilde{\mathsf{B}}_{y_j(E)}$ and extract the row corresponding to the vertex $y_j(E)$ \[ \mathsf{Q}_j:= (\operatorname{length}(E))^{-1} \tilde{\mathsf{Q}}_{y_j(E)}[y_j(E),:] \in \mathbb R^{1\times N}. \] Set \[ \mathsf{P}_1(I_H(E),[I_h(e_1)\dots I_h(e_N)]) = \mathsf{M}_E + \mathsf{Q}_1 -\mathsf{Q}_2 . \] \noindent \textbf{end} \subsubsection{Operator $\mathsf{P_2}$} \noindent \textbf{for} $E\in\mathring{\Delta}_1^H$ \textbf{do} Denote the matrices -- where indices $j,k$ run from $1$ to $\operatorname{card}(\Delta_1(\CT_H({\omega}_E^{\mathit{ext}})))$, $y$ through \linebreak[4]$\Delta_0(\CT_H({\omega}_E^{\mathit{ext}}))$, and $\ell=1,\ldots, N$ -- \begin{equation*} \mathsf{S}_E := \left[ \int_{{\omega}_E^{\mathit{ext}}} \curl \Vpsi_{E_j} \cdot\curl\Vpsi_{E_k}\,dx \right]_{j,k} \mathsf{T}_E := \left[ \int_{{\omega}_E^{\mathit{ext}}} \Vpsi_{E_j} \cdot\nabla\lambda_{y}\,dx \right]_{j,y} \end{equation*} and \begin{equation*} \mathsf{F}_E := \left[ \int_{{\omega}_E^{\mathit{ext}}} \curl \Vpsi_{E_j} \cdot\curl\Vpsi_{e_\ell}\,dx \right]_{j,\ell} \mathsf{G}_E := \left[ \int_{{\omega}_E^{\mathit{ext}}} \Vpsi_{e_\ell} \cdot\nabla\lambda_{y}\,dx \right]_{y, \ell} . \end{equation*} Solve the saddle-point system \begin{equation*} \begin{bmatrix} \mathsf{S} & \mathsf{T}^* \\ \mathsf{T} & 0 \end{bmatrix} \begin{bmatrix} \mathsf{U} \\ \mathsf{V} \end{bmatrix} = \begin{bmatrix} \mathsf{F} \\ \mathsf{G} \end{bmatrix} . \end{equation*} (This requires an additional one-dimensional gauge condition because the sum of the test functions $\sum_y\nabla\lambda_y$ equals zero.) Assemble the operator $S^1$ (locally) as described in the previous step and denote this matrix by $\mathsf{P}_1^{\mathit{loc}}$. Compute $\mathsf{U}- \mathsf{P}_1^{\mathit{loc}} \mathsf{U}$ and extract the line $\mathsf{X}$ corresponding to the edge $E$ \[ \mathsf{P_2}(I_H(E),[I_h(e_1)\dots I_h(e_N)]) = \mathsf{X} . \] \noindent \textbf{end} \section*{Conclusion} In this paper, we suggested a procedure for the numerical homogenization of $\VH(\curl)$-elliptic problems. The exact solution is decomposed into a coarse part, which is a good approximation in $\VH(\Div)^\prime$, and a corrector contribution by using the Falk-Winther interpolation operator. We showed that this decomposition gives an optimal order approximation in $\VH(\curl)$, independent of the regularity of the exact solution. Furthermore, the corrector operator can be localized to patches of macro elements, which allows for an efficient computation. This results in a generalized finite element method in the spirit of the Localized Orthogonal Decomposition which utilizes the bounded local cochain projection of the Falk-Winther as part of the algorithm. \section*{Acknowledgments} Main parts of this paper were written while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn. PH and BV acknowledge financial support by the DFG in the project OH 98/6-1 ``Wave propagation in periodic structures and negative refraction mechanisms''. DG acknowledges support by the DFG through CRC 1173 ``Wave phenomena: analysis and numerics'' and by the Baden-W\"urttemberg Stiftung (Eliteprogramm f\"ur Postdocs) through the project ``Mehrskalenmethoden für Wellenausbreitung in heterogenen Materialien und Metamaterialien''. \end{document}

\begin{document} \title[Bi-slant submersions] {On bi-slant submersions in complex geometry} \author[C. Sayar]{Cem Sayar$^1$} \address{$^1$Istanbul Technical University\\ Faculty of Science and Letters,\\ Department of Mathematics\\ 34469, Maslak /\.{I}stanbul Turkey} \email{[email protected]} \author[M. A. Akyol]{Mehmet Akif Akyol$^2$} \address{$^2$Bingol University\\ Faculty of Arts and Sciences,\\ Department of Mathematics\\ 12000, Bing\"{o}l, Turkey} \email{[email protected]} \author[R. Prasad]{Rajendra Prasad$^3$} \address{$^3$Lucknow University\\ Department of Mathematics and Astronomy\\ 226007, Uttar Pradesh, Lucknow, India} \email{[email protected]} \subjclass{Primary 53C15, 53B20} \keywords{Riemannian submersion, bi-slant submersion, horizontal distribution, Kaehler manifold} \date{January 1, 2004} \begin{abstract} In the present paper, we introduce bi-slant submersions from almost Hermitian manifolds onto Riemannian manifolds as a generalization of invariant, anti-invariant, semi-invariant, slant, semi-slant and hemi-slant Riemannian submersions. We mainly focus on bi-slant submersions from Kaehler manifolds. We provide a proper example of bi-slant submersion, investigate the geometry of foliations determined by vertical and horizontal distributions, and obtain the geometry of leaves of these distributions. Moreover, we obtain curvature relations between the base space, the total space and the fibres, and find geometric implications of these relations. \end{abstract} \maketitle \section{Introduction} The notion of a slant submanifold was introduced by B.-Y. Chen in \cite{Chen0} and first results on slant submanifolds were collected in his book \cite{Chen2}. After he defined that notion, many geometers were inspired by that fact and have obtained many results on the notion in the different total space. As a generalization of the notion, J. L. Cabrerizo et. al. defined the notion of bi-slant submanifold in \cite{Cabre} and see also \cite{Carri}. On the other hand, as an analogue of isometric immersion (Riemannian submanifold), the notion of Riemannian submersion was first introduced by B. O'Neill \cite{O} and A. Gray \cite{Gra} between two Riemannian manifolds. This notion has some aplications in physics and in mathematics. More precisely, Riemannian submersions have applications in supergravity and superstring theories \cite{IV1,M}, Kaluza-Klein theory \cite{BL,IV} and the Yang-Mills theory \cite{BL1,W1}. B. Watson \cite{Wat} considered submersions between almost Hermitian manifolds by taking account of almost complex structure of total manifold. In this case, the vertical and horizontal distributions are invariant. Afterwards, almost Hermitian submersions have been extensively studied different subclasses of almost Hermitian manifolds, for example; see \cite{Fa}. Inspried by B. Watson's article, B. \c{S}ahin introduced anti invariant submersions from almost Hermitian manifolds onto Riemannian manifolds \cite{Sah}. This notion has opened a new original and effective area in the theory of Riemannian submersions. That paper has been a source of inspiration to so many geometers. For example, as a special case of anti-invariant submersion, Lagrangian submersion was studied by H. M. Tastan \cite{Ta}. Later, several new types of Riemannian submersions were defined and studied such as semi-invariant submersion \cite{Akyol4,cem,Sa}, slant submersion \cite{Er, Gun, Gun1, Sa1}, hemi-slant submersion \cite{Ta3}, semi-slant submersion \cite{Akyol3, Park}, pointwise slant submersion \cite{Lee, Se}, quasi bi-slant submersion \cite{Prasad}, conformal slant submersion \cite{Akyol0,Akyol1} and conformal semi-slant submersion \cite{Akyol2}. Also, these kinds of submersions were considered in different kinds of structures such as cosymplectic, Sasakian, Kenmotsu, nearly Kaehler, almost product, para-contact, and et al. Recent developments in the theory of submersion can be found in the book \cite{baykit}. Recently, the first author of the paper and et.al. define Generic submersion in the sense of G. B. Ronsse (see: \cite{Rons}) for the complex context in \cite{Cemp}. We are motivated to fill a gap in the literature by giving the notion of bi-slant submersions in which the fibres consist of two slant distributions. In the present paper, as a special case of the above notion and generalization of invariant, anti-invariant, semi-invariant, slant, semi-slant and hemi-slant Riemannian submersions we introduce bi-slant submersion and investigate the geometry of base space, the total space and the fibres. The paper is organized as follows. Section 2 includes preliminaries. In section 3 contains the definition of bi-slant submersions, a proper example, the geometry of foliations determined by vertical and horizontal distributions and the geometry of leaves of these distributions. The last section of this paper includes curvature relations between the base space, the total space and the fibres, and find geometric implications of these relations. \section{Riemannian submersions} In this section, we give necessary background for Riemannian submersions.\\ Let $(M,g)$ and $(N,g_{\text{\tiny$N$}})$ be Riemannian manifolds, where $\dim(M)$ is greater than $\dim(N)$. A surjective mapping $\pi:(M,g)\rightarrow(N,g_{N})$ is called a \emph{Riemannian submersion} \cite{O} if\\ \textbf{(S1)} $\pi$ has maximal rank, and \\ \textbf{(S2)} $\pi_{*}$, restricted to $\ker\pi_{*}^{\bot},$ is a linear isometry.\\ In this case, for each $q\in N$, $\pi^{-1}(q)$ is a $k$-dimensional submanifold of $M$ and called a \emph{fiber}, where $k=\dim(M)-\dim(N).$ A vector field on $M$ is called \emph{vertical} (resp. \emph{horizontal}) if it is always tangent (resp. orthogonal) to fibers. A vector field $X$ on $M$ is called \emph{basic} if $X$ is horizontal and $\pi$-related to a vector field $X_{*}$ on $N,$ i.e., $\pi_{*}X_{p}=X_{*\pi(p)}$ for all $p\in M.$ We will denote by $\mathcal{V}$ and $\mathcal{H}$ the projections on the vertical distribution $\ker\pi_{*}$, and the horizontal distribution $\ker\pi_{*}^{\bot},$ respectively. As usual, the manifold $(M,g)$ is called \emph{total manifold} and the manifold $(N,g_{N})$ is called \emph{base manifold} of the submersion $\pi:(M,g)\rightarrow(N,g_{N})$. The geometry of Riemannian submersions is characterized by O'Neill's tensors $\mathcal{T}$ and $\mathcal{A}$, defined as follows: \begin{equation}\label{testequationn} \mathcal{T}_{U}{V}=\mathcal{V}\nabla_{\mathcal{V}{U}}\mathcal{H}{V}+\mathcal{H}\nabla_{\mathcal{V}{U}}\mathcal{V}{V}, \end{equation} \begin{equation}\label{testequationnn} \mathcal{A}_{U}{V}=\mathcal{V}\nabla_{\mathcal{H}{U}}\mathcal{H}{ V}+\mathcal{H}\nabla_{\mathcal{H}{U}}\mathcal{V}{V} \end{equation} for any vector fields ${U}$ and ${V}$ on $M,$ where $\nabla$ is the Levi-Civita connection of $g$. It is easy to see that $\mathcal{T}_{{U}}$ and $\mathcal{A}_{{U}}$ are skew-symmetric operators on the tangent bundle of $M$ reversing the vertical and the horizontal distributions. We now summarize the properties of the tensor fields $\mathcal{T}$ and $\mathcal{A}$. Let $V,W$ be vertical and $X,Y$ be horizontal vector fields on $M$, then we have \begin{equation}\label{testequation111} \mathcal{T}_{V}W=\mathcal{T}_{W}V, \end{equation} \begin{equation}\label{testequation00} \mathcal{A}_{X}Y=-\mathcal{A}_{Y}X=\frac{1}{2}\mathcal{V}[X,Y]. \end{equation} On the other hand, from (\ref{testequationn}) and (\ref{testequationnn}), we obtain \begin{equation}\label{testequation09} \nabla_{V}W=\mathcal{T}_{V}W+\hat{\nabla}_{V}W, \end{equation} \begin{equation}\label{testequation11} \nabla_{V}X=\mathrm{T}_{V}X+\mathcal{H}\nabla_{V}X, \end{equation} \begin{equation}\label{testequation} \nabla_{X}V=\mathcal{A}_{X}V+\mathcal{V}\nabla_{X}V, \end{equation} \begin{equation}\label{testequation123} \nabla_{X}Y=\mathcal{H}\nabla_{X}Y+\mathcal{A}_{X}Y, \end{equation} where $\hat{\nabla}_{V}W=\mathcal{V}\nabla_{V}W$. If $X$ is basic \[\mathcal{H}\nabla_{V}X=\mathcal{A}_{X}V.\] \begin{rem}\label{remark1} In this paper, we will assume all horizontal vector fields as basic vector fields. \end{rem} It is not difficult to observe that $\mathcal{T}$ acts on the fibers as the second fundamental form while $\mathcal{A}$ acts on the horizontal distribution and measures of the obstruction to the integrability of this distribution. For details on Riemannian submersions, we refer to O'Neill's paper \cite{O} and to the book \cite{Fa}. \section{Bi-slant Submersions} A manifold $M$ is called an \textit{almost Hermitian manifold} \cite{Yan} if it admits a tensor field $J$ of type (1,1) on itself such that, for any $X,Y \in TM$ \begin{equation} \label{e9} J^{2}=-I,\quad g(X,Y)=g(JX,JY). \end{equation} An almost Hermitian manifold $M$ is called \textit{Kaehler manifold} \cite{Yan} \\if $\forall X,Y \in TM$, \begin{equation} \label{e10} (\nabla_{X}J)Y=0, \end{equation} where $\nabla$ is the Levi-Civita connection with respect to the Riemannian metric $g$ and $I$ is the identity operator on the tangent bundle $TM$.\\ \begin{definition}\label{dfnbislant} Let $(M,g,J)$ be a Kaehler manifold and $(N,g_{\text{\tiny$N$}})$ be a Riemannian manifold. A Riemannian submersion $\pi : (M,g,J)\rightarrow (N,g_{N})$ is called a \textit{bi-slant submersion}, if there are two slant distributions $\mathcal{D}^{\theta_{1}}\subset ker\pi_{*}$ and $\mathcal{D}^{\theta_{2}}\subset ker\pi_{*}$ such that \begin{equation}\label{eqnbislant1} ker\pi_{*}=\mathcal{D}^{\theta_{1}}\oplus \mathcal{D}^{\theta_{2}}, \end{equation} where, $\mathcal{D}^{\theta_{1}}$ and $\mathcal{D}^{\theta_{2}}$ has slant angles $\theta_{1}$ and $\theta_{2}$, respectively. \end{definition} Suppose the dimension of distribution of $\mathcal{D}^{\theta_{1}}$ (resp. $\mathcal{D}^{\theta_{2}}$) is $m_1$ (resp. $m_2$). Then we easily see the following particular cases. \begin{enumerate} \item[(a)] If $m_1=0$ and $\theta_2=0$, then $\pi$ is an invariant submersion. \item[(b)]If $m_1=0$ and $\theta=\frac{\pi}{2},$ then $\pi$ is an anti-invariant submersion. \item[(c)] If $m_1\neq m_2\neq0,$ $\theta_1=0$ and $\theta_2=\frac{\pi}{2},$ then $\pi$ is a semi-invariant submersion. \item[(d)] If $m_1=0$ and $0<\theta_2<\frac{\pi}{2},$ then $\pi$ is a proper slant submersion. \item[(e)] If $m_1\neq m_2\neq0,$ $\theta_1=0$ and $0<\theta_2<\frac{\pi}{2},$ then $\pi$ is a semi-slant submersion. \item[(e)] If $m_1\neq m_2\neq0,$ $\theta_1=\frac{\pi}{2}$ and $0<\theta_2<\frac{\pi}{2},$ then $\pi$ is a hemi-slant submersion. \end{enumerate} If each slant angles are different from either zero or $\frac{\pi}{2}$, then the bi-slant submersion is called a \textit{proper bi-slant submersion}. Now, we present a non-trivial example of bi-slant submersions and demonstrate that the method presented in this paper is effective. \begin{rem} In present paper, we assume bi-slant submersion as proper bi-slant submersion i.e. slant angles are from either zero or $\frac{\pi}{2}$. \end{rem} \begin{example} Let $\mathbb{R}^{8}$ be $8-dimensional$ Euclidean space. $\mathbb{R}^{8},J,g$ is a Kaehler manifold with Euclidean metric $g$ on $\mathbb{R}^{8}$ and canonical complex structure $J$. Consider the map $\pi : \mathbb{R}^{8} \rightarrow \mathbb{R}^{4}$ with \begin{equation*} \pi(x_{1},x_{2},...x_{8})\mapsto (\frac{-x_{1}+x_{4}}{\sqrt{2}},-x_{2},\frac{-\sqrt{3}x_{5}+x_{8}}{2},-x_{6}). \end{equation*} Then, we have the Jacobian matrix of $\pi$ has rank $4$. That means $\pi$ is a submersion. So, with some calculations we observe that \begin{equation*} ker\pi_{*}=\mathcal{D}^{\theta_{1}}\oplus \mathcal{D}^{\theta_{2}}, \end{equation*} where \begin{equation*} \mathcal{D}^{\theta_{1}}=span \{V_{1}=\frac{1}{\sqrt{2}}(\partial x_{1}+\partial x_{4}), V_{2}=\partial x_{3}\} \end{equation*} and \begin{equation*} \mathcal{D}^{\theta_{2}}=span \{V_{3}=\frac{1}{2}\partial x_{5}+\frac{\sqrt{3}}{2}\partial x_{8}, V_{4}=\partial x_{7}\}. \end{equation*} Moreover, the slant angle of $\mathcal{D}^{\theta_{1}}$ is $\theta_{1}=\frac{\pi}{4}$ and the slant angle of $\mathcal{D}^{\theta_{2}}$ is $\theta_{2}=\frac{\pi}{3}$. \end{example} Let $\pi : (M,g,J)\rightarrow (N,g_{N})$ be a bi-slant submersion from a Kaehlerian manifold $M$ onto a Riemannian manifold $N$. Then, for any $V \in ker\pi_{*}$, we put \begin{equation}\label{decompvervec} JV=PV+FV, \end{equation} where $PV \in ker\pi_{*}$ and $FV \in ker\pi_{*}^{\perp}$. Also, for any $\xi \in ker\pi_{*}^{\perp}$, we put \begin{equation}\label{decomphorvec} J\xi=\phi \xi +\omega \xi, \end{equation} where $\phi \xi \in ker\pi_{*}$ and $\omega \xi \in ker\pi_{*}^{\perp}$. In this case, the horizontal distribution $ker\pi_{*}^{\perp}$ can be decomposed as follows \begin{equation}\label{eqnbislant2} ker\pi_{*}^{\perp}=F\mathcal{D}^{\theta_{1}}\oplus F\mathcal{D}^{\theta_{2}} \oplus \mu, \end{equation} where $\mu$ is the orthogonal complementary of $F\mathcal{D}^{\theta_{1}}\oplus F\mathcal{D}^{\theta_{2}}$ in $ ker\pi_{*}^{\perp}$, and it is invariant with respect to the complex structure $J$. \\ By using \eqref{decompvervec} and \eqref{decomphorvec}, we obtain the followings. \begin{lemma} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we have \begin{equation*} \textbf{(a)}P\mathcal{D}^{\theta_{1}}\subset \mathcal{D}^{\theta_{1}},\quad \textbf{ (b)}P\mathcal{D}^{\theta_{2}}\subset \mathcal{D}^{\theta_{2}},\quad \textbf{(c)}\phi \mu=\{0\},\quad \textbf{(d)}\omega \mu = \mu. \end{equation*} \end{lemma} With the help of \eqref{e9}, \eqref{decompvervec} and \eqref{decomphorvec} we obtain the following Lemma. \begin{lemma}\label{general} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we have \begin{equation*} \textbf{(a)}\, P^{2}X=-\cos^{2}\theta_{1}X, \quad \textbf{(b)}\, P^{2}U=-\cos^{2}\theta_{2}U, \end{equation*} \begin{equation*} \textbf{(c)}\, \phi FX=-\sin ^{2}\theta_{1}X, \quad \textbf{(d)}\, \phi FU=-\sin ^{2}\theta_{2}U, \end{equation*} \begin{equation*} \textbf{(e)}\, P^{2}X+\phi FX=-X, \quad \textbf{(f)}\, P^{2}U+\phi FU=-U, \end{equation*} \begin{equation*} \textbf{(g)}\, FPX+\omega FX=0, \quad \textbf{(h)}\, FPU+\omega FU=0, \end{equation*} for any vector field $X \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$. \end{lemma} We investigate the relation between complex structure $J$ and O'Neill tensors $\mathcal{T}$ and $\mathcal{A}$. \begin{lemma}\label{lemmagenel} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we have \begin{equation}\label{eq1} \phi \mathcal{T}_{X}Y+P\hat{\nabla}_{X}Y=\hat{\nabla}_{X}PY+\mathcal{T}_{X}FY, \end{equation} \begin{equation}\label{eq2} \omega \mathcal{T}_{X}Y+F\hat{\nabla}_{X}Y=\mathcal{T}_{X}PY+\mathcal{A}_{FY}X, \end{equation} \begin{equation}\label{eq3} P\mathcal{T}_{X}\xi+\phi \mathcal{A}_{\xi}X=\hat{\nabla}_{X}\phi \xi+\mathcal{T}_{X}\omega \xi, \end{equation} \begin{equation}\label{eq4} F\mathcal{T}_{X}\xi+\omega \mathcal{A}_{\xi}X=\mathcal{T}_{X}\phi \xi+\mathcal{A}_{\omega \xi}X, \end{equation} \begin{equation}\label{eq5} \phi \mathcal{H}\nabla_{\xi}\eta+P\mathcal{A}_{\xi}\eta=\mathcal{V}\nabla_{\xi}\phi \eta+\mathcal{A}_{\xi}\eta, \end{equation} \begin{equation}\label{eq6} \omega \mathcal{H}\nabla_{\xi}\eta+F\mathcal{A}_{\xi}\eta=\mathcal{A}_{\xi}\phi \eta+\mathcal{H}\nabla_{\xi}\omega \eta, \end{equation} for any $U,V \in ker\pi_{*}$ and $\xi, \eta \in ker\pi_{*}^{\perp}$. \end{lemma} \begin{proof} Let $U$ and $V$ be in $ker\pi_{*}$. Since $M$ is Kaehlerian manifold, we have $J\nabla_{U}V=\nabla_{U}JV$. From \eqref{testequation09}, \eqref{testequation11}, \eqref{decompvervec} and \eqref{decomphorvec}, we obtain \begin{eqnarray*} J\nabla_{U}V&=&\nabla_{U}PV+\nabla_{U}FV\\ \Rightarrow J(\mathcal{T}_{U}V+\hat{\nabla}_{U}V)&=&\mathcal{T}_{U}PV+\hat{\nabla}_{U}PV\\ &+&\mathcal{T}_{U}FV+\mathcal{H}\mathcal{\nabla}_{U}FV. \end{eqnarray*} \begin{eqnarray*} \Rightarrow \phi \mathcal{T}_{U}V+\omega \mathcal{T}_{U}V+P\hat{\nabla}_{U}V+F\hat{\nabla}_{U}V&=& \mathcal{T}_{U}PV+\hat{\nabla}_{U}PV\\ &+&\mathcal{T}_{U}FV+\mathcal{H}\mathcal{\nabla}_{U}FV. \end{eqnarray*} Then, in the view of Remark \ref{remark1}, considering the vertical and horizontal parts of the last equation gives us \eqref{eq1} and \eqref{eq2}. For the rest of the equations, the same way could be applied. \end{proof} Now, we obtain equations which mean Gauss and Weingarten equations for bi-slant submersions. \begin{lemma}\label{GauWei} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, for any $X,Y \in \mathcal{D}^{\theta_{1}}$ and $U,V \in \mathcal{D}^{\theta_{2}}$, we have \begin{equation}\label{GauWei1} g(\nabla_{X}Y,U)=\csc^{2}\theta_{1}\,g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X), \end{equation} \begin{equation}\label{GauWei2} g(\nabla_{U}V,X)=\csc^{2}\theta_{2}\,g(\mathcal{T}_{PX}FV-\mathcal{T}_{X}FPV+\mathcal{A}_{FX}FV,U). \end{equation} \end{lemma} \begin{proof} Assume that $X,Y$ be in $\mathcal{D}^{\theta_{1}}$ and $U,V$ be in $\mathcal{D}^{\theta_{2}}$. Then, from \eqref{e9}, \eqref{e10} and \eqref{decompvervec}, we have \begin{eqnarray*} g(\nabla_{X}Y,U)&=&g(\nabla_{X}JY,JU)\\ &=&g(\nabla_{X}PY,JU)+g(\nabla_{X}FY,JU). \end{eqnarray*} With the help of \eqref{e9} and \eqref{decompvervec}, we obtain \begin{eqnarray*} \Rightarrow g(\nabla_{X}Y,U)&=&-g(\nabla_{X}P^{2}Y,U)-g(\nabla_{X}FPY,U)\\ &+&g(\nabla_{X}FY,PU)+g(\nabla_{X}FY,FY). \end{eqnarray*} By Lemma \ref{general}-(a), Remark \ref{remark1}, \eqref{testequation09} and \eqref{testequation11}, we get \begin{eqnarray*} \Rightarrow g(\nabla_{X}Y,U)&=&\cos^{2}\theta_{1}\,g(\nabla_{X}Y,U)-g(\mathcal{T}_{X}FPY,U)\\ &+&g(\mathcal{T}_{X}FY,PU)+g(\mathcal{A}_{FY},FU). \end{eqnarray*} If we edit the last equation and take into account the properties of O'Neill tensors $\mathcal{T}$ and $\mathcal{A}$, we get \eqref{GauWei1}. To obtain \eqref{GauWei2}, the same idea can be used. \end{proof} \subsection{Integrability} In this section, we investigate the integrability of the distributions which are mentioned in the definition of bi-slant submersion. \begin{theorem} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the slant distribution $\mathcal{D}^{\theta_{1}}$ is integrable if and only if \begin{equation*} g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X)=g(\mathcal{T}_{PU}FX-\mathcal{T}_{U}FPX+\mathcal{A}_{FU}FX,Y), \end{equation*} where $X,Y \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$. \end{theorem} \begin{proof} Let $X,Y \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$. Then, by \eqref{GauWei1}, we get \begin{eqnarray*} g([X,Y],U)&=&g(\nabla_{X}Y,U)-g(\nabla_{Y}X,U)\\ &=&\csc^{2}\theta_{1}\big\{ g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X)\\ &-&g(\mathcal{T}_{PU}FX-\mathcal{T}_{U}FPX+\mathcal{A}_{FU}FX,Y)\big\}. \end{eqnarray*} Therefore, the slant distribution $\mathcal{D}^{\theta_{1}}$ is integrable if and only if $[X,Y] \in \mathcal{D}^{\theta_{1}}$, for any $X,Y \in \mathcal{D}^{\theta_{1}}$. So we obtain the assertion. \end{proof} \begin{theorem} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the slant distribution $\mathcal{D}^{\theta_{2}}$ is integrable if and only if \begin{equation*} g(\mathcal{T}_{PX}FU-\mathcal{T}_{X}FPU+\mathcal{A}_{FX}FU,V)=g(\mathcal{T}_{PX}FV-\mathcal{T}_{X}FPV+\mathcal{A}_{FX}FV,U), \end{equation*} where $X \in \mathcal{D}^{\theta_{1}}$ and $U,V \in \mathcal{D}^{\theta_{2}}$. \end{theorem} \begin{proof} Let $X \in \mathcal{D}^{\theta_{1}}$ and $U,V \in \mathcal{D}^{\theta_{2}}$. Then, from \eqref{GauWei2}, we get \begin{eqnarray*} g([U,V],X)&=&g(\nabla_{U}V,X)-g(\nabla_{V}U,X)\\ &=&\csc^{2}\theta_{2}\big\{ g(\mathcal{T}_{PX}FV-\mathcal{T}_{X}FPV+\mathcal{A}_{FX}FV,U)\\ &-&g(\mathcal{T}_{PX}FU-\mathcal{T}_{X}FPU+\mathcal{A}_{FX}FU,V)\big\}. \end{eqnarray*} So, the assertion is obtained. \end{proof} \subsection{Totally and Mixed Geodesicness} In this section, we investigate the geometry of the fibers, vertical distribution and horizontal distribution for a bi-slant submersion. \begin{theorem}\label{GEOD1} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the slant distribution $\mathcal{D}^{\theta_{1}}$ defines a totally geodesic foliation on $ker\pi_{*}$ if and only if the following condition holds; \begin{equation}\label{geod1} g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X)=0, \end{equation} where $X,Y \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$. \end{theorem} \begin{proof} Let $X,Y \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$. From \eqref{testequation09} and \eqref{GauWei1}, we have \begin{eqnarray*} g(\hat{\nabla}_{X}Y,U)&=&g(\nabla_{X}Y,U)\\ &=&\csc^{2}\theta_{1}\,g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X). \end{eqnarray*} So, the slant distribution $\mathcal{D}^{\theta_{1}}$ defines a totally geodesic foliation on $ker\pi_{*}$ if and only if $\hat{\nabla}_{X}Y \in \mathcal{D}^{\theta_{1}}$ i.e. $g(\mathcal{T}_{PU}FY-\mathcal{T}_{U}FPY+\mathcal{A}_{FU}FY,X)$. \end{proof} \begin{theorem}\label{GEOD2} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the slant distribution $\mathcal{D}^{\theta_{2}}$ defines a totally geodesic foliation on $ker\pi_{*}$ if and only if the following condition holds; \begin{equation}\label{geod2} g(\mathcal{T}_{PX}FV-\mathcal{T}_{X}FPV+\mathcal{A}_{FX}FV,U)=0, \end{equation} where $X \in \mathcal{D}^{\theta_{1}}$ and $U,V \in \mathcal{D}^{\theta_{2}}$. \end{theorem} \begin{proof} Let $X$ be in $\mathcal{D}^{\theta_{1}}$ and $U$ and $V$ be in $\mathcal{D}^{\theta_{2}}$. Thus, with the help of \eqref{testequation09} and \eqref{GauWei2}, we obtain \begin{eqnarray*} g(\hat{\nabla}_{U}V,X)&=&g(\nabla_{U}V,X)\\ &=&\csc^{2}\theta_{2}\,g(\mathcal{T}_{PX}FV-\mathcal{T}_{X}FPV+\mathcal{A}_{FX}FV,U). \end{eqnarray*} Therefore, we obtain the assertion. \end{proof} In the view of Theorem \ref{GEOD1} and Theorem \ref{GEOD2}, we have the following result. \begin{corollary} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the vertical distribution $ker\pi_{*}$ is a locally product $M_{\mathcal{D}^{\theta_{1}}}\times M_{\mathcal{D}^{\theta_{2}}}$ if and only if \eqref{geod1} and \eqref{geod2} hold, where $M_{\mathcal{D}^{\theta_{1}}}$ and $M_{\mathcal{D}^{\theta_{2}}}$ are integral manifolds of the distributions $\mathcal{D}^{\theta_{1}}$ and $\mathcal{D}^{\theta_{2}}$, respectively. \end{corollary} \begin{theorem}\label{VERGEODESIC} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, $ker\pi_{*}$ defines a totally geodesic foliation if and only if \begin{equation}\label{vergeodesic} \omega (\mathcal{T}_{W}PZ+\mathcal{A}_{FZ}W)+F(\hat{\nabla}_{W}PZ+\mathcal{T}_{W}FZ)=0,\\ \end{equation} where $W,Z \in ker\pi_{*}$. \end{theorem} \begin{proof} Let $W$ and $Z$ be in $ker\pi_{*}$. Then, from \eqref{testequation09}, \eqref{testequation11}, \eqref{e9}, \eqref{decompvervec} and \eqref{decomphorvec}, we obtain \begin{eqnarray*} \nabla_{W}Z&=&-J\nabla_{W}JZ=-J(\nabla_{W}PZ+\nabla_{W}FZ)\\ &=&-J(\mathcal{T}_{W}PZ+\hat{\nabla}_{W}PZ+\mathcal{T}_{W}FZ+\mathcal{A}_{FZ}W)\\ &=&-\phi \mathcal{T}_{W}PZ-\omega \mathcal{T}_{W}PZ-P\hat{\nabla}_{W}PZ-F\hat{\nabla}_{W}PZ\\ & &-P\mathcal{T}_{W}FZ+F\mathcal{T}_{W}FZ-\phi \mathcal{A}_{FZ}W-\omega \mathcal{A}_{FZ}W. \end{eqnarray*} Thus, it is known that $ker\pi_{*}$ defines a totally geodesic foliation if and only if $\nabla_{W}Z \in ker\pi_{*}$. So, we get the assertion. \end{proof} \begin{theorem}\label{HORGEODESIC} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, $ker\pi_{*}^{\perp}$ defines a totally geodesic foliation if and only if \begin{equation}\label{horgeodesic} \phi (\mathcal{A}_{\xi}\phi \eta +\mathcal{H}\nabla_{\xi}\omega \eta)+P(\mathcal{A}_{\xi}\omega \eta+\mathcal{V}\nabla_{\xi}\phi \eta)=0 \end{equation} for any $\xi , \eta \in ker\pi_{*}^{\perp}$. \end{theorem} \begin{proof} Let $\xi , \eta \in ker\pi_{*}^{\perp}$. With the help of the equations \eqref{testequation}, \eqref{testequation111}, \eqref{e9}, \eqref{decompvervec} and \eqref{decomphorvec}, we get \begin{eqnarray*} \nabla_{\xi}\eta&=&-J\nabla_{\xi}J\eta=-J(\nabla_{\xi}\phi \eta+\nabla_{\xi}\omega \eta)\\ &=&-J(\mathcal{A}_{\xi} \phi \eta +\mathcal{V}\nabla_{\xi}\phi \eta +\mathcal{H}\nabla_{\xi}\omega \eta+\mathcal{A}_{\xi}\omega \eta)\\ &=& -\phi \mathcal{A}_{\xi} \phi \eta - \omega \mathcal{A}_{\xi} \phi \eta - P\mathcal{V}\nabla_{\xi}\phi \eta -F\mathcal{V}\nabla_{\xi}\phi \eta\\ &-& \phi \mathcal{H}\nabla_{\xi}\omega \eta - \omega \mathcal{H}\nabla_{\xi}\omega \eta - P \mathcal{A}_{\xi}\omega \eta- F \mathcal{A}_{\xi}\omega \eta. \end{eqnarray*} Therefore, from the last equation, $ker\pi_{*}^{\perp}$ defines a totally geodesic foliation if and only if $\phi (\mathcal{A}_{\xi}\phi \eta +\mathcal{H}\nabla_{\xi}\omega \eta)+P(\mathcal{A}_{\xi}\omega \eta+\mathcal{V}\nabla_{\xi}\phi \eta)=0$. \end{proof} In the view of Theorem \ref{VERGEODESIC} and Theorem \ref{HORGEODESIC}, we give the following result. \begin{corollary} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, the following three facts are equal to each other: \begin{eqnarray*} &\textbf{(i)}& \text{M is a locally product } M_{ker\pi_{*}} \times M_{ker\pi_{*}^{\perp}}, \\ &\textbf{(ii)}& \pi \text{ is a totally geodesic map}, \\ &\textbf{(iii)}& \eqref{vergeodesic} \text{ and } \eqref{horgeodesic} \text{ hold}, \\ \end{eqnarray*} where $M_{ker\pi_{*}}$ and $M_{ker\pi_{*}^{\perp}}$ are integral manifolds of distributions $ker\pi_{*}$ and $ker\pi_{*}^{*}$, respectively. \end{corollary} \subsection{Parallelism of Canonical Structures} In this section, we investigate the parallelism of the canonical structures for a bi-slant submersion.\\ Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we define \begin{eqnarray} (\nabla_{W}P)Z&=&\hat{\nabla}_{W}PZ-P\hat{\nabla}_{W}Z, \label{pparallel}\\ (\nabla_{W}F)Z&=&\mathcal{H}\nabla_{W}FZ-F\hat{\nabla}_{W}Z,\label{fparallel}\\ (\nabla_{W}\phi)\xi &=&\hat{\nabla}_{W}\phi \xi-\phi \mathcal{H}\nabla_{W}\xi, \label{fiparallel}\\ (\nabla_{W}\omega)\xi &=&\mathcal{H}\nabla_{W}\omega \xi - \omega \mathcal{H}\nabla_{W}\xi, \label{omegaparallel} \end{eqnarray} where $W,Z \in ker\pi_{*}$ and $\xi \in ker\pi_{*}^{\perp}$. Then, it is said that \begin{itemize} \item $P$ is \textit{parallel} $\Leftrightarrow$ $\nabla P\equiv 0$, \item $F$ is \textit{parallel} $\Leftrightarrow$ $\nabla F\equiv 0$, \item $\phi$ is \textit{parallel} $\Leftrightarrow$ $\nabla \phi \equiv 0$, \item $\omega$ is \textit{parallel} $\Leftrightarrow$ $\nabla \omega \equiv 0$. \end{itemize} In the view of Lemma \ref{lemmagenel} and \eqref{pparallel}$\sim$\eqref{omegaparallel}, we have the following lemma. \begin{lemma}\label{paralleldefn} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, for any $W,Z \in ker\pi_{*}$ and $\xi \in ker\pi_{*}^{\perp}$, we get \begin{eqnarray} (\nabla_{W}P)Z&=&\phi \mathcal{T}_{W}Z-\mathcal{T}_{W}FZ,\label{pparallel2}\\ (\nabla_{W}F)Z&=&\omega \mathcal{T}_{W}Z-\mathcal{T}_{W}PZ, \label{fparallel2}\\ (\nabla_{W}\phi)\xi &=&P\mathcal{T}_{W}\xi - \mathcal{T}_{W}\omega \xi,\label{fiparallel2}\\ (\nabla_{W}\omega)\xi &=& F \mathcal{T}_{W}\xi - \mathcal{T}_{W}\phi \xi\label{omegaparallel2}. \end{eqnarray} \end{lemma} \begin{theorem} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, $F$ is parallel if and only if $\phi$ is parallel. \end{theorem} \begin{proof} Let $F$ be parallel. Then, for any $W,Z \in ker\pi_{*}$, from \eqref{fiparallel2} we have $\omega \mathcal{T}_{W}Z=\mathcal{T}_{W}PZ$. By using \eqref{e9}, \eqref{decompvervec} and fundamental properties of O'Neill tensor $\mathcal{T}$, we get \begin{eqnarray*} g(P\mathcal{T}_{W}\xi,Z)&=&g(J\mathcal{T}_{W}\xi,Z)=-g(\mathcal{T}_{W}\xi,JZ)\\ &=&-g(\mathcal{T}_{W}\xi,PZ)=g(\mathcal{T}_{W}PZ, \xi). \end{eqnarray*} In the view of the fact of parallelism of $F$, we obtain \begin{eqnarray*} g(P\mathcal{T}_{W}\xi,Z)&=&g(\mathcal{T}_{W}PZ, \xi)=g(\omega \mathcal{T}_{W}Z, \xi)\\ &=&g(J\mathcal{T}_{W}Z, \xi)=-g(\mathcal{T}_{W}Z, \omega \xi)=g(\mathcal{T}_{W} \omega \xi,Z). \end{eqnarray*} So, we have for any $Z \in ker\pi_{*}$ $g(P\mathcal{T}_{W}\xi,Z)=g(\mathcal{T}_{W} \omega \xi,Z)$ i.e. $\phi$ is parallel. \end{proof} It is said that the fiber is \textit{$\mathcal{D}^{\theta_{1}}\!-\!\mathcal{D}^{\theta_{2}}$-mixed geodesic}, for any two distributions $\mathcal{D}^{\theta_{1}}$ and $\mathcal{D}^{\theta_{2}}$ defined on the fiber of a Riemannian submersion, if for any $X \in \mathcal{D}^{\theta_{1}}$ and $U \in \mathcal{D}^{\theta_{2}}$, $\mathcal{T}_{X}U=0$. \begin{theorem} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$ with parallel canonical structure $F$. Then, the fibers are $\mathcal{D}^{\theta_{1}}\!-\!\mathcal{D}^{\theta_{2}}$-mixed geodesic. \end{theorem} \begin{proof} Let $X$ be in $\mathcal{D}^{\theta_{1}}$ and $U$ in $\mathcal{D}^{\theta_{2}}$. Then, from Lemma \ref{general}-(b) and \eqref{fparallel2}, we obtain \begin{equation*} \omega^{2}\mathcal{T}_{X}U=\omega(\omega \mathcal{T}_{X}U)=\omega \mathcal{T}_{X}PU=\mathcal{T}_{X}P^{2}U=-\cos^{2}\theta_{2}\mathcal{T}_{X}U. \end{equation*} On the other hand, from Lemma \ref{general}-(a) and \eqref{fparallel2}, we get \begin{equation*} \omega^{2}\mathcal{T}_{X}U=\omega^{2}\mathcal{T}_{U}X=\omega(\mathcal{T}_{U}PX)=\mathcal{T}_{U}P^{2}X=-\cos^{2}\theta_{1}\mathcal{T}_{U}X. \end{equation*} Therefore, we obtain \begin{equation*} -\cos^{2}\theta_{2}\mathcal{T}_{X}U=-\cos^{2}\theta_{1}\mathcal{T}_{X}U. \end{equation*} Since $\cos^{2}\theta_{2}\mathcal{T}_{X}U=\cos^{2}\theta_{1}\mathcal{T}_{X}U$, we have $\mathcal{T}_{X}U=$. That implies the fibers are $\mathcal{D}^{\theta_{1}}\!-\!\mathcal{D}^{\theta_{2}}$-mixed geodesic. \end{proof} \section{Curvature Relations} In this section, the sectional curvatures of the total space, base space and the fibers of a bi-slant submersion are investigated.\\ Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. We denote the Riemannian curvature tensors of $M$, $N$ and any fiber of the submersion with $R$, $R^{*}$ and $\hat{R}$, respectively. Also, we denote the sectional curvatures of $M$, $N$ and any fiber of the submersion with $K$, $K^{*}$ and $\hat{K}$, respectively. It is known that the sectional curvature for a Riemannian submersion is defined, for any pair of non-zero orthogonal vectors $U$ and $V$ \cite{O} \begin{equation}\label{sectionalcurvature} K(U,V)= \frac{R(U,V,V,U)}{g(U,U)g(V,V)}. \end{equation} For any $e_{1},e_{2}\in ker\pi_{*}$ and $E_{1}, E_{2} \in ker\pi^{\perp}_{*} $ the Riemannian curvature tensor $R$ is given by \cite{O} \begin{eqnarray} R(e_{1},e_{2},e_{3},e_{4})&=&\hat{R}(e_{1},e_{2},e_{3},e_{4})-g(\mathcal{T}_{e_{1}}e_{4},\mathcal{T}_{e_{2}}e_{3})\nonumber \\ & &+g(\mathcal{T}_{e_{2}}e_{4},\mathcal{T}_{e_{1}}e_{3}), \label{r1} \end{eqnarray} \begin{eqnarray} R(e_{1},e_{2},e_{3},E_{1})&=&g((\nabla_{e_{1}}\mathcal{T})(e_{2},e_{3}),E_{1})-g((\nabla_{e_{2}}\mathcal{T})(e_{1},e_{3}),E_{1}),\label{r2} \end{eqnarray} \begin{eqnarray} R(E_{1},E_{2},E_{3},e_{1})&=&-g((\nabla_{E_{3}}\mathcal{A})(E_{1},E_{2}),e_{1})-g(\mathcal{A}_{E_{1}}E_{2},\mathcal{T}_{e_{1}}E_{3})\nonumber \\ & &g(\mathcal{A}_{E_{2}}E_{3},\mathcal{T}_{e_{1}}E_{1})+g(\mathcal{A}_{E_{3}}E_{1},\mathcal{T}_{e_{1}}E_{2}),\label{r3} \end{eqnarray} \begin{eqnarray} R(E_{1},E_{2},E_{3},E_{4})&=&R^{*}(E_{1},E_{2},E_{3},E_{4})+2g(\mathcal{A}_{E_{1}}E_{2},\mathcal{A}_{E_{3}}E_{4})\nonumber \\ & &-g(\mathcal{A}_{E_{2}}E_{3},\mathcal{A}_{E_{1}}E_{4})+g(\mathcal{A}_{E_{1}}E_{3},\mathcal{A}_{E_{2}}E_{4}),\label{r4} \end{eqnarray} \begin{eqnarray} R(E_{1},E_{2},e_{1},e_{2})&=&-g((\nabla_{e_{1}}\mathcal{A})(E_{1},E_{2}),e_{2})+g((\nabla_{e_{2}}\mathcal{A})(E_{1},E_{2}),e_{1})\nonumber \\ & &-g(\mathcal{A}_{E_{1}}e_{1},\mathcal{A}_{E_{2}}e_{2})+g(\mathcal{A}_{E_{1}}e_{2},\mathcal{A}_{E_{2}}e_{1})\nonumber \\ & &+g(\mathcal{T}_{e_{1}}E_{1},\mathcal{T}_{e_{2}}E_{2})-g(\mathcal{T}_{e_{2}}E_{1},\mathcal{T}_{e_{1}}E_{2}),\label{r5} \end{eqnarray} \begin{eqnarray} R(E_{1},e_{1},E_{2},e_{2})&=&-g((\nabla_{E_{1}}\mathcal{T})(e_{1},e_{2}),E_{2})-g((\nabla_{e_{1}}\mathcal{A})(E_{1},E_{2}),e_{2})\nonumber \\ & & g(\mathcal{T}_{e_{1}}E_{1},\mathcal{T}_{e_{2}}E_{2})-g(\mathcal{A}_{E_{1}}e_{1},\mathcal{A}_{E_{2}}e_{2}),\label{r6} \end{eqnarray} where $R$, $R^{*}$ and $\hat{R}$ is Riemannian curvature of $M$, $N$ and fiber, respectively.\\ Furthermore, let $\pi$ be submersion from a Riemannian manifold $M$ onto a Riemannian manifold $N$. Then, the followings are given \cite{O}: \begin{eqnarray} K(e_{1},e_{2})=\hat{K}(e_{1},e_{2})-g(\mathcal{T}_{e_{1}}e_{1},\mathcal{T}_{e_{2}}e_{2})+\|\mathcal{T}_{e_{1}}e_{2}\|^{2},\label{curv1} \end{eqnarray} \begin{eqnarray} K(E_{1},e_{1})=g((\nabla_{E_{1}}\mathcal{T})(e_{1},e_{1}),E_{1})+\|\mathcal{A}_{E_{1}}e_{1}\|^{2}-\|\mathcal{T}_{e_{1}}E_{1}\|^{2},\label{curv2} \end{eqnarray} \begin{eqnarray} K(E_{1},E_{2})=K^{*}(E_{1},E_{2})-3\|\mathcal{A}_{E_{1}}E_{2}\|^{2},\label{curv3} \end{eqnarray} where $e_{1},e_{2}\in ker\pi_{*}$ and $E_{1}, E_{2} \in ker\pi^{\perp}_{*} $ orthonormal vector fields. \begin{theorem} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we obtain \begin{eqnarray}\label{sect1} K(e_{1},e_{2})&=&\hat{K}(Pe_{1},Pe_{2})\|Pe_{1}\|^{-2}\|Pe_{2}\|^{-2}+K^{*}(F{e_1},Fe_{2})\|Fe_{1}\|^{-2}\|Fe_{2}\|^{-2}\nonumber\\ & &-g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{Pe_{2}}Pe_{2})+\|\mathcal{A}_{Fe_{1}}Pe_{2}\|^{2}\nonumber\\ & &+g((\nabla_{Fe_{2}}\mathcal{T})(Pe_{1},Pe_{2}),Fe_{2})-\|\mathcal{T}_{Pe_{1}}Fe_{2}\|^{2}\nonumber\\ & &-3\|\mathcal{A}_{Fe_{1}}Fe_{2}\|^{2}+\|\mathcal{T}_{Pe_{2}}Pe_{1}\|^{2}, \end{eqnarray} \begin{eqnarray}\label{sect2} K(e_{1},E_{1})&=&\hat{K}(Pe_{1},\phi E_{1})\|Pe_{1}\|^{-2}\|\phi E_{1}\|^{-2}+K^{*}(Fe_{1},\omega E_{1})\|Fe_{1}\|^{-2}\|\omega E_{1}\|^{-2}\nonumber\\ & &-\|\mathcal{T}_{\phi E_{1}}Pe_{1}\|^{2}-\|\mathcal{T}_{P e_{1}}\omega E_{1}\|^{2}-3\|\mathcal{A}_{Fe_{1}}\omega E_{1}\|^{2} \nonumber \\ & &+ \|\mathcal{A}_{\omega E_{1}}Pe_{1}\|^{2}-\|\mathcal{T}_{\phi E_{1}}Fe_{1}\|^{2}-g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{\phi E_{1}}\phi E_{1}) \nonumber \\ & & +\|\mathcal{A}_{Fe_{1}}\phi E_{1}\|^{2}+g((\nabla_{\omega E_{1}}\mathcal{T})(Pe_{1},Pe_{1}),\omega E_{1}) \nonumber \\ & & +g((\nabla_{Fe_{1}}\mathcal{T})(\phi E_{1},\phi E_{1}),Fe_{1}), \end{eqnarray} \begin{eqnarray}\label{sect3} K(E_{1},E_{2})&=&\hat{K}(\phi E_{1},\phi E_{2})\|\phi E_{1}\|^{-2}\|\phi E_{2}\|^{-2}+K^{*}(\omega E_{1},\omega E_{2})\|\omega E_{1}\|^{-2}\|\omega E_{2}\|^{-2}\nonumber \\ & &+\|\mathcal{T}_{\phi E_{2}}\phi E_{1}\|^{2}-g(\mathcal{T}_{\phi E_{1}}\phi E_{1},\mathcal{T}_{\phi E_{2}}\phi E_{2}) \nonumber \\ & & +g((\nabla_{\omega E_{2}}\mathcal{T})(\phi E_{1},\phi E_{2}),\omega E_{2})-\|\mathcal{T}_{\phi E_{1}}\omega E_{2}\|^{2}\nonumber \\ & & +\|\mathcal{A}_{\omega E_{2}}\phi E_{1}\|^{2}+g((\nabla_{\omega E_{1}}\mathcal{T})(\phi E_{2},\phi E_{2}),\omega E_{1})\nonumber \\ & & -\|\mathcal{T}_{\phi E_{2}}\omega E_{1}\|^{2}+\|\mathcal{A}_{\omega E_{1}}\phi E_{2}\|^{2}-3\|\mathcal{A}_{\omega E_{1}}\omega E_{2}\|^{2}. \end{eqnarray} \end{theorem} \begin{proof} Let $e_{1},e_{2}\in ker\pi_{*}$ and $E_{1}, E_{2} \in ker\pi^{\perp}_{*} $ be orthonormal vector fields. Then, by the fact that $K(e_{1},e_{2})=K(Je_{1},Je_{2})$, \eqref{decompvervec} and \eqref{decomphorvec}, we get \begin{eqnarray*} K(e_{1},e_{2})=K(Je_{1},Je_{2})&=&K(Pe_{1},Pe_{2})+K(Pe_{1},Fe_{2})\\ & &+K(Fe_{1},Pe_{2})+K(Fe_{1},Fe_{2}). \end{eqnarray*} By the definition of the sectional curvature, we obtain \begin{eqnarray*} \Rightarrow K(e_{1},e_{2})&=& R(Pe_{1},Pe_{2},Pe_{2},Pe_{1})+R(Pe_{1},Fe_{2},Fe_{2},Pe_{1})\\ & &R(Fe_{1},Pe_{2},Pe_{2},Fe_{1})+R(Fe_{1},Fe_{2},Fe_{2},Fe_{1}). \end{eqnarray*} Thus, with the help of \eqref{r1}$\sim$\eqref{r6}, we have \begin{eqnarray*} \Rightarrow K(e_{1},e_{2})&=&\hat{R}(Pe_{1},Pe_{2},Pe_{2},Pe_{1})-g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{Pe_{2}}Pe_{2})+\|\mathcal{T}_{Pe_{1}}Pe_{2}\|^{2}\\ & & +g((\nabla_{Fe_{2}}\mathcal{T})(Pe_{1},Pe_{1}),Fe_{2})-\|\mathcal{T}_{Pe_{1}}Fe_{2}\|^{2}+\|\mathcal{A}_{Fe_{2}}Pe_{1}\|^{2}\\ & &+g((\nabla_{Fe_{1}}\mathcal{T})(Pe_{2},Pe_{2}),Fe_{1})-\|\mathcal{T}_{Pe_{2}}Fe_{1}\|^{2}+\|\mathcal{A}_{Fe_{1}}Pe_{2}\|^{2}\\ & & +R^{*}(Pe_{1},Pe_{2},Pe_{2},Pe_{1})-3\|\mathcal{A}_{Fe_{1}}Fe_{2}\|^{2}. \end{eqnarray*} Since, \begin{equation*} \hat{R}(Pe_{1},Pe_{2},Pe_{2},Pe_{1})=\hat{K}(Pe_{1},Pe_{2})\|Pe_{1}\|^{-2}\|Pe_{2}\|^{-2} \end{equation*} \[and\] \begin{equation*} R^{*}(Pe_{1},Pe_{2},Pe_{2},Pe_{1})=K^{*}(F{e_1},Fe_{2})\|Fe_{1}\|^{-2}\|Fe_{2}\|^{-2} \end{equation*} \eqref{sect1} is obtained. \eqref{sect2} and \eqref{sect3} can be obtained with a similar way. \end{proof} Now, we give some inequalities for sectional curvatures of total manifold, base manifold and fibers. \begin{corollary} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we have\\ \[ \begin{array}{ccc} \hat{K}(Pe_{1},Pe_{2})\|Pe_{1}\|^{-2}\|Pe_{2}\|^{-2} & &g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{Pe_{2}}Pe_{2}) \\ +K^{*}(F{e_1},Fe_{2})\|Fe_{1}\|^{-2}\|Fe_{2}\|^{-2}& \leq & +\|\mathcal{T}_{Pe_{1}}Fe_{2}\|^{2}, \\ -\hat{K}(e_{1},e_{2}) & & \\ \end{array} \] \end{corollary} \begin{proof} Let $e_{1},e_{2}\in ker\pi_{*}$ be orthonormal vector fields. Then, by \eqref{curv1} and \eqref{sect1}, we get \begin{eqnarray*} \hat{K}(e_{1},e_{2})-g(\mathcal{T}_{e_{1}}e_{1},\mathcal{T}_{e_{2}}e_{2})+\|\mathcal{T}_{e_{1}}e_{2}\|^{2}&=&\hat{K}(Pe_{1},Pe_{2})\|Pe_{1}\|^{-2}\|Pe_{2}\|^{-2}\nonumber \\ & &+K^{*}(F{e_1},Fe_{2})\|Fe_{1}\|^{-2}\|Fe_{2}\|^{-2} \nonumber \\ & & -g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{Pe_{2}}Pe_{2})\nonumber \\ & &+\|\mathcal{A}_{Fe_{1}}Pe_{2}\|^{2} -\|\mathcal{T}_{Pe_{1}}Fe_{2}\|^{2}\nonumber \\ & &+g((\nabla_{Fe_{2}}\mathcal{T})(Pe_{1},Pe_{2}),Fe_{2})\nonumber \\ & & -3\|\mathcal{A}_{Fe_{1}}Fe_{2}\|^{2}+\|\mathcal{T}_{Pe_{2}}Pe_{1}\|^{2}. \end{eqnarray*} Thus, we obtain the assertion. \end{proof} \begin{corollary} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, \[ \begin{array}{ccc} \hat{K}(Pe_{1},\phi E_{1}) & & g((\nabla_{E_{1}}\mathcal{T})(e_{1},e_{1}),E_{1})+\|\mathcal{A}_{E_{1}}e_{1}\|^{2}\\ +K^{*}(Fe_{1},\omega E_{1}) & \leq &+\|\mathcal{T}_{P e_{1}}\omega E_{1}\|^{2}+\|\mathcal{T}_{\phi E_{1}}Pe_{1}\|^{2} \\ & & +3\|\mathcal{A}_{Fe_{1}}\omega E_{1}\|^{2}+g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{\phi E_{1}}\phi E_{1}), \\ \end{array} \] where $e_{1}\in ker\pi_{*}$ and $E_{1} \in ker\pi^{\perp}_{*} $ orthonormal vector fields. \end{corollary} \begin{proof} Let $e_{1}\in ker\pi_{*}$ and $E_{1} \in ker\pi^{\perp}_{*} $ be orthonormal vector fields. Then, by \eqref{curv2} and \eqref{sect2}, we have \begin{eqnarray*} g((\nabla_{E_{1}}\mathcal{T})(e_{1},e_{1}),E_{1})+\|\mathcal{A}_{E_{1}}e_{1}\|^{2}&=&\hat{K}(Pe_{1},\phi E_{1})\|Pe_{1}\|^{-2}\|\phi E_{1}\|^{-2}\nonumber \\ & & +K^{*}(Fe_{1},\omega E_{1})\|Fe_{1}\|^{-2}\|\omega E_{1}\|^{-2} \nonumber \\ & & -\|\mathcal{T}_{\phi E_{1}}Pe_{1}\|^{2}-\|\mathcal{T}_{P e_{1}}\omega E_{1}\|^{2}\nonumber \\ & &-3\|\mathcal{A}_{Fe_{1}}\omega E_{1}\|^{2}+\|\mathcal{A}_{\omega E_{1}}Pe_{1}\|^{2} \nonumber \\ & &-\|\mathcal{T}_{\phi E_{1}}Fe_{1}\|^{2}-g(\mathcal{T}_{Pe_{1}}Pe_{1},\mathcal{T}_{\phi E_{1}}\phi E_{1})\nonumber \\ & &+\|\mathcal{A}_{Fe_{1}}\phi E_{1}\|^{2}+\|\mathcal{T}_{e_{1}}E_{1}\|^{2}\nonumber \\ & &+g((\nabla_{\omega E_{1}}\mathcal{T})(Pe_{1},Pe_{1}),\omega E_{1}) \nonumber \\ & &+g((\nabla_{Fe_{1}}\mathcal{T})(\phi E_{1},\phi E_{1}),Fe_{1}). \end{eqnarray*} Therefore, the assertion is obtained. \end{proof} \begin{corollary} Let $\pi$ be a bi-slant submersion from a Kaehlerian manifold $(M,g,J)$ onto a Riemannian manifold $(N,g_{N})$. Then, we obtain \[ \begin{array}{ccc} \hat{K}(\phi E_{1},\phi E_{2})\|\phi E_{1}\|^{-2}\|\phi E_{2}\|^{-2} & & g(\mathcal{T}_{\phi E_{1}}\phi E_{1},\mathcal{T}_{\phi E_{2}}\phi E_{2})+\|\mathcal{T}_{\phi E_{1}}\omega E_{2}\|^{2} \\ +K^{*}(\omega E_{1},\omega E_{2})\|\omega E_{1}\|^{-2}\|\omega E_{2}\|^{-2} & \leq & + \|\mathcal{T}_{\phi E_{2}}\omega E_{1}\|^{2}+3\|\mathcal{A}_{\omega E_{1}}\omega E_{2}\|^{2} \\ -K^{*}(E_{1},E_{2}) & & \\ \end{array} \] \end{corollary} \begin{proof} Let $E_{1}, E_{2} \in ker\pi^{\perp}_{*} $ be orthonormal vector fields. From \eqref{curv3} and \eqref{sect3}, we get \begin{eqnarray*} K^{*}(E_{1},E_{2})-3\|\mathcal{A}_{E_{1}}E_{2}\|^{2}&=&\hat{K}(\phi E_{1},\phi E_{2})\|\phi E_{1}\|^{-2}\|\phi E_{2}\|^{-2}\nonumber \\ & &+K^{*}(\omega E_{1},\omega E_{2})\|\omega E_{1}\|^{-2}\|\omega E_{2}\|^{-2}\nonumber \\ & &+\|\mathcal{T}_{\phi E_{2}}\phi E_{1}\|^{2}-g(\mathcal{T}_{\phi E_{1}}\phi E_{1},\mathcal{T}_{\phi E_{2}}\phi E_{2})\nonumber \\ & &+g((\nabla_{\omega E_{2}}\mathcal{T})(\phi E_{1},\phi E_{2}),\omega E_{2})-\|\mathcal{T}_{\phi E_{1}}\omega E_{2}\|^{2}\nonumber \\ & &+\|\mathcal{A}_{\omega E_{2}}\phi E_{1}\|^{2}+g((\nabla_{\omega E_{1}}\mathcal{T})(\phi E_{2},\phi E_{2}),\omega E_{1}) \nonumber \\ & &-\|\mathcal{T}_{\phi E_{2}}\omega E_{1}\|^{2}+\|\mathcal{A}_{\omega E_{1}}\phi E_{2}\|^{2}-3\|\mathcal{A}_{\omega E_{1}}\omega E_{2}\|^{2}. \end{eqnarray*} Hence, the assertion is obtained. \end{proof} \end{document}

\begin{document} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \title{Comment on ``On Visibility in the Afshar Two-Slit Experiment"} \author{Tabish Qureshi} \institute{Department of Physics, Jamia Millia Islamia\\ New Delhi-110025, India.\\ \email{[email protected]}} \maketitle \begin{abstract} Recently Kastner has analyzed the issue of visibility in a modified two-slit experiment carried out by Afshar et al, which has been a subject of much debate. Kastner describes a thought experiment which is claimed to show interference with hundred percent visibility and also an ``apparent" which-slit information. We argue that this thought experiment does not show interference at all, and is thus not applicable to the Afshar experiment. \keywords{Complementarity \and Two-slit experiment \and Wave-particle duality} \PACS{PACS 03.65.Ud ; 03.65.Ta} \end{abstract} An experiment which claims to violate Bohr's complementarity principle, proposed and carried out by Afshar et al \cite{afsharfp}, is a subject of current debate. Basically, it consists of a standard two-slit experiment, with a converging lens behind the conventional screen for obtaining the interference pattern. Although If the screen is removed, the light passes through the lens and produces two images of the slits, which are captured on two detectors $D_A$ and $D_B$ respectively. Opening only slit $A$ results in only detector $D_A$ clicking, and opening only slit $B$ leads to only $D_B$ clicking. Afshar argues that the detectors $D_A$ and $D_B$ yield information about which slit, $A$ or $B$, the particle initially passed through. If one places a screen before the lens, the interference pattern is visible. Conventionally, if one tries to observe the interference pattern, one cannot get the which-way information. Afshar has a clever scheme for establishing the existence of the interference pattern without actually observing it. First the exact location of the dark fringes are noted by observing the interference pattern. Then, thin wires are placed in the exact locations of the dark fringes. The argument is that if the interference pattern exists, sliding in wires through the dark fringes will not affect the intensity of light on the two detectors. If the interference pattern is not there, some photons are bound to hit the wires, and get scattered, thus reducing the photon count at the two detectors. This way, the existence of the interference pattern can be established without actually disturbing the photons in any way. Afshar et al carried out the experiment and found that sliding in wires in the expected locations of the dark fringes, doesn't lead to any significant reduction of intensity at the detectors. Hence they claim that they have demonstrated a violation of complementarity. Recently, Kastner has addressed the issue of interference visibility in the Afshar experiment \cite{kastner09}. Kastner believes that the essence of the Afshar experiment is captured by a thought experiment discussed by Srikanth \cite{srikanth} in the context of complementarity. Kastner analyzed this two-slit experiment in which there is an additional internal degree of freedom of the detector elements (which can be considered a “vibrational” component). The particle + detector state evolves from the slits to the final screen with initial detector state $|0\rangle$. The detector spatial basis states $|\phi_x\rangle$ and vibrational basis states $|v_U\rangle$ and $|v_L\rangle$ (corresponding to the particle passing through the upper and lower slit, respectively) are activated. This evolution, from the initial state to the detected particle, is given by \begin{equation} {1\over \sqrt{2}}(|U\rangle+|L\rangle)|0\rangle \rightarrow \sum_x |x\rangle \left\{a_x |\phi_x\rangle|v_U\rangle + b_x |\phi_x\rangle|v_L\rangle\right\}, \end{equation} where amplitudes $a_x$ and $b_x$ depend on wave number, distance, and slit of origin, and $|x\rangle$ are final particle basis states. Upon detection at a particular location $x$, one term remains from the sum on the right-hand side of (1): \begin{equation} |x\rangle \left\{a_x |\phi_x\rangle|v_U\rangle + b_x |\phi_x\rangle|v_L\rangle\right\}. \end{equation} Kastner argues that the result of this experiment is even more dramatic than that of the Afshar experiment, because visibility is hundred percent since a fully articulated interference pattern has been irreversibly recorded - not just indicated indirectly - and yet a measurement can be performed later, that seems to reveal “which slit” the photon went through. However, this argument is not correct, as can be seen from the following. Suppose there were no ``vibrational states", then the term which remains from the sum in (1) would be given by \begin{equation} |x\rangle \left\{a_x |\phi_x\rangle + b_x |\phi_x\rangle\right\}. \end{equation} The probability density of detecting the particle at position $x$ is then given by \begin{equation} P(x) = \left\{|a_x|^2 + |b_x|^2 + a_x^*b_x + a_xb_x^* \right\} \langle\phi_x|\phi_x\rangle, \end{equation} where the last two terms in the curly brackets denote interference. One the other hand, the probability density of detecting the particle at position $x$, in the presence of ``vibrational states" is given by \begin{eqnarray} P(x) &=& \{|a_x|^2\langle v_U|v_U\rangle + |b_x|^2\langle v_L|v_L\rangle + a_x^*b_x\langle v_U|v_L\rangle + a_xb_x^*\langle v_L|v_U\rangle \} \langle\phi_x|\phi_x\rangle \nonumber\\ &=& \left\{|a_x|^2 + |b_x|^2 \right\} \langle\phi_x|\phi_x\rangle, \end{eqnarray} where the interference terms are killed by the orthogonality of $|v_U\rangle$ and $|v_L\rangle$. So, contrary to the claim in \cite{kastner09}, this experiment does not show any interference, although the ``vibrational states" do provide which-way information. This is in perfect agreement with Bohr's complementarity principle. It can show interference if $|v_U\rangle$ and $|v_L\rangle$ are not strictly orthogonal. However, in that case one cannot extract any which-way information. In conlcusion, we have shown that the thought experiment, described by Kastner, does not show interference at all. What the experiment does show is that if there exists which-way information in the state, there is no interference pattern on the screen, in agreement with Bohr's complementarity principle. \end{document}

\begin{document} \title{Effective Mass Dirac-Morse Problem with any $\kappa$-value} \author{\small Altuð Arda} \email[E-mail: ]{[email protected]}\affiliation{Department of Physics Education, Hacettepe University, 06800, Ankara,Turkey} \author{\small Ramazan Sever} \email[E-mail: ]{[email protected]}\affiliation{Department of Physics, Middle East Technical University, 06531, Ankara,Turkey} \author{\small Cevdet Tezcan} \email[E-mail: ]{[email protected]}\affiliation{Faculty of Engineering, Baþkent University, Baglýca Campus, Ankara,Turkey} \author{\small H\"{u}seyin Akçay} \email[E-mail: ]{[email protected]}\affiliation{Faculty of Engineering, Baþkent University, Baglýca Campus, Ankara,Turkey} \date{\today} \begin{abstract} The Dirac-Morse problem are investigated within the framework of an approximation to the term proportional to $1/r^2$ in the view of the position-dependent mass formalism. The energy eigenvalues and corresponding wave functions are obtained by using the parametric generalization of the Nikiforov-Uvarov method for any $\kappa$-value. It is also studied the approximate energy eigenvalues, and corresponding wave functions in the case of the constant-mass for pseudospin, and spin cases, respectively.\\ Keywords: generalized Morse potential, Dirac equation, Position-Dependent Mass, Nikiforov-Uvarov Method, Spin Symmetry, Pseudospin Symmetry \end{abstract} \pacs{03.65.-w; 03.65.Ge; 12.39.Fd} \maketitle The investigation of the solutions for quantum mechanical systems having certain potentials in the case of position-dependent mass (PDM) [1, 2] has been received great attentions. Many authors have studied the solutions of different potentials for spatially-dependent mass, such as hypergeometric type potentials [3], Coulomb potential [4], $PT$-symmetric kink-like, and inversely linear plus linear potentials [5]. It is well known that the theory based on the effective-mass Schr\"{o}dinger equation is a useful ground for investigation of some physical systems, such as semiconductor heterostructures [6], the impurities in crystals [7-9], and electric properties of quantum wells, and quantum dots [10]. In the present work, we tend to solve the Dirac-Morse problem within the PDM formalism. The pseudospin symmetry is an interesting result appearing in Dirac equation of a particle moving in an external scalar, and vector potentials in the case of it when the sum of the potentials is nearly zero. It was observed that the single particle states have a quasidegeneracy labeled with the quantum numbers $\tilde{\ell}$, and $\tilde{s}$, which are called the pseudo-orbital angular momentum, and pseudospin angular momentum quantum numbers, respectively [11-16]. The concept of pseudospin symmetry has received great attentions in nuclear theory because of being a ground to investigate deformation, and superdeformation in nuclei [17, 18], and to build an effective shell-model coupling scheme [19, 20]. The symmetry appears in that case, when the magnitude of scalar potential is nearly equal to the magnitude of vector potential with opposite sign [14, 21-25] and the Dirac equation has the pseudospin symmetry, when the sum of the vector, and scalar potentials is a constant, i.e., $\Sigma(r)=V_v(r)+V_s(r)=const.$ or $d\Sigma(r)/dr=0$ [16]. The spin symmetry is another important symmetry occurring in Dirac theory in the presence of external scalar, and vector potentials. The spin symmetry appears in the Dirac equation, when the difference of scalar, and vector potentials is a constant, i.e., $\Delta(r)=V_{v}(r)-V_{s}(r)=const.$ [14, 16]. Recently, the pseudospin and/or spin symmetry have been studied by many authors for some potentials, such as Morse potential [26-28], Woods-Saxon potential [29], Coulomb [30], and harmonic potentials [31-33], Eckart potential [34-36], P\"{o}schl-Teller potential[37, 38], Hulth\'{e}n potential [39], and Kratzer potential [40]. In Ref. [41], the bound-state solutions of Dirac equation are studied for generalized Hulth\'{e}n potential with spin-orbit quantum number $\kappa$ in the position-dependent mass background. In this letter, we tend to show that the new scheme of the Nikiforov-Uvarov (NU) method could be used to find the energy spectra, and the corresponding eigenspinors within the framework of an approximation to the term proportional to $1/r^2$ for arbitray spin-orbit quantum number $\kappa$, i.e. $\kappa\neq 0$, when the mass depends on position. The NU method is a powerful tool to solve of a second order differential equation by turning it into a hypergeometric type equation [42]. Dirac equation for a spin-$\frac{1}{2}$ particle with mass $m$ moving in scalar $V_s(r)$, and vector potential $V_v(r)$ can be written as (in $\hbar=c=1$ unit) \begin{eqnarray} [\alpha\,.\,\textbf{P}+\beta(m+V_s(r))]\,\Psi_{n\kappa}(r)=[E-V_v(r)]\,\Psi_{n\kappa}(r)\,. \end{eqnarray} where $E$ is the relativistic energy of the particle, $\textbf{P}$ is three-momentum, $\alpha$ and $\beta$ are $4 \times 4$ Dirac matrices, which have the forms of $\alpha=\Bigg(\begin{array}{cc} 0 & \sigma \\ \sigma & 0 \end{array}\Bigg)$ and $\beta=\Bigg(\begin{array}{cc} 0 & I \\ -I & 0 \end{array}\Bigg)$, respectively, [43]. Here, $\sigma$ is a three-vector whose components are Pauli matrices and $I$ denotes the $2 \times 2$ unit matrix. $\textbf{J}$ denotes the total angular momentum , and $\hat{K}=-\beta(\sigma.\textbf{L}+1)$ corresponds to the spin-orbit operator of the Dirac particle in a spherically symmetric potential, where $\textbf{L}$ is the orbital angular momentum operator of the particle. The eigenvalues of the spin-orbit operator $\hat{K}$ are given as $\kappa=\pm(j+1/2)$, where $\kappa=-(j+1/2)<0$ correspond to the aligned spin $j=\ell+1/2$, and $\kappa=(j+1/2)>0$ correspond to the unaligned spin $j=\ell-1/2$. The total angular momentum quantum number of the particle is described as $j=\tilde{\ell}+\tilde{s}$\,,where $\tilde{\ell}=\ell+1$ is the pseudo-orbital angular momentum quantum number, and $\tilde{s}=1/2$ is the pseudospin angular momentum quantum number. For a given $\kappa=\pm1, \pm2, \ldots$, the relation between the spin-orbit quantum number $\kappa$\,, and "two" orbital angular momentum quantum numbers are given by $\kappa(\kappa+1)=\ell(\ell+1)$, and $\kappa(\kappa-1)=\tilde{\ell}(\tilde{\ell}+1)$. The Dirac spinor in spherically symmetric potential can be written in terms of upper and lower components as \begin{eqnarray} \Psi_{n \kappa}(r)=\,\frac{1}{r}\,\Bigg(\begin{array}{c} \,\chi_{n \kappa}\,(r)Y_{jm}^{\ell}(\theta,\phi) \\ i\phi_{n \kappa}\,(r)Y_{jm}^{\tilde{\ell}}(\theta,\phi) \end{array}\Bigg)\,, \end{eqnarray} where $Y_{jm}^{\ell}(\theta,\phi)$, and $Y_{jm}^{\tilde{\ell}}(\theta,\phi)$ are the spherical harmonics, and $\chi_{n \kappa}\,(r)/r$, and $\phi_{n \kappa}\,(r)/r$ are radial part of the upper and lower components. Substituting Eq. (2) into Eq. (1) enable us to write the Dirac equation as a set of two couple differential equations in terms of $\chi_{n \kappa}\,(r)$ and $\phi_{n \kappa}\,(r)$. By eliminating $\chi_{n \kappa}\,(r)$ or $\phi_{n \kappa}\,(r)$ in these coupled equations, we obtain \begin{eqnarray} \Big\{\,\frac{d^2}{dr^2}-\,\frac{\kappa(\kappa+1)}{r^2}\, +\,\frac{1}{M_{\Delta}(r)}\Big(\frac{dm(r)}{dr} -\frac{d\Delta(r)}{dr}\Big)\,(\frac{d}{dr}\,+\,\frac{\kappa}{r})\Big\}\chi_{n\kappa}(r)= M_{\Delta}(r)M_{\Sigma}(r)\chi_{n\kappa}(r)\,, \end{eqnarray} \begin{eqnarray} \Big\{\,\frac{d^2}{dr^2}-\,\frac{\kappa(\kappa-1)}{r^2}\, -\,\frac{1}{M_{\Sigma}(r)}\Big(\frac{dm(r)}{dr}+ \frac{d\Sigma(r)}{dr}\Big)\,(\frac{d}{dr}\,-\,\frac{\kappa}{r})\Big\}\phi_{n\kappa}(r)= M_{\Delta}(r)M_{\Sigma}(r)\phi_{n\kappa}(r)\,, \end{eqnarray} where $M_{\Delta}(r)=m+E_{n\kappa}-\Delta(r)$\,, $M_{\Sigma}(r)=m-E_{n\kappa}+\Sigma(r)$, and $\Delta(r)=V_{v}\,(r)-V_s\,(r)$, $\Sigma(r)=V_{v}\,(r)+V_s\,(r)$. In the NU-method, the Schr\"{o}dinger equation is transformed by using an appropriate coordinate transformation \begin{eqnarray} \sigma^{2}(s)\Psi''(s)+\sigma(s)\tilde{\tau}(s) \Psi'(s)+\tilde{\sigma}(s)\Psi(s)=0\,, \end{eqnarray} where $\sigma(s)$, $\tilde{\sigma}(s)$ are polynomials, at most second degree, and $\tilde{\tau}(s)$ is a first degree polynomial. The polynomial $\pi(s)$, and the parameter $k$ are required in the method \begin{eqnarray} \pi(s)=\frac{1}{2}\,[\sigma^{\prime}(s)-\tilde{\tau}(s)]\pm \sqrt{\frac{1}{4}\,[\sigma^{\prime}(s)-\tilde{\tau}(s)]^2- \tilde{\sigma}(s)+k\sigma(s)}, \end{eqnarray} \begin{eqnarray} \lambda=k+\pi^{\prime}(s ), \end{eqnarray} where $\lambda$ is a constant. The function under the square root in the polynomial in $\pi(s)$ in Eq. (6) must be square of a polynomial in order that $\pi(s)$ be a first degree polynomial. Replacing $k$ into Eq. (6), we define \begin{eqnarray} \tau(s)=\tilde{\tau}(s)+2\pi(s). \end{eqnarray} where the derivative of $\tau(s)$ should be negative [42]. Eq. (5) has a particular solution with degree $n$, if $\lambda$ in Eq. (7) satisfies \begin{eqnarray} \lambda=\lambda_{n}=-n\tau^{\prime}-\frac{\left[n(n-1)\sigma^{\prime\prime}\right]}{2}, \quad n=0,1,2,\ldots \end{eqnarray} To obtain the solution of Eq. (5) it is assumed that the solution is a product of two independent parts as $\Psi(s)=\phi(s)~y(s)$, where $y(s)$ can be written as \begin{eqnarray} y_{n}(s)\sim \frac{1}{\rho(s)}\frac{d^{n}}{ds^{n}} \left[\sigma^{n}(s)~\rho(s)\right], \end{eqnarray} where the function $\rho(s)$ is the weight function, and should satisfy the condition \begin{eqnarray} \left[\sigma(s)~\rho(s)\right]'=\tau(s)~\rho(s)\,, \end{eqnarray} and the other factor is defined as \begin{eqnarray} \frac{1}{\phi(s)}\frac{d\phi(s)}{ds}=\frac{\pi(s)}{\sigma(s)}. \end{eqnarray} In order to clarify the parametric generalization of the NU method, let us take the following general form of a Schr\"{o}dinger-like equation written for any potential, \begin{eqnarray} \left\{\frac{d^{2}}{ds^{2}}+\frac{\alpha_{1}-\alpha_{2}s}{s(1-\alpha_{3}s)} \frac{d}{ds}+\frac{-\xi_{1}s^{2}+\xi_{2}s-\xi_{3}}{[s(1-\alpha_{3}s)]^{2}}\right\}\Psi(s)=0. \end{eqnarray} When Eq. (13) is compared with Eq. (5), we obtain \begin{eqnarray} \tilde{\tau}(s)=\alpha_{1}-\alpha_{2}s\,\,\,;\,\,\sigma(s)=s(1-\alpha_{3}s)\,\,\,;\,\, \tilde{\sigma}(s)=-\xi_{1}s^{2}+\xi_{2}s-\xi_{3}\,. \end{eqnarray} Substituting these into Eq. (6) \begin{eqnarray} \pi(s)=\alpha_{4}+\alpha_{5}s\pm\sqrt{(\alpha_{6}-k\alpha_{3})s^{2}+(\alpha_{7}+k)s+\alpha_{8}}\,, \end{eqnarray} where the parameter set are \begin{eqnarray} \begin{array}{lll} \alpha_{4}=\frac{1}{2}\,(1-\alpha_{1})\,, & \alpha_{5}=\frac{1}{2}\,(\alpha_{2}-2\alpha_{3})\,, & \alpha_{6}=\alpha_{5}^{2}+\xi_{1} \\ \alpha_{7}=2\alpha_{4}\alpha_{5}-\xi_{2}\,, & \alpha_{8}=\alpha_{4}^{2}+\xi_{3}\,. & \end{array} \end{eqnarray} In NU-method, the function under the square root in Eq. (15) must be the square of a polynomial [42], which gives the following roots of the parameter $k$ \begin{eqnarray} k_{1,2}=-(\alpha_{7}+2\alpha_{3}\alpha_{8})\pm2\sqrt{\alpha_{8}\alpha_{9}}\,, \end{eqnarray} where $\alpha_{9}=\alpha_{3}\alpha_{7}+\alpha_{3}^{2}\alpha_{8}+\alpha_{6}$\,. We obtain the polynomials $\pi(s)$ and $\tau(s)$ for $k=-(\alpha_{7}+2\alpha_{3}\alpha_{8})-2\sqrt{\alpha_{8}\alpha_{9}}$, respectively \begin{eqnarray} \pi(s)=\alpha_{4}+\alpha_{5}s-\left[(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}}\,)s-\sqrt{\alpha_{8}}\,\right]\,, \end{eqnarray} \begin{eqnarray} \tau(s)=\alpha_{1}+2\alpha_{4}-(\alpha_{2}-2\alpha_{5})s-2\left[(\sqrt{\alpha_{9}} +\alpha_{3}\sqrt{\alpha_{8}}\,)s-\sqrt{\alpha_{8}}\,\right]. \end{eqnarray} Thus, we impose the following for satisfying the condition that the derivative of the function $\tau(s)$ should be negative in the method \begin{eqnarray} \tau^{\prime}(s)&=&-(\alpha_{2}-2\alpha_{5})-2(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}}\,) \nonumber \\ &=&-2\alpha_{3}-2(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}}\,)\quad<0. \end{eqnarray} From Eqs. (7), (8), (19), and (20), and equating Eq. (7) with the condition that $\lambda$ should satisfy given by Eq. (9), we find the eigenvalue equation \begin{eqnarray} \alpha_{2}n-(2n+1)\alpha_{5}&+&(2n+1)(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}}\,)+n(n-1)\alpha_{3}\nonumber\\ &+&\alpha_{7}+2\alpha_{3}\alpha_{8}+2\sqrt{\alpha_{8}\alpha_{9}}=0. \end{eqnarray} We obtain from Eq. (11) the polynomial $\rho(s)$ as $\rho(s)=s^{\alpha_{10}-1}(1-\alpha_{3}s)^{\frac{\alpha_{11}}{\alpha_{3}}-\alpha_{10}-1}$ and substituting it into Eq. (10) gives \begin{eqnarray} y_{n}(s)=P_{n}^{(\alpha_{10}-1,\frac{\alpha_{11}}{\alpha_{3}}-\alpha_{10}-1)}(1-2\alpha_{3}s)\,, \end{eqnarray} where $\alpha_{10}=\alpha_{1}+2\alpha_{4}+2\sqrt{\alpha_{8}}$, $\alpha_{11}=\alpha_{2}-2\alpha_{5}+2(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}})$ and $P_{n}^{(\alpha,\beta)}(1-2\alpha_{3}s)$ are the Jacobi polynomials. From Eq. (12), one obtaines \begin{eqnarray} \phi(s)=s^{\alpha_{12}}(1-\alpha_{3}s)^{-\alpha_{12}-\frac{\alpha_{13}}{\alpha_{3}}}\,, \end{eqnarray} then the general solution $\Psi(s)=\phi(s)y(s)$ becomes \begin{eqnarray} \Psi(s)=s^{\alpha_{12}}(1-\alpha_{3}s)^{-\alpha_{12}-\frac{\alpha_{13}}{\alpha_{3}}} P_{n}^{(\alpha_{10}-1,\frac{\alpha_{11}}{\alpha_{3}}-\alpha_{10}-1)}(1-2\alpha_{3}s). \end{eqnarray} where $\alpha_{12}=\alpha_{4}+\sqrt{\alpha_{8}}$ and $\alpha_{13}=\alpha_{5}-(\sqrt{\alpha_{9}}+\alpha_{3}\sqrt{\alpha_{8}}\,)$. Let us study the case where the parameter $\alpha_3=0$. In this type of problems, the eigenfunctions become \begin{eqnarray} \Psi(s)=s^{\alpha_{12}}\,e^{\alpha_{13}s}\,L^{\alpha_{10}-1}_{n}(\alpha_{11}s)\,, \end{eqnarray} when the limits $lim_{\alpha_3 \rightarrow 0}\,P^{(\alpha_{10}-1\,,\frac{\alpha_{11}}{\alpha_{3}}-\alpha_{10}-1)}_{n}(1-\alpha_{3}s)= L^{\alpha_{10}-1}_{n}(\alpha_{11}s)$ and $lim_{\alpha_3 \rightarrow 0}\,(1-\alpha_{3}s)^{-\,\alpha_{12}-\frac{\alpha_{13}}{\alpha_{3}}}= e^{\alpha_{13}s}$ are satisfied and the corresponding energy spectrum is \begin{eqnarray} \alpha_{2}n-2\alpha_{5}n+(2n+1)(\sqrt{\alpha_{9}\,}&-&\alpha_{3}\sqrt{\alpha_{8}\,}\,)+n(n-1)\alpha_{3} +\alpha_{7}\nonumber\\&+&2\alpha_{3}\alpha_{8}-2\sqrt{\alpha_{8}\alpha_{9}\,}+\alpha_{5}=0\,. \end{eqnarray} The generalized Morse potential is given by [44] \begin{eqnarray} V_M(x)=De^{-2\beta x}-2De^{-\beta x}\,, \end{eqnarray} where $x=(r/r_0)-1$\,,\,$\beta=\alpha r_0$\,,\,$D$ is the dissociation energy, $r_0$ is the equilibrium distance, and $\alpha$ is the potential width. The term proportional to $1/r^2$ in Eq. (4) can be expanded about $x=0$ [45] \begin{eqnarray} V_M(x)=\,\frac{\kappa(\kappa-1)}{r^2}=\,\frac{a_{0}}{(1+x)^2}=a_{0}(1-2x+3x^2+\ldots)\,;\,\, a_{0}=\,\frac{\kappa(\kappa-1)}{r_0^2}\,, \end{eqnarray} Instead, we now replace $V_M(x)$ by the potential [45] \begin{eqnarray} \tilde{V}_M(x)=a_{0}(a_{1}+a_{2}e^{-\beta x}+a_{3}e^{-2\beta x})\,, \end{eqnarray} Expanding the potential $\tilde{V}_M(x)$ around $x=0$, and combining equal powers with Eq. (28), one can find the arbitrary constants in the new form of the potential as \begin{eqnarray} a_{1}=1-\,\frac{3}{\beta}\,+\,\frac{3}{\beta^2}\,\,;\,\,\,a_{2}=\,\frac{4}{\beta}\,-\,\frac{6}{\beta^2}\,\,;\,\,\, a_{3}=-\,\frac{1}{\beta}\,+\,\frac{3}{\beta^2}\,. \end{eqnarray} Eq. (4) can not be solved analytically because of the last term in the equation, we prefer to use a mathematical identity such as $dm(r)/dr=-d\Sigma(r)/dr$ to eliminate this term. We obtain the mass function from the identity as \begin{eqnarray} m(x)=m_{0}+m_{1}e^{-\beta x}+m_{2}e^{-2\beta x}\,, \end{eqnarray} where $m_{0}$ corresponds to the integral constant, and the parameters $m_{1}$, and $m_{2}$ are $2D$, and $-D$, respectively. The parameter $m_{0}$ will denote the rest mass of the Dirac particle. By using the potential form given by Eq. (29) replaced by Eq. (28), inserting the mass function in Eq. (31), setting the "difference" potential $\Delta(r)$ to generalized Morse potential in Eq. (27) and using the new variable $s=e^{-\beta x}$, we have \begin{eqnarray} \Big\{\,\frac{d^2}{ds^2}\,+\,\frac{1}{s}\,\frac{d}{ds}\,&+&\frac{1}{s^2}\Big[ -\delta^2(a_{0}a_{1}+m^2_{0}-E^2)-\delta^2[a_{0}a_{2}+(m_{0}-E)(m_{1}+2D)] s\nonumber\\&-&\delta^2[a_{0}a_{3}+(m_{0}-E)(m_{2}-D]s^2\Big]\Big\}\phi_{n\kappa}(s)=0\,. \end{eqnarray} Comparing Eq. (32) with Eq. (13) gives the parameter set \begin{eqnarray} \begin{array}{ll} \alpha_1=1\,, & -\xi_1=-\delta^2[a_{0}a_{3}+(m_{0}-E)(m_{2}-D] \\ \alpha_2=0\,, & \xi_2=-\delta^2[a_{0}a_{2}+(m_{0}-E)(m_{1}+2D)] \\ \alpha_3=0\,, & -\xi_3=-\delta^2(a_{0}a_{1}+m^2_{0}-E^2) \\ \alpha_4=0\,, & \alpha_5=0 \\ \alpha_6=\xi_1\,, & \alpha_7=-\xi_2 \\ \alpha_8=\xi_3\,, & \alpha_9=\xi_1 \\ \alpha_{10}=1+2\sqrt{\xi_3}\,, & \alpha_{11}=2\sqrt{\xi_1} \\ \alpha_{12}=\sqrt{\xi_3}\,, & \alpha_{13}=-\sqrt{\xi_1} \end{array} \end{eqnarray} where $\delta=1/\alpha$. We write the energy eigenvalue equation of the generalized Morse potential by using Eq. (26) \begin{eqnarray} 2\delta\sqrt{a_{0}a_{1}+m^2_{0}-E^2\,} -\delta\,\frac{a_{0}a_{2}+(m_{0}-E)(m_{1}+2D)}{\sqrt{a_{0}a_{3}+(m_{0}-E)(m_{2}-D)\,}} =2n+1\,. \end{eqnarray} Since the negative energy eigenstates exist in the case of the pseudospin symmetry [14, 15, 16], so we choose the negative energy solutions in Eq. (46). In Table I, we give some numerical values of the negative bound state energies obtained from Eq. (46) for $CO$ molecule in atomic units, where we use the input parameter set as $D=11.2256$ eV, $r_{0}=1.1283$ $\AA$, $m_{0}=6.8606719$ amu, and $a=2.59441$ [46], and summarize our results for different $\tilde{\ell}$, and $n$ values. The corresponding lower spinor component can be written by using Eq. (25) \begin{eqnarray} \phi(s)=s^{w_{1}}\,e^{-\,w_{2}s}L^{2w_{1}}_{n}(2w_{2}s)\,, \end{eqnarray} where $w_{1}=\delta\sqrt{a_{0}a_{1}+m^2_{0}-E^2\,}$, and $w_{2}=\delta\sqrt{a_{0}a_{3}+(m_{0}-E)(m_{2}-D)\,}$. Let us study the two special limits, pseudospin and spin symmetry cases, respectively, in the case of the constant mass. \subsubsection{Pseudospin Case} The Dirac equation has the exact pseudospin symmetry if the "sum" potential could satisfy the condition that $d\Sigma(r)/dr=0$, i.e. $\Sigma(r)=A (const.)$ [14]. The parameters in our formalism become $m_{1}=m_{2}=0$. Setting the "difference" potential $\Delta(r)$ to the generalized Morse potential in Eq. (27), using Eq. (29) for the term proportional to $1/r^2$, and using the new variable $s=e^{-\beta x}$, we have from Eq. (4) \begin{eqnarray} \Big\{\,\frac{d^2}{ds^2}\,+\,\frac{1}{s}\,\frac{d}{ds}\,&+&\frac{1}{s^2}\Big[ -\delta^2[a_{0}a_{1}+M(m_{0}+E)]-\delta^2(2MD+a_{0}a_{2}) s\nonumber\\&+&\delta^2(MD-a_{0}a_{3})s^2\Big]\Big\}\phi(s)=0\,. \end{eqnarray} where $M=m_{0}+A-E$. By following the same procedure, the energy eigenvalue equation for the exact pseudospin symmetry in the case of constant mass is written \begin{eqnarray} 2\sqrt{a_{0}a_{1}+M(m_{0}+E)\,}=\frac{a_{0}a_{2}+2DM}{\sqrt{a_{0}a_{3}-DM\,}}+\alpha(2n+1)\,. \end{eqnarray} and the corresponding wave functions read as \begin{eqnarray} \phi^{m_{1}=m_{2}=0}(s)=s^{w'_{1}}\,e^{-\,w'_{2}s}L^{2w'_{1}}_{n}(2w'_{2}s)\,, \end{eqnarray} where $w\,'_{1}=\delta\sqrt{a_{0}a_{1}+M(m_{0}+E)\,}$\,, and $w\,'_{2}=\delta\sqrt{a_{0}a_{3}-DM\,}$\,. We must consideration the negative bound states solutions in Eq. (37) because there exist only the negative eigenvalues in the exact pseudospin symmetry [14, 15, 16]. \subsubsection{Spin Case} The spin symmetry appears in the Dirac equation if the condition is satisfied that $\Delta(r)=V_{v}(r)-V_{s}(r)=A(const.)$. In this case, we have from Eq. (3) \begin{eqnarray} \Big\{\frac{d^2}{dr^2}-\frac{\kappa(\kappa+1)}{r^2}-(m_{0}+E-A)(m_{0}-E-\Sigma(r))\Big\} \chi(r)=0\,, \end{eqnarray} where we set the "sum" potential as generalized Morse potential given in Eq. (27), and use approximation for the term proportional to $1/r^2$ in Eq. (29) [45] \begin{eqnarray} \tilde{V}_M(x)=b_{0}(b_{1}+b_{2}e^{-\beta x}+b_{3}e^{-2\beta x})\,, \end{eqnarray} where $b_{0}=\kappa(\kappa+1)/r^2_{0}$\,, and the parameters $b_{i} (i=1, 2, 3)$ are given in Eq. (30). Using the variable $s=e^{-\beta x}$, and inserting Eq. (40) into Eq. (39), we obtain \begin{eqnarray} \Big\{\,\frac{d^2}{ds^2}\,+\,\frac{1}{s}\,\frac{d}{ds}\,&+&\frac{1}{s^2}\Big[ -\delta^2[b_{0}b_{1}+M'(m_{0}-E)]+\delta^2(2DM\,'-b_{0}b_{2}) s\nonumber\\&-&\delta^2(b_{0}b_{3}+DM\,')s^2\Big]\Big\}\chi(s)=0\,. \end{eqnarray} where $M\,'=m_{0}+E-A$. We write the energy eigenvalue equation, and corresponding wave equations in the spin symmetry limit, respectively, \begin{eqnarray} \frac{\delta[2DM\,'-b_{0}b_{2}]}{\sqrt{b_{0}b_{3}+DM\,'\,}} +2\delta\sqrt{b_{0}b_{1}+M\,'(m_{0}-E)\,}=2n+1\,, \end{eqnarray} and \begin{eqnarray} \chi^{m_{1}=m_{2}=0}(s)=s^{w''_{1}}\,e^{-\,w''_{2}s}L^{2w''_{1}}_{n}(2w''_{2}s)\,, \end{eqnarray} where $w\,''_{1}=\delta\sqrt{b_{0}b_{1}+M'(m_{0}-E)\,}$\,, and $w\,''_{2}=\delta\sqrt{b_{0}b_{3}+DM'\,}$\,. We must take into account the positive energy solutions in Eq. (42) in the case of the exact spin symmetry [14, 15, 16]. In Summary, we have approximately solved the effective mass Dirac equation for the generalized Morse potential for arbitrary spin-orbit quantum number $\kappa$ in the position-dependent mass background. We have found the eigenvalue equation, and corresponding two-component spinors in terms of Legendre polynomials by using the parametric NU-method within the framework of an approximation to the term proportional to $1/r^2$\,. We have also obtained the energy eigenvalue equations, and corresponding wave functions for exact pseudospin, and spin symmetry limits in the case of constant mass. We have observed that our analytical results in the case of the pseudospin symmetry are good agreement with the ones obtained in the literature. \begin{table} \begin{ruledtabular} \caption{Energy eigenvalues for the $CO$ molecule for different values of $\tilde{\ell}$ and $(n,\kappa)$ in the case of position dependent mass.} \begin{tabular}{ccccc} $\tilde{\ell}$ & $n$ & $\kappa$ & state & $E<0$ \\ \hline 1 & 1 & -1 & $1s_{1/2}$ & 6.15913020 \\ 2 & 1 & -2 & $1p_{3/2}$ & 6.52968379 \\ 3 & 1 & -3 & $1d_{5/2}$ & 6.89146288 \\ 4 & 1 & -4 & $1f_{7/2}$ & 7.24974882 \\ \end{tabular} \end{ruledtabular} \end{table} \end{document}

\begin{document} \title{ Randomisation of Pulse Phases for Unambiguous and Robust Quantum Sensing } \author{Zhen-Yu Wang$^{1,\dagger}$} \email{E-mail: [email protected]} \author{Jacob E. Lang$^{2}$} \thanks{These authors contributed equally to this work} \author{Simon Schmitt$^{3}$} \thanks{These authors contributed equally to this work} \author{Johannes Lang$^{3}$, \\Jorge Casanova$^{4,5}$, Liam McGuinness$^{3}$, Tania S. Monteiro$^{2}$, Fedor Jelezko$^{3}$} \author{Martin B. Plenio$^{1}$} \affiliation{1. Institut f\"ur Theoretische Physik und IQST, Albert-Einstein-Allee 11, Universit\"at Ulm, D-89069 Ulm, Germany} \affiliation{2. Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom} \affiliation{3. Institute of Quantum Optics, Albert-Einstein-Allee 11, Universit\"at Ulm, D-89069 Ulm, Germany} \affiliation{4. Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain} \affiliation{5. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Spain} \begin{abstract} We develop theoretically and demonstrate experimentally a universal dynamical decoupling method for robust quantum sensing with unambiguous signal identification. Our method uses randomisation of control pulses to suppress simultaneously two types of errors in the measured spectra that would otherwise lead to false signal identification. These are spurious responses due to finite-width $\pi$ pulses, as well as signal distortion caused by $\pi$ pulse imperfections. For the cases of nanoscale nuclear spin sensing and AC magnetometry, we benchmark the performance of the protocol with a single nitrogen vacancy centre in diamond against widely used non-randomised pulse sequences. Our method is general and can be combined with existing multipulse quantum sensing sequences to enhance their performance. \end{abstract} \maketitle \emph{Introduction.--} The nitrogen-vacancy (NV) centre~\cite{doherty2013} in diamond has demonstrated excellent sensitivity and nanoscale resolution in a range of quantum sensing experiments~\cite{schirhagl2014nitrogen,rondin2014magnetometry,suter2016single,wu2016diamond}. In particular, under dynamical decoupling (DD) control~\cite{souza2012robust} the NV centre can be protected against environmental noise~\cite{ryan2010robust,deLange2010universal,naydenov2011dynamical} while at the same time being made sensitive to an AC magnetic field of a particular frequency~\cite{deLange2011single}. This makes the NV centre a highly promising probe for nanoscale nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI)~\cite{staudacher2013nuclear,rugar2015proton,deVience2015nanoscale,shi2015single,schmitt2017sub,boss2017quantum,glenn2018high,rosskopf2017quantum,pham2016nmr}. Moreover, NV centers under DD control can be used to detect, identify, and control nearby single nuclear spins~\cite{taminiau2012detection,kolkowitz2012sensing,zhao2012sensing,Muller2014,lang2015dynamical,sasaki2018determination,zopes2018nc,zopes2018prl,pfender2019high} and spin clusters~\cite{zhao2011atomic,shi2014sensing,wang2016positioning,wang2017delayed,abobeih2018one}, for applications in quantum sensing~\cite{degen2017quantum}, quantum information processing~\cite{casanova2016noise,casanova2017arbitrary}, quantum simulations~\cite{cai2013a}, and quantum networks~\cite{humphreys2018deterministic,perlin2019noise}. Errors in the DD control pulses are unavoidable in experiments and limit performance especially for larger number of pulses. To compensate for detuning and amplitude errors in control pulses, robust DD sequences that include several pulse phases~\cite{gullion1990new,ryan2010robust,casanova2015robust,genov2017arbitrarily} were developed. However, these robust sequences still require good pulse-phase control and, more importantly, they introduce spurious harmonic response \cite{loretz2015spurious} due to the finite length of the control pulses. This spurious response leads to false signal identification, e.g. the misidentification of $^{13}$C nuclei for $^{1}$H nuclei, and hence impact negatively the reliability and reproducibility of quantum sensing experiments. Under special circumstances it is possible to control some of these spurious peaks~\cite{haase2016pulse,lang2017enhanced,shu2017unambiguous}. However, it is highly desirable to design a systematic and reliable method to suppress any spurious response and to improve robustness of all existing DD sequences, such as the routinely used XY family of sequences~\cite{gullion1990new}, the universally robust (UR) sequences~\cite{genov2017arbitrarily}, and other DD sequences leading to enhanced nuclear selectivity~\cite{casanova2015robust,haase2018soft}. \begin{figure}\label{Fig1} \end{figure} In this Letter, we demonstrate that phase randomisation upon repetition of a basic pulse unit of DD sequences is a generic tool that improves their robustness and eliminates spurious response whilst maintaining the desired signal. This is achieved by, firstly, adding a global phase to the applied $\pi$ pulses within one elemental unit and, secondly, randomly changing this phase each time the unit is repeated. Our method is universal, that is, it can be directly incorporated to arbitrary DD sequences and is applicable for any physical realisation of a qubit sensor. \begin{figure} \caption{Quantum spectroscopy with DD. (a) Simulated averaged population signal as a function of the DD frequency [$1/(2\tau)$ for pulse spacing $\tau$]. One $^{1}{\rm H}$ spin and one $^{13}{\rm C}$ spin are coupled to the NV centre via the hyperfine-field components~\cite{casanova2015robust,seeSM} $(A_{\perp},A_{\parallel})=2\pi\times(2,1)$ kHz and $2\pi\times(5,50)$ kHz, respectively. The orange dashed line (blue solid line) is the signal obtained by a standard XY8 (randomised XY8) sequence using rectangular $\pi$ pulses with a time duration of $200$ ns and $M=200$. The presence of the $^{13}{\rm C}$ distorts the proton spin signal centred at the proton spin frequency (see the vertical dashed lines indicating the target $^{1}{\rm H}$ and the spurious $^{13}{\rm C}$ resonance frequencies for a magnetic field 450~G). The randomised XY8 sequence significantly reduces the signal distortion due to non-instantaneous control and reveals the real proton signal (see the green dash-dotted line for the signal obtained by a perfect XY8 sequence). (b) As (a) but adding $5\%$ (in terms of the ideal Rabi frequency) of errors in both driving amplitude and frequency detuning to the $\pi$ pulses. (c) and (d) [(e) and (f)] are the same as (a) and (b) but for the YY8~\cite{shu2017unambiguous} [UR8~\cite{genov2017arbitrarily}] sequence. Despite the YY8 sequence - which uses single-axis control to mitigate the spurious peak in the $^{13}{\rm C}$ spectrum when there is no pulse error - the presence of the $^{13}{\rm C}$ still distorts the proton spin signal centred at the proton spin frequency. In all cases, the randomised protocol reduces the signal distortion due to non-instantaneous control and control errors.} \label{FigH} \end{figure} \emph{DD-based quantum sensing.--} Whilst our method is applicable to any qubit sensor, we illustrate it here with single NV centres. For all experiments in this work a bias magnetic field between 400 and 500 Gauss aligned with the NV-axis splits the degenerate $m_s=\pm 1$ spin states allowing the selective addressing of the $m_s=0 \leftrightarrow m_s=-1$ transition, which represents our sensor qubit with the qubit states $|0(1)\rangle$ [see Fig.~\ref{Fig1}(a) and \cite{seeSM} for details of the experimental set-up]. The sensor qubit and its environmental interaction takes the general form $\hat{H}^\prime(t)=\frac{1}{2}\hat{\sigma}_{z}\hat{E}(t)$. Here $\hat{\sigma}_{z}=|0\rangle\langle 0|-|1\rangle\langle 1|$ is the Pauli operator of the sensor qubit, and $\hat{E}(t)$ is an operator that includes the target signal which oscillates at a particular frequency as well as the presence of noisy environmental fluctuations. In the case of nuclear-spin sensing, $\hat{E}(t)$ contains target and bath nuclear spin operators oscillating at their Larmor frequencies. For AC magnetometry, $\hat{E}(t)$ describe classical oscillating magnetic fields. The aim of quantum sensing is to detect a target such as a single nuclear spin via the control of the quantum sensor with a sequence of DD $\pi$ pulses. The latter often corresponds to a periodic repetition of a basic pulse unit which has a time duration $T$ and a number $N$ of pulses [see Fig.~\ref{Fig1}(b)]. The propagator of a single $\pi$ pulse unit in a general form reads $\hat{U}_{\rm unit}(\{\phi_{j}\}) = \hat{\mathtt{f}}_{N+1}\hat{P}(\phi_N)\hat{\mathtt{f}}_{N}\cdots \hat{P}(\phi_2)\hat{\mathtt{f}}_{2}\hat{P}(\phi_1)\hat{\mathtt{f}}_{1}$, where $\hat{\mathtt{f}}_{j}$ are the free evolutions separated by the control $\pi$ pulses with the propagator $\hat{P}(\phi_j)$. Errors in the control are included in $\hat{P}(\phi_j)$, while the different pulse phases $\phi_j$ are used by robust DD sequences to mitigate the effect of detuning and amplitude errors of the $\pi$ pulses. Using $M$ repetitions of the basic DD unit [see Fig.~\ref{Fig1}(c) for the case of a standard construction] allows for $M$-fold increased signal accumulation time $T_{\rm total}=M T$ which enhances the acquired contrast of the weak signal as $\propto M^2$~\cite{zhao2011atomic} and improves the fundamental frequency resolution to $\sim 1/T_{\rm total}$. To see how a target signal is sensed, we write the Hamiltonian $H^\prime(t)$ in the interaction picture of the DD control as~\cite{seeSM} \begin{equation} \hat{H}(t) = \frac{1}{2}F_z(t) \hat{\sigma}_z \hat{E}(t) + \frac{1}{2}[F_{\perp}(t) \hat{\sigma}_{-}+{\rm H.c.}]\hat{E}(t), \label{HInt} \end{equation} where $\hat{\sigma}_{-}=|0\rangle\langle 1|$. For ideal instantaneous $\pi$ pulses, $F_{\perp}(t)=0$ vanishes [see Fig.~\ref{Fig1} (b) which shows how the $F_{\perp}(t)$ vanishes between the $\pi$ pulses] and the modulation function $F_{z}(t)$ is the stepped modulation function widely used in the literature, that is, $F_{z}(t)=(-1)^{m}$ when $m$ $\pi$-pulses have been applied up to the moment $t$. The role of a DD based quantum sensing sequence is to tailor $F_{z}(t)$ such that it oscillates at the same frequency as the target signal in $\hat{E}(t)$, allowing resonant coherent coupling between the sensor and the target. \begin{figure} \caption{Removing spurious response with the phase randomisation protocol. (a) In the measured spectrum of an AC magnetic field sensed by a standard repetition of the XY8 sequence (see orange diamonds), the non-instantaneous $\pi$ pulses produce spurious peaks at the frequencies $2\nu_{0}$ and $4\nu_{0}$. Repeating the XY8 sequence with phase randomisation (see blue bullets) preserves the desired signal centred at $\nu_{0}$ and efficiently suppresses all the spurious peaks. The XY8 unit was repeated $M=25$ times in the upper panel and $M=125$ in the lower panel for a longer sensing time. (b) Detection of proton spins using the XY8 sequence. For the measured spectrum obtained by the standard protocol, the $^{13}{\rm C}$ nuclear spins naturally in diamond produce a strong and wide spurious peak that hinders proton spin detection. Using the randomisation protocol, the spurious $^{13}{\rm C}$ peak has been suppressed, revealing the proton spin signal centred around a frequency of 1.9 MHz.} \label{FigS} \end{figure} In realistic situations, where the $\pi$ pulses are not instantaneous due to limited control power, the function $F_{\perp}(t)$ has a non-zero value during $\pi$ pulse execution and $F_z(t)$ deviates from $\pm 1$~\cite{lang2017enhanced,lang2019non} [see Fig.~\ref{Fig1} (b) for the example of XY8 sequences]. While it is possible to eliminate the effect of deviation in $F_z(t)$ by pulse shaping technique~\cite{casanova2018shaped}, the presence of non-zero $F_{\perp}(t)$ may still alter the expected signal or cause spurious peaks to appear~\cite{loretz2015spurious}. In general, an oscillating component with a frequency $k/T_{\rm total}$ ($k$ being an integer) in $\hat{E}(t)$, not resonant with $F_z(t)$, will create spurious response when the Fourier amplitude~\cite{lang2017enhanced,seeSM} \begin{equation} f^{\perp}_{k}=\frac{1}{T_{\rm total}}\int_{0}^{T_{\rm total}}F_{\perp}(t)\exp(-i 2\pi k t/T_{\rm total})dt \end{equation} of $F_{\perp}(t)$ is non-zero. This spurious response can cause false signal identification, e.g., a wrong conclusion on the detected nuclear species~\cite{loretz2015spurious}, exemplified in Figs.~\ref{FigH} and \ref{FigS}. Suppressing the spurious response from $^{13}\rm{C}$ nuclei is especially critical, as it allows reliable nanoscale NMR or MRI without the use of hard to manufacture and consequently expensive, highly isotopically $^{12}\rm{C}$ purified diamond. However, as shown in Fig.~\ref{FigH} (c),(d), even for a YY8 sequence (designed to remove spurious resonances~\cite{shu2017unambiguous}) the target proton signal is still perturbed by other nuclear species ($^{13}\rm{C}$ in this case). In the presence of amplitude and detuning errors, standard strategies perform even worse. To remove all spurious peaks, one seeks to design a DD sequence that minimises the effect of $F_{\perp}(t)$ in a robust manner. We observe that by introducing a global phase to all the $\pi$ pulses, the form of $F_{z}(t)$ is unchanged but a phase factor is added to $F_{\perp}(t)$. This motivates the following method to preserve $F_{z}(t)$ and to suppress the effect of $F_{\perp}(t)$ by phase randomisation. \emph{Phase randomisation.--} In the randomisation protocol, a random global phase $\Phi_{r,m}$ (where the subscript $r$ means a random value) is added to all the pulses within each unit $m$, as shown in Fig.~\ref{Fig1}(d). The propagator of $M$ DD units with independent global phases reads $\hat{U}_{r} = \prod_{m=1}^{M}\hat{U}_{\rm unit}(\{\phi_j+\Phi_{r,m}\})$. If one sets all the random phases $\Phi_{r,m}$ to the same value (e.g. zero) the original DD sequence can be recovered [Fig.~\ref{Fig1}(c)]. Since each of the global phases does not change the internal structure (i.e., the relative phases among $\pi$ pulses) of the basic unit, the robustness of the basic DD sequence is preserved. On the other hand, as we will show in the following, these random global phases prevent control imperfections from accumulating. \emph{Universal suppression of spurious response.--} The randomisation protocol provides a universal method to suppress spurious response. For the sequence with randomisation, one can find that the Fourier amplitude reads $f^{\perp}_{k}=Z_{r,M} \tilde{f}^{\perp}_{k/M}$, where $\tilde{f}^{\perp}_{k/M}=\frac{1}{T}\int_{0}^{T}F_{\perp}(t)\exp(-i\frac{2\pi k t}{M T})dt$ is the Fourier component defined over a single period $T$~\cite{seeSM}. For random phases $\{\Phi_{r,m}\}$, the factor \begin{equation} Z_{r,M} =\frac{1}{M} \sum_{m=1}^{M} \exp(i \Phi_{r,m}), \label{eq:phaseAverage} \end{equation} captures the effect of the randomisation protocol. Due to the random values of the phases $\Phi_{r,m}$, $Z_{r,M}$ becomes a (normalised) 2D random walk with $\langle|Z_{r,M}|^2\rangle=1/M$ thus suppressing the contrast of spurious response by a factor of $1/(2M)$ compared with the standard protocol~\cite{seeSM}. Here, we note that one can design a set of specific (i.e. not random) phases $\Phi_{r,m}$ that minimise a certain $f^{\perp}_{k}$ completely. However, this set of phases would be specific to one $k$-value (i.e. it does not suppress all spurious peaks simultaneously). In this respect, the power of our method is that it is simple to implement and fully universal, suppressing all spurious peaks produced by any sequence whilst still retaining the ideal signal, as shown in Fig.~\ref{FigH}. To experimentally benchmark the performance, we carried out nanoscale detection of a classical AC magnetic field [Fig.~\ref{FigS} (a)] and, separately, the nanoscale NMR detection of an ensemble of proton spins with a natural $^{13}$C abundance ($1.1\%$) diamond [Fig.~\ref{FigS} (b)]. The standard repetition of the XY8 sequence, which was widely used in various sensing and sensing based applications (e.g., see Refs.~\cite{staudacher2013nuclear,rugar2015proton,deVience2015nanoscale,shi2015single,glenn2018high,abobeih2018one,humphreys2018deterministic,rosskopf2017quantum,loretz2015spurious,pham2016nmr}), produces spurious peaks when the duration of $\pi$ pulses is non-zero. In contrast, the randomisation protocol suppresses all the spurious peaks in the spectrum efficiently, and the spurious background noise from a $^{13}{\rm C}$ nuclear spin bath in diamond was removed while the desired proton signal was unaffected, demonstrating a clear and unambiguous proton spin detection without the use of $^{12}$C isotopically pure diamonds. In the experiments, we have repeated the randomisation protocol with $K=10$ samples of the random phase sequences $\{\Phi_{r,m}\}$ and averaged out the measured signals. This reduces the fluctuations of the (suppressed) spurious peaks, introduced by the applied random phases, because the variance of $|Z_{r,M}|^2$ (which is $(M-1)/M^3$) is further reduced by a factor of $1/K$~\cite{seeSM}. Removing the spurious response also improves the accuracy, for example, in measuring the depth of individual NV centres~\cite{pham2016nmr}. By falsely assuming that all the signal around $1.9$ MHz obtained by the standard XY8 sequences originates from hydrogen spins, the computed NV centre depth would be $5.88\pm0.52$~nm, instead of $7.62\pm0.29$~nm obtained by the randomised XY8 - a deviation of about 30~$\%$ [see Fig.~\ref{FigS} (b)]. \begin{figure} \caption{Experimental enhancement of sequence robustness with the phase randomisation protocol. (a) The fidelity of XY8 sequences as a function of detuning and Rabi frequency errors for randomisation (upper panels) and standard (lower panels) protocols. The control errors are measured in terms of the ideal Rabi frequency $\Omega_{\rm ideal}= 2\pi\times 32.8 $ MHz. The sequences have inter-pulse spacing $200$ ns and $M=25$ XY8 units. (b) The fidelity of XY8 sequences with respect to a static phase error between the X and Y pulses and the inter-pulse time interval $\tau$, for randomised (upper panels) and standard (lower panels) protocols with $M=12$. Resonant microwave $\pi$ pulses are used with a Rabi frequency $\Omega_{\rm ideal}= 2\pi\times 66.6 $ MHz. } \label{FigR} \end{figure} \emph{Enhancement on control robustness.--} As indicated in Fig.~\ref{FigH}, the randomisation protocol also enhances the robustness of the whole DD sequence. For simplicity, in the following discussion we neglect the effect of the environment and concentrate on static control imperfections. The latter introduce errors in the form of non-zero matrix elements $\langle 0|\hat{U}_{\rm unit}|1\rangle = C \epsilon + O(\epsilon^2)$, where $\epsilon$ is a small parameter and $C$ is a prefactor depending on the explicit form of control (see~\cite{seeSM} for details). For the standard protocol where the same $\hat{U}_{\rm unit}$ block is repeated, the static errors accumulate coherently, yielding $\langle 0|(\hat{U}_{\rm unit})^{M}|1\rangle = MC \epsilon+ O(\epsilon^2)$. The random phases in the randomisation protocol avoids this coherent error accumulation and one can find $\langle 0|\hat{U}_{r}|1\rangle = Z_{r,M} MC \epsilon+ O(\epsilon^2)$, where the error is suppressed by the factor $Z_{r,M}$ which is given by Eq.~(\ref{eq:phaseAverage}) for random phases~\cite{seeSM}. Compared with the suppression of control imperfections by deterministic phases, the randomisation protocol is universal and achieves both suppression of spurious response and enhancement of robustness, without loss of sensitivity to target signals as shown in Figs.~\ref{FigH} and \ref{FigS}. In Fig.~\ref{FigR} (a), we show the robustness of the widely used XY8 sequence, with respect to amplitude bias and frequency detuning of the microwave pulses, for the randomisation and standard protocols. The simulation and experiment demonstrate robustness improvement after applying phase randomisation. As shown in Fig.~\ref{FigR} (b), the randomisation protocol also suppresses errors in pulse phases. The latter is especially relevant for digital pulsing devices where the signal from a microwave source is split-up and the phase in one arm is shifted by suitable equipments. On top of errors due to the working accuracy of these devices, different cable lengths in both arms can sum up to errors in the relative phase. \emph{Conclusion.--} We present a randomisation protocol for DD sequences that efficiently and universally suppresses spurious response whilst maintaining the expected signal. This method is simple to implement, only requiring additional random control-pulse phases, and is valid for all DD sequence choices. The protocol functions equally well for quantum and classical signals, allowing clear and unambiguous AC field and nuclear spin detection, e.g., with the widely used XY family of sequences. Furthermore, the protocol also enhances the robustness of the whole pulse sequences. For sensing experiments with NV centres, the protocol reduces the reliance on hard to manufacture, expensive, highly isotopically purified diamond. The method has a general character being equally applicable to other quantum platforms and other DD applications. For example, it could be used to improve correlation spectroscopy~\cite{laraoui2013,ma2016proposal,wang2017delayed,rosskopf2017quantum} in quantum sensing and fast quantum gates in trapped ions~\cite{arrazola2018arrazola,manovitz2017fast} where DD has been used as an important ingredient. \emph{Acknowledgements.--} M.~B.~P. and Z.-Y.~W. acknowledge support by the ERC Synergy grant BioQ (Grant No. 319130), the EU project HYPERDIAMOND and AsteriQs, the QuantERA project NanoSpin, the BMBF project DiaPol, the state of Baden-W{\"u}rttemberg through bwHPC, and the German Research Foundation (DFG) through Grant No. INST 40/467-1 FUGG. J.~E.~L. is funded by an EPSRC Doctoral Prize Fellowship. F.~J., S.~S., L.~M., and J.~L. acknowledge support of Q-Magine of the QUANTERA, DFG (FOR 1493, SPP 1923, JE 290/18-1 and SFB 1279), BMBF (13N14438, 16KIS0832, and 13N14810), ERC (BioQ 319130), VW Stiftung and Landesstiftung BW. 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For the nanoscale NMR experiments [Fig. 3(b) of the main text] a $^{13}$C natural abundance diamond was implanted with $^{15}N$ ions using an energy of 1.5\,keV and a dose of $2 \times 10^9\mathrm{\frac{^{15}N^+}{cm^2}}$. Subsequent annealing in vacuum at 1000$^\circ$C for 3 hours created shallow single NV centres with depths around $5\pm 1$ \,nm. For the experiments measuring the classical AC fields [Fig. 3 (a)] we used a different diamond, which was polished into a solid immersion lens. In order to create NV centres in this diamond, the flat surface was overgrown with an about 100\,nm thick layer of isotopically enriched $^{12}$C (99.999$\%$) using the plasma enhanced chemical vapor deposition method, with parameters as in \cite{sm:Osterkamp}. The same diamond was used for the experiments showing the improved robustness of the randomisation protocol (Fig. 4). The experiments presented in \ref{add_exp} were measured with an about 4\,$\mu$m deep NV in a flat diamond with 0.1\% $^{13}$C content. Before experiments, all diamonds were boiled in a 1:1:1 tri-acid mixture (H$_2$SO$_4$:HNO$_3$:HClO$_4$) for 4 hours at 130$^{\circ}$C. \subsection{Setup} Using a home-built confocal setup, read-out and initialisation (into the $|0\rangle$ spin state) of the NV center was performed using a 532\,nm laser. The laser beam was chopped using an acousto optical modulator into pulses of 3\,$\mu$s duration. The spin-dependent fluorescence from the NV spin states was detected using an avalanche photodiode. The first 500\,ns of the every laser pulse yield the spin population while the fluorescence between 1.5\,$\mu$s and 2.5\,$\mu$s was used to normalise the data. Magnetic bias fields between 400\,G and 500\,G were used to lift the degeneracy of the $|-1\rangle, |+1\rangle$ spin states and create an effective qubit. Microwave pulses resonant with the NV centre spin were applied using a 20\,$\mu$m diameter copper wire placed on the diamond surface as an antenna. The pulses were generated with an Arbitrary Waveform Generator (Tektronix AWG70001A, sampling rate 50GSamples/s) and amplified to give Rabi frequencies between 5-70\,MHz. The same wire was used to apply classical radio-frequency fields generated by a Gigatronics 2520B signal generators. For the classical AC field detection, background magnetic noise at the frequencies detected was determined to be at least 100 fold weaker than the measured signals. \subsection{Measurement protocol} All experiments were performed using the QuDi software suite \cite{sm:qudi}. For the randomised protocols the standard versions were modified by adding a random global phase to all $\pi$ pulses in a basic unit, as described in the main text. These phases were generated using the Python package 'random' with a uniform distribution between 0 and 2$\pi$. Before applying the dynamical decoupling protocols, the spin of the NV centre is initialized in a coherent superposition ($\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle$)). Therefore, additionally to the laser pulse, a $\pi$/2 pulse is applied. Also, before the readout the acquired sensor phase is mapped into a population difference by an additional $\pi$/2 pulse. For our experiments showing the improved robustness we intentionally introduce pulse errors to the $\pi$ pulses. Those errors were calibrated in terms of the real Rabi frequency. Thereby, the two $\pi$/2 pulses were always applied error-free. Every experiment was repeated several times under identical experimental conditions, but with different sets of phases, and the resulting data was averaged. \section{Hamiltonian under dynamical decoupling control } As stated in the main text, the Hamiltonian without dynamical decoupling (DD) control has the general form \begin{equation} \hat{H}^{\prime}(t)=\frac{1}{2}\hat{\sigma}_{z}\hat{E}(t), \end{equation} where $\hat{\sigma}_{z}=|0\rangle\langle0|-|1\rangle\langle1|$ is the Pauli operator of the sensor. The environment operator $\hat{E}(t)$ includes both the target signal to be sensed and environmental noise. For the relevant case of nuclear spin sensing, $\hat{E}(t)=\frac{1}{2}\sum_{n}\left[\left(A_{n}^{\perp}\hat{I}_{n}^{+}e^{-i\omega_{n}t}+{\rm h.c.}\right)+A_{n}^{\parallel}\hat{I}_{n}^{z}\right],$where $\hat{I}_{n}^{\alpha}$ ($\alpha=x,y,z$) are spin operators for the $n$th nuclear spin. $A_{n}^{\perp}$ and $A_{n}^{\parallel}$ are components of hyperfine field at the position of the nuclear spin. The nuclear spin precession frequency $\omega_{n}$ is the Larmor frequency of the nuclear spin shifted by the hyperfine field at the location of the nuclear spin. For the case of a classical AC field, $\hat{E}(t)$ takes the form $\sum_{n}b_{n}\cos(\omega_{n}t+\phi_{n})$. A sequence of applied microwave pulses yields the control Hamiltonian \begin{equation} \hat{H}_{{\rm ctrl}}(t)=\frac{1}{2}\Omega(t)\left[\hat{\sigma}_{x}\cos\phi(t)+\hat{\sigma}_{y}\cos\phi(t)\right]. \end{equation} In the rotating frame with respect to the control $\hat{H}_{{\rm ctrl}}(t)$, the Hamiltonian $\hat{H}^{\prime}(t)$ becomes \begin{equation} \hat{H}(t)=\frac{1}{2}\hat{\sigma}(t)\hat{E}(t), \end{equation} where $\hat{\sigma}(t)$ is $\hat{\sigma}_{z}$ in the Heisenberg picture with respect to $\hat{H}_{{\rm ctrl}}(t)$. In the following, we derive $\hat{\sigma}(t)$ and hence Eq. (2) in the main text. If a $\pi$ pulse is applied at time $t_j$, the evolution driven by $\hat{H}_{{\rm ctrl}}(t)$ reads $\hat{P}_{j}(\theta) = \exp\left[-i\frac{1}{2} \theta \left( \hat{\sigma}_{x} \cos\phi_{j} + \hat{\sigma}_{y} \cos\phi_{j} \right) \right]$, where $\theta=\theta(t)\in[0,\pi]$ is the angle of rotation and $\phi(t_j)=\phi_j$. Defining $\hat{P}_{j}(\pi)\equiv \hat{P}_{j}$ as the propagator of a single $\pi$ pulse, the propagator for $2n+1$ ($j=0,1,\ldots$) pulses \begin{align} \hat{U}_{2n+1} & =\hat{P}_{2n+1}\cdots\hat{P}_{2}\hat{P}_{1}\\ & =(-1)^{n+1}e^{i\frac{\pi}{2}}\exp\left(-i\varphi_{2n+1}\right)|0\rangle\langle1|+{\rm h.c}., \end{align} and for $2n$ pulses \begin{align} \hat{U}_{2n} & =\hat{P}_{2j}\cdots\hat{P}_{2}\hat{P}_{1}\\ & =(-1)^{n}\exp\left(i\varphi_{2n}\right)|0\rangle\langle0|+(-1)^{n}\exp\left(-i\varphi_{2n}\right)|1\rangle\langle1|, \end{align} where $\varphi_{2n+1}=-\sum_{l=1}^{2n+1}(-1)^{l}\phi_{l}$ and $\varphi_{2n}=-\sum_{l=1}^{2j}(-1)^{l}\phi_{l}$. Using $\hat{U}_{2n+1}$ and $\hat{U}_{2n}$, we find $\hat{\sigma}_{z}$ in the rotating frame of the control during the $j$th pulse \begin{align} \hat{\sigma}(t) & =[\hat{P}_{j}(\theta)\hat{U}_{j-1}]^{\dagger}\hat{\sigma}_{z}[\hat{P}_{j}(\theta)\hat{U}_{j-1}]\\ & =F_{z}(t)\hat{\sigma}_{z}+\left[F_{\perp}(t)|1\rangle\langle0|+{\rm h.c.}\right], \end{align} where the modulation functions are \begin{equation} F_{z}(t)=(-1)^{j-1}\cos\theta\label{sm:eq:Fz} \end{equation} \begin{equation} F_{\perp}(t)=i(-1)^{j-1}\exp\left\{-i[2\sum_{l=1}^{j-1}(-1)^{l}\phi_{l}+(-1)^{j}\phi_{j}]\right\}\sin\theta.\label{sm:eq:Fx} \end{equation} Because $\theta=\theta(t)$ in Eqs. (\ref{sm:eq:Fz}) and (\ref{sm:eq:Fx}) is the pulse area that the $j$th pulse has rotated at the moment $t$, for instantaneous pulses $F_{\perp}(t)$ has no effect (because $\sin\theta=0$ for all time $t$) and $F_{\perp}(t) \in \{\pm 1\}$. For the realistic case that the pulses are not instantaneous, $F_{\perp}(t)$ is non-zero during the $\pi$ pulses. \section{Fourier amplitudes of the modulation functions} For DD sequences that are $M$ periodic repetitions of a basic pulse unit with period $T$ and $F_{\alpha}(t+T)=F_{\alpha}(t)$ ($\alpha=z,\perp$), the $k$th Fourier amplitude of the modulation functions (over the total sequence time $T_{{\rm total}}=MT$) is \begin{align} f^{\alpha}_{k} & \equiv\frac{1}{MT}\int_{0}^{MT}F_{\alpha}(t)\exp\left(-i\frac{2\pi kt}{MT}\right)dt\\ & =\frac{1}{MT}\sum_{m=1}^{M}\int_{(m-1)T}^{mT}F_{\alpha}(t)\exp\left(-i\frac{2\pi kt}{MT}\right)dt\\ & =c_{k,M}\tilde{f}^{\alpha}_{k/M} \end{align} where \begin{equation} \tilde{f}^{\alpha}_{k/M} \equiv \frac{1}{T}\int_{0}^{T}F_{\alpha}(t)\exp\left(-i\frac{2\pi k t}{M T}\right) dt, \end{equation} and $c_{k,M}=\frac{1}{M}\sum_{m=1}^{M}\exp\left(-i\frac{2\pi k(m-1)}{M}\right)$. When $k/M$ is not an integer $Mc_{k,M}$ is a sum over roots of unity so it cancels to zero. When $k/M$ is an integer however the sum gives $c_{k,M}=1$. Therefore for standard repetitions of a basic pulse unit, we obtain (for $k=1,2,\ldots$) \begin{equation} f^{\alpha}_{k}=\begin{cases} \tilde{f}^{\alpha}_{k/M} & {\rm if}\ k/M\in\mathbb{Z},\\ 0 & {\rm otherwise}. \end{cases} \end{equation} Under the randomisation protocol a random phase is added to all pulses in the $m$-th repetition of the basic unit, so a set of $M$ random phases is generated, $\{\Phi_{r,m}|m=1,\ldots,M\}$. This transformation does not affect $F_{z}(t)$ but alters $F_{\perp}(t)\rightarrow F_{\perp}(t)e^{i\Phi_{r,m}}$ for the $m$-th unit of the sequence. The Fourier amplitudes $f^{z}_{k}$ are thus unaffected but we have \begin{align} f^{\perp}_{k} & =\frac{1}{MT}\int_{0}^{MT}F_{\perp}(t)\exp\left(-i\frac{2\pi kt}{MT}\right)dt\\ & =\frac{1}{MT}\sum_{m=1}^{M}\int_{(m-1)T}^{mT}F_{\perp}(t)e^{-i\Phi_{r,m}}\exp\left(-i\frac{2\pi kt}{MT}\right)dt\\ & =Z_{r,M}\tilde{f}^{\perp}_{k/M}, \end{align} where $Z_{r,M}=\frac{1}{M}\sum_{m=1}^{M}e^{i[\Phi_{r,m}-2\pi k(m-1)/M]}$. Because $\Phi_{r,m}$ is chosen randomly, $\Phi_{r,m}-2\pi k(n-1)/M$ is also random and we can write \begin{equation} Z_{r,M}=\frac{1}{M}\sum_{m=1}^{M}\exp(i\Phi_{r,m}), \end{equation} which is Eq. (5) in the main text. Here $Z_{r,M}$ is a sum of random complex phases and represents a 2D random walk. It can be shown that $|Z_{r,M}|^2$ has the average $\langle|Z_{r,M}|^2\rangle=1/M$ and the variance $\langle(|Z_{r,M}|^2- \langle|Z_{r,M}|^2\rangle)^2\rangle =(M-1)/M^3$. For example, the average can be obtained as follows. By definition, \begin{align} |Z_{r,M}|^2 & = \frac{1}{M^2}\sum_{m,n=1}^{M}\exp[i(\Phi_{r,m}-\Phi_{r,n})] \\ & = \frac{1}{M^2} \left[M + \sum_{m\neq n}^{M}e^{i(\Phi_{r,m}-\Phi_{r,n})}\right]. \end{align} Therefore, $\langle|Z_{r,M}|^2\rangle=1/M$ because the average of independent random phases is zero. Similarly, one obtains the variance of $|Z_{r,M}|^2$. Consider the signal of a single nuclear spin. The population signal of expected resonances is given by $P=\cos^2(\frac{1}{2} |f^{z}_{k}|A_{\perp}MT)$, where $A_\perp$ is the perpendicular coupling strength to a single spin-half~\cite{sm:lang2017enhanced}. When the signals are weak this can be approximated by $P=1- (\frac{1}{2}|f^{z}_{k}|A_{\perp}MT)^2$ thus the signal contrast is proportional to $M^2$. This is unaffected by the addition of the random phase as $F_z(t)$ is insensitive to the pulse phases. The spurious signal of a nuclear spin is given by $P =1-\sin^2(\frac{1}{2}A_{\perp}|f^{\perp}_{k}|MT)\cos^2(\phi^{\perp}_{k})$, where $\phi^{\perp}_{k}$ is the complex phase of $f^{\perp}_{k}$~\cite{sm:lang2017enhanced}. For the standard protocol, we have $P= 1-\sin^2(\frac{1}{2}A_{\perp}|\tilde{f}^{\perp}_{k/M}|MT)\cos^2(\phi^{\perp}_{k})$. When the signal is weak this can be approximated by $P \approx 1-(\frac{1}{2}A_{\perp}|\tilde{f}^{\perp}_{k/M}|MT)^2\cos^2(\phi^{\perp}_{k})$ so when no random phase is added the spurious signal contrast is proportional to $M^2$. When the random phase is added the expected value of the signal contrast is given by $P \approx 1 - \frac{1}{8}M(T A_{\perp}|\tilde{f}^{\perp}_{k/M}|)^2$ (using $\langle |f^{\perp}_{k}|^2 \rangle = \langle |Z_{r,M}\tilde{f}^{\perp}_{k/M}|^2 \rangle = |\tilde{f}^{\perp}_{k/M}|^2/M$ and $\langle\cos^2(\phi_\perp^k)\rangle = 1/2$). Compared with standard repetitions of a basic pulse unit, this contrast only grows proportional to $M/2$ thus providing a significant suppression of spurious signals completely independent of the used pulse sequence. As shown above, the variance of the spurious signal due to random phases is determined by the variance of $|Z_{r,M}|^2$ (which is $(M-1)/M^3$). When one repeats the randomisation protocol with $K$ realizations of the random phase sequences $\{\Phi_{r,m}\}$ and average out the measure signals, the variance is further reduced by a factor of $1/K$ according to the central limit theorem. \section{Enhancing sequence robustness} For simplicity, in the following discussion we neglect the effect of the environment and concentrate on static control imperfections. \subsection{Evolution operator of a basic pulse unit} The evolution driven by a single $\pi$ pulse with control errors takes the general form \begin{equation} \hat{U}_{\pi}(\phi)=\left(\begin{array}{cc} e^{-i\alpha}\sin\epsilon & ie^{-i(\beta+\phi)}\cos\epsilon\\ ie^{i(\beta+\phi)}\cos\epsilon & e^{i\alpha}\sin\epsilon \end{array}\right). \end{equation} We assume that each pulse has the same static errors, that is, $\alpha$, $\beta$, $\epsilon$ are the same for all pulses. The pulse phase $\phi$ determined by the initial phase of the driving field is a controllable parameter. When $\epsilon = 0$ and $\beta=0$, $\hat{U}_{\pi}(\phi)$ describes a perfect $\pi$ pulse. Consider a basic unit with $N$ $\pi$ pulses applied at $t_{j}$ $(j=1,\ldots,N)$ with phases $\phi_{j}$. For simplicity, we use the transformation $t_{j+1}-t_{j}=\tau_{j}+\tau_{j+1}$ with $\tau_{0}\equiv0$. This transformation splits $t_{j+1}-t_{j}$ into two parts where $\tau_{j}$ ($\tau_{j+1}$) is associate with the $j$th ($(j+1)$th) pulses. From the definition, we have \begin{align} \tau_{N+1} & =(t_{N+1}-t_{N})-\tau_{N}\\ & =(-1)^{N}\sum_{j=0}^{N}(-1)^{j}(t_{j+1}-t_{j}). \end{align} by recursively using $\tau_{j+1}=(t_{j+1}-t_{j})-\tau_{j}$. Because a a basic DD unit is designed to eliminate static dephasing noise, the timing of the sequence satisfy $\sum_{j=0}^{N}(-1)^{j}(t_{j+1}-t_{j})=0$. In other words, $\tau_{N+1}=0$ for a basic pulse unit. With $\tau_{0}=\tau_{N+1}=0$ and that a detuning $\Delta$ of the control field introduces a control phase error $\Delta(t_{j+1}-t_{j})=\Delta(\tau_{j+1}+\tau_{j})$ during the times $t_{j}$ and $t_{j+1}$, the propagator of a basic pulse unit can be written as \begin{equation} \hat{U}_{{\rm unit}}=\hat{U}_{N}\hat{U}_{N-1}\cdots \hat{U}_{2}\hat{U}_{1}, \end{equation} by combining the contribution of a $\pi$ pulse and the free evolution we obtain \begin{align} \hat{U}_{j} & =\left(\begin{array}{cc} e^{-i[\alpha+(\tau_{j}+\tau_{j-1})\Delta]}\sin\epsilon & ie^{-i[\beta+\phi_{j}-(\tau_{j}+\tau_{j-1})\Delta]}\cos\epsilon\\ ie^{i[\beta+\phi_{j}-(\tau_{j}+\tau_{j-1})\Delta]}\cos\epsilon & e^{i[\alpha+(\tau_{j}+\tau_{j-1})\Delta]}\sin\epsilon \end{array}\right).\nonumber\\ & =\left(\begin{array}{cc} e^{-i[\alpha+(\tau_{j}+\tau_{j-1})\Delta]}\epsilon & ie^{-i[\beta+\phi_{j}-(\tau_{j}+\tau_{j-1})\Delta]}\\ ie^{i[\beta+\phi_{j}-(\tau_{j}+\tau_{j-1})\Delta]} & e^{i[\alpha+(\tau_{j}+\tau_{j-1})\Delta]}\epsilon \end{array}\right)+O(\epsilon^2), \end{align} \begin{figure} \caption{Spectra of a single NV centre coupled to both individual $^{13}$C spins and the background $^{13}$C spin bath. a) The readout in x and -x-bases highlights the saturation feature typical for a bath. b) Comparison of standard XY8 and its randomisation version. The randomisation of the $\pi$ pulse phases suppresses the spurious signals efficiently.} \label{fig_bath_single} \end{figure} For two pulses, we find \begin{equation} \hat{U}_{j+1}\hat{U}_{j}=\left(\begin{array}{cc} e^{i\varphi_{j}} & ic_{j}\epsilon\\ ic_{j}^{*}\epsilon & -e^{-i\varphi_{j}} \end{array}\right)+O(\epsilon^{2}), \end{equation} where \begin{equation} \varphi_{j}= \Delta(\tau_{j+1}-\tau_{j-1})-(\phi_{j+1}-\phi_{j})+\pi, \end{equation} and \begin{equation} c_{j}=e^{-i[\beta+\phi_{j}+\alpha+\Delta(\tau_{j+1}-\tau_{j-1})]}+e^{-i[\beta+\phi_{j+1}-\alpha-\Delta(\tau_{j+1}+2\tau_{j}+\tau_{j-1})]}, \end{equation} is a sum of phase factors where each term has a $\phi_{j}$ or $\phi_{j+1}$. Timing the $U_{j}$ recursively and using $\tau_{0}=\tau_{N+1}=0$, we obtain for even $N$ \begin{equation} \hat{U}_{{\rm unit}}=\left(\begin{array}{cc} e^{i\varphi} & iC\epsilon\\ iC^{*}\epsilon & e^{-i\varphi} \end{array}\right)+O(\epsilon^{2}), \label{sm:eq:UunitEven} \end{equation} where \begin{equation} \varphi=\sum_{j=1}^{N/2}\left[\phi_{2j-1}-\phi_{2j}+\pi\right], \end{equation} and $C$ is a sum of phase factors where each term has an independent sum of the phases $\phi_{j}$. In deed, Eq.~(\ref{sm:eq:UunitEven}) has the general form of a pulse sequence with an even number of $\pi$ pulses with respect to the leading order error $\epsilon$~\cite{sm:genov2017arbitrarily}. Similarly, we have for odd $N$, \begin{equation} \hat{U}_{{\rm unit}}=\left(\begin{array}{cc} C^{\prime*}\epsilon & ie^{-i(\varphi+\beta)}\\ ie^{i(\varphi+\beta)} & C^{\prime}\epsilon \end{array}\right)+O(\epsilon^{2}),\label{sm:eq:UunitOdd} \end{equation} where \begin{equation} \varphi=\sum_{j=1}^{(N-1)/2} \left[\phi_{2j-1}-\phi_{2j}+\pi\right]+\phi_{N}, \end{equation} and $C^\prime$ is a sum of phase factors where each term has an independent sum of the phases $\phi_{j}$. For the case that the lower-order errors of single $\pi$ pulses have been compensated by a robust sequence, one can still write the propagator in terms of the leading order error that has not been compensated by the sequence. The evolution operator of a single pulse sequence unit still has a general form given by Eq.~(\ref{sm:eq:UunitEven}) or (\ref{sm:eq:UunitOdd}), but may have another error $\epsilon_{\varphi}$ added to $\varphi$. For many sequences, such as the CP~\cite{sm:carr1954effects}, XY8~\cite{sm:gullion1990new}, AXY8~\cite{sm:casanova2015robust}, YY8~\cite{sm:shu2017unambiguous}, and UR-($4n+2$) ($n=1,2,\ldots$)~\cite{sm:genov2017arbitrarily} sequences, $\epsilon_{\varphi}$ is a higher-order error compared with $\epsilon$ and therefore can be neglected in the leading order error analysis. \subsection{Standard protocol} It is obvious that the control errors coherently accumulate in the standard protocol where the basic pulse unit is repeated $M$ times as $\hat{U} = (\hat{U}_{\rm unit})^M$. For example, for even $N$ and $\varphi=0$, \begin{equation} \hat{U} = \left(\begin{array}{cc} 1 & i M C \epsilon \\ i M C^{*} \epsilon & 1 \end{array}\right)+O(\epsilon^{2}), \end{equation} where the error $MC\epsilon$ scales linearly with $M$. \subsection{Randomisation protocol} When one adds a random global phase $\Phi_{r,m}$ on all the $\pi$ pulses in a basic DD unit, each $\hat{U}_{{\rm unit}}$ becomes \begin{equation} \hat{U}_{{\rm unit}}(\Phi_{r,m})=\left(\begin{array}{cc} e^{i\varphi} & iCe^{-i\Phi_{r,m}}\epsilon\\ iC^{*}e^{i\Phi_{r,m}}\epsilon & e^{-i\varphi} \end{array}\right)+O(\epsilon^{2}). \end{equation} For two $U_{{\rm unit}}$, we have \begin{equation} \hat{U}_{{\rm unit}}(\Phi_{r,m+1})\hat{U}_{{\rm unit}}(\Phi_{r,m})=\left(\begin{array}{cc} e^{2i\varphi} & iZ_m\epsilon\\ iZ_m^{*}\epsilon & e^{-2i\varphi} \end{array}\right)+O(\epsilon^{2}), \end{equation} where $Z_m=e^{-i\varphi} C (e^{-i\Phi_{r,m+1}}+e^{-i(\Phi_{r,m}-2\varphi)})$ is a sum of two phase factors and can be equally written as $Z_j=e^{-i\varphi} C (e^{-i\Phi_{r,m+1}}+e^{-i\Phi_{r,m}})$ for random phases $\Phi_{r,m}$ and $\Phi_{r,m+1}$. By mathematical induction, the evolution operator of $M$ DD units with random phases $\{\Phi_{r,m}\}$ is \begin{align} \hat{U}_{M} & =\hat{U}_{{\rm unit}}(\Phi_{r,M})\cdots \hat{U}_{{\rm unit}}(\Phi_{r,2})\hat{U}_{{\rm unit}}(\Phi_{r,1}),\\ & =\left(\begin{array}{cc} e^{iM\varphi} & i Z_{r,M}MC\epsilon\\ iZ_{r,M}^{*}MC^{*}\epsilon & e^{-iM\varphi} \end{array}\right)+O(\epsilon^{2}), \end{align} where the error $MC\epsilon$ is suppressed by the factor $Z_{r,M}=\frac{1}{M}\sum_{m=1}^{M}\exp(i\Phi_{r,m})$ for the random phases $\{\Phi_{r,m}\}$. This result is valid for an odd number $N$ of pulses as well. \section{Additional experiments} \label{add_exp} One of the most important advantages of quantum sensors is the possibility to measure quantum signals, such as hyperfine fields of single spins. This is highly relevant for the characterization of quantum systems. The randomisation protocol efficiently suppresses both spurious harmonics from a bath as well as from single spins. In Fig. \ref{fig_bath_single}(a) we show the spectrum of an NV center that couples to both individual $^{13}$C spins and the background $^{13}$C spin bath. The signal of the bath is centered around the bare Larmor of $^{13}$C at this bias field and shows the typical saturation highlighted by measuring the spectra for both x-basis and -x-basis readout. The signal of at least one strongly coupled spin is shifted to higher frequencies due to the hyperfine coupling and it overlaps for the different readout bases. In Fig. \ref{fig_bath_single}(b) we compare the spectra measured with standard XY8 and the randomisation version. The identical signal shape and amplitude of the non-spurious signals verify that the randomized version does not alter the signal accumulation. In order to amplify the spurious harmonics, we use larger number of $\pi$-pulses ($M$=60 and 100). We observe peaks at 2$\nu_0$ and 4$\nu_0$ for the standard XY8 method. For the same bias field the Larmor frequency of $^1$H is about 1.81\,MHz, what would make a differentiation very difficult. These spurious signals can be efficiently suppressed with the randomisation protocol. \end{document}

\begin{document} \begin{abstract} This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal K_\lambda=-\Delta+\frac\lambda{|x|^2}$$ in the energy space $H_\lambda^1:=\{f\in L^2,\quad\sqrt{\mathcal{K}_\lambda}f\in L^2\}$. In this paper, we develop a well-posedness theory and investigate the blow-up of solutions in $H_\lambda^1$. Furthermore we present a dichotomy between energy bounded and non-global existence of solutions under the ground state threshold. To this end, we use Caffarelli-Kohn-Nirenberg weighted interpolation inequalities and some equivalent norms considering $\mathcal K_\lambda$, which make it possible to control the non-linearity involving the singularity $|x|^{-\tau}$ as well as the inverse square potential. The novelty here is the investigation of the energy critical regime which remains still open and the challenge is to deal with three technical problems: a non-local source term, an inhomogeneous singular term $|\cdot|^{-\tau}$, and the presence of an inverse square potential. \end{abstract} \maketitle \tableofcontents \renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}} \section{Introduction} In this paper we are concerned with the Cauchy problem for the inhomogeneous generalized Hartree equation with inverse square potential \begin{equation} \begin{cases}\label{S} i\partial_t u-\mathcal K_\lambda u=\epsilon |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u,\\ u(x,0)=u_0(x), \quad (x,t)\in \mathbb{R}^n \times \mathbb{R}, \end{cases} \end{equation} where $p>2$, $\epsilon=\pm1$, and $\mathcal K_\lambda:=-\Delta+\frac\lambda{|x|^{2}}$ satisfying $\lambda>-\frac{(n-2)^2}{4}$. Here the case $\epsilon =1$ is \textit{defocusing}, while the case $\epsilon =-1$ is \textit{focusing}. The Riesz potential is defined on $\mathbb{R}^n$ by $$I_\alpha:=\frac{\Gamma(\frac{n-\alpha}2)}{\Gamma(\frac\alpha2)\pi^\frac{n}22^\alpha}\,|\cdot|^{\alpha-n},\quad 0<\alpha<n.$$ The assumption on $\lambda$ comes from the sharp Hardy inequality \cite{abde}, \begin{equation}\label{prt} \frac{(n-2)^2}4\int_{\mathbb{R}^n}\frac{|f(x)|^2}{|x|^2}\,dx\leq \int_{\mathbb{R}^n}|\nabla f(x)|^2 dx, \end{equation} which guarantees that $\mathcal K_\lambda$ is thepositive self-adjoint extension of $-\Delta+\lambda/|x|^{-2}$. It is known that in the range $-\frac{(n-2)^2}4 <\lambda< 1-\frac{(n-2)^2}4$, the extension is not unique (see \cite{ksww,ect}). In such a case, one picks the Friedrichs extension (see \cite{ksww,pst}). The problem \eqref{S} arises in various physical contexts. In the linear regime ($\epsilon=0$), the considered Schr\"odinger equation models quantum mechanics \cite{ksww,haa}. In the non-linear regime without potentials, namely $\lambda=0\neq\epsilon$, the equation \eqref{S} is of interest in the mean-field limit of large systems of non-relativistic bosonic atoms and molecules in a regime where the number of bosons is very large, but the interactions between them are weak \cite{fl,hs,pg,mpt}. The homogeneous problem associated to the considered problem \eqref{S}, specifically, when $\lambda=\tau=0$, has several physical origins such as quantum mechanics \cite{pg,pgl} and Hartree-Fock theory \cite{ll}. The particular case $p=2$ and $\lambda=\tau=0$ is called standard Hartree equation. It is a classical limit of a field equation describing a quantum mechanical non-relativistic many-boson system interacting through a two body potential \cite{gvl}. Now, let us return to the mathematical aspects of the generalized Hartree equation \eqref{S}. Recall the critical Sobolev index. If $u(x,t)$ is a solution of \eqref{S}, so is the family $$u_\delta(x,t):=\delta^{\frac{2-2\tau+\alpha}{2(p-1)}} u(\delta x, \delta^2 t),$$ with the re-scaled initial data $u_{\delta,0}:=u_{\delta}(x,0)$ for all $\delta>0$. Then, it follows that \begin{equation*} \|u_{\delta,0}\|_{\dot H^1}=\delta^{1-\frac n2 +\frac{2-2\tau+\alpha}{2(p-1)}}\|u_0\|_{\dot H^1}. \end{equation*} If $p=1+\frac{2-2\tau+\alpha}{n-2}$, the scaling preserves the $\dot H^1$ norm of $u_0$, and in this case, \eqref{S} is referred as the energy-critical inhomogeneous generalized Hartree equation. Moreover, the solution to \eqref{S} satisfies the mass and energy conservation, where the mass conservation is \begin{equation} \mathcal M[u(t)]:=\int_{\mathbb{R}^n} |u(x,t)|^2 dx = \mathcal M [u_0], \end{equation} and the energy conservation is \begin{equation} \mathcal E[u(t)]:= \int_{\mathbb{R}^n}\Big(|\nabla u|^2 +\lambda |x|^{-2} |u|^2\Big)\,dx+ \frac{\epsilon}{p}\mathcal P[u(t)]=\mathcal E[u_0], \end{equation} where the potential energy reads $$\mathcal P[u(t)]:=\int_{\mathbb{R}^n} |x|^{-\tau}\big(I_\alpha *|\cdot|^{-\tau}|u|^p\big)|u|^p dx.$$ To the best of our knowledge, this paper is the first one dealing with the energy-critical inhomogeneous Hatree equation with inverse square potential, precisely \eqref{S} with $\lambda\neq0$. The main contribution is to develop a local well-posedness theory in the energy-critical case, as well as to investigate the blow-up of the solution in energy space for the inhomogeneous generalized Hartree equation \eqref{S}. Precisely, the local theory is based on the standard contraction mapping argument via the availability of Strichartz estimates. More interestingly, we take advantage of some equivalent norms considering the operator $\mathcal{K}_\lambda$, namely $\|\sqrt{\mathcal{K}_\lambda}u\|_r\simeq\|u\|_{\dot W^{1,r}}$, which makes it possible to apply the contraction mapping principle without directly handling with the operator. In the repulsive regime($\epsilon=-1$), we prove that the solution blows up in finite time without assuming the classical assumption such as radially symmetric or $|x|u_0 \in L^2$. The blow-up phenomenon is expressed in terms of the non-conserved potential energy, which may give a criteria in the spirit of \cite{vdd}, which implies in particular the classical phenomena under the ground state threshold in the spirit of \cite{km}. In this paper, we deal with three technical problems by the equation \eqref{S}, a non-local source term, the inhomogeneous singular term $|\cdot|^{-\tau}$, and the presence of an inverse square potential. Indeed, in order to deal with the singular term $|\cdot|^{-\tau}$ in Lebesgue spaces, the method used in the literature decomposes the integrals on the unit ball and it's complementary (see, for example, \cite{mt}). However, this is no more sufficient to conclude in the energy critical case. For $\lambda=0$, the first author used some Lorentz spaces with the useful property $|\cdot|^{-\tau}\in L^{\frac{n}{\tau},\infty}$. To overcome these difficulties, we make use of some Caffarelli-Kohn-Nirenberg weighted interpolation inequalities which is different from the existing approaches. Before stating our results, we introduce some Sobolev spaces defined in terms of the operator $\mathcal K_\lambda$ as the completion of $C^\infty_0(\mathbb{R}^n)$ with respect to the norms \begin{align*} \|u\|_{\dot W^{1,r}_\lambda}&:=\|\sqrt{\mathcal K_\lambda} u\|_{L^r} \quad \textnormal{and} \quad \|u\|_{W^{1,r}_\lambda}:=\|\langle \sqrt{\mathcal K_\lambda}\rangle u\|_{L^r}, \end{align*} where $\langle \cdot\rangle:=(1+|\cdot|^2)^{1/2}$ and $L^r:=L^r(\mathbb{R}^n)$. We denote also the particular Hilbert cases $\dot W^{1,2}_\lambda=\dot H^1_\lambda$ and $W^{1,2}_\lambda=H^1_\lambda$. We note that by the definition of the operator $\mathcal K_\lambda$ and Hardy estimate \eqref{prt}, one has \begin{align*} \|u\|_{\dot H^1_\lambda}&:=\|\sqrt{\mathcal K_\lambda}u\|=\big(\|\nabla u\|^2+\lambda\||x|^{-1}u\|^2\big)^\frac12\simeq \|u\|_{\dot H^1}, \end{align*} where we write for simplicity $\|\cdot\|:=\|\cdot\|_{L^2(\mathbb{R}^n)}$. \subsection{Well-posedness in the energy-critical case} The theory of well-poseddness for the inhomogeneous Hartree equation ($\lambda=0$ in \eqref{S}) has been extensively studied in recent several years and is partially understood. (See, for examples, \cite{mt,sa, kls, sk} and references therein). For related results on the scattering theory, see also \cite{sx} for spherically symmetric datum and \cite{cx} in the general case. Our first result is the following well-posedness in the energy-critical case. \begin{thm}\label{loc} Let $n\ge3$, $\lambda >-\frac{(n-2)^2}{4}$ and $2\kappa=n-2-\sqrt{(n-2)^2+4\lambda}$. Assume that \begin{equation}\label{1.6} 0<\alpha<n, \quad 2\kappa < n-2-\frac{2(n-2)}{3n-2+2\sqrt{9n^2+8n-16}} \end{equation} and \begin{equation}\label{1.7} \frac{\alpha}{2}-\frac{n+2+\sqrt{9n^2+8n-16}}{2}<\tau < \frac{\alpha}{2}-\max\{\frac{n-4}{2}, \frac{n-4}{n},\frac{\kappa}{n-2-2\kappa}-\frac{n}{4}\}. \end{equation} Then, for $u_0 \in H_{\lambda}^1(\mathbb{R}^n)$, there exist $T>0$ and a unique solution $$u\in C([0,T]; H_\lambda^1) \cap L^q([0,T];W_{\lambda}^{1,r})$$ to \eqref{S} with $p=1+\frac{2-2\tau+\alpha}{n-2}$ for any admissible pair $(q,r)$ in Definition \ref{dms}. Furthermore, the continuous dependence on the initial data holds. \end{thm} We also provide the small data global well-posedness and scattering results as follows: \begin{thm}\label{glb} Under the same conditions as in Theorem \ref{loc} and the smallness assumption on $\|u_0\|_{H_{\lambda}^1}$, there exists a unique global solution $$u\in C([0,\infty); H_\lambda^1) \cap L^q([0,\infty);W_{\lambda}^{1,r})$$ to \eqref{S} with $p=1+\frac{2-2\tau+\alpha}{n-2}$ for any admissible pair $(q,r)$. Furthermore, the solution scatters in $H_\lambda^1$, i.e., there exists $\phi\in H_\lambda^1$ such that $$\lim_{t\to\infty} \|u(t)-e^{-it\mathcal{K}_{\lambda}}\phi\|_{H_{\lambda}^1}=0.$$ \end{thm} \subsection{Blow-up of energy solutions} We now turn to our attention to blow-up of the solution to \eqref{S} under the ground state threshold, in the focusing regime. A particular global solution of \eqref{S} with $\epsilon=-1$ is the stationary solution to \eqref{S}, namely \begin{equation}\label{E} -\Delta \varphi+\frac{\lambda}{|x|^2}\varphi=|x|^{-\tau}|\varphi|^{p-2}(I_\alpha *|\cdot|^{-\tau}|\varphi|^p)\varphi,\quad 0\neq \varphi\in {H^1_\lambda}. \end{equation} Such a solution called ground state plays an essential role in the focusing regime. The following result is the existence of ground states to \eqref{E}. \begin{thm}\label{gag} {Let $n\geq3$, $\lambda>-\frac{(n-2)^2}4$ and $p=1+\frac{2-2\tau+\alpha}{n-2}$. Assume that \begin{equation}\label{as} 0<\alpha<n \quad \text{and} \quad 0<\tau< 1+\frac{\alpha}{n}. \end{equation}\label{inte} Then, the following inequality holds: \begin{equation}\label{gagg} \int_{\mathbb{R}^n} |x|^{-\tau}|u|^p (I_\alpha \ast |\cdot|^{-\tau}|u|^p) \leq C_{N,\tau,\alpha,\lambda} \big\|\sqrt{\mathcal{K}_{\lambda} }u\big\|^{2p}. \end{equation}} Moreover, there exists $\varphi\in H_{\lambda}^1$ a ground state solution to \eqref{E}, which is a minimizing of the problem \begin{equation}\label{min} \frac1{C_{N,\tau,\alpha,\lambda}}=\inf\Big\{\frac{\|\sqrt{\mathcal K_\lambda} u\|^{2p}}{\mathcal P[u]},\quad0\neq u\in H^1_\lambda\Big\}. \end{equation} \end{thm} \begin{rems} \begin{enumerate} \item[1.] Theorem \ref{gag} does not require to assume that $p\geq2$; \item[2.] $C_{N,\tau,\alpha,\lambda}$ denotes the best constant in the inequality \eqref{gagg}; \item[3.] compared with the homogeneous regime $\tau=0$, the minimizing \eqref{min} is never reached for $\lambda>0$, see \cite{kmvzz}. \end{enumerate} \end{rems} Here and hereafter, we denote $\varphi$ a ground state solution of \eqref{E} and the scale invariant quantities \begin{align*} \mathcal{ME}[u_0]:=\frac{\mathcal E[u_0]}{\mathcal E[\varphi]},\quad \mathcal{MG}[u_0]:=\frac{\|\sqrt{\mathcal K_\lambda} u_0\|}{\|\sqrt{\mathcal K_\lambda} \varphi\|},\quad \mathcal{MP}[u_0]:=\frac{\mathcal P[u_0]}{\mathcal P[\varphi]}. \end{align*} The next theorem gives a blow-up phenomenon in the energy-critical focusing regime under the ground state threshold. \begin{thm}\label{t1} Under the assumptions in \ref{loc} and $\epsilon=-1$, let $\varphi$ be a ground state solution to \eqref{E} and $u\in C_{T^*}(H^1_\lambda)$ be a maximal solution of the focusing problem \eqref{S}. If \begin{equation} \label{ss'} \sup_{t\in[0,T^*)}\mathcal I[u(t)]<0, \end{equation} then $u$ blows-up in finite or infinite time. Here, $\mathcal I[u]:=\|\sqrt{\mathcal K_\lambda} u\|^2-\mathcal{P}[u]$. \end{thm} \begin{rems} \begin{enumerate} \item[1.] $u$ blows-up in infinite time means that it is global and there is $t_n\to\infty$ such that $\|\sqrt{\mathcal K_\lambda} u(t_n)\|\to\infty$; \item[2.] the threshold is expressed in terms of the potential energy $\mathcal P[u]$, which is a non conserved quantity; \item[3.] the theorem here doesn't require the classical assumptions such as spherically symmetric data or $|x|u_0\in L^2$; \item[4.] a direct consequence of the variance identity is that the energy solution to \eqref{S} blows-up in finite time if $|x|u_0\in L^2$ and \eqref{ss'} is satisfied; \end{enumerate} \end{rems} The next result is a consequence of Theorem \ref{t1}. \begin{cor}\label{t2} Under the assumptions in Theorem \ref{loc} and $\epsilon=-1$, Let $\varphi$ be a ground state of \eqref{E} and $u_0\in H^1_\lambda$ such that \begin{equation} \label{t11} \mathcal{ME}[u_0]<1. \end{equation} If we assume that \begin{equation}\label{t13} \mathcal{MG}[u_0]>1,\end{equation} then the energy solution of \eqref{S} blows-up in finite or infinite time \end{cor} \begin{rems} \begin{enumerate} \item[1.] The assumptions of the above result are more simple to check than \eqref{ss'}, because they are expressed in terms of conserved quantities; \item[2.] the above ground state threshold has a deep influence in the NLS context since the pioneering papers \cite{km,Holmer}; \end{enumerate} \end{rems} Finally, we close this subsection with some additional results which gives the boundedness of the energy solution. \begin{prop}\label{s} Under the assumptions in Theorem \ref{loc} and $\epsilon=-1$, let $\varphi$ be a ground state solution to \eqref{E} and $u\in C_{T^*}(H^1_\lambda)$ be a maximal solution of the focusing problem \eqref{S}. If \begin{equation} \label{ss} \sup_{t\in[0,T^*)}\mathcal{MP}[u(t)]<1, \end{equation} then $u$ is bounded in $H^1_\lambda$. \end{prop} The next is a consequence of Proposition \ref{s} \begin{cor}\label{s2} Under the assumptions in Theorem \ref{loc} and $\epsilon=-1$. Let $\varphi$ be a ground state of \eqref{E} and $u_0\in H^1_\lambda$ satisfying \eqref{t11} If \begin{equation}\label{t12} \mathcal{MG}[u_0]<1, \end{equation} then the energy solution of \eqref{S} is bounded. \end{cor} \begin{rem} \begin{enumerate} \item[1.] the global existence and energy scattering under the assumptions \eqref{ss} in Proposition \ref{s} or \eqref{t11}-\eqref{t12} in Corollary \ref{s2} is investigated in a paper in progress. \end{enumerate} \end{rem} The rest of this paper is organized as follows. In Section 2 we introduce some useful properties that we need. Section 3 develops a local theory and a global one for small datum. In section 4, the existence of ground states is established. Section 5 establishes blow-up of solutions under the ground state threshold and the boundedness of energy solutions. In the appendix, a Morawetz type estimate is proved. Throughout this paper, the letter $C$ stands for a positive constant which may be different at each occurrence. We also denote $A \lesssim B$ to mean $A \leq CB$ with unspecified constants $C>0$. \section{Preliminaries} In this section, we introduce some useful properties which will be utilized throughout this paper. We also recall the Strichartz estimates. Let us start with the Hardy-Littlewood-Sobolev inequality \cite{el} which is suitable for dealing with non-local source term in \eqref{S}: \begin{lem}\label{hls} Let $n\geq1$ and $0 <\alpha < n$. \begin{enumerate} \item[1.] Let $s\geq1$ and $r>1$ such that $\frac1r=\frac1s+\frac\alpha n$. Then, $$\|I_\alpha*g\|_{L^s}\leq C_{n,s,\alpha}\|g\|_{L^r}.$$ \item[2.] Let $1<s,r,t<\infty$ be such that $\frac1r +\frac1s=\frac1t +\frac\alpha n$. Then, $$\|f(I_\alpha*g)\|_{L^t}\leq C_{n,t,\alpha}\|f\|_{L^r}\|g\|_{L^s}.$$ \end{enumerate} \end{lem} The following lemma is a weighted version of the Sobolev embedding, that is, a special case of Caffarelli-Kohn-Nirenberg weighted interpolation inequalities \cite{sgw,csl}: {\begin{lem}\label{ckn} Let $n\geq1$ and $$1< p\leq q<\infty, \quad -\frac nq<b\leq a<\frac n{p'} \quad \text{and} \quad a-b-1=n\Big(\frac1q-\frac1p\Big).$$ Then, $$\||x|^{b}f\|_{L^q}\leq C\||x|^a\nabla f\|_{L^p}.$$ \end{lem}} Now, we describe several properties related to the operator $\mathcal K_\lambda.$ Since $\|f\|_{H^1} \simeq \|f\|_{H_\lambda^1}$, one has the following compact Sobolev injection (\cite[Lemma 3.1]{cg}): \begin{lem}\label{compact} Let $n\geq3$, $0<\tau<2$ and $2<r<\frac{2(n-\tau)}{n-2}$. Then, the following injection is compact: $$H^1_\lambda\hookrightarrow\hookrightarrow L^{r}(|x|^{-\tau}\,dx) .$$ \end{lem} We also have the following equivalent norms to Sobolev ones, see \cite{kmvzz} and \cite[Remark 2.2]{cg}: \begin{lem}\label{2.2} Let $n\geq3$, $\lambda>-\frac{(n-2)^2}4$, $1<r<\infty$ and $2\kappa=n-2-\sqrt{(n-2)^2+4\lambda}$. Then, \begin{enumerate} \item[1.] if $\frac{1+\kappa}n<\frac1r<\min\{1,1-\frac\kappa n\}$, thus, $\|f\|_{\dot W^{1,r}}\lesssim \|f\|_{\dot W_\lambda^{1,r}}$ \item[2.] if $\max\{\frac{1}n,\frac{\kappa}n\}<\frac1r<\min\{1,1-\frac\kappa N\}$, thus, $\|f\|_{\dot W_\lambda^{1,r}}\lesssim\|f\|_{\dot W^{1,r}}$ \end{enumerate} \end{lem} Finally, we recall the Strichartz estimates. As we shall see, the availability of these estimates is the key role in the proof of Theorem \ref{loc}. \begin{defi}\label{dms} Let $n\ge3$. We say that $(q,r)$ is an admissible pair if it satisfies \begin{equation*} 2\le q \le \infty, \quad 2\le r \le \frac{2n}{n-2} \quad \text{and} \quad \frac2q+\frac{n}{r}=\frac{n}{2}. \end{equation*} \end{defi} \begin{prop}\cite{bpst,zz,df}\label{str} Let $n\geq3$, $\lambda>-\frac{(n-2)^2}4$. Then, there exists $C>0$ such that \begin{enumerate} \item[1.] $\|e^{-it\mathcal K_\lambda}f\|_{L_t^q(L_x^r)}\leq C\|f\|,$ \item[2.] $\|\int_0^{t_1}e^{-i(t-t_1)\mathcal K_\lambda}F(\cdot,t_1)dt_1\|_{L_t^q(L_x^r)}\leq C\|F\|_{L_t^{\tilde q'}L_x^{\tilde r'}}.$ \end{enumerate} \end{prop} Finally, one gives a classical Morawetz estimate proved in the appendix. Let $\phi:\mathbb{R}^n\to\mathbb{R}$ be a smooth function and define the variance potential $$V_\phi(t):=\int_{\mathbb{R}^n}\phi(x)|u(x,t)|^2\,dx,$$ and the Morawetz action $$M_\phi(t):=2\Im\int_{\mathbb{R}^n} \bar u(\xi_ju_j)\,dx=2\Im\int_{\mathbb{R}^n} \bar u(\nabla\phi\cdot\nabla u)\,dx.$$ \begin{prop}\label{mrwz} Let $\phi:\mathbb{R}^N \rightarrow \mathbb{R}$ be a radial, real-valued multiplier, $\phi=\phi(|x|)$. Then, for any solution $u\in C([0,T];H_\lambda^1)$ of the generalized Hartree equation \eqref{S} in the focusing sign with initial data $u_0\in H_{\lambda}^{1}$, the following virial-type identities hold: \begin{equation*} V'_\phi(t)= 2\Im\int_{\mathbb{R}^n} \bar u\nabla\phi\cdot\nabla u dx \end{equation*} and \begin{align*} V''_\phi(t)=M_\phi'(t)&=4\sum_{k,l=1}^{N}\int_{\mathbb{R}^n}\partial_l\partial_k\phi\Re(\partial_ku\partial_l\bar u)dx-\int_{\mathbb{R}^n}\Delta^2\phi|u|^2dx+4\lambda\int_{\mathbb{R}^n}\nabla\phi\cdot x\frac{|u|^2}{|x|^4}dx\\ &\qquad-\frac{2(p-2)}{p}\int_{\mathbb{R}^n}\Delta\phi|x|^{-\tau}|u|^p(I_\alpha*|\cdot|^{-\tau}|u|^{p})dx\\ &\qquad\qquad -\frac{4\tau}p\int_{\mathbb{R}^n}x\cdot\nabla\phi|x|^{-\tau-2}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\\ &\qquad\qquad -\frac{4(N-\alpha)}p\sum_{k=1}^N\int_{\mathbb{R}^n}|x|^{-\tau}|u|^{p}\partial_k\phi(\frac{x_k}{|\cdot|^2}I_\alpha*|\cdot|^{-\tau}|u|^p)dx. \end{align*} \end{prop} {\section{Well-posedness in the energy space}} In this section, we develop a local theory in the energy space $H_{\lambda}^1$, Theorem \ref{loc}. Moreover, we prove Theorem \ref{glb} about the global theory for small datum. Let us first denote the source term \begin{align*} \mathcal N&:=\mathcal N[u]:=|x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u. \end{align*} \subsection{Nonlinear estimates} We first establish some nonlinear estimates for $\mathcal{N}[u]$. These nonlinear estimates will play an important role in proving the well-posedness results applying the contraction mapping principle. Before stating the nonlinear estimates, we introduce some notations. We set $$\mathcal A=\{(q,r):(q,r) \,\,\text{is}\,\, \text{admissible}\},$$ and then define the norm $$\|u\|_{\Lambda(I)}=\sup_{(q,r)\in \mathcal A}\|u\|_{L_t^q(I;L_x^r)}$$ and its dual weighted norm $$\|u\|_{\Lambda'(I)}=\sup_{(\tilde q,\tilde r)\in \mathcal A}\|u\|_{L_t^{\tilde q'}(I;L_x^{\tilde r'})}$$ for any interval $I\subset \mathbb{R}.$ \begin{lem}\label{non} Let $n\ge3$, $\lambda >-\frac{(n-2)^2}{4}$ and $p=1+\frac{2-2\tau+\alpha}{n-2}$. Assume that \begin{equation}\label{ass1} 0<\alpha<n, \quad 2\kappa < \frac{5n-4-\sqrt{9n^2+8n-16}}{2} \end{equation} and \begin{equation}\label{ass2} \frac{\alpha}{2}-\frac{n+2+\sqrt{9n^2+8n-16}}{2}<\tau < \frac{\alpha}{2}-\max\{\frac{n-4}{2}, \frac{n-4}{n},\frac{\kappa}{n-2-2\kappa}-\frac{n}{4}\}. \end{equation} Then, {there is an admissible pair $(q,r)$, such that} \begin{equation}\label{non1} \|\sqrt{\mathcal{K}_{\lambda}}\,\mathcal{N}[u]\|_{\Lambda'(0,T)} \leq C \|u\|_{L_t^{q}(I;\dot W_{\lambda}^{1,r})}^{2p-1} \end{equation} and \begin{equation}\label{non2} \|\mathcal{N}[u]-\mathcal{N}[v]\|_{\Lambda'(0,T)} \leq C (\|u\|_{L_t^q(I;\dot W_{\lambda}^{1,r})}^{2p-2} +\|u\|_{L_t^q(I;\dot W_{\lambda}^{1,r})}^{2p-2})\|u-v\|_{L_t^q(I;L_x^r)}. \end{equation} \end{lem} \begin{proof} It is sufficient to show that there exist $(q,r)$ for which \begin{equation}\label{non11} \|\nabla\mathcal{N}[u]\|_{L_t^2(I;L_x^{\frac{2n}{n+2}})} \leq C \|\nabla u\|_{L_t^{q}(I;L_x^r)}^{2p-1} \end{equation} and \begin{equation}\label{non22} \|\mathcal{N}[u]-\mathcal{N}[v]\|_{L_t^2(I;L_x^{\frac{2n}{n+2}})} \leq C \|\nabla u\|_{L_t^q(I;L_x^r)}^{2p-2} \|u-v\|_{L_t^q(I;L_x^r)} \end{equation} hold for $\lambda, \alpha, \kappa, \tau, p$ given as in the lemma. Indeed, by applying the equivalent norms to Sobolev ones (see Lemma \ref{2.2}) we obtain the desired estimates \eqref{non1} and \eqref{non2} if \begin{equation}\label{ka} \max\{\frac1n,\frac{\kappa}n\}<\frac{n+2}{2n}<\min\{1,\frac{n-\kappa}n\}, \quad \frac{1+\kappa}{n}<\frac1{r}<\min\{1,\frac{n-\kappa}{n}\}, \end{equation} Here, one can easily see that the first condition in \eqref{ka} is always satisfied. Now we start to prove \eqref{non11}. Let us set \begin{equation}\label{st} \frac1q=\frac{1}{2(2p-1)}, \quad \frac{n-2}{2n}\leq \frac1r \leq \frac12, \quad \frac{2}q+\frac{n}{r}=\frac{n}{2}, \quad 0<\alpha<n. \end{equation} We first see that \begin{align*} \|&\nabla\mathcal{N}[u]\|_{L_x^{\frac{2n}{n+2}}}\\ &\lesssim\big\||x|^{-\tau-1}|u|^{p-1}(I_\alpha\ast|\cdot|^{-\tau}|u|^p)\big\|_{L_x^{\frac{2n}{n+2}}}+\big\||x|^{-\tau}|u|^{p-1}(I_\alpha \ast|\cdot|^{-\tau-1}|u|^{p})\big\|_{L_x^{\frac{2n}{n+2}}}\\ &+\||x|^{-\tau}|u|^{p-2}|\nabla u|(I_\alpha\ast|\cdot|^{-\tau}|u|^p)\|_{L_x^{\frac{2n}{n+2}}}+\||x|^{-\tau}|u|^{p-1}(I_\alpha\ast|\cdot|^{-\tau}|u|^{p-1}\nabla u)\|_{L_x^{\frac{2n}{n+2}}}\\ &:=A_1+A_2+A_3+A_4. \end{align*} The first term $A_1$ is bounded by using Lemmas \ref{hls} and \ref{ckn}, in turn, \begin{align*} \big\||x|^{-\tau-1}|u|^{p-1}|I_\alpha\ast|\cdot|^{-\tau}|u|^p|\big\|_{L_x^{\frac{2n}{n+2}}} &\lesssim \||x|^{-\tau-1}|u|^{p-1}\|_{L_x^{a_1}} \||x|^{-\tau}|u|^p\|_{L_x^{b_1}} \\ &=\||x|^{-\frac{\tau+1}{p-1}}u\|^{p-1}_{L_x^{(p-1)a_1}}\||x|^{-\frac{\tau}{p}}u\|^{p}_{L_x^{pb_1}}\\ &\lesssim \|\nabla u\|_{L_x^r}^{2p-1}, \end{align*} if \begin{equation}\label{c7} 0<\frac1{a_1},\frac1{b_1}<1, \quad \frac1{a_1}+\frac1{b_1}=\frac{n+2}{2n}+\frac{\alpha}{n}, \end{equation} \begin{equation}\label{c8} 0<\frac{1}{(p-1)a_1}\leq \frac1r\leq1,\quad 0\leq\frac{\tau+1}{p-1}<\frac{n}{(p-1)a_1} ,\quad \frac{\tau+1}{p-1}-1=\frac{n}{(p-1)a_1}-\frac{n}{r} \end{equation} \begin{equation}\label{c9} 0<\frac{1}{pb_1}\leq \frac1r\leq 1,\quad 0 \leq \frac{\tau}{p}<\frac{n}{pb_1},\quad \frac{\tau}{p}-1=\frac{n}{pb_1}-\frac{n}{r}. \end{equation} Similarly, \begin{align*} A_2&=\big\||x|^{-\tau}|u|^{p-1}(I_\alpha \ast|\cdot|^{-\tau-1}|u|^{p})\big\|_{L_x^{\frac{2n}{n+2}}}\\ &\leq \||x|^{-\tau}|u|^{p-1}\|_{L_x^{a_2}}\||x|^{-\tau-1}|u|^p\|_{L_x^{b_2}}\\ &\lesssim \|\nabla u\|^{2p-1}_{L_x^r} \end{align*} if \begin{equation}\label{c10} 0<\frac1{a_2}, \frac{1}{b_2}<1, \quad \frac{1}{a_2}+\frac{1}{b_2}=\frac{n+2}{2n}+\frac{\alpha}{n}, \end{equation} \begin{equation}\label{c11} 0<\frac1{(p-1)a_2}\leq \frac1r\leq 1, \quad 0\leq \frac{\tau}{p-1}<\frac{n}{(p-1)a_2}, \quad \frac{\tau}{p-1}-1=\frac{n}{(p-1)a_2}-\frac{n}{r}, \end{equation} \begin{equation}\label{c12} 0<\frac{1}{pb_2}\leq \frac1r \leq 1, \quad 0\leq \frac{\tau+1}{p}<\frac{n}{pb_2}, \quad \frac{\tau+1}{p}-1=\frac{n}{pb_2}-\frac{n}{r}. \end{equation} The third term $A_3$ is bounded by using Lemma \ref{hls}, H\"older's inequality and Lemma \ref{ckn} in turn as \begin{align*} \big\||x|^{-\tau}|u|^{p-2}|\nabla u||I_\alpha \ast|\cdot|^{-\tau}|u|^{p}|\big\|_{L_x^{\frac{2n}{n+2}}} &\lesssim \||x|^{-\tau}|u|^{p-2}|\nabla u|\|_{L_x^{a_1}} \||x|^{-\tau}|u|^p\|_{L_x^{b_1}}\\ &\leq \||x|^{-\tau}|u|^{p-2}\|_{L_x^{a_3}}\|\nabla u\|_{L_x^r} \||x|^{-\tau}|u|^p\|_{L_x^{b_1}}\\ &=\||x|^{-\frac{\tau}{p-2}}u\|^{p-2}_{L_x^{(p-2)a_3}} \|\nabla u\|_{L_x^r} \||x|^{-\frac{\tau}{p}}u\|^p_{L_x^{pb_1}}\\ &\lesssim \|\nabla u\|_{L_x^r}^{2p-1} \end{align*} if \begin{equation}\label{c1} 0<\frac1{a_1}, \frac1{b_1}<1, \quad \frac{1}{a_1}+\frac{1}{b_1}=\frac{n+2}{2n}+\frac{\alpha}{n},\quad \frac{1}{a_1}=\frac{1}{a_3}+\frac{1}{r}, \end{equation} \begin{equation}\label{c2} 0<\frac{1}{(p-2)a_3} \leq \frac1r\leq 1, \quad 0\leq \frac{\tau}{p-2}<\frac{n}{(p-2)a_3}, \quad \frac{\tau}{p-2}-1=\frac{n}{(p-2)a_3}-\frac{n}{r}, \end{equation} \begin{equation}\label{c3} 0<\frac1{pb_1}\leq \frac1r\leq 1, \quad 0\leq \frac{\tau}{p}<\frac{n}{pb_1}, \quad \frac{\tau}{p}-1=\frac{n}{pb_1}-\frac{n}{r}. \end{equation} Similarly, \begin{align*} A_4&=\||x|^{-\tau}|u|^{p-1}(I_\alpha\ast|\cdot|^{-\tau}|u|^{p-1}\nabla u)\|_{L_x^{\frac{2n}{n+2}}}\\ &\lesssim \||x|^{-\tau}|u|^{p-1}\|_{L_x^{a_2}}\||x|^{-\tau}|u|^{p-1}\nabla u\|_{L_x^{b_2}}\\ &\leq \||x|^{-\frac{\tau}{p-1}}|u|\|^{p-1}_{L_x^{(p-1)a_2}}\||x|^{-\frac{\tau}{p-1}}u\|^{p-1}_{L_x^{(p-1)b_4}}\|\nabla u\|_{L_x^r}\\ &\lesssim \|\nabla u\|_{L_x^r}^{2p-1} \end{align*} if \begin{equation}\label{c4} 0<\frac1{a_2}, \frac{1}{b_2}<1,\quad \frac1{a_2}+\frac1{b_2}=\frac{n+2}{2n}+\frac{\alpha}{n},\quad \frac{1}{b_2}=\frac{1}{b_4}+\frac1r \end{equation} \begin{equation}\label{c5} 0<\frac{1}{(p-1)a_2}\leq \frac1r\leq 1, \quad 0\leq \frac{\tau}{p-1}\leq \frac{n}{(p-1)a_2}, \quad \frac{\tau}{p-1}-1=\frac{n}{(p-1)a_2}-\frac{n}{r}, \end{equation} \begin{equation}\label{c6} 0<\frac1{(p-1)b_4}\leq \frac1r\leq 1, \quad 0\leq \frac{\tau}{p-1}<\frac{n}{(p-1)b_4}, \quad \frac{\tau}{p-1}-1=\frac{n}{(p-1)b_4}-\frac{n}{r}. \end{equation} On the other hand, in order to show \eqref{non22}, we first use the following simple inequality \begin{align*} \big|\mathcal N[u]-\mathcal N[v]\big| &\lesssim \Big||x|^{-\tau}(|u|^{p-2}+|v|^{p-2})|u-v|(I_\alpha\ast|\cdot|^{-\tau}|u|^p)\Big| \\&\qquad\quad+\Big||x|^{-\tau}|v|^{p-1}\big(I_\alpha \ast|\cdot|^{-\tau}(|u|^{p-1}+|v|^{p-1})|u-v|\big)\Big|. \end{align*} From this, we see that \begin{align*} \|\mathcal N[u]-\mathcal N[v]\|_{L_x^{\frac{2n}{n+2}}} &\leq \||x|^{-\tau}(|u|^{p-2}+|v|^{p-2})|u-v|(I_\alpha\ast|\cdot|^{-\tau}|u|^p)\|_{L_x^{\frac{2n}{n+2}}}\\ &\qquad+\||x|^{-\tau}|v|^{p-1}\big(I_\alpha \ast|\cdot|^{-\tau}(|u|^{p-1}+|v|^{p-1})|u-v|\big)\|_{L_x^{\frac{2n}{n+2}}}\\ &:=B_1+B_2. \end{align*} Replacing $\nabla u$ with $u-v$ in the process of dealing with $A_3$, we get \begin{align*} B_1&=\||x|^{-\tau}(|u|^{p-2}+|v|^{p-2})|u-v|(I_\alpha\ast|\cdot|^{-\tau}|u|^p)\|_{L_x^{\frac{2n}{n+2}}}\\ &\lesssim \Big(\||x|^{-\tau}|u|^{p-2}|u-v|\|_{L_x^{a_1}}+\||x|^{-\tau}|v|^{p-2}|u-v|\|_{L_x^{a_1}}\Big)\||x|^{-\tau}|u|^p\|_{L_x^{b_1}}\\ &\lesssim \Big(\||x|^{-\frac{\tau}{p-2}}u\|_{L_x^{(p-2)a_3}}^{p-2}+ \||x|^{-\frac{\tau}{p-2}}u\|_{L_x^{(p-2)a_3}}^{p-2}\Big)\|u-v\|_{L_x^r}\||x|^{-\frac{\tau}{p}}u\|_{L_x^{pb_1}}^p\\ &\lesssim \big(\|\nabla u\|_{L_x^{r}}^{2p-2}+\|\nabla v\|_{L_x^{r}}^{2p-2}\big)\|u-v\|_{L_x^r} \end{align*} under the conditions \eqref{c1}, \eqref{c2} and \eqref{c3}. Similarly, replacing $\nabla u$ with $u-v$ in estimating $A_4$, we also have \begin{align*} B_2&=\||x|^{-\tau}|u|^{p-1}(I_\alpha\ast|\cdot|^{-\tau}(|u|^{p-1}+|v|^{p-1})|u-v|)\|_{L_x^{\frac{2n}{n+2}}}\\ &\lesssim \||x|^{-\tau}|u|^{p-1}\|_{L_x^{a_2}}\||x|^{-\tau}(|u|^{p-1}+|v|^{p-1})|u-v|\|_{L_x^{b_2}}\\ &\lesssim \||x|^{-\frac{\tau}{p-1}}u\|_{L_x^{(p-1)a_2}}^{p-1} \Big(\||x|^{-\frac{\tau}{p-1}}u\|_{L_x^{(p-1)b_4}}^p+\||x|^{-\frac{\tau}{p-1}}v\|_{L_x^{(p-1)b_4}}^p\Big)\|u-v\|_{L_x^r}\\ &\lesssim \big(\|\nabla u\|_{L_x^{r}}^{2p-2}+\|\nabla v\|_{L_x^{r}}^{2p-2}\big)\|u-v\|_{L_x^r} \end{align*} under the conditions \eqref{c4}, \eqref{c5} and \eqref{c6}. Now it remains to eliminate some redundant pairs, we then show that there exists an admissible pair $(q,r)$ satisfying the assumptions in the lemma. The third conditions of \eqref{c8} and \eqref{c9} can be rewritten with respect to $a$ and $b$, respectively, as follow: \begin{equation}\label{c19} \frac{n}{a_1}=\frac{(p-1)n}{r}+\tau-p+2, \quad \frac{n}{b_1}=\frac{pn}{r}+\tau-p. \end{equation} Inserting these into the second condition of \eqref{c7} implies \begin{equation}\label{a} \frac{(2p-1)n}{r}=\alpha+2p-2\tau-1+\frac{n}{2}. \end{equation} Here, we note that this equation is equivalent to the second condition of \eqref{c10} by using \eqref{c18}. Inserting \eqref{c19} into the conditions in \eqref{c7}, \eqref{c8} and \eqref{c9}, these conditions are summarized as follows: \begin{equation}\label{c15} \frac{p-\tau-2}{p-1}<\frac{n}{r}<\frac{p-\tau-2+n}{p-1}, \quad \frac{p-\tau}{p}<\frac{n}{r}<\frac{p-\tau+n}{p}, \end{equation} \begin{equation}\label{c13} \tau-p+2\leq 0, \quad \frac{n}{r}\leq n, \quad \frac{p-\tau-2}{p-1} \leq 1 <\frac{n}{r} \end{equation} \begin{equation}\label{c14} \tau-p\leq 0 , \quad \frac{p-\tau}{p}\leq 1 < \frac{n}{r} \end{equation} Since $\tau>0$, the first inequalities of the last conditions in \eqref{c13} and \eqref{c14} are redundant. The first condition in \eqref{c14} is also redundant by the first one in \eqref{c13}. Also, the third conditions of \eqref{c11} and \eqref{c12} can be rewritten as \begin{equation}\label{c18} \frac{n}{a_2}=\frac{(p-1)n}{r}+\tau-p+1, \quad \frac{n}{b_2}=\frac{pn}{r}+\tau-p+1 \end{equation} Inserting these into the conditions \eqref{c10}, \eqref{c11} and \eqref{c12}, these conditions are summarized as \begin{equation}\label{c16} \frac{p-\tau-1}{p-1}<\frac{n}{r}<\frac{p-\tau-1+n}{p-1}, \quad \frac{p-\tau-1}{p}<\frac{n}{r}<\frac{p-\tau-1+n}{p} \end{equation} \begin{equation}\label{c17} \tau-p+1\leq 0, \quad \frac{n}{r}\leq n, \quad \frac{p-\tau-1}{p-1}\leq 1 <\frac{n}{r}, \quad \frac{p-\tau-1}{p}\leq 1 < \frac{n}{ r}. \end{equation} Here, since $\tau>0$, the first inequalities of the last two conditions in \eqref{c17} are redundant. The first conditions in \eqref{c17} is also eliminated by the first one of \eqref{c13}. Finally, the first two conditions of \eqref{c2} and \eqref{c6} are summarized by inserting the third conditions of \eqref{c2} and \eqref{c6} as \begin{equation}\label{c20} \frac{p-\tau-2}{p-2}<\frac{n}r\leq n, \quad \tau-p+2\leq0, \quad \frac{p-\tau-2}{p-2}\leq 1<\frac{n}{r}, \end{equation} \begin{equation}\label{c21} \frac{p-\tau-1}{p-1}<\frac{n}{r}\leq n, \quad \tau-p+1\leq 0, \quad \frac{p-\tau-1}{p-1}\leq 1 <\frac{n}{r}. \end{equation} Here, the second condition in \eqref{c21} is eliminated by the second one in \eqref{c20}. Since $p>2$ and $\tau>0$, all lower bounds of $n/r$ in \eqref{c15}, \eqref{c16}, \eqref{c20} and \eqref{c21} are eliminated by $1$. Moreover, by using $p>2$ the upper bounds of $n/r$ in the second condition of \eqref{c15} and the first one of \eqref{c16} are also eliminated by the upper one of $n/r$ in the second condition of \eqref{c16}. As a result, combining all the above conditions, we get \begin{equation}\label{rr} 1<\frac{n}{r}<\min\Big\{\frac{p-\tau-2+n}{p-1}, \frac{p-\tau-1+n}{p}\Big\}, \quad 0<\tau\leq p-2. \end{equation} On the other hand, substituting the first condition into the third one in \eqref{st} implies \begin{equation}\label{r} \frac{n}{r}=\frac{n}{2}-\frac1{2p-1}. \end{equation} Note that \eqref{a} is exactly same as $p=1+\frac{2-2\tau+\alpha}{n-2}$ by substituting \eqref{r} into \eqref{a}. Eliminating $r$ by inserting \eqref{r} into the second conditios of \eqref{ka} and \eqref{st}, the first one of \eqref{rr}, we then get \begin{equation}\label{c25} 1+\kappa < \frac{n}{2}-\frac1{2p-1}<\min\{n,n-\kappa\}, \quad \frac{n-2}{2}\leq \frac{n}{2}-\frac1{2p-1}\leq \frac{n}2, \end{equation} \begin{equation}\label{c22} p>\frac{n}{2(n-2)}, \quad \tau <n-1+\min\Big\{(p-1)\Big(\frac{2p}{2p-1}-\frac{n}{2}\Big), p\Big(\frac{2p}{2p-1}-\frac{n}{2}\Big)\Big\}. \end{equation} Here, the first condition in \eqref{c25} can be divided into two inequalities \begin{equation}\label{c26} p>\frac12+\frac{1}{n-2-2\kappa}, \quad \max\big\{-\frac{n}{2},-\frac{n}{2}+\kappa\big\}<\frac1{2p-1}, \end{equation} in which the second condition is redundant since the maximum value is always negative. Since $p>2$ and $n\ge3$, the last condition in \eqref{c25} and the first condition in \eqref{c22} are redundant. Moreover, since $\frac{2p}{2p-1}-\frac{n}{2}<0$, the last condition in \eqref{c22} is reduced \begin{equation}\label{c23} \tau <n-1+ p\Big(\frac{2p}{2p-1}-\frac{n}{2}\Big). \end{equation} In order to eliminate $\alpha$, inserting $p=1+\frac{2-2\tau+\alpha}{n-2}$ into the last condition in \eqref{st}, we also have \begin{equation}\label{c24} \frac{n}{2}-\frac{(n-2)p}{2}<\tau<n-\frac{(n-2)p}{2}. \end{equation} Now we make the lower bounds of $\tau$ less than the upper ones of $\tau$ in \eqref{rr}, \eqref{c23} and \eqref{c24} to obtain \begin{align*} \nonumber \max\Big\{2, &\frac{5n-4-\sqrt{9n^2+8n-16}}{4(n-2)},\frac{n+4}{n}, \frac{n-2}{2(n-1)}\Big\} \leq \\ &\qquad\qquad\qquad\qquad\quad p \leq \min\Big\{\frac{2n}{n-2}, \frac{5n-4+\sqrt{9n^2+8n-16}}{4(n-2)} \Big\}, \end{align*} which is reduced \begin{equation}\label{c27} \max\Big\{2,\frac{n+4}{n},\frac12+\frac1{n-2-2\kappa}\Big\} < p < \frac{5n-4+\sqrt{9n^2+8n-16}}{4(n-2)} \end{equation} by using $p>2$, $n\ge3$ and combining the first condition in \eqref{c26}. The assumption \eqref{ass2} follows from inserting $p=1+\frac{2-2\tau+\alpha}{n-2}$ into \eqref{c27}. In fact, \eqref{c27} is expressed with respect to $\tau$, as follows: \begin{equation} \frac{\alpha}{2}-\frac{n-4+\sqrt{9n^2+8n-16}}{8}<\tau < \frac{\alpha}{2}-\max\{\frac{n-4}{2}, \frac{n-4}{n},\frac{\kappa}{n-2-2\kappa}-\frac{n}{4}\}. \end{equation} Finally, we make the lower bound of $\tau$ less than the upper ones of $\tau$ to deduce \begin{equation*} 2\kappa < \frac{5n-4-\sqrt{9n^2+8n-16}}{2}, \end{equation*} which implies the assumption \eqref{ass1}. Indeed, to obtain \eqref{ass1}, we can compute as follows: \begin{eqnarray*} &&\frac{n-4+\sqrt{9n^2+8n-16}}{8}>\frac{\kappa}{n-2-2\kappa}-\frac{n}{4}\\ &\Leftrightarrow&\frac{n-2-2\kappa}{8}>\frac{\kappa}{3n-4+\sqrt{9n^2+8n-16}}\times\frac{\sqrt{9n^2+8n-16}-(3n-4)}{\sqrt{9n^2+8n-16}-(3n-4)}\\ &\Leftrightarrow&n-2>\frac{\sqrt{9n^2+8n-16}+5n-4}{4(n-1)}\kappa. \end{eqnarray*} This is equivalent to \begin{align*} \kappa<\frac{4(n-1)(n-2)}{5n-4+\sqrt{9n^2+8n-16}}&=\frac{4(n-1)(n-2)\big\{5n-4-\sqrt{9n^2+8n-16}\big\}}{(5n-4)^2-9n^2-8n+16}\\ &=\frac{5n-4-\sqrt{9n^2+8n-16}}{4}. \end{align*} This ends the proof. \end{proof} \subsection{Local well-posedness in the energy space} By Duhamel's principle, we first write the solution of the Cauchy problem \eqref{S} as fix points of the function $$\Phi(u)=e^{-it\mathcal{K}_{\lambda}}u_0 + i\epsilon\int_0^t e^{-i(t-s)\mathcal{K}_{\lambda}} \mathcal N[u](s,\cdot) ds$$ where $\mathcal N[u]=|x|^{-\tau}|u|^{p-2}(J_\alpha \ast |\cdot|^{-\tau}|u|^p)u$. For appropriate values of $T,M,N>0$, we shall show that $\Phi$ defines a contraction map on $$X(T,M,N)=\{u \in C_t(I;H_\lambda^1) \cap L_t^{q}(I;W^{1,r}_{\lambda}): \sup_{t\in I} \|u\|_{H_{\lambda}^1}\leq M, \|u\|_{\mathcal W_{\lambda}(I)}\leq N\}$$ equipped with the distance $$d(u,v)=\|u-v\|_{\Lambda(I)}.$$ Here, $I=[0,T]$ and $(q,r)$ is given as in Proposition \ref{str}. We also define $$\|u\|_{\mathcal W_{\lambda}(I)}:= \|u\|_{\Lambda(I)} + \|\sqrt{\mathcal K_{\lambda}}u\|_{\Lambda(I)}$$ and $$\|u\|_{\mathcal W{'}_{\lambda}(I)}:= \|u\|_{\Lambda'(I)} + \|\sqrt{\mathcal K_{\lambda}}u\|_{\Lambda'(I)}.$$ We now show that $\Phi$ is well defined on $X$. By Proposition \ref{str}, we get \begin{equation}\label{w1} \|\Phi(u)\|_{\mathcal W_{\lambda}(I)}\leq C\|e^{-it\mathcal K_{\lambda}}u_0\|_{\mathcal W_{\lambda}(I)} +C\big\|\mathcal N[u]\big\|_{\mathcal W{'}_{\lambda}(I)} \end{equation} and \begin{equation*} \sup_{t\in I}\|\Phi(u)\|_{H_{\lambda}^1}\leq \|u_0\|_{H_{\lambda}^1}+\sup_{t \in I}\Big\|\int_0^t e^{-i(t-s)\mathcal{K}_{\lambda}} \mathcal N[u](\cdot,s) ds\Big\|_{H_{\lambda}^1}. \end{equation*} Here, for the second inequality we used the fact that $e^{it\mathcal K_{\lambda}}$ is an unitary on $L^2$. Since $\|\langle \sqrt{\mathcal K_{\lambda}} \rangle u \|\lesssim \|u\| + \|\sqrt{\mathcal K_{\lambda}}u\|$, using the fact $e^{it\mathcal K_{\lambda}}$ is an unitary on $L^2$ again, and then applying the dual estimate of the first one in Proposition \ref{str}, we see that $$ \sup_{t\in I} \Big\| \int_0^t e^{-i(t-s)\mathcal{K}_{\lambda}} \mathcal N[u](\cdot,s) ds \Big\|_{H_{\lambda}^1} \lesssim \|\mathcal{N}[u]\|_{\Lambda'(I)} +\|\sqrt{\mathcal{K}_{\lambda}}\mathcal{N}[u]\|_{\Lambda'(I)}. $$ Hence, \begin{equation*} \sup_{t\in I}\|\Phi(u)\|_{H_\lambda^1} \leq C \|u_0\|_{H_{\lambda}^1}+C\|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}'(I)}. \end{equation*} On the other hand, using Lemma \ref{non}, we get \begin{align}\label{w2} \nonumber \|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}'(I)}&\leq C \|\sqrt{\mathcal K_{\lambda}}u\|_{\Lambda(I)}^{2p-1} + \|\sqrt{\mathcal K_{\lambda}}u\|_{\Lambda(I)}^{2p-2}\|u\|_{\Lambda(I)} \\ \nonumber &\leq C \|\sqrt{\mathcal K_{\lambda}}u\|_{\Lambda(I)}^{2p-2}\|u\|_{\mathcal{W}_{\lambda}(I)}\\ &\leq C N^{2p-1} \end{align} if $u \in X$, and for some $\varepsilon>0$ small enough which will be chosen later we get \begin{equation}\label{sm} \|e^{it\mathcal{K}_{\lambda}}u_0\|_{\mathcal{W}_{\lambda}(I)}\leq \varepsilon \end{equation} which holds for a sufficiently small $T>0$ by the dominated convergence theorem. We now conclude that \begin{equation*} \|\Phi(u)\|_{\mathcal{W}_{\lambda}(I)} \leq \varepsilon + CN^{2p-1} \quad \textnormal{and} \quad \sup_{t \in I}\|\Phi(u)\|_{H_{\lambda}^1} \leq C \|u_0\|_{H_\lambda^1} + CN^{2p-1}. \end{equation*} Hence we get $\Phi(u)\in X$ for $u \in X$ if \begin{equation}\label{w3} \varepsilon + CN^{2p-1} \leq N \quad \textnormal \quad C\|u_0\|_{H_{\lambda}^1} + CN^{2p-1} \leq M. \end{equation} Next we show that $\Phi$ is a contraction on $X$. Using the same argument used in \eqref{w1}, we see \begin{equation*} \|\Phi(u)-\Phi(v)\|_{\Lambda(I)} \leq C \|\mathcal{N}[u]-\mathcal{N}[v]\|_{\Lambda'(I)}. \end{equation*} By applying Lemma \ref{non} (see \eqref{non2}), we see \begin{align*} \|\mathcal{N}[u]-\mathcal{N}[v]\|_{\Lambda'(I)}&\leq C\big(\|\sqrt{\mathcal{K}_{\lambda}} u\|_{\Lambda(I)}^{2p-1}+\|\sqrt{\mathcal{K}_{\lambda}} v\|_{\Lambda(I)}^{2p-2}\big)\|u-v\|_{\Lambda(I)} \\ &\leq C N^{2p-2}\|u-v\|_{\Lambda(I)} \end{align*} as in \eqref{w2}. Hence, for $u,v \in X$ we obtain $d(\Phi(u), \Phi(v))\leq C N^{2p-2}d(u,v)$. Now by taking $M=2C\|u_0\|_{H_\lambda^1}$ and $N=2\varepsilon$ and then choosing $\varepsilon>0$ small enough so that \eqref{w3} holds and $CN^{2p-2}\leq 1/2$, it follows that $\Phi$ is a contraction on $X$. Therefore, we have proved that there exists a unique local solution with $u \in C_t(I;H_{\lambda}^1) \cap L_t^q(I;W_{\lambda}^{1,r})$ for any admissible pair $(q,r)$. \subsection{Global well-posedness in the energy space for small data} Using the first estimate in Proposition \ref{str}, we observe that \eqref{sm} is satisfied also if $\|u_0\|_{H_{\lambda}^1}$ is sufficiently small, \begin{equation*} \|e^{-it\mathcal K_{\lambda}}u_0\|_{\mathcal{W}_\lambda(I)} \leq C \|u_0\|_{H_{\lambda}^1}\leq \varepsilon \end{equation*} from which one can take $T=\infty$ in the above argument to obtain a global unique solution. The continuous dependence of the solution $u$ with respect to the initial data $u_0$ follows clearly in the same way: \begin{align*} d(u,v) &\lesssim d(e^{-it\mathcal{K}_{\lambda}}u_0,e^{-it\mathcal{K}_{\lambda}}v_0) + d\Big(\int_0^t e^{-i(t-s)\mathcal{K}_{\lambda}}\mathcal{N}[u]ds,\int_0^t e^{-i(t-s)\mathcal{K}_{\lambda}}\mathcal{N}[v]ds \Big) \\ &\lesssim \|u_0 - v_0\| + \frac12 d(u,v) \end{align*} which implies \begin{align*} d(u,v) \lesssim \|u_0-v_0\|_{H_{\lambda}^1}. \end{align*} Here, $u,v$ are the corresponding solutions for initial data $u_0, v_0$, respectively. \subsection{Scattering in the energy space for small data} To prove the scattering property, we first note that \begin{align*} \|e^{it_2 \mathcal{K}_{\lambda}}u(t_2)-e^{it_2 \mathcal{K}_{\lambda}}u(t_1)\|_{H_{\lambda}^1} = \Big\|\int_{t_1}^{t_2} e^{is\mathcal{K}_{\lambda}}\mathcal{N}[u]\Big\|_{H_{\lambda}^1}\\ \lesssim \|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}{'}([t_1 ,t_2])} \\ \lesssim \|u\|^{2p-1}_{\mathcal{W}_{\lambda}([t_1,t_2])} \quad \rightarrow\quad 0 \end{align*} as $t_1, t_2 \rightarrow {\infty}$. This implies that $\phi :=\lim_{t\rightarrow {\infty}} e^{it\mathcal{K}_{\lambda}}u(t)$ exists in $H_\lambda^1$. Furthermore, $$u(t)-e^{-it\mathcal{K}_{\lambda}}\phi=i\int_t^{\infty} e^{i(t-s)\mathcal{K}_{\lambda}}\mathcal{N}[u]ds,$$ and hence \begin{align*} \|u(t)-e^{-it{\mathcal K}_\lambda}\phi\|_{H^1_\lambda}&=\Big\|\int_t^{\infty} e^{i(t-s)\mathcal{K}_{\lambda}}\mathcal{N}[u] ds\Big\|_{H_{\lambda}^1}\\ &\lesssim\|\mathcal{N}[u]\|_{\mathcal{W}_{\lambda}'([t,\infty])} \\ &\lesssim \|u\|^{2p-1}_{\mathcal{W}_{\lambda}([t,\infty])} \quad \rightarrow \quad 0 \end{align*} as $t\rightarrow {\infty}$. {The scattering is proved.} \section{Ground states and Gagliardo-Nirenberg estimate} In this section, we prove Theorem \ref{gag} dealing with the existence of ground states solutions to \eqref{E} and the Gagliardo-Nirenberg type estimate \eqref{gagg}. \subsection{Gagliardo-Nirenberg estimate} Using the Hardy-Littlewood-Sobolev inequality (Lemma \ref{hls}), we first see \begin{equation}\label{ie} \int_{\mathbb{R}^n} |x|^{-\tau}|u|^p (I_\alpha \ast |\cdot|^{-\tau}|u|^p) dx \lesssim \big\||x|^{-\tau}|u|^p\big\|^2_{{\frac{2n}{\alpha+n}}} \end{equation} if $0<\alpha<n.$ Applying Lemma \ref{ckn} to the right-hand side of \eqref{ie} with $b=-\frac{\tau}{p}$, $q=\frac{2np}{\alpha+n}$, $a=0$ and $p=2$, we get \begin{equation}\label{5.100'} \big\||x|^{-\tau}|u|^p\big\|^2_{{\frac{2n}{\alpha+n}}}= \big\||x|^{-\frac{\tau}{p}}u\big\|^{2p}_{{\frac{2np}{\alpha+n}}} \lesssim \|\nabla u\|^{2p} \end{equation} if \begin{equation}\label{iee} 0<\frac{\alpha+n}{2np} \leq \frac12 <1, \quad -\frac{\alpha+n}{2p}<-\frac{\tau}{p} \leq 0, \quad \frac{\tau}{p}-1=\frac{\alpha+n}{2p}-\frac{n}2. \end{equation} Finally, using the equivalent norm to Sobolev one (see Lemma \ref{2.2}), we obtain the desired estimate \eqref{inte} if $\frac{1+\kappa}{n}<\frac12<\min\{1, 1-\frac{\kappa}{n}\}$ which does not affect the assumptions in \eqref{as}. Now it remains to derive the assumptions in \eqref{as}. We note that the last equality in \eqref{iee} is equivalent to $p=1+\frac{2-2\tau+\alpha}{n-2}$. Using $p=1+\frac{2-2\tau+\alpha}{n-2}$, the requirements \eqref{iee} can be written as \begin{equation}\label{ie1} \alpha+n>0, \quad \alpha+n \ge n\tau, \quad 0\leq 2\tau<\alpha+n, \end{equation} which are reduced to $0<\tau\leq 1+\frac{\alpha}{n}$ since $\alpha+n\geq n\tau>2\tau>0$, as desired. \subsection{Existence of ground states} By using \eqref{gagg}, we first set $J(u)=\|\sqrt{\mathcal K_\lambda}u\|^{2p}/\|\mathcal P[u]\|$ and take a sequence $\{u_n\}_{n\in\mathbb{N}}$ in $H_{\lambda}^1$ such that \begin{align*} \gamma := \frac1{C_{n,\tau,\alpha,\lambda}} =\lim_{n\rightarrow \infty} \frac{\|\sqrt{\mathcal{K}_{\lambda}}u_n\|^{2p}}{\mathcal{P}[u_n]}. \end{align*} By the scaling $u(x) \mapsto u^{\delta,\mu}(x)=\delta u(\mu x)$ for $\delta, \mu \in \mathbb{R}$, we have \begin{align*} \|u^{\delta,\mu}\|^2&=\delta^2 \mu^{-n}\|u\|^2 \\ \|\sqrt{\mathcal K_{\lambda}}u^{\delta,\mu}\|^2&= \|\nabla u^{\delta,\mu}\|^2+\lambda\big\|\frac{u^{\delta,\mu}}{|x|}\big\|^2 \\ &=\delta^2 \mu^{2-n}\Big(\|\nabla u\|^2+\lambda\big\|\frac{u}{|x|}\big\|^2\Big)=\delta^2 \mu^{2-n}\|\sqrt{\mathcal K_\lambda}u\|^2 \\ \int_{\mathbb{R}^n}|x|^{-\tau}|u^{\delta,\mu}|^p(I_\alpha \ast |\cdot|^{-\tau}|u^{\delta,\mu}|^p) dx &= \delta^{2p}\mu^{2\tau-n-\alpha}\int_{\mathbb{R}^n}|x|^{-\tau}|u|^p(I_\alpha \ast |\cdot|^{-\tau}|u|^p) dx, \end{align*} which implies that $J(u^{\delta,\mu})=J(u)$ by $p=1+\frac{2-2\tau+\alpha}{n-2}$. Let $\psi_n = u_n^{\delta_n,\mu_n}$ where $$\delta_n= \frac{\|u_n\|^{\frac{n}2-1}}{\|\sqrt{\mathcal K_\lambda} u_n\|^{\frac{n}{2}}}, \quad \mu_n=\frac{\|u_n\|}{\|\sqrt{\mathcal K_\lambda} u_n\|}.$$ Then, we have $$\|\psi_n\|=\|\sqrt{\mathcal K_\lambda} \psi_n\|=1 \quad \textnormal{and} \quad \gamma=\lim_{n\rightarrow\infty} J(\psi_n)=\lim_{n\rightarrow \infty} \frac1{\mathcal P[\psi_n]}.$$ Now we take $\psi \in H_\lambda^1$ so that $\psi_n \rightharpoonup \psi$ in $H_\lambda^1$ and we will show that $$\frac{1}{\mathcal P[\psi_n]} \rightarrow \frac1{\mathcal P[\psi]} \quad \textnormal{as} \quad n\rightarrow \infty.$$ By using Lemma \ref{hls} {via \eqref{5.100'}}, we have \begin{align} \nonumber &\int_{\mathbb{R}^n} |x|^{-\tau}|\psi_n|^p (I_\alpha \ast |\cdot|^{-\tau}|\psi_n|^p)- |x|^{-\tau}|\psi|^p (I_\alpha \ast |\cdot|^{-\tau}|\psi|^p)dx \\ \nonumber &\quad =\int_{\mathbb{R}^n} |x|^{-\tau}|\psi|^p \big(I_\alpha \ast |\cdot|^{-\tau}(|\psi_n|^p - |\psi|^p)\big) dx\\ \nonumber &\qquad \qquad \qquad+ \int_{\mathbb{R}^n}|x|^{-\tau}(|\psi_n|^p-|\psi|^p)(I_\alpha \ast |\cdot|^{-\tau}|\psi_n|^p)dx\\ \nonumber &\quad\lesssim \Big(\big\||x|^{-\tau}|\psi|^p\big\|_{{\frac{2n}{\alpha+n}}} +\big\||x|^{-\tau}|\psi_n|^p\big\|_{{\frac{2n}{\alpha+n}}}\Big) \big\||x|^{-\tau}(|\psi_n|^p-|\psi|^p)\big\|_{{\frac{2n}{\alpha+n}}}\\ \label{dif} &\quad\lesssim \big\||x|^{-\tau}(|\psi_n|^p-|\psi|^p)\big\|_{{\frac{2n}{\alpha+n}}}. \end{align} Using the following simple inequality $$|u|^p-|v|^p \lesssim |u-v|(|u|^{p-1}+|v|^{p-1}), \quad p\ge1$$ and H\"older's inequality, the last term in \eqref{dif} is bounded as \begin{align}\label{i} \nonumber \big\||x|^{-\tau}(|\psi_n|^p-|\psi|^p)\big\|_{{\frac{2n}{\alpha+n}}} &\lesssim (\|\psi_n\|_{{(p-1)a_1}}^{p-1}+\|\psi\|_{{(p-1)a_1}}^{p-1}) \||x|^{-\tau}|\psi-\psi_n|\|_{{a_2}}\\ &\lesssim (\|\psi\|_{H_\lambda^1}^{p-1}+\|\psi_n\|_{H_\lambda^1}^{p-1})\||x|^{-\tau}|\psi-\psi_n|\|_{{a_2}} \end{align} if $0<\tau<2$ and \begin{equation}\label{gagc} \frac{\alpha+n}{2n}=\frac{1}{a_1}+\frac{1}{a_2}, \quad \frac{n-2}{2n}\leq\frac{1}{(p-1)a_1}\leq\frac12. \end{equation} {Indeed, for the last inequality we used the Sobolev embedding, $H^1(\mathbb{R}^n) \hookrightarrow L^q({\mathbb{R}^n})$ for $2\leq q \leq \frac{2n}{n-2}$ if $n\ge3$.} Thanks to the compactness of the Sobolev injection, Lemma \ref{compact}, under the condition \begin{equation}\label{gagcc} \frac{n-2}{2(n-\tau)}<\frac{1}{a_2}<\frac12, \end{equation} we then get $1/{\mathcal P[\psi_n]} \rightarrow 1/{\mathcal P[\psi]}=\gamma$ as $n \rightarrow \infty$. We need to check that there exist $a_1$ and $a_2$ satisfying \eqref{gagc}, \eqref{gagcc} and the assumptions in Theorem \ref{gag}, but we will postpone this until the end of the proof. By the lower semi-continuity of the norm, we see $$\|\psi\|\leq 1 \quad \textnormal{and} \quad \|\sqrt{\mathcal K_\lambda} \psi\|\leq 1,$$ from which $J(\psi)<\gamma$, and hence $\|\psi\|=\|\sqrt{\mathcal K_\lambda}\psi\|=1.$ Consequently, $$\psi_n \rightarrow \psi \quad \textnormal{in} \quad H_\lambda^1 \quad \textnormal{and} \quad \gamma=J(\psi)=\frac1{\mathcal P[\psi]}.$$ $\psi$ satisfies \eqref{E} because the minimizer satisfies the Euler equation $$\partial_\epsilon J(\psi+\epsilon \eta)_{|\epsilon=0}=0, \quad \forall \eta \in C_0^{\infty} \cap H_\lambda^1.$$ It remains to check the existence of $a_1$ and $a_2$ satisfying the conditions \eqref{gagc}, \eqref{gagcc} under the assumptions in Theorem \ref{gag}. Substituting the first condition in \eqref{gagc} into the second one of \eqref{gagc} with $p=1+\frac{2-2\tau+\alpha}{n-2}$, we see {\begin{equation}\label{gagc2} \frac{\alpha+n}{2n}-\frac{2-2\tau+\alpha}{2(n-2)}\leq\frac1{a_2} \leq \frac{\alpha+n}{2n}-\frac{2-2\tau+\alpha}{2n}. \end{equation}} To eliminate $a_2$, we make the lower bounds of $1/a_2$ of \eqref{gagcc} and \eqref{gagc2} less than the upper ones of $1/a_2$ of \eqref{gagcc} and \eqref{gagc2}. {Indeed, starting the process from the lower bound in \eqref{gagc2}, we arrive at $n\tau<\alpha+n$ which is satisfied by the assumption \eqref{as}. Similarly from the lower bound in \eqref{gagcc}, we arrive at $0<\tau<\frac{n+2}{2}$, but this is eliminated by \eqref{as} using the facts that $n\ge3$ and $\tau<2$.} \section{Blow-up of the energy solutions} In this section, we prove Theorem \ref{t1} which provides a criterion for blow-up phenomena in the energy-critical focusing regime under the threshold of the ground state. As a consequence, we establish Corollary \ref{t2}. Moreover, we prove Proposition \ref{s} and Corollary \ref{s2} about energy bounded solutions. \subsection{Criterion for blow-up} In order to prove Theorem \ref{t1}, we use proof by contradiction through the following inequality which will be proved: \begin{equation}\label{qq} V_{R}''\leq4\,\mathcal I[u]+\frac C{R^{2\tau}}+\frac C{R^2}, \end{equation} where $\mathcal I [u] = \|\sqrt{\mathcal K_\lambda} u\|^2 - \mathcal P[u]$ and $R\gg1$. Indeed, taking $u_0\in H^1_\lambda$ with \eqref{ss'} and assuming that $u$ is global, for $R\gg1$, $$V_{R}''\leq4\,\mathcal I[u]+\frac C{R^{2\tau}}+\frac C{R^2}<-c<0$$ if there is no sequence $t_n\to\infty$ such that $\|\sqrt{\mathcal K_\lambda} u(t_n)\|\to \infty$, which is contradiction. Before starting to prove \eqref{qq}, we first define $\phi_R(\cdot):=R^2\phi(\frac{\cdot}{R})$, $R>0$, where the radial function $\phi\in C_0^\infty(\mathbb{R}^n)$ satisfies $$\phi(|x|)=\phi(r):=\left\{ \begin{array}{ll} \frac{r^2}2,\quad\mbox{if}\quad r\leq1 ;\\ 0,\quad\mbox{if}\quad r\geq2, \end{array} \right.\quad\mbox{and}\quad \phi''\leq1.$$ Then, $\phi_R$ satisfies $$\phi_R''\leq1,\quad \phi_R'(r)\leq r,\quad\Delta \phi_R\leq N$$ and, for $|x|\leq R$ \begin{align}\label{calc} \nabla\phi_R(x)=x,\quad\Delta\phi_R(x)=N. \end{align} By recalling the definition of $V(t)$ and $M(t)$ in Section 2, we denote the localized variance and Morawetz action as \begin{align*} V_R(t):=\int_{\mathbb{R}^n}\phi_R(x)|u(x,\cdot)|^2\,dx, \quad V_R'(t)=M_R(t):=2\Im\int_{\mathbb{R}^n}\bar u\nabla \phi_R \cdot \nabla udx. \end{align*} By Proposition \ref{mrwz}, we divide $M_R'$ into two parts, $A$ and $B$, as $M_R'(t)=A + B$ where $$A=4\sum_{k,l=1}^{N}\int_{\mathbb{R}^n}\partial_l\partial_k\phi_R\Re(\partial_ku\partial_l\bar u)dx-\int_{\mathbb{R}^n}\Delta^2\phi_R|u|^2dx+4\lambda\int_{\mathbb{R}^n}\nabla\phi_R\cdot x\frac{|u|^2}{|x|^4}dx$$ and \begin{align} \nonumber B&=-\frac{2(p-2)}{p}\int_{\mathbb{R}^n}\Delta\phi_R|x|^{-\tau}|u|^p(I_\alpha*|\cdot|^{-\tau}|u|^{p})dx\\ \nonumber &\quad\qquad -\frac{4\tau}p\int_{\mathbb{R}^n}x\cdot\nabla\phi_R|x|^{-\tau-2}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\\ \nonumber &\qquad\qquad\qquad -\frac{4(N-\alpha)}p\sum_{k=1}^N\int_{\mathbb{R}^n}|x|^{-\tau}|u|^{p}\partial_k\phi_R(\frac{x_k}{|\cdot|^2}I_\alpha*|\cdot|^{-\tau}|u|^p)dx\\ \label{123} &=:B_1 + B_2+ B_3. \end{align} Using the following radial relations \begin{equation}\label{''} \partial_k=\frac{x_k}r\partial_r,\quad\partial_l\partial_k=\Big(\frac{\delta_{lk}}r-\frac{x_lx_k}{r^3}\Big)\partial_r+\frac{x_lx_k}{r^2}\partial_r^2 \end{equation} and the Cauchy-Schwarz inequality via the properties of $\phi$, it follows that \begin{align} \nonumber A&= 4\int_{\mathbb{R}^N} |\nabla u|^2 \frac{\phi_R'}{r}dx + 4 \int_{\mathbb{R}^N} |x\cdot \nabla u|^2 \big(\frac{\phi_R''}{r^2}-\frac{\phi_R'}{r^3}\big) dx \\ \nonumber &\qquad \qquad\qquad \qquad\qquad \qquad\quad-\int_{\mathbb{R}^N} \Delta^2\phi_R|u|^2dx + 4 \int_{\mathbb{R}^N} \frac{|u|^2}{r^3}\phi_R' dx \\ \nonumber &\leq4\int_{\mathbb{R}^n}|\nabla u|^2\frac{\phi_R'}r\,dx+4\int_{\mathbb{R}^n}\frac{|x\cdot\nabla u|^2}{r^2}\big(1-\frac{\phi_R'}r\big)dx\\ \nonumber &\qquad \qquad \qquad \qquad\qquad \qquad\quad -\int_{\mathbb{R}^n}\Delta^2\phi_R|u|^2\,dx+4\lambda\int_{\mathbb{R}^n}\frac{|u|^2}{r^3}\phi_R'dx\nonumber\\ \label{(I)} &\leq4\int_{\mathbb{R}^n}|\nabla u|^2dx-\int_{\mathbb{R}^n}\Delta^2\phi_R|u|^2dx+4\lambda\int_{\mathbb{R}^n}\frac{|u|^2}{r^2}dx. \end{align} On the other hand, to handle the part $B$, we split the integrals in $B$ into the regions $|x|<R$ and $|x|>R$. Then, by \eqref{calc}, the first two terms in $B$ are written \begin{align} \nonumber B_1+B_2&=\frac{2N(2-p)-4\tau}{p}\int_{|x|<R}|x|^{-\tau}|u|^p(I_\alpha*|\cdot|^{-\tau}|u|^{p})dx\\ \nonumber &\qquad \qquad \qquad \qquad +O\bigg(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\bigg)\\ &=\frac{2(N(2-p)-2\tau)}p\mathcal{P}[u]+O\bigg(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\bigg).\label{12} \end{align} For the third term $B_3$, with calculus done in \cite[Lemma 4.5]{st4}, we have \begin{align} B_3&=\frac{2(\alpha-N)}{p}\int_{|y|<R}\int_{|x|<R}I_\alpha(x-y)|y|^{-\tau}|u(y)|^p|x|^{-\tau}|u(x)|^{p}\,dx\,dy\nonumber\\ &\qquad \qquad \qquad \qquad \qquad \qquad +O\bigg(\int_{|x|>R}(I_\alpha*|\cdot|^{-\tau}|u|^p)|x|^{-\tau}|u|^pdx\bigg)\nonumber\\ &=\frac{2(\alpha-N)}{p}\int_{|x|<R}|x|^{-\tau}|u(x)|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)dx \nonumber\\ &\qquad \qquad \qquad \qquad \qquad \qquad+O\bigg(\int_{|x|>R}(I_\alpha*|\cdot|^{-\tau}|u|^p)|x|^{-\tau}|u|^pdx\bigg)\nonumber\\ &=\frac{2(\alpha-N)}{p}\mathcal P[u]+O\bigg(\int_{|x|>R}|x|^{-\tau}|u|^p(I_\alpha*|\cdot|^{-\tau}|u|^p)dx\bigg).\label{372} \end{align} Combining \eqref{(I)}, \eqref{123}, \eqref{12} and \eqref{372}, we then obtain \begin{align} M_R'&\leq-\int_{\mathbb{R}^n}\Delta^2\phi_R|u|^2\,dx+4\int_{\mathbb{R}^n}|\nabla u|^2+4\lambda\int_{\mathbb{R}^n}\frac{|u|^2}{r^2}\,dx -4\mathcal P[u]\nonumber\\ &\qquad \qquad \qquad \qquad \qquad \qquad \quad +O\left(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)\,dx\right)\nonumber\\ &\leq4\Big(\|\sqrt{\mathcal K_\lambda} u\|^2-\mathcal P[u]\Big)+O\left(\int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)\,dx\right)+O(R^{-2}).\nonumber \end{align} Here, for the last inequality we used the fact that $|\partial^{\nu}\phi_R|\lesssim R^{2-|\nu|}$. Now, using Lemma \ref{hls} and Lemma \ref{ckn} with $b=-\frac{\tau}{p}$, $q=\frac{2Np}{\alpha+N}$, $a=0$ and $p=2$, we obtain \begin{eqnarray*} \int_{|x|>R}|x|^{-\tau}|u|^{p}(I_\alpha*|\cdot|^{-\tau}|u|^p)\,dx &\lesssim& \||x|^{-\tau}|u|^{p}\|^2_{\frac{2N}{\alpha+N}}\\ &\lesssim& R^{-2\tau}\|u\|_{\frac{2Np}{\alpha+N}}^{2p}\\ &\lesssim& R^{-2\tau}\|\nabla u\|^{2p} \end{eqnarray*} if \begin{equation}\label{q} 0<\frac{\alpha+n}{2np}\leq \frac12<1 , \quad -\frac{\alpha+n}{2p}<-\frac{\tau}{p}\leq 0, \quad \frac{\tau}{p}-1=\frac{\alpha+n}{2p}-\frac{n}{2}. \end{equation} Here we note that the last equality in \eqref{q} is equivalent to $p=1+\frac{2-2\tau+\alpha}{n-2}$. Using $p=1+\frac{2-2\tau+\alpha}{n-2}$, the requirement \eqref{q} can be written as \eqref{ie1} which are reduced to $0<\tau\leq 1+\frac{\alpha}{n}$ since $\alpha+n\geq n\tau>2\tau>0$. Consequently, for large $R\gg1$, using the equivalent norm to Sobolev one (see Lemma \ref{2.2}), we get \begin{equation*} M_R'\leq4\,\mathcal I[u]+\frac C{R^{2\tau}}\|\sqrt{\mathcal K_\lambda} u\|^{2p}+\frac C{R^2}, \end{equation*} as desired. \subsection{The boundedness of the energy solution} Now, we prove Proposition \ref{s}. Specifically, the energy bound is demonstrated by combining the conservation law with the following lemma known as coercivity (or energy trapping) results, obtained through the assumption \eqref{ss}. \begin{lem}\label{bnd'} Let $\varphi\in H_\lambda^1$ be a ground state solution to \eqref{E}. Assume that there is $0<c<1$ satisfying $$\mathcal P[u]<c \mathcal P[\varphi], \quad u\in H_\lambda^1.$$ Then there exists a constant $c_\varphi>0$ such that \begin{align*} \|\sqrt{\mathcal K_\lambda} u\|^2&<c_\varphi\mathcal E[u]. \end{align*} \end{lem} \begin{proof} Thanks to \eqref{E}, we first see \begin{equation}\label{p} \mathcal P[\varphi]:=\int_{\mathbb{R}^n} |x|^{-\tau}|\varphi|^p(I_\alpha *|\cdot|^{-\tau}|\varphi|^p)dx=\|\sqrt{\mathcal K_\lambda}\varphi\|^2. \end{equation} Applying the Gagliardo-Nirenberg inequality, Theorem \ref{gag}, we have \begin{align*} (\mathcal P[u])^p \leq (\mathcal P[u])^{p-1}\cdot C_{N,\tau,\alpha,\lambda} \|\sqrt{\mathcal K_\lambda} u\|^{2p}\leq \Big(\frac{\mathcal P[u]}{\mathcal P[\varphi]}\Big)^{p-1}\|\sqrt{\mathcal K_\lambda} u\|^{2p}. \end{align*} Here, for the second inequality, we used the fact that \begin{equation}\label{poh} C_{N,\tau,\alpha,\lambda}=\frac{\mathcal P(\varphi)}{\|\sqrt{\mathcal K_\lambda}\varphi\|^{2p}}=({\mathcal P[\varphi]})^{1-p} \end{equation} by using \eqref{p}. Therefore, we obtain \begin{eqnarray*} \mathcal P[u] &\leq&\Big(\frac{\mathcal P[u]}{\mathcal P[\varphi]}\Big)^\frac{p-1}p\|\sqrt{\mathcal K_\lambda} u\|^2, \end{eqnarray*} which implies that \begin{align*} \mathcal E[u]&=\|\sqrt{\mathcal K_\lambda} u\|^2-\frac1p\mathcal P[u]\\ &\geq\Big(1-\frac{1}p\Big(\frac{\mathcal P[u]}{\mathcal P[\varphi]}\Big)^\frac{p-1}p\Big)\|\sqrt{\mathcal K_\lambda} u\|^2\\ &\geq\Big(1-\frac{c^{(p-1)/p}}p\Big)\|\sqrt{\mathcal K_\lambda} u\|^2. \end{align*} {This concludes the proof.} \end{proof} \subsection{Energy bounded/non-global solutions} Finally, we prove Corollaries \ref{t2} and \ref{s2}, which presents the dichotomy of energy bounded/non-global existence of solutions. \subsubsection{Energy bounded solutions} First, Corollary \ref{s2} follows from the invariance of \eqref{t11} and \eqref{t12} under the flow of \eqref{S}. We first define a function $f:[0,T^\ast) \rightarrow \mathbb{R}$ as \begin{equation}\label{def} f(t)= t- \frac{C_{N,\tau,\alpha,\lambda}}{p} t^{p}. \end{equation} Since $p>1$, the function $f(t)$ has a maximum value $f(t_1)=\frac{p-1}{p}C_{N,\tau,\alpha,\lambda}^{-\frac1{p-1}}$ at $t_1= (C_{N,\tau,\alpha,\lambda})^{-\frac1{p-1}}$. We note here that $t_1=\|\sqrt{\mathcal K_\lambda} \varphi\|^2$ by \eqref{poh} and \eqref{p}. Using the Gagliardo-Nirenberg type inequality, Theorem \ref{gag}, we see \begin{align} \mathcal E[u]=\|\sqrt{\mathcal K_\lambda} u\|^2 - \frac1p \mathcal P[u] &\geq\|\sqrt{\mathcal K_\lambda} u\|^2-\frac{C_{N,\tau,\alpha,\lambda}}{p}\|\sqrt{\mathcal K_\lambda} u\|^{2p}\label{xxx}\\ &=f\big(\|\sqrt{\mathcal K_\lambda} u\|^2\big).\nonumber \end{align} By the assumption \eqref{t11} with \eqref{poh} and \eqref{p}, we also see \begin{equation}\label{x} \mathcal E[u_0]<\mathcal E[\varphi] =f(t_1), \end{equation} which implies \begin{equation}\label{xx} f\big(\|\sqrt{\mathcal K_\lambda}u\|^2\big) \leq \mathcal E[u]= \mathcal E[u_0] <f(t_1). \end{equation} Since $\|\sqrt{\mathcal K_\lambda}u_0\|^2<\|\sqrt{\mathcal K_\lambda}\varphi\|^2=t_1$ by the assumption \eqref{t12}, and the continuity in time with \eqref{xx}, we get $$\|\sqrt{\mathcal K_\lambda}u(t)\|^2 < t_1, \quad \forall t \in [0,T^\ast),$$ which is equivalent to $$\mathcal{MG}[u(t)]<1, \quad \forall t \in [0,T^\ast).$$ Therefore \eqref{t11} and \eqref{t12} are invariant under the flow \eqref{S} and this implies that $T^\ast=\infty$, which concludes the proof. \subsubsection{Blow-up } To prove Corollary \ref{t2}, we use the same function $f(t)$ defined as \eqref{def}. By the assumption \eqref{t13} with \eqref{poh} and \eqref{p}, we have $$\|\sqrt{\mathcal K_\lambda} u_0\|^2>\|\sqrt{\mathcal K_\lambda}\varphi\|^2=t_1.$$ Thus, the continuity in time with \eqref{xx} gives $$ \|\sqrt{\mathcal K_\lambda} u (t)\|^2>t_1,\quad \forall\, t\in [0,T^*) .$$ Hence, $\mathcal{MG}[u(t)]>1$ on $[0,T^*)$, and this and \eqref{t11} are invariant under flow \eqref{S}. Finally, by using $\mathcal{E}[u(t)]>1$, $\mathcal{MG}[u(t)]>1$ and the identity $p\mathcal E[\varphi]=(p-1)\|\sqrt{\mathcal K_\lambda}\varphi\|^2$, we obtain for all $t\in[0,T^\ast)$ \begin{align*} \mathcal I[u(t)]&=\|\sqrt{\mathcal K_\lambda} u\|^2-\mathcal P[u]\\ &=p\mathcal E[u]-(p-1)\|\sqrt{\mathcal K_\lambda} u\|^2\\ &< p\mathcal E[\varphi]-(p-1)\|\sqrt{\mathcal K_\lambda}\varphi\|^2<0, \end{align*} which concludes the proof by using Theorem \ref{t1}. \section{Appendix: Morawetz estimate} In this section, we present a virial identity (Proposition \eqref{mrwz}) that exhibits the convexity property in time for certain quantities associated with solutions of the generalized Hartree equation \eqref{S}. This identity serves as the basis for studying blow-up phenomena. The virial identity for the free nonlinear Schr\"odinger equation was first established by Zakharov \cite{Za} and Glassey \cite{G}. When the free equation is perturbed by an electromagnetic potential, Fanelly and Vega \cite{FV} derived the corresponding virial identities for the linear Schr\"odinger and linear wave equations. The proof relies on the standard technique of Morawetz multipliers, introduced in \cite{M} for the Klein-Gordon equation. The identity we present here is the same as that in \cite{sx}, with the addition of a term corresponding to the contribution from the inverse square potential. \begin{proof}[Proof of Proposition \eqref{mrwz}] Let $u \in C_t([0,T];H_\lambda^1)$ be a solution to the focusing case of equation \eqref{S} \begin{align}\label{S1} i\partial_t u &=-\Delta u + \frac\lambda{|x|^2}u -|x|^{-\tau}|u|^{p-2}\Big(I_\alpha*|\cdot|^{-\tau}|u|^p\Big)u \\ \nonumber &=-\Delta u + \frac\lambda{|x|^2}u -\mathcal N. \end{align} By multiplying $2 \bar u$ to \eqref{S1}, we obtain \begin{equation*} -2 \Im{(\bar u \Delta u)} = \partial_t(|u|^2) \end{equation*} Using this, we can compute $$V_\phi'(t)=2\Im\int_{\mathbb{R}^n} \bar u \nabla\phi\cdot\nabla u dx=2\sum_{k=1}^N \Im\int_{\mathbb{R}^n} \bar u \partial_k\phi\cdot \partial_k u dx.$$ In order to consider the second derivative of $V_\xi$, we need to compute \begin{equation}\label{sec} V''_{\phi}(t)=2\sum_{k=1}^N \int_{\mathbb{R}^n} \partial_k\phi\cdot \partial_t \Im(\bar u \partial_k u) dx. \end{equation} Using \eqref{S1}, we have \begin{align} \partial_t\Im(\bar u \partial_k u ) &=\Re(i\partial_t u \partial_k\bar u )-\Re(i\bar{u} \partial_k \partial_t u )\nonumber\\ &=\Re\big(\partial_k\bar u (-\Delta u + \frac\lambda{|x|^2}u -\mathcal N)\big)-\Re\big(\bar u \partial_k(-\Delta u + \frac\lambda{|x|^2}u -\mathcal N)\big)\nonumber\\ &=\Re\big(\bar u \partial_k\Delta u -\Delta u\partial_k\bar u \big)+\Re\big(\bar u \partial_k\mathcal N-\mathcal N\partial_k\bar u \big)+\lambda\Re\big(\frac{u}{|x|^2}\partial_k\bar u-\bar u \partial_k(\frac{u }{|x|^2}) \big).\label{vr} \end{align} Here, for the last term, we see \begin{equation}\label{aa} \Re\big(\bar u \partial_k(\frac{u }{|x|^2})-\frac{u}{|x|^2}\partial_k\bar u \big)=-2\frac{x_k}{|x|^4}{|u|^2}. \end{equation} For the first two term, we will apply the following lemma, omitted here, can be found in the proof of \cite[Proposition 2.12]{sx}. \begin{lem}\label{lem} Let $\phi:\mathbb{R}^N \rightarrow \mathbb{R}$ be a radial, real-valued multiplier with $\phi=\phi(|x|)$. Then, for $\mathcal N$ defined as $\mathcal N = -|x|^{-\tau}|u|^{p-2}\big(J_\alpha*|\cdot|^{-\tau}|u|^p\big)u$, we have \begin{align*} &\Re\int_{\mathbb{R}^N}(\bar u \partial_k\Delta u -\Delta u\partial_k\bar u )+(\bar u \partial_k\mathcal N-\mathcal N\partial_k\bar u ) dx\\ &\qquad\qquad =\sum_{l=1}^{N}2\int_{\mathbb{R}^n}\partial_l\partial_k\phi\,\Re(\partial_ku\partial_l\bar u)dx-\frac12\int_{\mathbb{R}^n}\Delta^2\phi|u|^2dx\\ &\qquad \qquad\qquad-\frac{(p-2)}{p}\int_{\mathbb{R}^n}\Delta\phi|x|^{-\tau}|u|^p(J_\alpha*|\cdot|^{-\tau}|u|^{p})dx\\ &\qquad\qquad \qquad \qquad -\frac{2\tau}p\int_{\mathbb{R}^n}x\cdot\nabla\phi|x|^{-\tau-2}|u|^{p}(J_\alpha*|\cdot|^{-\tau}|u|^p)dx\\ & \qquad \qquad\qquad \qquad \qquad -\frac{2(N-\alpha)}p\int_{\mathbb{R}^n}\partial_k \phi|x|^{-\tau}|u|^{p}(\frac{x_k}{|\cdot|^2}J_\alpha*|\cdot|^{-\tau}|u|^p)dx. \end{align*} \end{lem} Therefore, by combining \eqref{sec}, \eqref{vr}, Lemma \ref{lem} and \eqref{aa}, we finish the proof. \end{proof} \section{Declarations} $\!\!\!\!\!\!\bullet$ The authors have no relevant financial or non-financial interests to disclose.\\ $\bullet$ The authors have no competing interests to declare that are relevant to the content of this article.\\ $\bullet$ All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.\\ $\bullet$ The authors have no financial or proprietary interests in any material discussed in this article.\\ $\bullet$ The data that support the findings of this study are available from the corresponding author upon reasonable request. \end{document}

\begin{document} \title{Generalized description of the spatio-temporal biphoton state in spontaneous parametric down-conversion} \author{Baghdasar Baghdasaryan} \email{[email protected]} \affiliation{Theoretisch-Physikalisches Institut, Friedrich Schiller University Jena, 07743 Jena, Germany} \affiliation{Helmholtz-Institut Jena, 07743 Jena, Germany} \author{Carlos Sevilla-Gutiérrez} \affiliation{Fraunhofer Institute for Applied Optics and Precision Engineering IOF, 07745 Jena, Germany} \author{Fabian Steinlechner} \email{[email protected]} \affiliation{Fraunhofer Institute for Applied Optics and Precision Engineering IOF, 07745 Jena, Germany} \affiliation{Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany} \author{Stephan Fritzsche} \affiliation{Theoretisch-Physikalisches Institut, Friedrich Schiller University Jena, 07743 Jena, Germany} \affiliation{Helmholtz-Institut Jena, 07743 Jena, Germany} \affiliation{Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany} \date{\today} \begin{abstract} Spontaneous parametric down-conversion (SPDC) is a widely used source for photonic entanglement. Years of focused research have led to a solid understanding of the process, but a cohesive analytical description of the paraxial biphoton state has yet to be achieved. We derive a general expression for the spatio-temporal biphoton state that applies universally across common experimental settings and correctly describes the nonseparability of spatial and spectral modes. We formulate a criterion on how to decrease the coupling of the spatial from the spectral degree of freedom by taking into account the Gouy phase of interacting beams. This work provides new insights into the role of the Gouy phase in SPDC, and also into the preparation of engineered entangled states for multidimensional quantum information processing. \end{abstract} \maketitle \section{Introduction}\label{introduction} Photon pairs generated via spontaneous parametric down-conversion (SPDC) have provided an experimental platform for fundamental quantum science \cite{doi:10.1063/5.0023103} and figure prominently in applications in quantum information processing, including recent milestone experiments in photonic quantum computing \cite{doi:10.1126/science.abe8770}. Several works in recent years have addressed the challenge of tailoring the spectral and spatial properties of $signal$ and $idler$ photons generated via SPDC in theory and experiment. In the spatial domain, that is, the transverse momentum space, much of this work was motivated by the objective of improving fiber coupling efficiency \cite{PhysRevA.83.023810,srivastav2021characterising} or the dimensionality of spatial entanglement \cite{Krenn6243,PhysRevA.102.052412,PhysRevApplied.14.054069}. In the spectral domain, the motivation was usually to engineer pure spectral states, which are crucial for protocols based on multiphoton interference \cite{Caspani2017}. This has been performed either by tailoring the nonlinearity of the crystal \cite{Graffitti:18} or by using counterpropagating photon pair generation in periodically poled waveguides \cite{Luo:20}. The frequency degree of freedom (DOF) has also been used to generate entangled states via spatial shaping of the pump beam \cite{Francesconi2021} or by transferring polarization into color entanglement \cite{PhysRevLett.103.253601}. The spatial shaping of the pump beam has been also used in Hong-Ou-Mandel interference experiments, in order to control the two-photon interference behavior \cite{PhysRevLett.90.143601}. Closed expressions for the state emitted by SPDC in bulk crystals have been derived using very special techniques and approximations, such as the narrowband \cite{PhysRevA.83.033816}, thin-crystal \cite{Yao_2011,PhysRevA.103.063508} or plane wave approximations \cite{PhysRevLett.99.243601}, where either the spectral or spatial biphoton state is considered. However, from the $X$-shaped spatio-temporal correlations \cite{PhysRevLett.102.223601,PhysRevLett.109.243901}, the spatial and spectral properties of SPDC have been known to be coupled. The $X$-shaped spatio-temporal correlation implies that if the twin photons are collected from different positions, they are detected with a certain time delay. In contrast, if the photons are detected at the same position, the time delay is very short (a few nanoseconds) \cite{Zhang2017}. To date, models that address both spectrum and space together have been limited to approximate phase matching functions \cite{Osorio_2008} or numerical calculations \cite{PhysRevA.86.053803}. The work \cite{PhysRevLett.102.223601} investigated the quite general phase matching function, but the pump beam was limited to monochromatic plane wave. Here, we present a simple-to-use closed expression for the biphoton state. The approach describes the full spectral and spatial properties of all interacting beams and applies to a wide range of experimental settings. The analytical treatment of the biphoton state decomposed into discrete Laguerre Gaussian (LG) modes also provides a deeper insight into the role of the Guoy phase in PDC. Especially, the spectral response of spatial modes in SPDC is determined by the Gouy phase of the pump, signal and idler beams. We will also show that the Gouy phase can be used to control the coupling strength of spatial and spectral DOF in parametric down-conversion (PDC). Next to providing an intuitive understanding, we also demonstrate the utility of the expression for quantum state engineering in spatial DOF for multidimensional quantum information processing. \section{Theoretical Methods} Let us start with the basic expressions of SPDC. We can make use of the paraxial approximation, since typical optical apparatuses support only paraxial rays about a central axis. In the paraxial regime, the longitudinal and transverse components of the wave vector can be treated separately $\bm{k}=\bm{q}+k_z(\omega)\bm{z}$. Consequently, the biphoton state in the momentum space can be represented by the following expression \cite{PhysRevA.62.043816,WALBORN201087,Karan_2020} \begin{align}\label{SPDCstate} \ket{\Psi} = \iint & d\bm{q}_s \: d\bm{q}_i \:d\omega_s \: d\omega_i\: \Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)\nonumber\\& \hat{a}^{\dagger}_s(\bm{q}_s,\omega_s)\:\hat{a}^{\dagger}_i(\bm{q}_i,\omega_i)\ket{vac}. \end{align} Equation \eqref{SPDCstate} refers to the generation of photon pairs with energies $\omega_{s,i}$ and transverse momenta $\bm{q}_{s,i}$ from the vacuum state $\ket{vac}$. The biphoton mode function $\Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)$ contains the rich high-dimensional spatio-temporal structure of SPDC that arises from the coupling between the wave vectors of the pump, signal, and idler beams. \subsection{Biphoton state decomposed in Laguerre Gaussian basis} The transverse spatial \cite{WALBORN201087,PhysRevA.83.052325} and frequency DOF \cite{PhysRevA.105.052429} have been successfully used in continuous variable information processing. However, in practical experimental settings, the continuous variable space is more often discretized using a set of modes. The proper choice of a set reduces the number of dimensions needed to describe the state. Moreover, discrete modes are easy to manipulate and detect using efficient experimental techniques \cite{Bolduc2013,Eckstein2011}. Since the projection of the orbital angular momentum (OAM) is conserved in SPDC \cite{Mair2001}, it is convenient to decompose the biphoton state into LG modes $\ket{p,\ell,\omega}=\int d\bm{q}\, \mathrm{LG}_{p}^{\ell}(\bm{q})\, \hat{a}^{\dagger}(\bm{q},\omega) \ket{vac} $, which are eigenstates of OAM \cite{ doi:10.1126/science.1227193}: \begin{align}\label{decomposition} \ket{\Psi}= &\iint \:d\omega_s \: d\omega_i\: \nonumber \\& \sum_{p_s,p_i=0}^{\infty}\: \sum^{\infty}_{\ell_s,\ell_i=-\infty}C_{p_s,p_i}^{\ell_s,\ell_i} \ket{p_s,\ell_s,\omega_s}\ket{p_i,\ell_i,\omega_i}, \end{align} where the coincidence amplitudes are calculated from the overlap integral $ C^{\ell_s,\ell_i}_{p_s,p_i} = \braket{p_s,\ell_s,\omega_s;p_i,\ell_i,\omega_i |\Psi}$, \begin{align}\label{coe1} C^{\ell_s,\ell_i}_{p_s,p_i} = \iint d\bm{q}_s \: d\bm{q}_i \: \Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)\:[\mathrm{LG}_{p_s}^{\ell_s}(\bm{q}_s)]^*\nonumber&\\ \times [\mathrm{LG}_{p_i}^{\ell_i}(\bm{q}_i)]^*. \end{align} The angular distribution of an LG mode in the momentum space is given by \begin{align} \label{LG} \mathrm{LG}_{p}^{\ell}(\rho,\varphi) =&e^{\frac{-\rho^2\,w^2}{4}}\,e^{i\ell\,\varphi}\,\sum_{u=0}^p\, T_u^{p,\ell}\, \rho^{2k+\abs{\ell}} \end{align} with $ T_u^{p,\ell}$ being \begin{align*} T_u^{p,\ell}= &\sqrt{\frac{p!\,(p+|\ell|)!}{\pi}}\, \biggr(\frac{ w}{\sqrt{2}}\biggl)^{2u+|\ell|+1}\,\frac{(-1)^{p+u}(i)^{\ell}}{(p-u)!\,(\abs{\ell}+u)!\,u!}, \end{align*} and where $\rho$ and $\varphi$ stand for the cylindrical coordinates $\bm{q}=(\rho,\varphi)$. The summations in Eq. \eqref{decomposition} run over the LG mode numbers $p$ and $\ell$ associated with the radial momentum and the OAM projection, respectively. Except for the fact, that we now deal with summations instead of integrations, this discretization will also help us to understand the coupling of spatial and spectral DOF in the frame of the Gouy phase of LG modes. Note that we discretize only the transverse spatial DOF, but in principle, it is also possible to discretize the frequency DOF \cite{Gil-Lopez:21}. The construction of the biphoton state reduces to the calculation of the coincidence amplitudes $C^{\ell_s,\ell_i}_{p_s,p_i}$, which in turn depend on the mode function $\Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)$. A compact expression for the mode function can be derived if the experimental setup and geometry is fixed. \subsection{Geometry and mode function} Here, we consider the scenario when a coherent laser beam propagates along the $z$ axis and is focused in the middle of a nonlinear crystal placed at $z=0$. Signal and idler fields propagate close to the pump direction, known as the quasicollinear regime. The crystal and the pump beam have typical transverse cross sections in the order of millimeters and micrometers, respectively. Hence, we assume that the crystal compared to the pump beam is infinitely extended in the transverse direction, which enforces the conservation of the transverse momentum, $\bm{q}_p=\bm{q}_{s}+\bm{q}_{i}$ \cite{PhysRevA.62.043816}. Taking into account also the energy conservation $\omega_p=\omega_s+\omega_i$, the mode function can be written as \cite{Karan_2020} \begin{align} \label{phasefunction} \Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)=&N_0\,\mathrm{V}_p(\bm{q}_s+\bm{q}_i)\,\mathrm{S}_p(\omega_s+\omega_i)\nonumber\\&\times\int_{-L/2}^{L/2} dz\:\exp{\biggr[iz(k_{z,p}-k_{z,s}-k_{z,i})\biggl]}, \end{align} where $N_0$ is the normalization constant, $\mathrm{V}_p(\bm{q}_p)$ is the spatial and $\mathrm{S}_p(\omega_p)$ the spectral distribution of the pump beam, and $L$ is the length of the nonlinear crystal along the $z$ axis. The important component of the mode function \eqref{phasefunction} is the phase mismatch in the $z$ direction $\Delta k_z=k_{z,p}-k_{z,s}-k_{z,i}$, which characterizes the differences in the energies and momenta of the signal and idler photons. Therefore, careful calculation of $\Delta k_z$ is essential for the quantitative description of SPDC, which we will do next. Experimentally generated lights are usually not monochromatic and contain a frequency distribution. Therefore, except for the central frequencies that meet energy conservation condition $\omega_{0,p}=\omega_{0,s}+\omega_{0,i}$, we expect a deviation from them, $\omega=\omega_0+\Omega$ with the assumption $\Omega\ll\omega_0$. Furthermore, in the paraxial approximation, the transverse component of the momentum is much smaller than the longitudinal component $|\bm{q}|\ll k$. Hence, we can apply the Taylor series on $k_z$ (Fresnel approximation) to $|\bm{q}|/k$ and also to small $\Omega$: \begin{equation*} k_z=k(\Omega)\sqrt{1-\frac{|\bm{q}|^2}{k(\Omega)^2}}\approx k+\frac{\Omega}{u_g}+\frac{G\Omega^2}{2}-\frac{|\bm{q}|^2}{2k}, \end{equation*} where $u_g=1/(\partial k/\partial \Omega)$ is the group velocity and $G=\partial/\partial \Omega \,(1/u_g)$ is the group velocity dispersion, evaluated at the respective central frequency. Here, we also assume that the propagation is along a principal axis of the crystal, so we can ignore the Poynting vector walk-off of extraordinary beams in the crystal. Next, we insert the corresponding $k_z$ of the pump, signal, and idler into the phase mismatch $\Delta k_z$ and arrive at \begin{equation}\label{phaseMatching} \Delta k_z= \Delta_{\Omega}+\rho_{s}^2\frac{k_p-k_s}{2k_pk_s}+\rho_{i}^2\frac{k_p-k_i}{2k_pk_i} -\frac{\rho_{s}\rho_{i}}{k_p}\cos{(\varphi_i-\varphi_s)}, \end{equation} where the frequency part $\Delta_{\Omega}$ is given by \begin{align}\label{spectralStr} \Delta_{\Omega}=&\frac{\Omega_s+\Omega_i}{u_{g,p}}-\frac{\Omega_s}{u_{g,s}}-\frac{\Omega_i}{u_{g,i}}+\frac{G_p(\Omega_s+\Omega_i)^2}{2}\nonumber\\& -\frac{G_s\Omega_s^2}{2}-\frac{G_i\Omega_i^2}{2}. \end{align} We used in Eq. \eqref{phaseMatching} the relation $\rho_p^2=\rho^2_s+\rho^2_i+2\rho_s\rho_i\cos{(\varphi_i-\varphi_s)}$ and assumed momentum conservation for central frequencies $\Delta k=k_p-k_s-k_i=0$. The condition $\Delta k=0$ ensures constructive interference in the crystal between the pump, signal, and idler beams, which is usually performed with birefringent crystals \cite{Karan_2020} or more recently by periodic poling along the crystal axis, $k_p-k_s-k_i-2 \pi /\Lambda=0$, where $\Lambda$ is the poling period \cite{doi:10.1063/1.123408}. The remaining components of the mode function \eqref{phasefunction} that we should still fix are the pump characteristics. We model the angular distribution of the pump with an LG beam. The advantage of this choice is that an arbitrary paraxial optical field can be expressed as a sum of LG beams $\sum_n a_n \mathrm{LG}_{p_n}^{\ell_n} $ with $\sum_n |a_n|^2=1$ by using their completeness relation. Thus, the theory developed for the LG pump can be easily extended to SPDC with a particular pump. The amplitudes $\eqref{expression}$ can then be upgraded to revised amplitudes $\sum_n a_n C_n$, which follows from Eq. \eqref{coe1}. Finally, the temporal distribution is modeled with a Gaussian envelope of pulse duration $t_0$, $\mathrm{S}_p(\omega_p)=\exp{[-(\omega_p-\omega_{0,p})^2\,t_0^2/4]}\, t_0/\sqrt{\pi}$ \cite{PhysRevA.56.1627}, but which can be extended to any arbitrary pump spectrum. \subsection{Derivation of coincidence amplitudes} We can now substitute Eqs. \eqref{LG}-\eqref{spectralStr} into Eq. \eqref{coe1} and calculate the coincidence amplitudes: \begin{widetext} \begin{align} \label{expansionfirst} C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}=N_0\sum_{u=0}^p\,\sum_{s=0}^{p_s}\,\sum_{i=0}^{p_i}\,T_u^{p,\ell}\: (T_s^{p_s,\ell_s})^* \:(T_i^{p_i,\ell_i})^*\: \int\,dz\,d\rho_s\,d\rho_i\,d\varphi_s\,d\varphi_i\, \Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s) e^{i\ell\varphi_s}\,e^{i(-\ell_s\varphi_s-\ell_i\varphi_i)}, \end{align} where we used a revised notation for the coincidence amplitudes $C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}$ to indicate the mode numbers of the pump. The function $\Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s)$ is defined as \begin{align} \Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s)=& [\rho^2_s+\rho^2_i+2\rho_s\rho_i\cos{(\varphi_i-\varphi_s)}]^\frac{2u+(\abs{\ell}-\ell)}{2}\,\rho_s^{\,\abs{\ell_s}+2s+1}\, \rho_i^{\,\abs{\ell_i}+2i+1}\,(\rho_s+\rho_i\,e^{i(\varphi_i-\varphi_s)})^{\ell} \nonumber\\&\times \exp{\biggl[-\frac{[\rho^2_s+\rho^2_i+2\rho_s\rho_i\cos{(\varphi_i-\varphi_s)}]\,w^2}{4}-\frac{\rho_s^2\,w_s^2}{4}-\frac{\rho_i^2\,w_i^2}{4}\biggr]}\,\frac{t_0}{\sqrt{\pi}}e^{-\frac{t_0^2(\Omega_s+\Omega_i)^2}{4}} \nonumber \\& \times \exp{\biggr[iz\biggr(\Delta_{\Omega}+\rho_s^2\frac{k_p-k_s}{2k_pk_s}+\rho_i^2\frac{k_p-k_i}{2k_pk_i}-\cos{(\varphi_i-\varphi_s)}\frac{\rho_s\rho_i}{k_p}\biggl)\biggl]}\label{bigexpression}. \end{align} \end{widetext} In Eq. \eqref{bigexpression}, the polar angle $\varphi$ of the pump beam has been expressed as a function of signal and idler coordinates, \begin{equation*} e^{i\,\ell\,\varphi}=(\cos{\varphi}+i\sin{\varphi})^{\ell}=\frac{e^{i\ell\varphi_s}}{\rho_p^{\ell}}\,(\rho_s+\rho_i\,e^{i(\varphi_i-\varphi_s)})^{\ell}, \end{equation*} by taking into account the conservation of transverse momentum, \begin{equation*} \bm{q}_p=\bm{q}_{s}+\bm{q}_{i}= \begin{pmatrix} \rho_{s}\cos{\varphi_{s}}+\rho_{i}\cos{\varphi_{i}}\\ \rho_{s}\sin{\varphi_{s}}+\rho_{i}\sin{\varphi_{i}} \end{pmatrix}. \end{equation*} The presentation of the coincidence amplitudes $C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}$ in Eq. \eqref{expansionfirst} with the function $\Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s)$ follows the goal to show the OAM conservation in SDPC. To do so, we expand the function $\Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s)$ as superposition of plane waves with the phases $\exp{[i\ell^{'}(\varphi_i-\varphi_s)]} $(Fourier series with complex coefficients), \begin{equation}\label{phasedif} \Theta(z,\rho_s,\rho_i,\varphi_i-\varphi_s)=\sum_{\ell^{'}=-\infty}^{\infty}f_{\ell^{'}}(z,\rho_s,\rho_i)e^{i\ell^{'}(\varphi_i-\varphi_s)}. \end{equation} We substitute expression \eqref{phasedif} into Eq. \eqref{expansionfirst} and perform the integration over the polar angles $\varphi_s$ and $\varphi_i$: \begin{align} \sum^\infty_{\ell^{'}=-\infty}f_{\ell^{'}}(z,\rho_s,\rho_i)\int_0^{2\pi}\int_0^{2\pi} e^{i\ell\varphi_s}\,e^{i(-\ell_s\varphi_s-\ell_i\varphi_i)}\nonumber\\ \times e^{i\ell^{'}(\varphi_i-\varphi_s)}d\varphi_sd\varphi_i \propto \delta_{\ell^{'},\ell-\ell_s}\delta_{\ell^{'},\ell_i}.\label{delta} \end{align} As expected, the Kronecker delta functions appear in Eq. \eqref{delta} which enforce the conservation of OAM $\ell-\ell_s=\ell_i$. This conservation is not valid out of the quasicollinear regime \cite{MOLINATERRIZA2003155} because of the spin-orbital angular momentum coupling in the nonparaxial regime \cite{PhysRevA.99.023403}. In a non-collinear regime, the total angular momentum should remain conserved, which can be a future topic to study. Going back to expression \eqref{expansionfirst}, we now calculate the integration over polar coordinates $\varphi_{s,i}$ explicitly. For simplicity, we consider the coincidence amplitudes $ C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}$ for positive OAM number of the pump beam $\ell \geq 0$. The coincidence amplitude for $\ell< 0$ is then given by $ C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}= (C_{p,p_s,p_i}^{-\ell,-\ell_s,-\ell_i})^*$, which follows from Eq. \eqref{coe1}. Furthermore, the two brackets on the first line in Eq. \eqref{bigexpression} should be rewritten as finite sums by using the binomial formula. For instance, the first bracket is written as \begin{align*} [\rho^2_s+\rho^2_i+2\rho_s\rho_i\cos{(\varphi_i-\varphi_s)}]^u=\sum_{m=0}^{u} \binom{u}{m}(\rho^2_s+\rho^2_i)^{u-m}&\\ \times [2\rho_s\rho_i\cos{(\varphi_i-\varphi_s)}]^m. \end{align*} The \textit{cosine} function can be expressed as the sum of two exponential functions by using Euler's formula, which should be again expressed as a Binomial sum. After this step, the angular integration takes the form of the integral representation of the Bessel function of the first kind \cite{YOUSIF1997199} \begin{equation*} \frac{1}{2\pi}\int^{2\pi}_0 e^{i n\varphi \pm iz\cos{(\varphi-\varphi^{\prime})} }d\varphi= (\pm i)^n e^{i n\varphi^{\prime}} J_n(z). \end{equation*} Next, the sum representation of the Bessel function should be used \begin{equation} J_n(z)= \sum_{k=0}^{\infty} \frac{(-1)^k}{k!\, \Gamma (k+n+1)}\biggl(\frac{z}{2}\biggr)^{2k+n},\label{Bsum} \end{equation} which transforms the integration over the radial coordinates into \begin{equation*} \int_0^{\infty} d\rho\,\rho^n e^{-a \,\rho^2}=\frac{\Gamma(\frac{n+1}{2})}{2a^{\frac{n+1}{2}}}. \end{equation*} The final result is achieved via summing over $k$ from Eq. \eqref{Bsum} by using the definition of the \textit{Regularized} hypergeometric function \cite{Hypergeometric2F1}. The coincidence amplitudes read for $\ell \geq 0$ as \begin{align} \label{expression} C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i} = & N_0\,\pi^{3/2}\:t_0\: e^{-\frac{t_0^2(\Omega_s+\Omega_i)^2}{4}} \: \delta_{\ell,\ell_s+\ell_i} \nonumber\\ & \sum_{u=0}^{p}\sum_{s=0}^{p_s}\sum_{i=0}^{p_i} T_u^{p,\ell}\: (T_s^{p_s,\ell_s})^* \:(T_i^{p_i,\ell_i})^*\: \sum_{n=0}^{\ell}\sum_{m=0}^{u}\nonumber\\ & \binom{\ell}{n}\binom{u}{m}\sum_{f=0}^{u-m}\sum_{v=0}^{m}\: \binom{u-m}{f}\binom{m}{v} \Gamma[h]\:\Gamma[b]\nonumber\\ & \int_{-L/2}^{L/2}dz\:e^{i z\, \Delta_{\Omega}}\:\frac{D^{d}}{H^{h}\: B^{b}}\: {_2}{\Tilde{F}}_1\biggl[h,b, 1+d,\frac{D^2}{H \,B }\biggl] \end{align} and $ C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}= (C_{p,p_s,p_i}^{-\ell,-\ell_s,-\ell_i})^*$ for $\ell < 0$. The function ${_2}{\Tilde{F}}_1$ is known as the \textit{regularized} \textit{hypergeometric} function \cite{Hypergeometric2F1}. The missing coefficients of Eq. \eqref{expression} are given by \begin{eqnarray*} H &=& \frac{w_p^2}{4}+\frac{w_s^2}{4}-i z\frac{k_p-k_s}{2k_p k_s}, \qquad D = -\frac{ w_p^2}{4}-iz\frac{1}{2k_p}, \\[0.1cm] B& = & \frac{w_p^2}{4}+\frac{w_i^2}{4}-i z\frac{k_p-k_i}{2k_p k_i}, \qquad d =\ell_i+m-n-2v, \\[0.1cm] h & = & \frac{1}{2}(2+2s+\ell+\ell_i+2(-f+u)-2n-2v+\abs{\ell_s}), \\[0.1cm] b &=& \frac{1}{2}(2+2f+2i+\ell_i+2m-2v+\abs{\ell_i}), \end{eqnarray*} where $w_p$, $w_s$ and $w_i$ are the beam waists of the pump signal and the idler beams, respectively. Expression \eqref{expression} for the coincidence amplitudes as a function of the pump mode constitutes the main result of this work. It allows the spatial and spectral emission profiles to be reconstructed mode by mode and is applicable in any experimental setting that exhibits cylindrical symmetry. It can be readily used to calculate many characteristics of SPDC: joint spectral density, photon bandwidths, pair-collection probability, heralding ratio, spectral and spatial correlation, etc. Previously, these could only be achieved through numerical calculations or for special cases with a limited scope of applicability. The experimental demonstration of Eq. \eqref{expression} has already been presented in Ref. \cite{carlos}, where we also showed how the coupling of spatial and spectral DOF deteriorates the spatial entanglement but can be compensated directly by a proper choice of the collection mode. \subsection{Gouy phase and spatio-temporal coupling} The spatio-temporal coupling encoded in Eq. \eqref{expression} is a fundamental feature of SPDC. However, the usual applications in quantum optics utilize either the spatial or spectral DOF, neglecting the correlation between them. Nevertheless, this coupling remains a fundamental issue in many protocols based on entangled photon sources, where any distinguishability arising from not-considered DOF reduces the coherence of the state. Next, we will illustrate the utility of expression \eqref{expression} in the frame of possible decoupling of spatial and spectral DOF $\Phi(\bm{q}_s,\bm{q}_i,\omega_s,\omega_i)= \Phi_{\bm{q}}(\bm{q}_s,\bm{q}_i)\Phi_{\omega}(\omega_s,\omega_i)$. We will show that this decoupling is closely related to the Gouy phase of interacting beams. The role of the Gouy phase in nonlinear processes has been investigated before. For instance, in SPDC, the change of the Gouy phase $\psi_G(z)=(N+1)\arctan(z/z_R)$ within the propagation distance has been used to control the relative phase of two different LG modes of measurement basis \cite{PhysRevLett.101.050501,DEBRITO2021126989}. Here, $N$ is the combined LG mode number $N=2p+\abs{\ell}$ and $z_R$ is the Rayleigh length. In four-wave mixing (FWM), the conversion behavior between LG modes is strongly affected by the Gouy phase \cite{PhysRevA.103.L021502}. The authors observed that the existence of a relative Gouy phase between modes with different mode numbers $N$ leads to a reduced FWM efficiency. Here, we have a similar situation: pump, signal, and idler fields acquire different Gouy phases along with propagation in the crystal due to different mode numbers $N$, causing a reduced efficiency of mode down-conversion. We expect intuitively that the shape of the spectrum of spatial modes is affected by the relative Gouy phase of interacting beams. This is still a guess and requires proof. We consider for simplicity the scenario in which the Rayleigh lengths of the three beams are equal $z_{R,p}=z_{R,i}=z_{R,s}$ and fixed. This condition matches the Gouy angle $\arctan(z/z_R)$ for all beams. Hence, the relative Gouy phase can be written as \begin{equation*} \psi_{G,p}-\psi_{G,s}-\psi_{G,i}=(N_p-N_s-N_i-1)\arctan(z/z_R). \end{equation*} This implies that the Gouy phase is fully defined by the relative mode number $N_R= N_p-N_s-N_i$. If the Gouy phase is responsible for different spectral dependencies of the coincidence amplitudes $C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}(\Omega_s$,$\Omega_i)$, the shape of the spectrum should remain the same for fixed $N_R$. Assuming $k_p= 2 k_s$, Eq. \eqref{expression} transforms into \begin{equation} C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}(\Omega_s,\Omega_i)\propto \int_{-L/2}^{L/2}dz\:e^{i z\, \Delta_{\Omega}}\:\frac{(i2z+k_p w_p^2)^{N_R}}{(-i2z+k_p w_p^2)^{N_R+1}}. \label{gouy} \end{equation} We see from Eq. \eqref{gouy} that the spectral response of $C_{p,p_s,p_i}^{\ell,\ell_s,\ell_i}(\Omega_s,\Omega_i)$ encoded only in the term $e^{i z\, \Delta_{\Omega}}$ remains unaffected up to a constant if $N_R$ is fixed. On the other hand, $N_R$ can be rewritten as \begin{equation}\label{Rnumber} N_R=\frac{ \psi_{G,R}}{\arctan(z/z_R)}+1, \end{equation} where $\psi_{G,R}$ is the relative Gouy phase $\psi_{G,p}-\psi_{G,s}-\psi_{G,i}$. Therefore, it follows from Eqs. \eqref{gouy} and \eqref{Rnumber} that the spectral response of coincidence amplitudes is determined by the relative Gouy phase $\psi_{G,R}$ if the pump characteristics $z_R$, $w_p$, and $k_p$ are fixed. This is what we wanted to prove. Note that the simple form of Eq. \eqref{gouy} is due to the assumptions $k_p= 2 k_s$ and $z_{R,p}=z_{R,i}=z_{R,s}$. The analytical proof for the general case requires more effort, which we omit here. This proof brings us a step forward in the decoupling problem of spatial and spectral DOF: the decoupling can be achieved for a selected subspace of modes that possess the same relative Gouy phase. So, if a state is engineered that consists of modes with $N_R=const.$ assuming $z_{R,p}=z_{R,i}=z_{R,s}$, then the modes contributing to the state have the same spectrum, i.e., the state is separable. The question of decoupling of spatial and spectral DOF can be now reformulated: How do we engineer a state only consisting of modes with the same relative mode number $N_R$. \begin{figure*}\label{fig1.pdf} \end{figure*} \section{Engineering high-dimensional entangled states in OAM basis} The state engineering in spatial DOF has been investigated theoretically in the thin crystal regime \cite{PhysRevA.67.052313,https://doi.org/10.1002/qute.202100066} and also implemented experimentally \cite{PhysRevA.98.060301,PhysRevA.98.062316}. In particular, three-, four-, and five-dimensional entangled states in OAM basis have been generated in Ref. \cite{PhysRevA.98.060301} using a superposition of LG beams for the pump. The correct superposition for the pump has been determined with a simultaneous perturbation stochastic approximation algorithm \cite{705889}. We show in this section how to calculate the correct superposition of LG beams with Eq. \eqref{expression}, in order to generate entangled states in OAM basis including the states from Ref. \cite{PhysRevA.98.060301}. Our method is very straightforward and requires no optimization algorithm. In comparison to Refs. \cite{PhysRevA.67.052313,https://doi.org/10.1002/qute.202100066}, our results can be directly implemented in a real experiment, since we do not consider the thin crystal approximation. State engineering in the thin crystal regime is inefficient due to an infinite amount of spatial modes generated in the down-conversion. \subsection{Determination of pump beam} We consider the four-dimensional subspace $\ell_s,\ell_i=0,1,2,3$ and $p_s=p_i=0$, which we refer to as $S_4$, with associated notation $\ket{p_s=0,\ell_s,\omega_s}\ket{p_i=0,\ell_i,\omega_i}:=\ket{\ell_s(\omega_s),\ell_i(\omega_i)}$. The goal is to engineer a four-dimensional maximally entangled state in this subspace. We model the pump beam as a superposition of LG beams, \begin{eqnarray*} \mathrm{V}_p & = & \sum_{\ell}a_{\ell}\;\mathrm{LG}_{0}^{\ell}, \end{eqnarray*} where the range of summation is determined with the possible minimal and maximal OAM values in the subspace, $\ell=[\mathrm{min}(\ell_s+\ell_i),\mathrm{max}(\ell_s+\ell_i)]$. The correct choice of the expansion amplitudes $a_{\ell}$ is now our task. Since the pump function appears in Eq. \eqref{decomposition} linearly, the corresponding state in $S_4$ is given by \begin{align*} \ket{\Psi_4} = \sum^6_{\ell=0}a_{\ell} \:\sum^{3}_{\ell_s,\ell_i=0}C_{0,0,0}^{\ell,\ell_s,\ell_i} \ket{\ell_s,\ell_i}. \end{align*} The matrix representation of the state $\ket{\Psi_4}$ can clarify the right choice of the coefficients $a_{\ell}$. The matrix consists of $16$ elements and is given by the left-hand side of the following expression: \begin{align} \begin{pmatrix} \textcolor{blue}{a_0}\,C_{0,0} & \textcolor{red}{a_1}\,C_{1,0} &\textcolor{blue}{a_2}\,C_{2,0}& a_3\,C_{3,0}\\ \textcolor{red}{a_1\,}C_{0,1} & \textcolor{blue}{a_2}\,C_{1,1} & a_3\,C_{2,1}& \textcolor{blue}{a_4}\,C_{3,1}\\ \textcolor{blue}{a_2}\,C_{0,2} & a_3\,C_{1,2} & \textcolor{blue}{a_4}\,C_{2,2}&\textcolor{red}{a_5}\,C_{3,2}\\ a_3\,C_{0,3} & \textcolor{blue}{a_4}\,C_{1,3} & \textcolor{red}{a_5}\,C_{2,3}& \textcolor{blue}{a_6}\,C_{3,3} \end{pmatrix}\rightarrow \begin{pmatrix} 0 & \textcolor{red}{1} &0&0\\ \textcolor{red}{1} & 0 & 0& 0\\ 0& 0 & 0&\textcolor{red}{1}\\ 0& 0 & \textcolor{red}{1}& 0 \end{pmatrix},\label{matrix} \end{align} where we used the notation $ C_{0,0,0}^{\ell_s+\ell_i,\ell_s,\ell_i}= C_{\ell_i,\ell_s}$. The state becomes maximally entangled in this subspace if the matrix has exactly one entry of $1$ in each row and each column and $0$ elsewhere (permutation matrix). The right-hand side of expression \eqref{matrix} is such a state that can be engineered if we select $a_1=1/C_{0,1}\approx1/C_{1,0}$, $a_5=1/C_{2,3}\approx1/C_{3,2}$ and $a_0=a_2=a_3=a_4=a_6=0$, where we assumed degenerate SPDC $k_p\approx 2 k_s$. This choice leads to the state $\ket{\Psi_4}=\frac{1}{2}(\ket{0,1}+\ket{1,0}+\ket{2,3}+\ket{3,2})$. Thus, the state engineering is finished, where the coefficients of the pump superposition $\{a_1,a_5\}$ should be calculated with the expression \eqref{expression}. In the same way, the state $\ket{\Psi^{\prime}_4}=\frac{1}{2}(\ket{0,0}+\ket{1,1}+\ket{2,2}+\ket{3,3})$ from Ref. \cite{PhysRevA.98.060301} can also be engineered, if we select $\{a_0,a_2,a_4,a_6\}$ to be equal to $\{1/C_{0,0},1/C_{1,1},1/C_{2,2},1/C_{3,3}\}$ and $a_1=a_3=a_5=0$. The states $\ket{\Psi^{\prime}_4}$ and $\ket{\Psi_4}$ are presented in Figs. \ref{fig1.pdf}(a) and \ref{fig1.pdf}(b) with blue-colored bars on top. As we can see, the modes contributing to the states $\ket{\Psi^{\prime}_4}$ and $\ket{\Psi_4}$ represent just a part of the full OAM emission (spiral bandwidth). Therefore, the postselection should be the final step in the engineering process, where undesirable modes are sorted out. Next, we should calculate the Schmidt number and the purity of the presented states, in order to evaluate the efficiency of the state preparation in the subspace $S_4$. We will use for all our calculations the same experimental parameters as in Ref. \cite{PhysRevA.98.060301}: $15$-$mm$-thick periodically poled $\mathrm{KTiOPO}_4$ crystal designed for a collinear frequency degenerate type-II phase matching, continuous-wave laser of wavelength $405$ \textit{nm} with beam waist $w_p=25$ $\mu m$ and detection modes of radius $w_{s,i}=33$ $\mu m$. \subsection{Schmidt number and purity of subspace states} We compare first the azimuthal Schmidt numbers of the states $\ket{\Psi_4}$ and $\ket{\Psi^{\prime}_4}$ in the subspace $S_4$. Obviously, the diagonal modes $\{\ket{0,2},\ket{2,0},\ket{1,3},\ket{3,1}\}$ in Fig. \ref{fig1.pdf} (a) are non-desirable and lead to a decrease of entanglement in $S_4$. Consequently, the state $\ket{\Psi^{\prime}_4}$ has an azimuthal Schmidt number less than $4$, $K=2.04$, while the Schmidt number of the state $\ket{\Psi_4}$ equals $4$. Therefore, the preparation of the state $\ket{\Psi_4}$ is more efficient in $S_4$ than for $\ket{\Psi^{\prime}_4}$. $K=4$ is necessary, but not a sufficient condition for a four-dimensional state to be maximally entangled. Additionally, the state should be pure. Hence, the state $\ket{\Psi_4}$ can be called maximally entangled in $S_4$, if it is also spatially pure. In order to calculate the spatial purity of $\ket{\Psi_4}$, we need the reduced density matrix $\rho_{\bm{q}}$, which results from tracing over the frequency $\rho_{\bm{q}}=\mathrm{Tr}_{\Omega}(\rho)$. The fact of a continuous wave laser $\mathrm{S}_p(\omega_p)\propto\delta(\omega_p-\omega_{0,p})$ sets the condition $\Omega_s=-\Omega_i:=\Omega$, which transforms Eq. \eqref{decomposition} into \begin{align} \ket{\Psi}= &\iint \:d\Omega\: \sum^{\infty}_{\ell_s,\ell_i=-\infty}C_{\ell_s,\ell_i}(\Omega) \ket{\ell_s,\Omega}\ket{\ell_i,-\Omega}. \end{align} Now, we calculate the density matrix $\rho=|\Psi\rangle\langle\Psi|$ and then trace over the spectral domain, which yields: \begin{equation} \rho_{\bm{q}}=\sum_{\ell_s,\ell_i}\sum_{\Tilde{\ell_s},\Tilde{\ell_i}}A^{\Tilde{\ell_s},\Tilde{\ell_i}}_{\ell_s,\ell_i}|\ell_s,\ell_i\rangle\langle\Tilde{\ell_s},\Tilde{\ell_i}|,\label{density} \end{equation} where $A^{\Tilde{\ell_s},\Tilde{\ell_i}}_{\ell_s,\ell_i}=\int d\Omega\: C_{\ell_s,\ell_i}(\Omega)\:[C_{\Tilde{\ell_s},\Tilde{\ell_i}}(\Omega)]^*$ is the overlap integral of the spectra of the OAM modes. Equation \eqref{density} is very useful to calculate the spatial purity in small subspaces. We run summations in Eq. \eqref{density} over $\ell_s,\ell_i,\Tilde{\ell_s},\Tilde{\ell_i}=0,1,2,3$, renormalize the state, construct the density matrix of the subspace $\rho_{\bm{q},s}$ and calculate the purity $\mathrm{Tr}(\rho^2_{\bm{q},s})$. Here, the subscript $s$ indicates the consideration of the subspace $S_4$. In fact, the state $\ket{\Psi_4}$ is spatially pure, $\mathrm{Tr}(\rho^2_{\bm{q},s})=1$. The reason is very trivial: all modes that contribute to the state consist of only positive OAM numbers, which leads to the same $N_R=\abs{\ell}-\abs{\ell_s}-\abs{\ell_i}=0$ for all modes due to $\ell=\ell_s+\ell_i$. Moreover, the experimental parameters from Ref. \cite{PhysRevA.98.060301} satisfy the condition $z_{R,p}\approx z_{R,i}\approx z_{R,s}$. Hence, all modes have the same relative Gouy phase and consequently, the same spectrum, which is presented in Fig. \ref{fig1.pdf}(c) with the blue curve. This means that the spatial and spectral DOF are decoupled in $S_4$. Interestingly, even though the authors did not consider the spectral DOF, the prepared state from Ref. \cite{PhysRevA.98.060301} is also separable in space and frequency in the smaller subspace of only four modes $\{\ket{0,0},\ket{1,1},\ket{2,2},\ket{3,3}\}$. We suppose that the engineering of maximally entangled states in spatial DOF in a certain subspace enforces automatic decoupling in spatial and spectral DOF in that subspace. \subsection{Purity and Schmidt number of the full biphoton state} Obviously, the subspace $S_4$ is a part of the full SPDC emission. The first four OAM modes out of the subspace in Fig. \ref{fig1.pdf}(b), $\ket{2,-1}$ and $\ket{3,-2}$, possess different spectra in contrast to the modes in $S_4$, shown in Fig. \ref{fig1.pdf}(c) with dotted and dashed curves, respectively. The appearance of modes with distinguishable spectra indicates the inseparability of spatial and spectral DOF out of $S_4$. The more distinguishable modes contribute to the state, the stronger the spatio-temporal coupling. This, in turn, leads to reduced purity for the spatial biphoton state. Usually, narrowband filters are used in front of detectors to increase the purity of the spatial state. On one hand, the spectral filters improve the purity of the spatial state; on the other hand, they reduce the rate of entangled photons. We calculated the spatial purity $\mathrm{Tr}(\rho^2_{\bm{q},\mathrm{full}})$ \cite{Osorio_2008} of the full biphoton state \eqref{SPDCstate} depending on the filter bandwidth $\Delta \lambda$, in order to quantify the influence of spectral filters on the biphoton state. We chose as a pump the same beam, which leads to the state $\ket{\Psi_4}$. Very narrow filters are required to end up with a more or less pure state, as we can see from Fig. \ref{fig2}. For instance, a typical spectral filter with a bandwidth of $1$ \textit{nm} would leave the state in a mixed state of purity $0.33$. The Schmidt number of the full spatio-temporal biphoton state is also different in comparison to the subspace state. The total Schmidt number can be calculated from the reduced density matrix in space and frequency for the signal by tracing over the idler $\rho_{\mathrm{signal}}=\mathrm{Tr}_{\mathrm{idler}}(\rho)$\cite{Osorio_2008}. The Schmidt number is then given by $K=1/\mathrm{Tr}(\rho^2_{\mathrm{signal}})$ \cite{computing}. The number of both spatial and spectral Schmidt modes in the range of frequencies $810 \pm 10$ \textit{nm} equals $140$, where $810$ \textit{nm} is the central frequency for signal and idler photons. In comparison, the number of Schmidt modes generated only at central frequency $810$ $nm$ equals $5.8$. Finally, a small remark about the thin crystal regime: The spatio-temporal coupling is absent in the thin crystal regime $L \ll z_{R,p}$, since the biphoton state is independent of the crystal features. The problem with this regime is that it gives rise to a huge amount of spatial modes. Assume we keep all parameters the same as in Ref. \cite{PhysRevA.98.060301}, but change the crystal length to $L=1$ $\mu m$. The thin crystal regime is then well achieved according to Ref. \cite{PhysRevA.103.063508}. The state becomes spatially pure, but possesses a large amount of Schmidt modes, $10^7$. \begin{figure} \caption{Purity of the spatial biphoton state depending on the bandwidth of spectral filter.} \label{fig2} \end{figure} \section{CONCLUSION} In summary, we derived a closed analytical expression for the biphoton spatio-temporal state in terms of the LG mode amplitudes. The expression readily reveals the dependence of the modal decomposition on frequency and thus correctly describes spectral-spatial coupling, a quintessential feature of SPDC. The expression provides a new understanding of how the Gouy phase is related to the decoupling of spatial and spectral DOF: the relative Gouy phase of the interacting beams fully defines the shape of the spectrum of down-converted photons. Engineering the modal decomposition of the pump beam can be used to engineer a high-dimensional OAM entanglement. State engineering can also be used to decrease the coupling between the spatial and spectral DOF, leading to an increase of the correlation stored in the spatial DOF. We thus hope that it will aid experimenters in the design and quantitative modeling of challenging experiments based on PDC. The authors thank Egor Kovlakov and Darvin Wanisch for very helpful discussions. \end{document}

\begin{document} \title{Splitting the K\"unneth formula} \author{Laurence R. Taylor} \address{Department of Mathematics\newline \indent University of Notre Dame\newline \indent Notre Dame, IN 46556\newline \indent U.S.A.} \email{[email protected]} \begin{abstract} There is a description of the torsion product of two modules in terms of generators and relations given by Eilenberg and Mac Lane. With some additional data on the chain complexes there is a splitting of the map in the K\"unneth\ formula in terms of these generators. Different choices of this additional data determine a natural coset reminiscent of the indeterminacy in a Massey triple product. In one class of examples the coset actually is a Massey triple product. The explicit formulas for a splitting enable proofs of results on the behavior of the interchange map and the long exact sequence boundary map on all the terms in the K\"unneth\ formula. Information on the failure of naturality of the splitting is also obtained. \end{abstract} \maketitle \date{\today} \section{Introduction} Fix a principal ideal domain $R$ and let $\complex[1]_\ast$ and $\complex[3]_\ast$ be two chain complexes of $R$ modules. The K\"unneth\ formula states that if $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic then there is a short exact sequence \begin{xyMatrixLine} 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[1]_\ast) \tensor[R] H_\secondIndex(\complex[3]_\ast) \ar[r]^-{\cs{cross product}}& H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \ar[r]^-{\cs{to torsion product}}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_\secondIndex(\complex[3]_\ast)\to0 \end{xyMatrixLine} which is natural for pairs of chain maps and which is split. For a proof in this generality see for example Dold \cite{Dold}*{VI, 9.13}. Let $\cs{to torsion product}_{k}\def\secondInt{\ell}\def\totalInt{n,\secondInt}\colon H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \to H_{k}\def\secondInt{\ell}\def\totalInt{n}(\complex[1]_\ast)\tor[R] H_{\secondInt}(\complex[3]_\ast)$ denote $\cs{to torsion product}$ followed by projection. Say that a map $\sigma\colon H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast) \to H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[1]_\ast\tensor[R] \complex[3]_\ast)$ \emph{splits the K\"unneth\ formula at $(i}\def\secondIndex{j,\secondIndex)$} provided $\cs{to torsion product}_{k}\def\secondInt{\ell}\def\totalInt{n,\secondInt}\circ \sigma = \identyMap{H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast)}$ if $(k}\def\secondInt{\ell}\def\totalInt{n,\secondInt)=(i}\def\secondIndex{j,\secondIndex)$ and is $0$ otherwise. \section{The main idea}\sectionLabel{main idea} Suppose the $R$ modules in the complexes $\complex[1]_\ast$ and $\complex[3]_\ast$ are free, so the K\"unneth\ formula holds. The general case is discussed in \S \namedSection{general case}. In \cite{Eilenberg-Mac Lane}*{\S11} Eilenberg and Mac Lane gave a generators and relations description of the torsion product: $\complex[1]\tor[R]\complex[3]$ is the free $R$ module on symbols $\cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}$ where $\ringElement\in R$, $\element[1]\in \complex[1]$ with $\element[1]\moduleDot\ringElement = 0$ and $\element[2]\in \complex[3]$ with $\ringElement\moduleDot\element[2] = 0$ modulo four types of relations described below, (\ref{free cycle gives map}.1) -- (\ref{free cycle gives map}.4). The symbols $\cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}$ will be called \emph{elementary tors}. In what follows, given any complex $\complex[2]_{\ast}$, $\complexCycles[2]_\ast$ denotes the cycles and $\complexBoundaries[2]_\ast$ denotes the boundaries. Given any cycle $\elementCycle[4]$ of degree $\abs{\element[4]}$ in $\complex[2]_{\ast}$, write $\homologyClassOf{\elementCycle[4]}\in H_{\abs{\element[4]}}(\complex[2]_\ast)$ for the homology class $\elementCycle[4]$ represents. Let $\cyclesToHomology[2]\colon \complexCycles[2]_\ast \to H_\ast(\complex[2]_\ast)$ denote the canonical map. Mac Lane \cite{Mac Lane}*{Prop.~V.10.6} describes a cycle in $H_{\totalInt}(\complex[1]_\ast\tensor[R]\complex[3]_\ast)$ representing a given elementary tor in the range of $\cs{to torsion product}$. Mac Lane's cycle is defined as follows. Lift $\element[1]$ to a cycle, $\elementCycle[1]$, and $\element[2]$ to a cycle $\elementCycle[3]$. Since $\element[1]\moduleDot \ringElement = 0$, $\elementCycle[1]\moduleDot \ringElement$ is a boundary. Choose $\liftToChain[1]\in \complex[1]_{\abs{\element[1]}+1}$ so that $\boundary[1]_{\abs{\element[1]}+1}(\liftToChain[1] ) = \elementCycle[1] \moduleDot\ringElement$. Choose $\liftToChain[3]$ so that $\boundary[3]_{\abs{\element[3]}+1}(\liftToChain[3]) = \ringElement\moduleDot \elementCycle[3]$. Up to sign and notation, Mac Lane's cycle is given by \namedNumber[Formula]{Mac Lane cycle A} \begin{equation*}\tag{\ref{Mac Lane cycle A}} \cs{tor cycles into}\bigl( {\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr) = (-1)^{\abs{\element[1]}+1} \elementCycle[1] \tensor \liftToChain[3] + \liftToChain[1] \tensor \elementCycle[3] \end{equation*} Mac Lane puts the sign in front of the other term but then gets a sign when evaluating $\cs{to torsion product}$. Mac Lane also writes (\ref{Mac Lane cycle A}) as a Bockstein. The short exact sequence $\xyLine[@C10pt]{0\ar[r]&R \ar[rr]^-{\moduleDot\ringElement}&& R\ar[rr]^-{\rho^{\ringElement}}&&\ry{\ringElement}\ar[r]&0}$ gives rise to a long exact sequence whose boundary term is called the Bockstein associated to the sequence: \begin{math} \mathfrak b^{\ringElement}_{\totalInt}\colon H_{\totalInt}\bigl({\complex[2]_\ast} {\tensor[R]}\ry{\ringElement}\bigr) \to H_{\totalInt-1}({\complex[2]_{\ast}}) \end{math} In terms of the Bockstein and the pairing \begin{equation*} H_{k}\def\secondInt{\ell}\def\totalInt{n}\bigl(\complex[1]_\ast\tensor[R]\ry{\ringElement}\bigr) \cs{cross product} H_{\secondInt}\bigl(\complex[3]_\ast\tensor[R]\ry{\ringElement}\bigr) \to H_{k}\def\secondInt{\ell}\def\totalInt{n+\secondInt}\bigl(\complex[1]_\ast\tensor[R]\complex[3]_\ast \tensor[R]\ry{\ringElement}\bigr) \end{equation*} \vskip-10pt \namedNumber[Formula]{Mac Lane cycle B} \begin{equation*}\tag{\ref{Mac Lane cycle B}} \cs{tor cycles into}\bigl( {\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr) = (-1)^{\abs{\element[1]}+1}\mathfrak b^{\ringElement}_{\abs{\element[1]}+\abs{\element[3]}+2}\bigl( \liftToChain[1] \tensor \liftToChain[3] \bigr) \end{equation*} Given a different choice of cycle for $\elementCycle[1]$, say $\elementCycleB[1]$, $\elementCycleB[1] =\elementCycle[1] + \boundary[1]_{\abs{\element[1]}+1} (\liftDelta[1])$. Take $\liftToChainA[1]{1} = \liftToChain[1] + \liftDelta[1]\moduleDot \ringElement$. With a similar choice of lift on the right, $\cs{tor cycles into}\bigl( {\elementCycleB[1]}, \liftToChainA[1]{1}; \elementCycleB[3], \liftToChainA[3]{1}\bigr) - \cs{tor cycles into}\bigl( {\elementCycle[1]}, \liftToChain[1]; \elementCycle[3], \liftToChain[3]\bigr)$ is a boundary and so different choices of cycles give the same homology class. \vskip10pt Indeterminacy comes from the choices of $\liftToChain[1]$ and $\liftToChain[3]$. With $\elementCycle[1]$ and $\elementCycle[3]$ fixed, $\liftToChain[1]$ is determined up to a cycle. Let $\liftToChainA[1]{1} = \liftToChain[1] + \liftCycle[1]$ and let $\liftToChainA[3]{1} = \liftToChain[3] + \liftCycle[3]$. Then \begin{equation*} {\homologyClassOf[big]{ \cs{tor cycles into}\bigl( {\element[1]}, \liftToChainA[1]{1}; \element[2], \liftToChainA[3]{1}\bigr)}} = \homologyClassOf[big]{\cs{tor cycles into}\bigl( {\element[1]}, \liftToChain[1]; \element[2], \liftToChain[3]\bigr) + (-1)^{\abs{\liftCycle[1]}} \bigl(\element[1]\cs{cross product} \homologyClassOf{\liftCycle[3]}\bigr)} + \bigl(\homologyClassOf{\liftCycle[1]}\cs{cross product} \element[2]\bigr) \end{equation*} Since $\homologyClassOf{\liftCycle[1]}$ and $\homologyClassOf{\liftCycle[3]}$ can be chosen arbitrarily, any element in the coset $\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast} {\abs{\element[1]}}{\abs{\element[2]}}$ can be realized. Let \namedNumber[Formula]{double coset} \begin{equation*}\tag{\ref{double coset}} \cosetTor[{\element[1]}]{\ringElement}{\element[2]} \subset H_{\abs{\element[1]}+\abs{\element[2]}+1} (\complex[1]_\ast\tensor[R]\complex[3]_\ast) \end{equation*} \noindent denote the coset determined by any of the $\homologyClassOf[big]{ \cs{tor cycles into}\bigl( {\element[1]}, \liftToChainA[1]{1}; \element[2], \liftToChainA[3]{1}\bigr)}$. The above discussion and Proposition V.10.6 of \cite{Mac Lane} shows the following. \begin{ThmS}[Mac Lane main lemma]{Lemma} For two complexes of free $R$ modules, $R$ a PID, the element $\homologyClassOf[big]{ \cs{tor cycles into}\bigl( {\element[1]}, \liftToChain[1]; \element[2], \liftToChain[3]\bigr)}$ determines $\cosetTor[{\element[1]}]{\ringElement}{\element[2]} $ a well-defined coset of $\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast} {\abs{\element[1]}}{\abs{\element[2]}}$ such that \begin{equation*} \cs{to torsion product}_{s,t} \bigl(\cosetTor[{\element[1]}]{\ringElement}{\element[2]}\bigr) = \begin{cases} \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]} & s=\abs{\element[1]}, t = \abs{\element[2]}\\ 0&\text{otherwise}\\ \end{cases} \end{equation*} \end{ThmS} To get a splitting requires one more step. Since $R$ is a PID, the set of boundaries in a free chain complex is a free submodule and hence there is a splitting of the boundary maps. Choose splittings for the complexes being considered here: $\complexSplitting[1]\colon\complexBoundaries[1]\to \complex[1]_{\ast+1}$ and $\complexSplitting[3]\colon\complexBoundaries[3]\to \complex[3]_{\ast+1}$. Define \namedNumber{torsion product cycle} \newCS{torsion product cycle 1}{{\ref{torsion product cycle}.1}} \newCS{torsion product cycle 2}{{\ref{torsion product cycle}.2}} \newCS{env:torsion product cycle 1}{{Formula}} \newCS{env:torsion product cycle 2}{{Formula}} \begin{align*}\tag{\cs{torsion product cycle 1}} \cs{tor cycles into}\bigl( {\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3], \complexSplitting[3];\ringElement\bigr) =& (-1)^{\abs{\element[1]}+1} \elementCycle[1]\tensor \complexSplitting[3]\bigl(\ringElement \elementCycle[3]\bigr) + \complexSplitting[1]\bigl(\elementCycle[1] \ringElement\bigr) \tensor \elementCycle[3]\\ \tag{\cs{torsion product cycle 2}} \cs{tor cycles into}\bigl( {\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3], \complexSplitting[3];\ringElement\bigr) =& (-1)^{\abs{\element[1]}+1}\mathfrak b^{\ringElement}_{\abs{\element[1]}+\abs{\element[3]}+2} \Bigl(\complexSplitting[1]\bigl(\elementCycle[1] \ringElement\bigr) \tensor \complexSplitting[3]\bigl(\ringElement \elementCycle[3]\bigr)\Bigr) \end{align*} \begin{ThmS}[free splitting is independent of cycles]{Lemma} The homology class $\homologyClassOf[big]{\cs{tor cycles into}\bigl( {\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3], \complexSplitting[3];\ringElement\bigr)}$ is independent of the choice of cycles $\elementCycle[1]$ and $\elementCycle[3]$. \end{ThmS} \begin{proof} See the paragraph just below (\ref{Mac Lane cycle B}). \end{proof} Define \begin{equation*} \cs{homology splitting}[{\complexSplitting[1]}] {\complexSplitting[3]}_{\abs{\element[1]}, \abs{\element[2]}} \bigl( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]} \bigr) = \homologyClassOf[big]{\cs{tor cycles into}\bigl( {\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3], \complexSplitting[3]; \ringElement\bigr)} \end{equation*} \begin{ThmS}[free cycle gives map]{Theorem} For fixed splittings $\complexSplitting[1]$ and $\complexSplitting[3]$, the function $\cs{homology splitting}[{\complexSplitting[1]}] {\complexSplitting[1]}_{\abs{\element[1]}, \abs{\element[2]}}$ defined on elementary tors induces an $R$ module map \begin{equation*} \cs{homology splitting}[{\complexSplitting[1]}] {\complexSplitting[3]}_{\abs{\element[1]}, \abs{\element[2]}}\colon H_{\abs{\element[1]}}(\complex[1]_\ast)\tor[R] H_{\abs{\element[2]}}(\complex[3]_\ast) \to H_{\abs{\element[1]} + \abs{\element[2]}+1}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \end{equation*} which splits the K\"unneth\ formula at $\bigl(\abs{\element[1]},\abs{\element[2]}\bigr)$. \end{ThmS} \begin{proof} The splitting at $\bigl(\abs{\element[1]},\abs{\element[2]}\bigr)$ follows from \namedRef{Mac Lane main lemma}. Fix splittings and let $\localName{\element[1]}{\ringElement}{\element[2]} = \homologyClassOf{\cs{tor cycles into}\bigl( {\elementCycle[1]}, \complexSplitting[1]; \elementCycle[3], \complexSplitting[3];\ringElement\bigr)}$. By Eilenberg and Mac Lane \cite{Eilenberg-Mac Lane}*{\S 11}, to prove $\cs{homology splitting}$ is a module map, it suffices to prove the following \vskip 10pt \noindent(\ref{free cycle gives map}.1) \enumline{$\localName{\element[1]_1}{\ringElement}{\element[2]} + \localName {\element[1]_2}{\ringElement}{\element[2]} = \localName {\element[1]_1+\element[1]_2}{\ringElement}{\element[2]}$} {$\element[1]_{i}\def\secondIndex{j}\ringElement = 0$; $\ringElement\element[2]=0$} (\ref{free cycle gives map}.2) \enumline{$\localName {\element[1]}{\ringElement}{\element[2]_1} + \localName {\element[1]}{\ringElement}{\element[2]_2} = \localName {\element[1]}{\ringElement}{\element[2]_1+\element[2]_2}$} {$\element[1]\ringElement=0$; $\ringElement \element[2]_{i}\def\secondIndex{j}=0$} (\ref{free cycle gives map}.3) \Enumline{$\localName {\element[1]}{\ringElement_1\cdot \ringElement_2}{\element[2]} = \localName {\element[1] \ringElement_1}{\ringElement_2}{\element[2]}$} {$\element[1] \ringElement_1 \ringElement_2 = 0$; $\ringElement_2\element[2]=0$} (\ref{free cycle gives map}.4) \Enumline{$\localName {\element[1]}{\ringElement_1\cdot \ringElement_2}{\element[2]} = \localName {\element[1]}{\ringElement_1}{\ringElement_2\element[2]}$} {$\element[1]\ringElement_1=0$; $\ringElement_1 \ringElement_2\element[2]=0$} These formulas are easily verified at the chain level using (\cs{torsion product cycle 1}), \namedRef{free splitting is independent of cycles} and carefully chosen cycles. \end{proof} \begin{DefS}{Remark} Eilenberg and Mac Lane work over $\mathbb Z$ but, as pointed out explicitly in \cite{Mac Lane slides}*{about the middle of page 285}, the proof uses nothing more than that submodules of free modules are free and that finitely generated modules are direct sums of cyclic modules. Hence the results are valid for PID's. \end{DefS} \begin{DefS}{Remark} The data contained in a splitting is surely related to the structure introduced by Heller in \cite{Heller}. See also Section \namedSection{Bocksteins determine}. \end{DefS} \section{Free Approximations}\sectionLabel{free approximations} A result attributed to Dold by Mac Lane \cite{Mac Lane}*{Lemma 10.5} is that given any chain complex over a PID there exists a free chain complex with a quasi-isomorphic chain map to the original complex. In this paper any such complex and quasi-isomorphism will be called a \emph{free approximation}. \begin{DefS*}{Warning} Some authors also require the chain map to be surjective. \end{DefS*} Here is a review of a construction of a free approximation, mostly to establish notation. Some lemmas needed later are also proved here. \def\chainMap[1]+ \chainMap[2]{ } A \emph{weak splitting} of a chain complex $\complex[1]_\ast$ at an integer $\totalInt$ is a free resolution $\xyLine[@C20pt]{0\ar[r]& \freeBoundaries[1]_{\totalInt} \ar[r]^-{\iota^{\complex[1]}_{\totalInt}}& \freeCycles[1]_{\totalInt} \ar[rr]^-{\hat{\freeCyclesMap[0]_{ }}_{H_{\totalInt}(\complex[1]_\ast)}}&& H_{\totalInt}(\complex[1]_\ast)\ar[r]&0}$ and a pair of maps $\splitPair[1]_{\totalInt} = \{\freeCyclesMap[1]_{\totalInt}, \freeBoundariesMap[1]_{\totalInt} \}$ of the resolution into $\complex[1]_\ast$ where $\freeCyclesMap[1]_{\totalInt}\colon \freeCycles[1]_{\totalInt}\to \complexCycles[1]_{\totalInt}$ and $\freeBoundariesMap[1]_{\totalInt}\colon \freeBoundaries[1]_{\totalInt}\to \complex[1]_{\totalInt+1}$. It is further required that \\ \null\hskip5pt\begin{minipage}{0.30\textwidth} \begin{xyMatrix} \freeCycles[1]_{\totalInt}\ar[r]^-{\freeCyclesMap[1]_{\totalInt}} \ar[dr]_{\hat{\freeCyclesMap[0]_{ }}_{H_{\totalInt}(\complex[1]_\ast)}}& \ar[d]^-{\cyclesToHomology[1]_{\totalInt}} \complexCycles[1]_{\totalInt} \\ &H_{\totalInt}(\complex[1]_\ast) \end{xyMatrix} \end{minipage}\hbox to 0.8in{\hfil and\hfil} \begin{minipage}{0.35\textwidth} \begin{xyMatrix}[@C40pt] \freeBoundaries[1]_{\totalInt}\ar[r]^-{\iota^{\complex[1]}_{\totalInt}}\ar[d]^-{\freeBoundariesMap[1]_{\totalInt}} &\freeCycles[1]_{\totalInt}\ar[d]^-{\freeCyclesMap[1]_{\totalInt}}\\ \complex[1]_{\totalInt+1}\ar[r]^-{\boundary[1]_{\totalInt+1}}& \complexCycles[1]_{\totalInt}\\ \end{xyMatrix} \end{minipage} commute. The complex is said to be \emph{weakly split} if it is weakly split at $\totalInt$ for all integers $\totalInt$. Any module over a PID has a free resolution and any complex has a weak splitting. If the complex is free, a splitting as in \S\namedSection{main idea} is a weak splitting. \vskip 10pt Given a weakly split complex, define a complex whose groups are $\naturalFreeComplex[1]_{\totalInt} = \freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt}$ and whose boundary maps are the compositions \noindent \resizebox{\textwidth}{!}{{$ \xymatrix@C30pt{ \complexFreeBoundaryMap[1]_{\totalInt}\colon \naturalFreeComplex[1]_{\totalInt} = \freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt}\ar[r]& \freeBoundaries[1]_{\totalInt-1}\ar[r]^-{\iota^{\complex[1]}_{\totalInt-1}}& \freeCycles[1]_{\totalInt-1}\ar[r]& \freeBoundaries[1]_{\totalInt-2} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt-1} = \naturalFreeComplex[1]_{\totalInt-1} } $ }} \noindent The submodule $0\displaystyle\mathop{\oplus} \freeBoundaries[1]_{\totalInt-1}\subset \freeBoundaries[1]_{\totalInt-2} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt-1} = \naturalFreeComplex[1]_{\totalInt-1}$ is the image of $\complexFreeBoundaryMap[1]_{\totalInt}$ so one choice of splitting, called the \emph{canonical splitting}, is the composition \begin{xyMatrix} \naturalFreeSplitting[1]\colon \mathbf{B}_{\totalInt-1}(\naturalFreeComplex[1]_\ast)=0 \displaystyle\mathop{\oplus} \freeBoundaries[1]_{\totalInt-1}\ar[r]& \freeBoundaries[1]_{\totalInt-1}\ar[r]& \freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt} = \naturalFreeComplex[1]_{\totalInt} \end{xyMatrix} \begin{ThmS}[free approximations]{Lemma} The map \begin{equation*} \naturalFreeMap[1]_{\totalInt}=\freeBoundariesMap[1]_{\totalInt-1}+ \freeCyclesMap[1]_{\totalInt}\colon \naturalFreeComplex[1]_{\totalInt} = \freeBoundaries[1]_{\totalInt-1} \displaystyle\mathop{\oplus} \freeCycles[1]_{\totalInt} \to \complex[1]_{\totalInt} \end{equation*} is a chain map which is a quasi-isomorphism. If $\freeCyclesMap[1]_\ast\colon\freeCycles[1]_\ast \to \complexCycles[1]_\ast$ is onto then $\naturalFreeMap[1]_{\ast}$ is onto. It is always possible to choose $\freeCyclesMap[1]_\ast$ to be onto. \end{ThmS} The proofs of the claimed results are standard. \begin{ThmS}[cover surjective chain maps]{Lemma} Let $\chainMap[1]_\ast\colon\complex[1]_\ast\to\complex[3]_\ast$ be a surjective chain map and let $\naturalFreeMap[3]_{\ast}\colon \freeApproximation[3]_\ast\to\complex[3]_\ast$ be a free approximation. Then there exist free approximations $\naturalFreeMap[1]_{\ast}\colon \freeApproximation[1]_\ast\to\complex[1]_\ast$ and surjective chain maps $\freeApproximationChainMap[1]_\ast\colon \freeApproximation[1]_\ast \to \freeApproximation[3]_\ast$ making \begin{xyMatrix} \freeApproximation[1]_\ast\ar[r]^-{\freeApproximationChainMap[1]_\ast} \ar[d]^-{\naturalFreeMap[1]_{\ast}}& \freeApproximation[3]_\ast \ar[d]^-{\naturalFreeMap[3]_{\ast}}\\ \complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}& \complex[3]_\ast \end{xyMatrix} commute. \end{ThmS} \begin{proof} Let $\xymatrix{ P_\ast\ar[r]^-{\hat{\chainMap[1]_{ }}_\ast} \ar[d]^-{\zeta_{\ast}}& \freeApproximation[3]_\ast \ar[d]^-{\naturalFreeMap[3]_{\ast}}\\ \complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}& \complex[3]_\ast }$ be a pull back. \def\chainMap[1]+ \chainMap[2]{P} Since $\chainMap[1]_\ast$ is onto, so is $\hat{\chainMap[1]_{ }}_\ast$ and the kernel complexes are isomorphic. By the 5 Lemma, $\zeta_\ast$ is a quasi-isomorphism. Let $\naturalFreeMap[10]_\ast\colon \freeApproximation[1]_\ast\to P_\ast$ be a surjective free approximation. Then $\naturalFreeMap[1]_{\ast} = \zeta_\ast\circ \naturalFreeMap[10]_{\ast}$ and $\freeApproximationChainMap[1]_\ast = \hat{\chainMap[1]_{ }}_\ast\circ \naturalFreeMap[10]_\ast$ are the desired maps. \end{proof} \begin{ThmS}[short exact free approximation]{Lemma} If $ \xymatrix@1@C10pt{ 0\ar[r]& \complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&& \complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&& \complex[2]_\ast\ \ar[r]& 0}$ is exact, there exist free approximations making the diagram below commute. \begin{xyMatrix}[@C10pt] 0\ar[r]&\freeApproximation[1]_\ast\ar[rr]^-{\freeMapApproximation[1]_\ast} \ar[d]^-{\vertMap{\complex[1]}_\ast}&& \freeApproximation[3]_\ast\ar[rr]^-{\freeMapApproximation[2]_\ast} \ar[d]^-{\vertMap{\complex[3]}_\ast}&& \freeApproximation[2]_\ast \ar[d]^-{\vertMap{\complex[2]}_\ast} \ar[r]& 0\\ 0\ar[r]&\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&& \complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&& \complex[2]_\ast\ar[r]&0 \end{xyMatrix} \end{ThmS} \begin{proof} Use \namedRef{cover surjective chain maps} to get $\freeMapApproximation[2]$. Let $\freeApproximation[1]_\ast$ be the kernel complex, hence free. There is a unique map $\vertMap{\complex[1]}_{\ast}$ making the diagram commute. By the 5 Lemma, $\vertMap{\complex[1]}_{\ast}$ is a quasi-isomorphism. \end{proof} \begin{ThmS}[Dold splitting]{Lemma} Suppose $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic. Suppose $\naturalFreeMap[1]_{\ast}\colon \freeApproximation[1]_\ast\to\complex[1]_\ast$ and $\naturalFreeMap[3]_{\ast}\colon \freeApproximation[3]_\ast\to\complex[3]_\ast$ are free approximations. Then so is {\setlength\belowdisplayskip{-10pt} \begin{equation*} \naturalFreeMap[1]_{\ast} \tensor \naturalFreeMap[3]_{\ast}\colon \freeApproximation[1]_\ast\tensor[R]\freeApproximation[3]_\ast \to\complex[1]_\ast\tensor[R]\complex[3]_\ast \end{equation*} }\end{ThmS}\nointerlineskip \namedNumber{Dold splitting diagram} \begin{proof} The K\"unneth\ formula is natural for chain maps so \noindent\resizebox{\textwidth}{!}{{$\xymatrix{ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\freeApproximation[1]_\ast)\tensor[R] H_{\secondIndex}(\freeApproximation[3]_\ast) \ar[r]^-{\cs{cross product}} \ar[d]^-{\hbox{\tiny$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} \naturalFreeMap[1]_\ast \tensor \naturalFreeMap[3]_\ast$}}_-{\hbox to 1in{(\ref{Dold splitting diagram}) }} & H_{\totalInt}(\freeApproximation[1]_\ast \tensor[R] \freeApproximation[3]_\ast) \ar[r]^-{\cs{to torsion product}} \ar[d]^-{(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} H_{i}\def\secondIndex{j}(\freeApproximation[1]_\ast)\tor[R] H_{\secondIndex}(\freeApproximation[3]_\ast) \to 0 \ar[d]^-{\hbox{\tiny$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} \naturalFreeMap[1]_\ast\tor \naturalFreeMap[3]_\ast$}} \\ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tensor[R] H_{\secondIndex}(\complex[3]_\ast) \ar[r]^-{\cs{cross product}} & H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \ar[r]^-{\cs{to torsion product}} & \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast) \to 0 }$}} \noindent commutes. The left and right vertical maps are tensor and torsion products of isomorphisms and hence isomorphisms. The middle vertical map is an isomorphism by the 5 Lemma. \end{proof} \section{The general case}\sectionLabel{general case} With notation and hypotheses as in \namedRef{Dold splitting}, applying $(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast$ to the cycle in (\cs{torsion product cycle 1}) gives \namedNumber{torsion product cycle II} \newCS{torsion product cycle II 1}{{\ref{torsion product cycle II}.1}} \newCS{torsion product cycle II 2}{{\ref{torsion product cycle II}.2}} \newCS{env:torsion product cycle II 2}{{Formula}} \begin{equation*}\tag{\cs{torsion product cycle II 1}} \cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)\bigl(\elementCycle[1], \ringElement, \elementCycle[3]\bigr) = \epsilon\, \freeCyclesMap[1]_\ast(\elementCycle[1]) \tensor \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) + \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \tensor _\ast(\elementCycle[3])\\ \end{equation*} \noindent where $\elementCycle[1]\in \freeCycles[1]_\ast$ satisfies $\freeHomologyMap[1]_\ast(\elementCycle[1]) = \element[1]$, $\elementCycle[3]\in \freeCycles[3]_\ast$ satisfies $\freeHomologyMap[3]_\ast(\elementCycle[3]) = \element[2]$ and $\epsilon=(-1)^{\abs{\element[1]}+1}$. In general there is no analogue to (\cs{torsion product cycle 2}) because not all complexes have the necessary Bocksteins. If $\complex[1]_\ast$ and $\complex[3]_\ast$ are torsion free then the necessary Bocksteins exist and applying $(\naturalFreeMap[1]\tensor \naturalFreeMap[3])_\ast$ to (\cs{torsion product cycle 2}) gives \begin{align*} \tag{\cs{torsion product cycle II 2}} \cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)\bigl(\elementCycle[1], \ringElement, \elementCycle[3]\bigr) =& \epsilon\, \mathfrak b^{\ringElement}_{\abs{\element[1]}+\abs{\element[3]}+2}\Bigl( \freeBoundariesMap[1]_\ast\bigl(\elementCycle[1]\moduleDot\ringElement\bigr) \tensor \freeBoundariesMap[3]_\ast \bigl(\ringElement \moduleDot\elementCycle[3]\bigr) \Bigr) \end{align*} \begin{ThmS}{Lemma} The homology class $\homologyClassOf[big]{\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)(\elementCycle[1],\ringElement, \elementCycle[3])}$ is independent of the lifts $\elementCycle[1]$ and $\elementCycle[3]$. \end{ThmS} \begin{proof} The cycles $\elementCycle[1]$ and $\elementCycle[3]$ are cycles in $\naturalFreeComplex[1]_\ast$ and $\naturalFreeComplex[3]_\ast$ so the result is immediate from \namedRef{free splitting is independent of cycles} \end{proof} \begin{ThmS}[weak splittings give map]{Theorem} Assume $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic. For fixed weak splittings $\splitPair[1]_\ast$ and $\splitPair[3]_\ast$ taking the homology class of $\cs{tor cycles into}(\splitPair[1]_\ast, \splitPair[3]_\ast)(\elementCycle[1],\elementCycle[3])$ yields a map \begin{equation*} \cs{homology splitting}[{\splitPair[1]_\ast}] {\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\colon H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast) \to H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \end{equation*} which splits the K\"unneth\ formula at $(i}\def\secondIndex{j,\secondIndex)$. \end{ThmS} \begin{proof} The cycle \ref{torsion product cycle II}.1 is the image of the cycle \ref{torsion product cycle}.1 and so $\cs{homology splitting}$ is a map by \namedRef{free cycle gives map}. \namedNumberRef{Dold splitting} applies and (\ref{Dold splitting diagram}) has exact rows. The splitting result follows from \namedRef{free cycle gives map}. \end{proof} \begin{ThmS}[weak splitting map in correct coset]{Corollary} The map $\cs{homology splitting}[{\splitPair[1]_\ast}] {\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}$ will depend on the weak splittings. For any choices of weak splittings, $\cs{homology splitting}[{\splitPair[1]_\ast}] {\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]} \bigr)$ is in the same coset of $\fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast} {i}\def\secondIndex{j}{\secondIndex}$. Denote this coset by $\cosetTor[{\element[1]}]{\ringElement}{\element[2]}$. \end{ThmS} \begin{proof} Suppose given two weak splittings, $\splitPair[1]_{i}\def\secondIndex{j} = \{\freeCyclesMap[1]_{i}\def\secondIndex{j}, \freeBoundariesMap[1]_{i}\def\secondIndex{j}\}$ and $\splitPairA[1]_{i}\def\secondIndex{j} = \{ \freeCyclesMapA[1]_{i}\def\secondIndex{j}, \freeBoundariesMapA[1]_{i}\def\secondIndex{j}\}$. Then $\freeCyclesMapA[1]_{i}\def\secondIndex{j} - \freeCyclesMap[1]_{i}\def\secondIndex{j} \colon\freeCycles[1]_{i}\def\secondIndex{j} \to \complexCycles[1]_{i}\def\secondIndex{j} \to H_{i}\def\secondIndex{j}(\complex[1]_\ast)$ is trivial so $\freeCyclesMapA[1]_{i}\def\secondIndex{j} - \freeCyclesMap[1]_{i}\def\secondIndex{j} \colon\freeCycles[1]_{i}\def\secondIndex{j} \to \complexBoundaries[1]_{i}\def\secondIndex{j}$. Since $\freeCycles[1]_{i}\def\secondIndex{j}$ is free, there exists a lift $\Psi_{i}\def\secondIndex{j}\colon \freeCycles[1]_{i}\def\secondIndex{j} \to \complex[1]_{i}\def\secondIndex{j+1}$. Next consider $\xyLine{ \boundary[1]_{i}\def\secondIndex{j+1}\bigl(\Psi_{i}\def\secondIndex{j} - (\freeBoundariesMapA[1]_{i}\def\secondIndex{j} - \freeBoundariesMap[1]_{i}\def\secondIndex{j}) \bigr) \colon \freeCycles[1]_{i}\def\secondIndex{j} \to \complex[1]_{i}\def\secondIndex{j+1} \ar[r]^-{\boundary[1]_{i}\def\secondIndex{j+1}}& \complex[1]_{i}\def\secondIndex{j}}$. This map is also trivial so there is a unique map $\freeCycles[1]_{i}\def\secondIndex{j} \to \complexCycles[1]_{i}\def\secondIndex{j+1}$ and hence a unique map $\Phi_{i}\def\secondIndex{j}\colon \freeCycles[1]_{i}\def\secondIndex{j} \to \complexCycles[1]_{i}\def\secondIndex{j+1} \to H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)$. Then $\cs{homology splitting}[{\splitPairA[1]_\ast}] {\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]} \bigr) - \cs{homology splitting}[{\splitPair[1]_\ast}] {\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]} \bigr) = (-1)^{i}\def\secondIndex{j+1} \homologyClassOf[big]{\Phi_{i}\def\secondIndex{j}(\elementCycle[1]) \cs{cross product} \elementCycle[3]} \in H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)\cs{cross product} \element[3]$. A similar calculation shows the variation in the other variable lies in $\element[1]\cs{cross product}H_{\secondIndex+1}(\complex[3]_\ast)$. \end{proof} \section{Splitting via Universal Coefficients}\sectionLabel{Bocksteins determine} In the torsion free case, \cs{env:torsion product cycle II 2} \cs{torsion product cycle II 2} suggests another way to produce a splitting. The Universal Coefficients formula says that for a torsion-free complex $\complex[2]_\ast$, there exists a natural short exact sequence which is unnaturally split: \noindent\resizebox{\textwidth}{!}{{$\xymatrix@C18pt{ 0\ar[r]&H_{\totalInt}\bigl(\complex[2]_\ast\bigr) \tensor[R]\ry{\ringElement[4]}\ar[rr]&& H_{\totalInt}\bigl(\complex[2]_\ast \tensor[R]\ry{\ringElement[4]}\bigr)\ar[rr]^-{\universalCoefficientsMap{2}{4}{\totalInt}}&& \rtorsion{\ringElement[4]}{H_{\totalInt-1}(\complex[2]_{\ast})}\ar[r]&0 }$}} \noindent where for a fixed $\ringElement$ in a PID $R$ and an $R$ module $\@ifnextchar_{\LRT@P}{P}$, $\rtorsion{\ringElement}{\@ifnextchar_{\LRT@P}{P}} = P\tor[R]\ry{\ringElement}$ denotes the submodule of elements annihilated by $\ringElement$. \noindent The Bockstein $\mathfrak b^{\ringElement[4]}_{\totalInt}$ is the composition {\setlength{\abovedisplayskip}{0pt} \setlength{\belowdisplayskip}{-10pt} \begin{equation*} \xymatrix@C18pt{ H_{\totalInt}\bigl(\complex[2]_\ast \tensor[R]\ry{\ringElement[4]}\bigr)\ar[rr]^-{\universalCoefficientsMap{2}{4}{\totalInt}}&& \rtorsion{\ringElement[4]}{H_{\totalInt-1}(\complex[2]_{\ast})} \subset H_{\totalInt-1}(\complex[2]_{\ast}) } \end{equation*} } \begin{ThmS}[Bocksteins in correct coset]{Theorem} Let $\complex[1]_\ast$ and $\complex[3]_\ast$ be torsion-free complexes. Given $\element[1]\in H_{i}\def\secondIndex{j}(\complex[1]_\ast)$ pick $\elementR[1]\in H_{i}\def\secondIndex{j+1}\bigl(\complex[1]_\ast\tensor[R]\ry{\ringElement}\bigr)$ such that $\universalCoefficientsMap{1}{0}{i}\def\secondIndex{j+1}(\elementR[1]) = \element[1]$. Given $\element[3]\in H_{\secondIndex}(\complex[3]_\ast)$ pick $\elementR[3]\in H_{\secondIndex+1}\bigl(\complex[3]_\ast\tensor[R]\ry{\ringElement}\bigr)$ such that $\universalCoefficientsMap{3}{0}{\secondIndex+1}(\elementR[3]) = \element[3]$. On elementary tors $\cs{elementary tor} {\element[1]}{\ringElement}{\element[3]}$ define \begin{equation*} \splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor} {\element[1]}{\ringElement}{\element[3]}\bigr) = (-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \elementR[1] \tensor \elementR[3] \bigr) \end{equation*} Then $\splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor} {\element[1]}{\ringElement}{\element[3]}\bigr) \in\cosetTor[{\element[1]}]{\ringElement}{\element[2]}$. \end{ThmS} \begin{proof} From \namedRef{weak splitting map in correct coset}, $(-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}(\elementR[1]\tensor \elementR[3])$ lies in $\cosetTor[{\element[1]}]{\ringElement}{\element[2]}$ if the splittings used are ones from a weak splitting. Any other choice of splitting for $\complex[1]_\ast$ is of the form $\elementR[1] + X_{\element[1]}$ for $X_{\element[1]}\in H_{i}\def\secondIndex{j+1}(\complex[1]_\ast)$ and any other choice of splitting for $\complex[3]_\ast$ is of the form $\elementR[3] + X_{\element[3]}$ for $X_{\element[3]}\in H_{\secondIndex+1}(\complex[3]_\ast)$. Then \begin{equation*} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( (\elementR[1]+ X_{\element[1]})\tensor( \elementR[3]+X_{\element[3]})\bigr) = \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}(\elementR[1]\tensor \elementR[3]) + X_{\element[1]}\cs{cross product} \element[3] + (-1)^{i}\def\secondIndex{j+1} \element[1]\cs{cross product} X_{\element[3]} \end{equation*} The result follows. \end{proof} If the Universal Coefficients splittings are chosen arbitrarily the map on the elementary tors may not descend to a map on the torsion product. This problem is overcome as follows. A family of splittings \begin{equation*} \splitBocksteinHomology^{\complex[1],\ringElement}_{\totalInt}\colon \rtorsion{\ringElement}{ H_{\totalInt}(\complex[1]_{\ast})} \to H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{\ringElement}\bigr)\\ \end{equation*} one for each non-zero $\ringElement\inR$ is \emph{a compatible family of splittings of $\complex[1]_\ast$ at $\totalInt$} provided, for all non-zero elements $\ringElement[4]_1$, $\ringElement[4]_2\inR$ the diagram \noindent\resizebox{\textwidth}{!}{{$\xymatrix{ \rtorsion{\ringElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)} \ar[r]^-{\subset}\ar[d]^-{\splitBocksteinHomology^{\complex[1],\ringElement[4]_2}_\totalInt}& \rtorsion{\ringElement[4]_1\moduleDot\ringElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)} \ar[r]^-{\moduleDot[1]\ringElement[4]_2}\ar[d]^-{\splitBocksteinHomology^{\complex[1],\ringElement[4]_1\moduleDot\ringElement[4]_2}_\totalInt}& \rtorsion{\ringElement[4]_1}{H_{\totalInt}(\complex[1]_\ast)} \ar[d]^-{\splitBocksteinHomology^{\complex[1],\ringElement[4]_1}_\totalInt}\\ H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{\ringElement[4]_2}\bigr) \ar[r]^-{\ringElement[4]_1\moduleDot[1]} \ar[d]^-{\universalCoefficientsMapA{1}{\ringElement[4]_2}{\totalInt+1}}& H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{\ringElement[4]_1\moduleDot\ringElement[4]_2}\bigr) \ar[r]^-{\rho^{\ringElement[4]_1}} \ar[d]^-{\universalCoefficientsMapA{1}{\ringElement[4]_1\moduleDot[1]\ringElement[4]_2}{\totalInt+1}}& H_{\totalInt+1}\bigl(\complex[1]_{\ast}\tensor[R] \ry{\ringElement[4]_1}\bigr) \ar[d]^-{\universalCoefficientsMapA{1}{\ringElement[4]_1}{\totalInt+1}}\\ \rtorsion{\ringElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)} \ar[r]^-{\subset}& \rtorsion{\ringElement[4]_1\moduleDot\ringElement[4]_2}{H_{\totalInt}(\complex[1]_\ast)} \ar[r]^-{\moduleDot[1]\ringElement[4]_2}& \rtorsion{\ringElement[4]_1}{H_{\totalInt}(\complex[1]_\ast)} \\ }$}} \noindent commutes, where the horizontal maps are induced from the short exact sequence of modules $\xyLine[@C30pt]{0\to\ry{\ringElement[4]_2}\ar[r]^-{\ringElement[4]_1\moduleDot[1]}& \ry{\ringElement[4]_1\moduleDot\ringElement[4]_2}\ar[r]^-{\rho^{\ringElement[4]_1}}& \ry{\ringElement[4]_1}} \to 0$ and the rows are exact. The diagram consisting of the bottom two rows always commutes and the vertical maps from the first row to the third are the identity. If the splittings come from a weak splitting of $\complex[1]_\ast$ then they are compatible for any $\totalInt$. \begin{ThmS}{Theorem} Suppose $\complex[1]_\ast$ and $\complex[3]_\ast$ are torsion-free. Given a compatible family of splittings of $\complex[1]_\ast$ at $i}\def\secondIndex{j$ and a compatible family of splittings of $\complex[3]_\ast$ at $\secondIndex$, the formula \begin{equation*} \splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}}{\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}\bigl(\cs{elementary tor} {\element[1]}{\ringElement}{\element[3]}\bigr) = (-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \elementRr[1]{i}\def\secondIndex{j} \cs{cross product} \elementRr[3]{\secondIndex} \bigr) \end{equation*} defines a map from $H_{i}\def\secondIndex{j}(\complex[1]_\ast) \tor[R] H_{{\secondIndex}}(\complex[3]_\ast)$ to $H_{{i}\def\secondIndex{j}+{\secondIndex}+1} (\complex[1]_\ast\tensor[R]\complex[3]_\ast)$ splitting the K\"unneth\ formula at $({i}\def\secondIndex{j},{\secondIndex})$. \end{ThmS} \begin{proof} It follows from \namedRef{Bocksteins in correct coset} that if $\splitByCrossProductOfBocksteins_{i}\def\secondIndex{j,\secondIndex}$ is a map then it splits the K\"unneth\ formula at $(i}\def\secondIndex{j,\secondIndex)$. To show $\splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}} {\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}$ is a map, it suffices to show that (\ref{free cycle gives map}.1-\ref{free cycle gives map}.4) hold. Equations (\ref{free cycle gives map}.1) and (\ref{free cycle gives map}.2) hold whether the splittings are compatible or not since the cross product, and hence $\splitByCrossProductOfBocksteinsA{\elementRr[1]{i}\def\secondIndex{j}} {\elementRr[3]{\secondIndex}}_{i}\def\secondIndex{j,\secondIndex}$ is bilinear. \newcommand{\elementT}[3][0]{\splitBocksteinHomology^{\complex[#1],#3}_{#2}(\element[#1])} \newcommand{\elementS}[4][0]{\splitBocksteinHomology^{\complex[#1],#3}_{#2}(#4)} To verify (\ref{free cycle gives map}.3) it suffices to show \namedNumber{equal Bocksteins} \begin{equation*}\tag{\ref{equal Bocksteins}} \mathfrak b^{\ringElement_1 \moduleDot \ringElement_2}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \elementT[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} \cs{cross product} \elementT[3]{\secondIndex}{\ringElement_1\moduleDot \ringElement_2} \bigr) = \mathfrak b^{\ringElement_2}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \elementS[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot \ringElement_1} \cs{cross product} \elementT[3]{\secondIndex}{\ringElement_2} \bigr) \end{equation*} To compute a Bockstein of a homology class, $\element[4]\in H_{\totalInt}\bigl( \complex[2]_\ast\tensor[R]\ry{\ringElement[4]}\bigr)$, first lift to a chain, $\hat{\element[4]}\in \complex[2]_{\totalInt}$ and then $\boundary[2]_{\totalInt}(\hat{\element[4]}) = \ringElement[4] \element[100]$. The class $\element[100]$ is unique because $\complex[2]_{\totalInt}$ is torsion-free and $\mathfrak b^{\ringElement[4]}_{\totalInt}(\element[4])$ is the homology class represented by $\element[100]$. \newcommand{\elementTC}[3][0]{\splitBocksteinChains^{\complex[#1],#3}_{#2}(\element[#1])} \newcommand{\elementSC}[4][0]{\splitBocksteinChains^{\complex[#1],#3}_{#2}(#4)} There are four homology classes in (\ref{equal Bocksteins}). For uniform notation, given $\elementS[2]{\totalInt}{\ringElement[4]}{\element[4]}$, let $\elementSC[2]{\totalInt}{\ringElement[4]}{\element[4]}$ be a lift to a representing chain. The cross product of homology classes is represented by the tensor product of chains so $C_1 = \elementTC[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} \tensor \elementTC[3]{\secondIndex}{\ringElement_1\moduleDot \ringElement_2} $ is a chain to compute the left hand side of (\ref{equal Bocksteins}) and $C_2 = \elementSC[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot \ringElement_1} \tensor \elementTC[3]{\secondIndex}{\ringElement_2} $ is a chain to compute the right hand side of (\ref{equal Bocksteins}). Note $\homologyClassOf[Big]{\boundary[1]_{i}\def\secondIndex{j}\bigl( \elementTC[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} \bigr)} = \elementT[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} (\ringElement_1\moduleDot \ringElement_2) $ and $\homologyClassOf[Big]{\boundary[1]_{i}\def\secondIndex{j}\bigl( \elementSC[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot\ringElement_1} \bigr)} = \elementS[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot\ringElement_1} (\ringElement_1\moduleDot \ringElement_2) $. If the splittings are compatible, $\elementT[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} = \elementS[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot\ringElement_1}$ so choose $\elementTC[1]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} = \elementSC[1]{i}\def\secondIndex{j}{\ringElement_2}{\element[1]\moduleDot\ringElement_1}$. Also $\homologyClassOf[Big]{\boundary[3]_{i}\def\secondIndex{j}\bigl( \elementTC[3]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} \bigr)} = (\ringElement_1\moduleDot \ringElement_2) \elementT[3]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} $ whereas $\homologyClassOf[Big]{\boundary[3]_{i}\def\secondIndex{j}\bigl( \elementSC[3]{i}\def\secondIndex{j}{\ringElement_2}{\element[3]} \bigr)} = \ringElement_2\elementS[3]{i}\def\secondIndex{j}{\ringElement_2}{\element[3]} $. If the splittings are compatible, $\elementT[3]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} = \elementS[3]{i}\def\secondIndex{j}{\ringElement_2}{\element[3]}$ so choose $\elementTC[3]{i}\def\secondIndex{j}{\ringElement_1\moduleDot \ringElement_2} = \ringElement_1 \elementSC[3]{i}\def\secondIndex{j}{\ringElement_2}{\element[3]}$. It follows that $C_1 = \ringElement_1\moduleDot C_2$. Since \begin{xyMatrix}[@C12pt] 0\ar[r]&R\ar[rr]^-{\ringElement_2} \ar[d]_-{\identyMap{R}} &&R\ar[rr]^-{\rho^{\ringElement_2}} \ar[d]^-{\ringElement_1\moduleDot[1]} &&\ry{\ringElement_2}\ar[r] \ar[d]^{\ringElement_1\moduleDot[1]} &0 \\ 0\ar[r]&R\ar[rr]^-{\ringElement_1\moduleDot[1]\ringElement_2}&&R\ar[rr]^-{\rho^{\ringElement_1\moduleDot[1]\ringElement_2}} &&\ry{\ringElement_1\moduleDot\ringElement_2}\ar[r]&0 \end{xyMatrix} commutes, $\mathfrak b^{\ringElement_1 \moduleDot \ringElement_2}_{i}\def\secondIndex{j+\secondIndex+2}(C_1) = \mathfrak b^{\ringElement_2}_{i}\def\secondIndex{j+\secondIndex+2}(\ringElement_1 C_2) $ as required. \end{proof} \section{Naturality of the splitting}\sectionLabel{Naturality} \namedNumber{p0} \namedNumber{p1} \namedNumber{p2} Fix a chain map $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$ between two weakly split chain maps. Pick a map $\freeCyclesChainMap[1]_{\totalInt}\colon \freeCycles[1]_{\totalInt} \to \freeCycles[2]_{\totalInt}$ satisfying \begin{equation*}\tag{\ref{p0}} \cyclesToHomology[2]_{\totalInt}\circ \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} = \chainMap[1]_{\totalInt}\circ \cyclesToHomology[1]_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\colon \freeCycles[1]_{\totalInt} \to H_{\totalInt}(\complex[2]_\ast) \end{equation*} Since the right hand square in the diagram below commutes \begin{xyMatrix}[@C1pt] \freeBoundaries[1]_{\totalInt}\ \subset \ar@<-12pt>@{.>}[d]^-{\freeBoundariesChainMap[1]_{\totalInt}} & \ar@<-2pt>[d]^-{\freeCyclesChainMap[1]_{\totalInt}} \freeCycles[1]_{\totalInt} \ar[rrrrrrr]^-{\cyclesToHomology[1]_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}} &&&&&&& \ar[d]^-{\chainMap[1]_{\totalInt}} H_{\totalInt}(\complex[1]_\ast) \ar[d]^-{\chainMap[1]_{\totalInt}}\\ \freeBoundaries[2]_{\totalInt}\ \subset & \freeCycles[2]_{\totalInt} \ar[rrrrrrr]^-{\cyclesToHomology[2]_{\totalInt}\circ \freeCyclesMap[2]_{\totalInt}} &&&&&&& H_{\totalInt}(\complex[2]_\ast) \\ \end{xyMatrix} \noindent there exists a unique map $\freeBoundariesChainMap[1]_{\totalInt} \colon \freeBoundaries[1]_{\totalInt} \to \freeBoundaries[2]_{\totalInt}$ making the left hand square commute. The set of choices for $\freeCyclesChainMap[1]_{\totalInt}$ consists of any one choice plus any map $L_{\totalInt}\colon \freeCycles[1]_\ast\to \freeBoundaries[2]_{\totalInt}$. The restricted map is $\freeBoundariesChainMap[1]_{\totalInt}$ plus the restriction of $L_{\totalInt}$. \vskip10pt The maps $ \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}$ and $ \chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt} $ have domain $\freeCycles[1]_{\totalInt}$ and range $\complexCycles[2]_{\totalInt}$ and they represent the same homology class. Hence $ \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} - \chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}$ lands in $\complexBoundaries[2]_{\totalInt}$. Since $\freeCycles[1]_{\totalInt}$ is free, there is a lift of this difference to a map $\weakMap[1]_{\totalInt}\colon\freeCycles[1]_{\totalInt} \to \complex[2]_{\totalInt+1}$ satisfying \begin{equation*}\tag{\ref{p1}} \boundary[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} = \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} - \chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt} \end{equation*} If $\freeCyclesChainMap[1]_{\totalInt}$ is replaced by $\freeCyclesChainMap[1]_{\totalInt} + L_{\totalInt}$, a choice for the new $\weakMap[1]_{\totalInt}$ is $\weakMap[1]_{\totalInt} + \freeBoundariesMap[2]\circ L_{\totalInt}$. The set of solutions to (\ref{p1}) consists of one solution, $\weakMap[1]_{\totalInt}$, plus any map of the form $\Lambda_{\totalInt}\colon \freeCycles[1]_{\totalInt} \to \complexCycles[2]_{\totalInt+1}\subset \complex[2]_{\totalInt+1}$. Given a fixed solution to (\ref{p1}) consider \begin{equation*} \xi = \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl( \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt} \bigr)\colon \freeBoundaries[1]_{\totalInt} \to \complex[2]_{\totalInt+1} \end{equation*} \noindent Notice if $\freeCyclesChainMap[1]_{\totalInt}$ is replaced by $\freeCyclesChainMap[1]_{\totalInt} + L_{\totalInt}$, the new $\xi$ is the same map as the old $\xi$. The image of $\xi$ is contained in the cycles of $\complex[2]_{\totalInt+1}$ and so gives a map \begin{equation*}\tag{\ref{p2}} \weakHomologyMap[1]_{\totalInt} = \bigl( \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt} \bigr) - \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} \colon \freeBoundaries[1]_{\totalInt} \to H_{\totalInt+1}(\complex[2]_{\ast}) \end{equation*} which does not depend on the choice of $\freeCyclesChainMap[1]_\ast$. The map $\weakHomologyMap[1]_\ast$ induces a map \begin{equation*} \weakTorsionHomologyMap[1]<1>_{\totalInt}\colon \rtorsion{\ringElement[1]}{H_{\totalInt}(\complex[1]_\ast) \to H_{\totalInt+1}(\complex[2]_\ast)}\tensor \ry{r} \end{equation*} defined as follows. Given $\element[1]\in {}\rtorsion{\ringElement[1]}{H_{\totalInt}(\complex[1]_\ast)}$ pick $\elementCycle[1]\in \freeCycles[1]_{\totalInt}$ so that $\homologyClassOf{\freeCyclesMap[1]_{\totalInt}(\elementCycle[1])} = \element[1]$. Then $\elementCycle[1]\moduleDot \ringElement[1] \in\freeBoundaries[1]_{\totalInt}$ so let $\weakTorsionHomologyMap[1]<1>_{\totalInt}(\element[1])$ be the homology class represented by $\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\moduleDot \ringElement[1])$ reduced mod $\ringElement[1]$. \begin{ThmS}{Proposition} Given a chain map $\chainMap[1]_\ast\colon \complex[1]_\ast\to\complex[2]_\ast$ between two weakly split chain complexes over a PID $R$, the map \begin{equation*} \weakTorsionHomologyMap[1]<1>_{\totalInt} \colon \rtorsion{\ringElement[1]}{H_{\totalInt}(\complex[1]_\ast) \to H_{\totalInt+1}(\complex[2]_\ast)}\tensor \ry{r} \end{equation*} is well-defined regardless of the choices made in (\ref{p0}) and (\ref{p1}). \end{ThmS} \begin{proof} Any other choice of element in $\freeCycles[1]_{\totalInt}$ has the form $\elementCycle[1] + b$ for $b\in \freeBoundaries[1]_{\totalInt}$. Then ${ \weakHomologyMap[1]_{\totalInt}\Bigl(\bigl(\elementCycle[1] + b\bigr)\moduleDot \ringElement[1]\Bigr) = \weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\moduleDot \ringElement[1] ) + \weakHomologyMap[1]_{\totalInt} \bigl(b\moduleDot \ringElement[1] \bigr) = \weakHomologyMap[1]_{\totalInt}(\elementCycle[1] \moduleDot \ringElement[1] ) + \weakHomologyMap[1]_{\totalInt}\bigl(b\bigr) \moduleDot \ringElement[1] }$ since \penalty-1000 $b\in\freeBoundaries[1]_{\totalInt}$. Hence $\weakHomologyMap[1]_{\totalInt}\Bigl(\bigl(\elementCycle[1] + b\bigr)\moduleDot \ringElement[1]\Bigr)$ and $\weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\moduleDot \ringElement[1])$ represent the same element in $H_{\totalInt+1}(\complex[2]_\ast)\tensor \ry{r}$ and therefore $\weakTorsionHomologyMap[1]<1>_{\totalInt}$ is well-define. Since $\weakHomologyMap[1]_{\totalInt}$ is an $R$ module map, so is $\weakTorsionHomologyMap[1]<1>_{\totalInt}$. Given a second lift, it has the form $\weakMap[1]_{\totalInt} + \Lambda$ where $\Lambda\colon \freeCycles[1]_{\totalInt} \to \complexCycles[2]_{\totalInt+1}$ and the new $\weakHomologyMap$ is $\weakHomologyMap[1]_{\totalInt} - \Lambda$. Compute \begin{math} \bigl(\weakHomologyMap[1]_{\totalInt} - \Lambda\bigr) (\elementCycle[1] \moduleDot \ringElement[1] ) = \weakHomologyMap[1]_{\totalInt}(\elementCycle[1] \moduleDot \ringElement[1] ) - \Lambda(\elementCycle[1]\moduleDot \ringElement[1]) \end{math} But $\Lambda$ is defined on all of $\freeCycles[1]_{\totalInt}$ so \begin{math} \bigl(\weakHomologyMap[1]_{\totalInt} - \Lambda\bigr) (\elementCycle[1]\moduleDot \ringElement[1] ) = \weakHomologyMap[1]_{\totalInt}(\elementCycle[1]\moduleDot \ringElement[1]) - \Lambda(\elementCycle[1])\moduleDot \ringElement[1] \end{math} and $\weakTorsionHomologyMap[1]<1>_{\totalInt}$ is independent of the lift. \end{proof} \begin{DefS*}{Remark} A similar result holds for left $R$ modules. \end{DefS*} \begin{DefS}[weak split chain map definition]{Definition} A \emph{weak split chain map} between two weakly split chain complexes $\{\complex[1]_\ast, \splitPair[1]_\ast\}$ and $\{\complex[2]_\ast, \splitPair[2]_\ast\}$ consists of a chain map $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$, a map $\freeCyclesChainMap[1]_\ast\colon \freeCycles[1]_\ast \to \freeCycles[2]_\ast$ satisfying (\ref{p0}) and a map $\weakMap[1]_{\totalInt}\colon\freeCycles[1]_{\totalInt} \to \complex[2]_{\totalInt+1}$ satisfying (\ref{p1}). From the above discussion, given any two weakly split chain complexes and a chain map between them, this data can be completed to a weakly split chain map. The map $\weakTorsionHomologyMap[1]<1>_{\ast}$ is independent of this completion. \end{DefS} \begin{ThmS}[deviation from naturality in Kunneth formula]{Theorem} Suppose given four weakly split complexes and weakly split chain maps $\chainMap[3]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$ and $\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[4]_\ast$. If $\cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}\in H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast)$ then \begin{align*} \cs{homology splitting}[{\splitPair[2]_\ast}]{\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex} \bigl(\cs{elementary tor}{\chainMap[1](\element[1])}{\ringElement}{\chainMap[2](\element[2])} \bigr) &=\ \bigl(\chainMap[1]\tensor\chainMap[2]\bigr)_\ast\bigl( \cs{homology splitting}[{\splitPair[1]_\ast}]{\splitPair[3]_\ast}_{i}\def\secondIndex{j,\secondIndex} (\cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}) \bigr) +\\ \noalign{\vskip 10pt}&\hskip-40pt (-1)^{i}\def\secondIndex{j} \chainMap[1](\element[1])\cs{cross product} \weakTorsionHomologyMap[2]<1>_{\secondIndex}(\element[2]) + \weakTorsionHomologyMap[1]<1>_{i}\def\secondIndex{j}(\element[1])\cs{cross product} \chainMap[2](\element[2]) \end{align*} \end{ThmS} \begin{DefS}{Remark} The $\weakTorsionHomologyMap$ maps take values in $H_\ast(\,\_\,)\tensor \ry{\ringElement[1]}$ but since the other factor in the cross product is $\ringElement[1]$-torsion, each cross product is well-defined in $H_{i}\def\secondIndex{j+\secondIndex+1}(\complex[2]_\ast\tensor[R] \complex[4]_\ast)$. \end{DefS} \begin{proof} It suffices to check the formula on elementary tors so fix $\cs{elementary tor}{\element[1]}{\ringElement}{\element[3]}$. The corresponding cycle \ref{torsion product cycle II} is \begin{equation*} X_0 = (-1)^{\abs{\element[1]}+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) \tensor \freeBoundariesMap[3]_{\secondIndex}\bigl(\ringElement \elementCycle[3]\bigr) + \freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr) \tensor \freeCyclesMap[3]_{\secondIndex}(\elementCycle[3]) \end{equation*} Evaluating $\chainMap[1]\otimes\chainMap[2]$ on $X_0$ gives \begin{equation*} X_1=(-1)^{{i}\def\secondIndex{j}+1} \chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{{i}\def\secondIndex{j}}(\elementCycle[1])\bigr) \tensor \chainMap[2]_{\secondIndex+1} \Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr) + \chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr)\Bigr) \tensor \chainMap[2]_{\secondIndex} \bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr) \end{equation*} and a chain representing $\cs{homology splitting}[{\splitPair[2]_\ast}] {\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex}\bigl( \cs{elementary tor}{\chainMap[1](\element[1])} {\ringElement}{\chainMap[2](\element[2])} \bigr) $ is \noindent\resizebox{\textwidth}{!}{{ $X_2 = (-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) \tensor \Bigl(\freeBoundariesMap[4]_{\secondIndex} \bigl(\ringElement \freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr)\Bigr) + \Bigl(\freeBoundariesMap[2]_{\secondIndex} \bigl(\freeCyclesChainMap[1]_{\secondIndex}(\elementCycle[1]) \ringElement\bigr) \Bigr) \tensor \freeCyclesMap[4]_{\secondIndex} \bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) $}} It suffices to prove the theorem for $\chainMap[1]_\ast\tensor \identyMap{\complex[3]_\ast}$ and then for $\identyMap{\complex[2]_\ast} \tensor\chainMap[2]_\ast$ and these calculations are straightforward. \end{proof} \begin{math check} \vskip10pt It suffices to prove the theorem for $\chainMap[1]_\ast\tensor \identyMap{\complex[3]_\ast}$ and then for $\identyMap{\complex[2]_\ast} \tensor\chainMap[2]_\ast$. Here is the proof for $\chainMap[1]_\ast\tensor \identyMap{\complex[3]_\ast}$. In this special case, $X_1$ and $X_2$ become \begin{align*} Y_1=& (-1)^{i}\def\secondIndex{j+1} \chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) \tensor \freeBoundariesMap[3]_{\secondIndex}\bigl(\ringElement \elementCycle[3]\bigr) + \chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr)\Bigr) \tensor \freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\\ Y_2=& (-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) \tensor \freeBoundariesMap[3]_{\secondIndex}\bigl(\ringElement \elementCycle[3]\bigr) + \freeBoundariesMap[2]_{i}\def\secondIndex{j} \bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) \ringElement\bigr) \tensor \freeCyclesMap[3]_{\secondIndex}(\elementCycle[3]) \end{align*} By (\ref{p1}) \begin{equation*} \freeCyclesMap[2]_{i}\def\secondIndex{j}\bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) = \chainMap[1]_{i}\def\secondIndex{j}\bigl(\freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) + \boundary[2]_{i}\def\secondIndex{j+1}\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr) \end{equation*} By (\ref{p2}) \begin{equation*} \freeBoundariesMap[2]_{i}\def\secondIndex{j} \bigl(\freeCyclesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) \ringElement\bigr) = \freeBoundariesMap[2]_{i}\def\secondIndex{j} \bigl(\freeBoundariesChainMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] \ringElement)\bigr) = \chainMap[1]_{i}\def\secondIndex{j+1}\Bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr)\Bigr) + \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] \ringElement) + \weakHomologyMap[1]_{i}\def\secondIndex{j} (\elementCycle[1]\moduleDot\ringElement[1]) \end{equation*} Hence \alignLine{ Y_2 - Y_1 =& (-1)^{i}\def\secondIndex{j+1} \Bigl(\boundary[2]_{i}\def\secondIndex{j+1} \bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor \Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr) + \Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] \ringElement) + \weakHomologyMap[1]_{i}\def\secondIndex{j} (\elementCycle[1]\moduleDot\ringElement[1]) \Bigr)\tensor \freeCyclesMap[3]_{\secondIndex}(\elementCycle[3]) = \\& (-1)^{i}\def\secondIndex{j+1}\boundary[5]_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor \freeBoundariesMap[3]_{\secondIndex}\bigl(\ringElement \elementCycle[3]\bigr) \bigr) + \weakTorsionHomologyMap[1]<1>_{i}\def\secondIndex{j}(\element[1])\tensor \freeCyclesMap[3]_{\secondIndex}(\elementCycle[3]) } since \begin{align*} \boundary[5]_{i}\def\secondIndex{j+\secondIndex+2}& \Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor \freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr) =\\& \Bigl(\boundary[2]_{i}\def\secondIndex{j+1} \bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor \Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr)+ (-1)^{i}\def\secondIndex{j+1} \Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] )\Bigr)\tensor \Bigl(\ringElement\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\Bigr)=\\& \Bigl(\boundary[2]_{i}\def\secondIndex{j+1} \bigl( \weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\bigr)\Bigr)\tensor \Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr)+ (-1)^{i}\def\secondIndex{j+1} \Bigl(\weakMap[1]_{i}\def\secondIndex{j}(\elementCycle[1] \ringElement)\Bigr)\tensor \Bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\Bigr) \end{align*} \vskip10pt For the other case $X_1$ and $X_2$ become \begin{align*} Y_1=&(-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) \tensor \chainMap[2]_{\secondIndex+1} \Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\ringElement \elementCycle[3]\bigr)\Bigr) + \freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr) \tensor \chainMap[2]_{\secondIndex}\bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr)\\ Y_2=&(-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1]) \tensor \freeBoundariesMap[4]_{\secondIndex} \bigl(\ringElement \freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) + \freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr) \tensor \freeCyclesMap[4]_{\secondIndex} \bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) \end{align*} By (\ref{p1}) \begin{equation*} \freeCyclesMap[4]_{\secondIndex} \bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3])\bigr) = \chainMap[2]_{\secondIndex} \bigl(\freeCyclesMap[3]_{\secondIndex}(\elementCycle[3])\bigr) + \boundary[4]_{\secondIndex+1} \bigl( \weakMap[2]_{\secondIndex}(\elementCycle[3])\bigr) \end{equation*} By (\ref{p2}) \begin{equation*} \freeBoundariesMap[4]_{\secondIndex} \bigl(\freeCyclesChainMap[2]_{\secondIndex}(\elementCycle[3]) \ringElement\bigr) = \freeBoundariesMap[4]_{\secondIndex} \bigl(\freeBoundariesChainMap[2]_{\secondIndex}(\elementCycle[3] \ringElement)\bigr) = \chainMap[2]_{\secondIndex+1}\Bigl(\freeBoundariesMap[3]_{\secondIndex} \bigl(\elementCycle[3] \ringElement\bigr)\Bigr) + \weakMap[2]_{\secondIndex}(\elementCycle[3] \ringElement) - \weakHomologyMap[2]_{\secondIndex} (\elementCycle[3]\moduleDot\ringElement[1]) \end{equation*} Hence \begin{align*} Y_2-Y_1=& (-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor \bigl( \weakMap[2]_{\secondIndex}(\elementCycle[3] \ringElement) + \weakHomologyMap[2]_{\secondIndex} (\elementCycle[3]\moduleDot\ringElement[1]) \bigr) + \freeBoundariesMap[1]_{i}\def\secondIndex{j} \bigl(\elementCycle[1] \ringElement\bigr) \tensor \boundary[4]_{\secondIndex+1} \bigl( \weakMap[2]_{\secondIndex}(\elementCycle[3])\bigr) =\\& \boundary[6]_{i}\def\secondIndex{j+\secondIndex+2}\bigl(\freeBoundariesMap[1]_{i}\def\secondIndex{j} (\elementCycle[1] \ringElement) \tensor \weakMap[2]_{\secondIndex}(\elementCycle[3] \ringElement)\bigr) +(-1)^{i}\def\secondIndex{j+1} \freeCyclesMap[1]_{i}\def\secondIndex{j}(\elementCycle[1])\tensor \weakHomologyMap[2]_{\secondIndex} (\elementCycle[3]\moduleDot\ringElement[1] \end{align*} \end{math check} \begin{ThmS}[naturality of cosets]{Corollary} Given chain maps $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[2]_\ast$ and $\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[4]_\ast$ \begin{equation*} \bigl(\chainMap[1]_\ast\tensor \chainMap[2]_\ast\bigr)_\ast\bigl( \cosetTor[{\element[1]}]{\ringElement}{\element[2]} \bigr)\subset \cosetTor[{\chainMap[1]_\ast(\element[1])}]{\ringElement} {\chainMap[2]_\ast(\element[2])} \end{equation*} In words, the cosets are natural and do not depend on the weak splittings of the complexes. \end{ThmS} \begin{proof} First check that the $0$-cosets behave correctly: \noindent\mathLine{ \bigl(\chainMap[1]_\ast\tensor \chainMap[2]_\ast\bigr)_\ast\Bigl( \fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[3]_\ast} {i}\def\secondIndex{j}{\secondIndex}\Bigr) \subset \fundamentalCoset{\chainMap[1]_\ast(\element[1])} {\chainMap[2]_\ast(\element[2])} {\complex[2]_\ast}{\complex[4]_\ast} {i}\def\secondIndex{j}{\secondIndex}} By \namedRef{deviation from naturality in Kunneth formula} $\cs{homology splitting}[{\splitPair[2]_\ast}]{\splitPair[4]_\ast}_{i}\def\secondIndex{j,\secondIndex} \bigl(\cs{elementary tor}{\chainMap[1](\element[1])}{\ringElement} {\chainMap[2](\element[2])} \bigr) \subset \cosetTor[{\chainMap[1]_\ast(\element[1])}]{\ringElement} {\chainMap[2]_\ast(\element[2])}$. One application of \namedRef{deviation from naturality in Kunneth formula} is to the case in which $\chainMap[1]_\ast$ is the identity but the weak splittings change. Hence changing the weak splittings does not change the cosets. The result follows. \end{proof} \section{The interchange map and the K\"unneth\ formula} There are natural isomorphisms $I\colon \complex[1]\tensor[R] \complex[3] \cong \complex[3]\tensor[R] \complex[1]$ and $I\colon \complex[1]\tor[R] \complex[3] \cong \complex[3]\tor[R] \complex[1]$. On elementary tensors, $I(\element[1]\tensor \element[2]) = \element[2]\tensor \element[1]$ and $I(\cs{elementary tor} {\element[1]}{\ringElement}{\element[2]})=\cs{elementary tor} {\element[2]}{\ringElement}{\element[1]}$. Applying $I$ to the tensor product of two chain complexes is not a chain map: a sign is required. The usual choice is \begin{equation*} T\colon \complex[1]_\ast\tensor[R] \complex[3]_\ast \to \complex[3]_\ast\tensor[R] \complex[1]_\ast \end{equation*} defined on elementary tensors by $T(\element[1] \tensor \element[2]) = (-1)^{\abs{\element[1]}\abs{\element[2]}} \element[2] \tensor \element[1]$. It follows that the cross product map satisfies \begin{equation*} T_\ast(\element[1] \cs{cross product} \element[2]) = (-1)^{\abs{\element[1]}\abs{\element[2]}} \element[2] \cs{cross product} \element[1] \end{equation*} for all $\element[1]\in H_{\abs{\element[1]}}(\complex[1]_\ast)$ and $\element[2]\in H_{\abs{\element[2]}}(\complex[3]_\ast)$. \begin{ThmS}[flip theorem I]{Theorem} For all $\element[1]\in H_{i}\def\secondIndex{j}(\complex[1]_\ast)$ and $\element[2]\in H_{\secondIndex}(\complex[3]_\ast)$ {\setlength\belowdisplayskip{-10pt} \begin{equation*} T_\ast\Bigl(\cs{homology splitting}[{\splitPair[1]}] {\splitPair[3]}_{i}\def\secondIndex{j+\secondIndex+1}\bigr( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}\bigr)\Bigr) = (-1)^{i}\def\secondIndex{j\cdot\secondIndex+1} \cs{homology splitting}[{\splitPair[3]}]{\splitPair[1]}_{i}\def\secondIndex{j+\secondIndex+1} \bigl( \cs{elementary tor}{\element[2]}{\ringElement}{\element[1]}\bigr) \end{equation*} } \end{ThmS} \begin{proof} Apply $T$ to the cycle in \cs{torsion product cycle II 1}. \end{proof} \begin{math check} $\epsilon\, \freeCyclesMap[1]_\ast(\elementCycle[1]) \tensor \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) + \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \tensor \freeCyclesMap[3]_\ast(\elementCycle[3]) $. \begin{align*} T\Bigl(& \epsilon\, \freeCyclesMap[1]_\ast(\elementCycle[1]) \tensor \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) + \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \tensor \freeCyclesMap[3]_\ast(\elementCycle[3])\Bigr) = \\& (-1)^{i}\def\secondIndex{j(\secondIndex+1)}\Bigl( (-1)^{i}\def\secondIndex{j+1} \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) \tensor \freeCyclesMap[1]_\ast(\elementCycle[1]) \Bigr) + (-1)^{(i}\def\secondIndex{j+1)\secondIndex}\Bigl( \freeCyclesMap[3]_\ast(\elementCycle[3]) \tensor \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \Bigr)=\\& (-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl( \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) \tensor \freeCyclesMap[1]_\ast(\elementCycle[1]) + (-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl( (-1)^{\secondIndex+1} \freeCyclesMap[3]_\ast(\elementCycle[3]) \tensor \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \Bigr)=\\& (-1)^{i}\def\secondIndex{j\secondIndex+1}\Bigl( (-1)^{\secondIndex+1} \freeBoundariesMap[3]_\ast\bigl(\ringElement \elementCycle[3]\bigr) \tensor \freeCyclesMap[1]_\ast(\elementCycle[1]) + \freeCyclesMap[3]_\ast(\elementCycle[3]) \tensor \freeBoundariesMap[1]_\ast \bigl(\elementCycle[1] \ringElement\bigr) \Bigr) \end{align*} This cycle represents $(-1)^{i}\def\secondIndex{j\cdot\secondIndex+1} \cs{homology splitting}[{\splitPair[3]}]{\splitPair[1]}_{i}\def\secondIndex{j+\secondIndex+1} \bigl( \cs{elementary tor}{\element[2]}{\ringElement}{\element[1]}\bigr) $. \end{math check} \begin{ThmS}[flip and Kunneth]{Corollary} If $R$ is a PID and if $\complex[1]_\ast\tor[R]\complex[3]_\ast$ is acyclic \noindent\resizebox{\textwidth}{!}{{$\xymatrix{ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tensor[R] H_{\secondIndex}(\complex[3]_\ast) \ar[r]^-{\cs{cross product}} \ar[d]^-{\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt}(-1)^{i}\def\secondIndex{j \secondIndex} I} & H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[3]_\ast) \ar[r]^-{\cs{to torsion product}} \ar[d]^-{T_\ast}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} H_{i}\def\secondIndex{j}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[3]_\ast)\to0 \ar[d]\ar[d]_-{\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} (-1)^{i}\def\secondIndex{j \secondIndex + 1} I} \\ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[3]_\ast)\tensor[R] H_{\secondIndex}(\complex[1]_\ast) \ar[r]^-{\cs{cross product}}& H_{\totalInt}(\complex[3]_\ast\tensor[R] \complex[1]_\ast) \ar[r]^-{\cs{to torsion product}}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt-1} H_{i}\def\secondIndex{j}(\complex[3]_\ast)\tor[R] H_{\secondIndex}(\complex[1]_\ast)\to0 }$}} \noindent commutes. The splittings can be chosen to make the diagram commute. \end{ThmS} \section{The boundary map and the K\"unneth\ formula} The boundary map in question is the map associated with the long exact homology sequence for a short exact sequence of chain complexes. Before stating the result some preliminaries are needed. \begin{DefS}{Definition} A pair of composable chain maps $\xymatrix@1@C12pt{\complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&& \complex[3]_\ast}$ and $\xymatrix@1@C12pt{\complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&& \complex[2]_\ast}$ form \emph{a weak exact sequence} provided there exists a short exact sequence of free approximations and chain maps making (\ref{weak exact sequence diagram}) below commute. \namedNumber{weak exact sequence diagram} {\setlength\belowdisplayskip{-10pt} \begin{equation*}\tag{\ref{weak exact sequence diagram}} \xymatrix@C10pt{ 0\ar[r]&\freeApproximation[1]_\ast\ar[rr]^-{\freeApproximationChainMap[1]_\ast} \ar[d]^-{\vertMap{\complex[1]}_\ast}&& \freeApproximation[3]_\ast\ar[rr]^-{\freeApproximationChainMap[2]_\ast} \ar[d]^-{\vertMap{\complex[3]}_\ast}&& \freeApproximation[2]_\ast \ar[d]^-{\vertMap{\complex[2]}_\ast} \ar[r]& 0\\ & \complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&& \complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&& \complex[2]_\ast} \end{equation*} } \end{DefS} Given a weak exact sequence there is a long exact homology sequence coming from the long exact sequence of the top row of (\ref{weak exact sequence diagram}): \begin{equation*} \xymatrix@C28pt{\cdots\to H_{i}\def\secondIndex{j+1}(\complex[2]_\ast)\ar[r]^-{\boldsymbol{\partial}_{i}\def\secondIndex{j+1}}& H_{i}\def\secondIndex{j}(\complex[1]_\ast)\ar[r]^-{\chainMap[1]_\ast}& H_{i}\def\secondIndex{j}(\complex[3]_\ast)\ar[r]^{\chainMap[2]_\ast}& H_{i}\def\secondIndex{j}(\complex[2]_\ast)\ar[r]^-{\boldsymbol{\partial}_{i}\def\secondIndex{j}}&\cdots }\] The boundary $\boldsymbol{\partial}_{i}\def\secondIndex{j+1} = \vertMap{\complex[1]}_\ast \circ \partial_{i}\def\secondIndex{j+1}\circ (\vertMap{\complex[2]}_\ast)^{-1}$ where $\partial_{i}\def\secondIndex{j+1}$ is the usual boundary in the long exact homology sequence for the free complexes. \begin{ThmS}{Lemma} A short exact sequence of chain complexes \begin{equation*} \xyLine[@C10pt]{ 0\ar[r]& \complex[1]_\ast\ar[rr]^-{\chainMap[1]_\ast}&& \complex[3]_\ast\ar[rr]^-{\chainMap[2]_\ast}&& \complex[2]_\ast\ \ar[r]& 0} \end{equation*} is weak exact. The boundary $\boldsymbol{\partial}_{i}\def\secondIndex{j+1}$ is the usual boundary map. \end{ThmS} \begin{proof} The commutative diagram of free approximations (\ref{weak exact sequence diagram}) is given by \namedRef{short exact free approximation}. The description of the boundary map is immediate. \end{proof} \begin{ThmS}[weak exact is preserved by products]{Lemma} If $\complex[1]_\ast\tor[R]\complex[4]_\ast$, $\complex[3]_\ast\tor[R]\complex[4]_\ast$ and $\complex[2]_\ast\tor[R]\complex[4]_\ast$ are acyclic and if $\xymatrix@1{ \complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}& \complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}& \complex[2]_\ast }$ is weak exact, then so are {\setlength\abovedisplayskip{0pt} \setlength\belowdisplayskip{0pt} \begin{equation*} \xymatrix@C40pt@R10pt{ \complex[1]_\ast\tensor[R]\complex[4]_\ast \ar[r]^-{\chainMap[1]_\ast \tensor \identyMap{\complex[4]_\ast}}& \complex[3]_\ast\tensor[R]\complex[4]_\ast \ar[r]^-{\chainMap[2]_\ast \tensor \identyMap{\complex[4]_\ast}}& \complex[2]_\ast\tensor[R]\complex[4]_\ast \\ \complex[4]_\ast\tensor[R]\complex[1]_\ast \ar[r]^-{\identyMap{\complex[4]_\ast} \tensor \chainMap[1]_\ast}& \complex[4]_\ast\tensor[R]\complex[3]_\ast \ar[r]^-{\identyMap{\complex[4]_\ast}\tensor\chainMap[2]_\ast}& \complex[4]_\ast\tensor[R]\complex[2]_\ast }\end{equation*}} \end{ThmS} \begin{proof} Pick free approximations satisfying (\ref{weak exact sequence diagram}), $\vertMap{\complex[1]}_\ast$, $\vertMap{\complex[3]}_\ast$, $\vertMap{\complex[2]}_\ast$ and a free approximation $\vertMap{\complex[4]}_\ast$. By \namedRef{Dold splitting} the required free approximations are $\vertMap{\complex[1]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$, $\vertMap{\complex[3]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$, $\vertMap{\complex[2]}_{\ast} \tensor\vertMap{\complex[4]}_{\ast}$, or $\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[1]}_{\ast}$, $\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[3]}_{\ast}$, $\vertMap{\complex[4]}_{\ast} \tensor\vertMap{\complex[2]}_{\ast}$. \end{proof} \begin{DefS*}{Warning} Even if $\xymatrix@1{ \complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}& \complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}& \, \complex[2]_\ast }$ is short exact, the pair $\chainMap[1]_\ast\tensor \identyMap{\complex[4]_\ast}$ and $\chainMap[2]_\ast\tensor \identyMap{\complex[4]_\ast}$ may only be weak exact. For them to be short exact requires that either $\complex[2]_\ast$ or $\complex[4]_\ast$ be torsion free. \end{DefS*} \begin{ThmS}[boundary of elementary tor]{Theorem} Suppose $\complex[1]_\ast\tor[R]\complex[4]_\ast$, $\complex[3]_\ast\tor[R]\complex[4]_\ast$ and $\complex[2]_\ast\tor[R]\complex[4]_\ast$ are acyclic and suppose $\xymatrix@1{ \complex[1]_\ast\ar[r]^-{\chainMap[1]_\ast}& \complex[3]_\ast\ar[r]^-{\chainMap[2]_\ast}& \,\complex[2]_\ast }$ is weak exact. Then for $\element[1]\in H_{i}\def\secondIndex{j}(\complex[2]_\ast)$ and $\element[2]\in H_{\secondIndex}(\complex[4]_\ast)$ {\setlength\belowdisplayskip{-10pt} \begin{equation*} \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1} \bigl(\cosetTor[{\element[1]}]{\ringElement}{\element[2]}\bigr) \subset - \cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{\ringElement}{\element[2]} \end{equation*} } \end{ThmS}\nointerlineskip \begin{proof} By \namedRef{weak exact is preserved by products} it may be assumed that the complexes are all free. Pick compatible splittings for $\complex[1]_\ast$, $\complex[2]_\ast$ and $\complex[4]_\ast$. Recall that Bocksteins and long exact sequence boundary maps anti-commute and that in short exact sequences of free chain complexes $\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex}(\elementCycle[1]\otimes\elementCycle[4]) = \boldsymbol{\partial}_{i}\def\secondIndex{j}(\elementCycle[1])\otimes\elementCycle[4] $. A routine calculation completes the proof. \end{proof} \begin{math check} Then \begin{equation*} (-1)^{i}\def\secondIndex{j+1} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr) \in\cosetTor[{\element[1]}]{\ringElement}{\element[2]} \end{equation*} Since $\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1} \circ \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2} = - \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1}\circ \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+2}$ \begin{align*} \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl( (-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl(& \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr)\Bigr) =\\& (-1)^{i}\def\secondIndex{j} \biggl(\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1}\Bigr( \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr)\Bigr)\biggr)=\\& (-1)^{i}\def\secondIndex{j} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1}\Bigr( \boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1])\bigr) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \Bigr)\ . \end{align*} On the other side \begin{equation*} (-1)^{i}\def\secondIndex{j-1+1} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j-1+\secondIndex+2}\Bigl( \splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \Bigr) \in\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{\ringElement}{\element[2]} \end{equation*} Both $\splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)$ and $\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1])\bigr) $ are chains in $\complex[1]_{i}\def\secondIndex{j}$ which are cycles in $\complex[1]_{i}\def\secondIndex{j}\tensor[R]\ry{\ringElement}$. Applying Bocksteins shows $\mathfrak b^{\ringElement}_{i}\def\secondIndex{j}\Bigl( \splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)\Bigr) = \ringElement \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])$ and $\mathfrak b^{\ringElement}_{i}\def\secondIndex{j}\Bigl(\boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1])\bigr) \Bigr) = -\boldsymbol{\partial}_{i}\def\secondIndex{j} \Bigl(\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+1} \bigl(\splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \bigr)\Bigr) = -\boldsymbol{\partial}_{i}\def\secondIndex{j} (\ringElement \element[1]) = -\ringElement \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1]) $. Hence $Z = \splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr) + \boldsymbol{\partial}_{i}\def\secondIndex{j+1}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1])\bigr) $ is a cycle. Hence \begin{align*} &\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl( (-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr)\Bigr) =\\& \hskip 40pt(-1)^{i}\def\secondIndex{j} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1}\biggl( \Bigl(Z - \splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr)\Bigr) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \biggr) =\\& (-1)^{i}\def\secondIndex{j+1} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1} \Bigl(\splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \Bigr) +(-1)^{i}\def\secondIndex{j+1} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1}\bigl( Z \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr)=\\& \hskip 40pt-(-1)^{i}\def\secondIndex{j} \mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+1} \Bigl(\splitBocksteinHomology^{\complex[1],\ringElement}_{i}\def\secondIndex{j-1}\bigl( \boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\bigr) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \Bigr) +(-1)^{i}\def\secondIndex{j+1+i}\def\secondIndex{j} Z \cs{cross product} \element[2] \end{align*} and therefore \begin{equation*} \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl( (-1)^{i}\def\secondIndex{j+1}\mathfrak b^{\ringElement}_{i}\def\secondIndex{j+\secondIndex+2}\bigl( \splitBocksteinHomology^{\complex[2],\ringElement}_{i}\def\secondIndex{j}(\element[1]) \cs{cross product} \splitBocksteinHomology^{\complex[4],\ringElement}_{\secondIndex}(\element[2]) \bigr)\Bigr) \in -\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{\ringElement}{\element[2]} \end{equation*} Since one element of $\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1} \bigl(\cosetTor[{\element[1]}]{\ringElement}{\element[2]}\bigr) $ is in $-\cosetTor[{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}]{\ringElement}{\element[2]}$ and since \mathLine{\boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1}\Bigl( \fundamentalCoset{\element[1]}{\element[2]}{\complex[1]_\ast}{\complex[4]_\ast} {i}\def\secondIndex{j}{\secondIndex}\Bigr) \subset \bigl(\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])\cs{cross product} H_{\secondIndex+1}(\complex[4]_\ast)\bigr) \displaystyle\mathop{\oplus} \bigl(H_{i}\def\secondIndex{j}(\complex[1]_\ast)\cs{cross product}\element[2]\bigr) } the result follows. \end{math check} \begin{ThmS}{Corollary} With assumptions and notation as in \namedRef{boundary of elementary tor} {\setlength\belowdisplayskip{-10pt} \begin{equation*} \boldsymbol{\partial}_{i}\def\secondIndex{j+\secondIndex+1} \bigl(\cosetTor[{\element[2]}]{\ringElement}{\element[1]}\bigr) \subset (-1)^{\secondIndex+1} \cosetTor[{\element[2]}]{\ringElement}{{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}} \end{equation*}} \end{ThmS}\nointerlineskip \begin{proof} Apply the interchange map (\ref{flip theorem I}) to get to the situation of \namedRef{boundary of elementary tor} and then apply the interchange map again. \end{proof} \begin{ThmS}[boundary and Kunneth]{Corollary} With assumptions and notation as in \namedRef{boundary of elementary tor} let $\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)} \colon H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast) \to H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast) $ be the map defined by $\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}\bigl( \cs{elementary tor}{\element[1]}{\ringElement}{\element[2]}) = \cs{elementary tor}{\boldsymbol{\partial}_{i}\def\secondIndex{j}(\element[1])}{\ringElement}{\element[2]} $. Then \noindent\resizebox{\textwidth}{!}{{$\xymatrix@R30pt{ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1} H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tensor[R] H_{\secondIndex}(\complex[4]_\ast) \ar[r]^-{\cs{cross product}} \ar[d]^-{\hbox{\tiny{$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1} \boldsymbol{\partial}_{i}\def\secondIndex{j}\tensor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}$}}}& H_{\totalInt+1}(\complex[2]_\ast\tensor[R] \complex[4]_\ast) \ar[r]^-{\cs{to torsion product}} \ar[d]_-{\boldsymbol{\partial}_{\totalInt+1}}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j}(\complex[2]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast)\to0 \ar[d]\ar[d]_-{\hbox{\tiny{$\displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} -\boldsymbol{\partial}_{i}\def\secondIndex{j}\tor \identyMap{H_{\secondIndex}(\complex[4]_\ast)}$}}} \\ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1} H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tensor[R] H_{\secondIndex}(\complex[4]_\ast) \ar[r]^-{\cs{cross product}}& H_{\totalInt}(\complex[1]_\ast\tensor[R] \complex[4]_\ast) \ar[r]^-{\cs{to torsion product}}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H_{i}\def\secondIndex{j-1}(\complex[1]_\ast)\tor[R] H_{\secondIndex}(\complex[4]_\ast)\to0 }$}} \noindent commutes. \end{ThmS} \begin{proof}The proof is immediate. \end{proof} \section{The Massey triple product} Suppose $X$ and $Y$ are CW complexes with finitely many cells in each dimension. Then the cellular cochains are free $\mathbb Z$ modules and the K\"unneth\ formula plus the Eilenberg-Zilber chain homotopy equivalence yields a K\"unneth\ formula \mathLine{ \xymatrix{ 0\to \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt} H^{i}\def\secondIndex{j}(X)\otimes H^{\secondIndex}(Y) \ar[r]^-{\cs{cross product}}& H^{\totalInt}(X\times Y)\ar[r]^-{\cs{to torsion product}}& \displaystyle\mathop{\oplus}_{i}\def\secondIndex{j+\secondIndex=\totalInt+1} H^{i}\def\secondIndex{j}(X)\tor H^{\secondIndex}(Y)\to0 }} Given $u\in H^{i}\def\secondIndex{j}(X)$ define $\secondU{u}\in H^{i}\def\secondIndex{j}(X\times Y)$ by $\secondU{u}=p_X^\ast(u)$ where $p_X\colon X\times Y \to X$ is the projection. For $v\in H^{\secondIndex}(Y)$ define $\secondU{v}\in H^{\secondIndex}(X\times Y)$ similarly and recall $u\cs{cross product} v = \secondU{u} \cup \secondU{v}$ where $\cup$ denotes the cup product. \begin{ThmS}{Theorem} With notation as above and non-zero $m\in \mathbb Z$ \begin{equation*} \cosetTor[u]{m}{v} = \boldsymbol{\langle} \secondU{u}, \secondU{(m)},\secondU{v} \boldsymbol{\rangle} \end{equation*} where $\boldsymbol{\langle} \secondU{u}, \secondU{(m)},\secondU{v} \boldsymbol{\rangle} $ is the Massey triple product of the indicated cohomology classes where $\secondU{(m)}$ is $m$ times the multiplicative identity in $H^0(X\times Y)$. \end{ThmS} The proof is immediate from \namedRef{Mac Lane cycle A} and the definition of the Massey triple product. \section{Weakly split chain complexes} Heller's category in \cite{Heller} carries much the same information as weak splittings. \gdef\chainMap[1]+ \chainMap[2]{\chainMap[2]\circ\chainMap[1]} \begin{ThmS}{Proposition} If $\chainMap[1]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ and $\chainMap[2]_\ast\colon \complex[3]_\ast \to \complex[2]_\ast$ are weakly split chain maps, then $\chainMap[2]\circ\chainMap[1]$ is weakly split by $\freeCyclesChainMap[10]_{\totalInt}= \freeCyclesChainMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}$ and $\weakMap[10]_{\totalInt} = \chainMap[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} + \weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}$. With these choices \begin{equation*} \weakTorsionHomologyMap[10]<1>_{\totalInt} = (\chainMap[2]_{\totalInt+1}\tensor \identyMap{\ry{\ringElement}})\circ\weakTorsionHomologyMap[1]<1>_{\totalInt} + \weakTorsionHomologyMap[2]<1>_{\totalInt}\circ \chainMap[1]_{\totalInt} \end{equation*} \end{ThmS} \begin{proof} Formula (\ref{p0}) is immediate. Formula (\ref{p1}) is a routine calculation. It is straightforward to check $\weakHomologyMap[10]_{\totalInt} = \chainMap[2]_{\totalInt+1}\circ \weakHomologyMap[1]_{\totalInt} + \weakHomologyMap[2]_{\totalInt} \circ \freeBoundariesChainMap[1]_{\totalInt} $ from which the formula for the $\weakTorsionHomologyMap$ follows. \end{proof} \begin{DefS}{Remark} Composition can be checked to be associative. \def\chainMap[1]+ \chainMap[2]{\identyMap{{\complex[1]_{ }}_{\ast}}} The pair $\freeCyclesChainMap[10]_{\ast}=\chainMap[1]+ \chainMap[2]$ and $\weakMap[10]_{\ast}=0$ give the identity for any weak spitting of $\complex[1]_{\ast}$. Hence weakly split chain complexes and weakly split chain maps form a category. \end{DefS} \begin{math check} \begin{equation*} \boundary[2]_{\totalInt+1}\circ \weakMap[10]_{\totalInt} = \freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt} - \chainMap[2]_{\totalInt}\circ\chainMap[1]_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} \end{equation*} \begin{align*} \boundary[2]_{\totalInt+1}\bigl(&\chainMap[4]_{\totalInt+1}\circ \weakMap[1]_{\totalInt}+ \weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}\bigr) = \chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) + \boundary[2]_{\totalInt+1}\bigl(\weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}\bigr) =\\& \chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) + \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt} - \chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ \freeCyclesChainMap[1]_{\totalInt} =\\& \chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) + \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} \bigr) - \bigl(\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ \freeCyclesChainMap[1]_{\totalInt} =\\& \chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) + \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr) - \bigl(\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ \freeCyclesChainMap[1]_{\totalInt} =\\& \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+ \chainMap[4]_{\totalInt}\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt}\bigr) - \bigl(\chainMap[4]_{\totalInt}\circ \freeCyclesMap[3]_{\totalInt}\bigr)\circ \freeCyclesChainMap[1]_{\totalInt} =\\& \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+ \chainMap[4]\bigl(\boundary[3]_{\totalInt}\circ \weakMap[1]_{\totalInt} - \freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt}\bigr) =\\& \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)+ \chainMap[4]_{\totalInt}\bigl( -\chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\bigr) = \bigl(\freeCyclesMap[2]_{\totalInt}\circ \freeCyclesChainMap[10]_{\totalInt}\bigr)- \chainMap[4]_{\totalInt}\circ \chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt} \end{align*} The required formula has been verified. \begin{equation*} \weakHomologyMap[10]_{\totalInt} = \weakMap[10]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl( \freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[10]_{\totalInt} - (\chainMap[2]_{\ast}\circ \chainMap[1]_{\ast})_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} \bigr)\colon \freeBoundaries[1]_{\totalInt} \to \complex[4]_{\totalInt+1} \end{equation*} \begin{align*} \weakMap[10]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl( & \freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[10]_{\totalInt} - (\chainMap[2]_{\ast}\circ \chainMap[1]_{\ast})_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} \bigr) =\\& \bigl( \chainMap[2]_{\totalInt+1}\circ \weakMap[1]_{\totalInt} + \weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt}\bigr) \big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -\\&\hskip10pt \bigl( \freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[2]_{\totalInt+1}\circ \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} \bigr) =\\& \chainMap[2]_{\totalInt+1}\Bigl(\weakMap[1]_{\totalInt} \big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -\bigl( \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt}\bigr)\Bigr) +\\& \weakMap[2]_{\totalInt} \circ \freeCyclesChainMap[1]_{\totalInt} \big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl( \freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[2]_{\totalInt+1}\circ \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} \bigr)=\\& \chainMap[2]_{\totalInt+1}\Bigl(\weakMap[1]_{\totalInt} \big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -\bigl( \freeBoundariesMap[2]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt}\bigr)\Bigr) +\\& \Bigl(\weakMap[2]_{\totalInt} \big\vert_{_{\scriptstyle\freeBoundaries[2]_{\totalInt}}} - \bigl( \freeBoundariesMap[4]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt} - \chainMap[2]_{\totalInt+1}\circ \freeBoundariesMap[2]_{\totalInt}\bigr) \Bigr) \freeBoundariesChainMap[1]_{\totalInt}=\\& \chainMap[2]_{\totalInt+1}\circ \weakHomologyMap[1]_{\totalInt} + \weakHomologyMap[2]_{\totalInt} \circ \freeBoundariesChainMap[1]_{\totalInt} \end{align*} The result follows. \end{math check} \begin{ThmS}{Proposition} Let $\chainMap[3]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ be a weakly split chain map and suppose $\chainMap[4]_\ast\colon \complex[1]_\ast \to \complex[3]_\ast$ is a chain map chain homotopic to $\chainMap[3]$. Let $D_\ast\colon \complex[1]_\ast \to \complex[3]_{\ast+1}$ be a chain homotopy with \begin{equation*} \chainMap[4]_\ast - \chainMap[3]_\ast = \boundary[3]_{\ast+1}\circ D_\ast + D_{\ast-1}\circ \boundary[1]_\ast \end{equation*} Then $\chainMap[4]$ is weakly split by $\freeCyclesChainMap[2]_{\totalInt}=\freeCyclesChainMap[1]_{\totalInt}$ and \begin{equation*} \weakMap[2]_{\totalInt} = \weakMap[1] + D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} + \boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt} \end{equation*} With these choices, $\weakTorsionHomologyMap[2]<1>_{\totalInt}=\weakTorsionHomologyMap[1]<1>_{\totalInt}$ \end{ThmS} \begin{proof} Since chain homotopic maps induce the same map in homology, it is possible to take $\freeCyclesChainMap[1]_{\totalInt}=\freeCyclesChainMap[2]_{\totalInt}$ and then $\freeBoundariesChainMap[1]_{\totalInt}=\freeBoundariesChainMap[2]_{\totalInt}$ The required verifications are straightforward. \end{proof} \begin{math check} \begin{align*} \boundary[3]_{\totalInt+1}\circ \weakMap[2]_{\totalInt} = & \boundary[3]_{\totalInt+1}\bigl(\weakMap[1]_{\totalInt} + D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} + \boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt} \bigr)=\\& \bigl(\freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} - \chainMap_{\totalInt}\circ \freeCyclesMap[1]_{\totalInt}\bigr) + \bigl( \chainMap[4]_{\totalInt} - \chainMap[3]_{\totalInt} - D_{\totalInt-1}\boundary[1]_\ast \bigr)\circ \freeCyclesMap[1]_{\totalInt}=\\& \freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[1]_{\totalInt} - \chainMap[4]_{\totalInt} \circ\freeCyclesMap[1]_{\totalInt} = \freeCyclesMap[3]_{\totalInt}\circ \freeCyclesChainMap[2]_{\totalInt} - \chainMap[4]_{\totalInt} \circ\freeCyclesMap[1]_{\totalInt} \end{align*} \begin{align*} &\weakHomologyMap[2]_{\totalInt} = \weakMap[2]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} -\bigl( \freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt} - \chainMap[2]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt}\bigr) \bigr) = \\& \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} + D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} + \boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt} -\bigl( \freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[2]_{\totalInt} - \chainMap[2]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt}\bigr) \bigr) = \\& \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl(\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} \bigr) + D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} + \boundary[3]_{\totalInt+2}\circ D_{\totalInt+1}\circ\freeBoundariesMap[1]_{\totalInt} -\\&\hskip30pt \bigl( \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} - \chainMap[2]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt}\bigr) \bigr) =\\& \weakMap[1]_{\totalInt}\big\vert_{_{\scriptstyle\freeBoundaries[1]_{\totalInt}}} - \bigl(\freeBoundariesMap[3]_{\totalInt}\circ \freeBoundariesChainMap[1]_{\totalInt} - \chainMap[1]_{\totalInt+1} \circ \freeBoundariesMap[1]_{\totalInt} \bigr) + D_{\totalInt} \circ \freeCyclesMap[1]_{\totalInt} - D_{\totalInt}\circ \boundary[1]_{\totalInt+1}\circ \freeBoundariesMap[1]_{\totalInt} = \weakHomologyMap[1]_{\totalInt} \end{align*} The required formulas have been verified. \end{math check} The remaining results are routine verifications. \begin{ThmS}[weakly split direct sum]{Proposition} Given two weakly split chain complexes, $\{\complex[1]_\ast$, $\splitPair[1]_\ast\}$ and $\{\complex[2]_\ast$, $\splitPair[2]_\ast\}$, then $\complex[1]_\ast\displaystyle\mathop{\oplus} \complex[3]_\ast$ is weakly split by the following data:\\ \def\chainMap[1]+ \chainMap[2]{\complex[1]\oplus \complex[3]} $\freeCycles[100]_{\totalInt}= \freeCycles[1]_{\totalInt}\displaystyle\mathop{\oplus} \freeCycles[2]_{\totalInt}$, $\freeCyclesMap[100]_{\totalInt} = \freeCyclesMap[1]_{\totalInt}\displaystyle\mathop{\oplus}\freeCyclesMap[2]_{\totalInt} $. Then $\freeBoundaries[100]_{\totalInt}= \freeBoundaries[1]_{\totalInt}\displaystyle\mathop{\oplus} \freeBoundaries[2]_{\totalInt}$ so let $\freeBoundariesMap[100]_{\totalInt} = \freeBoundariesMap[1]_{\totalInt}\displaystyle\mathop{\oplus}\freeBoundariesMap[2]_{\totalInt} $. \end{ThmS} \begin{ThmS}{Proposition} Given weakly split chain maps $\chainMap[1]_\ast\colon\complex[1]_\ast \to\complex[2]_\ast$ and $\chainMap[2]_\ast\colon\complex[3]_\ast \to\complex[4]_\ast$ then $\chainMap[1]_\ast\displaystyle\mathop{\oplus}\chainMap[2]_\ast$ is weakly split by \def\chainMap[1]+ \chainMap[2]{\chainMap[1]\oplus \chainMap[2]} $\freeCyclesChainMap[10]_{\totalInt}= \freeCyclesChainMap[2]_{\totalInt}\displaystyle\mathop{\oplus} \freeCyclesChainMap[1]_{\totalInt}$ and $\weakMap[10]_{\totalInt} = \weakMap[1]_{\totalInt} \displaystyle\mathop{\oplus} \weakMap[2]_{\totalInt}$. With these choices \begin{equation*} \weakTorsionHomologyMap[10]<1>_{\totalInt} = \weakTorsionHomologyMap[1]<1>_{\totalInt} \displaystyle\mathop{\oplus} \weakTorsionHomologyMap[2]<1>_{\totalInt} \end{equation*} \end{ThmS} \begin{DefS}{Remark} The zero complex with its evident splitting is a zero for the direct sum operation. The zero chain map between any two weakly split complexes is weakly split by letting \def\chainMap[1]+ \chainMap[2]{0_\ast} $\freeCyclesChainMap[10]_{\totalInt}$ and $\weakMap[10]_{\totalInt}$ be trivial. Then $\weakTorsionHomologyMap[10]<1>_{\totalInt}$ is also trivial. \end{DefS} There is an internal sum result. \begin{ThmS}{Proposition} Given weakly split chain maps $\chainMap[1]_\ast\colon\complex[1]_\ast \to\complex[2]_\ast$ and $\chainMap[2]_\ast\colon\complex[1]_\ast \to\complex[2]_\ast$ then $\chainMap[1]_\ast+\chainMap[2]_\ast$ is weakly split by \def\chainMap[1]+ \chainMap[2]{\chainMap[1]+ \chainMap[2]} $\freeCyclesChainMap[10]_{\totalInt}= \freeCyclesChainMap[2]_{\totalInt}+ \freeCyclesChainMap[1]_{\totalInt}$ and $\weakMap[10]_{\totalInt} = \weakMap[1]_{\totalInt} + \weakMap[2]_{\totalInt}$. With these choices \begin{equation*} \weakTorsionHomologyMap[10]<1>_{\totalInt} = \weakTorsionHomologyMap[1]<1>_{\totalInt} + \weakTorsionHomologyMap[2]<1>_{\totalInt} \end{equation*} \end{ThmS} \begin{DefS}{Remark} Unlike the direct sum case (\ref{weakly split direct sum}), there does not seem to be an easy way to weakly split the tensor product. \end{DefS} \begin{references} \bib{Dold}{book}{ author={Dold, Albrecht}, title={Lectures on algebraic topology}, series={Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, volume={200}, edition={2}, publisher={Springer-Verlag, Berlin-New York}, date={1980}, pages={xi+377}, isbn={3-540-10369-4}, review={\MR{606196 (82c:55001)}}, } \bib{Eilenberg-Mac Lane}{article}{ author={Eilenberg, Samuel}, author={Mac Lane, Saunders}, title={On the groups $H(\Pi,n)$. II. Methods of computation}, journal={Ann. of Math. (2)}, volume={60}, date={1954}, pages={49--139}, issn={0003-486X}, review={\MR{0065162 (16,391a)}}, } \bib{Heller}{article}{ author={Heller, Alex}, title={On the K\"unneth theorem}, journal={Trans. Amer. Math. Soc.}, volume={98}, date={1961}, pages={450--458}, issn={0002-9947}, review={\MR{0126479 (23 \#A3775)}}, } \bib{Mac Lane slides}{article}{ author={MacLane, Saunders}, title={Slide and torsion products for modules}, journal={Univ. e Politec. Torino. Rend. Sem. Mat.}, volume={15}, date={1955--56}, pages={281--309}, review={\MR{0082488 (18,558b)}}, } \bib{MacLane}{book}{ author={Mac Lane, Saunders}, title={Homology}, edition={1}, note={Die Grundlehren der mathematischen Wissenschaften, Band 114}, publisher={Springer-Verlag, Berlin-New York}, date={1967}, pages={x+422}, review={\MR{0349792 (50 \#2285)}}, } \end{references} \end{document}

\begin{document} \title{Some properties of the inverse error function} \author{Diego Dominici } \address{Department of Mathematics\\ State University of New York at New Paltz\\ 75 S. Manheim Blvd. Suite 9\\ New Paltz, NY 12561-2443\\ USA\\ Phone: (845) 257-2607\\ Fax: (845) 257-3571} \email{[email protected]} \thanks{This work was partially supported by a Provost Research Award from SUNY New Paltz.} \subjclass{Primary 33B20; Secondary 30B10, 34K25} \date{June 4, 2007} \keywords{Inverse error function, asymptotic analysis, discrete ray method, differential-difference equations, Taylor series} \begin{abstract} The inverse of the error function, $\operatorname{inverf}(x),$ has applications in diffusion problems, chemical potentials, ultrasound imaging, etc. We analyze the derivatives $\left. \frac{d^{n}}{dz^{n}} \operatorname*{inverf}\left( z\right) \right\vert _{z=0}$, as $n\rightarrow \infty$ using nested derivatives and a discrete ray method. We obtain a very good approximation of $\operatorname{inverf}(x)$ through a high-order Taylor expansion around $x=0$. We give numerical results showing the accuracy of our formulas. \end{abstract} \maketitle \section{Introduction} The error function $\operatorname{erf}(z),$ defined by \[ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\int\limits_{0}^{z}\exp\left( -t^{2}\right) dt, \] occurs widely in almost every branch of applied mathematics and mathematical physics, e.g., probability and statistics \cite{MR0034250}, data analysis \cite{MR999553}, heat conduction \cite{MR0016873}, etc. It plays a fundamental role in asymptotic expansions \cite{MR1429619} and exponential asymptotics \cite{MR990851}. Its inverse, which we will denote by $\operatorname*{inverf}\left( z\right) ,$ \[ \operatorname*{inverf}\left( z\right) =\operatorname{erf}^{-1}(z), \] appears in multiple areas of mathematics and the natural sciences. A few examples include concentration-dependent diffusion problems \cite{MR0071876}, \cite{MR0281322}, solutions to Einstein's scalar-field equations \cite{PhysRevD.51.444}, chemical potentials \cite{MR2166352}, the distribution of lifetimes in coherent-noise models \cite{PhysRevE.59.R2512}, diffusion rates in tree-ring chemistry \cite{MR2142222} and $3D$ freehand ultrasound imaging \cite{san-joseMICCAI03}. Although some authors have studied the function $\operatorname*{inverf}\left( z\right) $ (see \cite{MR1986919} and references therein), little is known about its analytic properties$,$ the major work having been done in developing algorithms for numerical calculations \cite{MR0341812}. Dan Lozier, remarked the need for new techniques in the computation of $\operatorname*{inverf} \left( z\right) $ \cite{MR1393742}. In this paper, we analyze the asymptotic behavior of the derivatives $\left. \frac{d^{n}}{dz^{n}}\operatorname*{inverf}\left( z\right) \right\vert _{z=0}$ for large values of $n,$ using a discrete WKB method \cite{MR1373150}. In Section 2 we present some properties of the derivatives of $\operatorname*{inverf}\left( z\right) $ and review our previous work on nested derivatives. In Section 3 we study a family of polynomials $P_{n}(x)$ associated with the derivatives of $\operatorname*{inverf}\left( z\right) $, which were introduced by L. Carlitz in \cite{MR0153878}. Theorem \ref{theorem} contains our main result on the asymptotic analysis of $P_{n}(x).$ In Section 4 we give asymptotic approximations for $\left. \frac{d^{n}}{dz^{n} }\operatorname*{inverf}\left( z\right) \right\vert _{z=0}$ and some numerical results testing the accuracy of our formulas. \section{Derivatives} Let us denote the function $\operatorname*{inverf}\left( z\right) $ by $\mathfrak{I}(z)$ and its derivatives by \begin{equation} d_{n}=\left. \frac{d^{n}}{dz^{n}}\operatorname*{inverf}\left( z\right) \right\vert _{z=0},\quad n=0,1,\ldots. \label{dn} \end{equation} Since $\operatorname{erf}(z)$ tends to $\pm1$ as $z\rightarrow\pm\infty,$ it is clear that $\operatorname*{inverf}\left( z\right) $ is defined in the interval $\left( -1,1\right) $ and has singularities at the end points. \begin{proposition} The function $\mathfrak{I}(z)$ satisfies the nonlinear differential equation \begin{equation} \mathfrak{I}^{\prime\prime}-2\mathfrak{I}\left( \mathfrak{I}^{\prime}\right) ^{2}=0 \label{ODE} \end{equation} with initial conditions \begin{equation} \mathfrak{I}(0)=0,\quad\mathfrak{I}^{\prime}(0)=\frac{\sqrt{\pi}}{2}. \label{d0d1} \end{equation} \end{proposition} \begin{proof} It is clear that $\mathfrak{I}(0)=0,$ since $\operatorname{erf}(0)=0.$ Using the chain rule, we have \[ \mathfrak{I}^{\prime}\left[ \operatorname{erf}(z)\right] =\frac {1}{\operatorname{erf}^{\prime}(z)}=\frac{\sqrt{\pi}}{2}\exp\left\{ \mathfrak{I}^{2}\left[ \operatorname{erf}(z)\right] \right\} \] and therefore \begin{equation} \mathfrak{I}^{\prime}=\frac{\sqrt{\pi}}{2}\exp\left( \mathfrak{I}^{2}\right) . \label{I'} \end{equation} Setting $z=0$ we get $\mathfrak{I}^{\prime}(0)=\frac{\sqrt{\pi}}{2}$ and taking the logarithmic derivative of (\ref{I'}) the result follows. \end{proof} To compute higher derivatives of $\mathfrak{I}(z),$ we begin by establishing the following corollary. \begin{corollary} The function $\mathfrak{I}(z)$ satisfies the nonlinear differential-integral equation \begin{equation} \mathfrak{I}^{\prime}(z)\int\limits_{0}^{z}\mathfrak{I}(t)dt=-\frac{1} {2}+\frac{1}{\sqrt{\pi}}\mathfrak{I}^{\prime}(z). \label{int diff} \end{equation} \end{corollary} \begin{proof} Rewriting (\ref{ODE}) as \[ \mathfrak{I}=\frac{1}{2}\frac{\mathfrak{I}^{\prime\prime}}{\left( \mathfrak{I}^{\prime}\right) ^{2}} \] and integrating, we get \[ \int\limits_{0}^{z}\mathfrak{I}(t)dt=\frac{1}{2}\left[ -\frac{1} {\mathfrak{I}^{\prime}(z)}+\frac{1}{\mathfrak{I}^{\prime}(0)}\right] =\frac{1}{2}\left[ -\frac{1}{\mathfrak{I}^{\prime}(z)}+\frac{2}{\sqrt{\pi} }\right] \] and multiplying by $\mathfrak{I}^{\prime}(z)$ we obtain (\ref{int diff}). \end{proof} \begin{proposition} The derivatives of $\mathfrak{I}(z)$ satisfy the nonlinear recurrence \begin{equation} d_{n+1}=\sqrt{\pi}\sum\limits_{k=0}^{n-1}\binom{n}{k+1}d_{k}d_{n-k},\quad n=1,2,\ldots\label{recurrence} \end{equation} with $d_{0}=0$ and $d_{1}=\frac{\sqrt{\pi}}{2}.$ \end{proposition} \begin{proof} Using \[ \mathfrak{I}(z)=\sum\limits_{n=0}^{\infty}d_{n}\frac{z^{n}}{n!} \] and $d_{1}=\frac{\sqrt{\pi}}{2}$ in (\ref{int diff}), we have \[ \left[ \frac{\sqrt{\pi}}{2}+\sum\limits_{n=1}^{\infty}d_{n+1}\frac{z^{n}} {n!}\right] \left[ \sum\limits_{n=1}^{\infty}d_{n-1}\frac{z^{n}}{n!} -\frac{1}{\sqrt{\pi}}\right] =-\frac{1}{2} \] or \[ \frac{\sqrt{\pi}}{2}\sum\limits_{n=1}^{\infty}d_{n-1}\frac{z^{n}}{n!} +\sum\limits_{n=2}^{\infty}\left[ \sum\limits_{k=0}^{n-2}\binom{n}{k+1} d_{k}d_{n-k}\right] \frac{z^{n}}{n!}-\frac{1}{\sqrt{\pi}}\sum\limits_{n=1} ^{\infty}d_{n+1}\frac{z^{n}}{n!}=0. \] Comparing powers of $z^{n},$ we get \[ \frac{\sqrt{\pi}}{2}d_{n-1}+\sum\limits_{k=0}^{n-2}\binom{n}{k+1}d_{k} d_{n-k}-\frac{1}{\sqrt{\pi}}d_{n+1}=0 \] or \[ \sum\limits_{k=0}^{n-1}\binom{n}{k+1}d_{k}d_{n-k}-\frac{1}{\sqrt{\pi}} d_{n+1}=0. \] \end{proof} Although one could use (\ref{recurrence}) to compute the higher derivatives of $\operatorname*{inverf}\left( z\right) ,$ the nonlinearity of the recurrence makes it hard to analyze the asymptotic behavior of $d_{n}$ as $n\rightarrow \infty.$ Instead, we shall use an alternative technique that we developed in \cite{MR2031140} and we called the method of "nested derivatives". The following theorem contains the main result presented in \cite{MR2031140}. \begin{theorem} Let \[ H(x)=h^{-1}(x),\quad f(x)=\frac{1}{h^{\prime}(x)},\quad z_{0}=h(x_{0}),\text{ \ }\ \left\vert f(x_{0})\right\vert \in\left( 0,\infty\right) . \] $\ $ \ Then, \[ H(z)=x_{0}+f(x_{0})\sum\limits_{n=1}^{\infty}\mathfrak{D}^{n-1}[f]\,(x_{0} )\frac{(z-z_{0})^{n}}{n!}, \] where we define $\mathfrak{D}^{n}[f]$\thinspace$(x),$ \textit{the n}$^{th} $\textit{ nested derivative} of the function $f(x),$ by $\mathfrak{D} ^{0}[f]\,(x)=1$ and \begin{equation} \mathfrak{D}^{n+1}[f]\,(x)=\frac{d}{dx}\left[ f(x)\times\mathfrak{D} ^{n}[f]\,(x)\right] ,\quad n=0,1,\ldots. \label{nested} \end{equation} \end{theorem} The following proposition makes the computation of $\mathfrak{D} ^{n-1}[f]\,(x_{0})$ easier in some cases. \begin{proposition} Let \begin{equation} \mathfrak{D}^{n}[f]\,(x)=\sum\limits_{k=0}^{\infty}A_{k}^{n}\frac {(x-x_{0})^{k}}{k!},\qquad f(x)=\sum\limits_{k=0}^{\infty}B_{k}\frac {(x-x_{0})^{k}}{k!}. \label{AB} \end{equation} Then, \begin{equation} A_{k}^{n+1}=\left( k+1\right) \sum\limits_{j=0}^{k+1}A_{k+1-j}^{n}B_{j}. \label{A} \end{equation} \end{proposition} \begin{proof} From (\ref{AB}) we have \begin{equation} f(x)\mathfrak{D}^{n}[f]\,(x)=\sum\limits_{k=0}^{\infty}\alpha_{k}^{n} \frac{(x-x_{0})^{k}}{k!}, \label{Dfxf} \end{equation} with \begin{equation} \alpha_{k}^{n}=\sum\limits_{j=0}^{k}A_{k-j}^{n}B_{j}. \label{alpha} \end{equation} Using (\ref{AB}) and (\ref{Dfxf}) in (\ref{nested}), we obtain \[ \sum\limits_{k=0}^{\infty}A_{k}^{n+1}(x-x_{0})^{k}=\frac{d}{dx}\sum \limits_{k=0}^{\infty}\alpha_{k}^{n}(x-x_{0})^{k}=\sum\limits_{k=0}^{\infty }\left( k+1\right) \alpha_{k+1}^{n}(x-x_{0})^{k} \] and the result follows from (\ref{alpha}). \end{proof} To obtain a linear relation between successive nested derivatives, we start by establishing the following lemma. \begin{lemma} Let \begin{equation} g_{n}\left( x\right) =\frac{\mathfrak{D}^{n}[f]\,(x)}{f^{n}\left( x\right) }. \label{gn} \end{equation} Then, \begin{equation} g_{n+1}(x)=g_{n}^{\prime}\left( x\right) +\left( n+1\right) \frac {f^{\prime}(x)}{f(x)}g_{n}\left( x\right) ,\quad n=0,1,\ldots. \label{ddnested} \end{equation} \end{lemma} \begin{proof} Using (\ref{nested}) in (\ref{gn}), we have \begin{gather*} g_{n+1}\left( x\right) =\frac{\mathfrak{D}^{n+1}[f]\,(x)}{f^{n+1}\left( x\right) }=\frac{\frac{d}{dx}\left[ f(x)\times\mathfrak{D}^{n} [f]\,(x)\right] }{f^{n+1}\left( x\right) }\\ =\frac{\frac{d}{dx}\left[ g_{n}\left( x\right) f^{n+1}\left( x\right) \right] }{f^{n+1}\left( x\right) }=\frac{g_{n}^{\prime}\left( x\right) f^{n+1}\left( x\right) +g_{n}\left( x\right) (n+1)f^{n}\left( x\right) f^{\prime}(x)}{f^{n+1}\left( x\right) } \end{gather*} and the result follows. \end{proof} \begin{corollary} Let \[ H(x)=h^{-1}(x),\quad f(x)=\frac{1}{h^{\prime}(x)},\quad z_{0}=h(x_{0}),\text{ \ }\ \left\vert f(x_{0})\right\vert \in\left( 0,\infty\right) . \] Then,$\ $ \begin{equation} \frac{d^{n}H}{dz^{n}}(z_{0})=\left[ f(x_{0})\right] ^{n}g_{n-1}(x_{0}),\quad n=1,2,\ldots. \label{deriv} \end{equation} \end{corollary} For the function $h(x)=\operatorname{erf}(z),$ we have \begin{equation} f(x)=\frac{1}{h^{\prime}(x)}=\frac{\sqrt{\pi}}{2}\exp\left( x^{2}\right) , \label{f} \end{equation} and setting $x_{0}=0$ we obtain $z_{0}=\operatorname{erf}(0)=0.$ Using the Taylor series \[ \frac{\sqrt{\pi}}{2}\exp\left( x^{2}\right) =\frac{\sqrt{\pi}}{2} \sum\limits_{k=0}^{\infty}\frac{x^{2k}}{k!} \] in (\ref{A}), we get \[ A_{k}^{n+1}=\frac{\sqrt{\pi}}{2}\left( k+1\right) \sum\limits_{j=0} ^{\left\lfloor \frac{k+1}{2}\right\rfloor }\frac{A_{k+1-2j}^{n}}{j!}, \] with $A_{k}^{n}$ defined in (\ref{AB}). Using (\ref{f}) in (\ref{ddnested}), we have \begin{equation} g_{n+1}(x)=g_{n}^{\prime}\left( x\right) +2\left( n+1\right) xg_{n}\left( x\right) ,\quad n=0,1,\ldots, \label{g} \end{equation} while (\ref{deriv}) gives \begin{equation} d_{n}=\left( \frac{\sqrt{\pi}}{2}\right) ^{n}g_{n-1}(0),\quad n=1,2,\ldots. \label{deriv2} \end{equation} In the next section we shall find an asymptotic approximation for a family of polynomials closely related to $g_{n}\left( x\right) $. \section{The polynomials $P_{n}(x)$} We define the polynomials $P_{n}(x)$ by $P_{0}(x)=1$ and \begin{equation} P_{n}(x)=g_{n}\left( \frac{x}{\sqrt{2}}\right) 2^{-\frac{n}{2}}. \label{Pngn} \end{equation} \begin{equation} P_{n+1}(x)=P_{n}^{\prime}(x)+\left( n+1\right) xP_{n}(x), \label{diffdiff} \end{equation} The first few $P_{n}\left( x\right) $ are \[ P_{1}(x)=x,\quad P_{2}(x)=1+2x^{2},\quad P_{3}(x)=7x+6x^{3},~\ldots~. \] The following propositions describe some properties of $P_{n}\left( x\right) .$ \begin{proposition} Let \begin{equation} P_{n}(x)=\sum\limits_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }C_{k} ^{n}x^{n-2k}, \label{Pcnk} \end{equation} where $\left\lfloor \cdot\right\rfloor $ denotes the integer part function. Then, \begin{equation} C_{0}^{n}=n! \label{C01} \end{equation} and \begin{equation} C_{k}^{n}=n!\sum\limits_{j_{k}=0}^{n-1}\sum\limits_{j_{k-1}=0}^{j_{k}-1} \cdots\sum\limits_{j_{1}=0}^{j_{2}-1}\prod_{i=1}^{k}\frac{j_{i}-2i+2}{j_{i} +1},\quad k=1,\ldots,\left\lfloor \frac{n}{2}\right\rfloor . \label{cnk} \end{equation} \end{proposition} \begin{proof} Using (\ref{Pcnk}) in (\ref{diffdiff}) we have \begin{gather*} \sum\limits_{0\leq2k\leq n+1}^{{}}C_{k}^{n+1}x^{n+1-2k}=\sum\limits_{0\leq 2k\leq n}^{{}}C_{k}^{n}\left( n-2k\right) x^{n-2k-1}+\sum\limits_{0\leq 2k\leq n}^{{}}\left( n+1\right) C_{k}^{n}x^{n+1-2k}\\ =\sum\limits_{2\leq2k\leq n+2}^{{}}C_{k-1}^{n}\left( n-2k+2\right) x^{n+1-2k}+\sum\limits_{0\leq2k\leq n}^{{}}\left( n+1\right) C_{k} ^{n}x^{n+1-2k}. \end{gather*} Comparing coefficients in the equation above, we get \begin{equation} C_{0}^{n+1}=C_{0}^{n}, \label{C0} \end{equation} \begin{equation} C_{k}^{n+1}=\left( n-2k+2\right) C_{k-1}^{n}+\left( n+1\right) C_{k} ^{n},\quad k=1,\ldots,\left\lfloor \frac{n}{2}\right\rfloor \label{Cn+1} \end{equation} and for $n=2m-1,$ \[ C_{m}^{2m}=C_{m-1}^{2m-1},\quad m=1,2,\ldots. \] From (\ref{C0}) we immediately conclude that $C_{0}^{n}=n!,$ while (\ref{Cn+1}) gives \begin{equation} C_{k}^{n}=n!\sum\limits_{j=0}^{n-1}\frac{j-2k+2}{\left( j+1\right) !} C_{k-1}^{j},\quad n,k\geq1. \label{Cnk1} \end{equation} Setting $k=1$ in (\ref{Cnk1}) and using (\ref{C01}), we have \begin{equation} C_{1}^{n}=n!\sum\limits_{j=0}^{n-1}\frac{j}{\left( j+1\right) !}C_{0} ^{j}=n!\sum\limits_{j=0}^{n-1}\frac{j}{j+1}. \label{C1n} \end{equation} Similarly, setting $k=2$ in (\ref{Cnk1}) and using (\ref{C1n}), we get \[ C_{2}^{n}=n!\sum\limits_{j=0}^{n-1}\frac{j-2}{\left( j+1\right) !}\left[ j!\sum\limits_{i=0}^{j-1}\frac{i}{i+1}\right] =n!\sum\limits_{j=0}^{n-1} \sum\limits_{i=0}^{j-1}\frac{j-2}{j+1}\frac{i}{i+1} \] and continuing this way we obtain (\ref{cnk}). \end{proof} \begin{proposition} The zeros of the polynomials $P_{n}(x)$ are purely imaginary for $n\geq1.$ \end{proposition} \begin{proof} For $n=1$ the result is obviously true. Assuming that it is true for $n$ and that $P_{n}(x)$ is written in the form \begin{equation} P_{n}(x)=n! {\displaystyle\prod\limits_{k=1}^{n}} (z-z_{k}),\quad\operatorname{Re}(z_{k})=0,\quad1\leq k\leq n, \label{product} \end{equation} we have two possibilities for $z^{\ast},$ with$\ P_{n+1}(z^{\ast})=0$: \begin{enumerate} \item $z^{\ast}=z_{k}$, for some $1\leq k\leq n.$ In this case, $\operatorname{Re}(z^{\ast})=0$ and the proposition is proved. \item $z^{\ast}\neq z_{k}$, for all $1\leq k\leq n$. From (\ref{diffdiff}) and (\ref{product}) we get \[ \frac{P_{n+1}(x)}{P_{n}(x)}=\frac{d}{dx}\ln\left[ P_{n}(x)\right] +(n+1)x= {\displaystyle\sum\limits_{k=1}^{n}} \frac{1}{z-z_{k}}+(n+1)x. \] Evaluating at $z=z^{\ast},$ we obtain \[ 0= {\displaystyle\sum\limits_{k=1}^{n}} \frac{1}{z^{\ast}-z_{k}}+(n+1)z^{\ast} \] and taking $\operatorname{Re}(\bullet),$ we have \begin{gather*} 0=\operatorname{Re}\left[ {\displaystyle\sum\limits_{k=1}^{n}} \frac{1}{z^{\ast}-z_{k}}+(n+1)z^{\ast}\right] \\ = {\displaystyle\sum\limits_{k=1}^{n}} \frac{\operatorname{Re}\left( z^{\ast}-z_{k}\right) }{\left\vert z^{\ast }-z_{k}\right\vert ^{2}}+(n+1)\operatorname{Re}(z^{\ast})=\operatorname{Re} (z^{\ast})\left[ {\displaystyle\sum\limits_{k=1}^{n}} \frac{1}{\left\vert z^{\ast}-z_{k}\right\vert ^{2}}+n+1\right] \end{gather*} which implies that \ $\operatorname{Re}(z^{\ast})=0.$ \end{enumerate} \end{proof} \subsection{Asymptotic analysis of $P_{n}(x)$} We first consider solutions to (\ref{diffdiff}) of the form \begin{equation} P_{n}(x)=n!A^{\left( n+1\right) }(x), \label{Pn1} \end{equation} with $x>0.$ Replacing (\ref{Pn1}) in (\ref{diffdiff}) and simplifying the resulting expression, we obtain \[ A^{2}(x)=A^{\prime}(x)+xA(x), \] with solution \begin{equation} A(x)=\exp\left( -\frac{x^{2}}{2}\right) \left[ C-\sqrt{\frac{\pi}{2} }\operatorname{erf}\left( \frac{x}{\sqrt{2}}\right) \right] ^{-1}, \label{f1} \end{equation} for some constant $C.$ Note that (\ref{Pn1}) is not an exact solution of (\ref{diffdiff}), since it does not satisfy the initial condition $P_{0}(x)=1.$ To determine $C$ in(\ref{f1}), we observe from (\ref{C01}) that \begin{equation} P_{n}(x)\sim n!x^{n},\quad x\rightarrow\infty. \label{Pn4} \end{equation} As $x\rightarrow\infty,$ we get from (\ref{f1}) \[ \ln\left[ A(x)\right] \sim-\frac{x^{2}}{2}-\ln\left( C-\sqrt{\frac{\pi}{2} }\right) +\frac{\exp\left( -\frac{x^{2}}{2}\right) }{\left( C-\sqrt {\frac{\pi}{2}}\right) x},\quad x\rightarrow\infty, \] which is inconsistent with (\ref{Pn4}) unless $C=\sqrt{\frac{\pi}{2}}.$ In this case, we have \begin{equation} A(x)\sim x+\frac{1}{x},\quad x\rightarrow\infty, \label{A1} \end{equation} matching (\ref{Pn4}). Thus, \begin{equation} A(x)=\sqrt{\frac{2}{\pi}}\exp\left( -\frac{x^{2}}{2}\right) \left[ 1-\operatorname{erf}\left( \frac{x}{\sqrt{2}}\right) \right] ^{-1}. \label{psi2} \end{equation} Since (\ref{Pn1}) and (\ref{A1}) give \[ P_{n}(x)\sim n!x^{n+1},\quad x\rightarrow\infty, \] instead of (\ref{Pn4}), we need to consider \begin{equation} P_{n}(x)=n!A^{\left( n+1\right) }(x)B(x,n). \label{Pn2} \end{equation} Replacing (\ref{Pn2}) in (\ref{diffdiff}) and simplifying, we get \[ B(x,n+1)=B(x,n)+\frac{1}{A(x)(n+1)}\frac{\partial B}{\partial x}(x,n). \] Using the approximation \[ B(x,n+1)=B(x,n)+\frac{\partial B}{\partial n}(x,n)+\frac{1}{2}\frac {\partial^{2}B}{\partial n^{2}}(x,n)+\cdots, \] we obtain \[ \frac{\partial B}{\partial n}=\frac{1}{A(x)(n+1)}\frac{\partial B}{\partial x}, \] whose solution is \begin{equation} B(x,n)=F\left[ \frac{n+1}{1-\operatorname{erf}\left( \frac{x}{\sqrt{2} }\right) }\right] , \label{B1} \end{equation} for some function $F(u).$ Matching (\ref{Pn2}) with (\ref{Pn4}) requires \begin{equation} B(x,n)\sim\frac{1}{x},\quad x\rightarrow\infty. \label{B2} \end{equation} Since in the limit as $x\rightarrow\infty,$ with $n$ fixed we have \[ \ln\left[ \frac{n+1}{1-\operatorname{erf}\left( \frac{x}{\sqrt{2}}\right) }\right] \sim\frac{x^{2}}{2}, \] (\ref{B1})-(\ref{B2}) imply \[ F(u)=\frac{1}{\sqrt{2\ln(u)}}. \] Therefore, for $x>0,$ \begin{equation} P_{n}(x)\sim n!\Phi\left( x,n\right) ,\quad n\rightarrow\infty, \label{Pn5} \end{equation} with \[ \Phi\left( x,n\right) =\left[ \sqrt{\frac{2}{\pi}}\frac{\exp\left( -\frac{x^{2}}{2}\right) }{1-\operatorname{erf}\left( \frac{x}{\sqrt{2} }\right) }\right] ^{n+1}\left[ 2\ln\left( \frac{n+1}{1-\operatorname{erf} \left( \frac{x}{\sqrt{2}}\right) }\right) \right] ^{-\frac{1}{2}}. \] From (\ref{Pcnk}) we know that the polynomials $P_{n}(x)$ satisfy the reflection formula \begin{equation} P_{n}(-x)=\left( -1\right) ^{n}P_{n}(x). \label{reflection} \end{equation} Using (\ref{reflection}), we can extend (\ref{Pn5}) to the whole real line and write \begin{equation} P_{n}(x)\sim n!\left[ \Phi\left( x,n\right) +\left( -1\right) ^{n} \Phi\left( -x,n\right) \right] ,\quad n\rightarrow\infty. \label{Pn6} \end{equation} In Figure \ref{P10} we compare the values of $P_{10}(x)$ with the asymptotic approximation (\ref{Pn6}). \begin{figure}\label{P10} \end{figure} We see that the approximation is very good, even for small values of $n.$ We summarize our results of this section in the following theorem. \begin{theorem} \label{theorem}Let the polynomials $P_{n}(x)$ be defined by \[ P_{n+1}(x)=P_{n}^{\prime}(x)+\left( n+1\right) xP_{n}(x), \] with $P_{0}(x)=1.$ Then, we have \begin{equation} P_{n}(x)\sim P_{n}(x)\sim n!\left[ \Phi\left( x,n\right) +\left( -1\right) ^{n}\Phi\left( -x,n\right) \right] ,\quad n\rightarrow\infty, \label{Pasympt} \end{equation} where \begin{equation} \Phi\left( x,n\right) =\left[ \sqrt{\frac{2}{\pi}}\frac{\exp\left( -\frac{x^{2}}{2}\right) }{1-\operatorname{erf}\left( \frac{x}{\sqrt{2} }\right) }\right] ^{n+1}\left[ 2\ln\left( \frac{n+1}{1-\operatorname{erf} \left( \frac{x}{\sqrt{2}}\right) }\right) \right] ^{-\frac{1}{2}}. \label{Phi} \end{equation} \end{theorem} \section{Higher derivatives of $\operatorname*{inverf}\left( z\right) $} From (\ref{deriv2}) and (\ref{Pngn}), it follows that \begin{equation} d_{n}=\frac{1}{\sqrt{2}}\left( \sqrt{\frac{\pi}{2}}\right) ^{n} P_{n-1}(0),\quad n=1,2,\ldots, \label{deriv3} \end{equation} where $d_{n}$ was defined in (\ref{dn}). Using Theorem \ref{theorem} in (\ref{deriv3}), we have \[ d_{n}\sim\frac{1}{\sqrt{2}}\left( \sqrt{\frac{\pi}{2}}\right) ^{n} \Phi\left( 0,n-1\right) \left[ 1+\left( -1\right) ^{n-1}\right] , \] as $n\rightarrow\infty.$ Using (\ref{Phi}), we obtain \begin{equation} \frac{d_{n}}{n!}\sim\frac{1}{2n\sqrt{\ln(n)}}\left[ 1+\left( -1\right) ^{n-1}\right] ,\quad n\rightarrow\infty. \label{deriv5} \end{equation} Setting $n=2N+1$ in (\ref{deriv5}), we have \begin{equation} \frac{d_{2N+1}}{\left( 2N+1\right) !}\sim\frac{1}{\left( 2N+1\right) \sqrt{\ln(2N+1)}},\quad N\rightarrow\infty. \label{derivodd} \end{equation} \subsection{Numerical results} In this section we demonstrate the accuracy of the approximation (\ref{deriv5}) and construct a high order Taylor series for $\operatorname*{inverf}\left( x\right) .$ In Figure \ref{compare1} we compare the logarithm of the exact values of $\left. \frac{d^{2n+1} }{dz^{2n+1}}\operatorname*{inverf}\left( x\right) \right\vert _{x=0}$ and our asymptotic formula (\ref{deriv5}). We see that there is a very good agreement, even for moderate values of $n$. \begin{figure}\label{compare1} \end{figure} Using (\ref{recurrence}), we compute the exact values \[ d_{1}=\frac{1}{2}\pi^{\frac{1}{2}},\quad d_{3}=\frac{1}{4}\pi^{\frac{3}{2} },\quad d_{5}=\frac{7}{8}\pi^{\frac{5}{2}},\quad d_{7}=\frac{127}{16} \pi^{\frac{7}{2}},\quad d_{9}=\frac{4369}{32}\pi^{\frac{9}{2}} \] and form the polynomial Taylor approximation \[ T_{9}(x)=\sum\limits_{k=0}^{4}d_{2k+1}\frac{x^{2k+1}}{\left( 2k+1\right) !}. \] In Figure \ref{compare2} we graph $\frac{T_{9}(x)}{\operatorname*{inverf} \left( x\right) }$ \ and \ $\frac{T_{9}(x)+R_{N}(x)}{\operatorname*{inverf} \left( x\right) },$ for $N=10,20,$ where \begin{equation} R_{N}(x)=\sum\limits_{k=5}^{N}\frac{x^{2k+1}}{\left( 2N+1\right) \sqrt {\ln(2N+1)}},\quad N=5,6,\ldots. \label{T1} \end{equation} \begin{figure}\label{compare2} \end{figure} The functions are virtually identical in most of the interval $\left( -1,1\right) $ except for values close to $x=\pm1.$ We show the differences in detail in Figure \ref{compare3}. Clearly, the additional terms in $R_{20}(x)$ give a far better approximation for $x\simeq1.$ \begin{figure}\label{compare3} \end{figure} In the table below we compute the exact value of and optimal asymptotic approximation to $\operatorname*{inverf}\left( x\right) $ for some $x$: \[ \begin{tabular} [c]{|c|c|c|c|}\hline $x$ & $\operatorname*{inverf}\left( x\right) $ & $T_{9}(x)+R_{N}(x)$ & $N$\\\hline $0.7$ & $.732869$ & $.732751$ & $6$\\\hline $0.8$ & $.906194$ & $.905545$ & $7$\\\hline $0.9$ & $1.16309$ & $1.16274$ & $11$\\\hline $0.99$ & $1.82139$ & $1.82121$ & $57$\\\hline $0.999$ & $2.32675$ & $2.32676$ & $423$\\\hline $0.9999$ & $2.75106$ & $2.75105$ & $3685$\\\hline \end{tabular} \ \ . \] Clearly, (\ref{T1}) is still valid for $x\rightarrow1,$ but at the cost of having to compute many terms in the sum. In this region it is better to use the formula \cite{MR1986919} \[ \operatorname*{inverf}\left( x\right) \sim\sqrt{\frac{1}{2} \operatorname*{LW}\left[ \frac{2}{\pi\left( x-1\right) ^{2}}\right] },\quad x\rightarrow1^{-}, \] where $\operatorname*{LW}(\cdot)$ denotes the Lambert-W function \cite{MR1414285}, which satisfies \[ \operatorname*{LW}(x)\exp\left[ \operatorname*{LW}(x)\right] =x. \] \newcommand{\etalchar}[1]{$^{#1}$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } \providecommand{\MRhref}[2]{ \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \end{document}

\begin{document} \title{The Moduli of Klein Covers of Curves} \author{Charles Siegel} \address[Charles Siegel]{Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, the University of Tokyo} \email[Charles Siegel]{[email protected]} \urladdr{http://db.ipmu.jp/member/personal/2754en.html} \date{} \subjclass{} \keywords{} \begin{abstract} We study the moduli space $\KM{g}$ of Klein four covers of genus $g$ curves and its natural compactification. This requires the construction of a related space which has a choice of basis for the Klein four group. This space has the interesting property that the two components intersect along a component of the boundary. Further, we carry out a detailed analysis of the boundary, determining components, degrees of the components over their images in $\overline{\mathcal{M}_g}$, and computing the canonical divisor of $\overline{\KM{g}}$. \end{abstract} \maketitle \tableofcontents \section*{Introduction} Ever since the nineteenth century, unramified double covers have been an essential tool for studying curves. They correspond to square roots of the trivial line bundle which form a group. This formulation can be used to study surface groups and the mapping class group, and also theta characteristics, the square roots of the canonical line bundle. There are intricate relationships between double covers and theta characteristics, in particular, and the difference between them only become completely clear after passing to moduli. The moduli of double covers of curves has two components, one that is isomorphic to $\mathcal{M}_g$, where the double cover is a disconnected union of two copies of the base curve, and one where the double cover is nontrivial, traditionally denoted by $\mathcal{R}_g$. While the moduli of theta characteristics also has two components, neither one is isomorphic to $\mathcal{M}_g$. The components, $\mathcal{S}_g^\pm$, correspond to whether the theta characteristic has an even or odd dimensional space of global sections.\footnote{The notations come from the French for covering, rev\^etement and from the fact that curves with theta characteristics are often called spin curves, due to connections with the quantum mechanical notion of spin.} Another, slightly more subtle, connection between the two moduli spaces is that the theta characteristics on a curve correspond to quadratic forms on the ($\mathbb{F}_2$-vector space of) points of order two on the curve. The quadratic form is given by, if $L$ is a theta characteristic, $\mu\mapsto h^0(L\otimes \mu)-h^0(L)\mod 2$, and induces a skew-symmetric bilinear form on the points of order two. This bilinear form is independent of the theta characteristic chosen and is called the Weil pairing. The Weil pairing, however, is really an invariant of a Klein four subgroup of the Jacobian, as $\langle \mu,\nu\rangle=\langle\mu,\mu+\nu\rangle$, and in fact, if $\{0,\mu_1,\mu_2,\mu_3\}$ is a Klein four subgroup of $\mathcal{J}(C)[2]$, then the Weil pairing on the group has the value $h^0(L\otimes\mathscr{O}_C)+h^0(L\otimes \mu_1)+h^0(L\otimes \mu_2)+h^0(L\otimes \mu_3)\mod 2$, which is manifestly symmetric in the group elements, suggesting that it could be clarified by studying the moduli of Klein covers. The boundaries of these moduli spaces have been studied in detail, and the fibers of the natural projection to $\overline{\mathcal{M}_g}$ have been made very explicit. The approach originates in {\cite{MR1082361}} for $\overline{\mathcal{S}_g}^{\pm}$ and is pushed through in detail in {\cite{MR2007379}}, and adapted to $\overline{\mathcal{R}_g}$ in {\cite{MR2117416}}. More recently, this approach has been adapted to proving that pluricanonical forms extend to both $\overline{\mathcal{S}_g}^{\pm}$ {\cite{MR2551759}} and to $\overline{\mathcal{R}_g}$ {\cite{MR2639318}}, bringing the study of the birational geometry of these spaces into reach. \subsection*{This paper} In this paper, we extend the description of the boundary and pluricanonical forms to the moduli of Klein covers of curves. This, however, is difficult to do directly, so instead we introduce an intermediate moduli space, $\overline{\ZZM{g}}$, of pairs of Prym curves and study it, then use the relationship between it and the moduli of Klein covers $\overline{\KM{g}}$ to prove the results on this space. In fact, we use this relationship to define $\overline{\KM{g}}$. In section 1 of this paper, we recall relevant facts about double covers. In particular, the classification of points in the fiber over a stable curve from {\cite{MR2117416}}, and the relationship between two competing notations for the components of the boundary of $\overline{\mathcal{R}_g}$, used in, for instance, {\cite{MR903385}} and \cite{MR2976944}. In section 2, we initiate the study of $\overline{\ZZM{g}}$, focusing on the interior. We construct the space and show that there are two connected components, corresponding to Weil pairing 0, which was studied in \cite{1302.5946} under the notation $\mathcal{R}^2\mathcal{M}_g$, and Weil pairing 1, and the degree of each component over $\mathcal{M}_g$, reproducing a result in {\cite{1206.5498}}, which holds in the degenerate case where the dihedral group is only four elements. In section 3, we analyze the boundary of $\overline{\ZZM{g}}$ in detail, describing the fibers over $\overline{\mathcal{M}_g}$, then identify the boundary components and determine how many objects in each fiber lie in each component. Here, we note an interesting fact. Although $\ZZM{g}$ is an unramified covering of $\mathcal{M}_g$ and has two components, the natural compactification $\overline{\ZZM{g}}$ is in fact connected, and the boundaries of the two components intersect nicely along a single component. In section 4, we proceed to analyzing $\overline{\KM{g}}$ and its boundary. We do so by showing that the group action of $\PSL_2(\mathbb{F}_2)$ on $\ZZM{g}$ extends to the boundary of each of the two components separately, identifying several components of $\partial\overline{\ZZM{g}}$. This allows us to describe the (much simple) boundary of $\overline{\KM{g}}$and to show that the natural map $\overline{\KM{g}}\to \overline{\mathcal{M}_g}$ is simply ramified along a single boundary component. In the last section, we follow \cite{MR2639318}, \cite{MR2551759} and \cite{MR664324}, to extend the pluricanonical forms from the smooth locus to an arbitrary resolution of singularities. The main tool in this is the Reid--Shepherd-Barron--Tai criterion \cite{MR605348,MR763023}. We conclude with a slope criterion for $\overline{\KM{g}}^i$ to be of general type analagous to similar results for $\overline{\mathcal{M}}_g$ and $\overline{\mathcal{R}_g}$: \begin{reptheorem}{maintheorem} For any $g$, $\overline{\KM{g}}^i$ has general type if there exists a single effective divisor $D\equiv a\lambda-\sum_T b_{\Delta_T}\Delta_T$ where $T$ runs over all boundary components, such that all the ratios $\frac{a}{b_T}$ are less than $\frac{13}{2}$ and the ratios $\frac{a}{b_{II,III,III}}$, $\frac{a}{b_{1,g-1,1:g-1}}$, $\frac{a}{b_{1,1,1}}$, $\frac{a}{b_{g-1,g-1,g-1}}$, $\frac{a}{b_{1,1:g-1,1:g-1}}$, $\frac{a}{b_{g-1,1:g-1,1:g-1}}$, and $\frac{a}{b_{1:g-1,1:g-1,1:g-1}}$ are less than $\frac{13}{3}$. \end{reptheorem} \subsection*{Acknowledgments} I would like to thank Gavril Farkas, for suggesting that the birational geometry of this space might be interesting, as well as for conversations on the relationship between the Weil pairing, theta characteristics and Klein four curves. Also I would like to thank Angela Gibney, Joe Harris, Tyler Kelly and Angela Ortega for helpful discussions on the moduli of curves, coverings, the Weil pairing and birational geometry and Amir Aazami and Jesse Wolfson for comments on an earlier draft. This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. \section{Background} In this section, we will recall relevant facts about double covers and points of order two on curves. For $C$ a smooth projective curve over $\mathbb{C}$, we denote by $\mathcal{J}(C)$ the group of line bundles of degree $0$ on $C$. It has a natural subgroup $\mathcal{J}(C)[2]$ consisting of the elements whose square is trivial. \begin{lemma} \label{lemma:prymdefs} The following data are equivalent: \begin{enumerate} \item $\tilde{C}\to C$ an irreducible \'etale double cover, \item $\mu\in \mathcal{J}(C)[2]$ nonzero, and \item $\tilde{C}\in\mathcal{M}_{2g-1}$ with $\iota:\tilde{C}\to\tilde{C}$ a fixed point free involution. \end{enumerate} \end{lemma} \begin{proof} Given a point of order two, we get an unramified double cover by looking at $\underline{\Spec}(\mathscr{O}_C\oplus \mu)$. Conversely, given a double cover, there is a single point of order two that pulls back to zero. To get between 1 and 3, we note that $C\cong \tilde{C}/\iota$. \end{proof} By Lemma \ref{lemma:prymdefs}, we have an equivalence between $\mathcal{J}(C)[2]$ and the set of \'etale double covers, with 0 corresponding to the trivial double cover. This induces a group structure on double covers, and if $\tilde{C}_\mu,\tilde{C}_\nu$ correspond to $\mu,\nu$, then $\tilde{C}_{\mu+\nu}$ is given by $\tilde{C}_\mu\times_C\tilde{C}_\nu/(\iota_\mu,\iota_\nu)$. \begin{definition}[quasistable curve] A genus $g\geq 2$ curve $X$ is \emph{quasistable} if every smooth rational component has at least two nodes and no two of these components intersect. We call the stable curve $C$ obtained by removing these rational components and gluing the nodes together the stabilization of $X$, and the nodes of $C$ obtained this way are the exceptional nodes and the rational components of $X$ over them exceptional components. \end{definition} We define the nonexceptional curve to be the union of the nonexceptional components and denote it by $X_{ne}$. \begin{definition}[Prym curve] A \emph{Prym curve} is a triple $(X,\eta,\beta)$ where $X$ is quasistable, $\eta\in\mathcal{J}(X)$ such that for all exceptional components, $E$, we have $\eta_E\cong\mathscr{O}_E(1)$, and $\beta:\eta^{\otimes 2}\to\mathscr{O}_X$ is a homomorphism that is generically nonzero on each nonexceptional component. \end{definition} \begin{remark} In the notation we will use for other objects, a Prym curve would be called a $\mathbb{Z}/2\mathbb{Z}$ curve, but we will continue to refer to them as Prym curves. \end{remark} \begin{definition}[Isomorphism of Prym Curves] An isomorphism of Prym curves $(X,\eta,\beta)$ and $(X',\eta',\beta')$ is an isomorphism $\sigma:X\to X'$ such that there exists an isomorphism $\tau:\sigma^*(\eta')\to \eta$ such that the diagram commutes: \begin{center} \leavevmode \begin{xy} (0,0)*+{\sigma^*(\mathscr{O}_{X'})}="a"; (20,0)*+{\mathscr{O}_X}="b"; (0,20)*+{\sigma^*(\eta')^{\otimes 2}}="c"; (20,20)*+{\eta^{\otimes 2}}="d"; {\ar^{\sim} "a";"b"}; {\ar^{\tau^{\otimes 2}} "c";"d"}; {\ar_{\sigma^*(\beta')} "c";"a"}; {\ar^{\beta} "d";"b"}; \end{xy} \end{center} \end{definition} Note that the definition of isomorphism of Prym curves does not depend on what $\tau$ is chosen, only on $\sigma$. Any automorphism of a Prym curve which induces the identity on the stable model of $X$ will be called \textit{inessential}. The group of automorphisms will be denoted by $\Aut(X,\eta,\beta)$ and the inessential automorphisms will be $\Aut_0(X,\eta,\beta)$. For the rest of this paper, for any curve, denote by $\nu$ the normalization morphism for a curve, and denote by $g^\nu$ the geometric genus of the normalization: \begin{proposition}[{\cite[Proposition 11]{MR2117416}}] Let $X$ be a quasistable curve, $Z$ its stable model, $\Gamma_Z$ the dual graph of $Z$, and $\Delta_X$ the set of nodes not lying under an exceptional curve, and assume further that $\Delta_X^c$ is eulerian. Then there are $2^{2g^\nu+b_1(\Delta_X)}$ Prym curves supported on $X$ and each has multiplicity $2^{b_1(\Gamma_Z)-b_1(\Delta_X)}$ in the fiber of $\mathcal{R}_g\to\mathcal{M}_g$. \end{proposition} We will denote the moduli space of these curves by $\overline{\ZM{g}}$, and we note that it has two components. One, isomorphic to $\overline{\mathcal{M}}_g$ where the Prym curve has $\eta\cong \mathscr{O}_X$ over a stable base, and $\overline{\mathcal{R}}_g$, the nontrivial Prym curves. \begin{remark}[Notation] The boundary components of the Prym moduli space have several competing notations in the literature. For the $2^{2g}$ objects, we always have 1 that is the disconnected double cover, and in this setting, it lies over the stable curve itself. Additionally, Donagi \cite{MR903385} classified the nontrivial points of order two on an irreducible 1-nodal curve in terms of the vanishing cycle $\delta$. In his notation, $\Delta_I$ is the subset with the marked point $\mu$ being equal to $\delta$, $\Delta_{II}$ when $\langle \delta,\mu\rangle=0$ but $\mu\neq\delta$ and $\Delta_{III}$ being when $\langle \delta,\mu\rangle\neq 0$, under the Weil pairing, defined below. Alternately, these three components are denoted by $\Delta_0''$, $\Delta_0'$ and $\Delta_0^{ram}$ by Farkas \cite{MR2639318} and it is noted that $\Delta_{III}=\Delta_0^{ram}$ is precisely the set of Prym curves on the quasistable curve. In the rest of this article, however, we will follow Donagi's notation. Over the 1-nodal reducible curves, the notation agrees, and the components are $\Delta_i$, $\Delta_{g-i}$ and $\Delta_{i:g-i}$, for the Prym curves supported on the component of genus $i$, $g-i$ or both, respectively. \end{remark} \section{The space \texorpdfstring{$\overline{\ZZM{g}}$}{ZMg}} \begin{definition}[Weil Pairing] Let $\mu,\nu\in\mathcal{J}(C)[2]$, and let $\kappa\in\Pic(C)$ such that $\kappa^{\otimes 2}\cong K_C$. Then the Weil pairing on the curve $C$ is given by \[\langle\mu,\nu\rangle=h^0(\kappa)+h^0(\kappa\otimes\mu)+h^0(\kappa\otimes \nu)+h^0(\kappa\otimes\mu\otimes\nu)\mod 2.\] \end{definition} The Weil pairing is bilinear, skew-symmetric, and independent of the choice of $\kappa$, and therefore we can see that $\langle\mu,\nu\rangle=\langle\mu,\mu+\nu\rangle=\langle \nu,\mu+\nu\rangle$, and so it is an invariant of a rank 2 subgroup of $\mathcal{J}(C)[2]$. \begin{lemma} \label{lemma:Z22Mg} The following data are equivalent: \begin{enumerate} \item A curve $C\in\mathcal{M}_g$ along with $\tilde{C}_i\to C$, $i=1,2$ irreducible, nonisomorphic unramified double covers, \item A curve $C\in\mathcal{M}_g$ along with $\mu_1,\mu_2\in \mathcal{J}(C)[2]$, with $\mu_1\neq\mu_2$, and \item A pair of curves $C,D\in \mathcal{M}_{2g-1}$ along with involutions $\sigma_C,\sigma_D$ that act freely on $C,D$ respectively and which are such that $C/\sigma_C$ and $D/\sigma_D$ are isomorphic, but the pairs $(C,\sigma_C)$ and $(D,\sigma_D)$ are not. \end{enumerate} \end{lemma} \begin{proof} This is just an application of Lemma \ref{lemma:prymdefs}. \end{proof} A related, but slightly different result is the following, where we do not choose a basis for the Klein four group. \begin{lemma} \label{lemma:K2^2Mg} The following data are equivalent: \begin{enumerate} \item A curve $C\in \mathcal{M}_g$, and $\tilde{C}\to C$ an \'etale Klein $4$ cover, \item A curve $C\in \mathcal{M}_g$, and $\phi:V_{4}\to \mathcal{J}(C)[2]$ an injective homomorphism, and \item A curve $\tilde{C}\in \mathcal{M}_{4g-3}$ with a free action of $V_{4}$ on $\tilde{C}$. \end{enumerate} \end{lemma} \begin{proof} This is again just an application of Lemma \ref{lemma:prymdefs}. \end{proof} For the rest of this section, we will be working in the case with a basis, and later will return to the basis-free case. \begin{definition}[$\mathbb{Z}_2^2$ curve] A $\mathbb{Z}_2^2$ curve is $(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)$ where $(X_i,\eta_i,\beta_i)$ is a Prym curve for $i=1,2$ and $X_1$ and $X_2$ have isomorphic stabilizations. \end{definition} An isomorphism of $\mathbb{Z}_2^2$ curves is just a pair of isomorphisms of Prym curves that induce the same isomorphism on the stable model. Equivalently, an automorphism can be seen by looking at the quasistable curve with exceptional nodes the union of the exceptional nodes of $X_1$ and $X_2$. From this viewpoint, an isomorphism is an isomorphism of such quasi-stable curves which induces isomorphisms on the pullbacks of $\eta_i$ and $\eta_i'$. Given this, we can see that the moduli space of $\mathbb{Z}_2^2$ curves is just $\overline{\ZM{g}}\times_{\overline{\mathcal{M}}_g}\overline{\ZM{g}}$. It is easy to see that it has at least five components, depending on $\eta_1,\eta_2$. If both are trivial, we have a copy of $\overline{\mathcal{M}}_g$. If one is trivial but the other is nontrivial, then we get two copies of $\mathcal{R}_g$. Additionally, when $\eta_1\cong \eta_2$, we get another copy of $\overline{\mathcal{R}}_g$ leaving the remnant: \begin{definition}[$\overline{\ZZM{g}}$] We denote by $\overline{\ZZM{g}}$ the closure in $\overline{\ZM{g}}\times_{\overline{\mathcal{M}}_g}\overline{\ZM{g}}$ of the locus of $\mathbb{Z}_2^2$ curves with $\eta_1\not\cong\eta_2$ both nontrivial. The locus of smooth curves in $\overline{\ZZM{g}}$ will be denoted by $\ZZM{g}$. \end{definition} Geometrically, the most natural thing to study is the moduli space of Klein four subgroups, with no choice of basis. This, by Lemma \ref{lemma:K2^2Mg} is then the moduli space of Klein four covers of curves. However, it is much easier to construct the space with a choice of basis, which is $\overline{\ZZM{g}}$ (also note that the geometricity of the covers is less clear on the boundary). There is a natural $\PSL_2(\mathbb{F}_2)$ action on $\ZZM{g}$, permuting the ordered bases of the $\mathbb{Z}_2^2\subset \mathcal{J}(C)[2]$. Below, we will extend it to the boundary and construct $\overline{\KM{g}}$, and we will deduce many of its properties from those of $\overline{\ZZM{g}}$. Here, we note that $\Aut(C,\eta_1,\eta_2)$ is just $\Aut(X_1,\eta_1,\beta_1)\times_{\Aut(C)}\Aut(X_2,\eta_2,\beta_2)$ where $(X_i,\eta_i,\beta_i)$ are the two Prym structures. Thus, the results of \cite{MR2117416} apply without difficulty, so we have good behavior of the universal deformation, we have a subgroup of inessential automorphisms, etc. We will recall those facts we need in the last section of the paper. \begin{lemma} \label{DegZZM} The map $\ZZM{g}\to\mathcal{M}_g$ has degree $(2^{2g}-1)(2^{2g}-2)$, thus, so does $\overline{\ZZM{g}}\to\overline{\mathcal{M}_g}$. \end{lemma} \begin{proof} We know that the degree of $\ZM{g}\times_{\mathcal{M}_g}\ZM{g}\to\mathcal{M}_g$ is $2^{4g}$ and that it breaks down as $\mathcal{M}_g\cup\mathcal{R}_g\cup\mathcal{R}_g\cup\mathcal{R}_g\cup\ZZM{g}$. Each component is dominant and equidimensional, so we can compute the degree on $\ZZM{g}$ as $2^{4g}-1-3(2^{2g}-1)=(2^{2g}-1)(2^{2g}-2)$. \end{proof} The space $\ZZM{g}$ is not irreducible. We can see that there must be at least two components because the Weil pairing is deformation invariant; we will write $\ZZM{g}^0$ and $\ZZM{g}^1$ for the two components, with Weil pairing respectively 0 and 1. \begin{lemma} The spaces $\ZZM{g}^0$ and $\ZZM{g}^1$ are both irreducible. \end{lemma} \begin{proof} For any curve $C$ of genus $g$, the action of $\Sp(2g,\mathbb{F}_2)$ on the space $\mathcal{J}(C)[2]\times \mathcal{J}(C)[2]$ has two orbits, pairs of points that are orthogonal and pairs that are nonorthogonal, and this is the monodromy of $\ZZM{g}\to \mathcal{M}_g$. \end{proof} \begin{remark} Although slightly more complex, in the case where we look at points of order $n$ rather than points of order $2$, we get a similar theorem, where the number of components is indexed by $\mathbb{Z}_n$. Similarly, the Klein moduli space, which we will study below, will have components indexed by $\mathbb{Z}_n/\mathbb{Z}_n^\times$. In our case, both of these are $\mathbb{Z}_2$, and we will identify them with $\{0,1\}$. \end{remark} Before moving on to a detailed analysis of the boundary, we compute the degrees of the maps $\ZZM{g}^i\to\mathcal{M}_g$. \begin{proposition} \label{DegZZM01} We have natural maps to $\mathcal{M}_g$ forgetting the points of order two, and their degrees are: \begin{enumerate} \item $\ZZM{g}^0\to \mathcal{M}_g$ has degree $(2^{2g}-1)(2^{2g-1}-2)$ \item $\ZZM{g}^1\to \mathcal{M}_g$ has degree $(2^{2g}-1)2^{2g-1}$ \end{enumerate} \end{proposition} \begin{proof} Fix a smooth curve $C$ of genus $g$. Any element of $\ZZM{g}$ lying over $C$ is of the form $(C,\eta,\eta')$ where $\eta,\eta'\in\mathcal{J}(C)[2]$, which is an $\mathbb{F}_2$ vector space. To be in $\ZZM{g}^0$ they must be orthogonal under the Weil pairing, which is a nondegenerate form. Thus, for each $\eta$, we must choose $\eta'\in\eta\perp$, a hyperplane in $\mathcal{J}(C)[2]$. However, as they are linearly independent, we asset that $\eta'\notin\{0,\eta\}$. Thus, we have $2^{2g}-1$ choices for $\eta$, and given one of those, we have $2^{2g-1}-2$ choices of $\eta'$ satisfying these conditions, which computes the degree of $\ZZM{g}^0\to\mathcal{M}_g$. To determine the degree of $\ZZM{g}^1\to \mathcal{M}_g$, we note that we can again choose $\eta$ freely, and now $\eta'\in \mathcal{J}(C)[2]\setminus \eta^\perp$, giving us the degree claimed. \end{proof} \section{Geometry of the boundary} \begin{proposition} Let $Z$ be a stable curve with dual graph $\Gamma_Z$. The fiber over $Z$ in the $\overline{\ZZM{g}}$ consists of the following objects: \begin{enumerate} \item $(2^{2g^\nu+b_1(\Gamma_Z)}-1)(2^{2g^\nu+b_1(\Gamma_Z)}-2)$ objects of multiplicity 1 where both Prym curves are supported on $Z$. \item If $X$ is quasistable with stabilization $Z$ and $\Delta_X^c$ is Eulerian, then we get $2^{4g^\nu+b_1(\Delta_X)+b_1(\Gamma_Z)}-2^{2g^\nu+b_1(\Delta_X)}$ objects of multiplicity $2^{b_1(\Gamma_Z)-b_1(\Delta_X)}$ supported on each of $(X,Z)$ and $(Z,X)$. \item If $X$ is quasistable with stabilization $Z$ and $\Delta_X^c$ is Eulerian, then we get two types of objects supported on $(X,X)$: \begin{enumerate} \item $2^{4g^\nu+2b_1(\Delta_X)}-2^{2g^\nu+b_1(\Delta_X)}$ objects of multiplicity $2^{2b_1(\Gamma_Z)-2b_1(\Delta_X)}$ with two distinct Prym structures \item $2^{2g^\nu+b_1(\Delta_X)}$ objects of multiplicity $2^{2b_1(\Gamma_Z)-2b_1(\Delta_X)}-2^{b_1(\Gamma_Z)-b_1(\Delta_X)}$ with the same Prym structure. \end{enumerate} \item If $X_1$, $X_2$ are quasistable over $Z$ and $\Delta_{X_1}^c$ and $\Delta_{X_2}^c$ are Eulerian, then there exist $2^{4g^\nu+b_1(\Delta_X)+b_1(\Delta_X)}$ objects of multiplicity $2^{2b_1(\Gamma_Z)-b_1(\Delta_{X_1})-b_1(\Delta_{X_2})}$ on $(X_1,X_2)$ and again on $(X_2,X_1)$. \end{enumerate} \end{proposition} \begin{proof} The fiber is a subscheme of $R_Z\times R_Z$ which is the complement of $M_g\cup R_g\cup R_g\cup R_g$, components which split off completely over smooth base curves. Away from the diagonal, we simply remove anything where one of the components is the trivial line bundle. On the diagonal, the situation is somewhat more intricate. We look at $\overline{\ZM{g}}\times_{\overline{\mathcal{M}_g}}\overline{\ZM{g}}\setminus(\overline{\mathcal{M}_g}\cup\overline{\ZM{g}}\cup\overline{\ZM{g}}\cup\overline{\ZM{g}})$ and then take the closure. This leaves us with some points on the diagonal, which we can see because on any degeneration, there will be classes of Prym curves that will degenerate to the same thing, which is seen by noting the multiplicity greater than 1. Additionally, a straightforward computation summing these over all of the Euler paths gives us total degree $(2^{2g^\nu+2b_1(\Gamma_Z)}-1)(2^{2g^\nu+2b_1(\Gamma_Z)}-2)$, which is the degree of the moduli space, showing that nothing has been missed. \end{proof} Now, applying the above to the general point on the boundary, we see that for a reducible 1-nodal curve, all Prym curves on it are supported on the stable curve itself, yielding $(2^{2g}-1)(2^{2g}-2)$ objects of multiplicity 1. The case of an irreducible curve is a bit more complex: \begin{corollary} \label{deltairrlemma} Let $Z$ be a 1-nodal irreducible stable curve of genus $g$ and $\nu:Z^\nu\to Z$ its normalization. There is only one unstable quasistable curve, $X=Z^\nu\cup_{x,y}\mathbb{P}^1$, where $x,y$ are the preimages of the node in $Z^\nu$. Then the fiber of $\overline{\ZZM{g}}\to \overline{\mathcal{M}}_g$ over $Z$ consists of the following objects: \begin{enumerate} \item On $(Z,Z)$, we have $(2^{2g-1}-1)(2^{2g-1}-2)$ objects of multiplicity 1. \item On each $(Z,X)$ and $(X,Z)$, we have a total of $2^{4g-2}-2^{2g-1}$ objects of multiplicity 2 \item On $(X,X)$ we have two types of objects: \begin{enumerate} \item $2^{4g-4}-2^{2g-2}$ objects of multiplicity 4 with non-isomorphic projections to $\overline{\ZM{g}}$. \item $2^{2g-2}$ objects of multiplicity 2 with the same projections to $\overline{\ZM{g}}$. \end{enumerate} \end{enumerate} \end{corollary} Now, we must compute the list of boundary divisors. We begin by looking at the boundary of $\overline{\ZZM{g}}\times_{\overline{\mathcal{M}_g}}\overline{\ZZM{g}}$. Points on the boundary can be classified into products $\Delta_a\times \Delta_b$ where $a,b$ were in $\{I,II,III\}$ over an irreducible 1-nodal curve and $\{i,g-i,i:g-i\}$ over a reducible curve with components of genus $i$ and $g-i$. We denote the restrictions of these loci to $\overline{\ZZM{g}}$ by $\Delta_{a,b}$, and note that although some are, many of these are not irreducible. The components with nonirreducible restrictions to $\overline{\ZZM{g}}^i$ are $\Delta_{II,II}$, $\Delta_{III,III}$, $\Delta_{i,i:g-i}$, $\Delta_{i:g-i,i}$, $\Delta_{g-i,i:g-i}$, $\Delta_{i:g-i,g-i}$, and $\Delta_{i:g-i,i:g-i}$. Specifically, they break up as \begin{eqnarray*} \Delta_{II,II} &=& \Delta_{II,II}^{\pm}+\Delta_{II,II}'\\ \Delta_{III,III} &=& \Delta_{III,III}^{diag}+\Delta_{III,III}'\\ \Delta_{i,i:g-i} &=& \Delta_{i,i:g-i}^i+\Delta_{i,i:g-i}'\\ \Delta_{i:g-i,i} &=& \Delta_{i:g-i,i}^i+\Delta_{i:g-i,i}'\\ \Delta_{g-i,i:g-i} &=& \Delta_{g-i,i:g-i}^{g-i}+\Delta_{g-i,i:g-i}'\\ \Delta_{i:g-i,g-i} &=& \Delta_{i:g-i,g-i}^{g-i}+\Delta_{i:g-i,g-i}'\\ \Delta_{i:g-i,i:g-i} &=& \Delta_{i:g-i,i:g-i}^i+\Delta_{i:g-i,i:g-i}^{g-i}+\Delta_{i:g-i,i:g-i}' \end{eqnarray*} where (with equality meaning equal to the closure of) \begin{eqnarray*} \Delta_{II,II}^{\pm} &=& \{(\eta_1,\eta_2)|\nu^*\eta_1\cong\nu^*\eta_2\}\\ \Delta_{II,II}' &=& \{(\eta_1,\eta_2)|\nu^*\eta_1\not\cong\nu^*\eta_2\}\\ \Delta_{III,III}^{diag} &=& \{(\eta_1,\eta_2)|\eta_1\cong \eta_2\}\\ \Delta_{III,III}' &=& \{(\eta_1,\eta_2)|\eta_1\not\cong\eta_2\}\\ \Delta_{i,i:g-i}^i &=& \{(\eta_1,\eta_2)|\eta_1|_{C_1}\cong \eta_2|_{C_1}\}\\ \Delta_{i,i:g-i}' &=& \{(\eta_1,\eta_2)|\eta_1|_{C_1}\not\cong \eta_2|_{C_1}\}\\ \Delta_{i:g-i,i}^i &=& \{(\eta_1,\eta_2)|\eta_1|_{C_1}\cong \eta_2|_{C_1}\}\\ \Delta_{i:g-i,i}' &=& \{(\eta_1,\eta_2)|\eta_1|_{C_1}\not\cong \eta_2|_{C_1}\}\\ \Delta_{g-i,i:g-i}^{g-i} &=& \{(\eta_1,\eta_2)|\eta_1|_{C_2}\cong \eta_2|_{C_2}\}\\ \Delta_{g-i,i:g-i}' &=& \{(\eta_1,\eta_2)|\eta_1|_{C_2}\not\cong \eta_2|_{C_2}\}\\ \Delta_{i:g-i,g-i}^{g-i} &=& \{(\eta_1,\eta_2)|\eta_1|_{C_2}\cong \eta_2|_{C_2}\}\\ \Delta_{i:g-i,g-i}' &=& \{(\eta_1,\eta_2)|\eta_1|_{C_2}\not\cong \eta_2|_{C_2}\}\\ \Delta_{i:g-i,i:g-i}^i &=& \{(\eta_1,\eta_2)|\eta_1|_{C_1}\cong \eta_2|_{C_1}\}\\ \Delta_{i:g-i,i:g-i}^{g-i}&=& \{(\eta_1,\eta_2)|\eta_1|_{C_2}\cong \eta_2|_{C_2}\}\\ \Delta_{i:g-i,i:g-i}' &=& \{(\eta_1,\eta_2)|\eta_1|_{C_i}\not\cong \eta_2|_{C_i}\mbox{ for }i=1,2\} \end{eqnarray*} Now that we have a list of all of the components, we compute which objects are in which, a straightforward computation: \begin{proposition} \label{prop:1nidegrees} Over the locus of 1-nodal irreducible curves in $\overline{\mathcal{M}_g}$ of the boundary components of $\overline{\ZZM{g}}$ consist of \begin{eqnarray*} \Delta_{I,II} &=& 2^{2g-1}-2\mbox{ objects of multiplicity }1\\ \Delta_{II,I} &=& 2^{2g-1}-2\mbox{ objects of multiplicity }1\\ \Delta_{I,III} &=& 2^{2g-2}\mbox{ objects of multiplicity }2\\ \Delta_{III,I} &=& 2^{2g-2}\mbox{ objects of multiplicity }2\\ \Delta_{II,III} &=& 2^{2g-2}(2^{2g-1}-2)\mbox{ objects of multiplicity }2\\ \Delta_{III,II} &=& 2^{2g-2}(2^{2g-1}-2)\mbox{ objects of multiplicity }2\\ \Delta_{II,II}^{\pm} &=& 2^{2g-1}-2\mbox{ objects of multiplicity }1\\ \Delta_{II,II}' &=& (2^{2g-1}-2)(2^{2g-1}-4)\mbox{ objects of multiplicity }1\\ \Delta_{III,III}^{diag} &=& 2^{2g-2}\mbox{ objects of multiplicity }2\\ \Delta_{III,III}' &=& 2^{4g-4}-2^{2g-2}\mbox{ objects of multiplicity }4 \end{eqnarray*} \end{proposition} \begin{remark} \label{Intersection} It is interesting to note that away from $\Delta_{III,III}'$, the Weil pairing is well-defined by continuity. However, on this divisor, this fails. Let us look at a simple example. Let $C$ be a smooth genus 2 curve with $p_1,\ldots,p_6$ the fixed points of the hyperelliptic involution. Then any point of order two is $p_i-p_j$ for $i\neq j$, and $p_i-p_j\equiv p_j-p_i$. If the vanishing cycle of the degeneration is $p_1-p_2$, then the pairs $(p_1-p_3,p_1-p_4)$ and $(p_1-p_3,p_2-p_4)$ both degenerate to the same point of $\Delta_{III,III}'$. We note that, for a genus 2 curve, the Weil pairing can be described as the cardinality of the intersection of the set of indices appearing in this representation. Thus, $\langle p_1-p_3,p_1-p_4\rangle=1$ and $\langle p_1-p_3,p_2-p_4\rangle=0$. Thus, the two components of this moduli space intersect on the boundary! \end{remark} In fact, we can say a bit more about the intersection: \begin{theorem} \label{thm:intersection} The intersection $\Delta_{III,III}'=\overline{\ZZM{g}}^0\cap\overline{\ZZM{g}}^1$ is transverse, in the sense that if $(C,\eta_1,\eta_2)\in\Delta_{III,III}'$, then we have \[T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}}\cong T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}}^0\oplus_{T_{(C,\eta_1,\eta_2)}\Delta_{III,III}'}T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}}^1.\] \end{theorem} \begin{proof} To see that this is precisely the intersection, we look at a degeneration of Prym curves with vanishing cycle $\delta$. The fiber over the nodal curve only has points coming together over $\Delta_{III}$, which is the part of the fiber over the quasi-stable curve. There, every Prym curve structure is the limit of both $\eta$ and $\eta+\delta$ for $\eta$ some Prym curve structure on a smooth curve in the degeneration. The Weil pairing is well-defined on all components over $\Delta_{III}$ other than $\Delta_{III,III}'$, by linearity and the definitions of $\Delta_I$ and $\Delta_{II}$. However, on $\Delta_{III,III}$, we have $(\eta_1,\eta_2)$ two Prym structures giving a $\mathbb{Z}_2^2$ curve. This limit as $\delta$ vanishes, is the same as the limit of $(\eta_1,\eta_2+\delta)$, and also two other loci. However, the Weil pairing is linear, so $\langle\eta_1,\eta_2+\delta\rangle=\langle\eta_1,\eta_2\rangle+\langle\eta_1,\delta\rangle$, and because it is in $\Delta_{III}$, $\langle \eta_1,\delta\rangle=1$, so this point is a limit of families of Veil pairing both 0 and 1, and this holds for every point in $\Delta_{III,III}'$. Transversality follows by looking at first order deformations of $(C,\eta_1,\eta_2)$. The Weil pairing determines which component $(C,\eta_1,\eta_2)$ is on, except along $\Delta_{III,III}'$ where it is indeterminate. So we describe all of the space: \begin{eqnarray*} T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}} &=& \mbox{First order deformations with }\langle\eta_1,\eta_2\rangle\mbox{ undefined, } 0, \mbox{ or }1, \\ T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}}^0 &=& \mbox{First order deformations with }\langle\eta_1,\eta_2\rangle\mbox{ undefined, or } 0, \\ {T_{(C,\eta_1,\eta_2)}\Delta_{III,III}'} &=& \mbox{First order deformations with }\langle\eta_1,\eta_2\rangle\mbox{ undefined,} \\ T_{(C,\eta_1,\eta_2)}\overline{\ZZM{g}}^1 &=& \mbox{First order deformations with }\langle\eta_1,\eta_2\rangle\mbox{ undefined, or } , \end{eqnarray*} From these descriptions, transversality follows immediately. \end{proof} A similar computation to Proposition \ref{prop:1nidegrees} over reducible 1-nodal curves gives: \begin{proposition} \label{prop:1nrdegrees} Over the locus of 1-nodal reducible curves that are a union of a genus $i$ and a genus $g-i$ curve in $\overline{\mathcal{M}_g}$, the boundary components of $\overline{\ZZM{g}}$ are of the following degrees (and all objects are multiplicity 1): \begin{eqnarray*} \Delta_{i,i} &=& (2^{2i}-1)(2^{2i}-2)\\ \Delta_{g-i,g-i} &=& (2^{2(g-i)}-1)(2^{2(g-i)}-2)\\ \Delta_{i,g-i} &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{g-i,i} &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i,i:g-i}^i &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i,i:g-i}' &=& (2^{2i}-2)(2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,i}^i &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,i}' &=& (2^{2i}-2)(2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{g-i,i:g-i}^{g-i} &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{g-i,i:g-i}' &=& (2^{2(g-i)}-2)(2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,g-i}^{g-i} &=& (2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,g-i}' &=& (2^{2(g-i)}-2)(2^{2i}-1)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,i:g-i}^i &=& (2^{2i}-1)(2^{2(g-i)}-1)(2^{2(g-i)}-2)\\ \Delta_{i:g-i,i:g-i}^{g-i}&=& (2^{2i}-1)(2^{2i}-2)(2^{2(g-i)}-1)\\ \Delta_{i:g-i,i:g-i}' &=& (2^{2i}-1)(2^{2(g-i)}-1)\\&&\times((2^{2i}-1)(2^{2(g-i)}-1)-(2^{2i}-1)-(2^{2(g-i)}-1)) \end{eqnarray*} \end{proposition} \section{The moduli of Klein curves} In this section, we extend the results of the previous section to a compactification of $\KM{g}=\ZZM{g}/\PSL_2(\mathbb{F}_2)$, that is, the moduli space where we do not choose a basis of the Deck transformations. We will do so by extending the group action to the compactification $\overline{\ZZM{g}}$, and defining the quotient to be $\overline{\KM{g}}$. \begin{proposition} The group action $\PSL_2(\mathbb{F}_2)\times \ZZM{g}\to\ZZM{g}$ extends to each component of $\overline{\ZZM{g}}$. \end{proposition} \begin{proof} Let $D_1=\Delta_{I,III}\cup\Delta_{III,I}\cup\Delta_{III,III}^{\diag}$ and $D_2=\Delta_{III,III}'\cup \Delta_{II,III}\cup\Delta_{III,II}$. Then the extension is actually straightforward over $\overline{\ZZM{g}}\setminus (D_1\cup D_2)$, as over this locus, the fibers are reduced, and the action is just by change of basis on a Klein 4 group. Only in the cases where multiplicities are no longer 1, namely $D_1$ and $D_2$, will these fail to just be Klein four groups. Now, we take the orbit of a $\mathbb{Z}_2^2$-curve in the locus where we have the group action, and degenerate it to $D_1$. Then, the six objects of multiplicity 1 of this fiber degenerate to $(\eta,\eta)\in\Delta_{III,III}^{\diag}$, $(\eta,\mathscr{O}^-_X)\in \Delta_{III,I}$ and $(\mathscr{O}^-_X,\eta)\in\Delta_{I,III}$, where $\mathscr{O}^-_X$ is the Prym curve structure on $X$ lying in $\Delta_I\subset\overline{\ZM{g}}$. Each of these appears with multiplicity 2. Here, the group action can be seen most clearly by noting that $\PSL_2(\mathbb{F}_2)\cong S_3$ (and in fact, the change of basis on a Klein four group is just permuting the three nonzero elements) and seeing the action as being that of $S_3$ on the ordered set $(\mathscr{O}_X^-,\eta,\eta)$ followed by forgetting the last element. Deeper degeneration into the strata $D_1\setminus D_2$ can be handled in the same way, leaving only $D_2$ remaining. To extend the action to $D_2$, we must first restrict to the individual irreducible components of $\overline{\ZZM{g}}$. This is because $\Delta_{III,III}'$ is the intersection of the two components, by Theorem \ref{thm:intersection}. So, by transversality the fiber multiplicity of elements in the intersection must be split evenly between the components. Now, let $(\epsilon_1,\epsilon_2)\in\Delta_{III,III}'$. Then $\epsilon_2\otimes\epsilon^{-1}_1$ gives a Prym structure on the closure of the complement of the exceptional components. There are two different Prym structures on the stable curve that pull back to this under the stabilization map, but one of them lies on each componenet. We will denote by $\eta^i$ the Prym structure such that $(\eta^i,\epsilon_1),(\eta^i,\epsilon_2)\in\overline{\ZZM{g}}^i$. Then, the $\PSL_2(\mathbb{F}_2)\cong S_3$ action is given by permutation of $\epsilon_1,\epsilon_2,\eta^i$. \end{proof} This extension allows us to take the quotient, which constructs from the moduli of pairs of Prym structures, which is the same as the moduli of Klein four groups of Prym structures, with the moduli of Klein four groups of Prym covers without a basis. \begin{definition}[Moduli of Klein four covers] We define the space $\overline{\KM{g}}$ to be the quotient of $\overline{\ZZM{g}}$ by the relation described above, and we call it the \emph{moduli of Klein four covers of genus $g$ curves}. \end{definition} Given an orbit $\{(C,\eta_1^i,\eta_2^i)\}$ where $i$ runs over the elements of the orbit, we will denote by $(C,\{\eta^i_j\}_{i,j})$ the corresponding point of $\overline{\KM{g}}$, with $i$ running over the orbit and $j=1,2$. The boundary in the Klein moduli space simplifies significantly. Because the action of $\PSL_2(\mathbb{F}_2)$ exchanges some boundary components, we group them together and give names to their images (fixing the Weil pairing as either $0$ or $1$ in each case) in the following: \begin{eqnarray*} \Delta_{I,II}\cup \Delta_{II,I}\cup\Delta_{II,II}^\pm&\to&\Delta_{I,II,II}\\ \Delta_{I,III}\cup\Delta_{III,I}\cup\Delta_{III,III}^{diag}&\to&\Delta_{I,III,III}\\ \Delta_{II,III}\cup\Delta_{III,II}\cup\Delta_{III,III}'&\to&\Delta_{II,III,III}\\ \Delta_{II}'&\to&\Delta_{II,II,II}\\ \Delta_{i,g-i}\cup\Delta_{g-i,i}\cup\Delta_{i,i:g-i}^i\cup\Delta_{i:g-i,i}^i\cup\Delta_{g-i,i:g-i}^{g-i}\cup\Delta_{i:g-i,g-i}^{g-i}&\to&\Delta_{i,g-i,i:g-i}\\ \Delta_{i,i}&\to&\Delta_{i,i,i}\\ \Delta_{g-i,g-i}&\to&\Delta_{g-i,g-i,g-i}\\ \Delta_{i,i:g-i}'\cup \Delta_{i:g-i,i}' \cup\Delta_{i:g-i,i:g-i}^i&\to&\Delta_{i,i:g-i,i:g-i}\\ \Delta_{g-i,i:g-i}'\cup\Delta_{i:g-i,g-i}'\cup\Delta_{i:g-i,i:g-i}^{g-i}&\to&\Delta_{g-i,i:g-i,i:g-i}\\ \Delta_{i:g-i,i:g-i}'&\to&\Delta_{i:g-i,i:g-i,i:g-i} \end{eqnarray*} Between the degrees computed in the previous section and the maps above all being $\PSL_2(\mathbb{F}_2)$ quotients, we find the following structure on the boundary \begin{eqnarray*} \Delta_{I,II,II} &=& 2^{2g-2}-1\mbox{ objects of multiplicity }1\\ \Delta_{I,III,III} &=& 2^{2g-2}\mbox{ objects of multiplicity }1\\ \Delta_{II,III,III} &=& 2^{2g-2}(2^{2g-2}-1)\mbox{ objects of multiplicity }2\\ \Delta_{II,II,II} &=& \frac{(2^{2g-1}-2)(2^{2g-1}-4)}{6}\mbox{ objects of multiplicity }1\\ \Delta_{i,g-i,i:g-i} &=& (2^{2i}-1)(2^{2(g-i)}-1)\mbox{ objects of multiplicity }1\\ \Delta_{i,i,i} &=& \frac{(2^{2i}-1)(2^{2i}-2)}{6}\mbox{ objects of multiplicity }1\\ \Delta_{g-i,g-i,g-i} &=& \frac{(2^{2(g-i)}-1)(2^{2(g-i)}-2)}{6}\mbox{ objects of multiplicity }1\\ \Delta_{i,i:g-i,i:g-i} &=& \frac{(2^{2i}-1)(2^{2i}-2)(2^{2(g-i)}-1)}{2}\mbox{ objects of multiplicity }1\\ \Delta_{g-i,i:g-i,i:g-i} &=& \frac{(2^{2(g-i)}-1)(2^{2(g-i)}-2)(2^{2i}-1)}{2}\mbox{ objects of multiplicity }1\\ \Delta_{i:g-i,i:g-i,i:g-i} &=& \frac{(2^{2i}-1)(2^{2(g-i)}-1)(2^{2i}-2)(2^{2(g-i)}-2)}{6}\mbox{ objects of}\\ && \mbox{multiplicity }1 \end{eqnarray*} This gives us \begin{proposition} The morphsim $\overline{\KM{g}}\to\mathcal{M}_g$ has degree $\frac{(2^{2g}-1)(2^{2g}-2)}{6}$ and is simply ramified along $\Delta_{II,III,III}$. \end{proposition} Thus, we have \begin{corollary} The canonical divisor of $\overline{\KM{g}}$ is \begin{eqnarray*} K_{\overline{\KM{g}}}&=&13\lambda-2\Delta_{I,II,II}-2\Delta_{I,III,III}-2\Delta_{II,II,II}-3\Delta_{II,III,III}\\ &&-\Delta_{1,g-1,1:g-1}-\Delta_{1,1,1}-\Delta_{g-1,g-1,g-1}\\ &&-\Delta_{1,1:g-1,1:g-1}-\Delta_{g-1,1:g-1,1:g-1}-\Delta_{1:g-1,1:g-1,1:g-1}\\ &&-2\sum_{i=1}^{\lfloor g/2\rfloor}(\Delta_{i,g-i,i:g-i}+\Delta_{i,i,i}+\Delta_{g-i,g-i,g-i}+\Delta_{i,i:g-i,i:g-i}\\ &&+\Delta_{g-i,i:g-i,i:g-i}+\Delta_{i:g-i,i:g-i,i:g-i}). \end{eqnarray*} \end{corollary} \begin{proof} We use the Hurwitz formula, which tells us that $K_{\overline{\KM{g}}}=\pi^*K_{\overline{\mathcal{M}_g}}+\Delta_{II,III,III}$. The canonical divisor of $\overline{\mathcal{M}_g}$ is $13\lambda-2\delta_0-3\delta_1-2\delta_2-\ldots -2\delta_{\lfloor g/2\rfloor}$\cite{MR664324}. We note that $\pi^*(\Delta_i)=\Delta_{i,g-i,i:g-i}+\Delta_{i,i,i}+\Delta_{g-i,g-i,g-i}+\Delta_{i,i:g-i,i:g-i}+\Delta_{g-i,i:g-i,i:g-i}+\Delta_{i:g-i,i:g-i,i:g-i}$ and $\pi^*\Delta_0=\Delta_{I,II,II}+\Delta_{I,III,III}+\Delta_{II,II,II}+2\Delta_{II,III,III}$ and $\pi^*\lambda=\lambda$. \end{proof} Now we'll work out some numerics of the odd and even components of $\overline{\KM{g}}$ \begin{proposition} The degrees of the natural projection maps to $\overline{\mathcal{M}_g}$ are \begin{itemize} \item for $\overline{\KM{g}}$, $\frac{(2^{2g}-1)(2^{2g}-2)}{6}$. \item for $\overline{\KM{g}}^0$, $\frac{(2^{2g}-1)(2^{2g-1}-2)}{6}$. \item for $\overline{\KM{g}}^1$, $\frac{(2^{2g}-1)2^{2g-1}}{6}$. \end{itemize} \end{proposition} This follows directly from Proposition \ref{DegZZM01} and Lemma \ref{DegZZM}. As a final computation involving the degrees, we compute the content of the boundary divisors when restricted to $\overline{\KM{g}}^0$ and $\overline{\KM{g}}^1$. \begin{theorem} The fiber over the generic element $C$ of the boundary of $\overline{\mathcal{M}}_g$ in $\overline{\KM{g}}$ is: \begin{enumerate} \item if $C$ is irreducible 1-nodal, the fiber in $\overline{\KM{g}}^0$ is \begin{enumerate} \item $2^{2g-2}-1$ elements of $\Delta_{I,II,II}^0$ with multiplicity $1$, \item $\frac{(2^{2g-1}-2)(2^{2g-2}-4)}{6}$ elements of $\Delta_{II,II,II}^0$ with multiplicity $1$, \item ${(2^{2g-1}-2)2^{2g-4}}$ elements of $\Delta_{II,III,III}^0$ with multiplicity $2$, \end{enumerate} \item if $C$ is irreducible 1-nodal, the fiber in $\overline{\KM{g}}^1$ is \begin{enumerate} \item $\frac{(2^{2g-1}-2)2^{2g-2}}{6}$ elements of $\Delta_{II,II,II}^1$ with multiplicity $1$, \item ${(2^{2g-1}-2)2^{2g-4}}$ elements of $\Delta_{II,III,III}^1$ with multiplicity $2$, \item ${2^{2g-2}}$ elements of $\Delta_{I,III,III}^1$ with multiplicity $1$, \end{enumerate} \item if $C$ is reducible with components of genus $i$ and $g-i$, the fiber in $\overline{\KM{g}}^0$ is \begin{enumerate} \item $(2^{2i}-1)(2^{2(g-i)}-1)$ elements of $\Delta_{i,g-i,i:g-i}^0$ with multiplicity $1$, \item $\frac{(2^{2i}-1)(2^{2i-1}-2)}{6}$ elements of $\Delta_{i,i,i}^0$ with multiplicity $1$, \item $\frac{(2^{2(g-i)}-1)(2^{2(g-i)-1}-2)}{6}$ elements of $\Delta_{g-i,g-i,g-i}^0$ with multiplicity $1$, \item ${(2^{2i-1}-1)(2^{2i-2}-1)(2^{2(g-i)}-1)}$ elements of $\Delta_{i,i:g-i,i:g-i}^0$ with multiplicity $1$, \item ${(2^{2(g-i)-1}-1)(2^{2(g-i)-2}-1)(2^{2i}-1)}$ elements of $\Delta_{g-i,i:g-i,i:g-i}^0$ with multiplicity $1$, \item $\frac{(2^{2i}-1)(2^{2(g-i)}-1)((2^{2i-1}-2)(2^{2(g-i)-1}-2)+(2^{2i-1})(2^{2(g-i)-1}))}{6}$ elements of\\ $\Delta_{i:g-i,i:g-i,i:g-i}^0$ with multiplicity $1$, \end{enumerate} \item if $C$ is reducible with components of genus $i$ and $g-i$, the fiber in $\overline{\KM{g}}^1$ is \begin{enumerate} \item $\frac{(2^{2i}-1)2^{2i-1}}{6}$ elements of $\Delta_{i,i,i}^1$ with multiplicity $1$, \item $\frac{(2^{2(g-i)}-1)2^{2(g-i)-1}}{6}$ elements of $\Delta_{g-i,g-i,g-i}^1$ with multiplicity $1$, \item ${(2^{2i-1}-1)(2^{2i-1})(2^{2(g-i)}-2)}$ elements of $\Delta_{i,i:g-i,i:g-i}^1$ with multiplicity $1$, \item ${(2^{2(g-i)-1}-1)(2^{2(g-i)-2})(2^{2i}-1)}$ elements of $\Delta_{g-i,i:g-i,i:g-i}^1$ with\\ multiplicity $1$, \item $\frac{(2^{2i}-1)(2^{2(g-i)}-1)((2^{2i-1}-2)(2^{2i-1})+(2^{2(g-i)-1})(2^{2(g-i)-1}-2))}{6}$ elements of\\ $\Delta_{i:g-i,i:g-i,i:g-i}^1$ with multiplicity $1$. \end{enumerate} \end{enumerate} \end{theorem} \begin{proof} We will work out parts 1 and 3, parts 2 and 4 being analagous. With the exception of $\Delta_{II,III,III}$, we can compute the Weil pairing by choosing any pair of elements in the group. For $\Delta_{II,III,III}$, we note that it must be divided evenly between the components. The groups consist of two elements from $\Delta_{III}$ and the one element of $\Delta_{II}$, and can be chosen to either be glued by $+1$ or $-1$ at the node. These will correspond to Weil pairing $0$ and $1$, thus dividing $\Delta_{II,III,III}$ evenly. For $\Delta_{I,II,II}$, by definition, we must have $\Delta_{I,II,II}^0=\Delta_{I,II,II}$. Similarly, we can see that $\Delta_{I,III,III}^0=\emptyset$. We can finish by computing that $\Delta_{II,II,II}^0$ must be the correct size to, with the other components, add up to $\frac{(2^{2g}-1)(2^{2g-1}-2)}{6}$. However, we can compute this directly by choosing $\mu_1,\mu_2\in\Delta_{II}$ distinct and orthogonal. Then, if $\delta$ is the vanishing cycle, we have $\mu_1\in\delta^\perp\setminus(\delta)$ and $\mu_2\in (\delta,\mu_1)^\perp\setminus(\delta,\mu_1)$, which gives the appropriate number. Over a reducible curve $C=C_i\cup C_{g-i}$, although the expressions are more complex, the situation is simpler. We begin by noting that everything in $\Delta_{i,g-i,i:g-i}$ must be in $\Delta_{i,g-i,i:g-i}^0$, because the generators have no common support curve. As for $\Delta_{i,i,i}$ and $\Delta_{g-i,g-i,g-i}$, they will be precisely the fibers of the lower genus maps $\KM{i}^0\to\mathcal{M}_i$ and $\KM{g-i}^0\to \mathcal{M}_{g-i}$. On $\Delta_{i,i:g-i,i:g-i}$, we can have any nonzero square trivial line bundle on the component $C_i$, and the second generator can have any restriction to $C_{g-i}$, but the restriction to $C_i$ must be orthogonal to the first, and here, we only divide by two choices in $\Delta_{i:g-i}$ that can be basis elements. The next component, $\Delta_{g-i,i:g-i,i:g-i}$, can be computed in a similar way. The final component, $\Delta_{i:g-i,i:g-i,i:g-i}$ starts with an arbitary element of $\Delta_{i:g-i}$, and the second must either have both restrictions orthogonal to those of the first, or else both nonorthogonal, and then we divide by 6 from choices of basis, completing the computation. \end{proof} \section{Pluricanonical forms} In this section, we show that pluricanonical forms on $\overline{\KM{g}}$ extend to any smooth model, allowing us to compute the Kodaira dimension on $\overline{\KM{g}}$ itself, rather than having to work on the set of smooth models. As such, the goal of this section is to prove \begin{theorem} \label{extensiontheorem} Fix $g\geq 4$ and $i\in\{0,1\}$, and let $\widehat{\ZZM{g}}^i\to\overline{\ZZM{g}}^i$ be any resolution of the singularities. Then every pluricanonical form defined on $\overline{\ZZM{g}}^{i,reg}$, the smooth locus, extends holomorphically to $\widehat{\ZZM{g}}^i$. Specifically, for all integers $\ell\geq 0$, we have isomorphisms \[H^0(\overline{\ZZM{g}}^{i,reg},K_{\overline{\ZZM{g}}^{i,reg}}^{\otimes\ell})\cong H^0(\widehat{\ZZM{g}}^i,K_{\widehat{\ZZM{g}}^i}^{\otimes\ell})\] \end{theorem} Analogues of this theorem are known for all of the relevant related moduli spaces: $\overline{\mathcal{M}_g}$ is proved in \cite[Theorem 1]{MR664324}, $\overline{\mathcal{R}_g}$ is proved in \cite[Theorem 6.1]{MR2639318}, and the moduli of spin curves in \cite[Theorem 4.1]{MR2551759}. Our proof will very closely follow the one in \cite{MR2639318} for $\overline{\mathcal{R}_g}$, which can be expected as $\overline{\ZZM{g}}$ is two of the components of $\overline{\mathcal{R}_g}\times_{\overline{\mathcal{M}}_g}\overline{\mathcal{R}_g}$. Before we can proceed, we need to make a few remarks about the versal deformations of an object $X=(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)\in\overline{\ZZM{g}}$. Let $\mathbb{C}_t^{3g-3}$ be the versal deformation space of $Z$, the stabilization of $X_i$ and $\mathbb{C}_\alpha^{3g-3}$ the versal deformation space of $X$. There are compatible decompositions \begin{eqnarray*} \mathbb{C}_\alpha^{3g-3}&\cong&\bigoplus_{p_i\in\Delta_{X_1}^c\cap\Delta_{X_2}}\mathbb{C}_{\tau_i}\oplus\bigoplus_{p_i\in\Delta_{X_2}^c\cap\Delta_{X_1}}\mathbb{C}_{\tau_i}\oplus\bigoplus_{p_i\in\Delta_{X_1}^c\cap\Delta_{X_2}^c}\mathbb{C}_{\tau_i}\\ &&\oplus\bigoplus_{p_i\in\Delta_{X_1}\cap\Delta_{X_2}}\mathbb{C}_{\tau_i}\oplus\bigoplus_{C_j\subset C}H^1(C_j^\nu,T_{C_j^\nu}(-D_j))\\ \mathbb{C}_t^{3g-3}&\cong&\bigoplus_{p_i\in\Sing(C)} \mathbb{C}_{t_i}\oplus\bigoplus_{C_j\subset C}H^1(C_j^\nu,T_{C_j^\nu}(-D_j)) \end{eqnarray*} where $D_j$ is the sum of the preimages of the nodes under the normalization map. There is a natural map from the versal deformation space of a $\mathbb{Z}_2^2$ curve to that of the underlying stable curve, given by $t_i=\alpha_i^2$ if $t_i=0$ is the locus where the exceptional node $p_i\in\Delta_{X_1}^c\cup \Delta_{X_2}^c$ persists and $t_i=\alpha_i$ otherwise. Similarly to the discussion in Section 1.2 of \cite{MR2117416}, we can blow up along all of the exceptional components and extend $\eta_1,\eta_2$ using only those in $\Delta_{X_1}^c$ and $\Delta_{X_2}^c$ respectively. This description makes the rest of the work in Section 6 of \cite{MR2639318} relatively straightforward to generalize. Set $X_\Delta$ to be the quasi-stable curve with exceptional nodes $\Delta_{X_1}^c\cup\Delta_{X_2}^c$. \begin{definition}[Elliptic tail] Let $X$ be a quasi-stable curve, a component $C_j$ is an elliptic tail if it has arithmetic genus 1 and intersects the rest of the curve in a single point. That point is called an elliptic tail node, and any automorphism of $X$ that is the identity away from $C_j$ is an elliptic tail automorphism. \end{definition} \begin{proposition} Let $\sigma\in \Aut(X)$ be an automorphism in genus $g\geq 4$. Then $\sigma$ acts on $\mathbb{C}_\alpha^{3g-3}$ as a quasi-reflection if and only if $X_\Delta$ has an elliptic tail $C_j$ such that $\sigma$ is the elliptic tail involution with respect to $C_j$. \end{proposition} The proof of this proposition follows from the proof of \cite[6.6]{MR2639318}. It implies that the smooth locus of $\overline{\ZZM{g}}$ is the locus where the automorphism group is generated by elliptic tail involutions. Now that we have determined the smooth locus, we must determine the non-canonical locus. If $G$ acts on a vector space $V$ by quasi-reflections, then $V/G\cong V$, so we let $H\subset\Aut(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)$ be generated by automorphisms acting as quasi-reflections, that is elliptic tail involutions. Then $\mathbb{C}^{3g-3}_\alpha/H\cong \mathbb{C}^{3g-3}_\nu$ where $\nu_i=\alpha_i^2$ if $p_i$ is an elliptic tail node and $\nu_i=\alpha_i$ else. On $\mathbb{C}^{3g-3}_\nu$, the automorphisms act without quasi-reflections, so the Reid--Shepherd-Barron--Tai criterion can be applied. \begin{theorem}[Reid--Shepherd-Barron--Tai Criterion \cite{MR605348,MR763023}] Let $V$ be a vector space of dimension $d$, $G\subset \GL(V)$ a finite group and $V_0\subset V$ the open set where $G$ acts freely. Fix $g\in G$, and let $g$ be conjugate to a diagonal matrix with $\zeta^{a_i}$ for $i=1,\ldots,d$ on the diagonal for $\zeta$ a fixed $m^{th}$ root of unity and $0\leq a_i<m$. If for all $g$ and $\zeta$, we have $\frac{1}{m}\sum_{i=1}^d a_i\geq 1$, then any $n$-canonical form on $V_0/G$ extends holomorphically to a resolution $\widetilde{V/G}$. \end{theorem} It is straightforward to check that for $g\geq 4$, we have a noncanonical singularity if $X_\Delta$ has an elliptic tail $C_j$ with $j$-invariant $0$ such that $\eta_1,\eta_2$ are both trivial on $C_j$. This goes as in \cite{MR2639318}, where the action of $\sigma$ is determined to be as the square of a sixth root of unity in two coordinates for an automorphism of order $6$ and as a cube root of unity in those two coordinates for an order $3$ element. Both of these fail the Reid--Shepherd-Barron--Tai criterion. Now, assuming that we have a noncanonical singularity, then we have an automorphism $\sigma$ of order $n$ failing Reid--Shepherd-Barron--Tai. Our goal is to classify such things, and eventually show that only the examples above exist. Let $p_{i_0}$, $p_{i_1}=\sigma(p_{i_0})$,$\ldots$, $\sigma^{m-1}(p_{i_0})=p_{i_{m-1}}$ be distinct nodes of the stabilization, $C$, which are permuted by $\sigma$ and not elliptic tail nodes. The action on the subspace corresponding to these nodes is then given by a matrix \[\left(\begin{array}{cccc}0&c_1&&\\\vdots&&\ddots&\\0&&&c_{m-1}\\c_m&0&\ldots&0\end{array}\right)\] for some complex numbers $c_j$. We call the pair $(X,\sigma)$ \emph{singularity reduced} if $\prod_{j=1}^m c_j$ is not $1$. By \cite{MR664324} and \cite[Proposition 3.6]{MR2551759}, we know that there is a deformation $X'$ of $X$ such that $\sigma$ deforms to $\sigma'$, an automorphism of $X'$ such that every cycle of nodes with $\prod_{j=1}^m c_j=1$ is smoothed and the action of $\sigma$ and $\sigma'$ on $\mathbb{C}_\nu^{3g-3}$ and $\mathbb{C}_{\nu'}^{3g-3}$ have the same eigenvalues. In particular, one will satisfy Reid--Shepherd-Barron--Tai if and only if the other does. Now, we fix a pair $(X,\sigma)$ that is singularity reduced and fails the Reid--Shepherd-Barron--Tai inequality. On $C$, the stabilization, the induced automorphism $\sigma_C$ must either fix all of the nodes or else exchange a single pair of them. We look at what the action does on the components. In \cite[Proposition 6.9]{MR2639318} the proof of \cite[Proposition 3.8]{MR2551759} is adapted to the situation of $\overline{\ZM{g}}$, and this proof goes through verbatum, telling us that the action fixes each component of the stable model. Now, we recall that \begin{theorem}[{\cite[Page 36]{MR664324}}] Assume that $(X,\sigma)$ is singularity reduced and fails the Reid--Shepherd-Barron--Tai inequality. Denote by $\varphi_j$ the induced automorphism on the normalization $C_j^\nu$ of the irreducible component $C_j$ of the stabilization $C$ of $X$. Then the pair $(C_j^\nu,\varphi_j)$ is one of the following: \begin{enumerate} \item $C_j^\nu$ rational, and the order of $\varphi_j$ is 2 or 4, \item $C_j^\nu$ elliptic, and the order of $\varphi_j$ is 2,4,3 or 6, \item $C_j^\nu$ hyperelliptic of genus 2, and $\varphi_j$ is the hyperelliptic involution, \item $C_j^\nu$ bielliptic of genus 2, and $\varphi_j$ is the associated involution, \item $C_j^\nu$ hyperelliptic of genus 3, and $\varphi_j$ is the hyperelliptic involution, and \item $C_j^\nu$ arbitary, and $\varphi_j$ is the identity. \end{enumerate} \end{theorem} As pointed out in \cite[Proposition 3.10]{MR2551759}, this rules out the possibility of nodes being exchanged, so the automorphism must fix all nodes and all components on the stable curve. \begin{proposition}[{\cite[Proposition 6.12]{MR2639318}}] In the same situation as above, set $D_j$ to be the divisor of the marked points on $C_j^\nu$ that are preimages of nodes. Then the triples $(C_j^\nu,D_j,\varphi_j)$ are one of the following types, and the contribution to the left hand side of the Reid--Shepherd-Barron--Tai inequality are at least $w_j$: \begin{enumerate} \item $C_j^\nu$ arbitary, $\varphi_j$ is the identity, and $w_j=0$, \item Elliptic tails: $C_j^\nu$ is elliptic, $D=p_1^+$ which is fixed by $\varphi_j$, $\varphi_j$ has order 2,3,4 or 6, and $w_j$ is, respectively, $0$, $\frac{1}{3}$, $\frac{1}{2}$ and $\frac{1}{3}$. \item Elliptic ladder: $C_j^\nu$ is elliptic and $D=p_1^++p_2^+$, with both points fixed, the automorphism is of order $2$, $3$, or $4$ and $w_j$ is, respectively, $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{3}{4}$ \item Hyperelliptic tail: $C_j^\nu$ has genus 2, $\varphi_j$ is the hyperelliptic involution, and $D_j=p_1^+$ fixed by $\varphi_j$. Then $w_j=\frac{1}{2}$. \end{enumerate} \end{proposition} With a bit of case by case work, essentially \cite{MR2639318} Propositions 6.13, 6.14, 6.15 and 6.16, we can see that hyperelliptic tails, elliptic ladders, and elliptic tails of order 4 do not occur, and that there must, in fact, be at least one elliptic tail of order 3 or 6, giving us our restrictions on the curve. Now, we look to the line bundles. Because the automorphism must pull back the line bundle to itself on the elliptic curve, it must be trivial on the elliptic tail, and this must hold for both of the Prym line bundles. Thus, if we start with $(X,\sigma)$ failing Reid--Shepherd-Barron--Tai, then we can deform to a singularity reduced pair $(X',\sigma')$ such that the Reid--Shepherd-Barron--Tai value is constant. The pair $(X',\sigma')$ must have an elliptic tail with $j$ invariant $0$, the automorphism must be of order 3 or 6, and $\eta_1,\eta_2$ must both be trivial along it. Thus: \begin{proposition} Fix $g\geq 4$. A point $(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)\in\overline{\ZZM{g}}$ is a non-canonical singularity if and only if $X_\Delta$ has an elliptic tail $C_j$ with $j$-invariant $0$ and $\eta_1|_{C_j}\cong \eta_2|_{C_j}\cong \mathscr{O}_{C_j}$. \end{proposition} \begin{proof}[Proof of Theorem \ref{extensiontheorem}] Let $\omega$ be a pluricanonical form on $\overline{\ZZM{g}}^{i,reg}$. We want to show that it lifts to a desingularization of some neighborhood of any point $(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)\in \overline{\ZZM{g}}^i$. Because this can be done for canonical singularities, we assume that $(X_1,X_2,\eta_1,\eta_2,\beta_1,\beta_2)$ is a general non-canonical singularity, and thus $X_\Delta=C_1\cup_p C_2$ where $(C_1,p)\in \mathcal{M}_{g-1,1}$ and $(C_2,p)\in \mathcal{M}_{1,1}$ with $j(C_2)=0$. We also assume that $\eta_1|_{C_2}\cong \eta_2|_{C_2}\cong \mathscr{O}_{C_2}$ and $\eta_i|_{C_1}$ are two arbitrary line bundles on $C_1$, so that we are on a hypersurface in $\Delta_{g-1,g-1}$. We consider the pencil $\phi:\overline{\mathcal{M}_{1,1}}\to \overline{\ZZM{g}}^i$ given by $\phi(C,p)=C_1\cup_p C$ and line bundles $\eta'_i$ trivial on $C$ and isomorphic to $\eta_i|_{C_i}$ on $C_i$. As $\phi(\overline{\mathcal{M}_{1,1}})$ does not intersect the ramification locus, then just as in \cite{MR664324} pages 41-44, we can construct an open neighborhood of the pencil, $S$, such that the restriction of $\overline{\ZZM{g}}^i\to\overline{\mathcal{M}_g}$ to $S$ is an isomorphism and every pluricanonical form on the smooth locus extends to a resolution $\hat{S}$ of $S$. For the arbitrary case, with more than one node, $\omega$ will extend locally to a desingularization, just as in \cite[Theorem 4.1]{MR2551759}. \end{proof} Then, Theorem \ref{extensiontheorem} in fact implies the same result for $\overline{\KM{g}}^i$. This is because $\overline{\ZZM{g}}^i\to\overline{\KM{g}}^i$ is a quotient by $\PSL_2(\mathbb{F}_2)$. The action is free except for along $\Delta_{I,III}\cup\Delta_{III,I}\cup\Delta_{III,III}^{\diag}$, where the stabilizer of a point is $\mathbb{Z}/2\mathbb{Z}$. Looking at the Reid--Shepherd-Barron--Tai criterion for $m=2$, we find that either the pluricanonical forms extend or we have a quasi-reflection, in which case the pluricanonical forms will also extend. So, either way, we can see that what we get are the invariants: $H^0(\overline{\KM{g}}^{i,reg},K^{\otimes\ell})\cong H^0(\overline{\ZZM{g}}^i,K^{\otimes \ell})^{\PSL_2(\mathbb{F}_2)}$, and so, because we can also do this for partial resolutions of $\overline{\ZZM{g}}^i$, we can do this for any resolution $\widehat{\KM{g}}^i$. We conclude with a statement about the birational geometry of these moduli spaces, justified by the above \begin{theorem} \label{maintheorem} For any $g$, $\overline{\KM{g}}^i$ has general type if there exists a single effective divisor $D\equiv a\lambda-\sum_T b_{\Delta_T}\Delta_T$ where $T$ runs over all boundary components, such that all the ratios $\frac{a}{b_T}$ are less than $\frac{13}{2}$ and the ratios $\frac{a}{b_{II,III,III}}$, $\frac{a}{b_{1,g-1,1:g-1}}$, $\frac{a}{b_{1,1,1}}$, $\frac{a}{b_{g-1,g-1,g-1}}$, $\frac{a}{b_{1,1:g-1,1:g-1}}$, $\frac{a}{b_{g-1,1:g-1,1:g-1}}$, and $\frac{a}{b_{1:g-1,1:g-1,1:g-1}}$ are less than $\frac{13}{3}$. \end{theorem} This allows us to begin computing the classes of divisors on the Klein moduli space to determine its Kodaira dimension, and thus begin the study of the birational geometry of these spaces. \end{document}

\begin{document} \draft \twocolumn[\hsize\textwidth\columnwidth\hsize\csname@twocolumnfalse\endcsname \title{Thermal entanglement in three-qubit Heisenberg models} \author{Xiaoguang Wang$^{1,2}$ Hongchen Fu$^{3}$ Allan I Solomon$^{3}$} \address{1. Institute of Physics and Astronomy, University of Aarhus, DK-8000, Aarhus C, Denmark.} \address{2. Institute for Scientific Interchange (ISI) Foundation, Viale Settimio Severo 65, I-10133 Torino, Italy} \address{3. Quantum Processes Group, The Open University, Milton Keynes, MK7 6AA, United Kingdom.} \date{\today} \maketitle \begin{abstract} We study pairwise thermal entanglement in three-qubit Heisenberg models and obtain analytic expressions for the concurrence. We find that thermal entanglement is absent from both the antiferromagnetic $XXZ$ model, and the ferromagnetic $XXZ$ model with anisotropy parameter $\Delta\ge 1$. Conditions for the existence of thermal entanglement are discussed in detail, as is the role of degeneracy and the effects of magnetic fields on thermal entanglement and the quantum phase transition. Specifically, we find that the magnetic field can induce entanglement in the antiferromagnetic $XXX$ model, but cannot induce entanglement in the ferromagnetic $XXX$ model. \end{abstract} \pacs{PACS numbers: 03.65.Ud, 03.67.Lx, 75.10.Jm.} ] \narrowtext \section{Introduction} \label{sec:intro} Over the past few years much effort has been put into studying the entanglement of multipartite systems both qualitatively and quantitatively. Entangled states constitute a valuable resource in quantum information processing\cite{Bennett}. Quite recently, entanglement in quantum operations \cite {EO0,EO1,EO2} and entanglement in indistinguishable fermionic and bosonic systems\cite{EI1,EI2,EI3} have been considered. Entanglement in two-qubit states has been well studied in the literature, as have various kinds of three-qubit entangled states\cite {Dur,threeq,Rajagopal}. The three-qubit entangled states have been shown to possess advantages over the two-qubit states in quantum teleportation\cite {tele}, dense coding\cite{dense} and quantum cloning\cite{clone}. An interesting and natural type of entanglement, thermal entanglement, was introduced and analysed within the Heisenberg $XXX$\cite{Arnesen}, $XX$\cite{Wang1}, and $XXZ$\cite{Wang2} models as well as the Ising model in a magnetic field\cite{Ising}. The state of the system at thermal equilibrium is represented by the density operator $\rho (T)=\exp \left( -\frac H{kT}\right) /Z,$ where $Z=$tr$\left[ \exp \left( -\frac H{kT}\right) \right]$ is the partition function, $H$ the system Hamiltonian, $k$ is Boltzmann's constant which we henceforth take equal to 1, and $T$ the temperature. As $\rho (T)$ represents a thermal state, the entanglement in the state is called {\em thermal entanglement}\cite{Arnesen}. A complication in the analysis is that, although standard statistical physics is characterized by the partition function, determined by the eigenvalues of the system, thermal entanglement properties require in addition knowledge of the eigenstates. The Heisenberg model has been used to simulate a quantum computer\cite {Loss}, as well as quantum dots\cite{Loss}, nuclear spins\cite {Kane}, electronic spins\cite{Vrijen} and optical lattices\cite{Moelmer}. By suitable coding, the Heisenberg interaction alone can be used for quantum computation\cite{Loss2}. The entanglement in the ground state of the Heisenberg model has been discussed by O'Connor and Wootters\cite{Oconnor}. In previous studies of thermal entanglement analytical results were only available for two-qubit quantum spin models. In this paper we analyze the three-qubit case, i.e. we consider pairwise thermal entanglement in three-qubit Heisenberg models. A general 3-qubit Heisenberg $XYZ$ model in a non-uniform magnetic field {\bf $B$} is given by: \begin{eqnarray} H &=&H_{XYZ}+H_{\rm mag} \nonumber \\ H_{XYZ} &=&\sum_{n=1}^3\left( \frac{J_1}{2}\sigma _{n}^{x}\sigma _{n+1}^{x}+\frac{J_2}{2}\sigma _{n}^{y}\sigma _{n+1}^{y}+\frac{J_3}{2}\sigma _{n}^{z}\sigma _{n+1}^{z}\right) \nonumber \\ H_{\rm mag} &=&\sum_{n=1}^3 {B_n \sigma _n^z}. \label{eq:xyz} \end{eqnarray} We use the standard notation, detailed later, and assume a periodic boundary, identifying the subscript $4$ with $1$ in the above expressions. For the 3-qubit case even this most general scenario is susceptible to numerical analysis. However, in this paper we shall restrict ourselves to special cases of Eq.(\ref{eq:xyz}) for which we are able to provide a succinct analytic treatment. The 3-site Heisenberg models we will study in this paper are the following: \begin{enumerate} \item The $XX$ model, corresponding to $J_1=J_2,\;\;\; J_3=0$ and ${\bf B}=0$. \item The $XXZ$ model, for which $J_1=J_2, J_3\neq 0$ and ${\bf B}=0$. \item The $XXZ$ model with uniform magnetic field ($B_1=B_2=B_3$). \end{enumerate} We start in Sec. II by examining the three-qubit $XX$ model. In Sec. III, IV, and V, we study thermal entanglement in the $XX$ model, the $XXZ$ model and the $XXZ$ model in a magnetic field, respectively. During the course of the analysis it will become clear that degeneracy plays an important role in thermal entanglement, as does the presence of magnetic fields. We find the critical temperatures involved in the quantum phase transition associated with the existence of entanglement in these quantum spin models. \section{Three-qubit $XX$ model and its solution} The three-qubit $XX$ model is described by the Hamiltonian\cite{Lieb} \begin{eqnarray} H_{XX} &=&\frac J2\sum_{n=1}^3\left( \sigma _{n}^{x}\sigma _{n+1}^{x}+\sigma _{n}^{y}\sigma _{n+1}^{y}\right) \nonumber \\ &=&J\sum_{n=1}^3\left( \sigma _{n}^{+}\sigma _{n+1}^{-}+\sigma _{n}^{-}\sigma _{n+1}^{+}\right) , \label{xy1} \end{eqnarray} where $\sigma _n^\alpha $ $(\alpha =x,y,z)$ are the Pauli matrices of the $n$ -th qubit, $\sigma _n^{\pm }=\frac 12\left( \sigma _n^x\pm i\sigma _n^y\right) $ the raising and lowering operators, and $J$ is the exchange interaction constant. Positive (negative) $J$ corresponds to the antiferromagnetic (ferromagnetic) case. As signalled above, we adopt periodic boundary conditions; $\sigma_4^x=\sigma_1^x,$ $\sigma_4^y=\sigma_1^y.$ We are therefore considering a three-qubit Heisenberg ring. The $XX$ model was intensively investigated in 1960 by Lieb, Schultz, and Mattis\cite{Lieb}. More recently the $XX$ model has been realized in the quantum-Hall system\cite{Hall}, the cavity QED system\cite {Zheng} and quantum dot spins\cite{I} for a quantum computer. In order to study thermal entanglement, the first step is to obtain all the eigenvalues and eigenstates of the Hamiltonian Eq.(\ref{xy1}). The eigenvalues themselves do not suffice to calculate the entanglement. The eigenvalue problem of the $XX$ model can be exactly solved by the Jordan-Wigner transformation\cite{JW}. In the three-qubit case the eigenvalues are more simply obtained as \cite{Wang1} \begin{eqnarray} E_0 &=&E_7=0, \nonumber \\ E_1 &=&E_2=E_4=E_5=-J, \nonumber \\ E_3 &=&E_6=2J. \label{eq:eeeigen} \end{eqnarray} \newline and the corresponding eigenstates are explicitly given by \begin{eqnarray} |\psi _0\rangle &=&|000\rangle , \nonumber \\ |\psi _1\rangle &=&3^{-1/2}\left( q|001\rangle +q^2|010\rangle +|100\rangle \right) , \nonumber \\ |\psi _2\rangle &=&3^{-1/2}\left( q^2|001\rangle +q|010\rangle +|100\rangle \right) , \nonumber \\ |\psi _3\rangle &=&3^{-1/2}\left( |001\rangle +|010\rangle +|100\rangle \right) , \nonumber \\ |\psi _4\rangle &=&3^{-1/2}\left( q|110\rangle +q^2|101\rangle +|011\rangle \right) , \nonumber \\ |\psi _5\rangle &=&3^{-1/2}\left( q^2|110\rangle +q|101\rangle +|011\rangle \right) , \nonumber \\ |\psi _6\rangle &=&3^{-1/2}\left( |110\rangle +|101\rangle +|011\rangle \right) , \nonumber \\ |\psi _7\rangle &=&|111\rangle . \label{eq:estate} \end{eqnarray} with $q=\exp \left( i2\pi /3\right) $ satisfying \begin{eqnarray} q^3 &=&1, \nonumber \\ q^2+q+1 &=&0. \end{eqnarray} This set (\ref{eq:estate}) of three-qubit states is itself interesting. Rajagopal and Rendell\cite{Rajagopal} have considered a similar set of three-qubit states which they have classified by means of permutation symmetries. Here the states $|\psi _0\rangle ,|\psi _3\rangle ,$ $|\psi _6\rangle ,$ and $|\psi _7\rangle $ are symmetric in the permutation of any pair of particles. We define a cyclic shift operator $P$ by its action on the basis $|ijk\rangle $\cite{Schnack} \begin{equation} P|ijk\rangle =|kij\rangle . \end{equation} Obviously the four states $|\psi _0\rangle ,|\psi _3\rangle ,$ $|\psi _6\rangle ,$ and $|\psi _7\rangle $ are the eigenstates of $P$ with eigenvalue 1. The other four states in the set (\ref{eq:estate}) are also eigenstates of $P$ as follows: \begin{mathletters} \begin{eqnarray} P|\psi _i\rangle &=&q^2|\psi _i\rangle \text{ }(i=1,4), \\ P|\psi _j\rangle &=&q|\psi _j\rangle \text{ }(j=2,5). \end{eqnarray} \end{mathletters} This is not surprising since the Hamitonian $H_{XX}$ as well as the other Hamiltonians considered later are invariant under the cyclic shift operator. For $J>0$ ($J<0$) the ground state is four (two)-fold degenerate. We will see that the degeneracy of the system influences thermal entanglement greatly. There is no pairwise entanglement in the eigenstate $|\psi _0\rangle $ and $|\psi _7\rangle .\,$\thinspace Pairwise entanglement exists in the state $|\psi _i\rangle $ ($i=1,2,...,6$) and the concurrence between any two different qubits is given by 2/3\cite{Dur,Koashi}. \section{Thermal entanglement in the $XX$ model} We first recall the definition of {\em concurrence}\cite{Wootters1} between a pair of qubits. Let $\rho _{12}$ be the density matrix of the pair and it can be either pure or mixed. The concurrence corresponding to the density matrix is defined as \begin{equation} {\cal C}_{12}=\max \left\{ \lambda _1-\lambda _2-\lambda _3-\lambda _4,0\right\} , \label{eq:c1} \end{equation} where the quantities $\lambda _i$ are the square roots of the eigenvalues of the operator \begin{equation} \varrho _{12}=\rho _{12}(\sigma _1^y\otimes \sigma _2^y)\rho _{12}^{*}(\sigma _1^y\otimes \sigma_2^y) \label{eq:c2} \end{equation} in descending order. The eigenvalues of $\varrho _{12}$ are real and non-negative even though $\varrho _{12}$ is not necessarily Hermitian. The values of the concurrence range from zero, for an unentangled state, to one, for a maximally entangled state. The state at thermal equilibrium is described by the density matrix \begin{eqnarray} \rho (T) &=&\frac 1Z\exp \left( -\beta H\right) , \nonumber \\ &=&\frac 1Z\sum_{k=0}^7\exp \left( -\beta E_k\right) |\psi _k\rangle \langle \psi _k|. \label{eq:rhoo} \end{eqnarray} where $\beta =1/T.$ From Eq.(\ref{eq:eeeigen}), the partition function is obtained as \begin{equation} Z=2+4e^{\beta J}+2e^{-2\beta J}. \label{eq:z} \end{equation} >From Eqs.(\ref{eq:eeeigen}) and (\ref{eq:rhoo}), the density matrix can be written as \begin{eqnarray} \rho (T) &=&\frac 1Z[|\psi _0\rangle \langle \psi _0|+|\psi _7\rangle \langle \psi _7| \nonumber \\ &&+e^{\beta J}(|\psi _1\rangle \langle \psi _1|+|\psi _4\rangle \langle \psi _4| \nonumber \\ &&+|\psi _2\rangle \langle \psi _2|+|\psi _5\rangle \langle \psi _5|) \nonumber \\ &&+e^{-2\beta J}(|\psi _3\rangle \langle \psi _3|+|\psi _6\rangle \langle \psi _6|)]. \label{eq:re} \end{eqnarray} In this paper we consider only pairwise thermal entanglement, and so we need to calculate the reduced density matrix $\rho_{12}(T)=\text{tr}_3(\rho(T))$. We denote the reduced density matrix tr$_3[|\psi _{i_1}\rangle \langle \psi _{i_1}|+...+|\psi _{i_N}\rangle \langle \psi _{i_N}|]$ by $\rho _{12}^{(i_1i_2...i_N)}.$ From Eq.(\ref{eq:estate}), we obtain \begin{mathletters} \begin{eqnarray} \rho _{12}^{(07)} &=&\left( \begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) , \label{eq:aaa}\\ \rho _{12}^{(1245)} &=&\frac 23\left( \begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 2 & -1 & 0 \\ 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) , \label{eq:bbb}\\ \rho _{12}^{(36)} &=&\frac 23\left( \begin{array}{llll} \frac 12 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & \frac 12 \end{array} \right),\label{eq:ccczzz}\\ \rho _{12}^{(012)} &=&\frac 23\left( \begin{array}{llll} \frac 52 & 0 & 0 & 0 \\ 0 & 1 & -\frac 12 & 0 \\ 0 & -\frac 12 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) ,\label{eq:ddd}\\ \rho _{12}^{(12)} &=&\frac 23\left( \begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & -\frac 12 & 0 \\ 0 & -\frac 12 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right).\label{eq:eee} \end{eqnarray} The last two reduced density matrices will be used later. >From Eqs. (\ref{eq:re}) and (\ref{eq:aaa}-\ref{eq:ccczzz}), we obtain \end{mathletters} \begin{eqnarray} \rho _{12}(T) &=&\frac 1Z\left( \rho _{12}^{(07)}+e^{\beta J}\rho _{12}^{(1245)}+e^{-2\beta J}\rho _{12}^{(36)}\right) \nonumber \\ &=&\frac 2{3Z}\left( \begin{array}{llll} v & 0 & 0 & 0 \\ 0 & w & y & 0 \\ 0 & y & w & 0 \\ 0 & 0 & 0 & v \end{array} \right) \label{eq:vwy} \end{eqnarray} with \begin{eqnarray} v &=&\frac 32+e^{\beta J}+\frac 12e^{-2\beta J}, \nonumber \\ w &=&2e^{\beta J}+e^{-2\beta J}, \nonumber \\ y &=&e^{-2\beta J}-e^{\beta J}. \end{eqnarray} The square roots of the four eigenvalues of the operator $\varrho _{12}$ are \begin{eqnarray} \lambda _1 &=&\frac{2(w-y)}{3Z},\text{ }\lambda _2=\frac{2(w+y)}{3Z}, \nonumber \\ \lambda _3 &=&\lambda _4=\frac{2v}{3Z}. \label{eq:lam} \end{eqnarray} >From Eqs.(\ref{eq:c1}), (\ref{eq:z}), (\ref{eq:vwy}), and (\ref{eq:lam}), we obtain the concurrence\cite{Oconnor} \begin{eqnarray} {\cal C}&=&\frac 4{3Z}\max (|y|-v,0), \label{eq:ccc} \\ &=&\max \left[ \frac{2|e^{-2x}-e^x|-3-2e^x-e^{-2x}}{ 3(1+2e^x+e^{-2x})},0\right] \label{eq:cc1} \end{eqnarray} where $x \equiv \beta J = J/T$. The concurrence depends only on the {\em ratio} of $J$ and $T$. Due to symmetry under cyclic shifts, the value of the concurrence does not depend on the choice of the pair of qubits. >From (\ref{eq:cc1}) we see that entanglement appears only when \begin{equation} \frac{2|z^{-2}-z|-3-2z-z^{-2}}{ 3(1+2z+z^{-2})} >0, \end{equation} or in other words \begin{equation} \label{fenmu} 2|z^{-2}-z|- 3 - 2 z- z^{-2} >0, \end{equation} where $z=\exp(x)$. We now consider two different cases: {\it Case 1}. Antiferromagnetic system; $J>0$;$z^{-2}-z<0$. In this case relation (\ref{fenmu}) requires \begin{equation} z^{-2}<-1 \end{equation} which is impossible. So there is no entanglement when $J>0$. {\it Case 2}. Ferromagnetic system; $J<0$,$z^{-2}-z > 0$. Relation (\ref{fenmu}) becomes \begin{equation} z^{-2}-4 z -3 >0. \end{equation} or \begin{equation} f(z) \equiv 4z^3+3z^2-1<0. \label{yfunction} \end{equation} The function $f(z)$ is an increasing function of the positive real argument $z$ and relation (\ref{yfunction}) is valid iff $0<z<z_c$, where the critical value $z_c$ determined by $f(z_c)=0$ is 0.4554; that is, $ x<-0.7866 $. For fixed $J$, we obtain the critical temperature $T_c=1.21736|J|$, above which there is no thermal entanglement. The critical temperature depends linearly on the absolute value of $J.$ In the ferromagnetic case the concurrence \begin{equation} {\cal C}= \max\left[\frac{1-4z^3-3z^2}{3(1+2z^3+z^2)},0\right] \label{eq:cc111} \end{equation} reaches its maximum value of $1/3$ when $z \to 0$, that is when $x\to -\infty$. Since the entanglement is a monotonic increasing function of ${\cal C}$ this means that the entanglement attains its maximum value for zero temperature, when $J$ is finite and nonzero. For finite temperatures, this maximum is also attained when $J\to -\infty$. In summary, we find that \\ \\ {\bf Theorem 1:} {\it The XX model is thermally entangled if and only if $J<-0.7866 T$; maximum entanglement is attained when $T \to 0$ or $J\to -\infty$.} The above discussion shows that in our 3-qubit model pairwise thermal entanglement occurs only in the ferromagnetic case. This result differs from that for the two-qubit $XX$ model, for which thermal entanglement exists in both the antiferromagnetic and ferromagnetic cases\cite{Wang1}. For the ferromagnetic case the states $|\psi _3\rangle $ and $|\psi_6\rangle $ constitute a doubly-degenerate ground state. Eq.(\ref{eq:cc111}) shows that the concurrence ${\cal C}=1/3$ at zero temperature. As noted in the last section the concurrence for any two qubits in the state $|\psi _3\rangle $ or $|\psi _6\rangle $ is 2/3. Here the value 1/3 appears due to the degeneracy. In fact, at zero temperature, the thermal entanglement can be calculated from $\rho^{(36)}_{12}$(\ref{eq:eee}). After normalization it is easy to obtain the concurrence, which is just 1/3. \section{The Anisotropic Heisenberg $XXZ$ model} We now consider a more general Heisenberg model, the anisotropic Heisenberg $XXZ$ model, which is described by the Hamiltonian\cite{A} \begin{equation} H_{XXZ}=H_{XX}+\frac{\Delta J}2\sum_{n=1}^3(\sigma _n^z\sigma _{n+1}^z-1), \end{equation} where $\Delta $ is the anisotropy parameter. The model reduces to the $XX$ model when $\Delta =0,$ and the isotropic Heisenberg $XXX$ model when $\Delta =1.$ It is straightforward to check that the added anisotropic term $H_{XXZ}-H_{XX}$ commutes with $H_{XX}$. Therefore the eigenstates of the $XXZ$ model are still given by Eq. (\ref{eq:estate}), now with the different eigenvalues \begin{eqnarray} E_0 &=&E_7=0, \nonumber \\ E_1 &=&E_2=E_4=E_5=-2J(\Delta +\frac 12), \nonumber \\ E_3 &=&E_6=-2J(\Delta -1). \end{eqnarray} Following the procedure of the previous section, we obtain the concurrence, which is of the same form as Eq.(\ref{eq:ccc}) with however the parameters $v,w,y,$ and the partition function $Z$ now given by \begin{eqnarray} v &=&\frac 32+\frac 12z^{2\Delta }(2z+z^{-2}), \nonumber \\ w &=&z^{2\Delta }(2z+z^{-2}), \nonumber \\ y &=&z^{2\Delta }(z^{-2}-z), \nonumber \\ Z &=&2+2z^{2\Delta }(2z+z^{-2}). \label{eq:vvv} \end{eqnarray} As in the last section, since $Z$ is always positive, we need only consider \begin{eqnarray} f(\Delta,z) &\equiv& |y|-v \nonumber \\ &=& z^{2\Delta}|z^{-2}-z|-\frac{3}{2}- z^{2\Delta+1}-\frac{1}{2}z^{2\Delta-2} \end{eqnarray} to determine whether entanglement occurs or not. Again, we have to consider two different cases: {\em Case 1}. When $J>0$ ($z>1$), namely the antiferromagnetic $XXZ$ model, the condition on $f(\Delta,z)$ leads to \begin{equation} z^{2\Delta-2}=e^{2x(\Delta-1)}<-1, \end{equation} which is impossible. So there is no entanglement in this case, irrespective of $\Delta$. {\em Case 2}. When $J<0$ ($z<1$), namely the ferromagnetic $XXZ$ model, the condition $f(\Delta,z)>0$ gives \begin{equation} \label{Jsmaller0} z^{2\Delta-2}-4 z^{2\Delta+1} -3 >0. \end{equation} We consider some special values of $\Delta$. (1) $\Delta\ge 1$: For $\Delta=1$ the relation (\ref{Jsmaller0}) implies $z^3<-1/2$ which is impossible. So there is no entanglement in the $XXX$ model. We can further prove that there is no entanglement for $\Delta>1$. In fact, it is easy to see that \begin{equation} f(\Delta,z)<z^{2(\Delta-1)}-3<0, \end{equation} where we have used the inequalities $z^{2\Delta+1}>0$ and $z^{2(\Delta-1)}<1$ for $\Delta>1$ and $z<1$. This means ${\cal C}=0$ and thus there is no entanglement. (2) $\Delta=1/2$: In this case\cite{Plus} the entanglement condition is obtained as \begin{equation} 4z^3+3z-1<0, \end{equation} which is an increasing function of $z$. So the model is entangled iff $0<z<z_c\approx 0.298$, where $z_c$ is determined as a root of $4z^3+3z-1=0$. (3) $\Delta=-1/2$: This is an interesting case whose importance has been emphasized recently\cite{Minus}. >From the eigenvalues we see that the excited state of the system is 6-fold degenerate when $\Delta=-1/2$. The function $f(\Delta,z)$ now reduces to $z^{-3}-7$, from which the critical values $z_c$ and $T_c$ are obtained analytically as $z_c=7^{-1/3}, T_c={3}|J|/{\ln 7}\approx 1.5417|J|$. (4) The limit case $\Delta\to -\infty$: The critical value $z_c$ is now determined by $z^{-2}-4z=0$, i.e., $z_c=4^{-1/3}$. Therefore the critical temperature $T_c=3|J|/\ln 4\approx 2.164 |J|$. Finally, for more general values of the anisotropy parameter we need to resort to numerical calculations. Fig.1 is a plot of the critical temperature as a function of the anisotropy parameter $\Delta$. From this we see that the critical temperature decreases as $\Delta$ increases, and reaches the asymptotic value $T_c=2.1640|J|$ as $\Delta \rightarrow -\infty .$ \begin{figure} \caption{The critical temperature $T_c$ as a function of the anisotropy parameter $\Delta$. The exchange constant $J=-1$.} \end{figure} We now give further analytical results for the case $\Delta<1$ and $z<1$. Consider $f(\Delta, z)$ as a function of $\Delta$. Then, from \begin{eqnarray} \frac{\partial f(\Delta, z)}{\partial \Delta} &=&(\ln z) z^{2\Delta} (z^{-2}-4z) \nonumber\\ && \left\{\begin{array}{ll} =0, & \mbox{when } z=z_0\equiv 4^{-1/3}\approx 0.62996; \\ >0, & \mbox{when } z>z_0; \\ <0, & \mbox{when } z<z_0. \end{array} \right. \label{eq:gradients} \end{eqnarray} we see that $f(\Delta, z)$ is an increasing (decreasing) function when $z>z_0$ ($z<z_0$). We consider these cases separately. {\em Case 2a}. When $z=z_0$, $f(\Delta, z_0)=-3<0$. So there is no entanglement in this case. {\em Case 2b}. When $z>z_0$, the function $f(\Delta, z)$ is an increasing function which reaches its maximum when $\Delta\to 1$. Since we have seen that there is no entanglement when $\Delta=1$ \begin{equation} f(\Delta, z)<f(1, z)<0 \quad \mbox{for }z>z_0, \end{equation} which means that there is no entanglement when $z>z_0$. {\em Case 2c}. The case $z<z_0$. Define the $z$-dependent point $\Delta_z$ by $f(\Delta_z, z)=0$ where \begin{equation} \Delta_z=\frac{1}{2\beta J} \ln \left[ \frac{3}{z^{-2}-4z}\right]<1. \label{eq:delta} \end{equation} Thus from Eq.(\ref{eq:gradients}) we know that $f(\Delta, z)>0$ when $\Delta<\Delta_z$ for all $z<z_0$, which is just the condition for entanglement. In Fig.\,2 we give plots of $f(\Delta, z)$ for $z=0.6295, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1$. Note that $\Delta_z$ is a decreasing function of $z$ and that \begin{equation} \Delta_z \to \left\{ \begin{array}{ll} -\infty & \mbox{when } z\to z_0; \\ 1 & \mbox{when } z\to 0, \end{array} \right. \end{equation} as indicated in Fig.\,2. \begin{figure} \caption{The function $f(\Delta, z)$ with respect $\Delta$ for $z=0.6295, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1$ (from left to right).} \end{figure} In summary, we have \\ \\ {\bf Theorem 2} {\it The XXZ model exhibits thermal entanglement only when $z<z_0$, that is, $J<-.4621T$, and $\Delta<\Delta_z$.} Note that Theorem 2 is entirely consistent with Theorem 1, since $\Delta_z=0$ in Eq.(\ref{eq:delta}) corresponds to $J=-0.7866T.$ In Fig.3 we plot the concurrence as a function of the anisotropy parameter $\Delta $ and exchange constant $J$. The figure shows that there is no thermal entanglement for the antiferromagnetic ($J>0$) $XXZ$ model, nor for the ferromagnetic ($J<0$) $XXZ$ model when $\Delta \ge 1$. \begin{figure} \caption{The concurrence as a function of $\Delta$ and $J$. The temperature $T=1$.} \end{figure} To end this section we investigate the concurrence at zero temperature. Nonzero concurrence occurs for the case $\Delta<1, J<0$. In this case, the doubly-degenerate ground state consists of $|\psi_3\rangle $ and $|\psi_6\rangle $. We may calculate the concurrence ${\cal C} = 1/3$ directly from the density matrix $\rho^{(36)}_{12}$. \section{Effects of magnetic fields} In this section we consider the effect of magnetic fields on thermal entanglement. The $XXZ$ model with uniform magnetic field $B$ along the $z$ direction is given by \begin{equation} H_{XXZM}=H_{XXZ}+B\sum_{n=1}^3\sigma _n^z. \end{equation} It is easy to check that the added magnetic term commutes with the Hamiltonian $H_{XXZ}$. Therefore the eigenstates of the $XXZ$ model are given by Eq.(\ref{eq:estate}). The eigenvalues are now \begin{eqnarray} E_0 &=&-3B \nonumber \\ E_1 &=&E_2=-2J(\Delta +\frac 12)-B \nonumber \\ E_3 &=&-2J(\Delta -1)-B, \nonumber \\ E_4 &=&E_5=-2J(\Delta +\frac 12)+B, \nonumber \\ E_6 &=&-2J(\Delta -1)+B. \nonumber \\ E_7 &=&3B\label{eq:eigen} \end{eqnarray} We see that the magnetic field partly removes the degeneracy. With a derivation completely analogous to that of Sec. III and Sec. IV, the reduced density operator is \begin{equation} \rho _{12}=\frac 2{3Z}\left( \begin{array}{llll} u & 0 & 0 & 0 \\ 0 & w & y & 0 \\ 0 & y & w & 0 \\ 0 & 0 & 0 & v \end{array} \right) \end{equation} with \begin{eqnarray} u &=&\frac 32e^{3\beta B}+\frac 12 e^{\beta B}z^{2\Delta}\left( 2z+z^{-2}\right) , \nonumber \\ v &=&\frac 32e^{-3\beta B}+\frac 12e^{-\beta B}z^{2\Delta}\left( 2z+z^{-2}\right) , \nonumber \\ w &=&\cosh (\beta B)z^{2\Delta }\left( 2z+z^{-2}\right) , \nonumber \\ y &=&\cosh (\beta B)z^{2\Delta }\left(z^{-2}-z\right) , \nonumber \\ Z &=&2\cosh (3\beta B)+2\cosh (\beta B)z^{2\Delta} \nonumber \\ &&\times (2z+z^{-2}) \end{eqnarray} The concurrence is then given by \begin{equation} {\cal C}=\frac 4{3Z}\max (|y|-\sqrt{uv},0). \end{equation} As an immediate consequence we see that the concurrence is an even function of the magnetic field. As the quantities $Z, u, v$ are all positive, for convenience we consider the quantity $y^2-uv$ instead of $|y|-\sqrt{uv}$. Thermal entanglement occurs when \begin{equation} y^2-uv=h\cosh (2\beta B)-g>0, \end{equation} where \begin{eqnarray} &&g=\frac{1}{4}[9+z^{4(\Delta -1)}(2z^6+8z^3-1)], \nonumber \\ &&h=\frac{1}{2}z^{2\Delta }\left[ z^{2\Delta }(z^{-2}-z)^2-(6z+3z^{-2})\right] . \label{eq:yyuv} \end{eqnarray} We now consider the effect of a magnetic field on the thermal entanglement. We first consider the $XXX$ model, $\Delta =1$, which does not exhibit thermal entanglement when $B=0$. One might expect that the magnetic field would induce thermal entanglement. It is easy to see that \begin{equation} 2(y^2-uv)=\cosh (2\beta B)\left( z^6-8z^3-2\right) -(z^3+2)^2. \end{equation} If $z<(4+3\sqrt{2})^{1/3}\approx 2.02$, $z^6-8z^3-2<0$ and thus $y^2-uv<0$ for any $B$. So for this range of $z$ values there is no thermal entanglement no matter how strong the magnetic field is. However, when $z>(4+3\sqrt{2})^{1/3}$, $z^6-8z^3-2>0$ and the condition for entanglement becomes \begin{equation} \cosh (2\beta B)>\frac{(z^3+2)^2}{z^6-8z^3-2} \end{equation} which can be fulfilled for strong enough $B$. So a magnetic field can induce entanglement in the $XXX$ model when $z>(4+3\sqrt{2})^{1/3}$. Now consider the case $\Delta =-1/2$. >From Eq.(\ref{eq:yyuv}) we obtain \begin{eqnarray} h &=&\frac 12(p^2-5p-5), \\ g &=&\frac 14(11+8p-p^2), \\ h-g &=&\frac 14(3p^2-18p-21), \end{eqnarray} which are parabolas in $p\equiv z^{-3}$, as shown in Fig.\,4. We consider three different cases: {\em Case 1: } $p<p_1=5/2+3\sqrt{5}/2$, In this case $h<0,g>0,h-g<0$ and $y^2-uv<0$. So there is no thermal entanglement. {\em Case 2:} $p_1<p<p_2=7$. In this case $h>0,g>0,h-g<0$. So $y^2-uv>0$, and so entanglement appears if the magnetic field is strong enough. {\em Case 3:} $p_2<p$. In this case $h>0, h-g>0$ and $y^2-uv$ is always positive; that is, here the XXZ model exhibits thermal entanglement for any magnetic field. Note that $p_2 = z^{-3}_c$ where $z_c$ is the critical value given in last section. The above two models show that the magnetic field can either induce entanglement in a non-entangled system or extend the entanglement range for an already entangled system. \begin{figure} \caption{The functions $h$, $g$, and $h-g$ in terms of $p=z^{-3}$.} \end{figure} In Fig.5 we plot the concurrence as a function of the magnetic field $B$ and exchange constant $J.$ At $B=0$ there is no thermal entanglement. The entanglement increases with the magnetic field $|B|$ until it reaches a maximum value, then decreases and gradually disappears. We can clearly see that there is no thermal entanglement for the ferromagnetic case, while thermal entanglement exists for the antiferromagnetic case. In other words, we can induce entanglement in the antiferromagnetic $XXX$ system by introducing a magnetic field, but cannot induce entanglement in the ferromagnetic $XXX$ system for any strength of magnetic field. \begin{figure} \caption{ Concurrence as a function of the magnetic field $B$ and the exchange constant $J$. The temperature $T=1$ and the anisotropy parameter $\Delta=1$. } \end{figure} \begin{figure} \caption{ Concurrence as a function of $T$ for different magnetic fields $B=1$(solid line), 3/2(dashed line), and 2(circle point line). } \end{figure} Fig. 6 gives a plot of the concurrence as a function of the temperature for different magnetic fields. One can see that there exist critical temperatures above which the entanglement vanishes. It is also noteworthy that the critical temperature increases as the magnetic field $B$ increases. Consider the interesting case $B=2$. We observe that the concurrence is zero at zero temperature and there is a maximum value of concurrence at a finite temperature. The entanglement can be increased by increasing the temperature. The maximum value is due to the optimal mixing of all eigenstates in the system. When considering zero temperature we find that there are different limits for different magnetic fields. Actually a more general result exists \begin{eqnarray*} \lim_{T\rightarrow 0}{\cal C}(\Delta ,B,1,T) &=&\frac 13\text{ for }\Delta >|B|-1/2. \\ &=&\frac 29\text{ for }\Delta =|B|-1/2, \\ &=&0\text{ for }\Delta <|B|-1/2. \end{eqnarray*} The special point $T=0, \; \; \Delta=B-1/2$ ($B\ge 0$ is assumed without loss of generality), at which the entanglement undergoes a sudden change with adjustment of the parameters $\Delta$ and $B$, is the point of quantum phase transition\cite{QPT}. The quantum phase transition takes place at zero temperature due to the variation of interaction terms in the Hamiltonian. By examining the eigenvalues (\ref{eq:eigen}) we can understand the phase transition. When $\Delta=B-1/2$, the ground state contains the three-fold degenerate states $|\psi_0\rangle, |\psi_1\rangle$, and $|\psi_2\rangle$. One may calculate the thermal entanglement from the density matrix $\rho_{12}^{(012)}$(\ref{eq:ddd}) and find the concurrence to be $2/9$. When $\Delta>B-1/2$, the ground state contains the two-fold degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$. The concurrence has the value $1/3$ as calculated from $\rho_{12}^{(12)}$(\ref{eq:eee}). When $\Delta<B-1/2$, the ground state is $|\psi_0\rangle$ and not degenerate. And the concurrence is zero in this case. \section{Conclusions} Apart from being a fundamental property of quantum mechanics, it appears that entanglement may provide an important resource in quantum information processes. One source of entanglement is provided by magnetic systems, such as those modelled in this paper. Within the current state of knowledge, only measures for {\em pairwise} entanglement are available. Thus, in order to study the entanglement properties of systems more complex than those simply involving two qubits, it is necessary to adopt a procedure whereby one traces out a subsystem, leaving effectively only a two-qubit system for which we can calculate the {\em concurrence}, which in turn gives a measure of the entanglement. Using this procedure, we have studied pairwise thermal entanglement in the following Heisenberg models; the $XX$ model, the $XXZ$ model and the $XXZ$ model in a magnetic field. We obtained analytical expressions for the concurrence, which indicated no thermal entanglement for the antiferromagnetic $XXZ$ model, nor for the ferromagnetic $XXZ$ model when the anisotropy parameter $\Delta \ge 1$. Conditions for the existence of thermal entanglement were studied in detail. The effects of magnetic fields on entanglement were also considered. We found that the magnetic field can induce entanglement in the antiferromagnetic $XXX$ model, but cannot induce entanglement in the ferromagnetic $XXX$ model, no matter how strong the magnetic field is. In this paper we have extended previous work on thermal entanglement from two-qubit models to three qubit models, concentrating on those systems where the pairwise entanglement can be studied analytically. It would be an attractive proposition to extend further the investigation of such Heisenberg models to the $N$-qubit case, which are under consideration. \acknowledgments X. Wang thanks K. M\o lmer, A. S\o rensen, W. K. Wootters and Paolo Zanardi for many valuable discussions. He is supported by the Information Society Technologies Programme IST-1999-11053, EQUIP, action line 6-2-1 and European project Q-ACTA. 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\begin{document} \def\paragraph{Proof.}{\paragraph{Proof.}} \def $\square${ $\square$} \def $\square${ $\square$} \def{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}{{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}} \def{\mathbb Q}}\def\R{{\mathbb R}} \def\E{{\mathbb E}{{\mathbb Q}}\def\R{{\mathbb R}} \def\E{{\mathbb E}} \def{\mathbb P}{{\mathbb P}} \def{\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G}{{\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G}} \def{\cal A}}\def\cd{{\rm cd}}\def\mf{{\rm mf}{{\cal A}}\def\cd{{\rm cd}}\def\mf{{\rm mf}} \def{\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}{{\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}} \def{\cal H}}\def\Ker{{\rm Ker}{{\cal H}}\def\Ker{{\rm Ker}} \title{\bf{Artin groups of spherical type up to isomorphism}} \author{ \textsc{Luis Paris}} \date{\today} \maketitle \begin{abstract} We prove that two Artin groups of spherical type are isomorphic if and only if their defining Coxeter graphs are the same. \end{abstract} \noindent {\bf AMS Subject Classification:} Primary 20F36. \section{Introduction} Let $S$ be a finite set. Recall that a {\it Coxeter matrix} over $S$ is a matrix $M=(m_{s\,t})_{s,t \in S}$ indexed by the elements of $S$ such that $m_{s\,s}=1$ for all $s \in S$, and $m_{s\,t}=m_{t\,s} \in \{2, 3, 4, \dots, +\infty\}$ for all $s,t \in S$, $s \neq t$. A Coxeter matrix $M=(m_{s\,t})$ is usually represented by its {\it Coxeter graph}, $\Gamma$, which is defined as follows. The set of vertices of $\Gamma$ is $S$, two vertices $s,t$ are joined by an edge if $m_{s\,t}\ge 3$, and this edge is labelled by $m_{s\,t}$ if $m_{s\,t} \ge 4$. For $s,t \in S$ and $m \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}_{\ge 2}$, we denote by $w(s,t:m)$ the word $sts \dots$ of length $m$. The {\it Artin group} associated to $\Gamma$ is defined to be the group $G=G_\Gamma$ presented by $$ G= \langle S\ |\ w(s,t:m_{s\,t}) = w(t,s:m_{s\,t})\text{ for } s,t \in S,\ s \neq t\text{ and } m_{s\,t} < +\infty \rangle\,. $$ The {\it Coxeter group} $W=W_\Gamma$ associated to $\Gamma$ is the quotient of $G$ by the relations $s^2=1$, $s\in S$. We say that $\Gamma$ (or $G$) is of {\it spherical type} if $W$ is finite, that $\Gamma$ (or $G$) is {\it right-angled} if $m_{s\,t} \in \{2, +\infty \}$ for all $s,t \in S$, $s \neq t$, and that $G$ (or $W$) is {\it irreducible} if $\Gamma$ is connected. The number $n=|S|$ is called the {\it rank} of $G$ (or of $W$). One of the main question in the subject is the classification of Artin groups up to isomorphism (see \cite{Bes2}, Question 2.14). This problem is far from being completely solved as Artin groups are poorly understood in general. For example, we do not know whether all Artin groups are torsion free, and we do not know any general solution to the word problem for these groups. The only known results concerning this classification question are contained in a work by Brady, McCammond, M\"uhlherr, and Neumann \cite{BMMN}, where the authors determine a sort of transformation on Coxeter graphs which does not change the isomorphism class of the associated Artin groups, and a work by Droms \cite{Dro}, where it is proved that, if $\Gamma$ and $\Omega$ are two right-angled Coxeter graphs whose associated Artin groups are isomorphic, then $\Gamma= \Omega$. Notice that an Artin group is biorderable if and only if it is right-angled, hence a consequence of Droms' result is that, if $\Gamma$ is a right-angled Coxeter graph and $\Omega$ is any Coxeter graph, and if the Artin groups associated to $\Gamma$ and $\Omega$ are isomorphic, then $\Gamma=\Omega$. The fact that right-angled Artin groups are biorderable is proved in \cite{DuTh}. In order to show that the remainig Artin groups are not biorderable, one has only to observe that, if $2<m_{s\,t}< +\infty$, then $(st)^{m_{s\,t}} = (ts)^{m_{s\,t}}$ and $st \neq ts$, and that, in a biorderable group, two distinct elements cannot have a common $m$-th power for a fixed $m$. In this paper we answer the classification question in the restricted framework of spherical type Artin groups. More precisely, we prove the following. \begin{thm} Let $\Gamma$ and $\Omega$ be two spherical type Coxeter graphs, and let $G$ and $H$ be the Artin groups associated to $\Gamma$ and $\Omega$, respectively. If $G$ is isomorphic to $H$, then $\Gamma=\Omega$. \end{thm} \noindent {\bf Remark.} I do not know whether a non spherical type Artin group can be isomorphic to a spherical type Artin group. Artin groups were first introduced by Tits \cite{Tit2} as extensions of Coxeter groups. Later, Brieskorn \cite{Bri1} gave a topological interpretation of the Artin groups of spherical type in terms of complements of discriminantal varieties. Define a (real) {\it reflection group} of rank $n$ to be a finite subgroup $W$ of $GL(n,\R)$ generated by reflections. Such a group is called {\it essential} if there is no non-trivial subspace of $\R^n$ on which $W$ acts trivially. Let ${\cal A}}\def\cd{{\rm cd}}\def\mf{{\rm mf}$ be the set of reflecting hyperplanes of $W$, and, for $H \in {\cal A}}\def\cd{{\rm cd}}\def\mf{{\rm mf}$, let $H_\C$ denote the complexification of $H$, {\it i.e.} the complex hyperplane in $\C^n$ defined by the same equation as $H$. Then $W$ acts freely on $M(W)= \C^n \setminus \cup_{H \in {\cal A}}\def\cd{{\rm cd}}\def\mf{{\rm mf}} H_\C$, and, by \cite{Che}, $N(W)= M(W)/W$ is the complement in $\C^n$ of an algebraic variety, $D(W)$, called {\it discriminantal variety} of type $W$. Now, take a spherical type Coxeter graph $\Gamma$, and consider the associated Coxeter group $W=W_\Gamma$. By \cite{Cox}, the group $W$ can be represented as an essential reflection group in $GL(n,\R)$, where $n=|S|$ is the rank of $W$, and, conversely, any essential reflection group of rank $n$ can be uniquely obtained in this way. By \cite{Bri1}, $\pi_1(N(W))$ is the Artin group $G=G_\Gamma$ associated to $\Gamma$. So, a consequence of Theorem 1.1 is that $\pi_1(N(W))= \pi_1( \C^n \setminus D(W))$ completely determines the reflection group $W$ as well as the discriminantal variety $D(W)$. Since the work of Brieskorn and Saito \cite{BrSa} and that of Deligne \cite{Del}, the combinatorial theory of spherical type Artin groups has been well studied. In particular, these groups are know to be biautomatic (see \cite{Cha2}, \cite{Cha}), and torsion-free. This last result is a direct consequence of \cite{Del} and \cite{Bri1}, it is explicitely proved in \cite{Deh}, and it shall be of importance in the remainder of the paper. The first step in the proof of Theorem 1.1 consists of calculating some invariants for spherical type Artin groups (see Section 3). It actually happens that these invariants separate the irreducible Artin groups of spherical type (see Proposition 5.1). Afterwards, for a given isomorphism $\varphi: G \to H$ between spherical type Artin groups, we show that, up to some details, $\varphi$ sends each irreducible component of $G$ injectively into a unique irreducible component of $H$, and that both components have the same invariants. In order to do that, we first need to show that an irreducible Artin group $G$ cannot be decomposed as a product of two subgroups which commute, unless one of these subgroups lies in the center of $G$ (see Section 4). From now on, $\Gamma$ denotes a spherical type Coxeter graph, $G$ denotes its associated Artin group, and $W$ denotes its associated Coxeter group. \noindent {\bf Acknowledgments.} The idea of looking at centralizers of ``good'' elements in the proof of Proposition 4.2 is a suggestion of Benson Farb. I am grateful to him for this clever idea as well as for all his useful conversations. I am also grateful to Jean Michel who pointed out to me his work with Michel Brou\'e, and to John Crisp for so many discussions on everything concerning this paper. \section{Preliminaries} We recall in this section some well-known results on Coxeter groups and Artin groups. For a subset $X$ of $S$, we denote by $W_X$ the subgroup of $W$ generated by $X$, and by $G_X$ the subgroup of $G$ generated by $X$. Let $\Gamma_X$ be the full Coxeter subgraph of $\Gamma$ whose vertex set is $X$. Then $W_X$ is the Coxeter group associated to $\Gamma_X$ (see \cite{Bou}), and $G_X$ is the Artin group associated to $\Gamma_X$ (see \cite{Lek} and \cite{Par1}). The subgroup $W_X$ is called {\it standard parabolic subgroup} of $W$, and $G_X$ is called {\it standard parabolic subgroup} of $G$. For $w \in W$, we denote by ${\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (w)$ the word length of $w$ with respect to $S$. The group $W$ has a unique element of maximal length, $w_0$, which satisfies $w_0^2=1$ and $w_0 S w_0 =S$, and whose length is $m_1 + \dots + m_n$, where $m_1, m_2, \dots, m_n$ are the exponents of $W$. The connected spherical Coxeter graphs are exactly the graphs $A_n$ ($n \ge 1$), $B_n$ ($n\ge 2$), $D_n$ ($n\ge 4$), $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$, $I_2(p)$ ($p \ge 5$) represented in \cite{Bou}, Ch. IV, \S 4, Thm. 1. (Here we use the notation $I_2(6)$ for the Coxeter graph $G_2$. We may also use the notation $I_2(3)$ for $A_2$, and $I_2(4)$ for $B_2$.) Let $F:G \to W$ be the natural epimorphism which sends $s$ to $s$ for all $s \in S$. This epimorphism has a natural set-section $T: W \to G$ defined as follows. Let $w \in W$, and let $w=s_1s_2 \dots s_l$ be a reduced expression of $w$ ({\it i.e.} $l={\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G}(w)$). Then $T(w)=s_1s_2 \dots s_l \in G$. By Tits' solution to the word problem for Coxeter groups \cite{Tit}, the definition of $T(w)$ does not depend on the choice of the reduced expression. Define the {\it Artin monoid} associated to $\Gamma$ to be the (abstract) monoid $G^+$ presented by $$ G^+= \langle S\ |\ w(s,t: m_{s\,t}) = w(t,s: m_{s\,t}) \text{ for } s\neq t \text{ and } m_{s\,t} <+\infty \rangle^+\,. $$ By \cite{BrSa}, the natural homomorphism $G^+ \to G$ which sends $s$ to $s$ for all $s \in S$ is injective. Note that this fact is always true, even if $\Gamma$ is not assumed to be of spherical type (see \cite{Par2}). The {\it fundamental element} of $G$ is defined to be $\Delta= T(w_0)$, where $w_0$ denotes the element of $W$ of maximal length. For $X \subset S$, We denote by $w_X$ the element of $W_X$ of maximal length, and by $\Delta_X=T(w_X)$ the fundamental element of $G_X$. The defining relations of $G^+$ being homogeneous, we can define two partial orders $\le_L$ and $\le_R$ on $G^+$ as follows. $\bullet$ We set $a \le_L b$ if there exists $c \in G^+$ such that $b=ac$. $\bullet$ We set $a \le_R b$ if there exists $c \in G^+$ such that $b=ca$. Now, the following two propositions are a mixture of several well-known results from \cite{BrSa} and \cite{Del}. \begin{prop} (1) $G^+$ is cancellative. (2) $(G^+, \le_L)$ and $(G^+, \le_R)$ are lattices. (3) $\{a \in G^+; a\le_L \Delta\} = \{a \in G^+; a\le_R \Delta \} = T(W)$. $\square$ \end{prop} Note that the fact that $G^+$ is cancellative is true even if $\Gamma$ is not of spherical type (see \cite{Mic}). The elements of $T(W)$ are called {\it simple elements}. We shall denote the lattice operations of $(G^+, \le_L)$ by $\vee_L$ and $\wedge_L$, and the lattice operations of $(G^+, \le_R)$ by $\vee_R$ and $\wedge_R$. Define the {\it quasi-center} of $G$ to be the subgroup $QZ(G)=\{ a \in G; aSa^{-1} = S \}$. \begin{prop} Assume $\Gamma$ to be connected. (1) For $X \subset S$ we have $$ \vee_L \{s; s\in X\} = \vee_R \{s; s\in X\} = \Delta_X\,. $$ In particular, $$ \vee_L \{s; s\in S\} = \vee_R \{s; s\in S\} = \Delta\,. $$ (2) There exists a permutation $\mu: S \to S$ such that $\mu^2= \Id$ and $\Delta s = \mu(s) \Delta$ for all $s \in S$. (3) The quasi-center $QZ(G)$ of $G$ is an infinite cyclic subgroup generated by $\Delta$. (4) The center $Z(G)$ of $G$ is an infinite cyclic subgroup of $G$ generated either by $\delta= \Delta$ if $\mu=\Id$, or by $\delta= \Delta^2$ if $\mu\neq \Id$. $\square$ \end{prop} The generator $\delta$ of $Z(G)$ given in the above proposition shall be called the {\it standard generator} of $Z(G)$. Note also that the assumption ``$\Gamma$ is connected'' is not needed in (1) and (2). Let $\Gamma$ be connected. Then $\mu \neq \Id$ if and only if $\Gamma$ is either $A_n$, $n\ge 2$, or $D_{2n+1}$, $n\ge 2$, or $E_6$, or $I_2(2p+1)$, $p\ge 2$ (see \cite{BrSa}, Subsection 7.2). Now, the following result can be found in \cite{Cha}. \begin{prop}[Charney \cite{Cha}] Each $a \in G$ can be uniquely written as $a=bc^{-1}$ where $b,c \in G^+$ and $b \wedge_R c =1$. $\square$ \end{prop} The expression $a=bc^{-1}$ of the above proposition shall be called the {\it Charney form} of $a$. An easy observation shows that, if $s_1s_2 \dots s_l$ and $t_1 t_2 \dots t_l$ are two positive expressions of a same element $a \in G^+$, then the sets $\{s_1, \dots, s_l\}$ and $\{t_1, \dots, t_l\}$ are equal. In particular, if $a \in G_X^+$, then all the letters that appear in any positive expression of $a$ lie in $X$. A consequence of this fact is the following. \begin{lemma} Let $X$ be a subset of $S$, let $a\in G_X$, and let $a=bc^{-1}$ be the Charney form of $a$ in $G$. Then $b,c \in G_X^+$ and $a=bc^{-1}$ is the Charney form of $a$ in $G_X$. \end{lemma} \paragraph{Proof.} Let $\vee_{X,R}$ and $\wedge_{X,R}$ denote the lattice operations of $(G_X^+,\le_R)$. The above observation shows that, if $a\le_R b$ and $b \in G_X^+$, then $a \in G_X^+$. This implies that $b \wedge_{X,R} c = b \wedge_R c$ for all $b,c \in G_X^+$. Now, let $a \in G_X$ and let $a=bc^{-1}$ be the Charney form of $a$ in $G_X$. We have $b,c \in G_X^+ \subset G^+$ and $b \wedge_R c = b \wedge_{X,R} c = 1$, thus $a=bc^{-1}$ is also the Charney form of $a$ in $G$. $\square$ \begin{corollary} Let $X$ be a subset of $S$. Then $G_X \cap G^+ = G_X^+$. $\square$ \end{corollary} \begin{corollary} Let $X$ be a subset of $S$, $X \neq S$. Then $G_X \cap \langle \Delta \rangle = \{ 1 \}$. \end{corollary} \paragraph{Proof.} Take $s \in S \setminus X$. By Proposition 2.2, we have $s \le_R \Delta$, thus $\Delta \not \in G_X^+ = G_X \cap G^+$. $\square$ \section{Invariants} The purpose of the present section is to calculate some invariants of the spherical type Artin groups. The first invariant that we want to calculate is the cohomological dimension, denoted by $\cd (G)$. We assume the reader to be familiar with this notion, and we refer to \cite{Bro} for definitions and properties. Our result is the following. \begin{prop} Let $n=|S|$ be the rank of $G=G_\Gamma$. Then $\cd (G)=n$. \end{prop} \paragraph{Proof.} Recall the spaces $M(W)$ and $N(W)$ defined in the introduction. Recall also that $\pi_1(N(W))=G$, that $W$ acts freely on $M(W)$, and that $N(W)=M(W)/W$. In particular, $\pi_1(M(W))$ is a subgroup of $\pi_1(N(W))=G$ (it is actually the kernel of the epimorphism $F: G \to W$). Finally, recall the well-known fact that, if $H_1$ is a subgroup of a given group $H_2$, then $\cd (H_1) \le \cd (H_2)$. Deligne proved in \cite{Del} that $M(W)$ is aspherical, and Brieskorn proved in \cite{Bri2} that $H^n(M(W), {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C})$ is a free abelian group of rank $\prod_{i=1}^n m_i \neq 0$, where $m_1, m_2, \dots, m_n$ are the exponents of $W$, thus $n \le \cd( \pi_1( M(W)) \le \cd(G)$. On the other hand, Salvetti has constructed in \cite{Sal} an aspherical CW-complex of dimension $n$ whose fundamental group is $G$, therefore $\cd (G) \le n$. $\square$ The next invariant which interests us is denoted by $\mf (G)$ and is defined to be the maximal order of a finite subgroup of $G/Z(G)$, where $Z(G)$ denotes the center of $G$. Its calculation is based on Theorems 3.2 and 3.3 given below. Recall the permutation $\mu: S \to S$ of Proposition 2.2. This extends to an isomorphism $\mu: G^+ \to G^+$ which permutes the simple elements. Actually, $\mu(a)= \Delta a \Delta^{-1}$ for all $a \in G^+$. \begin{thm}[Bestvina \cite{Bes}] Assume $\Gamma$ to be connected. Let $\GG= G/ \langle \Delta^2 \rangle$, and let $H$ be a finite subgroup of $\GG$. Then $H$ is a cyclic group, and, up to conjugation, $H$ has one of the following two forms. \noindent {\bf Type 1:} The order of $H$ is even, say $2p$, and there exists a simple element $a \in T(W)$ such that $a^p=\Delta$, $\mu(a)=a$, and $\overline{a}$ generates $H$, where $\overline{a}$ denotes the element of $\GG$ represented by $a$. \noindent {\bf Type 2:} The order of $H$ is odd, say $2p+1$, and there exists a simple element $a \in T(W)$ such that $(a\, \mu(a))^{{p-1}\over 2} a = \Delta$ and $\overline{ a\,\mu(a)}$ generates $H$. $\square$ \end{thm} Now, recall the so-called {\it Coxeter number} $h$ of $W$ (see \cite{Hum}, Section 3.18). Recall also that this number is related to the length of $\Delta$ by the following formula $$ {{nh}\over 2} = m_1+ \dots +m_n= {\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (\Delta)\,, $$ where $n=|S|$ is the rank of $G$, and $m_1, \dots, m_n$ are the exponents of $W$. \begin{thm}[Brieskorn-Saito \cite{BrSa}] Choose any order $S=\{s_1,s_2, \dots, s_n\}$ of $S$ and write $\pi=s_1s_2 \dots s_n \in G$. Let $h$ be the Coxeter number of $W$. (1) If $\mu = \Id$, then $h$ is even and $\pi^{h \over 2} = \Delta$. (2) If $\mu \neq \Id$, then $\pi^h= \Delta^2$. $\square$ \end{thm} Now, we can calculate the invariant $\mf (G)$. \begin{prop} Assume $\Gamma$ to be connected, and let $h$ be the Coxeter number of $W$. (1) If $\mu = \Id$, then $\mf (G)= h/2$. (2) If $\mu \neq \Id$, then $\mf (G)=h$. \end{prop} \paragraph{Proof.} Assume $\mu=\Id$. Let $\GG^0= G/Z(G) = G/ \langle \Delta \rangle$. First, observe that $\mf(G) \ge {h \over 2}$ by Theorem 3.3. So, it remains to prove that $\mf(G) \le {h \over 2}$, namely, that $|H| \le {h \over 2}$ for any finite subgroup $H$ of $\GG^0$. Let $H$ be a finite subgroup of $\GG^0$. Consider the exact sequence $$ 1 \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}/ 2{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C} \to \GG \stackrel{\phi}{\rightarrow} \GG^0 \to 1\,, $$ where $\GG= G/ \langle \Delta^2 \rangle$, and set $\tilde H= \phi^{-1}(H)$. By Theorem 3.2, $\tilde H$ is a cyclic group and, up to conjugation, $\tilde H$ is either of Type 1 or of Type 2. The order of $\tilde H$ is even, say $2p$, thus $\tilde H$ is of Type 1, and there exists a simple element $a \in T(W)$ such that $a^p=\Delta$ and $\overline{a}$ generates $\tilde H$. Let $a=s_1 s_2 \dots s_r$ be an expression of $a$, and let $X=\{s_1, s_2, \dots, s_r\}$. We have $\Delta= a^p \in G_X$, thus, by Corollary~2.6, $X=S$ and $r={\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (a) \ge |S|=n$. Finally, $$ |H|={|\tilde H| \over 2} = p = {{\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (\Delta) \over {\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (a)} = \left( {nh \over 2} \right) /r \le {h \over 2}\,. $$ Now, assume $\mu \neq \Id$. Let $\GG= G/Z(G)= G/\langle \Delta^2 \rangle$. First, observe that $\mf (G) \ge h$ by Theorem 3.3. So, it remains to prove that $\mf (G) \le h$, namely, that $|H| \le h$ for any finite subgroup $H$ of $\GG$. Let $H$ be a finite subgroup of $\GG$. By Theorem 3.2, $H$ is cyclic and, up to conjugation, $H$ is either of Type 1 or of Type 2. Let $p$ be the order of $H$. In both cases, Type 1 and Type 2, there exists an element $b \in G^+$ such that $b^p=\Delta^2$ and $\overline{b}$ generates $H$ (take $b=a$ if $H$ is of Type 1, and $b=a\, \mu(a)$ if $H$ is of Type 2). Let $b=s_1 s_2 \dots s_r$ be an expression of $b$, and let $X=\{s_1, s_2, \dots, s_r\}$. We have $\Delta^2=b^p \in G_X$, thus, by Corollary~2.6, $X=S$ and $r={\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (b) \ge |S|=n$. It follows that $$ |H| = p = {{\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (\Delta^2) \over {\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (b)} = {nh \over r} \le h\,. $$ $\square$ The values of the Coxeter numbers of the irreducible Coxeter groups are well-known (see, for instance, \cite{Hum}, Section 3.18). Applying Proposition 3.4 to these values, one can easily compute the invariant $\mf (G)$ for each irreducible (spherical type) Artin group. The result is given in Table 1. \def\vrule height 0pt depth 8pt width 0pt{\vrule height 0pt depth 8pt width 0pt} \def\vrule height 16pt depth 0pt width 0pt{\vrule height 16pt depth 0pt width 0pt} \def\vrule height 16pt depth 8pt width 0pt{\vrule height 16pt depth 8pt width 0pt} $$\vbox{ \begin{tabular}{ccccccccccccccc} \hline \vrule height 16pt depth 0pt width 0pt&{\vline\kern -0.2 em \vline}&&\vline&&\vline&&\vline& $D_n,\, n\ge 4$ &\vline& $D_n,\, n\ge 5$ &\vline&&\vline\\ \vrule height 0pt depth 8pt width 0pt$\Gamma$ &{\vline\kern -0.2 em \vline}& $A_1$ &\vline& $A_n,\ n\ge 2$ &\vline& $B_n,\, n\ge 2$ &\vline& $n$ even &\vline& $n$ odd &\vline& $E_6$ &\vline\\ \hline \vrule height 16pt depth 8pt width 0pt$\mf (G)$ &{\vline\kern -0.2 em \vline}& 1 &\vline& $n+1$ &\vline& $n$ &\vline& $n-1$ &\vline& $2n-2$ &\vline& 12 &\vline\\ \hline \end{tabular} \par \begin{tabular}{ccccccccccccccccc} \hline \vrule height 16pt depth 0pt width 0pt&{\vline\kern -0.2 em \vline}&&\vline&&\vline&&\vline&&\vline&&\vline& $I_2(p),\, p\ge 6$ &\vline& $I_2(p),\, p\ge 5$ &\vline\\ \vrule height 0pt depth 8pt width 0pt$\Gamma$ &{\vline\kern -0.2 em \vline}& $E_7$ &\vline& $E_8$ &\vline& $F_4$ &\vline& $H_3$ &\vline& $H_4$ &\vline& $p$ even &\vline& $p$ odd &\vline\\ \hline \vrule height 16pt depth 8pt width 0pt$\mf (G)$ &{\vline\kern -0.2 em \vline}& 9 &\vline& 15 &\vline& 6 &\vline& 5 &\vline& 15 &\vline& $p/2$ &\vline& $p$ &\vline\\ \hline \end{tabular} }$$ \centerline{{\bf Table 1:} The invariant $\mf(G)$.} \noindent {\bf Remark.} Combining \cite{Bes}, Theorem 4.5, with \cite{BrMi}, Section 3, one can actually compute all the possible orders for a finite subgroup of $G/Z(G)$. The maximal order suffices for our purpose, thus we do not include this more complicate calculation in this paper. The next invariant that we want to compute is the rank of the abelianization of $G$ that we denote by ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab} (G)$. This invariant can be easily computed using the standard presentation of $G$, and the result is as follows. \begin{prop} Let $\Gamma_0$ be the (non-labelled) graph defined by the following data. $\bullet$ $S$ is the set of vertices of $\Gamma_0$; $\bullet$ two vertices $s,t$ are joined by an edge if $m_{s\,t}$ is odd. \noindent Then the abelianization of $G$ is a free abelian group of rank ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab} (G)$, the number of connected components of $\Gamma_0$. $\square$ \end{prop} The last invariant which interests us is the rank of the center of $G$ that we denote by $\rkZ (G)$. The following proposition is a straightforward consequence of Proposition 2.2. \begin{prop} The center of $G$ is a free abelian group of rank $\rkZ (G)$, the number of components of $\Gamma$. $\square$ \end{prop} \section{Irreducibility} Throughout this section, we assume that $G$ is irreducible (namely, that $\Gamma$ is connected). Let $H_1, H_2$ be two subgroups of $G$. Recall that $[H_1, H_2]$ denotes the subgroup of $G$ generated by $\{a_1^{-1} a_2^{-1} a_1a_2; a_1 \in H_1\text{ and } a_2 \in H_2\}$. Our goal in this section is to show that $G$ cannot be expressed as $G= H_1 \cdot H_2$ with $[H_1, H_2] = \{1\}$, unless either $H_1 \subset Z(G)$ or $H_2 \subset Z(G)$. This shall implies that $G$ cannot be a non-trivial direct product. Recall that $\delta$ denotes the standard generator of $Z(G)$. For $X \subset S$, we denote by $\delta_X$ the standard generator of $G_X$, and, for $a \in G$, we denote by $Z_G(a)$ the centralizer of $a$ in $G$. \begin{lemma} Let $t \in S$ such that $\Gamma_{S \setminus \{t\}}$ is connected and $\mu(t) \neq t$ if $\mu \neq \Id$. Then $Z_G( \delta_{S \setminus \{t\}})$ is generated by $G_{S \setminus \{t\}} \cup \{ \delta \}$ and is isomorphic to $G_{S \setminus \{t\}} \times \{ \delta \}$. \end{lemma} \paragraph{Proof.} Assume first that $\mu=\Id$ (in particular, $\delta=\Delta$). By \cite{Par1}, Theorem 5.2, $Z_G(\delta_{S\setminus \{t\}})$ is generated by $G_{S \setminus \{t\}} \cup \{ \Delta^2, \Delta \Delta_{S \setminus \{t\}}^{-1}\}$, thus $Z_G(\delta_{S \setminus \{t\}})$ is generated by $G_{S \setminus \{t\}} \cup \{ \delta \}$. Now, assume $\mu \neq \Id$ (in particular, $\delta = \Delta^2$ and $\mu(t) \neq t$). By \cite{Par1}, Theorem 5.2, $Z_G(\delta_{S \setminus \{t\}})$ is generated by $G_{S \setminus \{t\}} \cup \{ \Delta^2, \Delta \Delta_{S \setminus \{\mu(t)\}}^{-1} \Delta \Delta_{S \setminus \{t\}}^{-1} \}$. Observe that \break $\Delta \Delta_{S \setminus \{\mu(t)\}}^{-1} \Delta \Delta_{S \setminus \{t\}}^{-1} = \Delta^2 \Delta_{S \setminus \{t\}}^{-2}$, thus $Z_G(\delta_{S \setminus \{t\}})$ is generated by $G_{S \setminus \{t\}} \cup \{\delta\}$. By the above, we have an epimorphism $G_{S \setminus \{t\}} \times \langle \delta \rangle \to Z_G( \delta_{S \setminus \{t\}})$, and, by Corollary~2.6, the kernel of this epimorphism is $\{1\}$. $\square$ \noindent {\bf Remark.} It is an easy exercise to show (under the assumption that $\Gamma$ is connect) that there always exists $t \in S$ such that $\Gamma_{S \setminus \{t\}}$ is connected and $\mu(t) \neq t$ if $\mu \neq \Id$. \begin{prop} Let $H_1, H_2$ be two subgroups of $G$ such that $G=H_1 \cdot H_2$ and $[H_1,H_2]=\{1\}$. Then either $H_1 \subset Z(G)$ or $H_2 \subset Z(G)$. If, moreover, $H_1 \cap H_2 = \{1\}$, then either $H_1=\{1\}$ and $H_2=G$, or $H_1=G$ and $H_2= \{1\}$. \end{prop} \paragraph{Proof.} We argue by induction on $n= |S|$. If $n=1$, then $\Gamma= A_1$ and $G={\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$, and the conclusion of the proposition is well-known. Assume $n \ge 2$. For $i=1,2$, let $\tilde H_i$ denote the subgroup of $G$ generated by $H_i \cup \{ \delta\}$. We have $G= \tilde H_1 \cdot \tilde H_2$, $[\tilde H_1, \tilde H_2] = \{1\}$, $H_1 \subset \tilde H_1$, and $H_2 \subset \tilde H_2$. Observe also that $\tilde H_1 \cap \tilde H_2$ must be included in the center of $G$, and that $\delta \in \tilde H_1 \cap \tilde H_2$, thus $\tilde H_1 \cap \tilde H_2 = \langle \delta \rangle$. Take $t \in S$ such that $\Gamma_{S \setminus \{t\}}$ is connected and $\mu(t) \neq t$ if $\mu \neq \Id$, write $X=S \setminus \{t\}$, and choose $d_1 \in \tilde H_1$ and $d_2\in \tilde H_2$ such that $\delta_X=d_1d_2$. Let $a \in G_X$. Choose $a_1 \in \tilde H_1$ and $a_2 \in \tilde H_2$ such that $a=a_1a_2$. We have $$ 1= a^{-1} \delta_X^{-1} a \delta_X = a_1^{-1} d_1^{-1} a_1d_1 a_2^{-1} d_2^{-1} a_2d_2\,, $$ thus $$ a_1^{-1} d_1^{-1} a_1d_1 = d_2^{-1} a_2^{-1} d_2a_2 \in \tilde H_1 \cap \tilde H_2 = \langle \delta \rangle \,. $$ Let $k \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ such that $a_1^{-1} d_1^{-1} a_1d_1 = \delta^k$. Consider the homomorphism $\deg: G \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ which sends $s$ to $1$ for all $s \in S$. Then $$ 0= \deg( a_1^{-1} d_1^{-1} a_1d_1) = \deg( \delta^k)= k\,{\rm lg}}\def\Id{{\rm Id}}\def\GG{{\cal G} (\delta)\,, $$ thus $k=0$, hence $a_1$ and $d_1$ commute. Now, $a_1$ and $d_2$ also commute (since $a_1 \in \tilde H_1$ and $d_2 \in \tilde H_2$), thus $a_1$ commutes with $\delta_X = d_1d_2$. By Lemma 4.1, $a_1$ can be written as $a_1= b_1 \delta^{p_1}$, where $b_1 \in G_X$ and $p_1 \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$. Note also that $b_1 = a_1 \delta^{-p_1} \in \tilde H_1$, since $\delta \in \tilde H_1$, thus $b_1 \in G_X \cap \tilde H_1$. Similarly, $a_2$ can be written as $a_2=b_2 \delta^{p_2}$ where $b_2 \in G_X \cap \tilde H_2$ and $p_2 \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$. We have $\delta^{p_1+p_2} = a b_1^{-1} b_2^{-1} \in G_X \cap \langle \delta \rangle = \{1\}$ (by Corollary 2.6), thus $p_1+p_2=0$ and $a=b_1b_2$. So, we have $$ G_X= (G_X \cap \tilde H_1) \cdot (G_X \cap \tilde H_2)\,. $$ Moreover, by Corollary 2.6, $$ (G_X \cap \tilde H_1) \cap (G_X \cap \tilde H_2) = G_X \cap \langle \delta \rangle = \{1\}\,. $$ By the inductive hypothesis, it follows that, up to permutation of 1 and 2, we have $G_X \cap \tilde H_1 = G_X$ (namely, $G_X \subset \tilde H_1$), and $G_X \cap \tilde H_2 = \{1\}$. We turn now to show that $\tilde H_2 \subset \langle \delta \rangle = Z(G)$. Since $H_2 \subset \tilde H_2$, this shows that $H_2 \subset Z(G)$. Let $a \in \tilde H_2$. Since $\delta_X \in G_X \subset \tilde H_1$, $a$ and $\delta_X$ commute. By Lemma 4.1, $a$ can be written as $a=b \delta^p$, where $b \in G_X$ and $p \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$. Since $\delta \in \tilde H_2$, we also have $b = a \delta^{-p} \in \tilde H_2$, thus $b \in G_X \cap \tilde H_2 = \{1\}$, therefore $a = \delta^p \in \langle \delta \rangle$. Now, assume that $H_1 \cap H_2=\{1\}$. By the above, we may suppose that $H_2 \subset Z(G)= \langle \delta \rangle$. In particular, there exists $k \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ such that $H_2 = \langle \delta^k \rangle$. Choose any order $S=\{s_1, \dots, s_n\}$ of $S$, and write $\pi=s_1s_2 \dots s_n \in G$. Let $b \in H_1$ and $p \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ such that $\pi= b \delta^{pk}$. Observe that $b \neq 1$ since $\pi$ is not central in $G$. Let $h$ denote the Coxeter number of $W$. By Theorem~3.3, $\pi^h= b^h \delta^{phk} \in Z(G)$, thus $b^h \in Z(G)$. Moreover, $b^h \neq 1$ since $G$ is torsion free and $b \neq 1$. This implies that $Z(H_1)\neq\{1\}$. Now, observe that $Z(H_1) \subset Z(G)= \langle \delta \rangle$, thus there exists $l >0$ such that $Z(H_1)=\langle \delta^l \rangle$. Finally, $\delta^{lk} \in H_1 \cap H_2 = \{1\}$, thus $kl=0$, therefore $k=0$ (since $l\neq 0$) and $H_2=\{1\}$. Then we also have $H_1=G$. $\square$ \begin{prop} Assume $n=|S| \ge 2$. Let $H$ be a subgroup of $G$ such that $G=H \cdot \langle \delta \rangle$. Then $\cd (H)= \cd (G)$, $\mf(H) = \mf(G)$, and ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab} (H)= {\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(G)$. \end{prop} \paragraph{Proof.} For all $s \in S$, take $b_s \in H$ and $p_s \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ such that $s=b_s \delta^{p_s}$. We can and do suppose that $p_s=p_t$ if $s$ and $t$ are conjugate in $G$. Then the mapping $S \to H$, $s \mapsto b_s=s\delta^{-p_s}$ determines a homomorphism $\varphi: G \to H$. We show that $\varphi: G \to H$ is injective. Observe that the mapping $S \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$, $s \mapsto p_s$ determines a homomorphism $\eta: G \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$, and that $\varphi(a)= a \delta^{-\eta (a)}$ for all $a \in G$. In particular, if $a \in \Ker \varphi$, then $a=\delta^{\eta(a)} \in Z(G)$. Choose any order $S=\{s_1, \dots, s_n\}$ of $S$, and write $\pi=s_1s_2 \dots s_n \in G$. Note that $\varphi(\pi) \neq 1$, since $\pi$ is not central in $G$, and that, by Theorem 3.3, there exists $k>0$ such that $\pi^k = \delta$. Let $a \in \Ker \varphi$. Then $a = \delta^{\eta (a)} = \pi^{k \eta(a)}$, thus $1=\varphi(a)= \varphi(\pi)^{k \eta(a)}$. We have $\varphi(\pi) \neq 1$ and $G$ is torsion free, hence $\eta(a) =0$ (since $k>0$) and $a=1$. Now, recall that $\cd (H_1) \le \cd (H_2)$ if $H_1$ is a subgroup of a given group $H_2$. So, $$ \cd (G)= \cd (\varphi(G)) \le \cd(H) \le \cd (G)\,. $$ The equality $G=H \cdot \langle \delta \rangle = H \cdot Z(G)$ implies that $Z(H)= Z(G) \cap H$ and $G/Z(G)= H/Z(H)$. In particular, we have $\mf(H)= \mf(G)$. Let ${\cal H}}\def\Ker{{\rm Ker}$ be a group, let $g$ be a central element in ${\cal H}}\def\Ker{{\rm Ker}$, and let $p>0$. Let $\GG= ({\cal H}}\def\Ker{{\rm Ker} \times {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C})/ \langle (g,p) \rangle$. Then one can easily verify (using the Reidemeister-Schreier method, for example) that we have exact sequences $1 \to {\cal H}}\def\Ker{{\rm Ker} \to \GG \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}/p{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C} \to 1$ and $1 \to \Ab ({\cal H}}\def\Ker{{\rm Ker}) \to \Ab (\GG) \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}/p{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C} \to 1$, where $\Ab (\GG)$ (resp. $\Ab({\cal H}}\def\Ker{{\rm Ker})$) denotes the abelianization of $\GG$ (resp. ${\cal H}}\def\Ker{{\rm Ker}$). Now, recall the equality $G=H \cdot \langle \delta \rangle$. By Proposition 4.2, we have $H \cap \langle \delta \rangle \neq \{1\}$. So, there exists $p>0$ such that $H \cap \langle \delta \rangle = \langle \delta^p \rangle$. Write $d=\delta^p \in H$. Then $d$ is central in $H$ and $G \simeq (H \times {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C})/\langle (d,p) \rangle$. By the above observation, it follows that we have an exact sequence $1 \to \Ab (H) \to \Ab (G) \to {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}/p{\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C} \to 1$, thus $\Ab (H)$ is a free abelian group of rank ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab} (G)$. $\square$ \section{Proof of the main theorem} \begin{prop} Let $\Gamma$ and $\Omega$ be two connected spherical type Coxeter graphs, and let $G$ and $H$ be the Artin groups associated to $\Gamma$ and $\Omega$, respectively. If $\cd (G)= \cd (H)$, $\mf (G)= \mf (H)$, and ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(G)= {\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(H)$, then $\Gamma= \Omega$. \end{prop} \paragraph{Proof.} Let $n$ and $m$ be the numbers of vertices of $\Gamma$ and $\Omega$, respectively. By Proposition~3.1, we have $n= \cd(G)= \cd(H) = m$. Suppose $n=m=1$. Then $\Gamma= \Omega= A_1$. Suppose $n=m \ge 3$. Then one can easily verify in Table 1 that the equality $\mf (G)= \mf (H)$ implies $\Gamma= \Omega$. Suppose $n=m=2$. Let $p,q \ge 3$, such that $\Gamma= I_2(p)$ and $\Omega= I_2(q)$. By Proposition~3.5, either ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(G)={\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(H)=2$ and $p,q$ are both even, or ${\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(G)= {\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(H)=1$ and $p,q$ are both odd. If $p,q$ are both even, then, by Table 1, ${p \over 2} = \mf(G) = \mf(H)= {q \over 2}$, thus $p=q$ and $\Gamma=\Omega= I_2(p)$. If $p,q$ are both odd, then, by Table 1, $p= \mf (G)= \mf (H)=q$, thus $\Gamma=\Omega= I_2(p)$. $\square$ \begin{corollary} Let $\Gamma$ and $\Omega$ be two connected spherical type Coxeter graphs, and let $G$ and $H$ be the Artin groups associated to $\Gamma$ and $\Omega$, respectively. If $G$ is isomorphic to $H$, then $\Gamma=\Omega$. $\square$ \end{corollary} \noindent {\bf Proof of Theorem 1.1.} Let $\Gamma$ and $\Omega$ be two spherical type Coxeter graphs, and let $G$ and $H$ be the Artin groups associated to $\Gamma$ and $\Omega$, respectively. We assume that $G$ is isomorphic to $H$ and turn to prove that $\Gamma=\Omega$. Let $\Gamma_1, \dots, \Gamma_p$ be the connected components of $\Gamma$, and let $\Omega_1, \dots, \Omega_q$ be the connected components of $\Omega$. For $i=1, \dots, p$, we denote by $G_i$ the Artin group associated to $\Gamma_i$, and, for $j=1, \dots, q$, we denote by $H_j$ the Artin group associated to $\Omega_j$. We have $G= G_1 \times G_2 \times \dots \times G_p$ and $H=H_1 \times H_2 \times \dots \times H_q$. We may and do assume that there exists $x \in \{0,1, \dots, p\}$ such that $\Gamma_i \neq A_1$ for $i=1, \dots, x$, and $\Gamma_i=A_1$ for $i=x+1, \dots, p$. So, $G_1, \dots, G_x$ are non abelian irreducible Artin groups of rank $\ge 2$, and $G_{x+1}, \dots, G_p$ are all isomorphic to ${\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$. Similarly, we may and do assume that there exists $y \in \{0,1, \dots, q\}$ such that $\Omega_j \neq A_1$ for $j=1, \dots, y$, and $\Omega_j=A_1$ for $j=y+1, \dots, q$. We can also assume that $x \ge y$. A first observation is, by Proposition 3.6, that $$ p= \rkZ(G)= \rkZ(H)= q\,. $$ Now, fix an isomorphism $\varphi: G \to H$. For $1 \le i\le p$, let $\iota_i: G_i \to G$ be the natural embedding, for $1 \le j\le p$, let $\kappa_j: H \to H_j$ be the projection on the $j$-th component, and, for $1 \le i,j \le p$, let $\varphi_{i\,j} = \kappa_j \circ \varphi \circ \iota_i: G_i \to H_j$. Let $j\in \{1, \dots, y\}$. Observe that $H_j= \prod_{i=1}^p \varphi_{i\,j}(G_i)$, and that $[\varphi_{i\,j}(G_i), \varphi_{k\,j}(G_k)]=1$ for all $i,k \in \{1, \dots, p\}$, $i \neq k$. Let $\delta_j^H$ denote the standard generator of $Z(H_j)$, and, for $i\in\{1, \dots, p\}$, let $\tilde H_{i\,j}$ be the subgroup of $H_j$ generated by $\varphi_{i\,j}(G_i) \cup \{\delta_j^H\}$. By Proposition~4.2, there exists $\chi (j) \in \{1, \dots, p\}$ such that $H_j= \tilde H_{\chi(j)\,j}$, and $\tilde H_{i\,j} = Z(H_j)= \langle \delta_j^H \rangle$ for $i \neq \chi(j)$. Since $H_j$ is non abelian, $\chi(j)$ is unique and $\chi(j)\in \{1, \dots, x\}$. We turn now to show that the map $\chi: \{1, \dots, y\} \to \{1, \dots, x\}$ is surjective. Since $x \ge y$, it follows that $x=y$ and $\chi$ is a permutation. Let $i \in \{1, \dots, x\}$ such that $\chi(j) \neq i$ for all $j \in \{1, \dots, y\}$. Then $\varphi_{i\,j} (G_i) \subset Z(H_j)$ for all $j=1, \dots, p$, thus $\varphi(G_i) \subset Z(H)$. This contradicts the fact that $\varphi$ is injective and $G_i$ is non abelian. So, up to renumbering the $\Gamma_i$'s, we can suppose that $\chi(i)=i$ for all $i\in \{1, \dots, x\}$. We prove now that $\varphi_{i\,i}: G_i \to H_i$ is injective for all $i \in \{1, \dots, x\}$. Let $a \in \Ker \varphi_{i\,i}$. Since $\varphi_{i\,j} (a) \in Z(H_j)$ for all $j \neq i$, we have $\varphi(a) \in Z(H)$. Since $\varphi$ is injective, it follows that $a \in Z(G_i)$. Let $\{s_1, \dots, s_r\}$ be the set of vertices of $\Gamma_i$, and let $\pi= s_1s_2 \dots s_r \in G_i$. Observe that $\varphi_{i\,i}(\pi) \neq 1$ since $\pi$ is not central in $G_i$. Let $\delta_i^G$ be the standard generator of $Z(G_i)$. By Theorem 3.3, there exists $k >0$ such that $\pi^k= \delta_i^G$. On the other hand, since $a \in Z(G_i)$, there exists $l \in {\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}$ such that $a= (\delta_i^G)^l= \pi^{kl}$. Now, $1= \varphi_{i\,i}(a)= \varphi_{i\,i}(\pi)^{kl}$, $H_i$ is torsion free, and $\varphi_{i\,i}(\pi) \neq 1$, thus $kl=0$ and $a=\pi^{kl}=1$. Let $i \in \{1, \dots, x\}$. Recall that $\varphi_{i\,i}: G_i \to H_i$ is injective, and $H_i= \varphi_{i\,i} (G_i) \cdot \langle \delta_i^H \rangle$, where $\delta_i^H$ denotes the standard generator of $H_i$. By Proposition 4.3, it follows that $$ \cd(G_i)= \cd(H_i)\,,\quad \mf(G_i) =\mf(H_i)\,,\quad {\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(G_i)= {\rm rkAb}}\def\rkZ{{\rm rkZ}}\def\Ab{{\rm Ab}(H_i)\,, $$ thus, by Proposition 5.1, $\Gamma_i=\Omega_i$. Let $i \in \{x+1, \dots, p\}$. Then $\Gamma_i= \Omega_i = A_1$. So, $\Gamma= \Omega$. $\square$ \noindent {\bf Remark.} In the proof above, the homomorphism $\varphi_{i\,i}$ is injective but is not necessarily surjective as we show in the following example. Let $G_1= \langle s_1,s_2 | s_1s_2s_1= s_2s_1s_2 \rangle$ be the Artin group associated to $A_2$, let $G_2={\mathbb Z}}\def\N{{\mathbb N}} \def\C{{\mathbb C}= \langle t \rangle$, and let $G=G_1 \times G_2$. We denote by $\delta= (s_1s_2)^3$ the standard generator of $Z(G_1)$. Let $\varphi: G \to G$ be the homomorphism defined by $$ \varphi(s_1)= s_1 \delta t\,, \quad \varphi(s_2)= s_2 \delta t \,, \quad \varphi(t)= \delta t \,. $$ Then $\varphi$ is an isomorphism but $\varphi_{1\,1}$ is not surjective. The inverse $\varphi^{- 1}: G \to G$ is determined by $$ \varphi^{-1} (s_1)= s_1 t^{-1}\,, \quad \varphi^{-1} (s_2)= s_2 t^{-1}\,, \quad \varphi^{-1} (t)= \delta^{-1} t^7\,. $$ \noindent \halign{# \cr Luis Paris\cr Institut de Math\'ematiques de Bourgogne\cr Universit\'e de Bourgogne\cr UMR 5584 du CNRS, BP 47870\cr 21078 Dijon cedex\cr FRANCE\cr \noalign{ } \texttt{[email protected]}\cr} \end{document}

\begin{document} \title{Memory for Light as a Quantum Process} \author{M. Lobino,$^1$ C. Kupchak,$^1$ E. Figueroa,$^{1,2}$ and A. I. Lvovsky$^{1,}$} \email{[email protected]} \affiliation{$^1$ Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada} \affiliation{$^2$ Max-Planck-Institut f{\"u}r Quantenoptik, Hans-Kopfermann-Str.\ 1, 85748 Garching, Germany} \begin{abstract} We report complete characterization of an optical memory based on electromagnetically induced transparency. We recover the superoperator associated with the memory, under two different working conditions, by means of a quantum process tomography technique that involves storage of coherent states and their characterization upon retrieval. In this way, we can predict the quantum state retrieved from the memory for any input, for example, the squeezed vacuum or the Fock state. We employ the acquired superoperator to verify the nonclassicality benchmark for the storage of a Gaussian distributed set of coherent states. \end{abstract} \pacs{42.50.Ex, 03.67.-a, 32.80.Qk, 42.50.Dv} \maketitle \paragraph{Introduction} Quantum memory for light is an essential technology for long distance quantum communication \cite{DLCZ} and for any future optical quantum information processor. Recently, several experiments have shown the possibility to store and retrieve nonclassical states of light such as the single photon \cite{KuzmichSingle,LukinSingle}, entangled \cite{KimbleEntang} and squeezed vacuum \cite{KozumaStorage,LvovskyStorage} states using coherent interactions with an atomic ensemble. In order to evaluate the applicability of a quantum memory apparatus for practical quantum communication and computation, it is insufficient to know its performance for specific, however complex, optical states, because in different protocols, different optical states are used for encoding quantum information \cite{DLCZ,Lloyd}. Practical applications of memory require answering a more general question: how will an \emph{arbitrary} quantum state of light be preserved after storage in a memory apparatus? Here we answer this question by performing complete characterization of the quantum process associated with optical memory based on electromagnetic induced transparency (EIT) \cite{FleischhauerReview}. Memory characterization is achieved by storing coherent states (i.~e. highly attenuated laser pulses) of different amplitudes and subsequently measuring the quantum states of the retrieved pulses. Based on the acquired information, the retrieved state for any arbitrary input can be predicted and additionally, any theoretical benchmark on quantum memory performance can be readily verified. \paragraph{Coherent state quantum process tomography} We can define complete characterization of an optical quantum memory as the ability to predict the retrieved quantum state $\hat{\mathcal{E}}(\hat\rho)$ when the stored input state $\hat\rho$ is known. This is a particular case of the quantum ``black box" problem, which is approached through a procedure called quantum process tomography (QPT) \cite{MohseniQPT}. \begin{figure} \caption{(color online). Schematic of the experimental setup used to characterize the process associated with the quantum memory. PBS, polarizing beam splitter.} \label{fig.1} \end{figure} QPT is based on the fact that every quantum process (in our case, optical memory) is a linear map on the linear space $\mathbb{L}(\mathbb{H})$ of density matrices over the Hilbert space $\mathbb{H}$ on which the process is defined. The associated process can thus be characterized by constructing a spanning set of ``probe" states in $\mathbb{L}(\mathbb{H})$ and subjecting each of them to the action of the quantum ``black box". If we measure the process output $\hat{\mathcal{E}}(\hat\rho_i)$ for each member $\hat\rho_i$ of this spanning set, we can calculate the process output for any other state $\hat\rho=\sum_i a_i\hat\rho_i$ according to \begin{equation} \hat{\mathcal{E}}(\hat\rho)=\sum_i a_i\hat{\mathcal{E}}(\hat\rho_i). \label{Eq.linearity} \end{equation} The challenge associated with this approach is the construction of the appropriate spanning set, given the infinite dimension of the optical Hilbert space and the lack of techniques for universal optical state preparation. For this reason, characterizing memory for light, that is not limited to the qubit subspace, is much more difficult than memory for superconducting qubits, which has been reported recently \cite{NeeleyQPT}. Our group has recently developed a process characterization technique that overcomes these challenges \cite{LobinoQPT}. Any density matrix $\hat\rho$ of a quantum optical state can be written as a linear combination of density matrices of coherent states $|\alpha\rangle$ according to the optical equivalence theorem \begin{equation} \hat\rho=2\int P_{\hat\rho}(\alpha)|\alpha\rangle\langle\alpha|d^2\alpha, \label{Eq.Pfunction} \end{equation} where $P_{\hat\rho}(\alpha)$ is the state's Glauber-Sudarshan P-function and the integration is performed over the entire complex plane. Although the P-function is generally highly singular, any quantum state can be arbitrarily well approximated by a state with an infinitely smooth, rapidly decreasing P-function \cite{Klauder}. Therefore, by measuring how the process affects coherent states, one can predict its effect on any other state. The advantage of such approach (which we call coherent-state quantum process tomography or csQPT) is that it permits complete process reconstruction using a set of ``probe" states that are readily available from a laser. \paragraph{Experimental setup} We performed csQPT on optical memory \cite{LvovskyStorage} realized in a warm rubidium vapor by means of electromagnetically-induced transparency (Fig.~\ref{fig.1}). The atoms are $^{87}$Rb and the vapor temperature is kept constant at 65$^\circ$C. The signal field is resonant with the $|^5 S_{1/2}, F=1 \rangle\leftrightarrow|^5 P_{1/2}, F=1\rangle$ transition at 795 nm and is produced by a continuous-wave Ti:Sapphire laser. An external cavity diode laser, phase locked at 6834.68 MHz to the signal laser \cite{AppelPhaselock} serves as the EIT control field source, and is resonant with the $|^5 S_{1/2}, F=2 \rangle\leftrightarrow|^5 P_{1/2}, F=1\rangle$ transition. The fields are red detuned from resonance by 630 MHz in order to improve the storage efficiency. The control field power is 5 mW and the beam spatial profile is mode matched with the signal beam to a waist of 0.6 mm inside the rubidium cell. Signal and control fields are orthogonally polarized; they are mixed and separated using polarizing beam splitters. The two photon detuning $\Delta_2$ between the signal and control fields is modified by varying the frequency of the control field laser through the phase lock circuit, while an acousto-optical modulator (AOM) is used to switch on and off the control field intensity. We analyzed two different operative conditions characterized by $\Delta_2$ = 0 and 0.54 MHz. The input pulse is obtained by chopping the continuous-wave signal beam via an AOM to produce 1 $\mu$s pulses [Fig.~\ref{fig.2}(c)] with a 100 kHz repetition rate. A second AOM is used to compensate for the frequency shift generated by the first. Transfer of the light state into the atomic ground state superposition (atomic spin wave) is accomplished by switching the control field off for the storage duration of $\tau=1\ \mu$s when the input pulse is inside the rubidium cell. We performed full state reconstruction of both the input and retrieved fields by time domain homodyne tomography \cite{LvovskyReview}. A part of the Ti:Sapphire laser beam serves as a local oscillator for homodyne detection; while its phase is scanned via a piezoelectric transducer, the homodyne current is recorded with an oscilloscope. For every state, 50000 samples of phase and quadrature are measured and processed by the maximum likelihood algorithm \cite{LvovskyMaxLik,LvovskyMaxLik2}, estimating the state density matrix in the Fock basis. \begin{figure}\label{fig.2} \end{figure} \paragraph{Tomography of quantum memory} In order to determine the coherent state mapping necessary for reconstructing the process, we measured 10 different coherent states $|\alpha_i\rangle$ with mean photon numbers ranging from 0 to 285 along with their corresponding retrieved states $\hat{\mathcal{E}}(|\alpha_i\rangle\langle\alpha_i|)$ [Fig.~\ref{fig.2}(a) and (b)]. Subsequently, we applied polynomial interpolation to determine the value of $\hat{\mathcal{E}}(|\alpha\rangle\langle\alpha|)$ for any value of $\alpha$ in the range 0 to 16.9. Performing tomographic reconstruction for these highly displaced states requires good phase stability between the signal and local oscillator. Phase fluctuations produce an artefact in the reconstruction in the form of amplitude dependent increase in the phase quadrature variance. In our measurements, the reconstructed input states $| \alpha_i \rangle$ resemble theoretical coherent states with a fidelity higher than $0.999$ for mean photon values up to 150 [Fig.~\ref{fig.2}(a) and (b)]. By inspecting the Wigner functions of the input and retrieved states, one can clearly notice the detrimental effects of the memory. First, there is attenuation of the amplitude by a factor of $0.41\pm0.01$ for the signal field in two-photon resonance with the control, which increases to a factor of $0.33\pm0.02$ when a two-photon detuning of $\Delta_2$ = 0.54 MHz is introduced. This corresponds to a mean photon number attenuation by factors of $0.17\pm0.02$ and $0.09\pm0.01$, respectively. Note that in the case of nonzero two-photon detuning, the attenuation is greater than the factor of 0.14 obtained in classical intensity measurement [Fig.~ \ref{fig.2}(c)]. This is because the temporal mode of the retrieved state is slightly chirped, and could not be perfectly matched to the mode of the local oscillator. \begin{figure} \caption{(color online). The diagonal elements of the process tensor $\chi_{kk}^{mm}$, measured by csQPT in the Fock basis for $\Delta_2$ = 0 (a) and 0.54 MHz (b).} \label{fig.1n} \end{figure} Second, retrieved coherent states experience an increase in the phase quadrature variance that depends quadratically on the state amplitude. This effect produces an ellipticity in the retrieved state Wigner function (Fig.~ \ref{fig.2}(a) and (b)] and can be attributed to the noise in the phase lock between the signal and control lasers \cite{AppelPhaselock}. Fluctuations $\Delta\phi$ of the relative phases between the two interacting fields randomize the phase of the retrieved signal field with respect to the local oscillator. Assuming a Gaussian distribution for $\Delta\phi$ with zero mean and variance $\sigma^2_\phi$ the variance of the phase quadrature can be expressed as: \begin{equation} \sigma^2_q=\frac{1}{2}+\frac{q_0^2}{2}\left(1-e^{2\sigma^2_\phi}\right), \label{Eq.variance} \end{equation} where $q_0$ is the mean amplitude. We fit our experimental data with Eq.\ref{Eq.variance} and estimate an $11^\circ$ standard deviation for $\Delta\phi$ [Fig.~\ref{fig.2}(d)], in agreement with independent estimates \cite{AppelPhaselock}. The third detrimental effect preventing the atomic ensemble from behaving as a perfect memory is the population exchange between atomic ground states \cite{HetetMemory,FigueroaSlowlight}. Besides limiting the memory lifetime, this exchange generates spontaneously emitted photons in the signal field mode adding an extra noise that thermalizes the stored light by increasing the quadrature variance independently of the input amplitude and phase. We measured the extra noise from the quadrature variance of retrieved vacuum states $\hat{\mathcal{E}}(|0\rangle\langle 0|)$ and found it to equal 0.185 dB when both fields were tuned exactly at the two photon resonance, which corresponds to the mean photon number in the retrieved mode equal to $\overline{n}=Tr\left[\hat{n} \hat{\mathcal{E}}(|0\rangle\langle 0|) \right]=0.022$. This noise is reduced to 0.05 dB (corresponding to $\overline{n}=0.005$ ) in the presence of two photon detuning. For this reason, it is beneficial to implement storage of squeezed light in the presence of two-photon detuning, in spite of higher losses. \begin{figure} \caption{(color online). Comparison of the experimentally measured squeezed vacuum states retrieved from the quantum memory and those predicted with csQPT. For each case, the Wigner function and the quadrature variance as a function of the local oscillator phase are shown. (a), Experimental measurement \cite{LvovskyStorage} with $\Delta_2$ = 0.54 MHz. (b), Prediction with $\Delta_2$ = 0.54 MHz. (c), Experimental measurement with $\Delta_2$ = 0. (d), Prediction with $\Delta_2$ = 0. } \label{fig.3} \end{figure} In the presence of the two-photon detuning, the evolution of the atomic ground state superposition brings about a phase shift of the retrieved state with respect to the input by $2\pi\Delta_2\tau$ = $200^\circ$ as is visible in Fig.~\ref{fig.2}(a). Based on the information collected from the storage of coherent states, we reconstruct the memory process in the $\chi$-matrix representation, defined by \cite{NielsenBook,Chuang} \begin{equation} \hat{\mathcal{E}}(\hat\rho)=\sum_{k,l,m,n}\chi^{n,m}_{k,l}A_{l,n}\hat\rho A_{m,k}, \label{Eq.chi} \end{equation} where $\chi^{n,m}_{k,l}$ is the rank 4 tensor comprising full information about the process and ${A_{i,j}}$ is a set of operators that form a basis in the space of operators on $\mathbb{H}$. Since $\mathbb{H}$ is the Hilbert space associated with an electromagnetic oscillator, it is convenient to choose $A_{i,j}=|i\rangle\langle j|$, where $|i\rangle$ and $|j\rangle$ are the photon number states. The details of calculating the process tensor are described elsewhere \cite{LobinoQPT}; Fig.~\ref{fig.1n} displays the diagonal subset $\chi^{m,m}_{k,k}$ of the process tensor elements. \paragraph{Performance tests} In order to verify the accuracy of our process reconstruction, we have used it to calculate the effect of storage on squeezed vacuum with $\Delta_2$ = 0.54 MHz, as studied in a recent experiment of our group \cite{LvovskyStorage}, and with $\Delta_2$ = 0 MHz. We applied the superoperator tensor measured with csQPT to the squeezed vacuum produced by a subthreshold optical parametric amplifier with a noise reduction in the squeezed quadrature of $-1.86$ dB and noise amplification in the orthogonal quadrature of $5.38$ dB (i.e. the same state as used as the memory input in Ref.~\cite{LvovskyStorage}). In this way, we obtained a prediction for the state retrieved from the memory, which we then compared with the results of direct experiments. This comparison yields quantum mechanical fidelities of $0.9959\pm0.0002$ and $0.9929\pm0.0002$ for the two-photon detunings of $\Delta_2$ = 0.54 MHz and $\Delta_2$ = 0 respectively (Fig.~\ref{fig.3}). As discussed above, zero detuning warrants lower losses (thus higher amplitude of the noise variance) and no phase rotation, but higher excess noise (thus no squeezing in the retrieved state). Nevertheless the two photon resonant configuration offers a better fidelity if the single photon state is stored \cite{KuzmichSingle,LukinSingle}, as evidenced by comparing the superoperator element $\chi_{1,1}^{1,1}$ of Fig. \ref{fig.1n} (a) and (b). In addition to the ability to predict the output of the memory for any input state, our procedure can be used to estimate the performance of the memory according to any available benchmark. As an example, we analyze the performance of our memory with respect to the classical limit on average fidelity associated with the storage of coherent states with amplitudes distributed in phase space according to a Gaussian function of width $1/\lambda$ \cite{PolzikBenchmark}. This limit as a function of $\lambda$ is given by: \begin{equation} F(\lambda)=2\lambda \int_0^{+\infty}\exp{(-\lambda\alpha^2)\langle\alpha|\hat{\mathcal{E}}(|\alpha\rangle\langle\alpha|)|\alpha\rangle}\alpha d\alpha\leq\frac{1+\lambda}{2+\lambda}. \label{Eq.benchmark} \end{equation} From csQPT data, we evaluate the average fidelity associated with our memory for both values of $\Delta_2$ (Fig.~\ref{fig.4}). Both configurations show nonclassical behavior. The higher value of average fidelity correspond to $\Delta_2$ = 0 and is explained by a higher storage efficiency. \begin{figure} \caption{(color online). Average fidelity of the quantum memory for a Gaussian distributed set of coherent states. Blue empty (red filled) dots show the average fidelity calculated from the csQPT experimental data for $\Delta_2$ = 0 (0.54 MHz). The experimental uncertainty is 0.0002. The solid line shows the classical limit \cite{PolzikBenchmark}. } \label{fig.4} \end{figure} \paragraph{Conclusion} In summary, we have demonstrated complete characterization of an EIT-based quantum memory by csQPT. This procedure allows one to predict the effect of the memory on an arbitrary quantum-optical state, and thus provides the ``specification sheet" of quantum-memory devices for future applications in quantum information technology. Furthermore, our results offer insights into the detrimental effects that affect the storage performance and provide important feedback for the device optimization. We anticipate this procedure to become standard in evaluating the suitability of a memory apparatus for practical quantum telecommunication networks. \paragraph*{Acknowledgements} This work was supported by NSERC, iCORE, CFI, AIF, Quantum$Works$, iCORE (C.K.) and CIFAR (A.L.). \end{document}

\begin{document} \title[Convex ordering and quantification of quantumness]{Convex ordering and quantification of quantumness} \author{J Sperling} \address{Arbeitsgruppe Theoretische Quantenoptik, Institut f\"ur Physik, Universit\"at Rostock, D-18051 Rostock, Germany} \ead{[email protected]} \author{W Vogel} \address{Arbeitsgruppe Theoretische Quantenoptik, Institut f\"ur Physik, Universit\"at Rostock, D-18051 Rostock, Germany} \ead{[email protected]} \begin{abstract} The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be introduced which is based on the algebraic definition of classical states. This definition resolves the ambiguity of the quantumness quantification using topological distance measures. Classical operations on quantum states will be considered to further generalize the ordering prescription. Our technique can be used for a natural and unambiguous quantification of general quantum properties whose classical reference has a convex structure. We apply this method to typical scenarios in quantum optics and quantum information theory to study measures which are based on the fundamental quantum superposition principle. \end{abstract} \pacs{03.67.Mn, 42.50.-p, 02.40.Ft, 03.65.Fd} \begin{indented} \item[]\today,\submitto{\PS} \end{indented} {\it Keywords}: Convex geometry, convex ordering, quantumness measures \maketitle \ioptwocol \section{Introduction} Characterizing the differences between the quantum and classical domain of physics is of fundamental interest for uncovering the quantumness of nature. Typically there are quantum counterparts to classical physics, such as coherent states in the system of the harmonic oscillator, or product states in the field of compound systems. Using classical statistical mixing, these pure states may be generalized to mixed classical ones. Thus, we obtain convex sets of states having a classical analogue with respect to a given physical property. Different measures have been introduced for quantifying the amount of quantumness of states having no such classical correspondence. These measures induce an ordering prescription enabling us to compare the quantumness of different states. In the system of the harmonic oscillator, one of the early attempts to quantify the amount of nonclassicality has been given by the trace-distance of an arbitrary state to the set of all classical ones being mixtures of coherent states~\cite{TraceDist,TraceDist2}. This led to a number of distance based nonclassicality probes, e.g., Hilbert-Schmidt-norm~\cite{HSDist,HSDist2} or the Bures distance~\cite{BuresDist} measures. Some nonclassicality metrics are based on the amount of Gaussian noise which is needed for the elimination of any quantum interference within the corresponding phase-space representation~\cite{Gauss1,Gauss12,Gauss2} or they directly use the negativities within the quasiprobability distribution~\cite{NegWiegner,FM10} as an indicator of the amount of nonclassicality. Another method for the quantification of nonclassicality is given via the potential of a state to generate entanglement~\cite{EntPot}. This translates quantumness of a single-mode harmonic oscillator to the quantification of entanglement. The axiomatic definition of general entanglement measure is given in~\cite{AxiomEntM,AxiomEntM2,AxiomEntM3}. This definition is based on so-called local operations and classical communications mapping separable quantum states onto separable ones. Under all examples of entanglement measures, there is one which is of particular interest for our considerations: the Schmidt number~\cite{SchmNummer,TerhalSN}. It has been shown that this entanglement measure has some advantageous properties in relation to other measures~\cite{SchmidtUni}. In particular, the degree of nonclassicality of a single mode system is directly transformed into the same Schmidt number using linear optics~\cite{UniQuant}. For some applications not all states with the same amount of quantumness are equally useful. For example, it has been shown that states can be too entangled for quantum computation~\cite{Eisert-QC}. Consequently, operational nonclassicality and entanglement measures have been introduced~\cite{SchmidtUni,Gehrke}. In particular quantum information protocols require information related measures of quantum effects. For example, the Fischer information~\cite{Fischer,FKMMSV10} is such a proper operational probe. More generally, entropic measures have been intensively studied~\cite{EntropyInformation,Context}. It has been shown that entropic inequalities and tomographic information can determine quantum correlations~\cite{MM09, MM14a,MM14}. In general, a given operational, distance-based, or entropic metrics induces an ordering prescription, which yields a particular sorting of quantum states regarding their amount of quantumness for some applications. In the current contribution we will use the inverse approach, i.e.: a convex ordering prescription of quantum states will imply a canonic measure. It will be shown that distance measures are, in general, not completely suitable for ordering quantum states unambiguously. Studying the algebraic implications of the definition of convex sets, we rigorously formulate an ordering procedure which does not depend on a distinct topological distance. We expand this method to include classical operations being especially defined for a particular notion of quantumness under study. The obtained sorting procedure induces a corresponding quantumness measures in a natural way. We apply this method to basic examples, such as entanglement, nonclassicality, and quantum information, showing the importance of the quantum superposition principle for the quantification of different quantum features. The paper is structured as follows. In Sec.~\ref{Sec:Motivation} we motivate our treatment. An unambiguous convex ordering prescription will be proposed in Section~\ref{Sec:Ordering}. In Sec.~\ref{Sec:ClassicalOperations} we include classical operations to further enhance the ordering technique. We introduce an axiomatic quantification and we study measures that count quantum superpositions in Sec.~\ref{Sec:Quantification}. A summary and conclusions are given in Sec.~\ref{Sec:SumCon}. \section{Motivation}\label{Sec:Motivation} Let us consider the convex set of all (pure and mixed) quantum states, $\mathcal Q$, and a closed, non-empty, and convex subset $\mathcal C\subset\mathcal Q$. The elements of $\mathcal C$ are supposed to be states with a given classical property, e.g.: separable states, $\mathcal C_{\rm sep}={\rm conv}\{|a\rangle\langle a|\otimes|b\rangle\langle b|:|a\rangle\in\mathcal H_A \wedge |b\rangle\in\mathcal H_B\}$, or coherent states, $\mathcal C_{\rm coh}={\rm conv}\{|\alpha\rangle\langle \alpha|:\alpha\in\mathbb C\}$. The general task is the determination of the amount of quantumness of an arbitrary quantum state $\rho\in\mathcal Q$ with respect to the classical property under study. The convexity of the set $\mathcal C$ guarantees that a mixing of two classical states remains classical. This is important, because it ensures that statistical averaging cannot increase quantum correlations. The closure of $\mathcal C$ is motivated by the argument that a convergent sequence of classical states should have its limit in the classical domain too. These fundamental requirements ensure that a classical system remains classical employing classical operations and classical statistics. Let us note that the property of quantum discord does not meet these conditions, since a non-zero discord can be obtained from a classical mixing of two zero discord states~\cite{DiscordReview}. One way of ordering quantum states is given by the distance of these states to the set of classical states $\mathcal C$. Here we will show that sorting quantum states by a distance cannot lead to one distinct order of states. For the time being, let us assume a two dimensional convex set $\mathcal C$. Using an appropriate coordinate transformation, this classical set $\mathcal C$ can be assumed to be a sphere -- with respect to the Euclidean norm $\|\,\cdot\,\|_2$ -- in the form: \begin{eqnarray} \mathcal C=\{x\in\mathcal Q:\, \|x\|_2\leq 1/2\}. \end{eqnarray} Now we may choose two nonclassical elements $y_1,y_2\in\mathcal Q\setminus\mathcal C$, which are given in the standard basis: $y_1=(1,0)^{\rm T}$ and $y_2=\frac{1}{\sqrt{2}}(1,1)^{\rm T}$. The distance $d_p$ to the set of classical states $\mathcal C$ in $p$-norm is given by \begin{eqnarray} d_p(y,\mathcal C)=\inf_{x\in\mathcal C}\|y-x\|_p. \end{eqnarray} For all $p$-norms the minimal distance of $y_1$ and $y_2$ to the classical states is obtained for $x_1=\frac{1}{2}(1,0)^{\rm T}\in\mathcal C$ and $x_2=\frac{1}{2\sqrt 2}(1,1)^{\rm T}\in\mathcal C$, respectively, cf. Fig.~\ref{Fig:Norms}. Thus, we can calculate $d_p(y_1,\mathcal C)$ and $d_p(y_2,\mathcal C)$ for different values of $p$, \begin{eqnarray} d_p(y_1,\mathcal C)=\|y_1-x_1\|_p=\left[\left(\frac{1}{2}\right)^p+0^p\right]^{1/p}=\frac{1}{2}\\ \nonumber d_p(y_2,\mathcal C)=\|y_2-x_2\|_p\\ \phantom{d_p(y_2,\mathcal C)} =\left[\left(\frac{1}{2\sqrt 2}\right)^p+\left(\frac{1}{2\sqrt 2}\right)^p\right]^{1/p} =\frac{2^{1/p}}{2\sqrt 2}. \end{eqnarray} This result displays the paradox of the quantification of quantumness with distance measures in Fig.~\ref{Fig:Norms}. Depending on the choice of the norm, we can claim that: $y_1$ is more nonclassical than $y_2$ ($2<p\leq\infty$); or $y_1$ is less nonclassical than $y_2$ ($1\leq p<2$); or $y_1$ and $y_2$ have an equal nonclassicality ($p=2$). \begin{figure} \caption{(color online) The dark gray area represents $\mathcal C$, and both gray areas depict $\mathcal Q$. The upper point represents $y_1$, the other one represents $y_2$. The blue circles are the spheres in $2$-norm showing the distance to $\mathcal C$. The equal size of them implies an equal $2$-norm-distance for both points. The green squares represent the spheres around the considered points in $1$-norm. In the case of the $1$-norm, the square around $y_2$ is larger than those around $y_1$. Whereas for the $\infty$-norm spheres (red squares) the relation is the other way around. } \label{Fig:Norms} \end{figure} Let us note that this particular two-dimensional cut already provides the ambiguity of the distance-measure approach for any dimension of convex sets. Additionally, any monotonic function of a distance, for example entropies, will inherit this characteristic. While those metrics can be useful in an operational sense, they are not suitable for an unambiguous quantification of the quantumness property itself. In the following we will show that the convexity of the classical set serves as the key element to resolve this paradox. \section{Ordering Quantum States}\label{Sec:Ordering} A convex set $\mathcal C$ is characterized through its algebraic definition, \begin{eqnarray}\label{Eq:Convexity} \rho,\rho'\in\mathcal C \wedge \lambda\in[0,1] \Rightarrow \lambda \rho+(1-\lambda)\rho'\in\mathcal C. \end{eqnarray} The question whether a general element $\rho\in\mathcal Q$ is in the convex set $\mathcal C$, or not, is independent of the choice of a distance. In addition, we show in~\ref{App:Normalization} that the normalization to ${\rm tr}\,\rho=1$ can be neglected from the mathematical point of view. For the quantification, we start with the formulation of a preorder relation $\preceq$. \begin{definition}\label{Def:Preorder} Two quantum states $\rho,\rho'\in\mathcal Q$ can be compared by $\preceq$: \begin{eqnarray*} \rho\preceq \rho' \Leftrightarrow \exists \gamma\in\mathcal C\, \exists \lambda\in[0,1]: \rho=\lambda \rho'+(1-\lambda)\gamma. \end{eqnarray*} \end{definition} This means a quantum state $\rho$ has less or equal nonclassicality compared with another state $\rho'$, if $\rho$ can be written as a classical statistical mixture of $\rho'$ and a classical state $\gamma$. Let us prove, that this relation fulfills the requirements of a preorder. \paragraph*{Proof.} $\preceq$ is reflexive: $\rho=1\rho+(1-1)\gamma\Rightarrow \rho\preceq \rho$; $\preceq$ is transitive: $\rho_1\preceq \rho_2$ and $\rho_2\preceq \rho_3$ imply \begin{eqnarray*} &\rho_1=\lambda \rho_2+(1-\lambda)\gamma_1 \wedge \rho_2=\kappa \rho_3+(1-\kappa)\gamma_2 \Rightarrow\\ &\rho_1=\lambda\kappa \rho_3 +(1-\lambda\kappa)\gamma_3 \Rightarrow \rho_1\preceq \rho_3, \end{eqnarray*} with $\gamma_3=\frac{\lambda(1-\kappa)}{1-\lambda\kappa}\gamma_2+\frac{1-\lambda}{1-\lambda\kappa}\gamma_1$ and $\frac{\lambda(1-\kappa)}{1-\lambda\kappa}+\frac{1-\lambda}{1-\lambda\kappa}=1$. In conclusion, $\preceq$ is a preorder. $\blacksquare$ For generating an order from the preorder $\preceq$, we consider the following equivalence~$\cong$. \begin{definition}\label{Def:Eqivalence} Two quantum states $\rho,\rho'\in\mathcal Q$ have the same order of quantumness, if \begin{eqnarray*} \rho\cong \rho' \Leftrightarrow \rho\preceq \rho' \wedge \rho'\preceq \rho. \end{eqnarray*} \end{definition} \paragraph*{Proof.} $\cong$ is reflexive: $\rho\preceq\rho\wedge\rho\preceq\rho$; $\cong$ is symmetric: $\rho\preceq\rho'\wedge\rho'\preceq\rho\Leftrightarrow\rho'\preceq\rho\wedge\rho\preceq\rho'$; $\cong$ is transitive: $\rho_1\cong\rho_2$ and $\rho_2\cong\rho_3$ are equivalent to \begin{eqnarray*} \rho_1\preceq\rho_2\quad\wedge\quad\rho_2\preceq\rho_1\quad\wedge\quad\rho_3\preceq\rho_2\quad\wedge\quad\rho_2\preceq\rho_3. \end{eqnarray*} Using the transitivity of $\preceq$, we obtain $\rho_1\preceq\rho_3\wedge\rho_3\preceq\rho_1$. Thus, $\cong$ is an equivalence relation. $\blacksquare$ With respect to the equivalence $\cong$, the $\preceq$ preorder given in Definition~\ref{Def:Preorder} becomes an order. The missing property is that $\preceq$ must be antisymmetric, \begin{eqnarray} \rho\preceq \rho'\wedge\rho'\preceq \rho\Rightarrow\rho\cong\rho', \end{eqnarray} which is true, cf. Definition~\ref{Def:Eqivalence}. Thus, we have constructed a rigorous way to order quantum states. \begin{proposition}\label{Lem:MinClass} Classical states have a minimal and equal order, i.e.: \begin{eqnarray*} \gamma\in\mathcal C \wedge \rho\in\mathcal Q\Rightarrow \gamma\preceq\rho \mbox{ and } \gamma,\gamma'\in\mathcal C\Rightarrow\gamma\cong\gamma'. \end{eqnarray*} Any state $\rho\in\mathcal Q$ with a minimal order, $\rho\preceq\gamma\in\mathcal C$, is classical, $\rho\in\mathcal C$. \end{proposition} \paragraph*{Proof.} From $\gamma=0\rho+(1-0)\gamma$ and Definition~\ref{Def:Preorder} follows $\gamma\preceq\rho$. Hence we find for all classical states $\gamma,\gamma'\in\mathcal C$: $\gamma\preceq\gamma'\wedge\gamma'\preceq\gamma$; and therefore $\gamma\cong\gamma'$. If $\rho\preceq\gamma\in\mathcal C$, i.e. $\exists\gamma'\in\mathcal C,\lambda\in[0,1]:\rho=\lambda\gamma+(1-\lambda)\gamma'$ , then $\rho$ is a convex combination of classical states and therefore classical. $\blacksquare$ The Definitions~\ref{Def:Preorder}~and~\ref{Def:Eqivalence} provide an order of quantum states, which is solely based on the convex structure of $\mathcal C$. These definitions highlight the natural assumption that a statistical mixture of a nonclassical state with a classical one cannot become more nonclassical than the initial one, cf. Fig.~\ref{Fig:NonclOrder}. Further on, in Proposition~\ref{Lem:MinClass} it has been shown that this order implies that all classical states are the only ones with a minimal nonclassicality. The mixing property and the minimality property of classical states are essential for any quantification of nonclassicality. \begin{figure} \caption{(color online) The inner green area represents $\mathcal C$, and the complete area represents $\mathcal Q$. A nonclassical element $\rho$ is given. All elements $\rho_1$ in the red (triangular) area above $\rho$ fulfill: $\rho\preceq\rho_1$. All elements $\rho_2$ in the green and blue area below $\rho$ fulfill: $\rho_2\preceq\rho$. } \label{Fig:NonclOrder} \end{figure} \section{Classical Operations}\label{Sec:ClassicalOperations} A classical quantum state may evolves in an experiment or it propagates in a classical channel including noise effects. Thus we have to deal with operations which map our state within the set $\mathcal Q$. Operations with a classical counterpart must not increase the amount of quantumness. Therefore, we study transformations mapping classical states onto each other. \begin{definition} We call a linear operation $\Lambda:\mathcal Q\to\mathcal Q$ a classical one, if $\forall\gamma\in\mathcal C: \Lambda(\gamma)\in\mathcal C$. The set of all classical operations $\Lambda$ is denoted as $\mathcal{CO}$. \end{definition} \begin{proposition}\label{Prop:SemiGroup} The set $\mathcal{CO}$ is convex and a semi-group. \end{proposition} \paragraph*{Proof.} The convexity follows from the linearity of the operation space together with the convexity of the set of classical states, \begin{eqnarray*} (\lambda\Lambda_1+(1-\lambda)\Lambda_2)(\gamma)=\lambda\underbrace{\Lambda_1(\gamma)}_{\in\mathcal C}+(1-\lambda)\underbrace{\Lambda_2(\gamma)}_{\in\mathcal C}\in\mathcal C. \end{eqnarray*} The semi-group property is given by \begin{eqnarray*} \Lambda_1,\Lambda_2\in\mathcal{CO}: (\Lambda_1\circ\Lambda_2)(\gamma)=&\Lambda_1(\Lambda_2(\gamma))\in\mathcal C, \end{eqnarray*} with the identity ${\rm Id}(\gamma)=\gamma\in\mathcal C$, being classical. $\blacksquare$ These classical operations or channels can be considered as quantum physical systems having a classical analogue. This includes interactions which evolve states in a classical way, or mix them with classical noise. For special quantum tasks it might be also useful to consider only sub-semi-groups of $\mathcal{CO}$, for example one-way classical communications for entanglement or phase rotations and phase dispersion for coherent states. Now, we have to verify that classical operations do not change the previously defined order. Therefore we formulate the following proposition. \begin{proposition}\label{Theo:ClassOp} {\rm (i)} A classical operation does not change the order, $\rho\preceq\rho' \Rightarrow \Lambda(\rho)\preceq\Lambda(\rho')$. {\rm (ii)} Mixing a quantum state with a classical one, is a classical operation. \end{proposition} \paragraph*{Proof.} For (i) let us consider two states with $\rho\preceq\rho'$ and a classical operation $\Lambda$, $\rho=\lambda\rho'+(1-\lambda)\gamma$, which implies $\Lambda(\rho)=\lambda\Lambda(\rho')+(1-\lambda)\Lambda(\gamma)$. Together with $\Lambda(\gamma)\in\mathcal C$ and Definition~\ref{Def:Preorder} we obtain (i). For claim (ii), we consider that a state $\rho$ is mixed with a classical state $\gamma$, $\lambda\in[0,1]$, \begin{eqnarray*} \rho'&=\lambda\rho+(1-\lambda)\gamma\\ &=\lambda{\rm Id}(\rho)+(1-\lambda)({\rm tr}\rho)\gamma\\ &=\left[\lambda{\rm Id}(\,\cdot\,)+(1-\lambda)({\rm tr}(\,\cdot\,))\gamma\right](\rho)=\Lambda(\rho). \end{eqnarray*} The identical transformation ${\rm Id}$ and $({\rm tr}(\,\cdot\,))\gamma$ are classical, i.e., $\forall \gamma' \in\mathcal C:({\rm tr}\,\gamma')\gamma\in\mathcal C$, and the convex structure of $\mathcal{CO}$ implies that $\Lambda\in\mathcal{CO}$. $\blacksquare$ This means that classical operations are compatible with the order $\preceq$, and they cannot increase the quantumness of the initial state. Therefore, the order given in Definition~\ref{Def:Preorder} can be generalized by using Proposition~\ref{Theo:ClassOp}. \begin{definition}\label{Def:NO} A quantum state $\rho$ has a lower or equal order of nonclassicality than the state $\rho'$, $\rho\preceq\rho'$, iff $\exists\Lambda\in\mathcal{CO}:\rho=\Lambda(\rho').$ They have the same nonclassicality, $\rho\cong\rho'$, if $\rho\preceq\rho'\wedge\rho'\preceq\rho$. \end{definition} This quantumness ordering prescription naturally generalizes the previous convex ordering with respect to $\mathcal C$ by including classical operations $\mathcal{CO}$. Condition~(ii) in Proposition~\ref{Theo:ClassOp} proves that the ordering includes the previous Definition~\ref{Def:Preorder}. In addition, the Definition~\ref{Def:NO} implies that all quantum states below a given state $\rho$ can be written as $\Lambda(\rho)$ for a classical operation $\Lambda$. Therefore it simply follows \begin{eqnarray} \Lambda(\rho)\preceq\rho. \end{eqnarray} Let us stress again, that the minimal states are uniquely classical ones. Now we want to further study properties of classical operations. A subgroup of $\mathcal{CO}$ are classical invertible maps $\mathcal{CO}_{-1}$, defined by \begin{eqnarray} \Lambda\in\mathcal{CO}_{-1}\Leftrightarrow \Lambda\in\mathcal{CO}\wedge\exists \Lambda^{-1}\in\mathcal{CO}. \end{eqnarray} These are classical operations which can be reversed, and the inverse is again a classical operation. This group always exists, since the identical transformation is its own inverse, ${\rm Id}\in\mathcal{CO}_{-1}$. The importance of this group is that it yields classes of quantum states with an equivalent order. Let us assume a classical invertible $\Lambda\in\mathcal{CO}_{-1}$ and an arbitrary state $\rho\in\mathcal Q$. It follows from $\rho'=\Lambda(\rho)$ that $\rho=\Lambda^{-1}(\rho')$. Together with the Definitions~\ref{Def:Eqivalence}~and~\ref{Def:NO} \begin{eqnarray} \rho\preceq\rho'\wedge\rho'\preceq\rho \Leftrightarrow \rho\cong\rho'. \end{eqnarray} Hence, it is possible to identify quantum states with an equal order of quantumness applying the group $\mathcal{CO}_{-1}$. \begin{proposition} All quantum states $\rho,\rho'\in\mathcal Q$, with $\rho'=\Lambda(\rho)$ and $\Lambda\in\mathcal{CO}_{-1}$, have an equal order of quantumness, $\rho\cong\rho'$. $\blacksquare$ \end{proposition} Using the sphere shaped classical set in Fig.~\ref{Fig:NonclOrder}, we observe in this case that classical invertible maps are rotations around the center. This structure, in the generalized scenario, will lead subsequently to nested sets with increasing amount of quantum interferences. \section{Axiomatic Quantification of Nonclassicality}\label{Sec:Quantification} So far, we have introduced the algebraic quantumness ordering prescription $\preceq$ on arbitrary classical, convex sets $\mathcal C$ that are closed under classical statistical mixtures and operations. Hence, a distance independent ordering technique is obtained. Eventually, we will use this approach to quantify the amount of quantumness in a natural way. Let us stress again that the standard approach is formulated in the opposite direction, i.e., a measure is proposed which implies an sorting of states. Contrary, the approach under study starts from a convex geometric ordering. Using the derived ordering, we can properly define quantumness measures. This means that we can introduce functions $\mu$, which map a classical states, $\rho\in\mathcal C$, to a real number $\mu(\rho)$. \begin{definition}\label{Def:Measure} A function $\mu:\mathcal Q\to\mathbb R$ is a quantumness measure, if $\rho\preceq\rho'\Leftrightarrow \mu(\rho)\leq\mu(\rho')$. \end{definition} The definition says that the measure quantifies the ordering, which is given by the algebraic sorting $\preceq$. Since for all classical states $\gamma\in\mathcal C$ holds $\gamma\preceq\rho\in\mathcal Q$, we have $\mu(\rho)=\inf_{\gamma\in\mathcal C}\mu(\gamma)=:\mu_{\rm min}$ if and only if $\rho\in\mathcal C$, cf. Proposition~\ref{Lem:MinClass}. Typically, one uses the convention $\mu_{\rm min}=0$. From the definition also follows \begin{eqnarray} \mu(\rho)\geq\mu(\Lambda(\rho)), \end{eqnarray} for any classical operation $\Lambda\in\mathcal{CO}$. Moreover, equally ordered quantum states, $\rho\cong\rho'$, have an equivalent amount of quantumness, \begin{eqnarray} \rho\preceq\rho'\wedge\rho'\preceq\rho\,\Leftrightarrow\,\mu(\rho)\leq\mu(\rho')\wedge\mu(\rho')\leq\mu(\rho). \end{eqnarray} The here considered quantification of quantum states with nonclassical properties has been based only on the most elementary definition of statistical averaging (convexity of $\mathcal C$) and the physical need for classical transformations, $\mathcal{CO}$. We did not make any further assumption about the classical property itself. In the case of entanglement, Definition~\ref{Def:Measure} is equivalent to the axiomatic definition of entanglement measures~\cite{AxiomEntM,AxiomEntM2,AxiomEntM3} adding the compatibility with local invertible transformations. For nonclassicality in the notion of coherent states, Definition~\ref{Def:Measure} is equivalent to the algebraic approach in Refs.~\cite{UniQuant,Gehrke}. Note that the quantification procedure loses its generality if only subsets of $\mathcal{CO}$ are considered, as it is often done in entanglement theory by restricting the set of all separable operations to operational subset of so-called local operations and classical communication~\cite{RMP-Horo}. \subsection{Quantumness measures based on the quantum superposition principle}\label{Sec:Example} As an example, we will consider in the following a measure which relies on the quantum superposition principle. Superpositions are the origin of the most fundamental differences between classical and quantum physics. Therefore, let us start with a set $\mathcal C_0$ of pure classical states, $|c\rangle\in\mathcal C_0$. The elements of the convex set $\mathcal C$ of all classical states are given by \begin{eqnarray} \gamma=\int_{\mathcal C_0} dP_{\rm cl}(c) |c\rangle\langle c|, \end{eqnarray} for a classical probability distribution $P_{\rm cl}$. Hence, a general classical state is a statistical mixtures of pure classical ones. For nonclassical states, $\rho\in\mathcal Q\setminus\mathcal C$, such a $P_{\rm cl}$ does not exist. The typical situation in quantum physics is that a generalized $P$ exists, but it has negativities. This scenario is relevant for the representations of both: expanding nonclassical states using coherent ones $\mathcal C_{0,\rm coh}=\{|\alpha\rangle:\, \alpha\in\mathbb C\}$ with the Glauber-Sudarshan representation~\cite{GSRep2,GSRep1}; and expanding entangled states by factorized ones $\mathcal C_{0,\rm sep}=\{|a\rangle\otimes|b\rangle:\,|a\rangle\in\mathcal H_A\wedge|b\rangle\in\mathcal H_B\}$ using optimized entanglement quasi-probabilities~\cite{EntRep}. Let us consider a classical operation, which has the following form, \begin{eqnarray} \Lambda(\rho)=M\rho M^\dagger,\mbox{ with } M|c\rangle=g(c)|f(c)\rangle, \end{eqnarray} with a classical valued function $f$, i.e. $|f(c)\rangle\in\mathcal C_0$, and a complex valued function $g$. This operation is a classical one, \begin{eqnarray} \nonumber\Lambda(\gamma)=\int_{\mathcal C_0} dP_{\rm cl}(c) M|c\rangle\langle c|M^\dagger\\ \phantom{\Lambda(\gamma)}=\int_{\mathcal C_0} dP_{\rm cl}(c) |g(c)|^2|f(c)\rangle\langle f(c)|, \end{eqnarray} which is again (neglecting normalization, see~\ref{App:Normalization}) a statistical mixture of pure classical states. In case that $f$ is bijective and $g(c)\neq0$ for all $c$, we have a classical operation in $\mathcal{CO}_{-1}$, \begin{eqnarray} M^{-1}|c\rangle=\frac{1}{g(c)}|f^{-1}(c)\rangle. \end{eqnarray} Examples are local invertible maps $M=A\otimes B$ ($\exists A^{-1},B^{-1}$) for separable states, or, for coherent states, \begin{eqnarray} M=\exp[xa^\dagger a]\exp[ya]\exp[za^\dagger], \end{eqnarray} where $x,y,z\in\mathbb C$, the annihilation and creation operators $a$ and $a^\dagger$, respectively, and \begin{eqnarray} M|\alpha\rangle=\exp[xa^\dagger a]\exp[ya]\exp[za^\dagger]|\alpha\rangle\nonumber\\ \phantom{M|\alpha\rangle}=e^{\frac{|z+\alpha|^2-|\alpha|^2}{2}+y(z+\alpha)}|(\alpha+z)e^{x}\rangle\in\mathcal C_0. \end{eqnarray} It is worth to note that the convex set of all classical operations, $\Lambda\in\mathcal{CO}$, can be written in the form of operator-sum decompositions~\cite{OpSumRep}, also called Krauss operators, \begin{eqnarray}\label{Eq:Krauss} \Lambda(\rho)=\sum_i M_i\rho M_i^\dagger. \end{eqnarray} Now we want to analyze a pure nonclassical state, which may be written as \begin{eqnarray}\label{Eq:SuperPosClassical} |\psi\rangle=\sum_{k=1}^r \psi_k |c_k\rangle, \end{eqnarray} with $|c_k\rangle\in\mathcal C_0$ and $r$ being the minimal number which allows this decomposition. This representation is possible for any pure state, if $\mathcal C_0$ includes at least a basis of the Hilbert space. Therefore, the state $|\psi\rangle$ is a superposition of $r$ classical states. The classical operator $M$ acts like \begin{eqnarray} M|\psi\rangle=\sum_{k=1}^r \psi_k g(c_k)|f(c_k)\rangle. \end{eqnarray} It is important that $M$ can only decrease the number $r$, for example, in the case $g(c_k)=0$ for some $k$ or for $f(c_k)=f(c_{k'})$. If $M\,\cdot\,M^\dagger\in\mathcal{CO}_{-1}$, then $r$ remains even unchanged. Therefore, let us define this minimal number $r$ of superimposed classical states as $r(\psi)$, \begin{eqnarray} r(\psi)=\inf\left\{r:|\psi\rangle=\sum_{k=1}^r \psi_k |c_k\rangle \wedge |c_k\rangle\in\mathcal C_0\right\}. \end{eqnarray} Obviously this number is 1, iff the state is an element of $\mathcal C_0$, and greater than one for a nonclassical pure state. Now let us consider a mixed state $\rho\in\mathcal Q$. This state can be written in various forms as a convex combination of pure states, \begin{eqnarray}\label{Eq:Dec1} \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|, \end{eqnarray} with $p_i>0$ and $\sum_i p_i=1$. In this case $\mu(\rho)$ can be obtained from a convex roof construction of $r(\psi)$~\cite{uhlmann}. In a particular decomposition given in Eq.~(\ref{Eq:Dec1}) the largest number of superposition of a pure state $|\psi_i\rangle$ can be found as $\sup_i\{r(\psi_i)\}$. Under all decompositions of $\rho$, the desired one is that with a minimum of needed superpositions. Thus, $\mu(\rho)$ is given by \begin{eqnarray}\label{eq:superpsoMeasure} \mu(\rho)=\inf \left\{\sup_i\{r(\psi_i)\}:\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|\right\}-1. \end{eqnarray} This number is 0, iff the mixed state is classical and greater than zero for nonclassical states. The number can become infinity, if no finite number of superpositions yields the given state. Let us highlight that states with an amount of quantumness up to $r$ define nested convex sets, $\mathcal C_{\mu\leq r}=\{\rho\in\mathcal Q:\mu(\rho)\leq r\}$ with $\mathcal C_r\subset \mathcal C_{r'}$ for $r\leq r'$. For convenience, it is also possible to map $\mu(\rho)$ together with a monotonically increasing function to another measure $\mu'$, e.g., \begin{eqnarray} \mu'(\rho)= 1-\exp(-\mu(\rho))\in[0,1]. \end{eqnarray} We also point out that the measures $\mu$ and $\mu'$ are invariant under classical invertible maps, $\mathcal{CO}_{-1}$, which is important for being compatible with the unambiguous ordering prescription. As we mentioned in Sec.~\ref{Sec:Motivation}, this is not true for a distance-based quantumness measure. Since $\mathcal{CO}_{-1}$ maps can be considered as a transformation of the underlying metric, a distance is in general not preserved. This function $\mu(\rho)$ in Eq.~(\ref{eq:superpsoMeasure}) is found to be an example of a quantumness measure based on convex ordering, which additionally characterizes the fundamental quantum superposition principle. In the case of coherent states it counts the minimal number of superpositions of (classical) coherent states needed to generate the state under study~\cite{UniQuant,Gehrke}. In the case of entanglement it represents the Schmidt number~\cite{SchmidtUni}. Hence, the given approach unifies and generalizes the previously considered methods. States with at most $r$ superpositions define nested, convex sets $\mathcal C_{\mu\leq r}$, which is advantageous for the construction of quantumness witnesses; cf.~\cite{MelWitness} and~\cite{SNWitness} for the construction of degree of nonclassicality witnesses and Schmidt number witnesses, respectively. Let us note that the number of superpositions as a quantifier of quantumness in Eq.~(\ref{eq:superpsoMeasure}) may be further refined. For example the properties of the individual classical terms $|c_k\rangle$ in the superposition decomposition in Eq.~(\ref{Eq:SuperPosClassical}) could be taken into account. For certain practical applications, such as special quantum teleportation protocols, also the weighting coefficients $\psi_k$ can play a significant role. This, however, leads to operational quantumness measures, cf.~\cite{SchmidtUni}, which are important for quantifying the useful nonclassicality for particular applications. It might be also useful to use the purity of a quantum state $\rho$ to further refine quantumness measures. \subsection{Example: Bits versus qubits} Another application of the superposition number is related to quantum information processing. A classical sequence of $N$ bits $\boldsymbol i=(i_1,\dots,i_N)$, with truth values ``0'' and ``1'', has a classical counterpart in a compound qubit quantum system $(\mathbb C^{2})^{\otimes N}$ as \begin{equation} |\boldsymbol i\rangle=|i_1\rangle\otimes\dots\otimes|i_N\rangle\in\mathcal C_0, \end{equation} where $|0\rangle$ and $|1\rangle$ are the ground and excited state, respectively, of any two-level system being described by the individual Hamiltonians \begin{equation} H=\frac{\hbar\omega}{2} \sigma_{z}, \mbox{ with } \sigma_z=|1\rangle\langle 1|-|0\rangle\langle 0|. \end{equation} Using classical probabilities, we only have statistical mixtures of sequences of bits as \begin{equation} \gamma=\sum_{\boldsymbol i\in\{0,1\}^N} p_{\boldsymbol i} |\boldsymbol i\rangle\langle \boldsymbol i|\in\mathcal C. \end{equation} Classical computational operations are those which compute -- including statistical imperfections or errors -- from a given classical sequence $\boldsymbol i$ another classical string $\boldsymbol j$ of $N$ bits with the probability $p(\boldsymbol j|\boldsymbol i)$: \begin{equation} \Lambda(|\boldsymbol i\rangle\langle \boldsymbol i|)=\sum_{\boldsymbol j\in\{0,1\}^N} p(\boldsymbol j|\boldsymbol i)\, |\boldsymbol j\rangle\langle \boldsymbol j|. \end{equation} An example of a classical invertible map is the $N$-bit NOT operation, $\Lambda(\,\cdot\,)={\rm NOT}^{\otimes N}(\,\cdot\,){\rm NOT}^{\otimes N}$, with ${\rm NOT}={\rm NOT}^\dagger=\sigma_x=|1\rangle\langle 0|+|0\rangle\langle 1|$. Please also note that the free unitary evolution with the given Hamiltonian also maps any classical string onto itself, see also~\cite{CMMV13}. Having identified the classical regime, we may study the quantum regime. Here, the pure states can be decomposed as \begin{equation} |\psi\rangle=\sum_{\boldsymbol i\in\{0,1\}^N}\psi_{\boldsymbol i}|\boldsymbol i\rangle, \end{equation} which is quantified by the superposition number \begin{equation} r(\psi)=|\{\psi_{\boldsymbol i}\neq0\}|, \end{equation} being the cardinality of the non-vanishing expansion coefficients $\psi_{\boldsymbol i}$. For example, a coherent superposition in a GHZ-type configuration, $(|0,\dots,0\rangle +|1,\dots,1\rangle)/\sqrt 2$, has a quantumness of $r=2$. This result, $r>1$, quantifies that such a state is beyond the classical information approach. A particular effect which can destroy these quantum interferences is given by decoherence, being the map \begin{equation*} \Lambda_{\rm dc}(\rho)=\int_{-\pi}^{+\pi} \!\!\!d\varphi\, p(\varphi) (\exp[i\varphi\sigma_z])^{\otimes N}\rho(\exp[-i\varphi\sigma_z])^{\otimes N}, \end{equation*} for a classical phase distribution $p(\varphi)$. We observe that a full decoherence, i.e. a uniform distribution $p(\varphi)=1/(2\pi)$, maps any initial state onto the corresponding classical one, \begin{equation} \Lambda_{\rm dc}(|\psi\rangle\langle\psi|)=\sum_{\boldsymbol i\in\{0,1\}^N} |\psi_{\boldsymbol i}|^2|\boldsymbol i\rangle\langle \boldsymbol i|. \end{equation} Consistently our approach identifies that decoherence diminishes quantum properties. In the case of full decoherence we have $\mu(\Lambda_{\rm dc}(\rho))=0$ for any state $\rho\in\mathcal Q$, cf. Eq.~(\ref{eq:superpsoMeasure}). Therefore, our approach not only predicts an unambiguous order of quantumness in quantum information. It additionally characterizes the evolution of these quantum properties in realistic scenarios. \section{Summary and conclusions}\label{Sec:SumCon} We have studied the quantification of quantum properties with a convex classical reference. It was outlined that distances-based measures, in general, lead to an ambiguous quantification. The origin of such a paradox lies in the fact that the nature of quantumness is an algebraic rather than a topological one: The mixture of classical states yields a convex subset of all quantum states. Based on the conservation of a classical feature under mixing, we have proposed a general convex ordering method. For handling classical processes or channels, we have additionally considered classical operations. We have shown that these transformations can be used to generalize our sorting procedure. By quantifying this order, we have obtained quantumness measures in a canonic form. In particular, quantumness probes based on the determination of quantum superpositions have been examined. The technique has been applied to typical examples in quantum physics such as entanglement and nonclassicality in terms of the Glauber-Sudarshan representation. Moreover, the embedding of classical information processing into the quantum domain led to a measure of the amount of quantumness in quantum information. In case of decoherence, we consistently retrieved the classical domain through our quantification. In conclusion, the number of quantum superpositions represents a vital measure to quantify the quantum nature of a system. Known examples have been considered in this context and they have been generalized. Ambiguities, as observed for other measures, do not occur and the role of reversible classical operations has been outlined. Our approach characterizes the quantum nature of states in terms of the fundamental superposition principle, and it naturally relates classical correlations to statistical mixing of states. We believe that this approach will be useful for characterizing even so-far unknown quantum effects in a broader context and for the general understanding of the strength of quantum effects in physical systems. \appendix \section{Normalization}\label{App:Normalization} Let us consider the normalization. It is more convenient to use the following sets, \begin{eqnarray} \mathcal Q'=&\{\lambda\rho: \lambda\geq0 \wedge \rho\in\mathcal Q \},\\ \mathcal C'=&\{\lambda\rho: \lambda\geq0 \wedge \rho\in\mathcal C \}, \end{eqnarray} instead of the normalized states, i.e. states with a unit trace: ${\rm tr}\,\rho=1$. The sets $\mathcal Q'$ and $\mathcal C'$ represent a cone construction over the sets $\mathcal Q$ and $\mathcal C$, respectively. According to these definitions, an element $\rho_3$ is element in $\mathcal C'$, if it can be written as a positive ($\lambda_1,\lambda_2\geq0$) linear combination of elements $\rho_1,\rho_2\in\mathcal C$, \begin{eqnarray} \rho_3=\lambda_1\rho_1+\lambda_2\rho_2. \end{eqnarray} In general, this linear combination is given by neither normalized states nor in a convex form. However, it can be rewritten in such a form. With ${\rm tr}\,\rho_3=\lambda_1{\rm tr}\,\rho_1+\lambda_2{\rm tr}\,\rho_2$, we obtain \begin{eqnarray*} \frac{\rho_3}{{\rm tr}\,\rho_3}{=}\frac{\lambda_1{\rm tr}\,\rho_1}{\lambda_1{\rm tr}\,\rho_1+\lambda_2{\rm tr}\,\rho_2} \frac{\rho_1}{{\rm tr}\,\rho_1} {+}\frac{\lambda_2{\rm tr}\,\rho_2}{\lambda_1{\rm tr}\,\rho_1+\lambda_2{\rm tr}\,\rho_2}\frac{\rho_2}{{\rm tr}\,\rho_2}. \end{eqnarray*} This is obviously a convex combination of normalized states. Therefore we can neglect without any loss of generality the normalization of the quantum states and perform the normalization at the end of our treatment. \ack This work was supported by the Deutsche Forschungsgemeinschaft through SFB 652. The authors gratefully acknowledge many stimulating discussions with Margarita and Vladimir Man'ko. \section*{References} \end{document}

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\newcommand{Cost_f}{Cost_f} \newcommand{Meta-Cluster }{Meta-Cluster } \newcommand{Meta-Clusters }{Meta-Clusters } \newcommand{\facilityset_j}{\mathcal{F}_j} \newcommand{\facilityset_{j'}}{\mathcal{F}_{j'}} \newcommand{\distd}[2]{c'(#1,~#2)} \newcommand{\cliset_{full}}{\mathcal{C}_{full}} \newcommand{\cliset_{part}}{\mathcal{C}_{part}} \newcommand{\cliset^*}{\mathcal{C}^*} \newcommand{\mathcal{B}_j}{\mathcal{B}_j} \newcommand{\mathcal{D}_{\ell_{j'}}}{\mathcal{D}_{\ell_{j'}}} \newcommand{\mathcal{T}}{\mathcal{T}} \newcommand{\dlj}[1]{r\mathcal{F}_{j_{#1}}} \newcommand{\rFj}[1]{rF_{j}_{#1}} \newcommand{\level}[1]{\rFj_#1} \newcommand{\bar\rho}{\bar\rho} \newcommand{\lb}[1]{\mathcal{L}_{#1}} \newcommand{\AVG}[1]{\mathcal{A}_{#1}} \newcommand{\radj}[1]{\mathcal{R}_{#1}} \newcommand{\cluster}[1]{\mathcal{P}_{#1}} \newcommand{\F}[1]{\mathcal{F}_{#1}} \newcommand{i_j}{i_j} \newcommand{\N}[1]{\mathcal{N}_{#1}} \newcommand{\clientset_i}{\mathcal{C}_i} \newcommand{\Out}[1]{\mathcal{O}_{#1}} \newcommand{\R}[1]{\mathcal{R}_{#1}} \newcommand{rmax_j}{rmax_j} \newcommand{\T}[1]{\mathcal{T}_{#1}} \newcommand{\cli}[1]{j_{}} \newcommand{\ball}[1]{\mathcal{B}_{#1}} \newcommand{\floor}[1]{\lfloor{#1}\rfloor} \newcommand{\rtj}[1]{r\mathcal{T}_{j_{#1}}} \title{On Variants of Facility Location Problem with Outliers} \titlerunning{Variants of Facility Location Problem with Outliers} \author{Rajni Dabas\inst{1} \and Neelima Gupta\inst{1}} \authorrunning{R. Dabas and N. Gupta} \institute{Department of Computer Science, University of Delhi, India\\ \email{[email protected], [email protected]}} \maketitle \begin{abstract} In this work, we study the extension of two variants of the facility location problem (FL) to make them robust towards a few distantly located clients. First, $k$-facility location problem ($k$FL), a common generalization of FL and $k$ median problems, is a well studied problem in literature. In the second variant, lower bounded facility location (LBFL), we are given a bound on the minimum number of clients that an opened facility must serve. Lower bounds are required in many applications like profitability in commerce and load balancing in transportation problem. In both the cases, the cost of the solution may be increased grossly by a few distantly located clients, called the outliers. Thus, in this work, we extend $k$FL and LBFL to make them robust towards the outliers. For $k$FL with outliers ($k$FLO) we present the first (constant) factor approximation violating the cardinality requirement by +1. As a by-product, we also obtain the first approximation for FLO based on LP-rounding. For LBFLO, we present a tri-criteria solution with a trade-off between the violations in lower bounds and the number of outliers. With a violation of $1/2$ in lower bounds, we get a violation of $2$ in outliers. \keywords{ Facility Location \and Outliers \and Approximation \and Lower Bound \and $k$-Facility Location \and $k$-Median.} \end{abstract} \section{Introduction} Consider an e-retail company that wants to open warehouses in a city for home delivery of essential items. Each store has an associated opening cost depending on the location in the city. The aim of the company is to open these warehouses at locations such that the cost of opening the warehouses plus the cost servicing all the customers in the city from the nearest opened store is minimised. In literature, such problems are called {\em facility location problems}(FL) where warehouses are the facilities and customers are the clients. Formally, in FL we are given a set $\mathcal{F}$ of $n$ facilities and a set $\mathcal{C}$ of $m$ clients. Each facility $i \in \mathcal{F}$ has an opening cost $f_i$ and cost of servicing a client $j \in \mathcal{C}$ from a facility $i \in \mathcal{F}$ is $c(i,j)$ (we assume that the service costs are metric). The goal is to open a subset $\mathcal{F}' \subseteq \mathcal{F}$ of facilities such that the cost of opening the facilities and servicing the clients from the opened facilities is minimised. In a variant of FL, called $k$-facility location problem ($k$FL), we are given an additional bound $k$ on the maximum number of warehouses/facilities that can be opened i.e. $|\mathcal{F}'| \le k$. In our example this requirement may be imposed to maintain the budget constraints or to comply with government regulations. In another variant of the problem, we are required to serve some minimum number of customers/clients from an opened facility. Such a requirement is natural to ensure profitability in our example. This minimum requirement is captured as lower bounds in facility location problems. That is, in lower bounded FL (LBFL), we are also given a lower bound $\lb{i}$ on the minimum number of clients that an opened facility $i$ must serve. In the above scenarios, a few distant customers/clients can increase the cost of the solution disproportionately; such clients are called {\em outliers}. Problem of outliers was first introduced by Charikar \textit{et al}.~\cite{charikar2001algorithms} for the facility location and the $k$-median problems. In this paper we extend $k$-facility location and lower bounded facility location to deal with the outliers and denote them by $k$FLO and LBFLO respectively. Since FL is well known to be NP-hard, NP-hardness of $k$FLO and LBFLO follows. We present the first (constant factor) approximation for $k$FLO opening at most $k+1$ facilities. In particular, we present the following result: \begin{theorem} \label{thm_flo} There is a polynomial time algorithm that approximates $k$-facility location problem with outliers opening at most $(k+1)$ facilities within $11$ times the cost of the optimal solution. \end{theorem} Our result is obtained using LP rounding techniques. As a by product, we get first constant factor approximation for FLO using LP rounding techniques. FLO is shown to have an unbounded integrality gap~\cite{charikar2001algorithms} with solution to the standard LP. We get around this difficulty by guessing the most expensive facility opened in the optimal solution. In particular we get the following: \begin{corollary} \label{coro-flo} There is a polynomial time algorithm that approximates facility location problem with outliers within $11$ times the cost of the optimal solution. \end{corollary} We reduce LBFLO to FLO and use any algorithm to approximate FLO to obtain a tri-criteria solution for the problem. To the best of our knowledge, no result is known for LBFLO in literature. In particular, we present our result in Theorem~\ref{thm_LBkFLO} where a tri-criteria solution is defined as follows: \begin{definition} A tri-criteria solution for LBFLO is an $(\alpha, \beta, \gamma)$- approximation solution $S$ that violates lower bounds by a factor of $\alpha$ and outliers by a factor of $\beta$ with cost no more than $\gamma OPT$ where $OPT$ denotes the cost of an optimal solution of the problem, $\alpha<1$ and $\beta>1$. \end{definition} \begin{theorem} \label{thm_LBkFLO} A polynomial time $(\alpha,\frac{1}{1-\alpha}, \lambda(\frac{1+\alpha}{1-\alpha})$-approximation can be obtained for LBFLO problem where $\alpha=(0,1)$ is a constant and $\lambda$ is an approximation factor for the FLO problem. \end{theorem} Theorem~\ref{thm_LBkFLO} presents a trade-off between the violations in the lower bounds and that in the number of outliers. Violation in outliers can be made arbitrarily small by choosing $\alpha$ close to $0$. And, violation in lower bounds can be chosen close to $1$ at the cost of increased violation in the outliers. Similar result can be obtained for LB$k$FLO with $+1$ violation in cardinality using Theorem~\ref{thm_flo}. The violation in the cardinality comes from that in $k$FLO. \textbf{Our Techniques: } For $k$FLO, starting with an LP solution $\rho^* = <x^*, y^*>$, we first eliminate the $x^*_{ij}$ variables and work with an auxilliary linear programming (ALP) relaxation involving only $y_i$ variables. This is achieved by converting $\rho^*$ into a {\em complete solution} in which either $x^*_{ij}=y^*_{i}$ or $x^*_{ij}=0$. Using the ALP, we identify the set of facilities to open in our solution. ALP is solved using iterative rounding technique to give a pseudo-integral solution (a solution is said to be pseudo integral if there are at most two fractional facilities). We open both the facilities at $+1$ loss in cardinality and at a loss of factor $2$ in the cost by guessing the maximum opening cost of a facility in the optimal. Once we identify the set of facilities to open, we can greedily assign the first $m-t$ clients in the increasing order of distance from the nearest opened facility. Thus, in the rest of the paper, we only focus on identifying the set of facilities to open. For LBFLO, we construct an instance $I'$ of FLO by ignoring the lower bounds and defining new facility opening cost for each $i \in \mathcal{F}$. An approximate solution $AS'$ to $I'$ is obtained using any approximation algorithm for FLO. Facilities serving less than $\alpha\lb{i}$ clients are closed and their clients are either reassigned to the other opened facilities or are made outliers. This leads to violation in outliers that is bounded by $\frac{1}{1-\alpha}$. Facility opening costs in $I'$ are defined to capture the cost of reassignments. \textbf{Related Work:} The problems of facility location and $k$-median with outliers were first defined by Charikar~\textit{et al}.~\cite{charikar2001algorithms}. Both the problems were shown to have unbounded integrality gap~\cite{charikar2001algorithms} with their standard LPs. For FLO, they gave a $(3+\epsilon)$-approximation using primal dual technique by guessing the most expensive facility opened by the optimal solution. For a special case of the problem with uniform facility opening costs and doubling metrics, Friggstad~\textit{et al}.~\cite{FriggstadKR19} gave a PTAS using multiswap local search. For $k$MO, Charikar~\textit{et al}.~\cite{charikar2001algorithms} gave a $4(1+1/\epsilon)$-approximation with $(1+\epsilon)$-factor violation in outliers. Using local search techniques, Friggstad~\textit{et al}.~\cite{FriggstadKR19} gave $(3+\epsilon)$ and $(1+\epsilon)$-approximations with ($1+\epsilon$) violation in cardinality for general and doubling metric respectively. Chen~\cite{chen-kMO} gave the first true constant factor approximation for the problem using a combination of local search and primal dual. Their approximation factor is large and it was improved to $(7.081+\epsilon)$ by Krishnaswamy~\textit{et al}.~\cite{krishnaswamy-kMO} by strengthening the LP. They use iterative rounding framework and, their factor is the current best result for the problem. Lower bounds in FL were introduced by Karger and Minkoff~\cite{Minkoff_LBFL} and Guha~\textit{et al}.~\cite{Guha_LBFL}. They independently gave constant factor approximations with violation in lower bounds. The first true constant factor($448$) approximation was given by Zoya Svitkina~\cite{Zoya_LBFL} for uniform lower bounds. The factor was improved to $82.6$ by Ahmadian and Swamy \cite{Ahmadian_LBFL}. Shi Li~\cite{Li_NonUnifLBFL} gave the first constant factor approximation for general lower bounds, with the constant being large ($4000$). Han~\textit{et al}.~\cite{Han_LBkM} studied the general lower bounded $k$-facility location (LB$k$FL) violating the lower bounds. Same authors~\cite{Han_LBknapsackM} removed the violation in the lower bound for the $k$-Median problem. The only work that deals with lower bound and outliers together is by Ahmadian and Swamy~\cite{ahmadian_lboutliers}. They have given constant factor approximation for lower-bounded min-sum-of-radii with outliers and lower-bounded k-supplier with outliers problems using primal-dual technique. \textbf{Organisation of the paper:} A constant factor approximation for $k$FLO is given in Section~\ref{kFLPO} opening at most $(k+1)$ facilities. In Section~\ref{LBkFLO}, the tri-criteria solution for LBFLO is presented. Finally we conclude with future scope in Section~\ref{conclusion}. \label{tri-criteria1} In this section we present a tri-criteria approximation for LBFLO problem with $1/4$-factor violation in lower bound and $2$-factor violation in outliers at a constant factor loss in cost. Following is the LP relaxation for LBFLO problem where for each $i \in \mathcal{F}$, $y_i$ indicates if facility $i$ is opened, for each $i \in \mathcal{F}$ and $j \in \mathcal{C}$, $x_{ij}$ indicates if client $j$ is assigned to facility $i$ and for each $j \in \mathcal{C}$, $z_j$ indicates if client $j$ is an outlier. \label{{LBFLO}} $\text{Minimize}~\mathcal{C}ostLBFLO(x,y,z) = \sum_{j \in \mathcal{C}}\sum_{i \in \mathcal{F}}\dist{i}{j}x_{ij} + \sum_{i \in \mathcal{F}}f_iy_i $ \begin{eqnarray} \text{subject to} &\sum_{i \in \mathcal{F}}{} x_{ij} + z_j \geq 1 & \forall ~\textit{j} \in \mathcal{C} \label{LbkFLo_const1}\\ & \lb y_i \leq \sum_{j \in \mathcal{C}}x_{ij} & \forall~ \textit{i} \in \mathcal{F} \label{LbkFLo_const2}\\ & \sum_{j \in \mathcal{C}} z_{j} \leq t & \label{LbkFLo_const4}\\ & x_{ij} \leq y_i & \forall~ \textit{i} \in \mathcal{F} , ~\textit{j} \in \mathcal{C} \label{LbkFLo_const5}\\ & y_i,x_{ij},z_j \in [0,1] \label{LPFLP_const5} \end{eqnarray} Constraints~\ref{LbkFLo_const1} ensure that every client is either served or is an outlier. Constraints~\ref{LbkFLo_const2} and~\ref{LbkFLo_const4} satisfies the lower bounds and number of outliers respectively. Constraints~\ref{LbkFLo_const5} are standard facility location constraint saying that a client is assigned to an open facility only. Let $opt=<x^*,y^*,z^*>$ be the optimal LP solution for the above LP. A solution is said to be an {\em integral open solution} if all the facilities are either fully opened or fully closed, i.e, $y_i=[0,1]$ for all $i \in \mathcal{F}$. Next we will construct an integral open solution $S=<\bar{x}, \bar{y}, \bar{z}>$. The solution $S$ is created in two steps: ($1$) First we create solution $S_{out}= <x', y', z'>$ by removing the clients that are outliers to a large extent($\geq 1/\lambda$) in $opt$ where $\lambda$ is a parameter to be fixed later, ($2$) Use clustering and rounding techniques to obtain an integral open solution for the remaining clients. Formally, in step 1, for all $j \in \mathcal{C}$ and $i \in \mathcal{F}$, set $z'_{j}=1$ and $x'_{ij}=0$ if $z^*_{j} \geq 1/\lambda$. Otherwise, set $z'_{j}=0$ and $x'_{ij}=x^*_{ij}$. For all $i \in \mathcal{F}$, $y'_{i}=y^*_{i}$. Note that in step 1 we just incur at most $\lambda$-factor in outliers. Also, $CostLBFLO(S_{out}) \leq CostLBFLO(opt)$. Next we will describe the Step 2 in detail. Let $\AVG{j}$ be the average connection cost for a client $j \in \mathcal{C}$ after Step 1, that is, $\AVG{j}=\sum_{i \in \mathcal{F}} \dist{i}{j}x'_{ij}/\sum_{i \in \mathcal{F}} x'_{ij}$. The clients are now considered in increasing order of radius $\lambda\AVG{j}$. Let $j$ be a client in this order, remove all the clients $k$ such that $c(j,k) \leq 2\lambda max \{ \AVG{j},\AVG{k} \}$ and repeat the process with the left over clients. Let $\mathcal{C}' \subset \mathcal{C}$ be the set of remaining clients after all the clients have been considered. Note that for any clients $j, k \in \mathcal{C}'$ the following property is satisfied: $c(j,k) > 2\lambda max \{ \AVG{j},\AVG{k} \}$. The total extent up to which facilities are opened in $\F{j}$ after Step 1 is $\geq (1-1/\lambda)(\sum_{i \in \mathcal{F}} x'_{ij}) \geq (1-1/\lambda) 1/\lambda$ where the last inequality follows because every client is served to an extent of at least $1/\lambda$ after first step. To obtain an integral open solution, for all $j \in \mathcal{C}'$, we open the cheapest(lowest facility opening cost) facility say $i_j \in \F{j}$ and transfer all the assignments coming on to the facilities in $\cluster{j}$ to $i_j$. Formally, set $\bar{y}_{i_j}=1$, $\bar{x}_{i_j j} = \sum_{i \in \cluster{j}}x'_{ij}$ and $\bar{y}_{i}=0$, $\bar{x}_{ij}=0$ for all $j \in \mathcal{C}$ and $i \neq i_j$. Set $\bar{z}_{j} = z'_{j}$ for all $j \in \mathcal{C}$. \begin{lemma} The integral open solution $S=<\bar{x},\bar{y},\bar{z}>$ violates lower bound by $\alpha=1/4$ and outliers by $\gamma=2$ at constant(?) factor loss in cost. \end{lemma} \begin{proof} Set $\lambda=1/2$. \end{proof} \section{$(k+1)$ solution for $k$FLO} \label{kFLPO} The problem $k$FLO can be represented as the following integer program (IP): \label{{k-FLPO}} $Minimize ~\mathcal{C}ostkFLO(x,y) = \sum_{j \in \mathcal{C}}\sum_{i \in \mathcal{F}}\dist{i}{j}x_{ij} + \sum_{i \in \mathcal{F}}f_iy_i $ \begin{eqnarray} subject~ to &\sum_{i \in \mathcal{F}}{} x_{ij} \leq 1 & \forall ~\textit{j} \in \mathcal{C} \label{LPFLP_const1}\\ & x_{ij} \leq y_i & \forall~ \textit{i} \in \mathcal{F} , ~\textit{j} \in \mathcal{C} \label{LPFLP_const2}\\ & \sum_{i \in \mathcal{F}}y_{i} \leq k & \label{LPFLP_const3}\\ & \sum_{j \in \mathcal{C}} \sum_{i \in \mathcal{F}}x_{ij} \geq m-t & \label{LPFLP_const4}\\ & y_i,x_{ij} \in \left\lbrace 0,1 \right\rbrace \label{LPFLP_const5} \end{eqnarray} where variable $y_i$ denotes whether facility $i$ is open or not and $x_{ij}$ indicates if client $j$ is served by facility $i$ or not. Constraints \ref{LPFLP_const1} ensure that the extent to which a client is served is no more than $1$. Constraints \ref{LPFLP_const2} ensure that a client is assigned only to an open facility. Constraint \ref{LPFLP_const3} ensures that the total number of facilities opened are atmost $k$ and Constraint \ref{LPFLP_const4} ensures that total number of clients served are at least $m-t$. LP-Relaxation of the problem is obtained by allowing the variables $y_i, x_{ij} \in [0, 1]$. Let us call it $LP$. Let $\rho^{*} = <x^*, y^*>$ denote the optimal solution of $LP$ and $\opt{}$ denote the cost of $\rho^*$. A solution is said to be a {\em complete solution} either $x^*_{ij}= y^*_i$ or $x^*_{ij}=0$, $\forall i \in \mathcal{F}$ and $\forall j \in \mathcal{C}$. We first eliminate $x$ variables from our solution $\rho^*$ by making it complete. This is achieved by standard technique of splitting the openings and making collocated copies of facilities. For every client $j \in \mathcal{C}$, we will define a bundle, $\facilityset_j$ as the set of facilities that are serving $j$ in our complete solution. Formally, $\facilityset_j = \{ i \in \mathcal{F} : x^*_{ij}>0\}$. Let $\dlj{} = max_{i \in \facilityset_j}\distd{i}{j}$ be the distance of farthest facility in $\facilityset_j$ from $j$. See Fig.~\ref{FIG_FLO1}($a$). Note that the complete solution $<x^*, y^*>$ satisfies the following property: \begin{enumerate} \item \label{prop1} $\sum_{i \in \facilityset_j} y^*_i \leq 1~\forall j \in \mathcal{C}$ as $\sum_{i \in \facilityset_j} y^*_i = \sum_{i \in \mathcal{F}}x^*_{ij} \leq 1$. \item \label{prop2} $\sum_{i \in \mathcal{F}}y^*_i \leq k$ \item \label{prop3} $\sum_{j \in \mathcal{C}} \sum_{i \in \facilityset_j} y^*_i \geq m-t$ as $\sum_{i \in \facilityset_j} y^*_i = \sum_{i \in \mathcal{F}}x^*_{ij}$ and $\sum_{j \in \mathcal{C}} \sum_{i \in \mathcal{F}} x^*_{ij} \geq m-t$. \end{enumerate} \subsection{Auxiliary LP (ALP)} \begin{figure} \caption{ ($a$) Set $\facilityset_j$ corresponding to a client $j$, ($b$) Discretization of distances} \label{FIG_FLO1} \end{figure} We first discretize our distances $c(i,j)$, by rounding them to the nearest power of $2$. Let $\distd{i}{j} = 2^r$, where $r$ is smallest power of $2$ such that $\dist{i}{j} \le 2^r$. See Fig.~\ref{FIG_FLO1}($b$). Next, we identify a set $\cliset_{full} $ of clients that are going to be served fully in our solution. Ideally, we would like to open at least one facility in $\facilityset_j$ for every $j \in \cliset_{full}$. If all the $\facilityset_j$'s ($j \in \cliset_{full}$) were pair-wise disjoint, an LP constraint like $\sum_{i \in \facilityset_j} w^*_i \geq 1$ for all $j \in \cliset_{full}$, along with constraints~\ref{LPALP_const3}(for partially served clients, say clients in $\cliset_{part}$),~\ref{LPALP_const0}(for cardinality) and~\ref{LPALP_const4}(for outliers), is sufficient to get us a psuedo-integral solution. But this, in general, is not true. Thus we further identify a set $\cliset^* \subseteq \cliset_{full}$ so that we open one facility in $\facilityset_j$ for every $j \in \cliset^*$ and ($i$) $\facilityset_j$'s ($j \in \cliset^*$) are pair-wise disjoint \textit{(disjointness property)} ($ii$) for every $\cli{f} \in \cliset_{full} \setminus \cliset^*$, there is a close-by (within constant factor of $\dlj{}$ distance from $\cli{f}$) client in $\cliset^*$. On a close observation, we notice that instead of $ \facilityset_j$'s, we are rather interested in smaller sets: let $rmax_j$ be the (rounded) distance of the farthest facility in $\facilityset_j$ serving $j$ in our solution and $\T{j} = \{ i \in \facilityset_j : c'(i,j) \leq rmax_j\}$. Then we actually want $\T{j}$'s ($j \in \cliset^*$) to be pair-wise disjoint. As the distances are discretized, we have that $rmax_j$ is either $\dlj{}$ or is $\le \dlj{}/2$. Since we don't know $rmax_j$, once a client is identified to be in $\cliset_{full}$, we search for it by starting with $\T{j} = \facilityset_j$, $\rtj{} = \dlj{}$ and, shrinking it over iterations. Shrinking is done whenever, for $\mathcal{B}_j = \{ i \in \facilityset_j : c'(i,j) \leq rmax_j/2\}$ , we obtain $\sum_{i \in \mathcal{B}_j} w_i = 1$. Thus we add a constraint $\sum_{i \in \mathcal{B}_j} w_i \le 1$ in our ALP and arrive at the following auxiliary LP (ALP). Variable $w_i$ denotes whether facility $i$ is opened in the solution or not. Constraints (9) and (10) correspond to the requirements of cardinality and outliers. For $\cli{f}\in \cliset_{full}$, if the ALP doesn't open a facility within $\ball{\cli{f}}$, it bounds the cost of sending $\cli{f}$ up to a distance of $\rtj{}$. $\text{Min}~CostALP(w) = \sum_{j \in \cliset_{part}}\sum_{i \in \T{j}}\distd{i}{j} w_i + \sum_{j \in \cliset_{full}} [ \sum_{i \in \mathcal{B}_j} \distd{i}{j} w_i + (1 - \sum_{i \in \mathcal{B}_j} w_i)\rtj{}] + \sum_{i \in \mathcal{F}}f_i w_i$ \begin{eqnarray} subject~ to &\sum_{i \in \T{j}} w_i = 1 & \forall ~ j \in \cliset^* \label{LPALP_const1}\\ &\sum_{i \in \mathcal{B}_j} w_i \leq 1 & \forall ~ j \in \cliset_{full} \label{LPALP_const2}\\ &\sum_{i \in \T{j}} w_i \leq 1 & \forall ~ j \in \cliset_{part} \label{LPALP_const3}\\ &\sum_{i \in \mathcal{F}} w_i \leq k & \label{LPALP_const0}\\ &|\cliset_{full}| + \sum_{j \in \cliset_{part}} \sum_{i \in \T{j}} w_i \geq m-t & \label{LPALP_const4}\\ & 0 \leq w_i \leq 1 &\label{LPALP_const5} \end{eqnarray} The following lemma gives a feasible solution to ALP such that cost is bounded by LP optimal within a constant factor. \begin{lemma} \label{fs_kflo} A feasible solution $w'$ can be obtained to the ALP such that $CostALP(w') \leq 2 \opt{LP}$. \end{lemma} \begin{proof} Let $w'_i = y^*_{i}$. \begin{enumerate} \item {\em Feasibility:} Constraints \ref{LPALP_const1} and \ref{LPALP_const2} hold vacuously as $\cliset_{full}$ and hence $\cliset^*$ are empty. Constraints \ref{LPALP_const3}, \ref{LPALP_const0} and \ref{LPALP_const4} hold by properties~\ref{prop1},~\ref{prop2} and~\ref{prop3} respectively. \item {\em Cost Bound:} As $\T{j}=\facilityset_j$, $CostALP(w'_i) = \sum_{j \in \mathcal{C}} \sum_{i \in \facilityset_j} \distd{i}{j} y^*_{i} + \sum_{i \in \mathcal{F}} f_i y^*_i \leq 2 \sum_{j \in \mathcal{C}} \sum_{i \in \facilityset_j} \dist{i}{j} x^*_{ij} + \sum_{i \in \mathcal{F}} f_i y^*_i = 2\sum_{j \in \mathcal{C}} \sum_{i \in \mathcal{F}} \dist{i}{j} x^*_{ij} + \sum_{i \in \mathcal{F}} f_i y^*_i = 2\opt{}$. The inequality follows as $c'(i,j) \le 2c(i,j)$ and $x^*_{ij} = y^*_i$. \end{enumerate} \end{proof} \qed \subsection{Iterative Rounding} We next present an iterative rounding algorithm(IRA) for solving the ALP. In every iteration of IRA, we compute an extreme point solution $w^*$ to ALP and check whether any of the constraints \ref{LPALP_const2} or \ref{LPALP_const3} has become tight. If a constraint corresponding to $j \in \cliset_{part}$ gets tight, we move the client to $\cliset_{full}$ and remove it from $\cliset_{part}$. We also update $\cliset^*$ so that disjointness property is satisfied. If a constraint corresponding to $\cli{f} \in \cliset_{full}$ gets tight, we shrink $\T{\cli{f}}$ to $\ball{\cli{f}}$; update $\ball{\cli{f}}$ and $\cliset^*$ accordingly. The algorithm is formally stated in Algorithm \ref{IRA}. For $j \in \cliset_{full}$, let $resp(j)$ be the client $j' \in \cliset^*$ who takes the responsibility of getting $j$ served. Whenever $j$ is added to $\cliset^*, resp(j)$ is set to $j$ and whenever it is removed because of another client $j' \in\cliset^*$, $resp(j)$ is set to $j'$. If $j$ was never added to $\cliset^*$, then there must be a $j'$ because of which it was not added to $\cliset^*$ in lines $4$ and $5$. Such a $j'$ takes the responsibility of $j$ in that case. Note that a client $j$ may be added and removed several times from $C^*$ over the iterations of the algorithm as $\T{j}$ and $\ball{j}$ shrink (see Fig.~\ref{fig_ALP} for illustration). \begin{algorithm} \footnotesize \begin{algorithmic}[1] \STATE $\cliset_{full} \leftarrow \phi$, $\cliset_{part} \leftarrow \mathcal{C}$, $\cliset^* \leftarrow \phi$, $\T{j}=\facilityset_j$, $\rtj{} = \dlj{}$\\ \WHILE {true} \STATE Find an extreme point solution $w^*$ to ALP \IF{there exists some $j \in \cliset_{part}$ such that $\sum_{i \in \T{j}} w^*_i$ = 1} {\STATE $\cliset_{part} \leftarrow \cliset_{part} \setminus \{j\}, \cliset_{full} \leftarrow \cliset_{full} \cup \{ j\}, \mathcal{B}_j \leftarrow \{ i \in \T{j} : \distd{i}{j} \leq \floor{\rtj{}/2}\}$ \STATE $process-\cliset^*(j)$.} \ENDIF \IF{there exists $j \in \cliset_{full}$ such that $\sum_{i \in \mathcal{B}_j} w^*_i=1$} {\STATE $\T{j} \leftarrow \mathcal{B}_j, \rtj{} = \floor{\rtj{} /2}, \mathcal{B}_j \leftarrow \{ i \in \T{j} : \distd{i}{j} \leq \floor{\rtj{}/2} \}$ \STATE $process-\cliset^*(j)$} \ENDIF \ENDWHILE \STATE Return $w^*$ \STATE $process-C^*(j)$ \IF{there exists $j' \in \cliset^*$ with $\rtj{'} < \rtj{}$ and $\T{j} \cap \T{j'} \neq \phi$} {\STATE $resp(j) = j'$, if there are more than one such $j'$s, choose any arbitrarily.} \ELSE {\STATE \textbf{if} $j \in \cliset^*$ \textbf{then} update $\rtj{}$ to its new value} \STATE \ \ \ \ \ \ \ \ \ \ \ \ \ \textbf{else} Add $j$ to $\cliset^*$ with $\rtj{}$ and $resp(j) = j$. {\STATE Remove all $j'$ from $\cliset^*$ for which $\rtj{} < \rtj{'}$ and $\T{j} \cap \T{j'} \neq \phi$, $resp(j') = j$. } \ENDIF \end{algorithmic} \caption{Iterative Rounding Algorithm} \label{IRA} \end{algorithm} \begin{figure}\label{fig_ALP} \end{figure} Lemmas~\ref{FS_ALP},~\ref{pseudo_integral}, and~\ref{neartocfull} help us analyse our algorithm. Lemma~\ref{FS_ALP} shows that the solution obtained in an iteration is feasible for the ALP of the next iteration. We also prove that the cost of the solutions computed is non-increasing over iterations. \begin{lemma} \label{FS_ALP} Let $ALP_{t}$ and $ALP_{t+1}$ be the auxiliary LPs before and after iteration $t$ of IRA. Let $w^t$ be the extreme point solution obtained in $t^{th}$ iteration. Then $(i)$ $w^t$ is a feasible solution to $ALP_{t+1}$, $(ii)$ $CostALP_{t+1}(w^{t}) \leq CostALP_{t}(w^{t})$ and hence $CostALP_{t+1}(w^{t+1}) \leq CostALP_{t}(w^{t})$. \end{lemma} \begin{proof} Note that the feasibility and the cost can change only when one of constraints~(\ref{LPALP_const2}) or constraints~(\ref{LPALP_const3}) $w^t$ becomes tight, that is, either condition at step 4 or condition at step 6 of the algorithm is true. \begin{itemize} \item[($i$)] When one of constraints~(\ref{LPALP_const3}) corresponding to a client $j$ becomes tight i.e. $\sum_{i \in \T{j}} w^t =1$, we move client $j$ from $\cliset_{part}$ to $\cliset_{full}$ and define the set $\mathcal{B}_j$. Thus, $\sum_{i \in \mathcal{B}_j } w^t \leq \sum_{i \in \T{j} } w^t = 1$. Thus the new constraints added in constraints~\ref{LPALP_const2} and~\ref{LPALP_const1} (if $j$ is added to $\cliset^*$) are satisfied. Constraint~(\ref{LPALP_const4}) holds as $|\cliset_{full}|$ increases by $1$ and $\sum_{j \in \cliset_{part}} \sum_{i \in \T{j}} w^t$ decreases by $1$. There is no change in constraint~\ref{LPALP_const0}. Let one of the constraints~(\ref{LPALP_const2}) corresponding to a full client $j$ becomes tight i.e. $\sum_{i \in \mathcal{B}_j} w^t =1$. Two things happen here: ($i$) we shrink $\T{j}$ to $\mathcal{B}_j$, hence $\sum_{i \in \T{j}} w^t=1$. Thus constraint~\ref{LPALP_const1} is satisfied if $j$ is added to $\cliset^*$. ($ii$) shrink $\mathcal{B}_j$ to half its radius, thus $\sum_{i \in \mathcal{B}_j } w^t \leq \sum_{i \in \T{j} } w^t = 1$. Thus constraint~\ref{LPALP_const2} corresponding to $j$ continue to be satisfied with the shrunk $\mathcal{B}_j$. There is no change in constraints~\ref{LPALP_const0} and~\ref{LPALP_const4}. \item[($ii$)] For a client $j$, let $\dlj{}^{t}$, $\mathcal{B}_j^{t}$ and $\T{j}^{t}$ be the set $\dlj{}$, $\mathcal{B}_j$ and $\T{j}$ corresponding to client $j$ in $ALP_{t}$ and $\dlj{}^{t+1}$, $\mathcal{B}_j^{t+1}$ and $\T{j}^{t+1}$ be the respective values in $ALP_{t+1}$. \begin{enumerate} \item[a.] When $\T{j}$ and $\mathcal{B}_j$ shrink because constraint~\ref{LPALP_const2} becomes tight for a client $j$. Cost paid by $j$ in $w^t$ in the $t^{th}$ iteration $ = \sum_{i \in \mathcal{B}_j^{t}} c'(i,j) w^t $ because $\sum_{i \in \mathcal{B}_j^{t}} w^t=1$ in the $t^{th}$ iteration. Since $\mathcal{B}_j^{t}= \T{j}^{t+1}$, $ \sum_{i \in \mathcal{B}_j^{t}} c'(i,j) w^t = \sum_{i \in \T{j}^{t+1} : c'(i,j) \leq \dlj{}^{t+1}/2} c'(i,j) w^t + \sum_{i \in \T{j}^{t+1} : c'(i,j) = \dlj{}^{t+1}} c'(i,j) w^t = \\ \sum_{i \in \mathcal{B}_j^{t+1}} c'(i,j) w^t + (1- \sum_{i \in \mathcal{B}_j^{t+1}} w^t)\dlj{}^{t+1}=$ Cost paid by $j$ in $w^t$ in the $(t+1)^{th}$ iteration. Thus change in cost is $0$. \item[b.] When a client $j$ is moved from $\cliset_{part}$ to $\cliset_{full}$ because constraint~\ref{LPALP_const3} becomes tight. Cost paid by $j$ in $w^t$ in the $t^{th}$ iteration $= \sum_{i \in \T{j}^{t}} c'(i,j) w^t = \\ \sum_{i \in \T{j}^{t} : c'(i,j) \leq \rtj{}^{t}/2} c'(i,j) w^t + \sum_{i \in \T{j}^{t} : c'(i,j) = \rtj{}^{t}} c'(i,j) w^t = \\ \sum_{i \in \mathcal{B}_j^{t+1}} c'(i,j) w^t + (1-\sum_{i \in \mathcal{B}_j^{t+1}} w^t) \rtj{}^{t+1}=$ Cost paid by $j$ in $w^t$ in the $(t+1)^{th}$ iteration. Thus change in cost is $0$. \end{enumerate} \end{itemize} \end{proof} \qed Thus we have, $CostALP_{t+1}(w^{t+1}) \le CostALP_{t+1}(w^{t}) = CostALP_{t}(w^{t})$ where the first inequality follows because $w^{t+1}$ is an extreme point solution and $w^t$ is a feasible solution to $ALP_{t+1}$. Hence, if $n$ is the number of iterations of the IRA then $CostALP_n(w^n) \leq CostALP_{1}(w^1) \leq CostALP(w') \le 2\opt{LP}$ where the second last inequality follows as $w^1$ is an extreme point solution and $w'$ is a feasible solution for $ALP = ALP_1$, last inequality follows from Lemma~\ref{fs_kflo}. Let $w^*$ be the solution returned by the IRA, then $w^* = w^n$. Lemma~\ref{pseudo_integral} establishes that at the end of our IRA, solution $w^*$ is pseudo-integral. \begin{lemma} \label{pseudo_integral} $w^*$ returned by Algorithm \ref{IRA} has at most two fractionally opened facilities. \end{lemma} \begin{proof} At the termination of the algorithm constraints~\ref{LPALP_const2} and~\ref{LPALP_const3} will not be tight. Let $n_f$ be the number of fractional variables at the end of the algorithm. Then there are exactly $n_f$ number of independent tight constraints from~(\ref{LPALP_const1}),~(\ref{LPALP_const0}) and~(\ref{LPALP_const4}). Let $X$ be the number of tight constraints of type~\ref{LPALP_const1}. There must be at least $2$ fractional variables corresponding to each of these constraints. Also, there must be at least $2$ fractional variables corresponding to constraint~\ref{LPALP_const1}, different from those obtained constraints~\ref{LPALP_const4}. Thus, $n_f \ge 2X + 2$ i.e. $X \le n_f/2 -1$. Also, the number of tight constraints is at most $X + 2$ and hence is at most $n_f/2 + 1$ giving us $n_f \le n_f/2 + 1$ or $n_f \le 2$. \end{proof} \qed We open both the fractionally opened facilities at a loss of $+ f_{max}$ in the facility opening cost where $f_{max}$ is the guess of the most expensive facility opened by the optimal. In Lemma~\ref{neartocfull} we show that for a client $j$ in $\cliset_{full} \setminus \cliset^*$ there is some client in $ \cliset^*$, that is close to $j$, i.e. within $5\rtj{}$ distance of $j$. \begin{lemma} \label{neartocfull} At the conclusion of the algorithm, for every $j \in \cliset_{full}$, there exists at least 1 unit of open facilities within distance $5 \rtj{}$ from j. Formally, $\sum_{i:c'(i,j) \leq 5\rtj{}} w^*_i \geq 1$. \end{lemma} \begin{proof} Let $j \in \cliset_{full}$. If $resp(j) = j$, then this means that $j$ was added to $\cliset^*$ and was present in $\cliset^*$ at the end of the algorithm. Then, one unit is open in $\T{j}$ i.e. within a distance of $\rtj{}$ of $j$. If $resp(j) = j' (\ne j)$ then $j$ was either never added to $\cliset^*$ or was removed later. In either case responsibility of opening a facility in a close vicinity of $j$ was taken by $j'$. First we consider the case when $j$ was added to $\cliset^*$ but removed later. Let $j_0, j_1, \ldots j_r$ be the sequence of clients such that $resp(j_i) = j_{i-1}, i = 1 \ldots r, \ resp(j_0) = j_0$ and $j_r = j$. Since $resp(j_0) = j_0$, one unit is open in $\T{j_0}$ i.e. within a distance of $\rtj{0}$ of $j_0$. Clearly, $\rtj{i-1} \le \rtj{i}/2$. Thus $\rtj{i} \le (1/2)^{r -i} \rtj{r}$ for all $i = 0 \ldots r - 1$. Thus, $\distd{j_{r}}{j_{0}} \le \sum_{i = 1~to~r} \distd{j_{i}}{j_{i-1}} \le \sum_{i = 1~to~r} (\rtj{i} + \rtj{i - 1}) \le \rtj{0} + 2\sum_{i = 1~to~r-1} \rtj{i} + \rtj{r} \le \rtj{0} + 2\sum_{i = 1~to~r-1} (1/2)^{r -i} \rtj{r} + \rtj{r}$. Thus one unit of facility is open within a distance of $2\sum_{i = 0~to~r-1} (1/2)^{r -i} \rtj{r} \\ + \rtj{r} = \sum_{t = 1~to~r} (1/2)^{r - t} \rtj{r} + \rtj{r} \le 3 \rtj{r}$ from $j$. Next, let $j$ was never added to $\cliset^*$. Then since $resp(j) = j' (\ne j)$, $j'$ was added to $\cliset^*$ at some point of time. Thus, from above one unit of facility is opened within distance $3\rtj{'}$ of $j'$. Also, $\distd{j}{j'} \le \rtj{} + \rtj{'} \le 2\rtj{}$. Thus, one unit of facility is opened within distance $5\rtj{}$ of $j$. \end{proof} \qed We run the algorithm for all the guesses of $f_{max}$ and select the one with the minimum cost. \textbf{Combining Everything:} Let $\bar{w}$ be our final solution. $Cost(\bar{w}) \leq Cost(w^*)+f_{max} \leq 5 \cdot CostALP_n(w^*)+f_{max} \leq 5\cdot2\cdot\opt{LP} + f_{max} \leq 11OPT$ where $OPT$ is the cost of the optimal solution. \label{tri-criteria1} In this section we present a tri-criteria approximation for LBFLO problem with $1/4$-factor violation in lower bound and $2$-factor violation in outliers at a constant factor loss in cost. Following is the LP relaxation for LBFLO problem where for each $i \in \mathcal{F}$, $y_i$ indicates if facility $i$ is opened, for each $i \in \mathcal{F}$ and $j \in \mathcal{C}$, $x_{ij}$ indicates if client $j$ is assigned to facility $i$ and for each $j \in \mathcal{C}$, $z_j$ indicates if client $j$ is an outlier. \label{{LBFLO}} $\text{Minimize}~\mathcal{C}ostLBFLO(x,y,z) = \sum_{j \in \mathcal{C}}\sum_{i \in \mathcal{F}}\dist{i}{j}x_{ij} + \sum_{i \in \mathcal{F}}f_iy_i $ \begin{eqnarray} \text{subject to} &\sum_{i \in \mathcal{F}}{} x_{ij} + z_j \geq 1 & \forall ~\textit{j} \in \mathcal{C} \label{LbkFLo_const1}\\ & \lb y_i \leq \sum_{j \in \mathcal{C}}x_{ij} & \forall~ \textit{i} \in \mathcal{F} \label{LbkFLo_const2}\\ & \sum_{j \in \mathcal{C}} z_{j} \leq t & \label{LbkFLo_const4}\\ & x_{ij} \leq y_i & \forall~ \textit{i} \in \mathcal{F} , ~\textit{j} \in \mathcal{C} \label{LbkFLo_const5}\\ & y_i,x_{ij},z_j \in [0,1] \label{LPFLP_const5} \end{eqnarray} Constraints~\ref{LbkFLo_const1} ensure that every client is either served or is an outlier. Constraints~\ref{LbkFLo_const2} and~\ref{LbkFLo_const4} satisfies the lower bounds and number of outliers respectively. Constraints~\ref{LbkFLo_const5} are standard facility location constraint saying that a client is assigned to an open facility only. Let $opt=<x^*,y^*,z^*>$ be the optimal LP solution for the above LP. A solution is said to be an {\em integral open solution} if all the facilities are either fully opened or fully closed, i.e, $y_i=[0,1]$ for all $i \in \mathcal{F}$. Next we will construct an integral open solution $S=<\bar{x}, \bar{y}, \bar{z}>$. The solution $S$ is created in two steps: ($1$) First we create solution $S_{out}= <x', y', z'>$ by removing the clients that are outliers to a large extent($\geq 1/\lambda$) in $opt$ where $\lambda$ is a parameter to be fixed later, ($2$) Use clustering and rounding techniques to obtain an integral open solution for the remaining clients. Formally, in step 1, for all $j \in \mathcal{C}$ and $i \in \mathcal{F}$, set $z'_{j}=1$ and $x'_{ij}=0$ if $z^*_{j} \geq 1/\lambda$. Otherwise, set $z'_{j}=0$ and $x'_{ij}=x^*_{ij}$. For all $i \in \mathcal{F}$, $y'_{i}=y^*_{i}$. Note that in step 1 we just incur at most $\lambda$-factor in outliers. Also, $CostLBFLO(S_{out}) \leq CostLBFLO(opt)$. Next we will describe the Step 2 in detail. Let $\AVG{j}$ be the average connection cost for a client $j \in \mathcal{C}$ after Step 1, that is, $\AVG{j}=\sum_{i \in \mathcal{F}} \dist{i}{j}x'_{ij}/\sum_{i \in \mathcal{F}} x'_{ij}$. The clients are now considered in increasing order of radius $\lambda\AVG{j}$. Let $j$ be a client in this order, remove all the clients $k$ such that $c(j,k) \leq 2\lambda max \{ \AVG{j},\AVG{k} \}$ and repeat the process with the left over clients. Let $\mathcal{C}' \subset \mathcal{C}$ be the set of remaining clients after all the clients have been considered. Note that for any clients $j, k \in \mathcal{C}'$ the following property is satisfied: $c(j,k) > 2\lambda max \{ \AVG{j},\AVG{k} \}$. The total extent up to which facilities are opened in $\F{j}$ after Step 1 is $\geq (1-1/\lambda)(\sum_{i \in \mathcal{F}} x'_{ij}) \geq (1-1/\lambda) 1/\lambda$ where the last inequality follows because every client is served to an extent of at least $1/\lambda$ after first step. To obtain an integral open solution, for all $j \in \mathcal{C}'$, we open the cheapest(lowest facility opening cost) facility say $i_j \in \F{j}$ and transfer all the assignments coming on to the facilities in $\cluster{j}$ to $i_j$. Formally, set $\bar{y}_{i_j}=1$, $\bar{x}_{i_j j} = \sum_{i \in \cluster{j}}x'_{ij}$ and $\bar{y}_{i}=0$, $\bar{x}_{ij}=0$ for all $j \in \mathcal{C}$ and $i \neq i_j$. Set $\bar{z}_{j} = z'_{j}$ for all $j \in \mathcal{C}$. \begin{lemma} The integral open solution $S=<\bar{x},\bar{y},\bar{z}>$ violates lower bound by $\alpha=1/4$ and outliers by $\gamma=2$ at constant(?) factor loss in cost. \end{lemma} \begin{proof} Set $\lambda=1/2$. \end{proof} \section{Tri-criteria for LBFLO} \label{LBkFLO} In this section, we present a tri-criteria solution for LBFLO problem with $\alpha=(0,1)$-factor violation in lower bound and at most $\beta=(\frac{1}{1-\alpha})$-factor violation in outliers at ($\lambda(\frac{1+\alpha}{1-\alpha})$)-factor loss in cost where $\lambda$ is approximation for FLO. Let $I$ be an instance of LBFLO. For a facility $i$, let $\N{i}$ be the set of $\lb{i}$ nearest clients. We construct an instance $I'$ of FLO with lower bounds ignored and facility costs updated as follows: if a facility $i$ is opened in optimal solution of $I$, then it pays at least $\sum_{j \in \N{i}}c(i,j)$ cost for serving $\N{i}$ clients. Therefore, $f'(i)=f(i)+\delta \sum_{j \in \N{i}}c(i,j)$ where $\delta$ is a tunable parameter. \begin{lemma} \label{fs} Optimal solution of $I'$ is bounded by $ (\delta+1)Cost_I(O)$ where $O$ is the optimal solution of $I$. \end{lemma} \begin{proof} Clearly $O$ is a feasible solution for $I'$. Thus, service cost is same as that in $O$. And, $\sum_{i \in O} f'(i) = \sum_{i \in O}[f(i)+ \delta \sum_{j \in \N{i}}c(i,j)] \leq \delta Cost_I(O)$. Therefore, $Cost_{I'}(I') \leq (\delta+1)Cost_I(O)$. \end{proof} \qed Once we have an instance $I'$ of FLO, we use any algorithm for FLO to get a solution $AS'$ to $I'$ of cost no more than $\lambda Cost_{I'}(O')$ where $O'$ is the optimal solution of $I'$ and $\lambda$ is approximation solution for FLO. Note that a facility $i$ opened in solution $AS'$ might serve less than $\alpha\lb{i}$ clients as we ignored the lower bounds in instance $I'$. We close such facilities and do some reassignments to improve the violation in the lower bounds to $\alpha$; in the process we make some violation in the number of outliers. We convert the solution $AS'$ to a solution $AS$ of LBFLO. We close every facility $i$ that is serving less than $\alpha\lb{i}$ clients in $AS'$ and either reassign its clients to other opened facilities or decide to leave them unserved. Cost of reassignment is charged to the facility opening costs of the closed facilities. Consider a facility $i$ opened in $AS'$ that served less than $\alpha \lb{i}$ clients. Let $\clientset_i$ be the set of clients, in $\N{i}$, assigned to $i$ in $AS'$ and $\bar \clientset_i$ be the remaining clients in $\N{i}$. Since $i$ serves $<\alpha \lb{i}$ clients, $|\bar \clientset_i| \ge (1-\alpha )\lb{i}$. Some of the clients in $\bar \clientset_i$ are outliers in $AS'$ and some are assigned to other facilities. Let $\Out{i}$ be the clients in $\N{i}$ that are outliers and $\R{i}$ be the clients in $\N{i}$ assigned to some other facilities. See Fig.~\ref{division of clients}($a$). If $\R{i} \ne \phi$ then let $j' \in \R{i}$ be the nearest client to $i$. then, \begin{equation} \label{eq1} c(i,j') \leq \frac{\sum_{j \in \R{i}} c(i,j)}{|\R{i}|} \leq \frac{\sum_{j \in \N{i}} c(i,j)}{|\R{i}|} \end{equation} \begin{figure} \caption{ ($a$) Division of clients in $\N{i}$ for a facility $i$ opened in $AS'$ ($b$) $ c(j,i') \leq c(i,j) + c(i,j') + c(j',i')$ } \label{division of clients} \end{figure} Clients in $\clientset_i$ are assigned to the facilities serving the clients in $\R{i}$ and are made outliers proportionally. That is, we assign $\frac{|\R{i}|}{|\R{i}|+|\Out{i}|} |\clientset_i|$ clients in $\clientset_i$ to the nearest facility $i' \ne i$ opened in $AS'$ and leave $\frac{|\Out{i}|}{|\R{i}|+|\Out{i}|} |\clientset_i|$ clients in $\clientset_i$ unserved. If $|\R{i}| \neq 0$, then the total cost of reassignment is $\sum_{j \in \clientset_i} c(j,i') \leq \sum_{j \in \clientset_i} (c(i,j) + c(i,j') + c(j',i')) \text{(by triangle inequality, see Fig.~\ref{division of clients}($b$))}$ $\leq \sum_{j \in \clientset_i}c(i,j) + ( \frac{|\R{i}|}{|\R{i}|+|\Out{i}|} |\clientset_i| \cdot 2 c(i,j'))$ (as $j'$ was assigned to $i'$ and not to $i$ in $AS'$) $\leq \sum_{j \in \clientset_i}c(i,j) + ( \frac{2|\clientset_i|}{|\R{i}|+|\Out{i}|} \cdot \sum_{i \in \N{i}} c(i,j))$ (using~(\ref{eq1})) $\leq \sum_{j \in \clientset_i}c(i,j) + ( \frac{2\alpha\lb{i}}{(1-\alpha)\lb{i}} \sum_{i \in \N{i}} c(i,j))$ (As $|\clientset_i| \leq \alpha\lb{i}$ and $|\R{i}|+|\Out{i}| \geq (1-\alpha)\lb{i}$) $\leq \sum_{j \in \clientset_i}c(i,j) + f'(i)$ (for $\delta \geq \frac{2\alpha}{1-\alpha}$). Thus the additional cost of reassignment of clients in $\clientset_i$ is bounded by the facility opening cost of $i$. Violation in outliers is $\frac{|\Out{i}| + \frac{|\Out{i}|}{|\R{i}|+|\Out{i}|} |\clientset_i|}{|\Out{i}|} \leq 1+\frac{|\clientset_i|}{|\R{i}|+|\Out{i}|} \leq 1+\frac{\alpha}{1-\alpha} = \frac{1}{1-\alpha}$. \textbf{Overall Cost Bound:} It is easy to see that $Cost_I(AS)=Cost_{I'}(AS')$ as cost of solution $AS$ is sum of $(i)$ the original connection cost which is equal to the connection cost of $AS'$, ($ii$) the additional cost of reassignment, which is paid in $AS'$ by facilities that are closed in $AS$ and, ($iii$) the facility cost of the remaining facilities. Thus, $Cost_I(AS)=Cost_{I'}(AS') \leq \lambda Cost_{I'}(O') \leq \lambda (1+\delta)Cost_{I}(O) = \lambda (\frac{1+\alpha}{1-\alpha})Cost_{I}(O)$ for $\delta=\frac{2\alpha}{1-\alpha}$. Using $\lambda=(3+\epsilon)$-approximation of Charikar \textit{et al}.~\cite{charikar2001algorithms} for FLO, we get $(3+\epsilon)(\frac{1+\alpha}{1-\alpha})$ factor loss in cost for $\epsilon>0$. \section{Conclusion and Future Scope} \label{conclusion} In this paper, we first presented a $11$-factor approximation for $k$-facility location problem with outliers opening at most $k+1$ facilities. This also gives us the first constant factor approximation for FLO using LP rounding techniques. Our result can be extended to knapsack median problem with outliers with $(1+\epsilon)$ violation in budget using enumeration techniques. We also gave a tri-crtieria, $(\alpha, \frac{1}{1-\alpha},(3+\epsilon)\frac{1+\alpha}{1-\alpha})$-solution for general LBFLO where $\alpha=(0,1)$ and $\epsilon>0$. It will be interesting and challenging to see if we can reduce the violation in outliers to $<2$ maintaining $\alpha >1/2$. We believe that using pre-processing and strengthened LP techniques of Krishnaswamy \textit{et al}.~\cite{krishnaswamy-kMO} we can get rid of the $+1$ violation in cardinality for $k$FLO. This will also directly extend our tri-criteria solution to lower bounded $k$-facility location problem with outliers (LB$k$FLO). \end{document}

\begin{document} \title{Monotone Periodic Orbits for Torus Homeomorphisms} \author{Kamlesh Parwani} \date{September 1, 2003.} \maketitle \begin{abstract} Let $f$ be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector $(\frac{p}{q},\frac{r}{q})$, then $f$ has a topologically monotone periodic orbit with the same rotation vector. \end{abstract} \section*{Introduction} In this article we prove a theorem about the existence of topologically monotone periodic orbits on the torus. The concept of monotone orbits on the annulus is certainly not new; it goes back to Aubry and Mather's proof of the existence of orbits whose radial order is preserved by an area-preserving twist map of the annulus. These orbits, that have their radial order preserved by the map, are called Birkhoff orbits (see \cite {Katok}) or monotone orbits. This notion of monotone orbits inspired the definition of topologically monotone orbits in \cite{Boyland}, where Boyland proved that any homeomorphism of the annulus isotopic to the identity that has a periodic orbit with a non-zero rotation number $\frac{p}{q}$ also has a topologically monotone periodic orbit with the same rotation number. A topologically monotone periodic orbit has the property that the isotopy class of the map, keeping the periodic orbit fixed as a set, is of finite order. The main tool used in Boyland's proof is Nielsen-Thurston theory. In \cite{Llibre&Mackay}, Llibre and Mackay asked whether a similar result was true for torus homeomorphisms. The goal of this paper is to answer that question by proving the same theorem on the torus. \textbf{Main Theorem.} \textit{If }$f$\textit{\ is a torus homeomorphism isotopic to the identity that has a periodic orbit with a non-zero rotation vector }$ \left( \frac{p}{q},\frac{r}{q}\right) $\textit{, then }$f$\textit{\ also has a topologically monotone periodic orbit with the same rotation vector.} There are some immediate complications one encounters while trying to generalize the theorem to the torus. First, the torus has rotation vectors instead of rotation numbers. Then on the annulus, under certain restrictions to the rotation number, we can only get the $pA$ (pseudoAnosov) isotopy class or the finite order isotopy class, but for the torus, there is the reducible isotopy class to deal with also. These concepts will be introduced in Section 1, and then in Section 2 we prove the main theorem of this paper. It should be noted that LeCalvez has proved the existence of topologically monotone periodic orbits on the torus under the assumption that the maps are smooth by using variational techniques (see \cite{LeCalvez}). Since we are dealing with homeomorphisms, we rely solely on topological methods. \section{Definitions and important results} \subsection{Rotation vectors} Let $f$ be a homeomorphism of the torus which is isotopic to the identity and let $F$ be its lift to the universal cover $\widetilde{T^{2}}$, the plane. Let $()_{1}$ and $()_{2}$ be the projections of a point in the plane to the $x$-axis and the $y$-axis respectively and let $x$ be a point on the torus $T^{2}$ with $\widetilde{x}$ as its lift. Then the \textbf{rotation vector} of $\widetilde{x}$, with respect to a lift $F$, is defined as following if the limit exists. \begin{equation*} \rho (\widetilde{x},F)=\left( \lim_{n\to \infty }\left( \frac{F^{n}( \widetilde{x})-\widetilde{x}}{n}\right) _{1},\lim_{n\to \infty }\left( \frac{ F^{n}(\widetilde{x})-\widetilde{x}}{n}\right) _{2}\right) \end{equation*} If $x$ is a periodic point, say of period $q$, then the rotation vector is always well defined and can be written as $(\frac{p}{q},\frac{r}{q})$ for some integers $p$ and $r$. In fact, the rotation vector for any point on the orbit of $x$ is the same, and so, we can associate the vector $(\frac{p}{q}, \frac{r}{q})$ to the periodic orbit. Periodic orbits with rotation vector $(\frac{p}{q},\frac{r}{q})$ and least period $q$ will be called $(p,r,q)$ orbits. Note that a $(p,r,q)$ orbit has the same rotation vector as a $(pt,rt,qt)$ orbit, where $t$ is some positive integer. The covering space of the torus comes naturally equipped with two important covering translations. Define $X(\widetilde{x})=\widetilde{x}+(1,0)$ and $Y( \widetilde{x})=\widetilde{x}+(0,1)$. Clearly, the rotation vector of a point depends on the lift, and the relationship is $\rho (\widetilde{x} ,Y^{m}X^{n}F)=\rho (\widetilde{x},F)+(n,m)$, where $n$ and $m$ are integers. So when we discuss periodic orbits with a certain rotation vector in this paper, we assume the existence of some lift for which that rotation vector is realized. Also, when we start with a periodic orbit, say $(p,r,q)$ orbit, and then prove that another $(p,r,q)$ orbit exists, it is to be understood that both rotation vectors are calculated with respect to the same lift. \subsection{The Nielsen-Thurston classification theorem and braids} Every orientation preserving homeomorphism of an orientable surface with negative Euler characteristic is isotopic to a homeomorphism $g$ such that either a) $g$ is finite order, or b) $g$ is pseudoAnosov ($pA$), or c) $g$ is reducible. A map $g$ is said to be \textit{reducible} if there is a disjoint collection $C$ of non-parallel, non-peripheral simple disjoint curves such that $g$ leaves invariant the union of disjoint regular neighborhoods of curves in $C$ , and the first return map on each complementary component is either of finite order or $pA$. This classification theorem was first announced in \cite{Thurston} and the proofs appeared later in \cite{FLP} and \cite{Casson}. The torus doesn't have negative Euler characteristic but, following Handel as in \cite{Handel}, we will examine the isotopy class relative to a periodic orbit; this will introduce punctures and provide the negative Euler characteristic to apply the Nielsen-Thurston Classification Theorem. When the isotopy class relative to a given periodic orbit is of finite order, the periodic orbit is called a \textit{finite order periodic orbit}, and \textit{ reducible} and $pA$ \textit{periodic orbits} are defined similarly. The isotopy class relative to a periodic orbit is also referred to as the \textit{braid} of the periodic orbit. \begin{definition} Let $x$\ and $y$\ be two distinct periodic points of least period $n$\ for homeomorphisms $f$ and $g$ respectively of the same orientable surface $S$. Then the orbit of $x$\ ($O(x)$) and the orbit of $y$\ ($O(y)$) have the same \textit{braid} if there exists an orientation-preserving homeomorphism $h$\ of $S$ with the property that $h$\ maps $O(x)$\ onto $O(y)$ and the isotopy class of $h^{-1}fh$\ relative to the orbit of $y$\ is the same as the isotopy class of $g$\ relative to the orbit of $y$, that is, $ [h^{-1}fh]_{O(y)}=[f]_{O(y)}$. \end{definition} A periodic orbit has a \textbf{trivial} braid if the isotopy class relative to the periodic orbit is of finite order, that is, there exists a homeomorphism $g$ isotopic to $f$, relative to the periodic orbit, such that $g^{n}=identity$ for some $n$. In other words, finite order periodic orbits have trivial braids. These periodic orbits are considered to be topologically monotone. A periodic orbit has a \textbf{non-trivial} braid if the isotopy class relative to the periodic orbit is not of finite order. In other words, periodic orbits with non-trivial braids are either reducible periodic orbits or are $pA$ periodic orbits. These periodic orbits are not topologically monotone. Boyland defined a natural partial order $(\vartriangleright)$ into these braids. If $\alpha $ and $\beta $ are two braids of periodic orbits, then $ \alpha \vartriangleright \beta $ if and only if the existence of a periodic orbit with braid $\alpha $ in any homeomorphism $f$ on a given surface implies the existence of a periodic orbit with braid $\beta $ for the same $f$. The proof of the fact that this is an actual partial order is not easy and is in \cite{Boyland}. He also proved that a $pA$ periodic orbit is strictly above (in the partial ordering) all other periodic orbits that are present in the $pA$ representative of the isotopy class relative to the $pA$ periodic orbit. The existence of topologically monotone periodic orbits on the annulus in \cite{Boyland} is established by showing that periodic orbits with non-trivial braids force the existence of periodic orbits with trivial braids and the same rotation number (non-trivial $\vartriangleright $ trivial). \begin{theorem}[Boyland] Let f be a homeomorphism of the annulus isotopic to the identity. If f has a periodic orbit with non-zero rotation number $\frac{p}{q}$, then f also has a topologically monotone periodic orbit with the same rotation number. \end{theorem} Essentially, we follow the same strategy on the torus and the proof of the main theorem in Section 2 is broken into two parts, reducible $ \vartriangleright $ finite order and $pA$ $\vartriangleright $ finite order. We will also need the following result which can be obtained from the arguments in \cite{Hall&Boyl} and is also proved in \cite{thesis}. \begin{theorem} Let f be a homeomorphism of the torus that is isotopic to the identity and has a pA periodic orbit. Let g be the pA representative of the isotopy class relative to this orbit and let G be its lift to the plane that fixes the lifts of all the points in the pA orbit. Then G has a dense orbit. \end{theorem} We will use this theorem in the next section to prove $pA$ $\vartriangleright $ trivial. \section{Finite order periodic orbits on the torus} In this section we prove the main theorem by showing that finite order periodic orbits are on the bottom in the partial ordering of periodic orbits for torus homeomorphism isotopic to the identity, that is, the reducible $ \vartriangleright $ finite order and $pA\vartriangleright $ finite order. We restrict our attention to periodic orbits with the same non-zero rotation vector, say $(\frac{p}{q},\frac{r}{q})$, and assume that there are no common factors between $p$, $r$, and $q$. Later we reduce the general case, in which there may be a common factor between $p$, $r$, and $q$, to the case of no factors. \begin{theorem} Let $f:T^{2}\rightarrow T^{2}$ be a homeomorphism isotopic to the identity. Suppose there exists a $(p,r,q)$ orbit such that $gcd(p,r,q)=1$, then there exists a finite order $(p,r,q)$ orbit. \end{theorem} \begin{proof} If the given periodic orbit is already of finite order type, then we're done. If not, the proof breaks down in to two cases---the periodic orbit is either of reducible type or it is of $pA$ type. These are handled separately below. \textbf{Reducible Case}. We assume that we obtain a reducible isotopy class, keeping the periodic orbit fixed. A reducing curve can be of two types---essential or non-essential (this is with respect to the unpunctured torus). A reducing curve cannot be non-essential for this would imply that there is a common factor between $p,r,q$. In fact, the common factor would be exactly the number of punctures contained in the disc bounded by the non-essential curve. So any reducing curve must be essential. All the essential reducing curves are disjoint, and thus, split the torus into parallel annuli. These annuli have an equal number of punctures, say $n$ punctures, in their interiors and are permuted by the action of the map $g$, which represents this reducible isotopy class. In this case, we will show that we can obtain a finite order isotopy class, keeping a periodic orbit with the same period and rotation vector (it may not be the same periodic orbit that we start with). Let $ A_{0},A_{1},A_{2},...,A_{m-1}$ be the annuli that $g$ permutes, numbered so that $g(A_{k})=A_{k+1}$ mod $m$ and $nm=q$. \begin{figure} \caption{The Reducible Case} \end{figure} CASE 1. Suppose that the maps $g^{m}:A_{k}\rightarrow A_{k}$ relative to the punctures are all of finite order. It is easy to see that all these maps are conjugate to each other, and so if one is of finite order, then all are of finite order. Since all finite order maps on the annulus are conjugate to rotations, it follows that $g^{mn}$ (or $g^{q}$) is the identity in each annulus. We will now argue that this implies that $g^{q}$ is isotopic to the identity on the entire torus relative to the $(p,r,q)$ orbit. The complements of the interiors of annuli containing the punctures (the $A_{k}$'s) are closed annuli that do not contain any punctures. Observe that $g^{q}$ fixes the boundary components of these unpunctured annuli since it fixes the boundary components of the $A_{k}$'s. So if $g^{q}$ is isotopic to the identity in all these unpunctured annuli relative to their boundaries (keeping the boundaries fixed throughout the isotopy) then $g^{q}$ is isotopic to the identity on the entire torus relative to the $(p,r,q)$ orbit, because we already know that $g^{q}$ is the identity on the annuli containing the punctures. Now suppose $g^{q}$ is not isotopic to the identity in one of these unpunctured annuli relative to its boundary components, then $g^{q}$ must be isotopic to some non-trivial Dehn twists. It is easy to see that the maps on all these annuli are all conjugate to each other so $g^{q}$ is isotopic to the same Dehn twists in each unpunctured annulus. However, $g^{q}$ is isotopic to identity on the entire torus when the punctures are allowed to move, because $g$ is isotopic to $f$ and $f$ is isotopic to the identity by assumption. If we have non-trivial Dehn twists that don't cancel each other out (because they are identical), $g^{q}$ is not isotopic to identity, which is a contradiction. It follows that $g^{q}$ is isotopic to the identity on the entire torus relative to the $(p,r,q)$ orbit. To show that $(p,r,q)$ orbit is topologically monotone, we require a map isotopic to $g$ relative to the orbit that is of finite order. Such a map is guaranteed by Fenchel's solution to the Nielsen Realization problem for finite solvable groups (see Chapter 3 in \cite{Zieschang}) which provides a map $h$ isotopic to $g$, relative to the $ (p,r,q)$ orbit, such that $h^{q}$ is the identity. This shows that the $ (p,r,q)$ orbit is of finite order type, that is, it is topologically monotone. CASE 2. Suppose that the maps $g^{m}:A_{k}\rightarrow A_{k}$ relative to the punctures are all $pA$. It now follows from Boyland's proof of Theorem 1.2 in \cite{Boyland} that there exists a finite order periodic orbit with the same period and rotation number in each $A_{k}$. Since all these annulus maps are conjugate, the periodic orbits connect in the torus to give a periodic orbit with the same rotation vector and period as the originally punctured orbit. This reduces to Case 1 and we can find an isotopy relative to the new orbit such that the isotoped map is of finite order. Because the periodic orbits in the $pA$ components are unremovable (see \cite{Boyl:stability}), this periodic orbit existed in the original map $f$. We have actually established a stronger result. If we do obtain a finite order periodic orbit which is distinct from the one we began with, then it is strictly below the original orbit in the partial order. This is because the only way for the original orbit to not be of a finite order type is for the reducible components to be $pA$, that is, the maps $g^{m}:A_{k}\rightarrow A_{k}$ are $pA$. And $pA$ orbits are strictly above all periodic orbits that are present in the $pA$ representative of the isotopy class (see \cite{Boyland}). \textbf{\textit{pA} Case}. In Lemma 2.3, we will prove that any $pA$ $(p,r,q)$ orbit forces another $(p,r,q)$ orbit. Boyland proved that this other periodic orbit is strictly below the $pA$ orbit in the partial order (see \cite{Boyland}). Furthermore, there are only finitely many periodic orbits of any given period in any $pA$ map. So consider a minimal $(p,r,q)$ orbit in the partial order. If it's $pA$, there is another $(p,r,q)$ orbit below it---so it's not minimal. If it's reducible and not of finite order, then it forces another $(p,r,q)$ finite order orbit (by the argument above for the reducible case). Thus, any minimal $(p,r,q)$ orbit must be of finite order and there is at least one minimal orbit. \end{proof} It now remains to prove Lemma 2.3. We shall appeal to the following result in \cite{Fathi} for the existence of a fixed point of positive index. The proof is based on the ideas used to demonstrate the Brouwer Plane Translation Theorem (see \cite{Franks}). \begin{theorem}[Fathi] Let $G:R^{2}\rightarrow R^{2}$\ be an orientation preserving homeomorphism which possesses a non-wandering point, then G has a fixed point. If G has only isolated fixed points, then it has a fixed point of positive index. \end{theorem} \begin{lemma} Let $g:T^{2}\rightarrow T^{2}$\ be the $\mathit{pA}$\ representative obtained from $\mathit{f}$ keeping the $\mathit{(p,r,q)}$ orbit fixed throughout the isotopy. Then $\mathit{g}$\ has another $\mathit{ (p,r,q)}$\ periodic orbit. \end{lemma} \begin{proof} Let $g$ be the $pA$ representative obtained relative to the $(p,r,q)$ orbit. Let $G$ be the lift to the plane that realizes the rotation vector $(\frac{p}{q},\frac{r}{q})$ for the $pA$ orbit and then consider $X^{-p}Y^{-r}G^{q}$; call this map $H$. Then by Theorem 1.3, we know that $H$ has a dense orbit and so there is a non-wandering point. Since the periodic points (of any given period) are isolated in a $pA$ map (see \cite {FLP}), the fixed points of $H$ on the plane are also isolated. So, by Theorem 2.2, we have a fixed point with positive index for $H$, which is a $(p,r,q)$ orbit for $g$, and each point in the orbit is fixed with positive index for $g^{q}$. The $pA$ $(p,r,q)$ orbit is the location of all the one-prongs or needles and these needles have index zero. Since there is a fixed point for $g^{q}$ with positive index and because the indices have to add up to zero, we have actually shown that the $pA$ periodic orbit forces at least two other $(p,r,q)$ periodic orbits. \end{proof} \begin{proof}[Proof of the Main Theorem] Let $f$ be a homeomorphism of the torus isotopic to the identity and suppose there exists a periodic orbit with rotation vector $(\frac{p}{q},\frac{r}{q})$. Also assume that this is a $(pt,rt,qt)$ periodic orbit, where $t$ is a positive integer and there are no common factors between $p$, $r$, and $q$. Let $F$ be the lift to the plane and consider $X^{-p}Y^{-r}F^{q}$. We obtain a periodic point of period $t$ for $X^{-p}Y^{-r}F^{q}$. Then by Theorem 2.2, we also obtain a fixed point for $X^{-p}Y^{-r}F^{q}$, which has period $q$ for $f$ and rotation vector $(\frac{p}{q},\frac{r}{q})$, that is, it is a $(p,r,q)$ orbit. Thus, without loss of generality, we may assume that we have a $(p,r,q)$ orbit where there are no common factors between $p$, $r$, and $q$. Now, by Theorem 2.1, we obtain a $(p,r,q)$ orbit that is topologically monotone and it has the desired rotation vector $(\frac{p}{q},\frac{r}{q})$. \end{proof} It is natural to ask if there is a similar theorem for non-periodic orbits with irrational rotation vectors. This question is unanswered even for the annulus. A similar theorem about the existence of monotone periodic orbits has been proved for periodic orbits on surfaces of higher genus (see \cite {Parwani}). \section*{Acknowledgments} The author would like to thank John Franks and Philip Boyland for several useful and stimulating conversations. \end{document}

\begin{document} \newdateformat{mydate}{\THEDAY~\monthname~\THEYEAR} \title [Vanishing viscosity: observations] {Observations on the vanishing viscosity limit} \author{James P. Kelliher} \address{Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521} \curraddr{Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521} \email{[email protected]} \subjclass[2010]{Primary 76D05, 76B99, 76D10} \keywords{Vanishing viscosity, boundary layer theory} \begin{abstract} Whether, in the presence of a boundary, solutions of the Navier-Stokes equations converge to a solution to the Euler equations in the vanishing viscosity limit is unknown. In a seminal 1983 paper, Tosio Kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the Navier-Stokes velocity in a boundary layer of width proportional to the viscosity. In a 2008 paper, the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary. We present here several observations that follow on from these two papers. \Ignore{ We make several observations regarding the vanishing viscosity limit, primarily regarding the control of the total mass of vorticity and the conditions in Tosio Kato's seminal 1983 paper \cite{Kato1983} shown by him to be equivalent to the vanishing viscosity limit. } \end{abstract} \date{(compiled on {\dayofweekname{\day}{\month}{\year} \mydate\today)}} \maketitle \begin{small} \begin{flushright} Compiled on \textit{\textbf{\dayofweekname{\day}{\month}{\year} \mydate\today}} \end{flushright} \end{small} \renewcommand\contentsname{} \begin{small} \tableofcontents \end{small} \noindent The Navier-Stokes equations for a viscous incompressible fluid in a domain $\Omega \subseteq \ensuremath{\BB{R}}^d$, $d \ge 2$, with no-slip boundary conditions can be written, \begin{align*} (NS) \left\{ \begin{array}{rl} \ensuremath{\partial}_t u + u \cdot \ensuremath{\nabla} u + \ensuremath{\nabla} p = \nu \Delta u + f &\text{ in } \Omega, \\ \dv u = 0 &\text{ in } \Omega, \\ u = 0 &\text{ on } \Gamma := \ensuremath{\partial} \Omega. \end{array} \right. \end{align*} The Euler equations modeling inviscid incompressible flow on such a domain with no-penetration boundary conditions can be written, \begin{align*} (EE) \left\{ \begin{array}{rl} \ensuremath{\partial}_t \overline{u} + \overline{u} \cdot \ensuremath{\nabla} \overline{u} + \ensuremath{\nabla} p = \nu \Delta \overline{u} + \overline{f} &\text{ in } \Omega, \\ \dv \overline{u} = 0 &\text{ in } \Omega, \\ \overline{u} \cdot \bm{n} = 0 &\text{ on } \Gamma. \end{array} \right. \end{align*} Here, $u = u_\nu$ and $\overline{u}$ are velocity fields, while $p$ and $\overline{p}$ are pressure (scalar) fields. The external forces, $f$, $\overline{f}$, are vector fields. (We adopt here the notation of Kato in \cite{Kato1983}.) We assume throughout that $\Omega$ is bounded and $\Gamma$ has $C^2$ regularity, and write $\bm{n}$ for the outward unit normal vector. The limit, \begin{align*} (VV) \qquad u \to \overline{u} \text{ in } L^\ensuremath{\infty}(0, T; L^2(\Omega)), \end{align*} we refer to as the \textit{classical vanishing viscosity limit}. Whether it holds in general, or fails in any one instance, is a major open problem in mathematical fluids mechanics. In \cite{K2006Kato, K2008VVV} a number of conditions on the solution $u$ were shown to be equivalent to ($VV$). The focus in \cite{K2006Kato} was on the size of the vorticity or velocity in a layer near the boundary, while the focus in \cite{K2008VVV} was on the accumulation of vorticity on the boundary. The work we present here is in many ways a follow-on to \cite{K2006Kato, K2008VVV}, each of which, especially \cite{K2006Kato}, was itself an outgrowth of Tosio Kato's seminal paper \cite{Kato1983} on the vanishing viscosity limit, ($VV$). This paper is divided into two themes. The first theme concerns the accumulation of vorticity---on the boundary, in a boundary layer, or in the bulk of the fluid. It explores the consequences of having control of the total mass of vorticity or, more strongly, the $L^1$-norm of the vorticity for solutions to ($NS$). We re-express in a specifically 3D form the condition for vorticity accumulation on the boundary from \cite{K2008VVV} in \cref{S:3DVersion}. In \cref{S:LpNormsBlowUp}, we show that if ($VV$) holds then the $L^p$ norms of the vorticity for solutions to ($NS$) must blow up for all $p > 1$ as $\nu \to 0$ except in very special circumstances. This leaves only the possibility of control of the vorticity's $L^1$ norm. Assuming such control, we show in \cref{S:ImprovedConvergence} that when ($VV$) holds we can characterize the accumulation of vorticity on the boundary more strongly than in \cite{K2008VVV}. In \cref{S:BoundaryLayerWidth}, we show that if we measure the width of the boundary layer by the size of the $L^1$-norm of the vorticity then the layer has to be wider than that of Kato if ($VV$) holds. We push this analysis further in \cref{S:OptimalConvergenceRate} to obtain the theoretically optimal convergence rate when the initial vorticity has nonzero total mass, as is generic for non-compatible initial data. We turn a related observation into a conjecture concerning the connection between the vanishing viscosity limit and the applicability of the Prandtl theory. In \cref{S:SomeConvergence}, we show that the arguments in \cite{K2008VVV} lead to the conclusion that some kind of convergence of a subsequence of the solutions to ($NS$) always occurs in the limit as $\nu \to 0$, but not necessarily to a solution to the Euler equations. The second theme more directly addresses Tosio Kato's conditions from \cite{Kato1983} that are equivalent to ($VV$). We also deal with the closely related condition from \cite{K2006Kato} that uses vorticity in place of the gradient of the velocity that appears in one of Kato's conditions. We derive in \cref{S:EquivCondition} a condition on the solution to ($NS$) on the boundary that is equivalent in 2D to ($VV$), giving a number of examples to which this condition applies in \cref{S:Examples}. In \cref{S:BardosTiti} we discuss some interesting recent results of Bardos and Titi that they developed using dissipative solutions to the Euler Equations. We show how weaker, though still useful, 2D versions of these results can be obtained using direct elementary methods. We start, however, in \cref{S:Background} with the notation and definitions we will need, and a summary of the pertinent results of \cite{K2006Kato, K2008VVV, Kato1983}. \section{Definitions and past results}\label{S:Background} \noindent We define the classical function spaces of incompressible fluids, \begin{align*} H &= \set{u \in (L^2(\Omega))^d: \dv u = 0 \text{ in } \Omega, \, u \cdot \mathbf{n} = 0 \text{ on } \Gamma} \end{align*} with the $L^2$-norm and \begin{align*} V &= \set{u \in (H_0^1(\Omega))^d: \dv u = 0 \text{ in } \Omega} \end{align*} with the $H^1$-norm. We denote the $L^2$ or $H$ inner product by $(\cdot, \cdot)$. If $v$, $w$ are vector fields then $(v, w) = (v^i, w^i)$, where we use here and below the common convention of summing over repeated indices. Similarly, if $M$, $N$ are matrices of the same dimensions then $M \cdot N = M^{ij} N^{ij}$ and \begin{align*} (M, N) = (M^{ij}, N^{ij}) = \int_\Omega M \cdot N. \end{align*} We will assume that $u$ and $\overline{u}$ satisfy the same initial conditions, \begin{align*} u(0) = u_0, \quad \overline{u}(0) = u_0, \end{align*} and that $u_0$ is in $C^{k + \ensuremath{\epsilon}}(\Omega) \cap H$, $\ensuremath{\epsilon} > 0$, where $k = 1$ for two dimensions and $k = 2$ for 3 and higher dimensions, and that $f = \overline{f} \in C^1_{loc}(\ensuremath{\BB{R}}; C^1(\Omega))$. Then as shown in \cite{Koch2002} (Theorem 1 and the remarks on p. 508-509), there is some $T > 0$ for which there exists a unique solution, \begin{align}\label{e:ubarSmoothness} \overline{u} \text{ in } C^1([0, T]; C^{k + \ensuremath{\epsilon}}(\Omega)), \end{align} to ($EE$). In two dimensions, $T$ can be arbitrarily large, though it is only known that some positive $T$ exists in three and higher dimensions. With such initial velocities, we are assured that there are weak solutions to $(NS)$, unique in 2D. Uniqueness of these weak solutions is not known in three and higher dimensions, so by $u = u_\nu$ we mean any of these solutions chosen arbitrarily. We never employ strong or classical solutions to $(NS)$. \Ignore{ It follows, assuming that $f$ is in $L^1([0, T]; L^2(\Omega))$, that for such solutions, \begin{align}\label{e:NSVariationalIdentity} \begin{split} &(u(t), \phi(t)) - (u(0), \phi(0)) \\ &\qquad= \int_0^t \brac{(u, u \cdot \ensuremath{\nabla} \phi) - \nu (\ensuremath{\nabla} u, \ensuremath{\nabla} \phi) + (f, \phi) + (u, \ensuremath{\partial}_t \phi))} \, dt \end{split} \end{align} for all $\phi$ in $C^1([0, T] \times \Omega) \cap C^1([0, T]; V)$. } We define $\gamma_\mathbf{n}$ to be the boundary trace operator for the normal component of a vector field in $H$ and write \begin{align}\label{e:RadonMeasures} \Cal{M}(\overline{\Omega}) \text{ for the space of Radon measures on } \overline{\Omega}. \end{align} That is, $\Cal{M}(\overline{\Omega})$ is the dual space of $C(\overline{\Omega})$. We let $\mu$ in $\Cal{M}(\overline{\Omega})$ be the measure supported on $\Gamma$ for which $\mu\vert_\Gamma$ corresponds to Lebesgue measure on $\Gamma$ (arc length for $d = 2$, area for $d = 3$). Then $\mu$ is also a member of $H^1(\Omega)^*$, the dual space of $H^1(\Omega)$. We define the vorticity $\omega(u)$ to be the $d \times d$ antisymmetric matrix, \begin{align}\label{e:VorticityRd} \omega(u) = \frac{1}{2}\brac{\ensuremath{\nabla} u - (\ensuremath{\nabla} u)^T}, \end{align} where $\ensuremath{\nabla} u$ is the Jacobian matrix for $u$: $(\ensuremath{\nabla} u)^{ij} = \ensuremath{\partial}_j u^i$. When working specifically in two dimensions, we alternately define the vorticity as the scalar curl of $u$: \begin{align}\label{e:VorticityR2} \omega(u) = \ensuremath{\partial}_1 u^2 - \ensuremath{\partial}_2 u^1. \end{align} Letting $\omega = \omega(u)$ and $\overline{\omega} = \omega(\overline{u})$, we define the following conditions: \begingroup \allowdisplaybreaks \begin{align*} (A) & \qquad u \to \overline{u} \text{ weakly in } H \text{ uniformly on } [0, T], \\ (A') & \qquad u \to \overline{u} \text{ weakly in } (L^2(\Omega))^d \text{ uniformly on } [0, T], \\ (B) & \qquad u \to \overline{u} \text{ in } L^\ensuremath{\infty}([0, T]; H), \\ (C) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} \overline{u} - \innp{\gamma_\mathbf{n} \cdot, \overline{u} \mu} \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (D) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} \overline{u} \text{ in } (H^{-1}(\Omega))^{d \times d} \text{ uniformly on } [0, T], \\ (E) & \qquad \omega \to \overline{\omega} - \frac{1}{2} \innp{\gamma_\mathbf{n} (\cdot - \cdot^T), \overline{u} \mu} \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (F) & \qquad \omega \to \overline{\omega} \text{ in } (H^{-1}(\Omega))^{d \times d} \text{ uniformly on } [0, T]. \end{align*} \endgroup We stress that $(H^1(\Omega))^*$ is the dual space of $H^1(\Omega)$, in contrast to $H^{-1}(\Omega)$, which is the dual space of $H^1_0(\Omega)$. The condition in $(B)$ is the classical vanishing viscosity limit of ($VV$). We will make the most use of condition $(E)$, which more explicitly means that \begin{align}\label{e:EExplicit} (\omega(t), M) \to (\overline{\omega}(t), M) - \frac{1}{2}\int_{\Gamma} ((M - M^T) \cdot \mathbf{n}) \cdot \overline{u}(t) \text{ in } L^\ensuremath{\infty}([0, T]) \end{align} for any $M$ in $(H^1(\Omega))^{d \times d}$. In two dimensions, defining the vorticity as in \refE{VorticityR2}, we also define the following two conditions: \begin{align*} (E_2) & \qquad \omega \to \overline{\omega} - (\overline{u} \cdot \BoldTau) \mu \text{ in } (H^1(\Omega))^* \text{ uniformly on } [0, T], \\ (F_2) & \qquad \omega \to \overline{\omega} \text{ in } H^{-1}(\Omega) \text{ uniformly on } [0, T]. \end{align*} Here, $\BoldTau$ is the unit tangent vector on $\Gamma$ that is obtained by rotating the outward unit normal vector $\mathbf{n}$ counterclockwise by $90$ degrees. \Ignore{ Condition ($E_2$) means that \begin{align*} (\omega(t), f) \to (\overline{\omega}(t), f) - \int_{\Gamma} (\overline{u}(t) \cdot \BoldTau) f \text{ in } L^\ensuremath{\infty}([0, T]) \end{align*} for any $f$ in $H^1(\Omega)$. } \refT{VVEquiv} is proved in \cite{K2008VVV} ($(A) \implies (B)$ having been proved in \cite{Kato1983}), to which we refer the reader for more details. \begin{theorem}[\cite{K2008VVV}]\label{T:VVEquiv} Conditions ($A$), ($A'$), ($B$), ($C$), ($D$), and ($E$) are equivalent (and each implies condition ($F$)). In two dimensions, condition ($E_2$) and, when $\Omega$ is simply connected, ($F_2$) are equivalent to the other conditions.\footnote{The restriction that $\Omega$ be simply connected for the equivalence of ($F_2$) was not, but should have been, in the published version of \cite{K2008VVV}.} \end{theorem} \cref{T:VVEquiv} remains silent about rates of convergence, but examining the proof of it in \cite{K2008VVV} easily yields the following: \begin{theorem}\label{T:ROC} Assume that ($VV$) holds with \begin{align*} \norm{u - \overline{u}}_{L^\ensuremath{\infty}(0, T; L^2(\Omega))} \le F(\nu) \end{align*} for some fixed $T > 0$. Then \begin{align*} \norm{(u(t) - \overline{u}(t), v)}_{L^\ensuremath{\infty}([0, T])} \le F(\nu) \norm{v}_{L^2(\Omega)} \text{ for all } v \in (L^2(\Omega))^d \end{align*} and \begin{align*} \norm{(\omega(t) - \overline{\omega}(t), \varphi)}_{L^\ensuremath{\infty}([0, T])} \le F(\nu) \norm{\ensuremath{\nabla} \varphi}_{L^2} \text{ for all } \varphi \in H_0^1(\Omega). \end{align*} \end{theorem} \begin{remark}\label{R:ROCOthers} \cref{T:ROC} gives the rates of convergence for ($A$) and ($F_2$); the rates for ($C$), ($D$), ($E$), and ($E_2$) are like those given for ($F_2$) (though the test function, $\varphi$, will lie in different spaces). \end{remark} In \cite{Kato1983}, Tosio Kato showed that ($VV$) is equivalent to \begin{align*} \nu \int_0^T \norm{\ensuremath{\nabla} u(s)}_{L^2(\Omega)}^2 \, dt \to 0 \text{ as } \nu \to 0 \end{align*} and to \begin{align}\label{e:KatoCondition} \nu \int_0^T \norm{\ensuremath{\nabla} u(s)}_{L^2(\Gamma_{c \nu})}^2 \, dt \to 0 \text{ as } \nu \to 0. \end{align} Here, and in what follows, $\Gamma_\delta$ is a boundary layer in $\Omega$ of width $\delta > 0$. In \cite{K2006Kato} it is shown that in \cref{e:KatoCondition}, the gradient can be replaced by the vorticity, so ($VV$) is equivalent to \begin{align}\label{e:KellCondition} \nu \int_0^T \norm{\omega(s)}_{L^2(\Gamma_{c \nu})}^2 \, dt \to 0 \text{ as } \nu \to 0. \end{align} Note that the necessity of \cref{e:KellCondition} follows immediately from \cref{e:KatoCondition}, but the sufficiency does not, since on the inner boundary of $\Gamma_{c \nu}$ there is no boundary condition of any kind. We also mention the works \cite{TW1998, W2001}, which together establish conditions equivalent to \refE{KatoCondition}, with a boundary layer slightly larger than that of Kato, yet only involving the tangential derivatives of either the normal or tangential components of $u$ rather than the full gradient. These conditions will not be used in the present work, however. \Ignore{ The setup and notation are that of \cite{K2008VVV, K2006Kato}, and is largely inherited from \cite{Kato1983}: Weak solutions to the Navier-Stokes equations in a bounded domain, $\Omega$, having $C^2$-boundary, $\Gamma$, are denoted by $u$, the viscosity, $\nu > 0$, being implied by context. Weak (or often strong) solutions to the Euler equations are denoted by $\overline{u}$. Except in \refS{NavierBCs}, we use homogeneous Dirichlet conditions ($u = 0$) for the Navier-Stokes equations and we in any case always use no-penetration conditions ($u \cdot \bm{n} = 0$) for the Euler equations. Here, $\bm{n}$ is the outward normal to the boundary. We use $\omega = \omega(u)$ to be the curl of $u$, defined to be $\ensuremath{\partial}_1 u^2 - \ensuremath{\partial}_2 u^1$ in $2D$ and the antisymmetric part of $\ensuremath{\nabla} u$ in higher dimensions. Similarly for $\overline{\omega} = \omega(\overline{u})$. We denote the $L^2$-inner product by $(\cdot, \cdot)$, and write $V$ for the space of all divergence-free vector fields in $H_0^1(\Omega)$. We will also use the related function space $H$ of divergence-free vector fields $v$ in $L^2(\Omega)$ with $v \cdot \mathbf{n} = 0$ on $\Gamma$ in the sense of a trace. See \cite{K2008VVV, K2006Kato} for more details. } \Part{Theme I: Accumulation of vorticity} \section{A 3D version of vorticity accumulation on the boundary}\label{S:3DVersion} \noindent In \cref{T:VVEquiv}, the vorticity is defined to be the antisymmetric gradient, as in \cref{e:VorticityRd}. When working in 3D, it is usually more convenient to use the language of three-vectors in condition ($E$). This leads us to the condition $(E')$ in \cref{P:EquivE}. \begin{prop}\label{P:EquivE} The condition (E) in \cref{T:VVEquiv} is equivalent to \begin{align*} (E') \qquad \curl u \to \curl \overline{u} + (\overline{u} \times \bm{n}) \mu \text{ in } L^\ensuremath{\infty}((0, T; (H^1(\Omega)^3)^*). \end{align*} \end{prop} \begin{proof} If $A$ is an antisymmetric $3 \times 3$ matrix then \begin{align*} A \cdot M &= \frac{A \cdot M + A \cdot M}{2} = \frac{A \cdot M + A^T \cdot M^T}{2} = \frac{A \cdot M - A \cdot M^T}{2} \\ &= A \cdot \frac{M - M^T}{2}. \end{align*} Thus, since $\omega$ and $\overline{\omega}$ are antisymmetric, referring to \refE{EExplicit}, we see that ($E$) is equivalent to \begin{align*} (\omega(t), M) \to (\overline{\omega}(t), M) - \int_\Gamma (M \bm{n}) \cdot \overline{u}(t) \text{ in } L^\ensuremath{\infty}([0, T]) \end{align*} for all \textit{antisymmetric} matrices $M \in (H^1(\Omega))^{3 \times 3}$. Now, for any three vector $\varphi$ define \begin{align*} F(\varphi) &= \tmatrix{0 & -\varphi_3 & \varphi_2} {\varphi_3 & 0 & -\varphi_1} {-\varphi_2 & \varphi_1 & 0}. \end{align*} Then $F$ is a bijection from the vector space of three-vectors to the space of antisymmetric $3 \times 3$ matrices. Straightforward calculations show that \begin{align*} F(\varphi) \cdot F(\psi) = 2 \varphi \cdot \psi, \qquad F(\varphi) v = \varphi \times v \end{align*} for any three-vectors, $\varphi$, $\psi$, $v$. Also, $F(\curl u) = 2 \omega$ and $F(\curl \overline{u}) = 2 \overline{\omega}$. For any $\varphi \in (H^1(\Omega))^3$ let $M = F(\varphi)$. Then \begin{align*} (\omega, M) &= \frac{1}{2} \pr{F(\curl u), F(\varphi)} = \pr{\curl u, \varphi}, \\ (\overline{\omega}, M) &= \frac{1}{2} \pr{F(\curl \overline{u}), F(\varphi)} = \pr{\curl \overline{u}, \varphi}, \\ (M \bm{n}) \cdot \overline{u} &= (F(\varphi) \bm{n}) \cdot \overline{u} = (\varphi \times \bm{n}) \cdot \overline{u} = - (\overline{u} \times \bm{n}) \cdot \varphi. \end{align*} In the last equality, we used the scalar triple product identity $(a \times b) \cdot c = - a \cdot (c \times b)$. Because $F$ is a bijection, this gives the equivalence of ($E$) and ($E'$). \end{proof} \section{\texorpdfstring{$L^p$}{Lp}-norms of the vorticity blow up for \texorpdfstring{$p > 1$}{p > 1}}\label{S:LpNormsBlowUp} \noindent \begin{theorem}\label{T:VorticityNotBounded} Assume that $\overline{u}$ is not identically zero on $[0, T] \times \Gamma$. If any of the equivalent conditions of \cref{T:VVEquiv} holds then for all $p \in (1, \ensuremath{\infty}]$, \begin{align}\label{e:omegaBlowup} \limsup_{\nu \to 0^+} \norm{\omega}_{L^\ensuremath{\infty}([0, T]; L^p)} \to \ensuremath{\infty}. \end{align} \end{theorem} \begin{proof} We prove the contrapositive. Assume that the conclusion is not true. Then for some $q' \in (1, \ensuremath{\infty}]$ it must be that for some $C_0 > 0$ and $\nu_0 > 0$, \begin{align}\label{e:omegaBoundedCondition} \norm{\omega}_{L^\ensuremath{\infty}([0, T]; L^{q'})} \le C_0 \text{ for all } 0 < \nu \le \nu_0. \end{align} Since $\Omega$ is a bounded domain, if \cref{e:omegaBoundedCondition} holds for some $q' \in (1, \ensuremath{\infty}]$ it holds for all lower values of $q'$ in $(1, \ensuremath{\infty}]$, so we can assume without loss of generality that $q' \in (1, \ensuremath{\infty})$. Let $q = q'/(q' - 1) \in (1, \ensuremath{\infty})$ be \Holder conjugate to $q$ and $p = 2/q + 1 \in (1, 3)$. Then $p, q, q'$ satisfy the conditions of \cref{C:TraceCor} with $(p -1) q = 2$. Applying \cref{C:TraceCor} gives, for almost all $t \in [0, T]$, \begingroup \allowdisplaybreaks \begin{align*} &\norm{u(t) - \overline{u}(t)}_{L^p(\Gamma)} \le C \norm{u(t) - \overline{u}(t)}_{L^2(\Omega)} ^{1 - \frac{1}{p}} \norm{\ensuremath{\nabla} u(t) - \ensuremath{\nabla} \overline{u}(t)}_{L^{q'}(\Omega)} ^{\frac{1}{p}} \\ &\qquad \le C \norm{u(t) - \overline{u}(t)}_{L^2(\Omega)} ^{1 - \frac{1}{p}} \pr{\norm{\ensuremath{\nabla} u(t)}_{L^{q'}} + \norm{\ensuremath{\nabla} \overline{u}(t)}_{L^{q'}}} ^{\frac{1}{p}} \\ &\qquad \le C \norm{u(t) - \overline{u}(t)}_{L^2(\Omega)} ^{1 - \frac{1}{p}} \pr{C(q') \norm{\omega(t)}_{L^{q'}} + \norm{\ensuremath{\nabla} \overline{u}(t)}_{L^{q'}}} ^{\frac{1}{p}} \\ &\qquad \le C \norm{u(t) - \overline{u}(t)}_{L^2(\Omega)} ^{1 - \frac{1}{p}} \end{align*} \endgroup for all $0 < \nu \le \nu_0$. Here we used \cref{e:omegaBoundedCondition} and the inequality, $\norm{\ensuremath{\nabla} u}_{L^{q'}(\Omega)} \le C(q') \norm{\omega}_{L^{q'}(\Omega)}$ for all $q' \in (1, \ensuremath{\infty})$ of Yudovich \cite{Y1963}. Hence, \begin{align*} \norm{u - \overline{u}}_{L^\ensuremath{\infty}([0, T]; L^p(\Gamma))} \le C \norm{u - \overline{u}}_{L^\ensuremath{\infty}([0, T]; L^2(\Omega))} ^{1 - \frac{1}{p}} \to 0 \end{align*} as $\nu \to 0$. But, \begin{align*} \norm{u - \overline{u}}_{L^\ensuremath{\infty}([0, T]; L^p(\Gamma))} = \norm{\overline{u}}_{L^\ensuremath{\infty}([0, T]; L^p(\Gamma))} \ne 0, \end{align*} so condition (B) cannot hold and so neither can any of the equivalent conditions in \cref{T:VVEquiv}. \end{proof} \section{Improved convergence when vorticity bounded in \texorpdfstring{$L^1$}{L1}}\label{S:ImprovedConvergence} \noindent In \cref{S:LpNormsBlowUp} we showed that if the classical vanishing viscosity limit holds then the $L^p$ norms of $\omega$ must blow up as $\nu \to 0$ for all $p \in (1, \ensuremath{\infty}]$---unless the Eulerian velocity vanishes identically on the boundary. This leaves open the possibility that the $L^1$ norm of $\omega$ could remain bounded, however, and still have the classical vanishing viscosity limit. This happens, for instance, for radially symmetric vorticity in a disk (Examples 1a and 3 in \cref{S:Examples}), as shown in \cite{FLMT2008}. In fact, as we show in \cref{C:EquivConvMeasure}, when ($VV$) holds and the $L^1$ norm of $\omega$ remains bounded in $\nu$, the convergence in condition ($E$) is stronger; namely, $weak^*$ in measure (as in \cite{FLMT2008}). (See \cref{e:RadonMeasures} and the comments after it for the definitions of $\Cal{M}(\overline{\Omega})$ and $\mu$.) \begin{cor}\label{C:EquivConvMeasure} Suppose that $u \to \overline{u} \text{ in } L^\ensuremath{\infty}(0, T; H)$ and $\curl u$ is bounded in $L^\ensuremath{\infty}(0, T; L^1(\Omega))$ uniformly in $\ensuremath{\epsilon}$. Then in 3D, \begin{align}\label{e:BetterConvergence} \curl u \to \curl \overline{u} + (u_0 \times \bm{n}) \mu \quad \weak^* \text{ in } L^\ensuremath{\infty}(0, T; \Cal{M}(\overline{\Omega})). \end{align} Similarly, ($C$), ($E$), and ($E_2$) hold with $\weak^*$ convergences in $L^\ensuremath{\infty}(0, T; \Cal{M}(\overline{\Omega}))$ rather than uniformly in $(H^1(\Omega))^*$. \end{cor} \begin{proof} We prove \cref{e:BetterConvergence} explicitly for 3D solutions, the results for ($C$), ($E$), and ($E_2$) following in the same way. Let $\psi \in C(\overline{\Omega})$. What we must show is that \begin{align*} (\curl u(t) - \curl \overline{u}(t), \psi) \to \int_\Gamma (u_0(t) \times \bm{n}) \cdot \psi \text{ in } L^\ensuremath{\infty}([0, T]). \end{align*} So let $\ensuremath{\epsilon} > 0$ and choose $\varphi \in H^1(\Omega)^d$ with $\norm{\psi - \varphi}_{C(\overline{\Omega})} < \ensuremath{\epsilon}$. We can always find such a $\varphi$ because $H^1(\Omega)$ is dense in $C(\overline{\Omega})$. Let \begin{align*} M = \max \set{\norm{\curl u - \curl \overline{u}}_{L^\ensuremath{\infty}(0, T; L^1(\Omega))}, \norm{\overline{u}}_{L^\ensuremath{\infty}([0, T] \times \Omega)}}, \end{align*} which we note is finite since $\norm{\curl u}_{L^\ensuremath{\infty}(0, T; L^1(\Omega))}$ and $\norm{\curl \overline{u}}_{L^\ensuremath{\infty}(0, T; L^1(\Omega))}$ are both finite. Then \begingroup \allowdisplaybreaks \begin{align*} &\abs{(\curl u(t) - \curl \overline{u}(t), \psi) - \int_\Gamma (u_0(t) \times \bm{n}) \cdot \psi} \\ &\qquad \le \abs{(\curl u(t) - \curl \overline{u}(t), \psi - \varphi) - \int_\Gamma (u_0(t) \times \bm{n}) \cdot (\psi - \varphi)} \\ &\qquad\qquad + \abs{(\curl u(t) - \curl \overline{u}(t), \varphi) - \int_\Gamma (u_0(t) \times \bm{n}) \cdot \varphi} \\ &\qquad \le 2 M \ensuremath{\epsilon} + \abs{(\curl u(t) - \curl \overline{u}(t), \varphi) - \int_\Gamma (u_0(t) \times \bm{n}) \cdot \varphi}. \end{align*} \endgroup By \cref{P:EquivE}, we can make the last term above smaller than, say, $\ensuremath{\epsilon}$, by choosing $\nu$ sufficiently small, which is sufficient to give the result. \end{proof} \begin{remark} Suppose that we have the slightly stronger condition that $\ensuremath{\nabla} u$ is bounded in $L^\ensuremath{\infty}(0, T; L^1(\Omega))$ uniformly in $\ensuremath{\epsilon}$. If we are in 2D, $W^{1, 1}(\Omega)$ is compactly embedded in $L^2(\Omega)$. This is sufficient to conclude that ($VV$) holds, as shown in \cite{GKLMN14}. \end{remark} \section{Width of the boundary layer}\label{S:BoundaryLayerWidth} \noindent Working in two dimensions, make the assumptions on the initial velocity and on the forcing in \cref{T:VVEquiv}, and assume in addition that the total mass of the initial vorticity does not vanish; that is, \begin{align}\label{e:NonzeroMass} m := \int_\Omega \omega_0 = (\omega_0, 1) \ne 0. \end{align} (In particular, this means that $u_0$ is not in $V$.) The total mass of the Eulerian vorticity is conserved so \begin{align}\label{e:mEAllTime} (\overline{\omega}(t), 1) = m \text{ for all } t \in \ensuremath{\BB{R}}. \end{align} The Navier-Stokes velocity, however, is in $V$ for all positive time, so its total mass is zero; that is, \begin{align}\label{e:mNSAllTime} (\omega(t), 1) = 0 \text{ for all } t > 0. \end{align} Let us suppose that the vanishing viscosity limit holds. Fix $\delta > 0$ let $\varphi_\delta$ be a smooth cutoff function equal to $1$ on $\Gamma_\delta$ and equal to 0 on $\Omega \setminus \Gamma_{2 \delta}$. Then by ($F_2$) of \cref{T:VVEquiv} and using \cref{e:mEAllTime}, \begin{align*} \abs{(\omega, 1 - \varphi_\delta) - m} \to \abs{(\overline{\omega}, 1 - \varphi_\delta) - m} = \abs{m - (\overline{\omega}, \varphi_\delta) - m} \le C \delta, \end{align*} the convergence being uniform on $[0, T]$. Thus, for all sufficiently small $\nu$, \begin{align}\label{e:omega1phiLimit} \abs{(\omega, 1 - \varphi_\delta) - m} \le C \delta. \end{align} \Ignore { \begin{align}\label{e:E2VVV} \omega \to \overline{\omega} - (\overline{u} \cdot \BoldTau) \mu \text{ in } (H^1(\Omega))^* \text{ uniformly on } [0, T]. \end{align} Fix $\delta > 0$ let $\varphi_\delta$ be a smooth cutoff function equal to $1$ on $\Gamma_\delta$ and equal to 0 on $\Omega \setminus \Gamma_{2 \delta}$. Letting $\nu \to 0$, since $\varphi_\delta = 1$ on $\Gamma$, we have \begin{align*} (\omega, \varphi_\delta) &\to (\overline{\omega}, \varphi_\delta) - \int_\Gamma \overline{u} \cdot \BoldTau = (\overline{\omega}, \varphi_\delta) + \int_\Gamma \overline{u}^\perp \cdot \mathbf{n} \\ &= (\overline{\omega}, \varphi_\delta) + \int_\Omega \dv \overline{u}^\perp = (\overline{\omega}, \varphi_\delta) - \int_\Omega \overline{\omega} \\ &= (\overline{\omega}, \varphi_\delta) - \int_\Omega \overline{\omega}_0 = (\overline{\omega}, \varphi_\delta) - m. \end{align*} The convergence here is uniform over $[0, T]$. Now, \begin{align*} \abs{(\overline{\omega}, \varphi_\delta)} \le \norm{\overline{\omega}}_{L^\ensuremath{\infty}} \abs{\Gamma_{2 \delta}} = \norm{\overline{\omega}_0}_{L^\ensuremath{\infty}} \abs{\Gamma_{2 \delta}} \le C \delta. \end{align*} Thus, for all sufficiently small $\nu$, \begin{align}\label{e:omegaphiLimit} \abs{(\omega, \varphi_\delta) + m} \le C \delta. \end{align} For $t > 0$, $u$ is in $V$ so the total mass of $\omega$ is zero for all $t > 0$; that is, \begin{align*} \int_\Omega \omega = 0. \end{align*} It follows that for all sufficiently small $\nu$, \begin{align}\label{e:omega1phiLimit} \abs{(\omega, 1 - \varphi_\delta) - m} \le C \delta. \end{align} This reflects one of the consequences of \cref{T:VVEquiv} that \begin{align*} \omega \to \overline{\omega} \text{ in } H^{-1}(\Omega) \text{ uniformly on } [0, T], \end{align*} which represents a kind of weak internal convergence of the vorticity. } In \cref{e:omega1phiLimit} we must hold $\delta$ fixed as we let $\nu \to 0$, for that is all we can obtain from the weak convergence in ($F_2$). Rather, this is all we can obtain without making some assumptions about the rates of convergence, a matter we will return to in the next section. Still, it is natural to ask whether we can set $\delta = c \nu$ in \cref{e:omega1phiLimit}, this being the width of the boundary layer in Kato's seminal paper \cite{Kato1983} on the subject. If this could be shown to hold it would say that outside of Kato's layer the vorticity for solutions to ($NS$) converges in a (very) weak sense to the vorticity for the solution to ($E$). The price for such convergence, however, would be a buildup of vorticity inside the layer to satisfy the constraint in \cref{e:mNSAllTime}. In fact, however, this is not the case, at least not by a closely related measure of vorticity buildup near the boundary. The total mass of the vorticity (in fact, its $L^1$-norm) in any layer smaller than that of Kato goes to zero and, if the vanishing visocity limit holds, then the same holds for Kato's layer. Hence, if there is a layer in which vorticity accumulates, that layer is at least as wide as Kato's and is wider than Kato's if the vanishing viscosity limit holds. This is the content of the following theorem. \begin{theorem}\label{T:BoundaryLayerWidth} Make the assumptions on the initial velocity and on the forcing in \cref{T:VVEquiv}. For any positive function $\delta = \delta(\nu)$, \begin{align}\label{e:OmegaL1VanishGeneral} \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\delta(\nu)}))} \le C \pr{\frac{\delta(\nu)}{\nu}}^{1/2}. \end{align} If the vanishing viscosity limit holds and \begin{align*} \limsup_{\nu \to 0^+} \frac{\delta(\nu)}{\nu} < \ensuremath{\infty} \end{align*} then \begin{align}\label{e:OmegaL1Vanish} \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\delta(\nu)}))} \to 0 \text{ as } \nu \to 0. \end{align} \end{theorem} \begin{proof} By the Cauchy-Schwarz inequality, \begin{align*} \norm{\omega}_{L^1(\Gamma_{\delta(\nu)})} \le \norm{1}_{L^2(\Gamma_{\delta(\nu)})} \norm{\omega}_{L^2(\Gamma_{\delta(\nu)})} \le C \delta^{1/2} \norm{\omega}_{L^2(\Gamma_{\delta(\nu)})} \end{align*} so \begin{align*} \frac{C}{\delta} \norm{\omega}_{L^1(\Gamma_{\delta(\nu)})}^2 \le \norm{\omega}_{L^2(\Gamma_{\delta(\nu)})}^2 \end{align*} and \begin{align*} \frac{C \nu}{\delta} \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\delta(\nu)}))}^2 \le \nu \norm{\omega}_{L^2([0, T]; L^2(\Gamma_{\delta(\nu)}))}^2. \end{align*} By the basic energy inequality for the Navier-Stokes equations, the right-hand side is bounded, giving \refE{OmegaL1VanishGeneral}, and if the vanishing viscosity limit holds, the right-hand side goes to zero by \cref{e:KellCondition}, giving \refE{OmegaL1Vanish}. \end{proof} \begin{remark} In \refT{BoundaryLayerWidth}, we do not need the assumption in \refE{NonzeroMass} nor do we need to assume that we are in dimension two. The result is of most interest, however, when one makes these two assumptions. \end{remark} \begin{remark} \refE{OmegaL1Vanish} also follows from condition (iii'') in \cite{K2006Kato} using the Cauchy-Schwarz inequality in the manner above, but that is using a sledge hammer to prove a simple inequality. Note that \refE{OmegaL1Vanish} is necessary for the vanishing viscosity limit to hold, but is not (as far as we can show) sufficient. \end{remark} \Ignore{ \begin{theorem}\label{T:BoundaryLayerWidth} Make the assumptions on the initial velocity and on the forcing in \cref{T:VVEquiv}. Assume that the vanishing viscosity limit holds. For any nonnegative function $\delta = \delta(\nu)$, \begin{align}\label{e:OmegaMassVanishGeneral} \limsup_{\nu \to 0^+} \int_0^T \abs{\int_{\Gamma_{\delta(\nu)}} \omega} \le C T \lim_{\nu \to 0} \frac{\delta(\nu)}{\nu}. \end{align} If \begin{align*} \lim_{\nu \to 0} \frac{\delta(\nu)}{\nu} = 0 \end{align*} then \begin{align}\label{e:OmegaMassVanish} \int_0^T \abs{\int_{\Gamma_{\delta(\nu)}} \omega} \to 0 \text{ as } \nu \to 0. \end{align} \end{theorem} \begin{proof} \begin{align*} \int_{\Gamma_\delta} \omega = \int_{A_{L, \delta}} \omega + \int_{\Gamma_\delta \setminus A_{L, \delta}} \omega, \end{align*} where \begin{align*} A_{L, \delta}= \set{x \in \Gamma_\delta \colon \abs{\omega} \ge L}. \end{align*} Thus, \begin{align*} \int_{\Gamma_\delta} \omega \le \int_{A_{L, \delta}} \omega + L \abs{\Gamma_\delta} \le \int_{A_{L, \delta}} \omega + C \delta L. \end{align*} Let $L$ vary with $\nu$ at a rate we will specify later. Then, \begin{align*} \nu &\int_0^T \int_{\Gamma_\delta} \abs{\omega}^2 = \nu \int_0^T \int_{A_{L, \delta}} \abs{\omega}^2 + \nu \int_0^T \int_{\Gamma_\delta \setminus A_{L, \delta}} \abs{\omega}^2 \\ &\ge \nu \int_0^T \int_{A_{L, \delta}} L \abs{\omega} \ge L \nu \int_0^T \abs{\int_{A_{L, \delta}} \omega} \\ &\ge L \nu \brac{\int_0^T \abs{\int_{\Gamma_\delta} \omega} - \int_0^T C \delta L} = L \nu \int_0^T \abs{\int_{\Gamma_\delta} \omega} - C T \nu \delta L^2. \end{align*} Define \begin{align*} M(\nu) = \int_0^T \abs{\int_{\Gamma_{\delta(\nu)}} \omega}, \quad M = \limsup_{\nu \to 0^+} M(\nu). \end{align*} Letting $L = \nu^{-1}$, we have \begin{align*} \limsup_{\nu \to 0^+} &\, \nu \int_0^T \int_{\Gamma_{\delta(\nu)}} \abs{\omega}^2 \ge \limsup_{\nu \to 0^+} \brac{L_k \nu M(\nu) - CT \nu \delta(\nu) L^2} \\ &= M - CT \limsup_{\nu \to 0^+} \frac{\delta(\nu)}{\nu} = M. \end{align*} But because we have assumed that the vanishing viscosity limit holds, the left-hand side vanishes with $\nu$ regardless of how the function $\delta$ is chosen. Thus, \begin{align*} M \le CT \limsup_{\nu \to 0^+} \frac{\delta(\nu)}{\nu}, \end{align*} giving \refE{OmegaMassVanishGeneral} and also \refE{OmegaMassVanish}. \end{proof} } \section{Optimal convergence rate}\label{S:OptimalConvergenceRate} \noindent Still working in two dimensions, let us return to \cref{e:omega1phiLimit}, assuming as in the previous section that the vanishing viscosity limit holds, but bringing the rate of convergence function, $F$, of \cref{T:ROC} into the analysis. We will now make $\delta = \delta(\nu) \to 0$ as $\nu \to 0$, and choose $\varphi_\delta$ slightly differently, requiring that it equal $1$ on $\Gamma_{\delta^*}$ and vanish outside of $\Gamma_\delta$ for some $0 < \delta^* = \delta^*(\nu) < \delta$. We can see from the argument that led to \cref{e:omega1phiLimit}, incorporating the convergence rate for ($F_2$) given by \cref{T:ROC}, that \begin{align*} \abs{(\omega, 1 - \varphi_\delta) - m} \le C \delta + \norm{\ensuremath{\nabla} \varphi_\delta}_{L^2(\Omega)} F(\nu). \end{align*} Because $\ensuremath{\partial} \Omega$ is $C^2$, we can always choose $\varphi_\delta$ so that $\abs{\ensuremath{\nabla} \varphi_\delta} \le C(\delta - \delta^*)^{-1}$. Then for all sufficiently small $\delta$, \begin{align*} \norm{\ensuremath{\nabla} \varphi_\delta}_{L^2(\Omega)} \le \pr{\int_{\Gamma_\delta \setminus \Gamma_{\delta^*}} \pr{\frac{C}{\delta - \delta^*}}^2}^{\frac{1}{2}} = C \frac{(\delta - \delta^*)^{\frac{1}{2}}}{\delta - \delta^*} = C (\delta - \delta^*)^{-\frac{1}{2}}. \end{align*} We then have \begin{align}\label{e:mDiffEst} \abs{(\omega, 1 - \varphi_\delta) - m} \le C \brac{\delta + (\delta - \delta^*)^{-\frac{1}{2}} F(\nu)}. \end{align} For any measurable subset $\Omega'$ of $\Omega$, define \begin{align*} \mathbf{M}(\Omega') = \int_{\Omega'} \omega, \end{align*} the total mass of vorticity on $\Omega'$. Then \begin{align*} \mathbf{M}(\Gamma_\delta^C) = (\omega, 1 - \varphi_\delta) + \int_{\Gamma_\delta \setminus \Gamma_{\delta^*}} \varphi_\delta \omega \end{align*} so \begin{align}\label{e:MDiffEst} \begin{split} \abs{(\omega, 1 - \varphi_\delta) - \mathbf{M}(\Gamma_\delta^C)} &\le \norm{\omega}_{L^2(\Gamma_\delta \setminus \Gamma_{\delta^*})} \norm{\varphi_\delta}_{L^2(\Gamma_\delta \setminus \Gamma_{\delta^*})} \\ &\le C (\delta - \delta^*)^{\frac{1}{2}} \norm{\omega}_{L^2(\Gamma_{\delta})}. \end{split} \end{align} \Ignore{ To obtain any reasonable control on the total mass of vorticity, we certainly need $\delta, \delta^* \to 0$ as $\nu \to 0$, but more important, as we can see from \cref{e:mDiffEst}, we need \begin{align}\label{e:LayerReq1} (\delta - \delta^*)^{-\frac{1}{2}} F(\nu) \to 0 \text{ as } \nu \to 0. \end{align} In light of \cref{T:BoundaryLayerWidth} and its proof, we should also require at least that \begin{align}\label{e:LayerReq2} (\delta - \delta^*)^{\frac{1}{2}} \norm{\omega}_{L^2(0, T; L^2(\Gamma_\delta))} \to 0 \text{ as } \nu \to 0 \end{align} so that the bound in \cref{e:mDiffEst} will lead, via \cref{e:MDiffEst}, to a bound on the total mass of vorticity outside the boundary layer, $\Gamma_\delta$. Now, as in the proof of \cref{T:BoundaryLayerWidth}, if we let $\delta - \delta^* = O(\nu)$ then the condition in \cref{e:LayerReq2} will hold by \cref{e:KellCondition}. Then the requirement in \cref{e:LayerReq1} becomes \begin{align*} F(\nu) = o \pr{(\delta - \delta^*)^{\frac{1}{2}}} = o (\nu^{\frac{1}{2}}). \end{align*} } From these observations and those in the previous section, we have the following: \begin{theorem}\label{T:VorticityMassControl} \Ignore{ Assume that $\delta = \delta(\nu) \to 0$ as $\nu \to 0$ and define \begin{align*} M_\delta = \norm{\int_\Omega \omega_0 - \int_{\Gamma_\delta^C} \omega(t)}_{L^2([0, T])}. \end{align*} If the classical vanishing viscosity limit in ($VV$) holds with a rate that is $o(\nu^{\frac{1}{2}})$ then $M_{\delta(\nu)} \to 0$ as $\nu \to 0$. } Assume that the classical vanishing viscosity limit in ($VV$) holds with a rate of convergence, $F(\nu) = o(\nu^{1/2})$. Then in 2D the initial mass of the vorticity must be zero. \end{theorem} \begin{proof} From \cref{e:mDiffEst,e:MDiffEst}, \begin{align*} M_\delta &:= \abs{m - \mathbf{M}(\Gamma_\delta^C)} \le \abs{m - (\omega, 1 - \varphi_\delta)} + \abs{(\omega, 1 - \varphi_\delta) - \mathbf{M}(\Gamma_\delta^C)} \\ &\le C \brac{\delta + (\delta - \delta^*)^{-\frac{1}{2}} F(\nu)} + C (\delta - \delta^*)^{\frac{1}{2}} \norm{\omega}_{L^2(\Gamma_{\delta})}. \end{align*} Choosing $\delta(\nu) = \nu$, $\delta^*(\nu) = \nu/2$, we have \begin{align*} M_\nu &\le C \brac{\nu + \nu^{-\frac{1}{2}} o(\nu^{\frac{1}{2}})} + C \nu^{\frac{1}{2}} \norm{\omega}_{L^2(\Gamma_{\nu})}, \end{align*} uniformly over $[0, T]$. Squaring, integrating in time, and applying Young's inequality gives \begin{align*} \norm{M_\nu}_{L^2([0, T])}^2 = \int_0^T M_\nu^2 \le CT (\nu^2 + o(1)) + C \nu \int_0^T \norm{\omega}_{L^2(0, T; L^2(\Gamma_\nu))}^2 \to 0 \end{align*} as $\nu \to 0$ by \cref{e:KellCondition}. Then, \begin{align*} \norm{m - M(\Omega)}_{L^2([0, T])} &\le \norm{m - M(\Gamma_\nu^C)}_{L^2([0, T])} + \norm{M(\Gamma_\nu)}_{L^2([0, T])} \\ &\le \norm{M_\nu}_{L^2([0, T])} + \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\nu}))} \to 0 \end{align*} as $\nu \to 0$ by \cref{T:BoundaryLayerWidth}. But $u(t)$ lies in $V$ so $M(\Omega) = 0$ for all $t > 0$. Hence, the limit above is possible only if $m = 0$. \end{proof} For non-compatible initial data, that is for $u_0 \notin V$, the total mass of vorticity will generically not be zero, so $C \sqrt{\nu}$ should be considered a bound on the rate of convergence for non-compatible initial data. As we will see in \cref{R:ROC}, however, a rate of convergence as good as $C \sqrt{\nu}$ is almost impossible unless the initial data is fairly smooth, and even then it would only occur in special circumstances. Therefore, let us assume that the rate of convergence in ($VV$) is only $F(\nu) = C \nu^{1/4}$. As we will see in \cref{S:Examples}, this is a more typical rate of convergence for the simple examples for which ($VV$) is known to hold. Now \cref{e:mDiffEst} still gives a useful bound as long as $\delta - \delta^*$ is slightly larger than the Prandtl layer width of $C \sqrt{\nu}$ (though \cref{e:MDiffEst} then fails to tell us anything useful). So let us set $\delta = 2 \nu^{1/2 - \ensuremath{\epsilon}}$, $\delta^* = \nu^{1/2 - \ensuremath{\epsilon}}$, $\ensuremath{\epsilon} > 0$ arbitrarily small. We are building here to a conjecture, so for these purposes we will act as though $\ensuremath{\epsilon} = 0$. If the Prandtl theory is correct, then we should expect that $\mathbf{M}(\Gamma_\delta^C) \to m$ as $\nu \to 0$, since outside of the Prandtl layer $u$ matches $\overline{u}$. But the total mass of vorticity for all positive time is zero, and the total mass in the Kato Layer, $\Gamma_\nu$, goes to zero by \cref{T:BoundaryLayerWidth}. There would be no choice then but to have a total mass of vorticity between the Kato and Prandtl layers that approaches $-m$ as the viscosity vanishes. (Since the Kato layer is much smaller than the Prandtl layer, this does not require that there be any higher concentration of vorticity in any particular portion of the Prandtl layer, though.) Now suppose that the rate of convergence is even slower than $C \nu^{1/4}$. Then \cref{e:mDiffEst} gives a measure of $\mathbf{M}(\Gamma_\delta^C)$ converging to $m$ well outside the Prandtl layer. This does not directly contradict any tenet of the Prandtl theory, but it suggests that for small viscosity the solution to the Navier-Stokes equations matches the solution to the Euler equations only well outside the Prandtl layer. This leads us to the following conjecture: \begin{conj}\label{J:Prandtl} If the vanishing viscosity limit in ($VV$) holds at a rate slower than $C \nu^{\frac{1}{4}}$ in 2D then the Prandtl theory fails. \end{conj} We conjecture no further, however, as to whether the Prandtl equations become ill-posed or whether the formal asymptotics fail to hold rigorously. \section{Some kind of convergence always happens}\label{S:SomeConvergence} \noindent Assume that $v$ is a vector field lying in $L^\ensuremath{\infty}([0, T]; H^1(\Omega))$. An examination of the proof given in \cite{K2008VVV} of the chain of implications in \cref{T:VVEquiv} shows that all of the conditions except (B) are still equivalent with $\overline{u}$ replaced by $v$. That is, defining \begingroup \allowdisplaybreaks \begin{align*} (A_v) & \qquad u \to v \text{ weakly in } H \text{ uniformly on } [0, T], \\ (A'_v) & \qquad u \to v \text{ weakly in } (L^2(\Omega))^d \text{ uniformly on } [0, T], \\ (B_v) & \qquad u \to v \text{ in } L^\ensuremath{\infty}([0, T]; H), \\ (C_v) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} v - \innp{\gamma_\mathbf{n} \cdot, v \mu} \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (D_v) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} v \text{ in } (H^{-1}(\Omega))^{d \times d} \text{ uniformly on } [0, T], \\ (E_v) & \qquad \omega \to \omega(v) - \frac{1}{2} \innp{\gamma_\mathbf{n} (\cdot - \cdot^T), v \mu} \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (E_{2, v}) & \qquad \omega \to \omega(v) - (v \cdot \BoldTau) \mu \text{ in } (H^1(\Omega))^* \text{ uniformly on } [0, T], \\ (F_{2, v}) & \qquad \omega \to \omega(v) \text{ in } H^{-1}(\Omega) \text{ uniformly on } [0, T], \end{align*} \endgroup we have the following theorem: \begin{theorem}\label{T:MainResultv} Conditions ($A_v$), ($A'_v$), ($C_v$), ($D_v$), and ($E_v$) are equivalent. In 2D, conditions ($E_{2,v}$) and, when $\Omega$ is simply connected, ($F_{2,v}$) are equivalent to the other conditions. Also, $(B_v)$ implies all of the other conditions. Finally, the same equivalences hold if we replace each convergence above with the convergence of a subsequence. \end{theorem} But we also have the following: \begin{theorem}\label{T:SubsequenceConvergence} There exists $v$ in $L^\ensuremath{\infty}([0, T]; H)$ such that a subsequence $(u_\nu)$ converges weakly to $v$ in $L^\ensuremath{\infty}([0, T]; H)$. \end{theorem} \begin{proof} The argument for a simply connected domain in 2D is slightly simpler so we give it first. The sequence $(u_\nu)$ is bounded in $L^\ensuremath{\infty}([0, T]; H)$ by the basic energy inequality for the Navier-Stokes equations. Letting $\psi_\nu$ be the stream function for $u_\nu$ vanishing on $\Gamma$, it follows by the Poincare inequality that $(\psi_\nu)$ is bounded in $L^\ensuremath{\infty}([0, T]; H_0^1(\Omega))$. Hence, there exists a subsequence, which we relabel as $(\psi_\nu)$, converging strongly in $L^\ensuremath{\infty}([0, T]; L^2(\Omega))$ and weak-* in $L^\ensuremath{\infty}([0, T]; H_0^1(\Omega))$ to some $\psi$ lying in $L^\ensuremath{\infty}([0, T]; H_0^1(\Omega))$. Let $v = \ensuremath{\nabla}^\perp \psi$. Let $g$ be any element of $L^\ensuremath{\infty}([0, T]; H)$. Then \begin{align*} (u_\nu, g) &= (\ensuremath{\nabla}^\perp \psi_\nu, g) = - (\ensuremath{\nabla} \psi_\nu, g^\perp) = (\psi_\nu, - \dv g^\perp) = (\psi_\nu, \omega(g)) \\ &\to (\psi, \omega(g)) = (v, g). \end{align*} In the third equality we used the membership of $\psi_v$ in $H_0^1(\Omega)$ and the last equality follows in the same way as the first four. The convergence follows from the weak-* convergence of $\psi_\nu$ in in $L^\ensuremath{\infty}([0, T]; H_0^1(\Omega))$ and the membership of $\omega(g)$ in $H^{-1}(\Omega)$. In dimension $d \ge 3$, let $M_\nu$ in $(H_0^1(\Omega))^d$ satisfy $u_\nu = \dv M_\nu$; this is possible by Corollary 7.5 of \cite{K2008VVV}. Arguing as before it follows that there exists a subsequence, which we relabel as $(M_\nu)$, converging strongly in $L^\ensuremath{\infty}([0, T]; L^2(\Omega))$ and weak-* in $L^\ensuremath{\infty}([0, T]; H_0^1(\Omega))$ to some $M$ that lies in $L^\ensuremath{\infty}([0, T]; (H_0^1(\Omega))^{d \times d})$. Let $v = \dv M$. Let $g$ be any element of $L^\ensuremath{\infty}([0, T]; H)$. Then \begin{align*} (u_\nu, g) &= (\dv M_\nu, g) = -(M_\nu, \ensuremath{\nabla} g) \to - (M, \ensuremath{\nabla} g) = (v, g), \end{align*} establishing convergence as before. \end{proof} It follows from \refTAnd{MainResultv}{SubsequenceConvergence} that all of the convergences in \cref{T:VVEquiv} hold except for $(B)$, but for a subseqence of solutions and the convergence is to some velocity field $v$ lying only in $L^\ensuremath{\infty}([0, T]; H)$ and not necessarily in $L^\ensuremath{\infty}([0, T]; H \cap H^1(\Omega))$ . In particular, we do not know if $v$ is a solution to the Euler equations, and, in fact, there is no reason to expect that it is. \Ignore{ \begin{lemma}\label{L:H1Dual} $H^{-1}(\Omega)$ is the image under $\Delta$ of $H^1_0(\Omega)$ and the image under $\dv$ of $(L^2(\Omega))^d$. \end{lemma} \begin{proof} Let $w$ be in $H^{-1}(\Omega) = H^1_0(\Omega)^*$. By the density of $\Cal{D}(\Omega)$ in $H^1_0(\Omega)$ the value of $(w, \varphi)_{H_0^1(\Omega), H_0^1(\Omega)^*}$ on test functions $\varphi$ in $\Cal{D}(\Omega)$ is enough to uniquely determine $w$. By the Riesz representation theorem there exists a $u$ in $H^1_0(\Omega)$ such that for all $\varphi$ in $H^1_0(\Omega)$ and hence in $\Cal{D}(\Omega)$, \begin{align*} (w, \varphi)_{H_0^1(\Omega), H_0^1(\Omega)^*} &= \innp{u, \varphi} = \int_\Omega \ensuremath{\nabla} u \cdot \ensuremath{\nabla} \varphi = - \int_\Omega \Delta u \cdot \varphi \\ &= (-\Delta u, \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*}, \end{align*} which shows that $w$ as a linear functional is equal to $- \Delta u$ as a distribution, and the two can be identified. Because the identification of $w$ and $u$ in the Riesz representation theorem is bijective, $H^{-1}(\Omega) = \Delta H^1(\Omega)$. Since $\Delta = \dv \ensuremath{\nabla}$, it also follows that $H^{-1}(\Omega) \subseteq \dv (L^2(\Omega))^d$. To show the opposite containment, let $f$ be in $(L^2(\Omega))^d$. Then by the Hodge decomposition, we can write \begin{align*} f = \ensuremath{\nabla} u + g \end{align*} with $u$ in $H^1(\Omega)$ and $g$ in $(L^2(\Omega))^d$ with $\dv g = 0$ as a distribution. Then for any $\varphi$ in $\Cal{D}(\Omega)$, \begin{align*} &(\dv f, \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*} = - (f, \ensuremath{\nabla} \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*} \\ &\qquad= - (\ensuremath{\nabla} u, \ensuremath{\nabla} \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*} - (g, \ensuremath{\nabla} \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*} \\ &\qquad= - \innp{u, \varphi} + (\dv g, \varphi)_{\Cal{D}(\Omega), \Cal{D}(\Omega)^*} = \innp{-u, \varphi} = (w, \varphi)_{H_0^1(\Omega), H_0^1(\Omega)^*} \end{align*} for some $w$ in $H^1_0(\Omega)^*$ by the Riesz representation theorem. It follows that $\dv f$ and $w$ can be identified, using the same identification as before. What we have shown is that $\dv (L^2(\Omega))^d \subseteq H^{-1}(\Omega)$, completing the proof. \end{proof} } \Ignore{ \section{Convergence to another solution to the Euler equations?} \noindent One could imagine that the solutions $u = u_\nu$ to the Navier-Stokes equations converge, in the limit, to a solution to the Euler equations, but one different from $\overline{u}$ and possibly with lower regularity. Since such solutions are determined by their initial velocity, this means that the vector $v$ to which $(u_\nu)$ converges has initial velocity $v^0 \ne \overline{u}^0$. (This conclusion would be true even if $v$ had so little regularity that it had not been determined uniquely by its initial velocity.) Now, $\overline{u}(t)$ is continuous in $H$, since it is a strong solution, as too, if we restrict ourselves to two dimensions, is $u(t)$. If $v$ has bounded vorticity, say, then $v(t)$ is continuous in $H$ as well. It would seem ...... } \Ignore{ \section{Physical meaning of the vortex sheet on the boundary?} \noindent Calling the term $\omega^* := - (\overline{u} \cdot \BoldTau) \mu$ (in 2D) a \textit{vortex sheet} is misleading, and I regret referring to it that way in \cite{K2008VVV} without some words of explanation. The problem is that we cannot interpret $\omega^*$ as a distribution on $\Omega$ because applying it to any function in $\Cal{D}(\Omega)$ gives zero. And how could we recover the velocity associated to $\omega^*$? One natural, if unjustified, way to try to interpret $\omega^*$ is to extend it to the whole space so that it is a measure supported along the curve $\Gamma$. To determine the associated velocity $v$, let $\Omega_- = \Omega$ and $\Omega_+ = \Omega^C$ with $v_\pm = v|_{\Omega_\pm}$, and let $[v] = v_+ - v_-$. Then as on page 364 of \cite{MB2002}, we must have \begin{align*} [v] \cdot \mathbf{n} = 0, \quad [v] \cdot \BoldTau = - \overline{u} \cdot \BoldTau. \end{align*} That is, the normal component of the velocity is continuous across the boundary while the jump in the tangential component is the strength of the vortex sheet. Now, let us assume that the vanishing viscosity limit holds, so that the limiting vorticity is $\overline{\omega} - (\overline{u} \cdot \BoldTau) \mu = \overline{\omega} - \omega^*$. Since $u \to \overline{u}$ strongly with $\omega(\overline{u}) = \overline{\omega}$, the term $\overline{\omega}$ has to account for all of the kinetic energy of the fluid. If the limit is to be physically meaningful, certainly energy cannot be \textit{gained} (though it conceivably could be lost to diffusion, even in the limit). Thus, we would need to have the velocity $v$ associated with $\omega^*$ vanish in $\Omega$; in other words, $v_- \equiv 0$. This leads to $\omega(v_+) = \dv v_+ = 0$ in $\Omega_+$, $v_+ \cdot \mathbf{n} = 0$ on $\Gamma$, $v_+ \cdot \BoldTau = \overline{u} \cdot \BoldTau$ on $\Gamma$, with some conditions on $v_+$ at infinity. But this is an overdetermined set of equations. In fact, if $\Omega$ is simply connected then $\Omega_+$ is an exterior domain, and if we ignore the last equation, then up to a multiplicative constant there is a unique solution vanishing at infinity. This cannot, in general, be reconciled with the need for the last equation to hold. Actually, perhaps the correct physical interpretation of $\omega^*$ comes from the observation in the first paragraph of this section: that it has no physical effect at all since, as a distribution, it is zero. If the vanishing viscosity limit holds, it is reasonable to assume that if there is a boundary separation of the vorticity it weakens in magnitude as the viscosity vanishes and so contributes nothing in the limit. Or, looked at another way, if in looking for the velocity $v$ corresponding to the vortex sheet $\omega$ as we did above we assume that $v$ is zero outside $\Omega$, we would obtain \begin{align*} v \cdot \mathbf{n} = 0, \quad v \cdot \BoldTau = \overline{u} \cdot \BoldTau \end{align*} on the boundary. For a very small viscosity, then, $u$ has almost the same effect as $\overline{u}$ in the interior of $\Omega$, while the vortex sheet that is forming on the boundary as the viscosity vanishes has nearly the same effect as $\overline{u}$ on the boundary. } \Part{Theme II: Kato's Conditions} \section{An equivalent 2D condition on the boundary}\label{S:EquivCondition} \noindent \begin{theorem}\label{T:BoundaryIffCondition} For ($VV$) to hold in 2D it is necessary and sufficient that \begin{align}\label{e:BoundaryCondition2D} \nu \int_0^T \int_\Gamma \omega \, \overline{u} \cdot \BoldTau \to 0 \text{ as } \nu \to 0. \end{align} \end{theorem} \begin{proof} Since the solution is in 2D and $f \in L^2(0, T; H) \supseteq C^1_{loc}(\ensuremath{\BB{R}}; C^1(\Omega))$, Theorem III.3.10 of \cite{T2001} gives \begin{align}\label{e:RegTwoD} \begin{split} &\sqrt{t} u \in L^2(0, T; H^2(\Omega)) \cap L^\ensuremath{\infty}(0, T; V), \\ &\sqrt{t} \ensuremath{\partial}_t u \in L^2(0, T; H), \end{split} \end{align} so $\omega(t)$ is defined in the sense of a trace on the boundary. This shows that the condition in \cref{e:BoundaryCondition2D} is well-defined. For simplicity we give the argument with $f = 0$. We perform the calculations using the $d$-dimensional form of the vorticity in \cref{e:VorticityRd}, specializing to 2D only at the end. (The argument applies formally in higher dimensions; see \cref{R:BoundaryConditionInRd}.) Subtracting ($EE$) from ($NS$), multiplying by $w = u - \overline{u}$, integrating over $\Omega$, using \cref{L:TimeDerivAndIntegration} for the time derivative, and $u(t) \in H^2(\Omega)$, $t > 0$, for the spatial integrations by parts, leads to \begin{align}\label{e:BasicEnergyEq} \begin{split} \frac{1}{2} \diff{}{t} &\norm{w}_{L^2}^2 + \nu \norm{\ensuremath{\nabla} u}_{L^2}^2 \\ &= - (w \cdot \ensuremath{\nabla} \overline{u}, w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) - \nu \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u}. \end{split} \end{align} Now, \begin{align*} \begin{split} (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} &= 2 (\frac{\ensuremath{\nabla} u - (\ensuremath{\nabla} u)^T}{2} \cdot \mathbf{n}) \cdot \overline{u} + ((\ensuremath{\nabla} u)^T \cdot \bm{n}) \cdot \overline{u} \\ &= 2 (\omega(u) \cdot \mathbf{n}) \cdot \overline{u} + ((\ensuremath{\nabla} u)^T \cdot \bm{n}) \cdot \overline{u}. \end{split} \end{align*} But, \begin{align*} \int_\Gamma &((\ensuremath{\nabla} u)^T \cdot \bm{n}) \cdot \overline{u} = \int_\Gamma \ensuremath{\partial}_i u^j n^j \overline{u}^i = \frac{1}{2} \int_\Gamma \ensuremath{\partial}_i(u \cdot \bm{n}) \overline{u}^i \\ &= \frac{1}{2} \int_\Gamma \ensuremath{\nabla} (u \cdot \bm{n}) \cdot \overline{u} = 0, \end{align*} since $u \cdot \bm{n} = 0$ on $\Gamma$ and $\overline{u}$ is tangent to $\Gamma$. Hence, \begin{align}\label{e:gradunuolEq} \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} = 2 (\omega(u) \cdot \mathbf{n}) \cdot \overline{u} \end{align} and \begin{align*} \frac{1}{2} \diff{}{t} &\norm{w}_{L^2}^2 + \nu \norm{\ensuremath{\nabla} u}_{L^2}^2 \\ &= - (w \cdot \ensuremath{\nabla} \overline{u}, w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) - 2 \nu \int_\Gamma (\omega(u) \cdot \mathbf{n}) \cdot \overline{u}. \end{align*} By virtue of \cref{L:TimeDerivAndIntegration}, we can integrate over time to give \begin{align}\label{e:VVArg} \begin{split} &\norm{w(T)}_{L^2}^2 + 2 \nu \int_0^T \norm{\ensuremath{\nabla} u}_{L^2}^2 = - 2 \int_0^T (w \cdot \ensuremath{\nabla} \overline{u}, w) + 2 \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) \\ &\qquad - 2 \nu \int_0^T \int_\Gamma (\omega(u) \cdot \mathbf{n}) \cdot \overline{u}. \end{split} \end{align} In two dimensions, we have (see (4.2) of \cite{KNavier}) \begin{align}\label{e:gradunomega} (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} = ((\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \BoldTau) (\overline{u} \cdot \BoldTau) = \omega(u) \, \overline{u} \cdot \BoldTau, \end{align} and \cref{e:VVArg} can be written \begin{align}\label{e:VVArg2D} \begin{split} &\norm{w(T)}_{L^2}^2 + 2 \nu \int_0^T \norm{\ensuremath{\nabla} u}_{L^2}^2 = - 2 \int_0^T (w \cdot \ensuremath{\nabla} \overline{u}, w) + 2 \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) \\ &\qquad - \nu \int_0^T \int_\Gamma \omega(u) \, \overline{u} \cdot \BoldTau. \end{split} \end{align} The sufficiency of \refE{BoundaryCondition2D} for the vanishing viscosity limit ($VV$) to hold (and hence for the other conditions in \cref{T:VVEquiv} to hold) follows from the bounds, \begin{align*} \abs{(w \cdot \ensuremath{\nabla} \overline{u}, w)} &\le \norm{\ensuremath{\nabla} \overline{u}}_{L^\ensuremath{\infty}([0, T] \times \Omega)} \norm{w}_{L^2}^2 \le C \norm{w}_{L^2}^2, \\ \nu \int_0^T \abs{(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u})} &\le \sqrt{\nu} \norm{\ensuremath{\nabla} \overline{u}}_{L^2([0, T] \times \Omega)} \sqrt{\nu} \norm{\ensuremath{\nabla} u}_{L^2([0, T] \times \Omega)} \le C \sqrt{\nu}, \end{align*} and Gronwall's inequality. Proving the necessity of \refE{BoundaryCondition2D} is just as easy. Assume that $(VV)$ holds, so that $\norm{w}_{L^\ensuremath{\infty}([0, T]; L^2(\Omega))} \to 0$. Then by the two inequalities above, the first two terms on the right-hand side of \refE{VVArg2D} vanish with the viscosity as does the first term on the left-hand side. The second term on the left-hand side vanishes as proven in \cite{Kato1983} (it follows from a simple argument using the energy equalities for ($NS$) and ($E$)). It follows that, of necessity, \refE{BoundaryCondition2D} holds. \Ignore{ The reason this argument is formal is twofold. First, $w$ is not a valid test function in the weak formulation of the Navier-Stokes equations because it does not vanish on the boundary and because it varies in time. Beyond time zero the solution has as much regularity as the boundary allows \ToDo{But only up to a finite time; this is a factor to deal with}, so this is a problem only when trying to reach a conclusion after integrating in time down to time zero. This is the second reason the argument is formal: in obtaining \refE{VVArg} we act as though $w$ is strongly continuous in time down to time zero. This is true in 2D, where this part of the argument is not formal, but only weak continuity is known in higher dimensions. (This is also the reason we need assume no additional regularity for the initial velocity in 2D.) To get around these difficulties, we derive \refE{VVArg} rigorously. Choose a sequence $(h_n)$ of nonnegative functions in $C_0^\ensuremath{\infty}((0, T])$ such that $h_n \equiv 1$ on the interval $[n^{-1}, T]$ with $h_n$ strictly increasing on $[0, n^{-1}]$. Then $h'_n = g_n \ge 0$ with $g_n \equiv 0$ on $[n^{-1}, T]$. Observe that $\smallnorm{g_n}_{L^1([0, T])} = 1$. Letting $w = u - \overline{u}$ as before, because $h_n w$ vanishes at time zero we can legitimately subtract ($EE$) from ($NS$), multiply by $h_n w$, and integrate over $\Omega$ to obtain, in place of \refE{BasicEnergyEq}, \begingroup \allowdisplaybreaks \begin{align*} \begin{split} \frac{1}{2} \diff{}{t} &\smallnorm{h_n^{1/2} w}_{L^2}^2 - \frac{1}{2} \int_\Omega h_n'(t) \abs{w}^2 + \nu (\ensuremath{\nabla} u, \ensuremath{\nabla} (h_n u)) \\ &= - (w \cdot \ensuremath{\nabla} \overline{u}, h_n w) - (u \cdot \ensuremath{\nabla} w, h_n w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} (h_n \overline{u})) \\ &\qquad\qquad - \nu \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot (h_n \overline{u}) \\ &= - (w \cdot \ensuremath{\nabla} \overline{u}, h_n w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} (h_n \overline{u})) - \nu \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot (h_n \overline{u}), \end{split} \end{align*} \endgroup since $(u \cdot \ensuremath{\nabla} w, h_n w) = h_n(u \cdot \ensuremath{\nabla} w, w) = 0$. Integrating in time gives \begingroup \allowdisplaybreaks \begin{align*} &\smallnorm{w(T)}_{L^2}^2 - \smallnorm{h_n^{1/2} w(0)}_{L^2}^2 - \int_0^T \int_\Omega h_n' \abs{w}^2 + 2 \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} (h_n u)) \\ &\qquad = - 2 \int_0^T (w \cdot \ensuremath{\nabla} \overline{u}, h_n w) + 2 \int_0^T \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} (h_n \overline{u})) \\ &\qquad\qquad\qquad\qquad - 2 \int_0^T \nu \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot (h_n \overline{u}). \end{align*} \endgroup The second term on the left hand side vanishes because $h_n(0) = 0$. For the four terms containing $h_n$ without derivative, the $h_n$ becomes $1$ in the limit as $n \to \ensuremath{\infty}$. This leaves the one term containing $h_n'$. Now, $\overline{u}(t)$ is continuous in $H$ and in 2D $u(t)$ is also continuous in $H$. Hence, in 2D $w(t)$ is continuous in $H$. In 3D if we assume that $u_0 \in V$ then $u(t)$ is continuous in $H$ (in fact, in $V$) up to some finite time, $T^* > 0$. Hence, in 3D, $w(t)$ is continuous in $H$ on $[0, T^*)$; $T^*$ may depend on $\nu$, but we will take $n$ to 0 before taking $\nu$ to $0$, so this will not matter. Hence, $F(s) = \norm{w(s)}^2$ is continuous on $[0, T^*)$ with $T^* = T$ in 2D, so \ToDo{Does the $0 \le$ really hold? I don't think so.} \begin{align*} 0 &\le \lim_{n \to \ensuremath{\infty}} \int_0^T \int_\Omega h_n' \abs{w}^2 = \lim_{n \to \ensuremath{\infty}} \int_0^T g_n(s) F(s) \, ds \\ &= \lim_{n \to \ensuremath{\infty}} \int_0^{\frac{1}{n}} g_n(s) F(s) \, ds \le \norm{g_n}_{L^1} \norm{F}_{L^\ensuremath{\infty} \pr{0, \frac{1}{n}}} = 0. \end{align*} This gives us \refE{VVArg}. } \end{proof} \begin{remark}\label{R:ROC} It follows from the proof of \refT{BoundaryIffCondition} that in 2D, \begin{align*} \norm{u(t) - \overline{u}(t)} \le C \brac{\nu^{\frac{1}{4}} + \abs{\nu \int_0^T \int_\Gamma \omega \, \overline{u} \cdot \BoldTau}^{\frac{1}{2}}} e^{C t}. \end{align*} Suppose that $\overline{u}_0$ is smooth enough that $\Delta \overline{u} \in L^\ensuremath{\infty}([0, T] \times \Omega)$. Then before integrating to obtain \cref{e:BasicEnergyEq} we can replace the term $\nu (\Delta u, w)$ with $\nu (\Delta w, w) + \nu (\Delta \overline{u}, w)$. Integrating by parts gives \begin{align*} \nu (\Delta w, w) = \nu \norm{\ensuremath{\nabla} w}_{L^2}^2, \end{align*} and we also have, \begin{align*} \nu (\Delta \overline{u}, w) \le \nu \norm{\Delta \overline{u}}_{L^2} \norm{w}_{L^2} \le \frac{\nu^2}{2} \norm{\Delta \overline{u}}_{L^2}^2 + \frac{1}{2} \norm{w}_{L^2}^2. \end{align*} This leads to the bound, \begin{align*} \norm{u(t) - \overline{u}(t)}_{L^2} \le C \brac{\nu + \abs{\nu \int_0^T \int_\Gamma \omega \, \overline{u} \cdot \BoldTau}^{\frac{1}{2}}} e^{C t} \end{align*} (and also $\norm{u - \overline{u}}_{L^2(0, T; H^1)} \le C \nu^{1/2} e^{Ct}$). Thus, the bound we obtain on the rate of convergence in $\nu$ is never better than $O(\nu^{1/4})$ unless the initial data is smooth enough, in which case it is never better than $O(\nu)$. In any case, only in exceptional circumstances would the rate not be determined by the integral coming from the boundary term. \end{remark} \begin{remark}\label{R:BoundaryConditionInRd} Formally, the argument in the proof of \cref{T:BoundaryIffCondition} would give in any dimension the condition \begin{align*} \nu \int_0^T \int_\Gamma (\omega(u) \cdot \mathbf{n}) \cdot \overline{u} \to 0 \text{ as } \nu \to 0. \end{align*} In 3D, one has $\omega(u) \cdot \bm{n} = (1/2) \vec{\omega} \times \bm{n}$, so the condition could be written \begin{align*} \nu \int_0^T \int_\Gamma (\vec{\omega} \times \bm{n}) \cdot \overline{u} = \nu \int_0^T \int_\Gamma \vec{\omega} \cdot (\overline{u} \times \bm{n}) \to 0 \text{ as } \nu \to 0, \end{align*} where $\vec{\omega}$ is the 3-vector form of the curl of $u$. We can only be assured, however, that $u(t) \in V$ for all $t > 0$, which is insufficient to define $\vec{\omega}$ on the boundary. (The normal component could be defined, though, since both $\vec{\omega}(t)$ and $\dv \vec{\omega}(t) = 0$ lie in $L^2$.) Even assuming more compatible initial data in 3D, such as $u_0 \in V$, we can only conclude that $u(t) \in H^2$ for a short time, with that time decreasing to $0$ as $\nu \to 0$ (in the presence of forcing; see, for instance, Theorem 9.9.4 of \cite{FoiasConstantin1988}). \end{remark} \Ignore{ \begin{remark}\label{R:BoundaryCondition2DRd} Since $\overline{u} \times \bm{n}$ is a tangent vector, the second form of the condition in \refE{BoundaryCondition3D} shows that it is only the tangential components of $\vec{\omega}$ that matter in this condition. More specifically, only the tangential component perpendicular to $\overline{u}$ matters. \end{remark} } There is nothing deep about the condition in \refE{BoundaryCondition2D}, but what it says is that there are two mechanisms by which the vanishing viscosity limit can hold: Either the blowup of $\omega$ on the boundary happens slowly enough that \begin{align}\label{e:nuL1Bound} \nu \int_0^T \norm{\omega}_{L^1(\Gamma)} \to 0 \text{ as } \nu \to 0 \end{align} or the vorticity for ($NS$) is generated on the boundary in such a way as to oppose the sign of $\overline{u} \cdot \BoldTau$. (This latter line of reasoning is followed in \cite{CKV2014}, leading to a new condition in a boundary layer slightly thicker than that of Kato.) In the second case, it could well be that vorticity for $(NS)$ blows up fast enough that \refE{nuL1Bound} does not hold, but cancellation in the integral in \refE{BoundaryCondition2D} allows that condition to hold. \begin{lemma}\label{L:TimeDerivAndIntegration} Assume that $v \in L^\ensuremath{\infty}(0, T; V)$ with $\ensuremath{\partial}_t v \in L^2(0, T; V')$ as well as $\sqrt{t} \ensuremath{\partial}_t v \in L^2(0, T; H)$. Then $v \in C([0, T]; H)$, \begin{align*} \frac{1}{2} \diff{}{t} \norm{v}_{L^2}^2 = (\ensuremath{\partial}_t v, v) \text{ in } \Cal{D}'((0, T)) \text{ with } \sqrt{t} (\ensuremath{\partial}_t v, v) \in L^1(0, T), \end{align*} and \begin{align*} \int_0^T \diff{}{t} \norm{v(t)}_{L^2}^2 \, dt = \norm{v(T)}_{L^2}^2 - \norm{v(0)}_{L^2}^2. \end{align*} \end{lemma} \begin{proof} Having $v \in L^2(0, T; V)$ with $\ensuremath{\partial}_t v \in L^2(0, T; V')$ is enough to conclude that $(\ensuremath{\partial}_t v, v) = (1/2) (d/dt) \norm{v}_{L^2}^2$ in $\Cal{D}'((0, T))$ and $v \in C([0, T]; H)$ (see Lemma III.1.2 of \cite{T2001}). Let $T_0 \in (0, T)$. Our stronger assumptions also give $(d/dt) \norm{v}_{L^2}^2 = 2(\ensuremath{\partial}_t v, v) \in L^1(T_0, T)$. Hence, by the fundamental theorem of calculus for Lebesgue integration (Theorem 3.35 of \cite{Folland1999}) it follows that \begin{align*} \int_{T_0}^T \diff{}{t} \norm{v}_{L^2}^2 \, dt = \norm{v(T)}_{L^2}^2 - \norm{v(T_0)}_{L^2}^2. \end{align*} But $v$ is continuous in $H$ down to time zero, so taking $T_0$ to 0 completes the proof. \end{proof} \section{Examples where the 2D boundary condition holds}\label{S:Examples} \noindent All examples where the vanishing viscosity limit is known to hold have some kind of symmetry---in geometry of the domain or the initial data---or have some degree of analyticity. Since \refE{BoundaryCondition2D} is a necessary condition, it holds for all of these examples. But though it is also a sufficient condition, it is not always practicable to apply it to establish the limit. We give here examples in which it is practicable. This includes all known 2D examples having symmetry. In all explicit cases, the initial data is a stationary solution to the Euler equations. \Example{1} Let $\overline{u}$ be any solution to the Euler equations for which $\overline{u} = 0$ on the boundary. The integral in \refE{BoundaryCondition2D} then vanishes for all $\nu$. From \refR{ROC}, the rate of convergence (here, and below, in $\nu$) is $C \nu^{1/4}$ or, for smoother initial data, $C \nu$. \Example{1a} Example 1 is not explicit, since we immediately encounter the question of what (nonzero) examples of such steady solutions there are. As a first example, let $D$ be the disk of radius $R > 0$ centered at the origin and let $\omega_0 \in L^\ensuremath{\infty}(D)$ be radially symmetric. Then the associated velocity field, $u_0$, is given by the Biot-Savart law. By exploiting the radial symmetry, $u_0$ can be written, \begin{align}\label{e:u0Circular} u_0(x) &= \frac{x^\perp}{\abs{x}^2} \int_0^{\abs{x}} \omega_0(r) r \, dr, \quad \end{align} where $B({\abs{x}})$ is the ball of radius $\abs{x}$ centered at the origin and where we abuse notation a bit in the writing of $\omega_0(r)$. Since $u_0$ is perpendicular to $\ensuremath{\nabla} u_0$ it follows from the vorticity form of the Euler equations that $\overline{u} \equiv u_0$ is a stationary solution to the Euler equations. Now assume that the total mass of vorticity, \begin{align}\label{e:m} m := \int_{\ensuremath{\BB{R}}^2} \omega_0, \end{align} is zero. We see from \refE{u0Circular} that on $\Gamma$, $u_0 = m x^\perp R^{-1} = 0$, giving a steady solution to the Euler equations with velocity vanishing on the boundary. (Note that $m = 0$ is equivalent to $u_0$ lying in the space $V$ of divergence-free vector fields vanishing on the boundary.) \Example{1b} Let $\omega_0 \in L^1 \cap L^\ensuremath{\infty}(\ensuremath{\BB{R}}^2)$ be a compactly supported radially symmetric initial vorticity for which the total mass of vorticity vanishes; that is, $m = 0$. Then the expression for $u_0$ in \refE{u0Circular}, which continues to hold throughout all of $\ensuremath{\BB{R}}^2$, shows that $u_0$ vanishes outside of the support of its vorticity. If we now restrict such a radially symmetric $\omega_0$ so that its support lies inside a domain (even allowing the support of $\omega_0$ to touch the boundary of the domain) then the velocity $u_0$ will vanish on the boundary. In particular, $u_0 \cdot \bm{n} = 0$ so, in fact, $u_0$ is a stationary solution to the Euler equations in the domain, being already one in the whole plane. In fact, one can use a superposition of such radially symmetric vorticities, as long as their supports do not overlap, and one will still have a stationary solution to the Euler equations whose velocity vanishes on the boundary. Such a superposition is called a \textit{superposition of confined eddies} in \cite{FLZ1999A}, where their properties in the full plane, for lower regularity than we are considering, are analyzed. These superpositions provide a fairly wide variety of examples in which the vanishing viscosity limit holds. It might be interesting to investigate the precise manner in which the vorticity converges in the vanishing viscosity limit; that is, whether it is possible to do better than the ``vortex sheet''-convergence in condition $(E_2)$ of \cite{K2008VVV}. In \cite{Maekawa2013}, Maekawa considers initial vorticity supported away from the boundary in a half-plane. We note that the analogous result in a disk, even were it shown to hold, would not cover this Example 1b when the support of the vorticity touches the boundary. \Example{2 [2D shear flow]} Let $\phi$ solve the heat equation, \begin{align}\label{e:HeatShear} \left\{ \begin{array}{rl} \ensuremath{\partial}_t \phi(t, z) = \nu \ensuremath{\partial}_{zz} \phi(t, z) & \text{on } [0, \ensuremath{\infty}) \times [0, \ensuremath{\infty}), \\ \phi(t, 0) = 0 & \text{ for all } t > 0, \\ \phi(0) = \phi_0. & \end{array} \right. \end{align} Assume for simplicity that $\phi_0 \in W^{1, \ensuremath{\infty}}((0, \ensuremath{\infty})$. Let $u_0 = (\phi_0, 0)$ and $u(t, x) = (\phi(t, x_2), 0)$. Let $\Omega = [-L, L] \times (0, \ensuremath{\infty})$ be periodic in the $x_1$-direction. Then $u_0 \cdot \bm{n} = 0$ and $u(t) = 0$ for all $t > 0$ on $\ensuremath{\partial} \Omega$ and \begin{align*} \ensuremath{\partial}_t u(t, x) &= \nu(\ensuremath{\partial}_{x_2 x_2} \phi(t, x_2), 0) = \nu \Delta u(t, x), \\ (u \cdot \ensuremath{\nabla} u)(t, x) &= \matrix{\ensuremath{\partial}_1 u^1 & \ensuremath{\partial}_1 u^2} {\ensuremath{\partial}_2 u^1 & \ensuremath{\partial}_2 u^2} \matrix{u^1}{u^2} = \matrix{0 & 0}{\ensuremath{\partial}_2 \phi(t, x_2) & 0} \matrix{\phi(t, x_2)}{0} \\ &= \matrix{0}{\ensuremath{\partial}_2 \phi(t, x_2) \phi(t, x_2)} = \frac{1}{2} \ensuremath{\nabla} \phi(t, x_2). \end{align*} It follows that $u$ solves the Navier-Stokes equations on $\Omega$ with pressure, $p = - \frac{1}{2} \phi(t, x_2)$. Similarly, letting $\overline{u} \equiv u_0$, we have $\ensuremath{\partial}_t \overline{u} = 0$, $\overline{u} \cdot \ensuremath{\nabla} \overline{u} = \frac{1}{2} \ensuremath{\nabla} \phi_0$ so $\overline{u} \equiv u_0$ is a stationary solution to the Euler equations. Now, $\omega = \ensuremath{\partial}_1 u^2 - \ensuremath{\partial}_2 u^1 = - \ensuremath{\partial}_2 \phi(t, x_2)$ so \begin{align*} \int_\Gamma \omega \, \overline{u} \cdot \BoldTau &= - \int_\Gamma \ensuremath{\partial}_2 \phi(t, x_2)|_{x_2 = 0} \phi_0(0) = - \phi_0(0)\int_{-L}^L \ensuremath{\partial}_{x_2} \phi(t, x_2)|_{x_2 = 0} \, d x_1 \\ &= -L \phi_0(0) \ensuremath{\partial}_{x_2} \phi(t, x_2)|_{x_2 = 0}. \end{align*} The explicit solution to \refE{HeatShear} is \begin{align*} \phi(t, z) &= \frac{1}{\sqrt{4 \pi \nu t}} \int_0^\ensuremath{\infty} \brac{e^{-\frac{(z - y)^2}{4 \nu t}} - e^{-\frac{(z + y)^2}{4 \nu t}}} \phi_0(y) \, dy \end{align*} (see, for instance, Section 3.1 of \cite{StraussPDE}). Thus, \begingroup \allowdisplaybreaks \begin{align*} \ensuremath{\partial}_z \phi(t, z)|_{z = 0} &= -\frac{2}{4 \nu t \sqrt{4 \pi \nu t}} \int_0^\ensuremath{\infty} y \brac{e^{-\frac{y^2}{4 \nu t}} + e^{-\frac{y^2}{4 \nu t}}} \phi_0(y) \, dy \\ &= -\frac{1}{\nu t \sqrt{4 \pi \nu t}} \int_0^\ensuremath{\infty} y e^{-\frac{y^2}{4 \nu t}} \phi_0(y) \, dy \\ &= -\frac{1}{\nu t \sqrt{4 \pi \nu t}} \int_0^\ensuremath{\infty} (- 2 \nu t) \diff{}{y} e^{-\frac{y^2}{4 \nu t}} \phi_0(y) \, dy \\ &= -\frac{1}{\sqrt{\pi \nu t}} \int_0^\ensuremath{\infty} \diff{}{y} e^{-\frac{y^2}{4 \nu t}} \, \phi_0(y) \, dy \\ &= \frac{1}{\sqrt{\pi \nu t}} \int_0^\ensuremath{\infty} e^{-\frac{y^2}{4 \nu t}} \phi_0'(y) \, dy \end{align*} \endgroup so that \begin{align*} \abs{\ensuremath{\partial}_{x_2} \phi(t, x_2)|_{x_2 = 0}} \le \frac{C}{\sqrt{\nu t}}. \end{align*} We conclude that \begin{align*} \abs{\nu \int_0^T \int_\Gamma \omega \, \overline{u} \cdot \BoldTau} \le C \sqrt{\nu} \int_0^T t^{-1/2} \, dt = C \sqrt{\nu T}. \end{align*} The condition in \refE{BoundaryCondition2D} thus holds (as does \cref{e:nuL1Bound}). From \refR{ROC}, the rate of convergence is $C \nu^{\frac{1}{4}}$ (even for smoother initial data). \Example{3} Consider Example 1a of radially symmetric vorticity in the unit disk, but without the assumption that $m$ given by \refE{m} vanishes. This example goes back at least to Matsui in \cite{Matsui1994}. The convergence also follows from the sufficiency of the Kato-like conditions established in \cite{TW1998}, as pointed out in \cite{W2001}. A more general convergence result in which the disk is allowed to impulsively rotate for all time appears in \cite{FLMT2008}. A simple argument to show that the vanishing viscosity limit holds is given in Theorem 6.1 \cite{K2006Disk}, though without a rate of convergence. Here we prove it with a rate of convergence by showing that the condition in \refE{BoundaryCondition2D} holds. Because the nonlinear term disappears, the vorticity satisfies the heat equation, though with Dirichlet boundary conditions not on the vorticity but on the velocity: \begin{align}\label{e:RadialHeat} \left\{ \begin{array}{rl} \ensuremath{\partial}_t \omega = \nu \Delta \omega & \text{in } \Omega, \\ u = 0 & \text{on } \Gamma. \end{array} \right. \end{align} Unless $u_0 \in V$, however, $\omega \notin C([0, T]; L^2)$, so we cannot easily make sense of the initial condition this way. An orthonormal basis of eigenfunctions satisfying these boundary conditions is \begin{align*} u_k(r, \theta) &= \frac{J_1(j_{1k} r)}{\pi^{1/2}\abs{J_0(j_{1k})}} \ensuremath{\widehat}{e}_\theta, \quad \omega_k(r, \theta) = \frac{j_{1k} J_0(j_{1k} r)}{\pi^{1/2}\abs{J_0(j_{1k})}}, \end{align*} where $J_0$, $J_1$ are Bessel functions of the first kind and $j_{1k}$ is the $k$-th positive root of $J_1(x) = 0$. (See \cite{K2006Disk} or \cite{LR2002}.) The $(u_k)$ are complete in $H$ and in $V$ and are normalized so that\footnote{This differs from the normalization in \cite{K2006Disk}, where $\norm{u_k}_H = j_{1k}^{-1}$, $\norm{\omega_k}_{L^2} = 1$.} \begin{align*} \norm{u_k}_H = 1, \quad \norm{\omega_k}_{L^2} = j_{1k}. \end{align*} Assume that $u_0 \in H \cap H^1$. Then \begin{align*} u_0 = \sum_{k = 1}^\ensuremath{\infty} a_k u_k, \quad \smallnorm{u_0}_H^2 = \sum_{k = 1}^\ensuremath{\infty} a_k^2 < \ensuremath{\infty}. \end{align*} (But, \begin{align*} \smallnorm{u_0}_V^2 = \sum_{k = 1}^\ensuremath{\infty} a_k^2 j_{1k}^2 = \ensuremath{\infty} \end{align*} unless $u_0 \in V$.) We claim that \begin{align*} u(t) = \sum_{k = 1}^\ensuremath{\infty} a_k e^{- \nu j_{1k}^2 t} u_k \end{align*} provides a solution to the Navier-Stokes equations, ($NS$). To see this, first observe that $u \in C([0, T]; H)$, so $u(0) = u_0$ makes sense as an initial condition. Also, $u(t) \in V$ for all $t > 0$. Next observe that \begin{align*} \omega(t) := \omega(u(t)) = \sum_{k = 1}^\ensuremath{\infty} a_k e^{- \nu j_{1k}^2 t} \omega_k \end{align*} for all $t > 0$, this sum converging in $H^n$ for all $n \ge 0$. Since each term satisfies \cref{e:RadialHeat} so does the sum. Taken together, this shows that $\omega$ satisfies \cref{e:RadialHeat} and thus $u$ solves ($NS$). \Ignore{ \begin{align*} \sum_{k = 1}^\ensuremath{\infty} a_k^2 e^{- 2 \nu j_{1k}^2 t} \norm{\omega_k}_{L^2}^2 = \sum_{k = 1}^\ensuremath{\infty} a_k^2 j_{1k} e^{- 2 \nu j_{1k}^2 t} < \ensuremath{\infty} \end{align*} for all $t > 0$ } The condition in \refE{BoundaryCondition2D} becomes \begin{align*} \nu \int_0^T & \int_\Gamma \omega \, \overline{u} \cdot \BoldTau = \nu \sum_{k = 1}^\ensuremath{\infty} \int_0^T \int_\Gamma a_k e^{- \nu j_{1k}^2 t} \omega_k \, \overline{u} \cdot \BoldTau \, dt \\ &= \nu \sum_{k = 1}^\ensuremath{\infty} \int_0^T a_k e^{- \nu j_{1k}^2 t} \omega_k|_{r = 1} \int_\Gamma \overline{u} \cdot \BoldTau \, dt\\ &= m \nu \sum_{k = 1}^\ensuremath{\infty} a_k \frac{j_{1k} J_0(j_{1k})} {\pi^{1/2}\abs{J_0(j_{1k})}} \int_0^T e^{- \nu j_{1k}^2 t} \, dt. \end{align*} In the final equality, we used \begin{align*} \int_\Gamma \overline{u} \cdot \BoldTau = - \int_\Gamma \overline{u}^\perp \cdot \bm{n} = - \int_\Omega \dv \overline{u}^\perp = \int_\Omega \overline{\omega} = m. \end{align*} (Because vorticity is transported by the Eulerian flow, $m$ is constant in time.) Then, \begingroup \allowdisplaybreaks \begin{align*} &\abs{\nu \int_0^T \int_\Gamma \omega \, \overline{u} \cdot \BoldTau} \le \abs{m} \nu \sum_{k = 1}^\ensuremath{\infty} \frac{\abs{a_k}}{\pi^{1/2}} j_{1k} \int_0^T e^{- \nu j_{1k}^2 t} \, dt \\ &\qquad = \abs{m} \nu \sum_{k = 1}^\ensuremath{\infty} \frac{\abs{a_k}}{\pi^{1/2}} j_{1k} \frac{1 - e^{- \nu j_{1k}^2 T}}{\nu j_{1k}^2} \\ &\qquad \le \frac{\abs{m}}{\pi^{\frac{1}{2}}} \pr{\sum_{k = 1}^\ensuremath{\infty} a_k^2}^{\frac{1}{2}} \pr{\sum_{k = 1}^\ensuremath{\infty} \frac{(1 - e^{- \nu j_{1k}^2 T})^2}{j_{1k}^2}} ^{\frac{1}{2}} \\ &\qquad = \frac{\abs{m}}{\pi^{\frac{1}{2}}} \smallnorm{u_0}_H \pr{\sum_{k = 1}^\ensuremath{\infty} \frac{(1 - e^{- \nu j_{1k}^2 T})^2}{j_{1k}^2}} ^{\frac{1}{2}}. \end{align*} \endgroup Classical bounds on the zeros of Bessel functions give $1 + k < j_{1k} \le \pi(\frac{1}{2} + k)$ (see, for instance, Lemma A.3 of \cite{K2006Disk}). Hence, with $M = (\nu T)^{-\ensuremath{\alpha}}$, $\ensuremath{\alpha} > 0$ to be determined, we have \begingroup \allowdisplaybreaks \begin{align*} \sum_{k = 1}^\ensuremath{\infty} &\frac{(1 - e^{- \nu j_{1k}^2 T})^2}{j_{1k}^2} \le C \sum_{k = 1}^\ensuremath{\infty} \frac{(1 - e^{- \nu k^2 T})^2}{k^2} \\ &\le (1 - e^{- \nu T})^2 + \int_{k = 1}^M \frac{(1 - e^{- \nu x^2 T})^2}{x^2} \, dx + \int_{k = M + 1}^\ensuremath{\infty} \frac{(1 - e^{- \nu x^2 T})^2}{x^2} \, dx \\ &\le \nu^2 T^2 + \nu^2 T^2 \int_{k = 1}^M \frac{x^4}{x^2} \, dx + \int_{k = M + 1}^\ensuremath{\infty} \frac{1}{x^2} \, dx \\ &\le \nu^2 T^2 + \nu^2 T^2 \frac{1}{3} \pr{M^3 - 1} + \frac{1}{M} \le \nu^2 T^2 + \nu^2 T^2 M^3 + \frac{1}{M} \\ &= \nu^2 T^2 + \nu^2 T^2 \nu^{-3 \ensuremath{\alpha}} T^{- 3 \ensuremath{\alpha}} + (\nu T)^\ensuremath{\alpha} = \nu^2 T^2 + (\nu T)^{2 - 3 \ensuremath{\alpha}} + (\nu T)^\ensuremath{\alpha} \end{align*} \endgroup as long as $\nu M^2 T \le 1$ (used in the third inequality); that is, as long as \begin{align}\label{e:albetaReq} (\nu T)^{1 - 2 \ensuremath{\alpha}} \le 1. \end{align} Thus \cref{e:BoundaryCondition2D} holds (as does \cref{e:nuL1Bound}), so ($VV$) holds. The rate of convergence in ($VV$) is optimized when $(\nu T)^{2 - 3 \ensuremath{\alpha}} = (\nu T)^\ensuremath{\alpha}$, which occurs when $\ensuremath{\alpha} = \frac{1}{2}$. The condition in \refE{albetaReq} is then satisfied with equality. \refR{ROC} then gives a rate of convergence in the vanishing viscosity limit of $C \nu^{\frac{1}{4}}$ (even for smoother initial data), except in the special case $m = 0$, which we note reduces to Example 1a. \ReturnExample{1a} Let us apply our analysis of Example 3 to the special case of Example 1a, in which $u_0 \in V$. Now, on the boundary, \begin{align*} (\ensuremath{\partial}_t u + u \cdot \ensuremath{\nabla} u + \ensuremath{\nabla} p) \cdot \BoldTau = \nu \Delta u \cdot \BoldTau = \nu \Delta u^\perp \cdot (- \bm{n}) = - \nu \ensuremath{\nabla}^\perp \omega \cdot \bm{n}. \end{align*} But $\ensuremath{\nabla} p \equiv 0$ so the left-hand side vanishes. Hence, the vorticity satisfies homogeneous Neumann boundary conditions for positive time. (This is an instance of Lighthill's formula.) Since the nonlinear term vanishes, in fact, $\omega$ satisfies the heat equation, $\ensuremath{\partial}_t \omega = \nu \Delta \omega$ with homogeneous Neumann boundary conditions and hence $\omega \in C([0, T]; L^2(\Omega))$. Moreover, multiplying $\ensuremath{\partial}_t \omega = \nu \Delta \omega$ by $\omega$ and integrating gives \begin{align*} \norm{\omega(t)}_{L^2}^2 + 2 \nu \int_0^t \norm{\ensuremath{\nabla} \omega(s)}_{L^2}^2 \, ds = \norm{\ensuremath{\nabla} \omega_0}_{L^2}^2. \end{align*} We conclude that the $L^2$-norm of $\omega$, and so the $L^p$-norms for all $p \le 2$, are bounded in time uniformly in $\nu$. (In fact, this holds for all $p \in [1, \ensuremath{\infty}]$. This conclusion is not incompatible with \refT{VorticityNotBounded}, since $\overline{u} \equiv 0$ on $\Gamma$.) This argument for bounding the $L^p$-norms of the vorticity fails for Example 3 because the vorticity is no longer continuous in $L^2$ down to time zero unless $u_0 \in V$. It is shown in \cite{FLMT2008} (and see \cite{GKLMN14}) that such control is nonetheless obtained for the $L^1$ norm. \section{On a result of Bardos and Titi}\label{S:BardosTiti} \noindent Bardos and Titi in \cite{BardosTiti2013a, Bardos2014Private}, also starting from, essentially, \refE{VVArg} make the observation that, in fact, for the vanishing viscosity limit to hold, it is necessary and sufficient that $\nu \omega$ (or, equivalently, $\nu [\ensuremath{\partial}_{\bm{n}} u]_{\BoldTau}$) converge to zero on the boundary in a weak sense. In their result, the boundary is assumed to be $C^\ensuremath{\infty}$, but the initial velocity is assumed to only lie in $H$. Hence, the sufficiency condition does not follow immediately from \refE{VVArg}. Their proof of sufficiency involves the use of dissipative solutions to the Euler equations. (The use of dissipative solutions for the Euler equations in a domain with boundaries was initiated in \cite{BardosGolsePaillard2012}. See also \cite{BSW2014}.) We present here the weaker version of their results in 2D that can be obtained without employing dissipative solutions. The simple and elegant proof of necessity is as in \cite{Bardos2014Private}, simplified further because of the higher regularity of our initial data. \begin{theorem}[Bardos and Titi \cite{BardosTiti2013a, Bardos2014Private}]\label{T:BardosTiti} Working in 2D, assume that $\ensuremath{\partial} \Omega$ is $C^2$ and that $\overline{u} \in C^1([0, T; C^1(\Omega))$. Then for $u \to \overline{u}$ in $L^\ensuremath{\infty}(0, T; H)$ to hold it is necessary and sufficient that \begin{align}\label{e:BardosNecCond} \nu \int_0^T \int_\Gamma \omega \, \varphi \to 0 \text{ as } \nu \to 0 \text{ for any } \varphi \in C^1([0, T] \times \Gamma). \end{align} \end{theorem} \begin{proof} Sufficiency of the condition follows immediately from setting $\varphi = (\overline{u} \cdot \BoldTau)|_\Gamma$ in \refT{BoundaryIffCondition}. To prove necessity, let $\varphi \in C^1([0, T] \times \Gamma)$. We will need a divergence-free vector field $v_\delta \in C^1([0, T]; H \cap C^\ensuremath{\infty}(\Omega))$ such that $v_\delta \cdot \BoldTau = \varphi$. Moreover, we require of $v_\delta$ that it satisfy the same bounds as the boundary layer corrector of Kato in \cite{Kato1983}; in particular, \begin{align}\label{e:vBounds} \norm{\ensuremath{\partial}_t v_\delta}_{L^1([0, T]; L^2(\Omega))} \le C \delta^{1/2}, \qquad \norm{\ensuremath{\nabla} v_\delta}_{L^\ensuremath{\infty}([0, T]; L^2(\Omega))} \le C \delta^{-1/2}. \end{align} This vector field can be constructed in several ways: we detail one such construction at the end of this proof. The proof now proceeds very simply. We multiply the Navier-Stokes equations by $v_\delta$ and integrate over space and time to obtain \begin{align}\label{e:BardosNec} \begin{split} \int_0^T (\ensuremath{\partial}_t &u, v_\delta) + \int_0^T (u \cdot \ensuremath{\nabla} u, v_\delta) + \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} v_\delta) \\ &= \nu \int_0^T \int_\Gamma (\ensuremath{\nabla} u \cdot \bm{n}) \cdot v_\delta = \nu \int_0^T \int_\Gamma \omega \, v_\delta \cdot \BoldTau = \nu \int_0^T \int_\Gamma \omega \, \varphi. \end{split} \end{align} Here, we used \refE{gradunomega} with $v_\delta$ in place of $\overline{u}$, and we note that no integrations by parts were involved. Now, assuming that the vanishing viscosity limit holds, Kato shows in \cite{Kato1983} that setting $\delta = c \nu$---and using the bounds in \refE{vBounds}---each of the terms on the left hand side of \refE{BardosNec} vanishes as $\nu \to 0$. By necessity, then, so does the right hand side, giving the necessity of the condition in \refE{BardosNecCond}. It remains to construct $v_\delta$. To do so, we place coordinates on a tubular neighborhood, $\Sigma$, of $\Gamma$ as in the proof of \cref{L:Trace}. In $\Sigma$, define \begin{align*} \psi(s, r) = - r \varphi(s). \end{align*} Write $\ensuremath{\widehat}{r}$, $\ensuremath{\widehat}{s}$ for the unit vectors in the directions of increasing $r$ and $s$. Then $\ensuremath{\widehat}{r} \cdot \ensuremath{\widehat}{s} = 0$ and $\ensuremath{\widehat}{r} = - \bm{n}$ on $\Gamma$. Thus, on the boundary, \begin{align*} \ensuremath{\nabla} \psi(s, r) = -\varphi(s) \ensuremath{\widehat}{r} -r \varphi'(s) \ensuremath{\widehat}{s}. \end{align*} This gives \begin{align*} \ensuremath{\nabla} \psi(s) \cdot \bm{n} = -\varphi(s) \ensuremath{\widehat}{r} \cdot \bm{n} = \varphi(s). \end{align*} It also gives $\ensuremath{\nabla} \psi \in C^1([0, T]; C(\Sigma))$ so that $\psi \in \varphi \in C^1([0, T] \times \Sigma)$. We now follow the procedure in \cite{Kato1983}. Let $\zeta: [0, \ensuremath{\infty}) \to [0, 1]$ be a smooth cutoff function with $\zeta \equiv 1$ on $[0, 1/2]$ and $\zeta \equiv 0$ on $[1, \ensuremath{\infty}]$. Define $\zeta_\delta(\cdot) = \zeta(\cdot/\delta)$ and \begin{align*} v_\delta(x) = \ensuremath{\nabla}^\perp (\zeta_\delta(\dist(x, \ensuremath{\partial} \Omega)) \psi(x)). \end{align*} Note that $v_\delta$ is supported in a boundary layer of width proportional to $\delta$. The bounds in \refE{vBounds} follow as shown in \cite{K2006Kato}. \end{proof} To establish the necessity of the stronger condition in \refT{BardosTiti}, we used (based on Bardos's \cite{Bardos2014Private}) a vector field supported in a boundary layer of width $c \nu$, as in \cite{Kato1983}. We used it, however, to extend to the whole domain an arbitrary cutoff function defined on the boundary, rather than to correct the Eulerian velocity as in \cite{Kato1983}. \begin{remark} In this proof of \refT{BardosTiti} the time regularity in the test functions could be weakened slightly to assuming that $\ensuremath{\partial}_t \varphi \in L^1([0, T]; C(\Gamma))$, for this would still allow the first bound in \refE{vBounds} to be obtained. \end{remark} \begin{remark} Using the results of \cite{BardosTiti2013a, BSW2014} it is possible to change the condition in \refE{BardosNecCond} to apply to test functions $\varphi$ in $C^1([0, T]; C^\ensuremath{\infty}(\Gamma))$ (\cite{Bardos2014Private}). Moreover, this can be done without assuming time or spatial regularity of the solution to the Euler equations: only that the initial velocity lies in $H$. \end{remark} \Ignore{ \section{Speculation on another condition for the VV limit} \noindent There is nothing deep about the condition in \refE{BoundaryCondition2D}, but what it says is that there are two mechanisms by which the vanishing viscosity limit can hold. First, the blowup of $\omega$ on the boundary can happen slowly enough that \begin{align}\label{e:nuL1Bound} \nu \int_0^T \norm{\omega}_{L^1(\Gamma)} \to 0 \text{ as } \nu \to 0 \end{align} or, second, the vorticity for ($NS$) can be generated on the boundary in such a way as to oppose the sign of $\overline{u} \cdot \BoldTau$. In the second case, it could well be that vorticity for $(NS)$ blows up fast enough that \refE{nuL1Bound} does not hold, but cancellation in the integral in \refE{BoundaryCondition2D} allows that condition to hold. A natural question to ask is whether the condition, \begin{align*} (G) \qquad \nu \int_0^T \norm{\omega}_{L^1(\Gamma)} \to 0 \text{ as } \nu \to 0 \end{align*} is equivalent to the conditions in \cref{T:VVEquiv}. The sufficiency of this condition follows immediately, since it implies that \refE{BoundaryCondition2D} holds. To see why we might suspect that ($G$) is necessary for ($VV$) to hold, we start with the necessary and sufficient condition $(iii')$ of Theorem 1.2 of \cite{K2006Kato} that \begin{align*} \nu \int_0^T \norm{\omega}_{L^2(\Gamma_\nu)}^2 \to 0 \text{ as } \nu \to 0, \end{align*} where $\Gamma_\nu = \set{x \in \Omega \colon \dist(x, \Gamma) < \nu}$. For sufficiently regular $u_\nu^0$, for all $t > 0$, $\omega(t)$ will lie in $H^2(\Omega) \supseteq C(\overline{\Omega})$, and one might expect to have \begin{align}\label{e:ApproxIntegral} \nu \int_0^T \norm{\omega}_{L^2(\Gamma_\nu)}^2 &\cong \nu \int_0^T \int_0^\nu \norm{\omega}_{L^2(\Gamma)}^2 = \nu^2 \int_0^T \norm{\omega}_{L^2(\Gamma)}^2. \end{align} Then using \Holders inequality followed by Jensen's inequality, \begin{align}\label{e:HJBound} \pr{\frac{\nu}{T^{3/2}} \int_0^T \norm{\omega}_{L^1(\Gamma)}}^2 \le \pr{\frac{\nu}{T} \int_0^T \norm{\omega}_{L^2(\Gamma)}}^2 \le \frac{\nu^2}{T} \int_0^T \norm{\omega}_{L^2(\Gamma)}^2. \end{align} But the left-hand side of \refE{ApproxIntegral} must vanish, and so too must the left-hand side of \refE{HJBound}, implying that $(G$) holds. The problem with this argument, however, is that the best we can say rigorously is that from \refT{BoundaryLayerWidth} and the continuity of $\omega(t)$ for all $t > 0$, \begin{align*} \nu \int_0^T \norm{\omega}_{L^1(\Gamma)}^2 &= \nu \int_0^T \lim_{\delta \to 0} \frac{1}{\delta^2} \norm{\omega}_{L^1(\Gamma_{\delta})}^2 \le \nu \liminf_{\delta \to 0} \frac{1}{\delta^2} \int_0^T \norm{\omega}_{L^1(\Gamma_{\delta})}^2 \\ &\le \nu \lim_{\delta \to 0} \frac{1}{\delta^2} \frac{C \delta}{\nu} \le \ensuremath{\infty}, \end{align*} where in the first inequality we used Fatou's lemma. If we could improve this inequality to show that $\nu \int_0^T \norm{\omega}_{L^1(\Gamma)}^2$ is $o(1/\nu)$, then using \Holders inequality followed by Jensen's inequality, \begin{align*} \pr{\frac{\nu}{T} \int_0^T \norm{\omega}_{L^1(\Gamma)}}^2 \le \frac{\nu^2}{T} \int_0^T \norm{\omega}_{L^1(\Gamma)}^2 \to 0 \text{ as } \nu \to 0. \end{align*} \Ignore{ Letting $f = \overline{\omega}$ in condition ($E_2$) of \cref{T:VVEquiv} gives \begin{align*} (\omega, \overline{\omega}) \to \norm{\overline{\omega}}_{L^2}^2 - \int_\Gamma \overline{\omega} \, \overline{u} \cdot \BoldTau. \end{align*} But, \begin{align*} (\omega, \overline{\omega}) &= (\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) = - (\Delta u, \overline{u}) + \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u}, \end{align*} where we used Lemma 6.6 of \cite{K2008VVV} for scalar vorticity (in which the factor of 2 in that lemma does not appear). By Equation (4.2) of \cite{KNavier}, \begin{align*} (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} = ((\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \BoldTau) (\overline{u} \cdot \BoldTau) = \omega(u) \, \overline{u} \cdot \BoldTau. \end{align*} Thus, \begin{align*} - \nu (\Delta u, \overline{u}) + \nu \int_\Gamma \omega(u) \, \overline{u} \cdot \BoldTau \to \nu \norm{\overline{\omega}}_{L^2}^2 - \nu \int_\Gamma \overline{\omega} \, \overline{u} \cdot \BoldTau. \end{align*} The right-hand side vanishes with $\nu$ since $\overline{u}$ is in $C^{1 + \ensuremath{\epsilon}}$, so \begin{align*} \nu \int_\Gamma \omega(u) \, \overline{u} \cdot \BoldTau \to - \nu (\Delta u, \overline{u}). \end{align*} It remains to show that the right-hand side vanishes with $\nu$. Now, \begin{align*} \nu (\Delta u, \overline{u}) = (\nu \Delta u, \overline{u}) = \nu (\ensuremath{\partial}_t u, \overline{u}) + \nu (u \cdot \ensuremath{\nabla} u, \overline{u}) + \nu (\ensuremath{\nabla} p, \overline{u}) \end{align*} } \Ignore{ We make the assumptions on the initial velocity and on the forcing in \cref{T:VVEquiv}. \begin{theorem} The vanishing viscosity limit holds over any finite time interval $[0, T]$ if and only if $A_\nu \to 0$ as $\nu \to 0$, where \begin{align}\label{e:Anu} A_\nu = \nu \int_0^T \norm{\omega}_{L^1(\Gamma)}. \end{align} Moreover, \begin{align}\label{e:RateOfConvergence} \norm{u(t) - \overline{u}(t)}_{L^2}^2 \le (C\nu + C A_\nu + \smallnorm{u_\nu^0 - \overline{u}^0}_{L^2}^2)^{1/2} e^{Ct} \end{align} for all sufficiently small $\nu > 0$, with $C$ depending only upon the initial velocities and $T$. \end{theorem} \begin{proof} Subtracting ($EE$) from ($NS$), multiplying by $w = u - \overline{u}$, and integrating over $\Omega$ leads to \begin{align*} \frac{1}{2} \diff{}{t} &\norm{\omega}_{L^2}^2 + \nu \norm{\ensuremath{\nabla} u}_{L^2}^2 = - (w \cdot \ensuremath{\nabla} \overline{u}, w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) - \nu \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} \\ &= - (w \cdot \ensuremath{\nabla} \overline{u}, w) + \nu(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) - \nu \int_\Gamma \omega \, \overline{u} \cdot \BoldTau. \end{align*} Here we used Equation (4.2) of \cite{KNavier} to conclude that \begin{align*} (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \overline{u} = ((\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot \BoldTau) (\overline{u} \cdot \BoldTau) = \omega \, \overline{u} \cdot \BoldTau. \end{align*} Integrating over time gives \begin{align*} \frac{1}{2} &\norm{w(t)}_{L^2}^2 + \nu \int_0^t \norm{\ensuremath{\nabla} u}_{L^2}^2 = \norm{w(0)}_{L^2}^2 - \int_0^t (w \cdot \ensuremath{\nabla} \overline{u}, w) + \nu \int_0^t (\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u}) \\ &\qquad - \nu \int_0^t \int_\Gamma \omega \, \overline{u} \cdot \BoldTau. \end{align*} Using the bounds, \begin{align*} \abs{(w \cdot \ensuremath{\nabla} \overline{u}, w)} &\le \norm{\ensuremath{\nabla} \overline{u}}_{L^\ensuremath{\infty}([0, T] \times \Omega)} \norm{w}_{L^2}^2 \le C \norm{w}_{L^2}^2, \\ \nu \int_0^T \abs{(\ensuremath{\nabla} u, \ensuremath{\nabla} \overline{u})} &\le \nu \norm{\ensuremath{\nabla} \overline{u}}_{L^2([0, T] \times \Omega)} \norm{\ensuremath{\nabla} u}_{L^2([0, T] \times \Omega)} \\ &\le C \nu \norm{\ensuremath{\nabla} u}_{L^2([0, T] \times \Omega)} \\ &\le C \nu + \frac{\nu}{2} \norm{\ensuremath{\nabla} u}_{L^2([0, T] \times \Omega)}^2 , \\ - \nu \int_0^t \int_\Gamma \omega \, \overline{u} \cdot \BoldTau &\le \nu \norm{\overline{u}}_{L^\ensuremath{\infty}} \int_0^T \norm{\omega}_{L^1(\Gamma)} \le C \nu \int_0^T \norm{\omega}_{L^1(\Gamma)} \end{align*} gives \begin{align}\label{e:VVArg} \begin{split} &\norm{w(t)}_{L^2}^2 + \nu \int_0^t \norm{\ensuremath{\nabla} u}_{L^2}^2 \le \norm{w(0)}_{L^2}^2 + C \nu + C A_\nu + C \int_0^t \norm{w}_{L^2}^2. \end{split} \end{align} Applying Gronwall's inequality leads to \refE{RateOfConvergence} and shows that $A_\nu \to 0$ implies ($VV$). \end{proof} Let \begin{align*} \Gamma_\delta = \set{x \in \Omega \colon \dist(x, \Gamma) < \delta}, \end{align*} where we always assume that $\delta > 0$ is sufficiently small that $\Gamma_\delta$ is a tubular neighborhood of $\Gamma$. \begin{lemma}\label{L:BoundaryLayerWidth} For any sufficiently small $\delta > 0$ \begin{align}\label{e:OmegaL1VanishGeneral} \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\delta}))}^2 \le C \frac{\delta}{\nu} \end{align} for all sufficiently small $\delta(\nu)$. \end{lemma} \begin{proof} By the Cauchy-Schwarz inequality, \begin{align*} \norm{\omega}_{L^1(\Gamma_{\delta})} \le \norm{1}_{L^2(\Gamma_{\delta})} \norm{\omega}_{L^2(\Gamma_{\delta})} \le C \delta^{1/2} \norm{\omega}_{L^2(\Gamma_{\delta})} \end{align*} so \begin{align*} \norm{\omega}_{L^1(\Gamma_{\delta})}^2 \le C \delta \norm{\omega}_{L^2(\Gamma_{\delta})}^2 \end{align*} and \begin{align*} \frac{C \nu}{\delta} \norm{\omega}_{L^2([0, T]; L^1(\Gamma_{\delta}))}^2 \le \nu \norm{\omega}_{L^2([0, T]; L^2(\Gamma_{\delta}))}^2. \end{align*} By the basic energy inequality for the Navier-Stokes equations, the right-hand side is bounded, giving \refE{OmegaL1VanishGeneral}. \end{proof} \begin{theorem}\label{T:Anu} Assume that $\Gamma$ is $C^3$. If the vanishing viscosity limit holds then $A_\nu \to 0$ as $\nu \to 0$. \end{theorem} \begin{proof} Impose at first the extra regularity condition that $u_\nu^0$ lies in $H^3(\Omega)$, so that the $u(t)$ lies in $H^3(\Omega)$ for all $t > 0$. Then for all $t > 0$, $\omega(t)$ is in $H^2(\Omega)$ and hence $\omega(t)$ is continuous up to the boundary by Sobolev embedding. Thus,\begin{align}\label{e:BoundaryIntegralLimit} \norm{\omega(t)}_{L^1(\Gamma)}^2 = \lim_{\delta \to 0} \frac{1}{\delta^2} \norm{\omega(t)}_{L^1(\Gamma_{\delta})}^2. \end{align} It follows from Fatou's lemma that \begin{align*} \nu \int_0^T &\norm{\omega(t)}_{L^1(\Gamma)}^2 \, dt = \nu \int_0^T \lim_{\delta \to 0} \frac{1}{\delta^2} \norm{\omega(t)}_{L^1(\Gamma_{\delta})}^2 \, dt\\ &= \nu \int_0^T \liminf_{\delta \to 0} \frac{1}{\delta^2} \norm{\omega(t)}_{L^1(\Gamma_{\delta})}^2 \, dt \le \nu \liminf_{\delta \to 0} \int_0^T \frac{1}{\delta^2} \norm{\omega(t)}_{L^1(\Gamma_{\delta})}^2 \, dt\\ &\le \nu \liminf_{\delta \to 0} \frac{1}{\delta^2} C \frac{\delta}{\nu} = \liminf_{\delta \to 0} \frac{C}{\delta}. \end{align*} In the last inequality we used \refL{BoundaryLayerWidth}. \textbf{Of course, this is BAD!!!} Using \Holders and Jensen's inequalities it follows that \begin{align*} \pr{\frac{\nu}{T} \int_0^T \norm{\omega}_{L^1(\Gamma)}}^2 \le \frac{\nu^2}{T} \int_0^T \norm{\omega}_{L^1(\Gamma)}^2 \le \frac{C \nu}{T}, \end{align*} completing the proof. \end{proof} \begin{remark} In higher dimensions, we could attempt the same argument using $\ensuremath{\nabla} u$ in place of $\omega$. A problem remains, though, in that we cannot conclude that $\ensuremath{\nabla} u$ has sufficient space regularity over a finite time interval independent of the viscosity so that $\omega(t)$ is continuous. Weak solutions do have sufficient regularity so that the left-hand side of \refE{BoundaryIntegralLimit} (with $\ensuremath{\nabla} u$ in place of $\omega$) makes sense, but there is no reason to suppose that equality with the right-hand side holds. \end{remark} } } \Ignore{ \section{An alternate derivation of Kato's conditions}\label{S:AlternateDerivation} \noindent The argument that led to \refE{VVArg} in the proof of \refT{BoundaryIffCondition} is perhaps the first calculation that anyone who ever attempts to establish the vanishing viscosity limit makes. It is simple, direct, and natural. Because we were working in 2D it was easy to make the argument rigorous, but the essential idea is contained in the formal argument. Kato's introduction of a boundary layer corrector, on the other hand, handles the rigorous proof of the necessity and sufficiency of his conditions in higher dimensions while at the same time striving to give the motivation for those very conditions. In this way, it obscures to some extent the nature of the argument, and appears somewhat unmotivated. That is to say, one can follow the technical details easily enough, but it is hard to see what the plan is at the outset. (Kato uses the energy inequality for the Navier-Stokes equations in a way that avoids treating $w = u - \overline{u}$ as though it were a test function for the Navier-Stokes equations. This now classical technique is clearly explained in Section 2.2 of \cite{IftimieLopeses2009}.) We give a different derivation below, which starts with \refT{BoundaryIffCondition}. We give the formal argument, which is rigorous in two dimensions if we pay more attention to the regularity of the solutions. \begin{theorem} The condition in \cref{e:KellCondition} is necessary and sufficient for ($VV$) to hold. \end{theorem} \begin{proof} Let $v$ be the boundary layer velocity defined by Kato in \cite{Kato1983}, where $\delta = c \nu$: so $v$ is divergence-free, vanishes outside of $\Gamma_{c \nu}$, and $v = \overline{u}$ on $\Gamma$. (In all that follows, one can also refer to \cite{K2006Kato}, which gives Kato's argument using (almost) his same notation.) Since $v = \overline{u}$ on $\Gamma$, by \refT{BoundaryIffCondition}, and using \cref{e:gradunomega}, ($VV$) holds if and only if \begin{align*} \nu \int_0^T \int_\Gamma (\ensuremath{\nabla} u \cdot \mathbf{n}) \cdot v = \int_0^T (\nu \Delta u, v) + \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} v) \to 0 \end{align*} as $\nu \to 0$. Using Lemma A.2 of \cite{K2006Kato}, \begin{align*} \nu \int_0^T (\ensuremath{\nabla} u, \ensuremath{\nabla} v) &= 2 \nu \int_0^T (\omega(u), \omega(v)) \le 2 \nu \int_0^T \norm{\omega(u)}_{L^2(\Gamma_{c \nu})} \norm{\omega(v)}_{L^2} \\ &\le \sqrt{\nu} \norm{\ensuremath{\nabla} v}_{L^2([0, T] \times \Omega)} \sqrt{\nu} \norm{\omega(u)}_{L^2([0, T] \times \Gamma_{c \nu})} \\ &\le C \pr{\nu \int_0^T \norm{\omega(u)}_{L^2(\Gamma_{c \nu})}^2}^{1/2}, \end{align*} since $\norm{\ensuremath{\nabla} v}_{L^2([0, T] \times \Omega)} \le C \nu^{-1/2}$. Also, \begin{align*} \int_0^T (\nu \Delta u, v) = \int_0^T \brac{(\ensuremath{\partial}_t u, v) + (u \cdot \ensuremath{\nabla} u, v) + (\ensuremath{\nabla} p, v) - (f, v)}. \end{align*} The integral involving the pressure disappears, while \begin{align*} \int_0^T \abs{(f, v)} \le C \nu^{1/2} \int_0^T \norm{f}_{L^2(\Gamma_{c \nu})}, \end{align*} using the bound on $\norm{v}_{L^\ensuremath{\infty}([0, T]; L^2)}$ in \cite{Kato1983} (Equation (3.1) of \cite{K2006Kato}). This vanishes with the viscosity since $f$ lies in $L^1([0, T]; L^2)$. The integral involving $(u \cdot \ensuremath{\nabla} u, v)$ we bound the same way as in \cite{K2006Kato}. Using Lemma A.4 of \cite{K2006Kato}, \begingroup \allowdisplaybreaks \begin{align*} &\abs{\int_0^t (u \cdot \ensuremath{\nabla} u, v)} = 2 \abs{\int_0^t (v, u \cdot \omega(u))} \\ &\qquad \le 2 \norm{v}_{L^\ensuremath{\infty}([0, T] \times \Omega)} \int_0^t \norm{u}_{L^2(\Gamma_{c \nu})} \norm{\omega(u)}_{L^2(\Gamma_{c \nu})} \\ &\qquad \le C \nu \int_0^t \norm{\ensuremath{\nabla} u}_{L^2(\Gamma_{c \nu})} \norm{\omega(u)}_{L^2(\Gamma_{c \nu})} \\ &\qquad \le C \nu^{1/2} \norm{\ensuremath{\nabla} u}_{L^2([0, T]; L^2(\Gamma_{c \nu}))} \nu^{1/2} \norm{\omega(u)}_{L^2([0, T]; L^2(\Gamma_{c \nu}))} \\ &\qquad \le C \pr{\nu \int_0^t \norm{\omega(u)}_{L^2(\Gamma_{c \nu})}^2}^{1/2}. \end{align*} \endgroup Finally, \begin{align*} \int_0^T (\ensuremath{\partial}_t u, v) = \int_0^T \int_\Omega \ensuremath{\partial}_t (u v) + \int_0^T (u, \ensuremath{\partial}_t v). \end{align*} As in \cite{Kato1983}, \begin{align*} \abs{\int_0^t (u, \ensuremath{\partial}_t v)} &\le \int_0^t \norm{u}_{L^2(\Omega)} \norm{\ensuremath{\partial}_t v}_{L^2(\Omega)} \le C \nu^{1/2}. \end{align*} Also, \begin{align*} \int_0^T \int_\Omega &\ensuremath{\partial}_t (u v) = \int_0^T \diff{}{t} (u, v) = (u(T), v(T)) - (u_\nu^0, v(0)) \\ &\le \norm{u(T)}_{L^2} \norm{v}_{L^2} + \smallnorm{u_\nu^0}_{L^2} \norm{v(0)}_{L^2} \\ &\le C \smallnorm{u_0}_{L^2} \norm{v}_{L^\ensuremath{\infty}([0, T]; L^2)} \le C \sqrt{\nu}. \end{align*} We conclude from all these inequalities that \begin{align*} \nu \int_0^T \norm{\omega(u)}_{L^2(\Gamma_{c \nu})}^2 \to 0 \text{ as } \nu \to 0 \end{align*} is a sufficient condition for the vanishing viscosity limit to hold (as, too, is Kato's condition involving $\ensuremath{\nabla} u$ in place of $\omega(u)$). The necessity follows easily from the energy inequality. \end{proof} } \ifbool{IncludeNavierBCSection}{ \section{Navier boundary conditions in 2D}\label{S:NavierBCs} \refT{VorticityNotBounded} says that if the vanishing viscosity limit holds, then there cannot be a uniform (in $\nu$) bound on the $L^2$-norm of the vorticity. This is in stark contrast to the situation in the whole space, where such a bound holds, or for Navier boundary conditions in 2D, where such a bound holds for $L^p$, $p > 2$, as shown in \cite{FLP} and \cite{CMR}. For Navier boundary conditions in 2D, then, as long as the initial vorticity is in $L^p$ for $p > 2$ there will be a uniform bound on the $L^2$-norm of the vorticity, since the domain is bounded. In fact, for Navier boundary conditions in 2D the classical vanishing viscosity limit ($VV$) does hold, even for much weaker regularity on the initial velocity than that considered here (see \cite{KNavier}). The argument in the proof of \refT{VorticityNotBounded} then shows that \begin{align}\label{e:VelocityGammaConvergence} u \to \overline{u} \text{ in } L^\ensuremath{\infty}([0, T]; L^2(\Gamma)). \end{align} We also have weak$^*$ convergence of the vorticity in $\Cal{M}(\overline{\Omega})$, as we show in \cref{T:VorticityConvergenceNavier}. \begin{theorem}\label{T:VorticityConvergenceNavier} Assume that the solutions to $(NS)$ are with Navier boundary conditions in 2D, and that the initial vorticity $\omega_0 = \overline{\omega}_0$ is in $L^\ensuremath{\infty}$ (slightly weaker assumptions as in \cite{KNavier} can be made). Then all of the conditions in \cref{T:VVEquiv} hold, but with the three conditions below replacing conditions $(C)$, $(E)$, and $(E_2)$, respectively: \begin{align*} (C^N) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} \overline{u} \quad \weak^* \text{ in } L^\ensuremath{\infty}(0, T; \Cal{M}(\overline{\Omega})^{d \times d}), \\ (E^N) & \qquad \omega \to \overline{\omega} \quad \weak^* \text{ in } L^\ensuremath{\infty}(0, T; \Cal{M}(\overline{\Omega})^{d \times d}), \\ (E_2^N) & \qquad \omega \to \overline{\omega} \quad \weak^* \text{ in } L^\ensuremath{\infty}(0, T; \Cal{M}(\overline{\Omega})). \\ \end{align*} \end{theorem} \begin{proof} First observe that $(E^N)$ is just a reformulation of $(E_2^N)$ with vorticity viewed as a matrix. Also, it is sufficient to prove convergences in $(H^1(\Omega))^*$, using the same argument as in the proof of \cref{C:EquivConvMeasure}, since $\omega$ is bounded in all $L^p$ spaces, including $p = $. It is shown in \cite{KNavier} that condition $(B)$ holds, from which $(A)$ and $(A')$ follow immediately. Condition $(D)$ is weaker than $(C^N)$ and condition $(F_2)$ is weaker than conditions $(E_2^N)$, so it remains only to show that $(C^N)$ and $(E_2^N)$ hold. We show this by modifying slightly the argument in the proof of \cref{T:VVEquiv} given in \cite{K2008VVV}. \noindent $\mathbf{(A') \implies (C^N)}$: Assume that ($A'$) holds and let $M$ be in $(H^1(\Omega))^{d \times d}$. Then \begin{align*} (\ensuremath{\nabla} u, M) &= - (u, \dv M) + \int_\Gamma (M \cdot \mathbf{n}) \cdot u \\ &\to -(\overline{u}, \dv M) + \int_\Gamma (M \cdot \mathbf{n}) \cdot \overline{u} \text{ in } L^\ensuremath{\infty}([0, T]). \end{align*} The convergence follows from condition $(A')$ and \refE{VelocityGammaConvergence}. But, \begin{align*} -(\overline{u}, \dv M) = (\ensuremath{\nabla} \overline{u}, M) - \int_\Gamma (M \cdot \mathbf{n}) \cdot \overline{u}, \end{align*} giving ($C^N$). \noindent $\mathbf{(A') \implies (E_2^N)}$: Assume that ($A'$) holds and let $f$ be in $H^1(\Omega)$. Then \begin{align*} (\omega, f) &= - (\dv u^\perp, f) = (u^\perp, \ensuremath{\nabla} f) - \int_\Gamma (u^\perp \cdot \mathbf{n}) f \\ &= - (u, \ensuremath{\nabla}^\perp f) + \int_\Gamma (u \cdot \BoldTau) f \\ &\to -(\overline{u}, \ensuremath{\nabla}^\perp f) + \int_\Gamma (\overline{u} \cdot \BoldTau) f \text{ in } L^\ensuremath{\infty}([0, T]) \end{align*} where $u^\perp = -\innp{u^2, u^1}$ and we used the identity $\omega(u) = - \dv u^\perp$ and \refE{VelocityGammaConvergence}. But, \begin{align*} -(\overline{u}, &\ensuremath{\nabla}^\perp f) = (\overline{u}^\perp, \ensuremath{\nabla} f) = - (\dv \overline{u}^\perp, f) + \int_\Gamma (\overline{u}^\perp \cdot \mathbf{n}) f \\ &= - (\dv \overline{u}^\perp, f) - \int_\Gamma (\overline{u} \cdot \BoldTau) f = (\overline{\omega}, f) - \int_\Gamma (\overline{u} \cdot \BoldTau) f, \end{align*} giving ($E_2^N$). \end{proof} \begin{remark} If one could show that \refE{VelocityGammaConvergence} holds in dimension three then \refT{VorticityConvergenceNavier} would hold, with convergences in $(H^1(\Omega))^*$, in dimension three as well for initial velocities in $H^{5/2}(\Omega)$. This is because by \cite{IP2006} the vanishing viscosity limit holds for such initial velocities, and the argument in the proof of \refT{VorticityConvergenceNavier} would then carry over to three dimensions by making adaptations similar to those we made to the 2D arguments in \cite{K2008VVV}. Note that \refE{VelocityGammaConvergence} would follow, just as in 2D, from a uniform (in $\nu$) bound on the $L^p$-norm of the vorticity for some $p \ge 2$ if that could be shown to hold, though that seems unlikely. \end{remark} } { } \Ignore{ \section{High friction limit} \noindent Assume that $\overline{u}$ is a vector field lying in $L^\ensuremath{\infty}([0, T]; H)$ and let $u = u^\ensuremath{\alpha}$ be a vector field in $L^\ensuremath{\infty}([0, T]; H \cap H^1(\Omega)$ parameterized by $\ensuremath{\alpha}$, where $\ensuremath{\alpha} \to \ensuremath{\infty}$. This is the scenario that occurs in the high friction limit [\textbf{add references}], where $\overline{u}$ (which lies in $L^\ensuremath{\infty}([0, T]; V) \subseteq L^\ensuremath{\infty}([0, T]; H)$), a subject that we return to briefly at the end of this section. Define the conditions \begin{align*} (A_\ensuremath{\alpha}) & \qquad u \to \overline{u} \text{ weakly in } H \text{ uniformly on } [0, T], \\ (A'_\ensuremath{\alpha}) & \qquad u \to \overline{u} \text{ weakly in } (L^2(\Omega))^d \text{ uniformly on } [0, T], \\ (B_\ensuremath{\alpha}) & \qquad u \to \overline{u} \text{ in } L^\ensuremath{\infty}([0, T]; H), \\ (C_\ensuremath{\alpha}) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} \overline{u} \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (D_\ensuremath{\alpha}) & \qquad \ensuremath{\nabla} u \to \ensuremath{\nabla} \overline{u} \text{ in } (H^{-1}(\Omega))^{d \times d} \text{ uniformly on } [0, T], \\ (E_\ensuremath{\alpha}) & \qquad \omega \to \omega(\overline{u}) \text{ in } ((H^1(\Omega))^{d \times d})^* \text{ uniformly on } [0, T], \\ (E_{2, \ensuremath{\alpha}}) & \qquad \omega \to \omega(\overline{u}) \text{ in } (H^1(\Omega))^* \text{ uniformly on } [0, T], \\ (F_{2, \ensuremath{\alpha}}) & \qquad \omega \to \omega(\overline{u}) \text{ in } H^{-1}(\Omega) \text{ uniformly on } [0, T], \end{align*} we have the following theorem: \begin{theorem}\label{T:MainResultal} Assume that $u \to \overline{u}$ in $L^\ensuremath{\infty}([0, t]; L^2(\Gamma))$. Conditions ($A_\ensuremath{\alpha}$), ($A'_\ensuremath{\alpha}$), ($C_\ensuremath{\alpha}$), ($D_\ensuremath{\alpha}$), and ($E_\ensuremath{\alpha}$) are equivalent. In two dimensions, conditions ($E_{2, \ensuremath{\alpha}}$) and ($F_{2, \ensuremath{\alpha}}$) are equivalent to the other conditions when $\Omega$ is simply connected. Also, $(B_\ensuremath{\alpha})$ implies all of the other conditions. Finally, the same equivalences hold if we replace each convergence above with the convergence of a subsequence. \end{theorem} \begin{proof} $\mathbf{(A) \iff (A')}$: Let $v$ be in $(L^2(\Omega))^d$. By Lemma 7.3 of \cite{K2008VVV}, $v = w + \ensuremath{\nabla} p$, where $w$ is in $H$ and $p$ is in $H^1(\Omega)$. Then assuming $(A)$ holds, \begin{align*} (u(t), v) & = (u(t), w) \to (\overline{u}(t), w) = (\overline{u}(t), v) \end{align*} uniformly over $t$ in $[0, T]$, so $(A')$ holds. The converse is immediate. \noindent $\mathbf{(B) \implies (A)}$: This implication is immediate. \noindent $\mathbf{(A') \implies (C)}$: Assume that ($A'$) holds and let $M$ be in $(H^1(\Omega))^{d \times d}$. Then \begin{align*} (\ensuremath{\nabla} &u(t), M) = - (u(t), \dv M) + \int_\Gamma (M \cdot \mathbf{n}) u(t) \\ &\to -(\overline{u}(t), \dv M) + \int_\Gamma (M \cdot \mathbf{n}) \overline{u} (t) = (\ensuremath{\nabla} \overline{u}(t), M) \text{ in } L^\ensuremath{\infty}([0, T]). \end{align*} But, \begin{align*} -(\overline{u}(t), \dv M) = (\ensuremath{\nabla} \overline{u}(t), M) - \int_\Gamma (M \cdot \mathbf{n}) \cdot \overline{u}, \end{align*} giving ($C$). \noindent $\mathbf{(C) \implies (D)}$: This follows simply because $H^1_0(\Omega) \subseteq H^1(\Omega)$. \noindent $\mathbf{(D) \implies (A)}$: Assume ($D$) holds, and let $v$ be in $H$. Then $v = \dv M$ for some $M$ in $(H^1_0(\Omega))^{d \times d}$ by Corollary 7.5 of \cite{K2008VVV}, so \begin{align*} (u&(t), v) = (u(t), \dv M) = -(\ensuremath{\nabla} u(t), M) + \int_\Gamma (M \cdot \mathbf{n}) \cdot u(t) \\ & \to -(\ensuremath{\nabla} \overline{u}(t), M) + \int_\Gamma (M \cdot \mathbf{n}) \cdot \overline{u}(t) = (\overline{u}(t), \dv M) = (\overline{u}(t), v) \end{align*} uniformly over $[0, T]$. from which ($A$) follows. Now assume that $d = 2$. \noindent $\mathbf{(A') \implies (E_2)}$: Assume that ($A'$) holds and let $f$ be in $H^1(\Omega)$. Then \begin{align*} (\omega(t), f&) = - (\dv u^\perp(t), f) = (u^\perp(t), \ensuremath{\nabla} f) - \int_\Gamma (u^\perp \cdot \mathbf{n}) f \\ &= - (u(t), \ensuremath{\nabla}^\perp f) - \int_\Gamma (u^\perp \cdot \mathbf{n}) f \to -(\overline{u}(t), \ensuremath{\nabla}^\perp f) - \int_\Gamma (\overline{u}^\perp \cdot \mathbf{n}) f \\ &= (\overline{u}^\perp(t), \ensuremath{\nabla} f) - \int_\Gamma (\overline{u}^\perp \cdot \mathbf{n}) f = - (\dv \overline{u}^\perp(t), f) = (\overline{\omega}(t), f) \end{align*} in $L^\ensuremath{\infty}([0, T])$, giving ($E_2$). Here we used $u^\perp = -\innp{u^2, u^1}$ the identity, $\omega(u) = - \dv u^\perp$, and the fact that $\ensuremath{\nabla}^\perp f$ lies in $H$. \noindent $\mathbf{(E_2) \implies (F_2)}$: Follows for the same reason that $(C) \implies (D)$. \noindent $\mathbf{(F_2) \implies (A)}$: Assume ($F_2$) holds, and let $v$ be in $H$. Then $v = \ensuremath{\nabla}^\perp f$ for some $f$ in $H^1_0(\Omega)$ ($f$ is called the stream function for $v$), and \begin{align*} (u(t), &v) = (u(t), \ensuremath{\nabla}^\perp f) = - (u^\perp(t), \ensuremath{\nabla} f) = (\dv u^\perp(t), f) - \int_\Gamma (u^\perp(t) \cdot \mathbf{n}) f \\ &= - (\omega(t), f) - \int_\Gamma (u^\perp(t) \cdot \mathbf{n}) f \to - (\overline{\omega}(t), f) - \int_\Gamma (\overline{u}^\perp(t) \cdot \mathbf{n}) f \\ &= (\dv \overline{u}^\perp(t), f) - \int_\Gamma (u^\perp(t) \cdot \mathbf{n}) f = - (\overline{u}^\perp(t), \ensuremath{\nabla} f) = (\overline{u}(t), \ensuremath{\nabla}^\perp f) \\ &= (\overline{u}(t), v) \end{align*} in $L^\ensuremath{\infty}([0, T])$, which shows that ($A$) holds. What we have shown so far is that ($A$), ($A'$), ($B$), ($C$), and ($D$) are equivalent, as are $(E_2)$ and $(F_2)$ in two dimensions. It remains to show that $(E)$ is equivalent to these conditions as well. We do this by establishing the implications $(C) \implies (E) \implies (A)$. \noindent $\mathbf{(C) \implies (E)}$: Follows directly from the vorticity being the antisymmetric gradient. \noindent $\mathbf{(E) \implies (A)}$: Let $v$ be in $H$ and let $x$ be the vector field in $(H^2(\Omega) \cap H_0^1(\Omega))^d$ solving $\Delta x = v$ on $\Omega$ ($x$ exists and is unique by standard elliptic theory). Then, utilizing Lemma 7.6 of \cite{K2008VVV} twice (and suppressing the explicit dependence of $u$ and $\overline{u}$ on $t$), \begin{align}\label{e:EImpliesAEquality} \begin{split} (u, v) &= (u, \Delta x) = - (\ensuremath{\nabla} u, \ensuremath{\nabla} x) + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot u \\ &= -2 (\omega(u), \omega(x)) - \int_\Gamma (\ensuremath{\nabla} u x) \cdot \mathbf{n} + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot u \\ &= -2 (\omega(u), \omega(x)) + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot u \\ &\to -2(\omega(\overline{u}), \omega(x)) + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot \overline{u} \\ &= -(\ensuremath{\nabla} \overline{u}, \ensuremath{\nabla} x) + \int_\Gamma (\ensuremath{\nabla} \overline{u} x) \cdot \mathbf{n} + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot \overline{u} \\ &= -(\ensuremath{\nabla} \overline{u}, \ensuremath{\nabla} x) + \int_\Gamma (\ensuremath{\nabla} x \cdot \mathbf{n}) \cdot \overline{u} = (\overline{u}, \Delta x) = (\overline{u}, v), \end{split} \end{align} giving $(A)$. \end{proof} In the case of the high friction limit, at least in 2D, $(B_\ensuremath{\alpha})$ holds so all of the conditions hold. This means that the vorticities and gradients converge weakly in the sense of the conditions $(C_\ensuremath{\alpha})$ through $(F_{2, \ensuremath{\alpha}})$---convergence that does not include a vortex sheet on the boundary. } \addtocontents{toc}{\protect } \appendix \section{Some Lemmas} \noindent \refC{TraceCor}, which we used in the proof of \refT{VorticityNotBounded}, follows from \refL{Trace}. \begin{lemma}[Trace lemma]\label{L:Trace} Let $p \in (1, \ensuremath{\infty})$ and $q \in [1, \ensuremath{\infty}]$ be chosen arbitrarily, and let $q'$ be \Holder conjugate to $q$. There exists a constant $C = C(\Omega)$ such that for all $f \in W^{1, p}(\Omega)$, \begin{align*} \norm{f}_{L^p(\Gamma)} \le C \norm{f}_{L^{(p - 1) q}(\Omega)} ^{1 - \frac{1}{p}} \norm{f}_{W^{1, q'}(\Omega)} ^{\frac{1}{p}}. \end{align*} If $f \in W^{1, p}(\Omega)$ has mean zero or $f \in W^{1, p}_0(\Omega)$ then \begin{align*} \norm{f}_{L^p(\Gamma)} \le C \norm{f}_{L^{(p - 1) q}(\Omega)} ^{1 - \frac{1}{p}} \norm{\ensuremath{\nabla} f}_{L^{q'}(\Omega)} ^{\frac{1}{p}}. \end{align*} \end{lemma} \begin{proof} We prove this for $f \in C^\ensuremath{\infty}(\Omega)$, the result following by the density of $C^\ensuremath{\infty}(\Omega)$ in $W^{1, p}(\Omega)$. We also prove it explicitly in two dimensions, though the proof extends easily to any dimension greater than two. Let $\Sigma$ be a tubular neighborhood of $\Gamma$ of uniform width $\delta$, where $\delta$ is half of the maximum possible width. Place coordinates $(s, r)$ on $\Sigma$ where $s$ is arc length along $\Gamma$ and $r$ is the distance of a point in $\Sigma$ from $\Gamma$, with negative distances being inside of $\Omega$. Then $r$ ranges from $-\delta$ to $\delta$, with points $(s,0)$ lying on $\Gamma$. Also, because $\Sigma$ is only half the maximum possible width, $\abs{J}$ is bounded from below, where \begin{align*} J = \det \frac{\ensuremath{\partial}(x, y)}{\ensuremath{\partial} (s, r)} \end{align*} is the Jacobian of the transformation from $(x, y)$ coordinates to $(s, r)$ coordinates. Let $\varphi \in C^\ensuremath{\infty}(\Omega)$ equal 1 on $\Gamma$ and equal 0 on $\Omega \setminus \Sigma$. Then if $a$ is the arc length of $\Gamma$, \begingroup \allowdisplaybreaks \begin{align*} \norm{f}_{L^p(\Gamma)}^p &= \int_0^a \int_{-\delta}^0 \pdx{}{r} \brac{(\varphi f)(s, r)}^p \, dr \, ds \\ &\le \int_0^a \int_{-\delta}^0 \abs{\pdx{}{r} \brac{(\varphi f)(s, r)}^p} \, dr \, ds \\ &\le \int_0^a \int_{-\delta}^0 \abs{\ensuremath{\nabla} \brac{(\varphi f)(s, r)}^p} \, dr \, ds \\ &= \pr{\inf_{\supp \varphi} \abs{J}}^{-1} \int_0^a \int_{-\delta}^0 \abs{\ensuremath{\nabla} \brac{(\varphi f)(s, r)}^p} \inf_{\supp \varphi} \abs{J} \, dr \, ds \\ &\le \pr{\inf_{\supp \varphi} \abs{J}}^{-1} \int_0^a \int_{-\delta}^0 \abs{\ensuremath{\nabla} \brac{(\varphi f)(s, r)}^p} \abs{J} \, dr \, ds \\ &= C \int_{\Sigma \cap \Omega} \abs{\ensuremath{\nabla} \brac{(\varphi f)(x, y)}^p} \, dx \, dy \\ &\le C \norm{\ensuremath{\nabla} \brac{\varphi f}^p}_{L^1(\Omega)} \\ &= C p \norm{(\varphi f)^{p - 1} \ensuremath{\nabla} \brac{\varphi f}}_{L^1(\Omega)} \\ &\le C p \norm{(\varphi f)^{p - 1}}_{L^q} \norm{\ensuremath{\nabla} \brac{\varphi f}}_{L^{q'}(\Omega)} \\ &= C p \brac{\int_{\Omega}{(\varphi f)^{{(p - 1)} q}}} ^{\frac{1}{q}} \norm{\ensuremath{\nabla} \brac{\varphi f}}_{L^{q'}(\Omega)} \\ &= C p \norm{\varphi f}_{L^{(p - 1) q}(\Omega)} ^{p - 1} \norm{\varphi \ensuremath{\nabla} f + f \ensuremath{\nabla} \varphi} _{L^{q'}(\Omega)} \\ &\le C p \norm{f}_{L^{(p - 1) q}(\Omega)} ^{p - 1} \norm{f} _{W^{1, q'}(\Omega)}. \end{align*} \endgroup The first inequality then follows from raising both sides to the $\frac{1}{p}$ power and using $p^{1/p} \le e^{1/e}$. The second inequality follows from Poincare's inequality. \end{proof} \begin{remark} The trace inequality in \refL{Trace} is a folklore result, most commonly referenced in the special case where $p = q = q' = 2$. We proved it for completeness, since we could not find a proof (or even clear statement) in the literature. We also note that a simple, but incorrect, proof of it (for $p = q = q' = 2$) is to apply the \textit{invalid} trace inequality from $H^{\frac{1}{2}}(\Omega)$ to $L^2(\Gamma)$ then use Sobolev interpolation. \end{remark} Note that in \cref{L:Trace} it could be that $(p - 1) q \in (0, 1)$, though in our application of it in \cref{S:LpNormsBlowUp}, via \cref{C:TraceCor}, we have $(p - 1) q = 2$. Also, examining the last step in the proof, we see that for $p = 1$ the lemma reduces to $\norm{f}_{L^1(\Gamma)} \le C \norm{f}_{W^{1, q'}(\Omega)}$, which is not useful. \begin{cor}\label{C:TraceCor} Let $p, q, q'$ be as in \cref{L:Trace}. For any $v \in H$, \begin{align*} \norm{v}_{L^p(\Gamma)} \le C \norm{v}_{L^{(p - 1) q}(\Omega)} ^{1 - \frac{1}{p}} \norm{\ensuremath{\nabla} v}_{L^{q'}(\Omega)} ^{\frac{1}{p}} \end{align*} and for any $v \in V \cap H^2(\Omega)$, \begin{align*} \norm{\curl v}_{L^p(\Gamma)} \le C \norm{\curl v}_{L^{(p - 1) q}(\Omega)} ^{1 - \frac{1}{p}} \norm{\ensuremath{\nabla} \curl v}_{L^{q'}(\Omega)} ^{\frac{1}{p}}. \end{align*} \end{cor} \begin{proof} If $v \in H$, then \begin{align*} \int_\Omega v^i & = \int_\Omega v \cdot \ensuremath{\nabla} x_i = - \int_\Omega \dv v \, x_i + \int_\Gamma (v \cdot \bm{n}) x_i = 0. \end{align*} If $v \in V$ then \begin{align*} \int_\Omega \curl v &= - \int_\Omega \dv v^\perp = - \int_{\ensuremath{\partial} \Omega} v^\perp \cdot \bm{n} = 0. \end{align*} Thus, \refL{Trace} can be applied to $v_1, v_2$, and $\curl v$, giving the result. \end{proof} } { } \end{document}

\begin{document} \begin{abstract} We propose the notion of GAS numerical semigroup which generalizes both almost symmetric and 2-AGL numerical semigroups. Moreover, we introduce the concept of almost canonical ideal which generalizes the notion of canonical ideal in the same way almost symmetric numerical semigroups generalize symmetric ones. We prove that a numerical semigroup with maximal ideal $M$ and multiplicity $e$ is GAS if and only if $M-e$ is an almost canonical ideal of $M-M$. This generalizes a result of Barucci about almost symmetric semigroups and a theorem of Chau, Goto, Kumashiro, and Matsuoka about 2-AGL semigroups. We also study the transfer of the GAS property from a numerical semigroup to its gluing, numerical duplication and dilatation. \end{abstract} \keywords{Almost symmetric numerical semigroup, almost Gorenstein ring, 2-AGL semigroup, 2-AGL ring, canonical ideal.} \title{Almost canonical ideals and GAS numerical semigroups} \section*{Introduction} The notion of Gorenstein ring turned out to have great importance in commutative algebra, algebraic geometry and other mathematics areas and in the last decades many researchers have developed generalizations of this concept obtaining rings with similar properties in certain respects. With this aim, in 1997 Barucci and Fr\"oberg \cite{BF} introduced the notion of almost Gorenstein ring, inspired by numerical semigroup theory. We recall that a numerical semigroup $S$ is simply an additive submonoid of the set of the natural numbers $\mathbb{N}$ with finite complement in $\mathbb{N}$. The simplest way to relate it to ring theory is by associating with $S$ the ring $k[[S]]=k[[t^s \mid s \in S]]$, where $k$ is a field and $t$ is an indeterminate. Actually it is possible to associate a numerical semigroup $v(R)$ with every one-dimensional analytically irreducible ring $R$. In this case a celebrated result of Kunz \cite{K} ensures that $R$ is Gorenstein if and only if $v(R)$ is a symmetric semigroup, see also \cite[Theorem 4.4.8]{BH} for a proof in the particular case of $k[[S]]$. In \cite{BF} the notions of almost symmetric numerical semigroup and almost Gorenstein ring are introduced, where the latter is limited to analytically unramified rings. It turns out that $k[[S]]$ is almost Gorenstein if and only if $S$ is almost symmetric. More recently this notion has been generalized in the case of one-dimensional local ring \cite{GMP} and in higher dimension \cite{GTT}. Moreover, in \cite{CGKM} it is introduced the notion of $n$-AGL ring in order to stratify the Cohen-Macaulay rings. Indeed a ring is almost Gorenstein if and only if it is either $1$-AGL or $0$-AGL, with $0$-AGL equivalent to be Gorenstein. In this respect $2$-AGL rings are near to be almost Gorenstein and for this reason their properties have been deepened in \cite{CGKM,GIT}. In \cite{CGKM} it is also studied the numerical semigroup case, where $2$-AGL numerical semigroups are close to be almost symmetric. In this paper we introduce the class of {\em Generalized Almost Symmetric numerical semigroups}, briefly GAS numerical semigroups, that includes symmetric, almost symmetric and 2-AGL numerical semigroups, but not 3-AGL. Moreover, if $S$ has maximal embedding dimension and it is GAS, then it is either almost symmetric or 2-AGL. Our original motivation to introduce this class is a result on 2-AGL numerical semigroups that partially generalize a property of almost symmetric semigroups. More precisely, let $S$ be a numerical semigroup with multiplicity $e$ and let $M$ be its maximal ideal. In \cite[Corollary 8]{BF} it is proved that $M-M$ is symmetric if and only if $S$ is almost symmetric with maximal embedding dimension. If we do not assume that $S$ has maximal embedding dimension, it holds that $S$ is almost symmetric if and only if $M-e$ is a canonical ideal of $M-M$ (indeed $S$ has maximal embedding dimension exactly when $M-e=M-M$, see \cite[Theorem 5.2]{B}). In \cite[Corollary 5.4]{CGKM} it is shown that $S$ is 2-AGL if and only if $M-M$ is almost symmetric and not symmetric, provided that $S$ has maximal embedding dimension. Hence, it is natural to investigate what happens to $M-M$, for a 2-AGL semigroup, if we do not make any assumptions on its embedding dimension. It turns out that $M-e$ is an ideal of $M-M$ that satisfies some equivalent conditions, that are the analogue for ideals to the defining conditions of almost symmetric semigroup (cf. Definition 2.1 and Proposition \ref{almost canonical ideal}); for this reason we called the ideals in this class \emph{almost canonical ideals}. However the converse is not true: there exist numerical semigroups $S$ such that $M-e$ is an almost canonical ideal of $M-M$, but that are not 2-AGL. This fact lead us to look for those numerical semigroup satisfying this property, and we found that these semigroups naturally generalize 2-AGL semigroups (this is evident if we look at $2K\setminus K$, where $K$ is the canonical ideal of $S$, cf. Proposition 3.1 and Definition 3.2); moreover, as we said above this class coincides with the union of 2-AGL and almost symmetric semigroups, if we assume maximal embedding dimension; hence we called them Generalized Almost Symmetric (briefly GAS). It turns out that GAS semigroups are interesting under many aspects; for example, if $S$ is GAS, it is possible to control both the semigroup generated by its canonical ideal (that plays a fundamental role in \cite{CGKM}; cf. Theorem \ref{Livelli più alti}) and its pseudo-Frobenius numbers (cf. Proposition \ref{PF GAS}). Hence, in this paper, after recalling the basic definitions and notations, we introduce, in Section 2, the concept of almost canonical ideal. We show under which respect they are a generalization of canonical ideals and we notice that, similarly to the canonical case, a numerical semigroup $S$ is almost symmetric if and only if it is an almost canonical ideal of itself. Moreover, we prove several equivalent conditions for a semigroup ideal to be almost canonical (cf. Proposition \ref{almost canonical ideal}) and we show how to find all the almost canonical ideals of a numerical semigroup and to count them (Corollary \ref{Number of almost canonical ideals}). In Section 3 we develop the theory of GAS semigroups proving many equivalent conditions (see Proposition \ref{Characterizations GAS}), exploring their properties (cf. Theorem \ref{Livelli più alti} and Proposition \ref{PF GAS}) and relating them with other classes of numerical semigroups that have been recently introduced to generalize almost symmetric semigroups. The main result is Theorem \ref{T. Almost Canonical ideal of M-M}, where it is proved that $S$ is GAS if and only if $M-e$ is an almost canonical ideal of $M-M$. Finally in Section 4 we study the transfer of the GAS property from $S$ to some numerical semigroup constructions: gluing in Theorem \ref{gluing}, numerical duplication in Theorem \ref{Numerical duplication S-<K>} and dilatation in Proposition \ref{dilatation}. Several computations are performed by using the GAP system \cite{GAP} and, in particular, the NumericalSgps package \cite{DGM}. \section{Notation and basic definitions} A numerical semigroup $S$ is a submonoid of the natural numbers $\mathbb{N}$ such that $|\mathbb{N} \setminus S| < \infty$. Therefore, there exists the maximum of $\mathbb{N} \setminus S$ that is said to be the Frobenius number of $S$ and it is denoted by $\F(S)$. Given $s_1, \dots, s_{\nu} \in \mathbb{N}$ we set $\langle s_1, \dots, s_{\nu} \rangle=\{\lambda_1 s_1 + \dots + \lambda_{\nu} s_{\nu} \mid \lambda_1, \dots, \lambda_{\nu} \in \mathbb{N} \}$ which is a numerical semigroup if and only if $\gcd(s_1, \dots, s_{\nu})=1$. We say that $s_1, \dots, s_{\nu}$ are minimal generators of $\langle s_1, \dots, s_{\nu} \rangle$ if it is not possible to delete one of them obtaining the same semigroup. It is well-known that a numerical semigroup have a unique system of minimal generators, which is finite, and its cardinality is called embedding dimension of $S$. The minimum non-zero element of $S$ is said to be the multiplicity of $S$ and we denote it by $e$. It is always greater than or equal to the embedding dimension of $S$ and we say that $S$ has maximal embedding dimension if they are equal. Unless otherwise specified, we assume that $S \neq \mathbb{N}$. A set $I \subseteq \mathbb{Z}$ is said to be a relative ideal of $S$ if $I+S\subseteq I$ and there exists $z \in S$ such that $z+I \subseteq S$. If it is possible to chose $z=0$, i.e. $I \subseteq S$, we simply say that $I$ is an ideal of $S$. Two very important relative ideals are $M(S)=S\setminus \{0\}$, which is an ideal and it is called the maximal ideal of $S$, and $K(S)=\{x \in \mathbb{N} \mid \F(S)-x \notin S\}$. We refer to the latter as the standard canonical ideal of $S$ and we say that a relative ideal $I$ of $S$ is canonical if $I=x+K(S)$ for some $x \in \mathbb{Z}$. If the semigroup is clear from the context, we write $M$ and $K$ in place of $M(S)$ and $K(S)$. Given two relative ideals $I$ and $J$ of $S$, we set $I-J = \{x \in \mathbb{Z} \mid x+J \subseteq I\}$ which is a relative ideal of $S$. For every relative ideal $I$ it holds that $K-(K-I)=I$, in particular $K-(K-S)=S$. Moreover, an element $x$ is in $I$ if and only if $\F(S)-x \notin K-I$, see \cite[Hilfssatz 5]{J}. As a consequence we get that the cardinalities of $I$ and $K-I$ are equal. Also, if $I \subseteq J$ are two relative ideals, then $|J \setminus I|=|(K-I)\setminus (K-J)|$. We now collect some important definitions that we are going to generalize in the next section. \begin{definition} \label{Basic definitions} \rm Let $S$ be a numerical semigroup. \begin{enumerate} \item The {\it pseudo-Frobenius numbers} of $S$ are the elements of the set $\PF(S)=(S-M)\setminus S$. \item The {\it type} of $S$ is $t(S)=|\PF(S)|$. \item $S$ is {\it symmetric} if and only if $S=K$. \item $S$ is {\it almost symmetric} if and only if $S-M=K \cup \{\F(S)\}$. \end{enumerate} \end{definition} We note that $M-M=S \cup \PF(S)$. Given $0 \leq i \leq e-1$, let $\omega_i$ be the smallest element of $S$ that is congruent to $i$ modulo $e$. A fundamental tool in numerical semigroup theory is the so-called Ap\'ery set of $S$ that is defined as $\Ap(S)=\{\omega_0=0, \omega_1, \dots, \omega_{e-1}\}$. In $\Ap(S)$ we define the partial ordering $x \leq_S y$ if and only if $y= x+s$ for some $s \in S$ and we denote the maximal elements of $\Ap(S)$ with respect to $\leq_S$ by ${\rm Max}_{\leq_S}(\Ap(S))$. With this notation $\PF(S)=\{\omega -e \mid \omega \in {\rm Max}_{\leq S}(\Ap(S)) \}$, see \cite[Proposition 2.20]{RG}. We also recall that $S$ is symmetric if and only if $t(S)=1$, that is also equivalent to say that $k[[S]]$ has type $1$ for every field $k$, i.e. $k[[S]]$ is Gorenstein. Also for almost symmetric semigroups many useful characterizations are known, for instance it is easy to see that our definition is equivalent to $M+K \subseteq M$, but see also \cite[Theorem 2.4]{N} for another useful characterization related to the Ap\'ery set of $S$ and its pseudo-Frobenius numbers. \section{Almost canonical ideals of a numerical semigroup} If $I$ is a relative ideal of $S$, the set $\mathbb{Z}\setminus I$ has a maximum that we denote by $\F(I)$. We set $\widetilde{I}=I+(\F(S)-\F(I))$, that is the unique relative ideal $J$ isomorphic to $I$ for which $\F(S)=\F(J)$, and we note that $\widetilde{I} \subseteq K \subseteq \mathbb{N}$ for every $I$. The following is a generalization of Definition \ref{Basic definitions}. \begin{definition} \rm Let $I$ be a relative ideal of a numerical semigroup $S$. \begin{enumerate} \item The {\it pseudo-Frobenius numbers} of $I$ are the elements of the set $\PF(I)=(I-M)\setminus I$. \item The {\it type} of $I$ is $t(I)=|\PF(I)|$. \item $I$ is {\it canonical} if and only if $\widetilde{I}=K$. \item $I$ is {\it almost canonical} if and only if $\widetilde{I}-M=K \cup \{\F(S)\}$. \end{enumerate} \end{definition} \begin{remark} \rm \label{Rem as} {\bf 1.} $S$ is an almost canonical ideal of itself if and only if it is an almost symmetric semigroup. \\ {\bf 2.} $M$ is an almost canonical ideal of $S$ if and only if $S$ is an almost symmetric semigroup. Indeed, $M-M=S-M$, since $S \neq \mathbb{N}$. Moreover, $t(M)=t(S)+1$. \\ {\bf 3.} It holds that $K-M=K \cup \{\F(S)\}$. One containment is trivial, so let $x \in ((K-M) \setminus (K \cup \{\F(S)\}))$. Then $0 \neq \F(S)-x \in S$ and, thus, $\F(S)=(\F(S)-x)+x \in M+ (K-M) \subseteq K$ yields a contradiction. In particular, a canonical ideal is almost canonical. \\ {\bf 4.} Since $\F(S)=\F(\widetilde{I})$, it is always in $\widetilde{I}-M$. Moreover, we claim that $(\widetilde{I}-M) \subseteq K \cup \{\F(S)\}$. Indeed, if $x \in (\widetilde{I}-M)\setminus\{\F(S)\}$ and $x \notin K$, then $\F(S)-x \in M$ and $\F(\widetilde{I})=\F(S)=(\F(S)-x)+x \in \widetilde{I}$. In addition, $\widetilde{I}$ is always contained in $\widetilde{I}-M$ because it is a relative ideal of $S$. Hence, $I$ is an almost canonical ideal of $S$ if and only if $K \setminus \widetilde{I} \subseteq (\widetilde{I}-M)$. \end{remark} Given a relative ideal $I$ of $S$, the Ap\'ery set of $I$ is $\Ap(I)=\{i \in I \mid i-e \notin I\}$. As in the semigroup case, in $\Ap(I)$ we define the partial ordering $x \leq_S y$ if and only if $y= x+s$ for some $s \in S$ and we denote by ${\rm Max}_{\leq_S}(\Ap(I))$ the maximal elements of $\Ap(I)$ with respect to $\leq_S$. \begin{proposition} Let $I$ be a relative ideal of $S$. The following statements hold: \begin{enumerate} \item $\PF(I)= \{i-e \mid i \in {\rm Max}_{\leq_S}(\Ap(I)) \}$; \item $I$ is canonical if and only if its type is $1$. \end{enumerate} \end{proposition} \begin{proof} (1) An integer $i \in I$ is in ${\rm Max}_{\leq_S}(\Ap(I))$ if and only if $i-e \notin I$ and $s+i \notin \Ap(I)$, i.e. $s+i-e \in I$, for every $s \in M$. This is equivalent to say that $i-e \in (I-M)\setminus I=\PF(I)$. \\ (2) Since $\F(S)\in \widetilde I-M$, we have $t(\widetilde I)=t(I)=1$ if and only if $\widetilde{I}-M=\widetilde I \cup \{\F(S)\}$. Therefore, a canonical ideal has type 1 by Remark \ref{Rem as}.3. Conversely, assume that $t(\widetilde{I})=1$ and let $x \notin \widetilde{I}$. Since $\widetilde{I} \subseteq K$, we only need to prove that $x \notin K$. By (1), there is a unique maximal element in $\Ap(\widetilde{I})$ with respect to $\leq_S$ and, clearly, it is $\F(S)+e$. Let $0 \neq \lambda \in \mathbb{N}$ be such that $x+ \lambda e \in \Ap(\widetilde{I})$. Then, there exists $y \in S$ such that $x+\lambda e + y = \F(S)+e$ and $x=\F(S)-(y+(\lambda-1)e) \notin K$, since $y+(\lambda-1)e \in S$. \end{proof} Let $g(S)=|\mathbb{N}\setminus S|$ denote the genus of $S$ and let $g(I)=|\mathbb{N}\setminus \widetilde{I}|$. We recall that $2g(S) \geq \F(S) + t(S)$ and the equality holds if and only if $S$ is almost symmetric, see, e.g., \cite[Proposition 2.2 and Proposition-Definition 2.3]{N}. \begin{proposition} \label{almost canonical ideal} Let $I$ be a relative ideal of $S$. Then $g(I)+g(S) \geq \F(S) + t(I)$. Moreover, the following conditions are equivalent: \begin{enumerate} \item $I$ is almost canonical; \item $g(I)+g(S)=\F(S)+t(I)$; \item $\widetilde{I}-M=K-M$; \item $K-(M-M) \subseteq \widetilde{I}$; \item If $x \in \PF(I)\setminus \{\F(I)\}$, then $\F(I)-x \in \PF(S)$. \end{enumerate} \end{proposition} \begin{proof} Clearly, $t(I)=t(\widetilde{I})$ and $g(I)-t(\widetilde{I})=|\mathbb{N} \setminus \widetilde{I}|-|(\widetilde{I}-M)\setminus \widetilde{I}|=|\mathbb{N}\setminus (\widetilde{I}-M)|$. Moreover, since $\F(S)+1-g(S)$ is the number of the elements of $S$ smaller than $\F(S)+1$, it holds that $\F(S)-g(S)=|\mathbb{N}\setminus K|-1=|\mathbb{N} \setminus (K \cup {\F(S)})|$. We have $\widetilde{I}-M \subseteq K \cup \{\F(S)\}$ by Remark \ref{Rem as}.4, then $g(I)-t(I) \geq \F(S) -g(S)$ and the equality holds if and only if $\widetilde{I}-M = K \cup \{\F(S)\}$, i.e. $I$ is almost canonical. Hence, (1) $\Leftrightarrow$ (2). \\ (1) $\Leftrightarrow$ (3). We have already proved that $K-M=K \cup \{\F(S)\}$ in Remark \ref{Rem as}.3. \\ (1) $\Rightarrow$ (4). The thesis is equivalent to $M-M \supseteq K-\widetilde{I}$. Let $x \in K-\widetilde{I}$ and assume by contradiction that there exists $m \in M$ such that $x+m \notin M$. Then, $\F(S)-x-m \in K \cup \{\F(S)\}=\widetilde{I}-M$ and, so, $\F(S)-x \in \widetilde{I}$. Since $x \in K-\widetilde{I}$, this implies $\F(S) \in K$, that is a contradiction. \\ (4) $\Rightarrow$ (1). Let $x \in K$. It is enough to prove that $x \in \widetilde{I}-M$. Suppose by contradiction that there exists $m \in M$ such that $x+m \notin \widetilde{I}\supseteq K-(M-M)$. In particular, $x+m\notin K-(M-M)$ and so $\F(S)-(x+m) \in M-M$. This implies $\F(S)-x \in M$, that is a contradiction because $x \in K$. \\ (1) $\Rightarrow$ (5) We notice that $\PF(\widetilde{I})=\{x+\F(S)-\F(I) \mid x \in \PF(I)\}$. Let $x \in \PF(I)\setminus \{\F(I)\}$ and let $y=x+\F(S)-\F(I) \in \PF(\widetilde I) \setminus \{ \F(S)\}$. We first note that $\F(S)-y \notin S$, otherwise $\F(S)=y+(\F(S)-y) \in \widetilde I$. Assume by contradiction that $\F(S)-y \notin \PF(S)$, i.e. there exists $m \in M$ such that $\F(S)-y+m \notin S$. This implies that $y-m \in K \subseteq \widetilde{I}-M$ by (1) and, thus, $y=(y-m)+m \in \widetilde I$ yields a contradiction. Hence, $\F(I)-x=\F(S)-y \in \PF(S)$. \\ (5) $\Rightarrow$ (4) Assume by contradiction that there exists $x \in (K-(M-M))\setminus \widetilde I$. It easily follows from the definition that there is $s \in S$ such that $x+s \in \PF(\widetilde{I})$. Then, $\F(S)-x-s \in \PF(S) \cup \{0\} \subseteq M-M$ by (5) and $\F(S)-s=x +(\F(S)-x-s) \in (K-(M-M)) + (M-M) \subseteq K$ gives a contradiction. \end{proof} \begin{remark} \rm {\bf 1.} In \cite[Theorem 2.4]{N} it is proved that a numerical semigroup $S$ is almost symmetric if and only if $\F(S)-f \in \PF(S)$ for every $f \in \PF(S) \setminus \{\F(S)\}$. Hence, the last condition of Proposition \ref{almost canonical ideal} can be considered a generalization of this result. \\ {\bf 2.} Almost canonical ideals naturally arise characterizing the almost symmetry of the numerical duplication $S \! \Join^b \! I$ of $S$ with respect to the ideal $I$ and $b \in S$, a construction introduced in \cite{DS}. Indeed \cite[Theorem 4.3]{DS} says that $S \! \Join^b \! I$ is almost symmetric if and only if $I$ is almost canonical and $K-\widetilde{I}$ is a numerical semigroup. \\ {\bf 3.} Let $T$ be an almost symmetric numerical semigroup with odd Frobenius number (or, equivalently, odd type). Let $b$ be an odd integer such that $2b \in T$ and set $I=\{x \in \mathbb{Z} \mid 2x+b \in T\}$. Then, \cite[Proposition 3.3]{S} says that $T$ can be realized as a numerical duplication $T=S \! \Join^b \! I$, where $S=T/2=\{y \in \mathbb{Z} \mid 2y \in T\}$, while \cite[Theorem 3.7]{S} implies that $I$ is an almost canonical ideal of $S$. In general this is not true if the Frobenius number of $T$ is even. \end{remark} Since $\F(K-(M-M))= \F(\widetilde{I})=\F(K)$ and $\widetilde{I} \subseteq K$ for every relative ideal $I$, Condition (4) of Proposition \ref{almost canonical ideal} allows to find all the almost canonical ideals of a numerical semigroup. Clearly it is enough to focus on the relative ideals with Frobenius number $\F(S)$. \begin{corollary} \label{Number of almost canonical ideals} Let $S$ be a numerical semigroup with type $t$. If $I$ is almost canonical, then $t(I)\leq t+1$. Moreover, for every integer $i$ such that $1 \leq i \leq t+1$, there are exactly $\binom{t}{i-1}$ almost canonical ideals of $S$ with Frobenius number $\F(S)$ and type $i$. In particular, there are exactly $2^{t}$ almost canonical ideals of $S$ with Frobenius number $\F(S)$. \end{corollary} \begin{proof} Let $C=\{s \in S \mid s>\F(S)\}=K-\mathbb{N}$ be the conductor of $S$ and let $n(S)=|\{s \in S \mid s<\F(S)\}|$. It is straightforward to see that $g(S)+n(S)=\F(S)+1$. If $I$ is almost canonical, Proposition \ref{almost canonical ideal} implies that \begin{align*} t(I)&=g(I)+g(S)-\F(S)\leq |\mathbb{N}\setminus (K-(M-M))|-n(S)+1= \\ &=|(M-M)\setminus (K-\mathbb{N})|-n(S)+1 =|(M-M)\setminus C|-n(S)+1=\\ &=|(M-M)\setminus S|+|S \setminus C|-n(S)+1=t+n(S)-n(S)+1=t+1. \end{align*} By Proposition \ref{almost canonical ideal} an ideal $I$ with $\F(I)=\F(S)$ is almost canonical if and only if $K-(M-M) \subseteq I \subseteq K$ and we notice that $|K \setminus (K-(M-M))|=|(M-M)\setminus S|=t$. Let $A \subseteq (K \setminus (K-(M-M)))$ and consider $I=(K-(M-M)) \cup A$. We claim that $I$ is an ideal of $S$. Indeed, let $x \in A$, $m \in M$ and $y \in (M-M)$. It follows that $m+y \in M$ and, then, $x+m+y \in K$, since $K$ is an ideal. Therefore, $x+m \in K-(M-M)$ and $I$ is an ideal of $S$. Moreover, by \cite[Lemma 4.7]{DS}, $t(I)=|(K-I)\setminus S|+1=|K\setminus I|+1=t+1-|A|$ and the thesis follows, because there are $\binom{t}{i-1}$ subsets of $K\setminus (K-(M-M))$ with cardinality $t+1-i$. \end{proof} If $S$ is a symmetric semigroup, the only almost canonical ideals with Frobenius number $\F(S)$ are $M$ and $S$. In this case $t(M)=t(S)+1=2$. If $S$ is pseudo-symmetric, the four almost canonical ideals with Frobenius number $\F(S)$ are $M$, $S$, $M \cup \{\F(S)/2\}$ and $K$. In this case $t(M)=3$, $t(S)=t(M \cup \{\F(S)/2\})=2$ and $t(K)=1$. \section{GAS numerical semigroups} In \cite{CGKM} it is introduced the notion of $n$-almost Gorenstein local rings, briefly $n$-AGL rings, where $n$ is a non-negative integer. These rings generalize almost Gorenstein ones that are obtained when either $n=0$, in which case the ring is Gorenstein, or $n=1$. In particular, in \cite{CGKM} it is studied the case of the 2-AGL rings, that are closer to be almost Gorenstein, see also \cite{GIT}. Given a numerical semigroup $S$ with standard canonical ideal $K$ we denote by $\langle K \rangle$ the numerical semigroup generated by $K$. Following \cite{CGKM} we say that $S$ is $n$-AGL if $|\langle K \rangle \setminus K|=n$. It follows that $S$ is symmetric if and only if it is 0-AGL, whereas it is almost symmetric and not symmetric if and only if it is 1-AGL. It is easy to see that a numerical semigroup is 2-AGL if and only if $2K=3K$ and $|2K\setminus K|=2$, see \cite[Theorem 1.4]{CGKM} for a proof in a more general context. We now give another easy characterization that will lead us to generalize this class. \begin{proposition} A numerical semigroup $S$ is 2-AGL if and only if $2K=3K$ and $2K \setminus K=\{\F(S)-x, \F(S)\}$ for a minimal generator $x$ of $S$. \end{proposition} \begin{proof} One implication is trivial, so assume that $S$ is 2-AGL. Since $S$ is not symmetric, there exists $k \in \mathbb{N}$ such that $k$ and $\F(S)-k$ are in $K$ and so $\F(S) \in 2K \setminus K$. Let now $a \in (2K \setminus K) \setminus \{\F(S)\}$. Since $a \notin K$, we have $\F(S)-a \in S$. Assume that $\F(S)-a=s_1+s_2$ with $s_1,s_2 \in S \setminus \{0\}$. It follows that $\F(S)-s_1=a+s_2 \in 2K$, since $2K$ is a relative ideal, and by definition $\F(S)-s_1 \notin K$. Therefore, $\{a,\F(S)-s_1,\F(S)\} \subseteq 2K \setminus K$ and this is a contradiction, since $S$ is 2-AGL. Hence, $a=\F(S)-x$, where $x$ is a minimal generator of $S$. \end{proof} In light of the previous proposition we propose the following definition. \begin{definition} \rm We say that $S$ is a {\it generalized almost symmetric} numerical semigroup, briefly {\rm GAS} numerical semigroup, if either $2K=K$ or $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$ for some $r \geq 0$ and some minimal generators $x_1, \dots, x_r$ of $S$ such that $x_i-x_j \notin \PF(S)$ for every $i,j$. \end{definition} The last condition could seem less natural, but these semigroups have a better behaviour. For instance, in Theorem \ref{Livelli più alti} we will see that this condition ensures that every element in $ \langle K \rangle \setminus K$ can be written as $\F(S)-x$ for a minimal generator $x$ of $S$. We recall that $S$ is symmetric if and only if $2K=K$ and it is almost symmetric exactly when $2K \setminus K \subseteq \{\F(S)\}$. \begin{examples} \rm {\bf 1.} Let $S= \langle 9, 24, 39, 43, 77 \rangle$. Then, $\PF(S)=\{58, 73, 92, 107\}$ and $2K \setminus K=\{107-77,107-43,107-39,107-24,107-9,107\}$. Hence, $S$ is a GAS semigroup. \\ {\bf 2.} If $S=\langle 7,9,15 \rangle$, we have $2K=3K$ and $2K \setminus K=\{26-14,26-7,26\}$. Hence, $S$ is 3-AGL but it is not GAS because $14$ is not a minimal generator of $S$. \\ {\bf 3.} Consider the semigroup $S=\langle 8, 11, 14, 15, 17, 18, 20, 21 \rangle$. We have $2K \setminus K=\{13-11,13-8,13\}$, but $S$ is not GAS because $11-8 \in \PF(S)$. In this case $2K=3K$ and thus $S$ is 3-AGL. \end{examples} The last example shows that in a numerical semigroup $S$ with maximal embedding dimension there could be many minimal generators $x$ such that $\F(S) -x \in 2K\setminus K$. This is not the case if we assume that $S$ is GAS. \begin{proposition} \label{MED} If $S$ has maximal embedding dimension $e$ and it is {\rm GAS}, then it is either almost symmetric or {\rm 2-AGL} with $2K\setminus K=\{\F(S)-e,\F(S)\}$. \end{proposition} \begin{proof} Assume that $S$ is not almost symmetric and let $\F(S)-x=k_1+k_2 \in 2K\setminus K$ with $x\neq 0$ and $k_1,k_2 \in K$. Let $x\neq e$ and consider $\F(S)-e=k_1+k_2+x-e$. Since $x-e \leq \F(S)-e < \F(S)$ and $S$ has maximal embedding dimension, $x-e \in \PF(S) \setminus \{\F(S)\} \subseteq K$ and, therefore, $\F(S)-e \in 3K \setminus K$. Moreover, $\F(S)-e$ cannot be in $2K$, because $S$ is GAS and $x-e \in \PF(S)$, then, $k_1+x-e \in 2K \setminus K$. Hence, we have $\F(S)-(\F(S)-k_1-x+e) \in 2K\setminus K$ and, thus, $\F(S)-k_1-x+e$ is a minimal generator of $S$. Since $S$ has maximal embedding dimension, this implies that $\F(S)-k_1-x \in \PF(S)$ and, then, $\F(S)-k_1 \in S$ yields a contradiction, since $k_1 \in K$. This means that $x=e$ and $2K\setminus K=\{\F(S)-e,\F(S)\}$. Suppose by contradiction that $2K \neq 3K$ and let $\F(S)-y \in 3K \setminus 2K$. In particular, $\F(S)-y \notin K$ and, therefore, $y \in S$. If $\F(S)-y =k_1+k_2+ k_3$ with $k_i \in K$ for every $i$, then $k_1+k_2 \in 2K \setminus K$ and, thus, $k_1+k_2=\F(S)-e$. This implies that $\F(S)-e<\F(S)-y$, i.e. $y<e$, that is a contradiction. \end{proof} In particular, we note that in a 2-AGL semigroup with maximal embedding dimension it always holds that $2K \setminus K=\{\F(S)-e, \F(S)\}$. \begin{proposition} \label{Characterizations GAS} Given a numerical semigroup $S$, the following conditions are equivalent: \begin{enumerate} \item $S$ is {\rm GAS}; \item $x-y \notin (M-M)$ for every different $x,y \in M\setminus (S-K)$; \item either $S$ is symmetric or $2M \subseteq S-K \subseteq M$ and $M-M=((S-K)-M) \cup \{0\}$. \end{enumerate} \end{proposition} \begin{proof} If $S$ is symmetric, then $M \subseteq S-K$ and both (1) and (2) are true, so we assume $S \neq K$. \\ $(1) \Rightarrow (2)$ Note that $K-S=K$ and $K-(S-K)=K-((K-K)-K)=K-(K-2K)=2K$. Thus, $x \in S \setminus (S-K)$ if and only if $\F(S)-x \in (K-(S-K))\setminus (K-S)=2K \setminus K$. Hence, if $S$ is GAS, then $x-y \notin S \cup \PF(S)=M-M$ for every $x,y \in M\setminus (S-K)$. \\ $(2) \Rightarrow (1)$ If $x$, $y \in M \setminus (S-K)$, then $\F(S)-x$, $\F(S)-y \in 2K \setminus K$ and $x-y \notin \PF(S)$, since it is not in $M-M$. We only need to show that $x$ is a minimal generator of $S$. If by contradiction $x=s_1+s_2$, with $s_1$, $s_2 \in M$, it follows that also $s_1$ is in $M \setminus (S-K)$. Therefore, $s_2=x-s_1 \in M$ yields a contradiction since $x-s_1 \notin M-M$ by hypothesis. \\ $(2) \Rightarrow (3)$ Since $S$ is not symmetric, $S-K$ is contained in $M$. Moreover, if $2M$ is not in $S-K$, then there exist $m_1, m_2 \in M$ such that $m_1+m_2 \in 2M \setminus (S-K)$. Clearly also $m_1$ is not in $S-K$ and $(m_1+m_2)-m_1=m_2 \in M \subseteq M-M$ yields a contradiction. It always holds that $((S-K)-M) \cup \{0\} \subseteq M-M$, then given $x \in (M-M) \setminus \{0\}$ and $m \in M$, we only need to prove that $x+m \in S-K$. If $m \in M \setminus (S-K)$ and $x+m \notin S-K$, then $(x+m)-m=x \in M-M$ gives a contradiction. If $m \in (S-K) \setminus 2M$ and $k \in K$, then $0 \neq m+k \in S$ and, so, $x+m+k \in M$, that implies $x+m \in S-K$. Finally, if $m \in 2M$, then $x+m \in 2M \subseteq S-K$. \\ $(3) \Rightarrow (2)$ Let $x,y \in M \setminus (S-K)$ with $x \neq y$ and assume by contradiction that $x-y \in (M-M)=((S-K)-M)\cup \{0\}$. By hypothesis $y \in M$, then $x=(x-y)+y \in S-K$ yields a contradiction. \end{proof} In the definition of GAS semigroup we required that in $2K\setminus K$ there are only elements of the type $\F(S)-x$ with $x$ minimal generator of $S$. In general, this does not imply that the elements in $3K\setminus 2K$ are of the same type. For instance, consider $S=\langle 8,12,17,21,26,27,30,31 \rangle$, where $2K \setminus K=\{23-21,23-17,23-12,23-8,23\}$ and $3K\setminus 2K=\{23-20,23-16\}$. However, by Proposition \ref{MED}, this semigroup is not GAS. In fact, this never happens in a GAS semigroup as we are going to show in Theorem \ref{Livelli più alti}. First we need a lemma. \begin{lemma} \label{Lemma livelli più alti} Assume that $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$ with $x_1, \dots, x_r$ minimal generators of $S$. If $\F(S)-x \in nK \setminus (n-1)K$ for some $n>2$ and $x=s_1+s_2$ with $s_1$, $s_2 \in M$, then $\F(S)-s_1 \in (n-1)K$. \end{lemma} \begin{proof} Let $\F(S)-(s_1+s_2)=k_1 + \dots + k_n \in nK \setminus (n-1)K$ with $k_i \in K$ for $1 \leq i \leq n$. Since $\F(S)-(s_1+s_2) \notin (n-1)K$, we have $\F(S) \neq k_1+k_2 \in 2K \setminus K$ and, then, $\F(S)-(k_1+k_2)$ is a minimal generator of $S$. Since $\F(S)-(k_1+k_2)=s_1+s_2+k_3+ \dots + k_{n}$, this implies that $s_1+k_3 + \dots + k_{n}\notin S$, that is $k_1+k_2+s_2 =\F(S)-(s_1+k_3+\dots + k_{n}) \in K$. Therefore, $\F(S)-s_1=(k_1+k_2+s_2)+k_3+\dots+ k_{n} \in (n-1)K$ and the thesis follows. \end{proof} \begin{theorem} \label{Livelli più alti} Let $S$ be a {\rm GAS} numerical semigroup that is not symmetric. Then, $\langle K \rangle \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$ for some minimal generators $x_1, \dots, x_r$ with $r \geq 0$ and $x_i-x_j \notin \PF(S)$ for every $i$ and $j$. \end{theorem} \begin{proof} We first prove that $x_i-x_j \notin \PF(S)$ for every $i$ and $j$ without assuming that $x_i$ and $x_j$ are minimal generators. We can suppose that $x_i=x_1$ and $x_j=x_2$. Let $\F(S)-x_1=k_1+\dots +k_n \in nK \setminus (n-1)K$ with $k_i \in K$ for every $i$ and assume by contradiction that $x_1-x_2 \in \PF(S)$. We note that $\F(S)-x_2=k_1+\dots +k_n + (x_1-x_2)$ and $k_1+(x_1-x_2) \in K$. Indeed, if $\F(S)-k_1 -(x_1-x_2)=s \in S$, then $s \neq 0$ and $\F(S)-k_1=(x_1-x_2)+s \in S$ yields a contradiction. If $k_1+k_2+(x_1-x_2) \notin K$, then it is in $2K \setminus K$ and, since also $k_1+k_2 \in 2K \setminus K$, we get a contradiction because their difference is a pseudo-Frobenius number. Hence, $k_1+k_2+(x_1-x_2) \in K$. We proceed by induction on $n$. If $n=2$, it follows that $\F(S)-x_2=k_1+k_2+(x_1-x_2) \in K$, that is a contradiction. So, let $n \geq 3$ and let $i$ be the minimum index for which $k_1+ \dots + k_i + (x_1-x_2) \notin K$. It follows that $k_1+ \dots + k_i + (x_1-x_2) \in 2K\setminus K$ and, since also $k_1+k_2 \in 2K \setminus K$, this implies that $k_3+ \dots + k_i +(x_1-x_2) \notin \PF(S)$. Moreover, it cannot be in $S$, because it is the difference of two minimal generators, since $S$ is GAS. Therefore, there exists $m \in M$ such that $k_3+ \dots + k_i +(x_1-x_2)+ m \notin S$, that means $\F(S)-(k_3+ \dots + k_i +(x_1-x_2)+m)=k' \in K$. Thus, $\F(S)-((x_1-x_2)+m)=k'+k_3+ \dots + k_i \in jK \setminus K$ for some $1< j < n$. Moreover, $\F(S)-m=k'+k_3+ \dots + k_i +(x_1-x_2)\in \langle K \rangle \setminus K$ and by induction $(x_1-x_2)+m-m \notin \PF(S)$, that is a contradiction. Hence, $x_1-x_2 \notin \PF(S)$. Let now $h \geq 3$. To prove the theorem it is enough to show that, if $\F(S)-x \in hK \setminus (h-1)K$, then $x$ is a minimal generators of $S$. We proceed by induction on $h$. Using the GAS hypothesis, the case $h=3$ is very similar to the general case, so we omit it (the difference is that also $\F(S) \in 2K \setminus K$). Suppose by contradiction that $x=s_1+s_2$ and $\F(S)-(s_1+s_2)=k_1+ \dots +k_h \in hK \setminus (h-1)K$ with $k_1, \dots, k_h \in K$ and $s_1, s_2 \in M$. Clearly, $\F(S)-s_1 \notin K$ and by Lemma \ref{Lemma livelli più alti} we have $\F(S)-s_1 \in (h-1)K$; in particular, $s_1$ is a minimal generator of $S$ by induction. Let $1< i < h$ be such that $\F(S)-s_1 \in iK \setminus (i-1)K$. Since $\F(S)-(s_1+s_2) \notin (h-1)K$, we have $k_1+ \dots + k_i \in iK \setminus (i-1)K$ and, by induction, $\F(S)-(k_1+\dots+k_i)$ is a minimal generator of $S$ and $\F(S)-(k_1+\dots+k_i)-s_1 \notin \PF(S)$ by the first part of the proof. This means that there exists $s \in M$ such that $\F(S)-(k_1+\dots+k_i)-s_1+s \notin S$, i.e. $k_1+\dots+k_i+s_1-s \in K$. This implies that $\F(S)-(s_2+s)=(k_1 + \dots + k_i +s_1-s)+k_{i+1}+\dots+k_h \in (h-i+1)K$ and, since $h-i+1 <h$, the induction hypothesis yields a contradiction because $s_2+s$ is not a minimal generator of $S$. \end{proof} We recall that in an almost symmetric numerical semigroup $\F(S)-f \in \PF(S)$ for every $f \in \PF(S)\setminus \{\F(S)\}$, see \cite[Theorem 2.4]{N}. The following proposition generalizes this fact. \begin{proposition} \label{PF GAS} Let $S$ be a numerical semigroup with $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$, where $x_i$ is a minimal generator of $S$ for every $i$. \begin{enumerate} \item For every $i$, there exist $f_j, f_k \in \PF(S)$ such that $f_j+f_k=\F(S)+x_i$. \item For every $f \in \PF(S)\setminus \{\F(S)\}$, it holds either $\F(S)-f \in \PF(S)$ or $\F(S)-f+x_i \in \PF(S)$ for some $i$. \end{enumerate} \end{proposition} \begin{proof} Let $\F(S)-x_i=k_1+k_2 \in 2K\setminus K$ for some $k_1, k_2 \in K$ and let $s \in M$. Since $x_i+s \in S$, we have $\F(S)-x_i-s \notin K$ and then $\F(S)-x_i-s=k_1+k_2-s \notin 2K$ because $x_i+s$ is not a generator of $S$. In particular, $k_1-s$ and $k_2-s$ are not in $K$. This means that $\F(S)-k_1+s$ and $\F(S)-k_2+s$ are in $S$ and, thus, $\F(S)-k_1, \F(S)-k_2 \in \PF(S)$. Moreover, $\F(S)-k_1+\F(S)-k_2=2\F(S)-(\F(S)-x_i)=\F(S)+x_i$ and (1) holds. Let now $f \in \PF(S) \setminus \{\F(S)\}$ and assume that $\F(S)-f \notin \PF(S)$. Then, there exists $s \in M$ such that $\F(S)-f+s \in \PF(S)$. In particular, $f-s \in K$ and $\F(S)-s=(\F(S)-f)+(f-s) \in 2K \setminus K$; thus, $s$ has to be equal to $x_i$ for some $i$ and $\F(S)-f+x_i \in \PF(S)$. \end{proof} \begin{examples} \rm \label{Examples} {\bf 1.} Let $S=\langle 28,40,63,79,88\rangle$. We have $2K \setminus K=\{281-28,281\}$ and $S$ is 2-AGL. In this case $\PF(S)=\{100,132,177,209,281\}$ and $100+209=132+177=281+28$. \\ {\bf 2.} Consider $S= \langle 67, 69, 76, 78, 86 \rangle$. Here $2K \setminus K=\{485-86,485\}$ and the semigroup is 2-AGL. Moreover, $\PF(S)=\{218, 226, 249, 259, 267, 322, 485 \}$, $218+267=226+259=485$ and $249+322=485+86$. \\ {\bf 3.} If $S=\langle 9,10,12,13 \rangle$, then $2K \setminus K=\{17-13,17-12,17-10,17-9,17\}$ and $\PF(S)=\{11,14,15,16,17\}$. Hence, $S$ is GAS and, according to the previous proposition, we have \begin{align*} \F(S)+9&=11+15 &\F(S)+12=14+15&\\ \F(S)+10&=11+16 &\F(S)+13=14+16&. \end{align*} {\bf 4.} Conditions (1) and (2) in Proposition \ref{PF GAS} do not imply that every $x_i$ is a minimal generator. For instance, if we consider the numerical semigroup $S=\{15,16,19,20,24\}$, we have $2K \setminus K=\{42-40,42-36,42-32,42-24,42-20,42-19,42-16,42-15,42\}$ and $\PF(S)=\{28,29,33,37,41,42\}$. Moreover, \begin{align*} \F(S)+40&=41+41 &\F(S)+20=29+33 \\ \F(S)+36&=37+41 &\F(S)+19=28+33 \\ \F(S)+32&=37+37 &\F(S)+16=29+29 \\ \F(S)+24&=33+33 &\F(S)+15=28+29 \end{align*} and, so, it is straightforward to see that the conditions in Proposition \ref{PF GAS} hold, but $32$, $36$ and $40$ are not minimal generators. \end{examples} We recall that $\L(S)$ denotes the set of the gaps of the second type of $S$, i.e. the integers $x$ such that $x \notin S$ and $\F(S)-x \notin S$, i.e. $x \in K \setminus S$, and that $S$ is almost symmetric if and only if $\L(S) \subseteq \PF(S)$, see \cite{BF}. \begin{lemma} \label{Lemma L(S)} Let $S$ be a numerical semigroup with $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r,\F(S)\}$, where $x_i$ is a minimal generator of $S$ for every $i$. If $x \in \L(S)$ and $\F(S)-x \notin \PF(S)$, then both $x$ and $\F(S)-x+x_i$ are pseudo-Frobenius numbers of $S$ for some $i$. \end{lemma} \begin{proof} Assume by contradiction that $x \notin \PF(S)$. Therefore, there exists $s \in M$ such that $x+s \notin S$ and, then, $\F(S)-x-s \in K$. Moreover, since $\F(S)-x \notin \PF(S)$, there exists $t \in M$ such that $\F(S)-x+t \notin S$ and then $x-t \in K$. Consequently, $\F(S)-s-t=(\F(S)-x-s)+(x-t) \in 2K$ and $\F(S)-s-t\notin K$, since $s+t \in S$. This is a contradiction, because $s+t$ is not a minimal generator of $S$. Hence, $x \in \PF(S)$ and, since $\F(S)-x \notin \PF(S)$, Proposition \ref{PF GAS} implies that $\F(S)-x+x_i \in \PF(S)$ for some $i$. \end{proof} \begin{lemma} \label{difference} As ideal of $M-M$, it holds $\widetilde{M-e}=M-e$ and \[ K(M-M) \setminus (M-e) =\{x-e \mid x \in \L(S) \text{ and } \F(S) - x \notin \PF(S)\}. \] \end{lemma} \begin{proof} We notice that $\F(S)-e \notin (M-M)$ and, if $y > \F(S)-e$ and $m \in M$, we have $y+m >\F(S)-e+m \geq \F(S)$. Therefore, $\F(M-M)=\F(S)-e=\F(M-e)$ and, then, $\widetilde{M-e}=M-e$. We have $x-e \in K(M-M) \setminus (M-e)$ if and only if $x\notin M$ and $(\F(S)-e)-(x-e) \notin (M-M)$ that is in turn equivalent to $x \notin M$ and $\F(S)-x \notin S \cup \PF(S)$. Since $x \neq 0$, this means that $x \in \L(S)$ and $\F(S)-x \notin \PF(S)$. \end{proof} The following corollary was proved in \cite[Theorem 5.2]{B} in a different way. \begin{corollary} \label{canonical ideal} $S$ is almost symmetric if and only if $M-e$ is a canonical ideal of $M-M$. \end{corollary} \begin{proof} By definition $M-e$ is a canonical ideal of $M-M$ if and only if $K(M-M) = (M-e)$. In light of the previous lemma, this means that there are no $x \in \L(S)$ such that $\F(S)-x \notin \PF(S)$, that is equivalent to say that $\L(S)\subseteq \PF(S)$, i.e. $S$ is almost symmetric. \end{proof} In \cite[Corollary 8]{BF} it was first proved that $S$ is almost symmetric with maximal embedding dimension if and only if $M-M$ is a symmetric semigroup. In general it holds $M-M \subseteq M-e \subseteq K(M-M)$ and the first inclusion is an equality if and only if $S$ has maximal embedding dimension, whereas the previous corollary says that the second one is an equality if and only if $S$ is almost symmetric. Moreover, if $S$ has maximal embedding dimension, in \cite[Corollary 5.4]{CGKM} it is proved that $S$ is 2-AGL if and only if $M-M$ is an almost symmetric semigroup which is not symmetric. If we want to generalize this result in the same spirit of Corollary \ref{canonical ideal}, it is not enough to consider the 2-AGL semigroups, but we need that $S$ is GAS. More precisely, we have the following result. \begin{theorem} \label{T. Almost Canonical ideal of M-M} The semigroup $S$ is {\rm GAS} if and only if $M-e$ is an almost canonical ideal of the semigroup $M-M$. \end{theorem} \begin{proof} In the light of Remark \ref{Rem as}.4 and Lemma \ref{difference}, $M-e$ is an almost canonical ideal of $M-M$ if and only if \begin{equation} \label{Eq.Canonical Ideal of M-M} K(M-M) \setminus (M-e) \subseteq ((M-e)-((M-M)\setminus \{0\})). \end{equation} Assume that $S$ is GAS with $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$. By Lemma \ref{difference} the elements of $K(M-M) \setminus (M-e)$ can be written as $x-e$ with $x \in \L(S)$ and $\F(S)-x \notin \PF(S)$. In addition, Lemma \ref{Lemma L(S)} implies that both $x$ and $\F(S)-x+x_i$ are pseudo-Frobenius numbers of $S$ for some $i$. Let $0 \neq z\in (M-M)$. We need to show that $x-e+z\in M-e$, i.e. $x+z \in M$. Assume by contradiction $x+z \notin M$, which implies $\F(S)-x-z \in K$. Since $x+z \notin M$ and $x \in \PF(S)$, it follows that $z \notin M$ and, then, $z \in \PF(S)$; hence, $z+x_i \in M$ and $\F(S)-z-x_i \notin K$. We also have that $x-x_i \in K$, since $\F(S)-x+x_i \in \PF(S)$. Therefore, \[ \F(S)-z-x_i=(\F(S)-x-z)+(x-x_i) \in 2K \setminus K \] and this yields a contradiction because $(z+x_i)-x_i \in \PF(S)$ and $S$ is a GAS semigroup. Conversely, assume that the inclusion (\ref{Eq.Canonical Ideal of M-M}) holds. An element in $2K\setminus K$ can be written as $\F(S)-s$ for some $s \in S$, since it is not in $K$. Assume by contradiction that $s \neq 0$ is not a minimal generator of $S$, i.e. $\F(S)-s_1-s_2=k_1+k_2 \in 2K\setminus K$ for some $s_1,s_2 \in M$ and $k_1, k_2 \in K$. It follows that $\F(S)-k_1-s_1=k_2 + s_2 \notin S$, otherwise $\F(S)-s_1 \in K$. Moreover, $k_1+s_1 \notin \PF(S) \cup S$, since $k_1+s_1+s_2 = \F(S)-k_2 \notin S$. Hence, Lemma \ref{difference} and our hypothesis imply that \[k_2+s_2-e=\F(S)-k_1-s_1-e \in ((M-e)-((M-M)\setminus \{0\})).\] Therefore, $\F(S)-k_1-e=(k_2+s_2-e)+s_1 \in M-e$ and, thus, $k_1 \notin K$ yields a contradiction. This means that $2K \setminus K=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$ with $x_i$ minimal generator of $S$ for every $i$. Now, assume by contradiction that $z=x_i-x_j \in \PF(S)$ for some $i,j$ and let $\F(S)-x_i=\F(S)-x_j-z=k_1+k_2$ for some $k_1, k_2 \in K$. Since $k_2+z+x_j=\F(S)-k_1 \notin S$, it follows that $k_2+z \notin S \cup \PF(S)$. Moreover, $\F(S)-k_2-z \notin S$, otherwise $\F(S)-k_2\in S$. Therefore, Lemma \ref{difference} and inclusion (\ref{Eq.Canonical Ideal of M-M}) imply that $\F(S)-k_2-z-e \in ((M-e)-((M-M)\setminus \{0\}))$ and, since $z \in M-M$, it follows that $\F(S)-k_2 \in M$ which is a contradiction because $k_2 \in K$. \end{proof} \begin{example} \rm Consider $S=\langle 9,13,14,15,19 \rangle$, that is a GAS numerical semigroup with $2K \setminus K=\{25-15,25-13,25-9,25\}$. Then, $M-9$ is an almost canonical ideal of $M-M$ by the previous theorem. In fact \begin{equation*} \begin{split} &M-M=\{0,9,13,14,15,17, \rightarrow\}, \\ &K(M-M)=\{0,4,5,6,8,9,10,11,12,13,14,15,17 \rightarrow\},\\ &M-9=\{0,4,5,6,9,10,13,14,15,17 \rightarrow\}, \\ &(M-9)-((M-M)\setminus \{0\})=K(M-M) \cup \{16\}=\{0,4,5,6,8 \rightarrow\}. \end{split} \end{equation*} \end{example} \begin{remark} \rm If $S$ is {\rm GAS}, it is possible to compute the type of $M-e$ seen as an ideal of the semigroup $M-M$. In fact by Theorem \ref{T. Almost Canonical ideal of M-M} and Proposition \ref{almost canonical ideal} it follows that \begin{align*} t(M-e)&=g(M-e)+g(M-M)-\F(M-M)= \\ &=g(M)-e+g(S)-t(S)-\F(S)+e= 2g(S)+1-t(S)-\F(S). \end{align*} Moreover, we recall that $2g(S) \geq t(S)+\F(S)$ is always true and the equality holds exactly when $S$ is almost symmetric. Therefore, as $t(S)$ is a measure of how far $S$ is from being symmetric, $t(M-e)=t(M)$ (as ideal of $M-M$) can be seen as a measure of how far $S$ is from being almost symmetric. On the other hand, we note that the type of $M$ as an ideal of $S$ is simply $t(S)+1$. \end{remark} If $S$ has type 2 and $\PF(S)=\{f,\F(S)\}$, in \cite[Theorem 6.2]{CGKM} it is proved that $S$ is 2-AGL if and only if $3(\F(S)-f) \in S$ and $\F(S)=2f-x$ for some minimal generator $x$ of $S$. In the next proposition we generalize this result to the GAS case. \begin{proposition} \label{type 2} Assume that $S$ is not almost symmetric and that it has type 2, i.e. $\PF(S)=\{f,\F(S)\}$. Then, $S$ is {\rm GAS} if and only if $\F(S)=2f-x$ for some minimal generator $x$ of $S$. In this case, if $n$ is the minimum integer for which $n(\F(S)-f) \in S$, then $|2K \setminus K|=2$, $|3K \setminus 2K|= \dots = |(n-1)K \setminus (n-2)K|=1$ and $nK=(n-1)K$. \end{proposition} \begin{proof} Assume first that $S$ is GAS and let $\F(S)-x$, $\F(S)-y \in 2K \setminus K$. Proposition \ref{PF GAS} implies that $\F(S)+x=f_1+f_2$ and $\F(S)+y=f_3+f_4$ for some $f_1,f_2,f_3,f_4 \in \PF(S)$. Since $f_i$ has to be different from $\F(S)$ for all $i$, it follows that $\F(S)+x=\F(S)+y=2f$ and, then, $x=y$. In particular, $\F(S)=2f-x$. Assume now that $\F(S)=2f-x$ for some minimal generator $x$ of $S$. Clearly, $\F(S)-x=2(\F(S)-f) \in 2K \setminus K$. Let $y \neq 0,x$ be such that $\F(S)-y \in 2K \setminus K$. Since $2K \setminus K$ is finite, we may assume that $y$ is maximal among such elements with respect to $\leq_S$, that is $\F(S)-(y+m) \notin 2K\setminus K$ for every $m \in M$. Let $\F(S)-y=k_1+k_2$ with $k_1$, $k_2 \in K$. Since $\F(S)-y-m=k_1+k_2-m \notin 2K \setminus K$, then $k_1-m$ and $k_2-m$ are not in $K$, which is equivalent to $\F(S)-k_1+m \in S$ and $\F(S)-k_2+m \in S$ for every $m \in M$. This means that $\F(S)-k_1$, $\F(S)-k_2 \in \PF(S)\setminus \{\F(S)\}$ which implies $\F(S)-y=2(\F(S)-f)=\F(S)-x$ and, thus, $x=y$. Therefore, $|2K \setminus K|=2$ and $S$ is GAS. Moreover, if $S$ is GAS and $\F(S)-y =k_1+\dots+k_r \in rK \setminus (r-1)K$ with $r>2$ and $k_1, \dots, k_r \in K$, then $k_1= \dots=k_r=\F(S)-f$ because $k_i+k_j \in 2K \setminus K$ for every $i$ and $j$. Therefore, if $n(\F(S)-f) \in S$, then $nK=(n-1)K$. Assume that $r(\F(S)-f) \notin S$. Clearly, it is in $rK$ and we claim that it is not in $K$. In fact, if $r(\F(S)-f) \in K$, it follows that it is in $\L(S)$ and, if $\F(S)-r(\F(S)-f)=f$, then $(r-1)(\F(S)-f)=0 \in S$ yields a contradiction. Therefore, Lemma \ref{Lemma L(S)} implies that $\F(S)-r(\F(S)-f)+x =f$ and, again, $(r-1)(\F(S)-f)=x \in S$ gives a contradiction. This means that $r(\F(S)-f) \in rK \setminus K$. Moreover, if $r(\F(S)-f)=k_1+\dots+ k_{r'} \in r'K\setminus(r'-1)K$ with $1<r'<r$ and $k_1, \dots, k_{r'} \in K$, we get $k_1=\dots=k_{r'}=\F(S)-f$ as above, that is a contradiction. Hence, $|rK \setminus (r-1)K|=1$ for every $1<r<n$. \end{proof} \begin{example} \rm \label{GAS tipo 2} Consider $S=\langle 5,6,7\rangle$. In this case $f=8$ and $\F(S)=9$. Therefore, the equality $\F(S)=2f-7$ implies that $S$ is GAS. With the notation of the previous corollary we have $n=5$ and, in fact, $2K \setminus K=\{2,9\}$, $3K \setminus 2K=\{3\}$ and $4K \setminus 3K=\{4\}$. \end{example} In \cite{HHS} another generalization of almost Gorenstein ring is introduced. More precisely a Cohen-Macaulay local ring admitting a canonical module $\omega$ is said to be {\it nearly Gorenstein} if the trace of $\omega$ contains the maximal ideal. In the case of numerical semigroups it follows from \cite[Lemma 1.1]{HHS} that $S$ is nearly Gorenstein if and only if $M \subseteq K+(S-K)$, see also the arXiv version of \cite{HHS}. It is easy to see that an almost symmetric semigroup is nearly Gorenstein, but in \cite{CGKM} it is noted that a 2-AGL semigroup is never nearly Gorenstein (see also \cite[Remark 3.7]{BS} for an easy proof in the numerical semigroup case). This does not happen for GAS semigroups. \begin{corollary} Let $S$ be a {\rm GAS} semigroup, not almost symmetric, with $\PF(S)=\{f,\F(S)\}$. It is nearly Gorenstein if and only if $3f-2\F(S) \in S$. \end{corollary} \begin{proof} We will use the following characterization proved in \cite{MS}: $S$ is nearly Gorenstein if and only if for every minimal generator $y$ of $S$ there exists $g \in \PF(S)$ such that $g+y-g' \in S$ for every $g' \in \PF(S)\setminus \{g\}$. By Proposition \ref{type 2} it follows that $\F(S)=2f-x$ with $x$ minimal generator of $S$. Let $y \neq x$ another minimal generator of $S$ and assume by contradiction that $\F(S)+y-f \notin S$. Therefore, there exists $s \in S$ such that $\F(S)+y-f+s \in \PF(S)$. If it is equal to $\F(S)$, then $f=y+s \in S$ yields a contradiction. If $\F(S)+y-f+s=f$, then $y+s=2f-\F(S)=x$ by Proposition \ref{type 2} and this gives a contradiction, since $x \neq y$ is a minimal generator of $S$. Hence, $\F(S)+y-f \in S$ for every minimal generator $y \neq x$. On the other hand, $\F(S)+x-f=2f-x+x-f=f \notin S$ and, therefore, $S$ is nearly Gorenstein if and only if $f+x-\F(S)=3f-2\F(S)\in S$. \end{proof} \begin{examples} \rm {\bf 1.} In Example \ref{GAS tipo 2} we have $3f-2\F(S)=6 \in S$ and, then, the semigroup is both GAS and nearly Gorenstein. \\ {\bf 2.} Consider $S=\langle 9,17,67\rangle$ that has $\PF(S)=\{59,109\}$. Since $2*59-109=9$ and $3*59-2*109=-41 \notin S$, the semigroup is GAS but not nearly Gorenstein. \\ {\bf 3.} If $S=\langle 10,11,12,25 \rangle$, we have $\PF(S)=\{38,39\}$ and $2*38-39=37$ is not a minimal generators, thus, $S$ is not GAS. On the other hand, it is straightforward to check that this semigroup is nearly Gorenstein. \end{examples} \begin{remark} \rm In literature there are other two generalizations of almost Gorenstein ring. One is given by the so-called ring with canonical reduction, introduced in \cite{R}, which is a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak m)$ possessing a canonical ideal $I$ that is a reduction of $\mathfrak m$. When $R=k[[S]]$ is a numerical semigroup ring, this definition gives a generalization of almost symmetric semigroup and $R$ has a canonical reduction if and only if $e+\F(S)-g \in S$ for every $g \in \mathbb{N} \setminus S$, see \cite[Theorem 3.13]{R}. This notion is unrelated with the one of GAS semigroup, in fact it is easy to see that $S=\langle 4,7,9,10 \rangle$ is GAS and it doesn't have canonical reductions, while $S=\langle 8,9,10,22 \rangle$ is not GAS, but has a canonical reduction. Another generalization of the notion of almost Gorenstein ring is given by the so-called generalized Gorenstein ring, briefly GGL, introduced in \cite{GIKT,GK}. A Cohen-Macaulay local ring $(R,\mathfrak{m})$ with a canonical module $\omega$ is said to be GGL with respect to $\mathfrak{a}$ if either $R$ is Gorenstein or there exists an exact sequence of $R$-modules \[ 0 \xrightarrow{} R \xrightarrow{\varphi} \omega \xrightarrow{} C \xrightarrow{} 0 \] where $C$ is an Ulrich module of $R$ with respect to some $\mathfrak m$-primary ideal $\mathfrak a$ and $\varphi \otimes R/\mathfrak a$ is injective. We note that $R$ is almost Gorenstein and not Gorenstein if and only if it is GGL with respect to $\mathfrak m$. Let $S$ be a numerical semigroup and order $\PF(S)=\{f_1,f_2, \dots, f_t=\F(S)\}$ by the usual order in $\mathbb{N}$. Defining a numerical semigroup GGL if its associated ring is GGL, in \cite{T} it is proved a useful characterization: $S$ is GGL if either it is symmetric or the following properties hold: \begin{enumerate} \item there exists $x \in S$ such that $f_i+f_{t-i}=\F(S)+x$ for every $i=1, \dots, \lceil t/2 \rceil$; \item $((c-M) \cap S) \setminus c=\{x\}$, where $c=S-\langle K \rangle$. \end{enumerate} Using this characterization it is not difficult to see that also this notion is unrelated with the one of GAS semigroup. In fact, the semigroups in Examples \ref{Examples}.2 and \ref{Examples}.3 are GAS but do not satisfy (1), whereas the semigroup $S=\langle 5,9,12 \rangle$ is not GAS by Proposition \ref{type 2}, because $\PF(S)=\{13,16\}$, but it is easy to see that it is GGL with $x=10$. \end{remark} \section{Constructing GAS numerical semigroups} In this section we study the behaviour of the GAS property with respect to some constructions. In this way we will be able to construct many numerical semigroups satisfying this property. \subsection{Gluing of numerical semigroups} Let $S_1=\langle s_1, \dots, s_n \rangle$ and $S_2=\langle t_1, \dots, t_m \rangle$ be two numerical semigroups and assume that $s_1, \dots, s_n$ and $t_1, \dots, t_m$ are minimal generators of $S_1$ and $S_2$ respectively. Let also $a\in S_2$ and $b \in S_1$ be not minimal generators of $S_2$ and $S_1$ respectively and assume $\gcd(a,b)=1$. The numerical semigroup $\langle aS_1,bS_2 \rangle=\langle as_1, \dots, as_n, bt_1, \dots, bt_m \rangle$ is said to be the gluing of $S_1$ and $S_2$ with respect to $a$ and $b$. It is well-known that $as_1, \dots, as_n, bt_1, \dots, bt_m$ are its minimal generators, see \cite[Lemma 9.8]{RG}. Moreover, the pseudo-Frobenius numbers of $T=\langle aS_1,bS_2 \rangle$ are \[ \PF(T)=\{af_1+bf_2+ab \mid f_1 \in \PF(S_1), f_2 \in \PF(S_2)\}, \] see \cite[Proposition 6.6]{N}. In particular, $t(T)=t(S_1)t(S_2)$ and $\F(T)=a\F(S_1)+b\F(S_2)+ab$. Consequently, since $K(T)$ is generated by the elements $\F(T)-f$ with $f \in \PF(T)$, it is easy to see that $K(T)=\{ak_1+bk_2 \mid k_1 \in K(S_1), k_2 \in K(S_2) \}$. Since $t(T)=t(S_1)t(S_2)$, it follows that $T$ is symmetric if and only if both $S_1$ and $S_2$ are symmetric, so in the next theorem we exclude this case. \begin{theorem} \label{gluing} Let $T$ be a gluing of two numerical semigroups and assume that $T$ is not symmetric. The following are equivalent: \begin{enumerate} \item $T$ is {\rm GAS}; \item $T$ is {\rm 2-AGL}; \item $T=\langle 2S, b \mathbb{N} \rangle$ with $b \in S$ odd and $S$ is an almost symmetric semigroup, but not symmetric. \end{enumerate} \end{theorem} \begin{proof} (2) $\Rightarrow$ (1) True by definition. \\ (1) $\Rightarrow$ (3) Let $T=\langle aS_1, bS_2 \rangle$. Since $T$ is not symmetric, we can assume that $S_1$ is not symmetric and, then, $\F(S_1)=k_1+k_2$ for some $k_1$, $k_2 \in K(S_1)$. This implies that \[ \F(T)-b(\F(S_2)+a)=a\F(S_1)+b\F(S_2)+ab-b\F(S_2)-ab=ak_1+ak_2 \in 2K(T) \setminus K(T) \] because $\F(S_2)+a \in S_2$. Therefore, since $T$ is GAS, $\F(S_2)+a$ is a minimal generator of $S_2$. By definition of gluing, $a$ is not a minimal generator of $S_2$, so write $a=s+s'$ with $s$, $s' \in M(S_2)$. Since $\F(S_2)+s+s'$ is a minimal generator of $S_2$, we get $\F(S_2)+s=\F(S_2)+s'=0$, i.e. $\F(S_2)=-1$ and $a=s+s'=2$. This proves that $T=\langle 2S_1, b \mathbb{N} \rangle$. Clearly, $b$ is odd by definition of gluing, so we only need to prove that $S_1$ is almost symmetric. Assume by contradiction that it is not almost symmetric and let $s \in M(S_1)$ such that $\F(S_1)-s=k_1+k_2 \in 2K(S_1)\setminus K(S_1)$ with $k_1$, $k_2 \in K(S_1)$. Then \[ \F(T)-(2s+b)=2\F(S_1)-b+2b-2s-b=2k_1+2k_2 \in 2K(T) \setminus K(T) \] and $2s+b$ is not a minimal generator of $T$, contradiction. \\ (3) $\Rightarrow$ (2) Since $S$ is not symmetric, $\langle K(S) \rangle \setminus K(S)= 2K(S) \setminus K(S)=\{\F(S)\}$. Consider an element $z \in \langle K(T) \rangle \setminus K(T)$, that is $z=2k_1+b\lambda_1 + \dots + 2k_r + b\lambda_r = 2(k_1+\dots +k_r)+b(\lambda_1+ \dots +\lambda_r)$ for some $k_1, \dots, k_r \in K(S)$ and $\lambda_1, \dots, \lambda_r \in \mathbb{N}$. Since $z \notin K(T)$, then $k_1+ \dots +k_r \notin K(S)$ and so $k_1+\dots +k_r=\F(S)$. Therefore, $z=2\F(S)+b(\lambda_1+\dots + \lambda_r) \in 2K(T)\setminus K(T)$ and, since it is not in $K(T)$ and $\F(T)=2\F(S)+b$, it follows that either $z=2\F(S)$ or $z=2\F(S)+b$. Hence, $|\langle K(T) \rangle \setminus K(T)|=2$ and thus $T$ is 2-AGL. \end{proof} \subsection{Numerical Duplication} In the previous subsection we have shown that if a non-symmetric GAS semigroup is a gluing, then it can be written as $\langle 2S, b \mathbb{N}\rangle$. This kind of gluing can be seen as a particular case of another construction, the {\it numerical duplication}, introduced in \cite{DS}. Given a numerical semigroup $S$, a relative ideal $I$ of $S$ and an odd integer $b \in S$, the numerical duplication of $S$ with respect to $I$ and $b$ is defined as $S \! \Join^b \! I=2\cdot S \cup \{2 \cdot I +b\}$, where $2\cdot X=\{2x \mid x\in X\}$ for every set $X$. This is a numerical semigroup if and only if $I+I+b \subseteq S$. This is always true if $I$ is an ideal of $S$ and, since in the rest of the subsection $I$ will always be an ideal, we ignore this condition. In this case, if $S$ and $I$ are minimally generated by $\{s_1, \dots, s_\nu\}$ and $\{i_1, \dots, i_\mu\}$ respectively, then $S \! \Join^b \! I=\langle 2s_1, \dots, 2s_\nu, 2i_1+b, \dots, 2i_\mu+b \rangle$ and these generators are minimal. It follows that $\langle 2S, b \mathbb{N} \rangle = S\! \Join^b \!S$. \begin{remark} \label{PF duplication} \rm The Frobenius number of $S \! \Join^b \! I$ is equal to $2\F(I)+b$. Moreover, the odd pseudo-Frobenius numbers of $S \! \Join^b \! I$ are $\{2\lambda+b \mid \lambda \in \PF(I)\}$, whereas the even elements in $\PF(S \! \Join^b \! I)$ are exactly the doubles of the elements in $((M-M) \cap (I-I)) \setminus S$; see the proof of \cite[Proposition 3.5]{DS}. In particular, if $2f \in \PF(S \! \Join^b \! I)$, then $f \in \PF(S)$. \end{remark} In this subsection we write $K$ in place of $K(S)$. We note that $S-\langle K \rangle \subseteq S$ and $\F(S-\langle K \rangle)=\F(S)$. \begin{lemma} \label{Lemma Numerical Duplication} Let $S$ be a numerical semigroup, $b \in S$ be an odd integer, $I$ be an ideal of $S$ with $\F(I)=\F(S)$ and $T=S \! \Join^b \! I$. The following hold: \begin{enumerate} \item If $k\in K$, then both $2k$ and $2k+b$ are in $K(T)$. In particular, if $\F(S)-x \in iK \setminus K$, then $\F(T)-2x \in iK(T)\setminus K(T)$; \item Let $k \in K(T)$. If $k$ is odd, then $\frac{k-b}{2} \in K$, otherwise $\F(S)-\frac{k}{2} \notin I$; \item If $I=S-\langle K \rangle$ and $k \in K(T)$ is even, then $\frac{k}{2} \in jK$ for some $j \geq 1$. \item Let $I=S-\langle K \rangle$. If $\F(T)-2i-b \in \langle K(T) \rangle \setminus K(T)$, then $\F(S)-i \in \langle K \rangle \setminus K$ for every $i \in I$. Moreover, $\F(S)-x \in \langle K \rangle \setminus K$ if and only if $\F(T)-2x \in \langle K(T) \rangle \setminus K(T)$. \end{enumerate} \end{lemma} \begin{proof} (1) If $k \in K$, then $2k+b\in K(T)$, since $\F(T)-(2k+b) = 2(\F(S)-k)\notin 2 \cdot S$. Moreover, $\F(T)-2k=2(\F(S)-k)+b$ and $\F(S)-k \notin I$ because it is not in $S$, so $2k \in K(T)$. Therefore, if $\F(S)-x=k_1+\dots + k_i \in iK \setminus K$ with $k_1, \dots, k_i \in K$, then $\F(T)-2x=2k_1+ \dots +2k_{i-1}+(2k_i+b) \in iK(T)$ and, clearly, it is not in $K(T)$, since $2x \in T$. \\ (2) Let $k$ be odd. Since $2(\F(S)-\frac{k-b}{2})=2\F(S)+b-k=\F(T)-k \notin T$, it follows that $\F(S)-\frac{k-b}{2}\notin S$, i.e. $\frac{k-b}{2} \in K$. If $k$ is even, then $2(\F(S)-\frac{k}{2})+b=\F(T)-k \notin T$ and, thus, $\F(S)-\frac{k}{2}\notin I$.\\ (3) Since $\F(S)-\frac{k}{2} \notin S-\langle K \rangle$ by (2), there exist $i\geq 1$ and $a \in iK$ such that $\F(S)-\frac{k}{2}+a \notin S$, that is $\frac{k}{2}-a \in K$. Hence, $\frac{k}{2}=a+ (\frac{k}{2}-a) \in (i+1)K$. \\ (4) If $\F(T)-2i-b=k_1+ \dots + k_j + \dots k_n \in \langle K(T) \rangle \setminus K(T)$ with $k_1, \dots, k_j \in K(T)$ even and $k_{j+1}, \dots, k_n \in K(T)$ odd, then $\F(S)-i=\frac{k_1}{2}+\dots + \frac{k_j}{2} + \frac{k_{j+1}-b}{2} + \dots + \frac{k_{n}-b}{2} + \frac{(n-j)}{2}b \in \langle K \rangle \setminus K$ by (2) and (3). Using (1) the other statement is analogous. \end{proof} \begin{example} \rm \label{Example Numerical Duplication} {\bf 1.} In the previous lemma we cannot remove the hypothesis $\F(I)=\F(S)$. For instance, consider $S=\langle 3,10,11 \rangle$, $I=\langle 3,10 \rangle$ and $T=S \! \Join^3 \! I $. Then, $\F(I)=11\neq 8=\F(S)$ and we have $\F(S)-6 \in 2K \setminus K$, but $\F(T)-12 \notin \langle K(T) \rangle$. \\ {\bf 2.} In the third statement of the previous lemma, $j$ may be bigger than 1. For instance, consider $S=\langle 6,28,47,97\rangle$ and $T=S\! \Join^{47} \!(S-\langle K\rangle)=\langle 12,56,71,94,115,153,159,194,197,241 \rangle$. Then $88,126,170,182 \in K(T)$, while $44,63,91 \in 2K \setminus K$ and $85 \in 3K \setminus 2K$. \end{example} \begin{corollary} \label{Numerical duplication 2-AGL} Let $b \in S$ be odd and let $I=S-\langle K \rangle$. The following hold: \begin{enumerate} \item If $S$ is not almost symmetric, then $S\! \Join^b \!M$ is not {\rm GAS}; \item $S$ is n-{\rm AGL} if and only if $S \! \Join^b \! I$ is n-{\rm AGL}. \end{enumerate} \end{corollary} \begin{proof} (1) Let $T=S\! \Join^b \!M$ and let $x \neq 0$ be such that $\F(S)-x \in 2K \setminus K$. By Lemma \ref{Lemma Numerical Duplication} (1), $\F(T)-2x$ and $\F(T)-(2x+b)$ are in $2K(T) \setminus K(T)$. Even though $2x+b$ and $2x$ are minimal generators, their difference $b$ is a pseudo-Frobenius number of $T$ by Remark \ref{PF duplication}, because $0 \in \PF(M)$, hence $T$ is not GAS. \\ (2) Let $T=S \! \Join^b \! I$. By Lemma \ref{Lemma Numerical Duplication} (4) we have that $\F(S)-x \in \langle K\rangle \setminus K$ if and only if $\F(T)-2x \in \langle K(T) \rangle \setminus K(T)$. Moreover, if $\F(T)-(2i+b) \in \langle K(T) \rangle \setminus K(T)$, Lemma \ref{Lemma Numerical Duplication} (4) implies that $\F(S)-i \in \langle K \rangle$ and, since $i \in (S-\langle K \rangle)$, it follows that $\F(S) \in S$, that is a contradiction. Hence, $S$ is $n$-AGL if and only if $T$ is $n$-AGL. \end{proof} \begin{remark} \rm If $S$ is almost symmetric with type $t$, then $M=K-(M-M)$ and, consequently, $S\! \Join^b \!M$ is almost symmetric with type $2t+1$ by \cite[Theorem 4.3 and Proposition 4.8]{DS}. \end{remark} If $R$ is a one-dimensional Cohen-Macaulay local ring with a canonical module $\omega$ such that $R \subseteq \omega \subseteq \overline{R}$, in \cite[Theorem 4.2]{CGKM} it is proved that the idealization $R \ltimes (R:R[\omega])$ is 2-AGL if and only if $R$ is 2-AGL. The numerical duplication may be considered the analogous of the idealization in the numerical semigroup case, since they are both members of a family of rings that share many properties (see \cite{BDS}); therefore, Corollary \ref{Numerical duplication 2-AGL} (2) should not be surprising. In the following proposition we generalize this result for the GAS property. \begin{theorem} \label{Numerical duplication S-<K>} Let $S$ be a numerical semigroup, let $b \in S$ be an odd integer and let $I=S-\langle K \rangle$. The semigroup $T=S \! \Join^b \! I$ is {\rm GAS} if and only if $S$ is {\rm GAS}. \end{theorem} \begin{proof} Assume that $T$ is GAS and let $\F(S)-x \in 2K \setminus K$. By Lemma \ref{Lemma Numerical Duplication}, $\F(T)-2x\in 2K(T) \setminus K(T)$, so $2x$ is a minimal generator of $T$ and, thus, $x$ is a minimal generator of $S$. Now let $\F(S)-x$, $\F(S)-y \in 2K \setminus K$ and assume by contradiction that $x-y \in \PF(S)$. In particular, $S$ is not symmetric and, then, $I=M-\langle K \rangle$. Moreover, $\F(T)-2x$ and $\F(T)-2y$ are in $2K(T) \setminus K(T)$. We also notice that $x-y \in I-I$, indeed, if $i \in I$ and $a \in \langle K \rangle$, it follows that $(x-y)+i+a \in (x-y)+M \subseteq S$. Therefore, Remark \ref{PF duplication} implies that $2(x-y) \in \PF(T)$; contradiction. Conversely, assume that $S$ is GAS and let $\F(T)-z=k_1+k_2 \in 2K(T) \setminus K(T)$ with $k_1$, $k_2 \in K(T)$. If $z=2i+b$ is odd and both $k_1$ and $k_2$ are odd, then $i\in I$ and $\F(S)-i=(k_1-b)/2+(k_2-b)/2+b \in 2K$ by Lemma \ref{Lemma Numerical Duplication}.(2); on the other hand, if $k_1$ and $k_2$ are both even, $\F(S)-i=k_1/2+k_2/2 \in \langle K \rangle$ by Lemma \ref{Lemma Numerical Duplication}.3. Since $i \in (S-\langle K \rangle)$, in both cases we get $\F(S) \in S$, that is a contradiction. Hence, $z=2x$ is even. If $k_1$ is even and $k_2$ is odd, Lemma \ref{Lemma Numerical Duplication} implies that $\F(S)-x=k_1/2 + (k_2-b)/2 \in (j+1)K \setminus K$ for some $j\geq 1$ and, therefore, by Theorem \ref{Livelli più alti} it follows that $x$ is a minimal generator of $S$, i.e. $z=2x$ is a minimal generator of $T$. Moreover, let $\F(T)-2x$, $\F(T)-2y \in 2K(T)\setminus K(T)$ and assume by contradiction that $2x-2y \in \PF(T)$. Remark \ref{PF duplication} implies that $x-y \in \PF(S) \subseteq K \cup \{\F(S)\}$. Thus, if $\F(T)-2x=k_1+k_2$ with $k_1$, $k_2 \in K(T)$ and $k_1$ even, then $\F(S)-x=k_1/2+(k_2-b)/2 \in \langle K(S) \rangle \setminus K(S)$ by Lemma \ref{Lemma Numerical Duplication} and, so, $\F(S)-y=k_1/2+(k_2-b)/2+(x-y) \in \langle K(S) \rangle \setminus K(S)$. Hence, Theorem \ref{Livelli più alti} yields a contradiction, because $x-y \in \PF(S)$. \end{proof} \begin{example} \rm {\bf 1.} Consider the semigroup $S$ in Example \ref{Example Numerical Duplication}.2. It is GAS and, then, the previous theorem implies that also $T=S\! \Join^{47} \!(S-\langle K\rangle)$ is GAS. However, we notice that $2K\setminus K=\{44,63,91\}$, $3K \setminus 2K=\{85\}$ and $4K=3K$, while $2K(T) \setminus K(T)=\{135,173,217,229\}$ and $2K(T)=3K(T)$. \\ {\bf 2.} Despite Theorem \ref{Numerical duplication S-<K>}, if $S \! \Join^b \! I$ is GAS for an ideal $I$ different form $S-\langle K \rangle$, it is not true that also $S$ is GAS. For instance, the semigroup $S$ in Example \ref{Example Numerical Duplication}.1 is not GAS, but $S\! \Join^3 \! I$ is. \end{example} \subsection{Dilatations of numerical semigroups} We complete this section studying the transfer of the GAS property in a construction recently introduced in \cite{BS}: given $a \in M-2M$, the numerical semigroup $S+a=\{0\} \cup \{m+a \mid m \in M\}$ is called dilatation of $S$ with respect to $a$. \begin{proposition} \label{dilatation} Let $a \in M-2M$. The semigroup $S+a$ is {\rm GAS} if and only if $S$ is {\rm GAS}. \end{proposition} \begin{proof} We denote the semigroup $S+a$ by $T$. Recalling that $\F(T)=\F(S)+a$, by \cite[Lemma 3.1 and Lemma 3.4]{BS} follows that $2K(T)=2K(S)$ and \begin{equation*} \begin{split} &2K(S) \setminus K(S)=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}, \\ &2K(T) \setminus K(T)=\{\F(T)-(x_1+a), \dots, \F(T)-(x_r+a), \F(T)\} \end{split} \end{equation*} for some $x_1, \dots, x_r \in M$. Assume that $S$ is a GAS semigroup. Then, $x_i$ is a minimal generator of $S$ and it is straightforward to see that $x_i+a$ is a minimal generator of $T$. Moreover, if $(x_i+a)-(x_j+a) \in \PF(T)$, then for every $m \in M$ we have $x_i-x_j+m+a \in T$, i.e. $x_i-x_j+m \in M$, that is a contradiction, since $S$ is GAS. Now assume that $T$ is GAS. Suppose by contradiction that $x_i$ is not a minimal generator of $S$, that is $x_i=s_1+s_2$ for some $s_1$, $s_2 \in M$. We have $\F(S)-(s_1+s_2) \in 2K(S)\setminus K(S)$ and so $\F(S)-s_1 \in 2K(S)\setminus K(S)$, since $2K(S)$ is a relative ideal. Hence, $s_1=x_j$ for some $j$ and $(x_i+a)-(x_j+a)=s_2 \in S$. Since $x_i+a$ is a minimal generator, we have that $s_2 \notin T$. Moreover, for every $m+a \in M(T)$ we clearly have $s_2+m+a \in M(T)$, because $s_2 \in S$. This yields a contradiction because $(x_i+a)-(x_j+a)=s_2 \in \PF(T)$ and $T$ is GAS. Finally, if $x_i-x_j \in \PF(S)$, it is trivial to see that $x_i-x_j \in \PF(T)$. \end{proof} \begin{remark} \rm Suppose $2K(S+a) \setminus K(S+a)=\{\F(S+a)-(x_1+a), \dots, \F(S+a)-(x_r+a), \F(S+a)\}$ with $x_1+a, \dots, x_r+a$ minimal generators of $S+a$, but $S+a$ is not GAS. Then $2K(S) \setminus K(S)=\{\F(S)-x_1, \dots, \F(S)-x_r, \F(S)\}$, but it is not necessarily true that $x_1, \dots, x_r$ are minimal generators of $S$. For instance, consider $S=\langle 7,9,11 \rangle$ and $S+7=\langle 14, 16, 18, 21, 23, 25, 27, 29, 38, 40 \rangle$. In this case $2K(S+7) \setminus K(S+7)=\{33-29,33-18,33\}$ and $2K(S) \setminus K(S)=\{26-22, 26-11, 26\}$. \end{remark} \end{document}